Search is not available for this dataset
idx int64 0 20.8k | my_solu listlengths 1 1 | new_prompt stringlengths 4 4.76k | old_prompt stringlengths 104 4.12k | type stringclasses 9
values | solution stringlengths 15 9.26k | old_solution listlengths 1 1 |
|---|---|---|---|---|---|---|
4,709 | [
"<bos><start_of_turn>user\nA box of 100 personalized pencils costs $\\$30$. How many dollars does it cost to buy 2500 pencils?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can set up a proportion to find the cost of 1 pencil. ки\nStep 2: $\\frac{\\text{cost of 1 pencil}}{\\text{number... | <bos><start_of_turn>user
A box of 100 personalized pencils costs $\$30$. How many dollars does it cost to buy 2500 pencils?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can set up a proportion to find the cost of 1 pencil. ки
Step 2: $\frac{\text{cost of 1 pencil}}{\text{number of pencils}} ... | <bos><start_of_turn>user
A box of 100 personalized pencils costs $\$30$. How many dollars does it cost to buy 2500 pencils?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | The ratio of pencils to cost is $100:\$30$. Multiplying both parts of this ratio by 25 gives a ratio of $2500:\$750$, so 2500 pencils costs $\boxed{\$750}$. | [
"<bos><start_of_turn>user\nA box of 100 personalized pencils costs $\\$30$. How many dollars does it cost to buy 2500 pencils?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can set up a proportion to find the cost of 1 pencil. ки\nStep 2: $\\frac{\\text{cost of 1 pencil}}{\\text{number... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,710 | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We are looking for a number of the form $4a+3=... | <bos><start_of_turn>user
If a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the number we're looking for $n$.
According to the given information, $n$ gives a remainder of 3 when divided by 4, so $n \equiv 3 \pmod{4}$.
Similarly, $n$ gives a remainder of 4 when divided by 5, so $n \equiv 4 \pmod{5}$.
Finally, $n$ gives a remainder of 5 when divided by 6, so $n \equiv 5 \pmod{6}$... | [
"<bos><start_of_turn>user\nIf a number gives a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6, what is the least possible value of this number?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We are looking for a number of the fo... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,711 | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | <bos><start_of_turn>user
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\%$ of the pond algae-free?
If we know the answer to the above question is 28, what is the value of unknown variable X?
Pleas... | MATH_FOBAR | We want to find the value of $X$ in the given situation.
We are told that the amount of algae covering the pond doubled every day until it was completely covered on day $X$ of the month.
To find the day on which $75\%$ of the pond was algae-free, we need to determine the value of $X$.
If the pond was completely covered... | [
"<bos><start_of_turn>user\nThe amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day X of the month. On what day of that month was $75\\%$ of the pond algae-free?\nIf we know the answer to the above question is 28, what is the value of unknown variable ... |
4,712 | [
"<bos><start_of_turn>user\nUsing the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ... | <bos><start_of_turn>user
Using the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ки
| <bos><start_of_turn>user
Using the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find the result of $3 @ 5$, we substitute $a=3$ and $b=5$ into the given expression:
$$
3 @ 5 = 3 \times 3 - 3 \times 5
$$
Now we can simply evaluate the expression:
$$
3 \times 3 - 3 \times 5 = 9 - 15
$$
And finally simplify:
$$
9 - 15 = -6
$$
Therefore, $\boxed{-6}$ is the result of $3 @ 5$. The answer is: -6 | [
"<bos><start_of_turn>user\nUsing the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ... |
4,712 | [
"<bos><start_of_turn>user\nUsing the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ... | <bos><start_of_turn>user
Using the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ки
Step 2... | <bos><start_of_turn>user
Using the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find the result of $3 @ 5$, we substitute $a=3$ and $b=5$ into the given expression:
$$
3 @ 5 = 3 \times 3 - 3 \times 5
$$
Now we can simply evaluate the expression:
$$
3 \times 3 - 3 \times 5 = 9 - 15
$$
And finally simplify:
$$
9 - 15 = -6
$$
Therefore, $\boxed{-6}$ is the result of $3 @ 5$. The answer is: -6 | [
"<bos><start_of_turn>user\nUsing the special operation $a @ b$ where it represents thrice the number $a$ minus thrice the number $b$, find the result for $3 @ 5$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We substitute $a=3$ and $b=5$ into the expression $3@5$ to get $3@5=3(3)-3(5)$. ... |
4,713 | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ and $\\sqrt{... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can simplify each square root separately before multiplying: ки
| <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To simplify the expression, we can use the properties of radicals.
First, let's simplify the expressions under the radicals separately:
$\sqrt{7 \times 2} = \sqrt{14}$
$\sqrt{2^3 \times 7^3} = \sqrt{8 \times 343} = \sqrt{2744}$
Next, let's multiply the simplified expressions:
$\sqrt{14} \times \sqrt{2744} = \sqrt{... | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... |
4,713 | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can simplify each square root separately before multiplying: ки
Step 2: $\sqrt{7\times 2} = \sqrt{14}$ ки
| <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To simplify the expression, we can use the properties of radicals.
First, let's simplify the expressions under the radicals separately:
$\sqrt{7 \times 2} = \sqrt{14}$
$\sqrt{2^3 \times 7^3} = \sqrt{8 \times 343} = \sqrt{2744}$
Next, let's multiply the simplified expressions:
$\sqrt{14} \times \sqrt{2744} = \sqrt{... | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... |
4,713 | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can simplify each square root separately before multiplying: ки
Step 2: $\sqrt{7\times 2} = \sqrt{14}$ ки
Step 3: $\sqrt{2^3\times 7... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To simplify the expression, we can use the properties of radicals.
First, let's simplify the expressions under the radicals separately:
$\sqrt{7 \times 2} = \sqrt{14}$
$\sqrt{2^3 \times 7^3} = \sqrt{8 \times 343} = \sqrt{2744}$
Next, let's multiply the simplified expressions:
$\sqrt{14} \times \sqrt{2744} = \sqrt{... | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... |
4,713 | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can simplify each square root separately before multiplying: ки
Step 2: $\sqrt{7\times 2} = \sqrt{14}$ ки
Step 3: $\sqrt{2^3\times 7... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To simplify the expression, we can use the properties of radicals.
First, let's simplify the expressions under the radicals separately:
$\sqrt{7 \times 2} = \sqrt{14}$
$\sqrt{2^3 \times 7^3} = \sqrt{8 \times 343} = \sqrt{2744}$
Next, let's multiply the simplified expressions:
$\sqrt{14} \times \sqrt{2744} = \sqrt{... | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... |
4,713 | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can simplify each square root separately before multiplying: ки
Step 2: $\sqrt{7\times 2} = \sqrt{14}$ ки
Step 3: $\sqrt{2^3\times 7... | <bos><start_of_turn>user
What is the simplified result of $\sqrt{7\times 2}$ multiplied by $\sqrt{2^3\times 7^3}$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To simplify the expression, we can use the properties of radicals.
First, let's simplify the expressions under the radicals separately:
$\sqrt{7 \times 2} = \sqrt{14}$
$\sqrt{2^3 \times 7^3} = \sqrt{8 \times 343} = \sqrt{2744}$
Next, let's multiply the simplified expressions:
$\sqrt{14} \times \sqrt{2744} = \sqrt{... | [
"<bos><start_of_turn>user\nWhat is the simplified result of $\\sqrt{7\\times 2}$ multiplied by $\\sqrt{2^3\\times 7^3}$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can simplify each square root separately before multiplying: ки\nStep 2: $\\sqrt{7\\times 2} = \\sqrt{14}$ ки\nStep 3: ... |
4,714 | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^2 + (y-2)... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_Rephrased | We can rewrite the equation of the circle as $(x+3)^2 + (y-2)^2 = 4^2$.
This is the equation of a circle with center $(-3, 2)$ and radius 4.
To find the value of $c$, we can expand the equation and simplify:
$(x+3)^2 + (y-2)^2 = 16$
$x^2 + 6x + 9 + y^2 - 4y + 4 = 16$
$x^2 + y^2 + 6x - 4y - 3 = 0$
Comparing this with th... | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... |
4,714 | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^2 + (y-2)... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_Rephrased | We can rewrite the equation of the circle as $(x+3)^2 + (y-2)^2 = 4^2$.
This is the equation of a circle with center $(-3, 2)$ and radius 4.
To find the value of $c$, we can expand the equation and simplify:
$(x+3)^2 + (y-2)^2 = 16$
$x^2 + 6x + 9 + y^2 - 4y + 4 = 16$
$x^2 + y^2 + 6x - 4y - 3 = 0$
Comparing this with th... | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... |
4,714 | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^2 + (y-2)... | <bos><start_of_turn>user
What value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_Rephrased | We can rewrite the equation of the circle as $(x+3)^2 + (y-2)^2 = 4^2$.
This is the equation of a circle with center $(-3, 2)$ and radius 4.
To find the value of $c$, we can expand the equation and simplify:
$(x+3)^2 + (y-2)^2 = 16$
$x^2 + 6x + 9 + y^2 - 4y + 4 = 16$
$x^2 + y^2 + 6x - 4y - 3 = 0$
Comparing this with th... | [
"<bos><start_of_turn>user\nWhat value of $c$ is needed for the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ to have a radius of 4?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the equation as $(x^2 + 6x + 9) + (y^2 - 4y + 4) = -c + 9 + 4$, which simplifies to $(x+3)^... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,715 | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | <bos><start_of_turn>user
A set of numbers $d_1, d_2, d_3, \ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?
Please reason step by step.<end_of_turn>
<start... | gpt-3.5-turbo | Let's denote the sum of the first $n$ terms of the sequence as $S_n$. We are given that $S_{300} = 600$ and $S_{450} = 900$.
We can write the recursive equation for the sum of the first $n$ terms in the form of the sequence as:
$S_n = d_1 + d_2 + d_3 + \ldots + d_n$
Using the given recursive equation for the individ... | [
"<bos><start_of_turn>user\nA set of numbers $d_1, d_2, d_3, \\ldots$ is generated by the equation $d_n = d_{n-2} - 2d_{n-1}$ for all $n \\ge 3$. The sum of the first $300$ terms is $600$, and the sum of the first $450$ terms is $900$. What is the sum of the first $600$ terms?\nPlease reason step by step.<end_of_tur... |
4,716 | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: Now we have $\\log_5 (x+4) = 3$. Since $x=121$, we can substitute this ... | <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We know that $\log_{11} x = 2$ means $x = 11^2 = 121$. ки
| <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve for $x$, we will use the properties of logarithms.
Given that $\log_{11} x = 2$, we can rewrite this equation as $x = 11^2$.
Now, we can substitute the value of $x$ into the equation $\log_5 (x+4) = 3$.
$\log_5 (11^2 + 4) = 3$
Simplifying inside the logarithm:
$\log_5 (121 + 4) = 3$
$\log_5 (125) = 3$
S... | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: So the equation $\\log_5 (x+4) = 3$ becomes $\\log_5 (121+4) = 3$. ки\n... |
4,716 | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: So the equation $\\log_5 (x+4) = 3$ becomes $\\log_5 (121+4) = 3$. ки\n... | <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We know that $\log_{11} x = 2$ means $x = 11^2 = 121$. ки
Step 2: So the equation $\log_5 (x+4) = 3$ becomes $\log_5 (121+4) = 3$. ки
| <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve for $x$, we will use the properties of logarithms.
Given that $\log_{11} x = 2$, we can rewrite this equation as $x = 11^2$.
Now, we can substitute the value of $x$ into the equation $\log_5 (x+4) = 3$.
$\log_5 (11^2 + 4) = 3$
Simplifying inside the logarithm:
$\log_5 (121 + 4) = 3$
$\log_5 (125) = 3$
S... | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: So the equation $\\log_5 (x+4) = 3$ becomes $\\log_5 (121+4) = 3$. ки\n... |
4,716 | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: So the equation $\\log_5 (x+4) = 3$ becomes $\\log_5 (121+4) = 3$. ки\n... | <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We know that $\log_{11} x = 2$ means $x = 11^2 = 121$. ки
Step 2: So the equation $\log_5 (x+4) = 3$ becomes $\log_5 (121+4) = 3$. ки
Step 3: This can... | <bos><start_of_turn>user
Assume $\log_{11} x$ is 2. Calculate the value of x in $\log_5 (x+4) = 3$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve for $x$, we will use the properties of logarithms.
Given that $\log_{11} x = 2$, we can rewrite this equation as $x = 11^2$.
Now, we can substitute the value of $x$ into the equation $\log_5 (x+4) = 3$.
$\log_5 (11^2 + 4) = 3$
Simplifying inside the logarithm:
$\log_5 (121 + 4) = 3$
$\log_5 (125) = 3$
S... | [
"<bos><start_of_turn>user\nAssume $\\log_{11} x$ is 2. Calculate the value of x in $\\log_5 (x+4) = 3$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We know that $\\log_{11} x = 2$ means $x = 11^2 = 121$. ки\nStep 2: So the equation $\\log_5 (x+4) = 3$ becomes $\\log_5 (121+4) = 3$. ки\n... |
4,717 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Aunt Angela has 70 jellybeans in total. ки
| <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gsm8k | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... |
4,717 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Aunt Angela has 70 jellybeans in total. ки
Step 2: ... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gsm8k | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... |
4,717 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Aunt Angela has 70 jellybeans in total. ки
Step 2: ... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gsm8k | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... |
4,717 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Aunt Angela has 70 jellybeans in total. ки
Step 2: ... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gsm8k | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... |
4,717 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Aunt Angela has 70 jellybeans in total. ки
Step 2: ... | <bos><start_of_turn>user
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gsm8k | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 | [
"<bos><start_of_turn>user\nAunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Aunt Angela has 70 jellybeans in total. ки... |
4,718 | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding the right side gives $5x = 12x - 12$. ки\nStep 3:... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки
| <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve the equation $\frac{5x}{3(x-1)}=4$, we will follow these steps:
Step 1: Clear the fraction by multiplying both sides of the equation by $3(x-1)$.
\begin{align*}
\frac{5x}{3(x-1)} \cdot 3(x-1) &= 4 \cdot 3(x-1) \\
5x &= 12(x-1)
\end{align*}
Step 2: Distribute $12$ on the right side of the equation.
\begin{ali... | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... |
4,718 | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки
Step 2: Expanding, we have $5x = 12x - 12$. ки
| <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve the equation $\frac{5x}{3(x-1)}=4$, we will follow these steps:
Step 1: Clear the fraction by multiplying both sides of the equation by $3(x-1)$.
\begin{align*}
\frac{5x}{3(x-1)} \cdot 3(x-1) &= 4 \cdot 3(x-1) \\
5x &= 12(x-1)
\end{align*}
Step 2: Distribute $12$ on the right side of the equation.
\begin{ali... | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... |
4,718 | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки
Step 2: Expanding, we have $5x = 12x - 12$. ки
Step 3: Subtracting $12x$ from ... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve the equation $\frac{5x}{3(x-1)}=4$, we will follow these steps:
Step 1: Clear the fraction by multiplying both sides of the equation by $3(x-1)$.
\begin{align*}
\frac{5x}{3(x-1)} \cdot 3(x-1) &= 4 \cdot 3(x-1) \\
5x &= 12(x-1)
\end{align*}
Step 2: Distribute $12$ on the right side of the equation.
\begin{ali... | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... |
4,718 | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки
Step 2: Expanding, we have $5x = 12x - 12$. ки
Step 3: Subtracting $12x$ from ... | <bos><start_of_turn>user
Solve the equation $\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve the equation $\frac{5x}{3(x-1)}=4$, we will follow these steps:
Step 1: Clear the fraction by multiplying both sides of the equation by $3(x-1)$.
\begin{align*}
\frac{5x}{3(x-1)} \cdot 3(x-1) &= 4 \cdot 3(x-1) \\
5x &= 12(x-1)
\end{align*}
Step 2: Distribute $12$ on the right side of the equation.
\begin{ali... | [
"<bos><start_of_turn>user\nSolve the equation $\\frac{5x}{3(x-1)}=4$. Express your answer in simplest fractional form.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We begin by cross-multiplying to get $5x = 12(x-1)$. ки\nStep 2: Expanding, we have $5x = 12x - 12$. ки\nStep 3: Subtracting... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,719 | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Let the two integers be $x$ and $y$, where $... | <bos><start_of_turn>user
If you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | Let's call the larger integer $x$ and the smaller integer $y$. We are given two equations:
1. $x - y = 6$
2. $xy = 135$
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method. Rearranging equation 1 for $x$, we have $x = 6 + y$. We can substitu... | [
"<bos><start_of_turn>user\nIf you subtract one positive integer from another and the result is 6, and if you multiply these integers together their product equals 135, what is the larger of the two integers?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Let the two integers be $x$ and $y$... |
4,720 | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: Plugging in $x=0$ into the equati... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки
| <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find where the line $5x + 2y = 10$ crosses the $y$-axis, we need to find the point where $x = 0$.
Substituting $x = 0$ into the equation $5x + 2y = 10$, we have:
$5(0) + 2y = 10$
$2y = 10$
Dividing both sides by 2, we find:
$y = 5$
Therefore, the line $5x + 2y = 10$ crosses the $y$-axis at the point $\boxed{... | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... |
4,720 | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки
Step 2: When $x=0$, the equation becomes $5(0) + 2y... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find where the line $5x + 2y = 10$ crosses the $y$-axis, we need to find the point where $x = 0$.
Substituting $x = 0$ into the equation $5x + 2y = 10$, we have:
$5(0) + 2y = 10$
$2y = 10$
Dividing both sides by 2, we find:
$y = 5$
Therefore, the line $5x + 2y = 10$ crosses the $y$-axis at the point $\boxed{... | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... |
4,720 | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки
Step 2: When $x=0$, the equation becomes $5(0) + 2y... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find where the line $5x + 2y = 10$ crosses the $y$-axis, we need to find the point where $x = 0$.
Substituting $x = 0$ into the equation $5x + 2y = 10$, we have:
$5(0) + 2y = 10$
$2y = 10$
Dividing both sides by 2, we find:
$y = 5$
Therefore, the line $5x + 2y = 10$ crosses the $y$-axis at the point $\boxed{... | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... |
4,720 | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки
Step 2: When $x=0$, the equation becomes $5(0) + 2y... | <bos><start_of_turn>user
Where does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To find where the line $5x + 2y = 10$ crosses the $y$-axis, we need to find the point where $x = 0$.
Substituting $x = 0$ into the equation $5x + 2y = 10$, we have:
$5(0) + 2y = 10$
$2y = 10$
Dividing both sides by 2, we find:
$y = 5$
Therefore, the line $5x + 2y = 10$ crosses the $y$-axis at the point $\boxed{... | [
"<bos><start_of_turn>user\nWhere does the line $5x + 2y = 10$ cross the $y$-axis? Give your answer as a coordinate pair.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To find where the line crosses the $y$-axis, we set $x=0$ and solve for $y$. ки\nStep 2: When $x=0$, the equation becomes ... |
4,721 | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1:... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We are given a sum of complex numbers, each of which is of the form:
\[\exp\left(\frac{2k\pi i}{13}\right),\]
where $k$ ranges from $1$ to $12$.
To evaluate the sum, let's start by noticing that the number $13$ is prime. This is important because it will allow us to apply a useful property of complex numbers in the fo... | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... |
4,721 | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1:... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We are given a sum of complex numbers, each of which is of the form:
\[\exp\left(\frac{2k\pi i}{13}\right),\]
where $k$ ranges from $1$ to $12$.
To evaluate the sum, let's start by noticing that the number $13$ is prime. This is important because it will allow us to apply a useful property of complex numbers in the fo... | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... |
4,721 | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1:... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We are given a sum of complex numbers, each of which is of the form:
\[\exp\left(\frac{2k\pi i}{13}\right),\]
where $k$ ranges from $1$ to $12$.
To evaluate the sum, let's start by noticing that the number $13$ is prime. This is important because it will allow us to apply a useful property of complex numbers in the fo... | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... |
4,721 | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1:... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We are given a sum of complex numbers, each of which is of the form:
\[\exp\left(\frac{2k\pi i}{13}\right),\]
where $k$ ranges from $1$ to $12$.
To evaluate the sum, let's start by noticing that the number $13$ is prime. This is important because it will allow us to apply a useful property of complex numbers in the fo... | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... |
4,721 | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1:... | <bos><start_of_turn>user
Evaluate the cumulative sum of the complex numbers represented as
\[\exp\left(\frac{2\pi i}{13}\right) + \exp\left(\frac{4\pi i}{13}\right) + \exp\left(\frac{6\pi i}{13}\right) + \dots + \exp\left(\frac{24\pi i}{13}\right).\]
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We are given a sum of complex numbers, each of which is of the form:
\[\exp\left(\frac{2k\pi i}{13}\right),\]
where $k$ ranges from $1$ to $12$.
To evaluate the sum, let's start by noticing that the number $13$ is prime. This is important because it will allow us to apply a useful property of complex numbers in the fo... | [
"<bos><start_of_turn>user\nEvaluate the cumulative sum of the complex numbers represented as\n\\[\\exp\\left(\\frac{2\\pi i}{13}\\right) + \\exp\\left(\\frac{4\\pi i}{13}\\right) + \\exp\\left(\\frac{6\\pi i}{13}\\right) + \\dots + \\exp\\left(\\frac{24\\pi i}{13}\\right).\\]\nPlease reason step by step.<end_of_tur... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,722 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | <bos><start_of_turn>user
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?
Please reason s... | MATH_SV | ariable x is 5.
The answer is: 5 | [
"<bos><start_of_turn>user\nFifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of X or 7 or both? Express your answer as a common fraction. The answer is 25. What is the value of unknown variable X?\nPlease r... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
4,723 | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To convert from polar coordinates to rectangular coordinates, w... | <bos><start_of_turn>user
Convert the point $\left( X, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| MATH_SV | To convert a point from polar coordinates to rectangular coordinates, we can use the following formulas:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
In this case, we are given the polar coordinates $\left( X, \frac{7 \pi}{6} \right)$ and the value of y is given as -4.
Using the formula for y, we can set up the equation... | [
"<bos><start_of_turn>user\nConvert the point $\\left( X, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. The answer is -4. What is the value of unknown variable X?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To convert from polar coordinates to rectangular c... |
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