idx int64 0 20.8k | question stringlengths 16 4.03k | gt_cot stringclasses 1
value | gt stringclasses 1
value | type stringclasses 9
values | solution stringlengths 15 9.61k | my_solu sequencelengths 1 1 | code sequencelengths 1 1 | pred sequencelengths 1 1 | report sequencelengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|
0 | Honey earned $x a day. Every day, she spent part of her pay and saved the rest. After 20 days of work, she spent $1360. How much did Honey save in 20 days?
If we know the answer to the above question is 240, what is the value of unknown variable x? | GSM_FOBAR | Honey earned $x a day.
After 20 days of work, she spent $1360, which means she spent $1360 / 20 = $68 per day.
The amount she saved in 20 days is her total earnings minus the amount she spent: 20x - $1360.
We are given that the amount she saved is $240, so we can write: 20x - $1360 = $240.
Adding $1360 to both sides, w... | [
"<bos><start_of_turn>user\nHoney earned $x a day. Every day, she spent part of her pay and saved the rest. After 20 days of work, she spent $1360. How much did Honey save in 20 days?\nIf we know the answer to the above question is 240, what is the value of unknown variable x?<end_of_turn>\n<start_of_turn>model\nLet... | [
"Thus Honey earned $12 a day and thus earned \\boxed{12} dollars per day."
] | [
"12"
] | [
null
] | ||
1 | Solve for $x$: $\sqrt{3x+5}$ is to $\sqrt{6x+5}$ as $\sqrt{5}$ is to $3$. Your answer should be written as an irreducible fraction. | gpt-3.5-turbo | To solve for $x$, we need to equate the two ratios:
\begin{align*}
\frac{\sqrt{3x+5}}{\sqrt{6x+5}} &= \frac{\sqrt{5}}{3}
\end{align*}
First, let's eliminate the square root in the numerator by squaring both sides of the equation:
\begin{align*}
\left(\frac{\sqrt{3x+5}}{\sqrt{6x+5}}\right)^2 &= \left(\frac{\sqrt{5}}{... | [
"<bos><start_of_turn>user\nSolve for $x$: $\\sqrt{3x+5}$ is to $\\sqrt{6x+5}$ as $\\sqrt{5}$ is to $3$. Your answer should be written as an irreducible fraction.<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import *\n\nx = symbols('x')\n\n# Define the equation and solve it\n... | [
"So the answer is $\\boxed{-\\frac{650406504065041}{1000000000000000}}$."
] | [
"-\\frac{650406504065041}{1000000000000000}"
] | [
null
] | ||
2 | A car rental company charges a daily rate of $\$20.00$ plus $\$0.10$ per mile driven. How much, in dollars, does it cost to rent the car for 3 days and drive it for 150 miles? | gpt-3.5-turbo | To calculate the cost of renting the car for 3 days, we simply multiply the daily rate of $\$20.00$ by the number of days:
$3 \times \$20.00 = \$60.00$
Next, we need to calculate the cost of driving the car for 150 miles. The rate is $\$0.10$ per mile, so we multiply the number of miles driven by $\$0.10$:
$150 \tim... | [
"<bos><start_of_turn>user\nA car rental company charges a daily rate of $\\$20.00$ plus $\\$0.10$ per mile driven. How much, in dollars, does it cost to rent the car for 3 days and drive it for 150 miles?<end_of_turn>\n<start_of_turn>model\nWe can compute the total cost of renting and driving the car using sympy.\n... | [
"The total cost of renting and driving the car for 3 days and 150 miles is $\\boxed{75}$ dollars."
] | [
"75"
] | [
null
] | ||
3 | A bag contains ten balls: red, blue, yellow, green, and white. When two balls are drawn at random at the same time, the probability that both balls are not green is $\frac{14}{15}$. If the probability that both balls are red is $\frac{1}{15}$, the probability that both balls are blue is $\frac{2}{15}$, and the probabil... | gpt-3.5-turbo | Let's denote the number of white balls in the bag by $w$. Since there are $10$ balls in total, we have $w + 4$ balls that are not white.
The probability that both balls are not green is $\frac{14}{15}$. Since there are $5$ total balls that are not green (red, blue, yellow, and white), the probability that the first ba... | [
"<bos><start_of_turn>user\nA bag contains ten balls: red, blue, yellow, green, and white. When two balls are drawn at random at the same time, the probability that both balls are not green is $\\frac{14}{15}$. If the probability that both balls are red is $\\frac{1}{15}$, the probability that both balls are blue is... | [
"Unfortunately, I couldn't get the sympy library to solve the system of equations.\n\nIt may be due to an error in my formulation or a limitation of the library.\nIn any case, if we go back to the original equation, and use Python's sympy library, we can solve this system.\nTo do so, I am going to take a leap of fa... | [
"2"
] | [
null
] | ||
4 | A rectangular room has an area of $2^{12} \times 5^8$ square units and a length that is twice its width. What is the perimeter of the room? | gpt-3.5-turbo | Let the width of the room be $w$ units.
Since the length of the room is twice its width, we can write the length as $2w$ units.
We are given that the area of the room is $2^{12} \times 5^8$ square units.
The area of a rectangle is given by the formula $A = \text{{length}} \times \text{{width}}$. Substituting the gi... | [
"<bos><start_of_turn>user\nA rectangular room has an area of $2^{12} \\times 5^8$ square units and a length that is twice its width. What is the perimeter of the room?<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import *\n\nwidth = symbols('width')\nlength = 2 * width\n\nar... | [
"We can now see that the perimeter is $6\\sqrt{area}$ and put the value of area:\n$6\\sqrt{1600000000} = 6 * 10000 = 60000$.\nThe answer is $\\boxed{60000}$."
] | [
"60000"
] | [
null
] | ||
5 | In a certain arithmetic series, the combined value of the 2nd and 4th terms is $6$. Identify what the 3rd term in this series is. | gpt-3.5-turbo | Let's denote the first term in the arithmetic series as $a$ and the common difference between consecutive terms as $d$.
According to the problem, the combined value of the 2nd and 4th terms is $6$. Mathematically, we can express this as:
$(a + d) + (a + 3d) = 6$
Now, let's simplify this equation:
$2a + 4d = 6$
We ... | [
"<bos><start_of_turn>user\nIn a certain arithmetic series, the combined value of the 2nd and 4th terms is $6$. Identify what the 3rd term in this series is.<end_of_turn>\n<start_of_turn>model\nWe can define a function which returns the arithmetic progression term, given the common difference d, first term a and ter... | [
"We can now examine the result of the last iteration and see that d is 1, a is 2 and a3 is 3, so the answer is $\\boxed{3}$."
] | [
"3"
] | [
null
] | ||
6 | The sum of the $x$-coordinates of the vertices of a square in the Cartesian plane is $12$. Find the sum of the $x$-coordinates of the center of each side of the square. | gpt-3.5-turbo | Let's consider a square in the Cartesian plane. We know that the $x$-coordinates of the vertices are the same, so let's denote the $x$-coordinate of each vertex as $x$.
If we label the vertices of the square as $A$, $B$, $C$, and $D$ in counterclockwise order starting from the bottom-left vertex, then the $x$-coordin... | [
"<bos><start_of_turn>user\nThe sum of the $x$-coordinates of the vertices of a square in the Cartesian plane is $12$. Find the sum of the $x$-coordinates of the center of each side of the square.<end_of_turn>\n<start_of_turn>model\nWe can compute the sum of the x-coordinates of the center of each side using the fol... | [
"The sum of the x-coordinates of the center of each side is $\\boxed{48}$."
] | [
"48"
] | [
null
] | ||
7 | The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 4 classrooms. There are 56 boys and 44 girls. How many total students are in each classroom? | gsm8k | Each classroom will have 14 boys because 56 boys divided into 4 classrooms equals 14 boys per classroom.
Each classroom will have 11 girls because 44 girls divided into 4 classrooms equals <<44/4=11>>11 girls per classroom.
In total, each classroom will have 25 students because 14 boys plus 11 girls equals 25 total stu... | [
"<bos><start_of_turn>user\nThe school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 4 classrooms. There are 56 boys and 44 girls. How many total students are in each classroom?<end_of_turn>\n<start_of_turn>model\nLet's solve this pro... | [
"Thus there are \\boxed{25} students per classroom."
] | [
"25"
] | [
null
] | ||
8 | Among the numbers $\frac{7}{10}, \frac{4}{5}$ and $\frac{3}{4}$, which number is the average of the other two? | MATH_Rephrased | To find the average of two numbers, we add the two numbers and divide by 2.
So, the average of $\frac{7}{10}$ and $\frac{4}{5}$ is $\frac{1}{2} \left( \frac{7}{10} + \frac{4}{5} \right) = \frac{1}{2} \left( \frac{35}{50} + \frac{40}{50} \right) = \frac{1}{2} \cdot \frac{75}{50} = \frac{75}{100} = \frac{3}{4}$.
Since $\... | [
"<bos><start_of_turn>user\nAmong the numbers $\\frac{7}{10}, \\frac{4}{5}$ and $\\frac{3}{4}$, which number is the average of the other two?<end_of_turn>\n<start_of_turn>model\nWe can compute the average of the given numbers as follows:\n\n```python\nfrom sympy import Rational, Interval\n\n# Define the numbers\nnum... | [
"Therefore, the number \u0003frac{3}{4} is the average of the other two numbers. Hence the answer is $\\boxed{\\frac{3}{4}}$."
] | [
"\\frac{3}{4}"
] | [
null
] | ||
9 | What is the greatest common divisor of $2^{1998}-1$ and $2^{1989}-1$? | math | Let $m = 2^{1998} - 1$ and $n = 2^{1989}-1$. Then, $2^9n = 2^9(2^{1989}-1) = 2^{1998} - 2^9 = m - (2^9 - 1)$. By the Euclidean algorithm, it follows that \begin{align*}
\text{gcd}\,(m,n) &= \text{gcd}\,(n,m-2^9n) \\
&= \text{gcd}\,(n,2^9-1). \\
\end{align*}Since $9$ divides $1998$, by the difference of odd powers facto... | [
"<bos><start_of_turn>user\nWhat is the greatest common divisor of $2^{1998}-1$ and $2^{1989}-1$?<end_of_turn>\n<start_of_turn>model\nWe can compute the greatest common divisor of $2^{1998}-1$ and $2^{1989}-1$ using sympy as follows:\n```python\nfrom sympy import gcd\n\n# Compute the greatest common divisor\ngcd(2**... | [
"So the answer is $\\boxed{511}$."
] | [
"511"
] | [
null
] | ||
10 | In how many unique ways can the characters of the term 'MADAM' be rearranged, considering the repeated letters? | gpt-3.5-turbo | To find the number of unique ways to rearrange the characters of the term 'MADAM' considering the repeated letters, we need to calculate the number of permutations of the letters.
The term 'MADAM' contains 5 letters in total, with 2 'M's and 2 'A's.
The number of unique ways to rearrange these letters can be calculat... | [
"<bos><start_of_turn>user\nIn how many unique ways can the characters of the term 'MADAM' be rearranged, considering the repeated letters?<end_of_turn>\n<start_of_turn>model\nWe can compute the permutations of M A D A M, including the repeated letters.\n```python\nimport itertools\n\nans = len(list(itertools.permut... | [
"Hence, there are 120 such unique ways, hence the answer is $\\boxed{120}$."
] | [
"120"
] | [
null
] | ||
11 | Bill's roof can bear x pounds of weight. If 100 leaves fall on his roof every day, and 1000 leaves weighs 1 pound, It will take 5000 days for his roof to collapse. What is the value of unknown variable x? | GSM_SV | To solve this problem, we need to determine the value of x, which represents the weight-bearing capacity of Bill's roof in pounds.
We know that 1000 leaves weigh 1 pound. Therefore, each leaf weighs 1/1000 pounds.
If 100 leaves fall on Bill's roof every day, the weight of the leaves per day is 100/1000 = 1/10 pounds.
I... | [
"<bos><start_of_turn>user\nBill's roof can bear x pounds of weight. If 100 leaves fall on his roof every day, and 1000 leaves weighs 1 pound, It will take 5000 days for his roof to collapse. What is the value of unknown variable x?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```... | [
"Solving for x, we get:\n\\boxed{5} pounds."
] | [
"5"
] | [
null
] | ||
12 | Luna, the poodle, is supposed to eat 2 cups of kibble every day. But Luna's master, Mary, and her husband, Frank, sometimes feed Luna too much kibble. One day, starting with a new, 12-cup bag of kibble, Mary gave Luna 1 cup of kibble in the morning and 1 cup of kibble in the evening, But on the same day, Frank also g... | gsm8k | Mary fed Luna 1+1=2 cups of kibble.
Frank fed Luna 1-cup plus twice 1-cup, or 1+2=3 cups of kibble.
In total, they fed Luna 2+3=<<2+3=5>>5 cups of kibble.
Thus, if the new bag held 12 cups of kibble, the next morning, 12-5=<<12-5=7>>7 cups of kibble remained in the bag.
#### 7 | [
"<bos><start_of_turn>user\nLuna, the poodle, is supposed to eat 2 cups of kibble every day. But Luna's master, Mary, and her husband, Frank, sometimes feed Luna too much kibble. One day, starting with a new, 12-cup bag of kibble, Mary gave Luna 1 cup of kibble in the morning and 1 cup of kibble in the evening, Bu... | [
"Thus Mary will find \\boxed{7} cups of kibble remaining in the bag."
] | [
"7"
] | [
null
] | ||
13 | Convert $969_{10}$ to base 8. | gpt-3.5-turbo | To convert a decimal number to base 8, we can use the method of repeated division by 8.
We start by dividing 969 by 8, and write down the quotient and the remainder. The remainder will be the least significant digit in base 8. Then, we divide the quotient by 8 again, and write down the new remainder. We continue this ... | [
"<bos><start_of_turn>user\nConvert $969_{10}$ to base 8.<end_of_turn>\n<start_of_turn>model\nWe can use Python's sympy library to convert base-10 integer to any other base.\n```python\nfrom sympy import digamma\n\nn = 969\nbase = 8\n\nn_list = []\n# The idea is to repeatedly divide $n$ by $b$ and append the remaind... | [
"So $969_{10}$ in base 8 is $\\boxed{1711_8}$"
] | [
"1711_8"
] | [
null
] | ||
14 | What is the remainder when the base-12 integer $1531_{12}$ is divided by $8$? | MATH_Rephrased | First, we need to convert $1531_{12}$ to base-10.
$1531_{12} = 1 \cdot 12^3 + 5 \cdot 12^2 + 3 \cdot 12^1 + 1 \cdot 12^0 = 1 \cdot 1728 + 5 \cdot 144 + 3 \cdot 12 + 1 \cdot 1 = 1728 + 720 + 36 + 1 = 2485$.
Now, we divide $2485$ by $8$ to find the remainder.
$2485 = 8 \cdot 310 + 5$.
Therefore, the remainder is $\boxed{... | [
"<bos><start_of_turn>user\nWhat is the remainder when the base-12 integer $1531_{12}$ is divided by $8$?<end_of_turn>\n<start_of_turn>model\nWe can use Python's sympy library to do base conversion and modulo operations.\n```python\nfrom sympy import Rational\n\ndef base_to_dec(base, number):\n decimal = 0\n p... | [
"So the answer is \\boxed{5}."
] | [
"5"
] | [
null
] | ||
15 | In an educational circle game involving 42 contenders, including Lily, everyone takes turns using three particular calculators. These calculators have starting values of 1, 0, and -1. Lily goes first and performs the following on each: she raises the first calculator's display to the power of three, the second she leav... | gpt-3.5-turbo | Let's analyze the operations performed on the calculators.
For the first calculator, each contender raises its display to the power of three. We know that $$1^3 = 1$$ and $$(-1)^3 = -1$$. Therefore, after an odd number of contenders, the first calculator will display $-1$, and after an even number of contenders, it w... | [
"<bos><start_of_turn>user\nIn an educational circle game involving 42 contenders, including Lily, everyone takes turns using three particular calculators. These calculators have starting values of 1, 0, and -1. Lily goes first and performs the following on each: she raises the first calculator's display to the powe... | [
"So the answer is $\\boxed{0}$."
] | [
"0"
] | [
null
] | ||
16 | If Luigi purchased four pizzas for a total of $80 and each pizza was divided into 5 pieces, what is the cost of each individual piece of pizza, in dollars? | GSM_Rephrased | Luigi purchased four pizzas for a total of $80, so each pizza cost $80/4 = $20.
Each pizza was divided into 5 pieces, so the cost of each individual piece of pizza is $20/5 = $4.
#### 4
The answer is: 4 | [
"<bos><start_of_turn>user\nIf Luigi purchased four pizzas for a total of $80 and each pizza was divided into 5 pieces, what is the cost of each individual piece of pizza, in dollars?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```python\ntotal_cost = 80\ntotal_pizzas = 4\nper_pi... | [
"Thus the cost of each piece of pizza is \\boxed{4} dollars."
] | [
"4"
] | [
null
] | ||
17 | In the equation $\displaystyle\sqrt[3]{4 - \frac{p^2}{3}} = -2$, find the positive difference between the two values of p that satisfy the equation. | gpt-3.5-turbo | To solve the given equation, we start by isolating the term containing $p^2$. We can do this by cubing both sides of the equation:
$\left(\sqrt[3]{4 - \frac{p^2}{3}}\right)^3 = (-2)^3$
Simplifying, we get:
$4 - \frac{p^2}{3} = -8$
Next, we can solve this equation for $p^2$:
$-\frac{p^2}{3} = -12$
Dividing both si... | [
"<bos><start_of_turn>user\nIn the equation $\\displaystyle\\sqrt[3]{4 - \\frac{p^2}{3}} = -2$, find the positive difference between the two values of p that satisfy the equation.<end_of_turn>\n<start_of_turn>model\nWe can use sympy for solving equation of such kind.\n```python\nfrom sympy import symbols, solve\n\np... | [
"So there are no real solutions of given equation.\nThus the answer is $\\boxed{0}$."
] | [
"0"
] | [
null
] | ||
18 | What is the minimum number of rides you need to take every month so that the cost per ride of a 30-day pass, which costs $50, is less than the cost of a one-way ticket, which costs $2 per ride? | GSM_Rephrased | To find the minimum number of rides needed, we need to compare the cost per ride of the 30-day pass with the cost of a one-way ticket.
The cost per ride of the 30-day pass is $50 divided by the number of rides taken in a month. Let's call this number R.
So the cost per ride of the 30-day pass is $50/R.
The cost of a on... | [
"<bos><start_of_turn>user\nWhat is the minimum number of rides you need to take every month so that the cost per ride of a 30-day pass, which costs $50, is less than the cost of a one-way ticket, which costs $2 per ride?<end_of_turn>\n<start_of_turn>model\nWe can compute the minimum number of rides such that the to... | [
"Unfortunately sympy didn't give us an answer this time, but we can solve it manually. The minimum number of rides should be $\\boxed{16}$."
] | [
"16"
] | [
null
] | ||
19 | Assuming all letters are unique, how many possible combinations of the word COMBINATION can be formed? | gpt-3.5-turbo | To find the number of possible combinations of the word COMBINATION, we need to calculate the total number of permutations.
The word COMBINATION has 11 letters in total, with 2 O's and 2 N's repeated.
Therefore, the number of permutations is given by:
$\dfrac{11!}{2! \times 2!} = \dfrac{11 \times 10 \times 9 \times ... | [
"<bos><start_of_turn>user\nAssuming all letters are unique, how many possible combinations of the word COMBINATION can be formed?<end_of_turn>\n<start_of_turn>model\nWe can compute the answer by iterating over all possible permutations of the word COMBINATION. This will give us all possible combinations.\n\nThe fol... | [
"So the answer is $\\boxed{12}$."
] | [
"12"
] | [
null
] | ||
20 | Nikki's garden has a ratio of irises to roses to tulips of 2:5:3. She currently has 25 roses and 15 tulips. If she plans to add 20 roses and a certain number of irises to maintain the ratio of irises to roses while adding 10 tulips, how many irises will she have in total? | gpt-3.5-turbo | The current ratio of irises to roses to tulips in Nikki's garden is 2:5:3.
This means that for every 2 irises, there are 5 roses and 3 tulips.
We are given that Nikki currently has 25 roses and 15 tulips.
To maintain the ratio of irises to roses while adding 20 roses and 10 tulips, we need to find the number of iris... | [
"<bos><start_of_turn>user\nNikki's garden has a ratio of irises to roses to tulips of 2:5:3. She currently has 25 roses and 15 tulips. If she plans to add 20 roses and a certain number of irises to maintain the ratio of irises to roses while adding 10 tulips, how many irises will she have in total?<end_of_turn>\n<s... | [
"So the answer is \\boxed{28}."
] | [
"28"
] | [
null
] | ||
21 | Sarah, Greg, and Lisa went bowling together. Sarah's score was 40 points more than Greg's, and Lisa's score was 10 points less than Greg's. If the average of their three scores was 94, what was Sarah's score? | gpt-3.5-turbo | Let's denote Greg's score as $x$.
According to the problem, Sarah's score was 40 points more than Greg's, so Sarah's score would be $x + 40$.
Lisa's score was 10 points less than Greg's, so Lisa's score would be $x - 10$.
The sum of their scores would then be $x + (x + 40) + (x - 10) = 3x + 30$.
Since the average o... | [
"<bos><start_of_turn>user\nSarah, Greg, and Lisa went bowling together. Sarah's score was 40 points more than Greg's, and Lisa's score was 10 points less than Greg's. If the average of their three scores was 94, what was Sarah's score?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python's symp... | [
"Thus Sarah scored \\boxed{166} points."
] | [
"166"
] | [
null
] | ||
22 | A rectangular silver block has dimensions $n\times m\times p$ inches. Given that a two-inch cube ($2\times2\times2$) of silver weighs 3 pounds and is worth $\$200$, how much is the silver block worth when $n=4, m=6, p=2$? Round your answer to the nearest dollar. | gpt-3.5-turbo | To find the weight of the silver block, we need to find the volume of the block and then multiply it by the weight of a two-inch cube.
The volume of the rectangular silver block is given by the formula:
\[V = n \times m \times p\]
Substituting the given values $n=4$, $m=6$, and $p=2$, we have:
\[V = 4 \times 6 \ti... | [
"<bos><start_of_turn>user\nA rectangular silver block has dimensions $n\\times m\\times p$ inches. Given that a two-inch cube ($2\\times2\\times2$) of silver weighs 3 pounds and is worth $\\$200$, how much is the silver block worth when $n=4, m=6, p=2$? Round your answer to the nearest dollar.<end_of_turn>\n<start_... | [
"So the answer is \\boxed{$576}."
] | [
"576"
] | [
null
] | ||
23 | Let $\alpha$, $\beta$, and $\gamma$ be complex numbers such that $|\beta| = 1$, $|\gamma| = 2$, and $\overline{\alpha} \beta \neq 1.$ Find the maximum value of
\[\left| \frac{\gamma(\beta - \alpha)}{1 - \overline{\alpha} \beta} \right|.\] | gpt-3.5-turbo | We can write
\begin{align*}
\left| \frac{\gamma (\beta - \alpha)}{1 - \overline{\alpha} \beta} \right| &= \frac{|\gamma (\beta - \alpha)|}{|1 - \overline{\alpha} \beta|} \\
&= \frac{2 |\beta - \alpha|}{|1 - \overline{\alpha} \beta|} \\
&= \frac{2 |\alpha - \beta|}{|1 - \overline{\alpha} \beta|}.
\end{align*}By the Tria... | [
"<bos><start_of_turn>user\nLet $\\alpha$, $\\beta$, and $\\gamma$ be complex numbers such that $|\\beta| = 1$, $|\\gamma| = 2$, and $\\overline{\\alpha} \\beta \\neq 1.$ Find the maximum value of\n\\[\\left| \\frac{\\gamma(\\beta - \\alpha)}{1 - \\overline{\\alpha} \\beta} \\right|.\\]<end_of_turn>\n<start_of_turn... | [
"We can now examine the max value of the simplified expression, by examining $\\alpha$, $\\beta$, and $\\gamma$ one by one and utilizing sympy to simplify the expression, and looking at the output.\nWe find that the max value of the simplified expression is \\boxed{2}."
] | [
"2"
] | [
null
] | ||
24 | A rectangular prism has dimensions in the ratio 1:2:4 and the shortest edge is equal to an edge of a cube with volume $27 cm^3$. If each edge of the rectangular prism is increased by $50\%$, what will be the new volume of the prism? | gpt-3.5-turbo | Let the dimensions of the rectangular prism be $x$, $2x$, and $4x$, where $x$ is the length of the shortest edge.
The volume of the rectangular prism is given by the formula: $V = l \times w \times h$, where $l$, $w$, and $h$ are the lengths of the three dimensions. So, the initial volume of the prism is:
$V_1 = x \t... | [
"<bos><start_of_turn>user\nA rectangular prism has dimensions in the ratio 1:2:4 and the shortest edge is equal to an edge of a cube with volume $27 cm^3$. If each edge of the rectangular prism is increased by $50\\%$, what will be the new volume of the prism?<end_of_turn>\n<start_of_turn>model\nWe can compute the ... | [
"The new volume of the prism is $\\boxed{1458}$ cm${}^3$."
] | [
"1458"
] | [
null
] | ||
25 | What is the remainder when $2^{2013}$ is divided by 7? | gpt-3.5-turbo | To find the remainder when $2^{2013}$ is divided by 7, we can use modular arithmetic.
We start by finding the remainders when the powers of 2 are divided by 7:
\begin{align*}
2^1 &\equiv 2 \pmod{7} \\
2^2 &\equiv 4 \pmod{7} \\
2^3 &\equiv 1 \pmod{7} \\
2^4 &\equiv 2 \pmod{7} \\
2^5 &\equiv 4 \pmod{7} \\
2^6 &\equiv ... | [
"<bos><start_of_turn>user\nWhat is the remainder when $2^{2013}$ is divided by 7?<end_of_turn>\n<start_of_turn>model\nWe can compute the remainder of $2^{2013}$ when divided by 7 by using sympy.\n```python\nfrom sympy import Integer, Mod\n\nremainder = Mod(2**2013, 7)\nprint(remainder)\n```<end_of_turn>\n<start_of_... | [
"So the answer is $\\boxed{1}$."
] | [
"1"
] | [
null
] | ||
26 | Given $h(x) = 2x + b$ and $p(x) = bx - 1.$ If $h(p(x)) = 8x + e,$ what is the value of $e$? | gpt-3.5-turbo | To find the value of $e,$ we need to substitute the expression for $p(x)$ into $h(p(x)),$ and then compare the resulting expression with $8x + e.$
Starting with $h(p(x)),$ we substitute $p(x) = bx - 1$:
\[h(p(x)) = h(bx - 1) = 2(bx - 1) + b = 2bx - 2 + b.\]
Now we compare this expression with $8x + e.$ To do this, we... | [
"<bos><start_of_turn>user\nGiven $h(x) = 2x + b$ and $p(x) = bx - 1.$ If $h(p(x)) = 8x + e,$ what is the value of $e$?<end_of_turn>\n<start_of_turn>model\nWe can compute $e$ using sympy.\n```python\nimport sympy as sp\n\n# create symbols\nx, b, e = sp.symbols('x b e')\n\n# define the functions\nh = 2 * x + b\np = b... | [
"We can now see that if $b=1$ then $e=-1$. \nIf $b$ is a different value, let's say $b=3$, then the value of $e$ will be different. So $e$ takes different values for different values of $b$. \nTo get $e$ in terms of $b$ we need to use algebraic manipulation.\nLet's do some algebraic manipulations to get the equatio... | [
"8"
] | [
null
] | ||
27 | Billy has four horses. Each one eats x pounds of oats, twice a day. How many pounds of oats does he need to feed his horses for 3 days?
If we know the answer to the above question is 96, what is the value of unknown variable x? | GSM_FOBAR | Each horse eats x pounds of oats twice a day, so each horse eats 2*x pounds of oats per day.
For 3 days, each horse will eat 3 * 2*x = 6*x pounds of oats.
Billy has four horses, so the total amount of oats needed for 3 days is 4 * 6*x = 24*x pounds.
We are given that the total amount of oats needed is 96 pounds, so we ... | [
"<bos><start_of_turn>user\nBilly has four horses. Each one eats x pounds of oats, twice a day. How many pounds of oats does he need to feed his horses for 3 days?\nIf we know the answer to the above question is 96, what is the value of unknown variable x?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem... | [
"Thus Billy needs \\boxed{4} pounds of oats for his 4 horses."
] | [
"4"
] | [
null
] | ||
28 | Alex starts a new routine where he runs 2 miles on the first day and increases his distance by 2 miles each subsequent day. How many miles will he have run after 10 days? | gpt-3.5-turbo | To find out how many miles Alex will have run after 10 days, we need to find the sum of an arithmetic series.
The first term, $a$, is 2 miles.
The common difference, $d$, is 2 miles (since he increases his distance by 2 miles each subsequent day).
The number of terms, $n$, is 10 days.
The formula to find the sum of a... | [
"<bos><start_of_turn>user\nAlex starts a new routine where he runs 2 miles on the first day and increases his distance by 2 miles each subsequent day. How many miles will he have run after 10 days?<end_of_turn>\n<start_of_turn>model\nWe can define a function which returns the total distance run after any number of ... | [
"So Alex has ran \\boxed{110} miles after 10 days."
] | [
"110"
] | [
null
] | ||
29 | A park has x benches with a capacity of 4 people each. On a Sunday afternoon, 80 people were sitting on the benches. What's the number of available spaces on the benches that other people can sit on before the carrying capacity of the benches is full?
If we know the answer to the above question is 120, what is the valu... | GSM_FOBAR | Each bench has a capacity of 4 people, so the total capacity of x benches is 4x.
On a Sunday afternoon, 80 people were sitting on the benches.
The number of available spaces on the benches is the total capacity minus the number of people sitting on them: 4x - 80.
We are given that the number of available spaces is 120,... | [
"<bos><start_of_turn>user\nA park has x benches with a capacity of 4 people each. On a Sunday afternoon, 80 people were sitting on the benches. What's the number of available spaces on the benches that other people can sit on before the carrying capacity of the benches is full?\nIf we know the answer to the above q... | [
"So the answer is \\boxed{100}."
] | [
"100"
] | [
null
] | ||
30 | If Carlos randomly selects a chocolate from a box containing 50 chocolates, where 3 are caramels, twice as many are nougats, the number of truffles is equal to the number of caramels plus 6, and the remaining chocolates are peanut clusters, what is the percentage chance that he will pick a peanut cluster? | GSM_Rephrased | The number of caramels is 3.
The number of nougats is twice as many as caramels, so 2 * 3 = 6 nougats.
The number of truffles is equal to the number of caramels plus 6, so 3 + 6 = 9 truffles.
The total number of caramels, nougats, and truffles is 3 + 6 + 9 = 18 chocolates.
The remaining chocolates are peanut clusters, ... | [
"<bos><start_of_turn>user\nIf Carlos randomly selects a chocolate from a box containing 50 chocolates, where 3 are caramels, twice as many are nougats, the number of truffles is equal to the number of caramels plus 6, and the remaining chocolates are peanut clusters, what is the percentage chance that he will pick ... | [
"Thus the percentage chance of getting a peanut cluster is \\boxed{64}%"
] | [
"64"
] | [
null
] | ||
31 | Bill decides to bring donuts to work for a meeting that day. He buys a box of donuts that has x in total in them. He eats 2 on the ride in because he's hungry. When he gets to the office, the secretary takes another 4 out of the box when he's not looking. Lastly, right before the meeting Bill sets the box down on h... | GSM_SV | To solve this problem, we need to determine the value of x, which represents the total number of donuts that Bill initially bought.
Let's break down the information given:
Number of donuts Bill initially bought: x
Number of donuts Bill ate on the ride in: 2
Number of donuts the secretary took: 4
Number of donuts stolen... | [
"<bos><start_of_turn>user\nBill decides to bring donuts to work for a meeting that day. He buys a box of donuts that has x in total in them. He eats 2 on the ride in because he's hungry. When he gets to the office, the secretary takes another 4 out of the box when he's not looking. Lastly, right before the meet... | [
"Thus there were \\boxed{50} donuts in total."
] | [
"50"
] | [
null
] | ||
32 | Out of all the gingerbread cookies that Anna is making, some have red hats, some have blue boots, and some have both red hats and blue boots. What percentage of the gingerbread men have red hats? | GSM_Rephrased | We don't have the total number of gingerbread cookies, so let's assume it is 100 for simplicity.
Let's say there are 40 gingerbread men with red hats and 30 gingerbread men with blue boots.
We also know that some gingerbread men have both red hats and blue boots.
Let's assume there are 10 gingerbread men with both red ... | [
"<bos><start_of_turn>user\nOut of all the gingerbread cookies that Anna is making, some have red hats, some have blue boots, and some have both red hats and blue boots. What percentage of the gingerbread men have red hats?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```python\nt... | [
"Thus \\boxed{60}% of cookies have a red hat."
] | [
"60"
] | [
null
] | ||
33 | Find the difference between the larger and the smaller root of the quadratic equation $$(x-3)(x+3) = 21x - 63.$$ | gpt-3.5-turbo | To find the roots of the quadratic equation $(x-3)(x+3) = 21x - 63$, we begin by simplifying the equation on both sides:
Expanding the left side of the equation gives us $x^2 - 9 = 21x - 63$.
Bringing all terms to one side, we have $x^2 - 21x + 54 = 0$.
To find the roots, we can use the quadratic formula: $x = \f... | [
"<bos><start_of_turn>user\nFind the difference between the larger and the smaller root of the quadratic equation $$(x-3)(x+3) = 21x - 63.$$<end_of_turn>\n<start_of_turn>model\nWe can use sympy for solving quadratic equations and finding the roots.\n```python\nfrom sympy import symbols, solve\n\n# Define the equatio... | [
"So the difference between the larger and the smaller root is $18 - 3 = \\boxed{15}$."
] | [
"15"
] | [
null
] | ||
34 | If $f(x) = x + 5$ and $g(x) = x^2 - 4x$, find the value of $f(g(3))$. | gpt-3.5-turbo | First, let's find the value of $g(3)$.
We substitute $x = 3$ into the expression for $g(x)$:
$g(3) = (3)^2 - 4(3) = 9 - 12 = -3$
Next, let's find the value of $f(g(3))$.
We substitute $x = -3$ into the expression for $f(x)$:
$f(g(3)) = f(-3) = -3 + 5 = 2$
Therefore, the value of $f(g(3))$ is $\boxed{2}$. The answ... | [
"<bos><start_of_turn>user\nIf $f(x) = x + 5$ and $g(x) = x^2 - 4x$, find the value of $f(g(3))$.<end_of_turn>\n<start_of_turn>model\nWe can compute $f(g(3))$ by substituting $3$ into the expression $x^2 - 4x$ and then substituting the result into the expression $x + 5$.\n```python\ndef f(x):\n return x + 5\n\nde... | [
"So $f(g(3)) = \\boxed{2}$."
] | [
"2"
] | [
null
] | ||
35 | For how many integers $n$ does the condition $n^3 > -50$ and $n^3 < 50$ apply? | gpt-3.5-turbo | To solve this problem, we need to find the number of integer solutions for $n$ that satisfy the conditions $n^3 > -50$ and $n^3 < 50$.
For $n^3 > -50$, we can take the cube root of both sides to get $n > \sqrt[3]{-50}$.
For $n^3 < 50$, we can take the cube root of both sides to get $n < \sqrt[3]{50}$.
To find the nu... | [
"<bos><start_of_turn>user\nFor how many integers $n$ does the condition $n^3 > -50$ and $n^3 < 50$ apply?<end_of_turn>\n<start_of_turn>model\nWe can compute the intervals of $n$ for which $n^3$ is greater than -50 and less than 50, and count the number of integers in those intervals.\n```python\n# Define the constr... | [
"So the answer is $\\boxed{5}$."
] | [
"5"
] | [
null
] | ||
36 | Evaluate $\log_{\frac{1}{2}}8$. | gpt-3.5-turbo | To evaluate $\log_{\frac{1}{2}}8$, we want to find the exponent to which we raise $\frac{1}{2}$ in order to obtain $8$.
In other words, we want to find $x$ such that $\left(\frac{1}{2}\right)^x=8$.
We can rewrite $8$ as $\left(\frac{1}{2}\right)^{-3}$.
So, we have $\left(\frac{1}{2}\right)^x=\left(\frac{1}{2}\right)... | [
"<bos><start_of_turn>user\nEvaluate $\\log_{\\frac{1}{2}}8$.<end_of_turn>\n<start_of_turn>model\nWe can compute the answer by using logarithms identities as follows:\n\n```python\nfrom sympy import log, exp\n\n# Compute log_1/2(8) using logarithm identities\nresult = log(8) / log(1/2)\nresult\n```<end_of_turn>\n<st... | [
"Therefore, the answer is \\boxed{-3}."
] | [
"-3"
] | [
null
] | ||
37 | If Frankie and Carla played a total of 30 games of ping pong against each other, and Frankie won half as many games as Carla, how many games did Carla win? | GSM_Rephrased | Let's assume that Carla won x games.
Since Frankie won half as many games as Carla, Frankie won x/2 games.
The total number of games played is the sum of the games won by Carla and Frankie, which is x + x/2 = 30.
Combining like terms, we get 3x/2 = 30.
To solve for x, we can multiply both sides of the equation by 2/3, ... | [
"<bos><start_of_turn>user\nIf Frankie and Carla played a total of 30 games of ping pong against each other, and Frankie won half as many games as Carla, how many games did Carla win?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python's sympy library.\n```python\nimport sympy as sp\n\n# let's ... | [
"Thus Carla won \\boxed{20} games."
] | [
"20"
] | [
null
] | ||
38 | Given that a line is parameterized by a parameter $t$ and the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$, what is the vector on the line at $t = -7$? | MATH_Rephrased | The vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$.
The change in $t$ from 2 to 3 is 1, so the change in the vector is $\begin{pmatrix} 3 \\ -4 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix}... | [
"<bos><start_of_turn>user\nGiven that a line is parameterized by a parameter $t$ and the vector on the line at $t = 2$ is $\\begin{pmatrix} 1 \\\\ 4 \\end{pmatrix}$, and the vector on the line at $t = 3$ is $\\begin{pmatrix} 3 \\\\ -4 \\end{pmatrix}$, what is the vector on the line at $t = -7$?<end_of_turn>\n<start... | [
"So the answer is $\\boxed{\\begin{pmatrix}-17 \\\\ 76\\end{pmatrix}}$."
] | [
"\\begin{pmatrix}-17\\76\\end{pmatrix}"
] | [
null
] | ||
39 | Suppose $a,$ $b,$ and $c$ are real numbers with $a + 2b + c = 4.$ Determine the highest value of
\[ab + ac + bc.\] | gpt-3.5-turbo | We use Simon's Favorite Factoring Trick. We can write
\[ab + ac + bc = a(b + c) + bc = a(4 - a) + bc = a^2 - 4a + bc + 4.\]By AM-GM,
\[a^2 + bc \ge 2 \sqrt{a^2 \cdot bc} = 2a \sqrt{bc},\]so
\[ab + ac + bc = a^2 - 4a + bc + 4 \ge 2a \sqrt{bc} - 4a + 4 = 2a (\sqrt{bc} - 2) + 4.\]Also, by AM-GM,
\begin{align*}
a + 2 \sqr... | [
"<bos><start_of_turn>user\nSuppose $a,$ $b,$ and $c$ are real numbers with $a + 2b + c = 4.$ Determine the highest value of\n\\[ab + ac + bc.\\]<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import Poly\n\ndef max_value_for_quadratic(a, b, c, M):\n # Coefficients of the q... | [
"The maximum value is $\\boxed{12}$."
] | [
"12"
] | [
null
] | ||
40 | Calculate the number of trailing zeros in the number $2000^{100} \times 5^{50}$. | gpt-3.5-turbo | To calculate the number of trailing zeros in a number, we need to determine how many times the number can be divided evenly by 10. In other words, we want to determine the power of 10 that divides the number.
We can rewrite $2000^{100} \times 5^{50}$ as $(2^4 \times 10^3)^{100} \times (5^2 \times 10)^{50}$.
Expanding... | [
"<bos><start_of_turn>user\nCalculate the number of trailing zeros in the number $2000^{100} \\times 5^{50}$.<end_of_turn>\n<start_of_turn>model\nWe can compute the trailing zeros of the given number by noticing that each trailing zero is the result of a multiplication by $5$. Therefore, we need to know how many ti... | [
"So the answer is $\\boxed{350}$."
] | [
"350"
] | [
null
] | ||
41 | Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are X parrots total on Bird Island, how many of those parrots are blue?
If we know the answer to the above question is 23, what is the value of unknown variable X? | MATH_FOBAR | We are told that three-fourths of the parrots on Bird Island are green, and the remainder are blue.
To find the number of blue parrots, we can set up an equation using the given information.
We know that three-fourths of the parrots are green, so the number of green parrots is (3/4) * X.
The number of blue parrots is t... | [
"<bos><start_of_turn>user\nThree-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are X parrots total on Bird Island, how many of those parrots are blue?\nIf we know the answer to the above question is 23, what is the value of unknown variable X?<end_of_turn>\n<start_of_turn>mod... | [
"So the answer is $\\boxed{92}$."
] | [
"92"
] | [
null
] | ||
42 | Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the set of all real numbers. | math | The domain of the function is the set of all real numbers if and only if the denominator $-5x^2 + 2x + k$ is nonzero for all $x.$ In other words, the quadratic
\[-5x^2 + 2x + k = 0\]should not have any real solutions. This means that the discriminant is negative, i.e.
\[4 - 4(-5)(k) = 4 + 20k < 0.\]Solving, we find $... | [
"<bos><start_of_turn>user\nFind all values of $k$ so that the domain of\n\\[b(x) = \\frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\\]is the set of all real numbers.<end_of_turn>\n<start_of_turn>model\nWe can compute the roots of the denominator and ensure that they are not equal to zero. This will guarantee that the domain o... | [
"We can now examine all values of $k$ to see if the above solution is correct. The solutions seem to be correct.\n\nTherefore, the value of $k$ so that the domain of $b(x)$ is the set of all real numbers is \\boxed{0}."
] | [
"0"
] | [
null
] | ||
43 | Suppose that $g$ is a function and $g^{-1}$ is the inverse of $g$. If $g(4)=2$, $g(6)=3$, and $g(7)=8$, evaluate $g^{-1}\left(g^{-1}(8)+g^{-1}(2)\right)$. | gpt-3.5-turbo | Let's start by finding the values of $g^{-1}$ for $g(4)=2$, $g(6)=3$, and $g(7)=8$.
From $g(4)=2$, we know that $g^{-1}(2)=4$.
From $g(6)=3$, we know that $g^{-1}(3)=6$.
From $g(7)=8$, we know that $g^{-1}(8)=7$.
Now let's evaluate $g^{-1}(g^{-1}(8)+g^{-1}(2))$.
Substituting the values we found earlier, we have $g... | [
"<bos><start_of_turn>user\nSuppose that $g$ is a function and $g^{-1}$ is the inverse of $g$. If $g(4)=2$, $g(6)=3$, and $g(7)=8$, evaluate $g^{-1}\\left(g^{-1}(8)+g^{-1}(2)\\right)$.<end_of_turn>\n<start_of_turn>model\nWe can use a table to represent the values of $g$ and $g^{-1}$ and get an answer:\n```python\ng ... | [
"So the answer is $\\boxed{11}$."
] | [
"11"
] | [
null
] | ||
44 | Anna, Alison, and Jeff collect stamps. Anna had x stamps in her collection, Alison had 28 stamps in her collection, and Jeff had 31 stamps in his collection. Alison gave Anna half of her collection in exchange for a novel, and then Anna traded Jeff two bluebird stamps for one mountain stamp. Anna had 50 stamps in the e... | GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of stamps Anna initially had in her collection.
Let's break down the information given:
Number of stamps Anna initially had: x
Number of stamps Alison initially had: 28
Number of stamps Jeff initially had: 31
Number of stamps Anna h... | [
"<bos><start_of_turn>user\nAnna, Alison, and Jeff collect stamps. Anna had x stamps in her collection, Alison had 28 stamps in her collection, and Jeff had 31 stamps in his collection. Alison gave Anna half of her collection in exchange for a novel, and then Anna traded Jeff two bluebird stamps for one mountain sta... | [
"Unfortunately, I couldn't get sympy to solve this, but the logic should be clear. Here is how we can approach it, and you can try implementing it using a different language or library (or even without any code!).\n\nAnna had $x$ stamps at the start. After she got $\\frac{28}{2}$ stamps from Alison, she had $x + \\... | [
"18"
] | [
null
] | ||
45 | What is the result of the logarithmic expression: $\log_3(1/\sqrt{3})$? | gpt-3.5-turbo | To find the result of the logarithmic expression $\log_3(1/\sqrt{3})$, we can use the properties of logarithms.
First, let's simplify the expression inside the logarithm. We have $1/\sqrt{3}$.
To simplify this further, we can use the fact that $\sqrt{3} = 3^{1/2}$.
So, $1/\sqrt{3} = 1/3^{1/2}$.
Now, we can rewri... | [
"<bos><start_of_turn>user\nWhat is the result of the logarithmic expression: $\\log_3(1/\\sqrt{3})$?<end_of_turn>\n<start_of_turn>model\nWe can compute the answer by using logarithms and exponentials, but sympy offers a simpler solution.\n```python\nfrom sympy import log, sqrt\n\nn = 3\nexp = 1/sqrt(n)\nanswer = lo... | [
"So the answer is \\boxed{-\\frac{1}{2}}."
] | [
"-\\frac{1}{2}"
] | [
null
] | ||
46 | Ms. Mitsuko told the class that if the average score on their quiz is higher than 75% then they will get a pizza party. There are 25 students in the class. One student is absent and has to take the test the next day. The average score of the students who took the test was x%. The lowest grade the absent student can get... | GSM_SV | To solve this problem, we need to determine the value of x, which represents the average score of the students who took the test.
We know that there are 25 students in the class and one student is absent.
The lowest grade the absent student can get for the class to still get a pizza party is 27.
Let's break down the in... | [
"<bos><start_of_turn>user\nMs. Mitsuko told the class that if the average score on their quiz is higher than 75% then they will get a pizza party. There are 25 students in the class. One student is absent and has to take the test the next day. The average score of the students who took the test was x%. The lowest g... | [
"Thus the lowest grade for the absent student such that they get a pizza party is \\boxed{77}"
] | [
"77"
] | [
null
] | ||
47 | What is the coefficient of $x^5$ in the expansion of $(x+2)^{14}$? | gpt-3.5-turbo | To find the coefficient of $x^5$ in the expansion of $(x+2)^{14}$, we need to expand the binomial using the binomial theorem.
The binomial theorem states that for any real numbers $a$ and $b$ and a positive integer $n$, we have:
$$(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \... | [
"<bos><start_of_turn>user\nWhat is the coefficient of $x^5$ in the expansion of $(x+2)^{14}$?<end_of_turn>\n<start_of_turn>model\nWe can compute the desired coefficient using sympy and then use sympy's Poly function to see if we get the right answer.\n```python\nfrom sympy import Poly, symbols, expand\nx = symbols(... | [
"We can now examine the polynomial's coefficients to find the desired coefficient. We can see that the coefficient of $x^5$ is \\boxed{1025024}."
] | [
"1025024"
] | [
null
] | ||
48 | Consider a numbering system with base $b$. In this system, there are a total of 100 (one hundred in base 10) unique three-digit numbers that consist of distinct digits for each number. What is the value of the base $b$? | gpt-3.5-turbo | To solve this problem, we need to find the value of the base $b$ in the numbering system.
In a base $b$ numbering system, there are $b$ possible digits to choose from for each digit place.
For a three-digit number, the first digit can be chosen from $b-1$ possibilities (excluding zero). The second digit can be chosen... | [
"<bos><start_of_turn>user\nConsider a numbering system with base $b$. In this system, there are a total of 100 (one hundred in base 10) unique three-digit numbers that consist of distinct digits for each number. What is the value of the base $b$?<end_of_turn>\n<start_of_turn>model\nWe can compute the 100 numbers ma... | [
"Unfortunately, this solution takes too long to execute. Alternatively, we can consider the structure of the numbers in base $b$. In base $b$, each three digit number equals $d_2b^2 + d_1b + d_0$ for $d_0 < d_1 < d_2 < b$. If we count all possible permutations of three digits, we get $b(b-1)(b-2)$. However, this do... | [
"4"
] | [
null
] | ||
49 | Matt can make a batch of a dozen cookies using 2 pounds of flour. He uses 4 bags of flour each weighing 5 pounds. If Jim eats 15 cookies how many cookies are left? | gsm8k | Matt used 4*5=<<4*5=20>>20 pounds of flour
That means he made 20/2=<<20/2=10>>10 batches
So he cooked 10*12=<<10*12=120>>120 cookies
After Jim eats 15 there are 120-15=<<120-15=105>>105 cookies left
#### 105 | [
"<bos><start_of_turn>user\nMatt can make a batch of a dozen cookies using 2 pounds of flour. He uses 4 bags of flour each weighing 5 pounds. If Jim eats 15 cookies how many cookies are left?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```python\nflour_in_one_bag = 5\nnumber_of... | [
"Thus there are \\boxed{105} cookies left."
] | [
"105"
] | [
null
] | ||
50 | During the lunch rush, a taco truck is selling soft tacos for $2 each and hard shell tacos for $5 each. The first group of customers, a family, purchases four hard shell tacos and three soft tacos. The remaining customers in the lunch rush each buy two soft tacos. If there were a total of ten customers after the family... | GSM_Rephrased | The family purchased four hard shell tacos for $5 each, so they spent 4 * $5 = $<<4*5=20>>20 on hard shell tacos.
They also purchased three soft tacos for $2 each, so they spent 3 * $2 = $<<3*2=6>>6 on soft tacos.
In total, the family spent $20 + $6 = $<<20+6=26>>26.
The remaining customers each bought two soft tacos f... | [
"<bos><start_of_turn>user\nDuring the lunch rush, a taco truck is selling soft tacos for $2 each and hard shell tacos for $5 each. The first group of customers, a family, purchases four hard shell tacos and three soft tacos. The remaining customers in the lunch rush each buy two soft tacos. If there were a total of... | [
"Thus the total amount of money made was \\boxed{50} dollars."
] | [
"50"
] | [
null
] | ||
51 | A right triangle has a hypotenuse of 15 inches and one of the non-right angles measures $30^{\circ}$. What is the area of this triangle? | gpt-3.5-turbo | To find the area of the triangle, let's first find the lengths of the two legs.
Let the length of the shorter leg be $x$ inches. Since one of the non-right angles measures $30^{\circ}$, we can use trigonometric ratios to find the length of the longer leg.
Using the sine function, we have:
$$\sin (30^{\circ}) = \frac... | [
"<bos><start_of_turn>user\nA right triangle has a hypotenuse of 15 inches and one of the non-right angles measures $30^{\\circ}$. What is the area of this triangle?<end_of_turn>\n<start_of_turn>model\nWe can use the 30-60-90 triangle properties to immediately deduce that $x = 7.5$ and $y = 15\\sqrt{3}/2$.\nWe will ... | [
"So the area of the triangle is $\\boxed{28.125\\sqrt{3}}$."
] | [
"28.125\\sqrt{3}"
] | [
null
] | ||
52 | In the first team, Beth scored 12 while Jan scored 10. In the second team, Judy scored 8 while Angel scored 11. How many more points did the first team get than the second team? | gsm8k | Beth and Jan scored a total of 12 + 10 = <<12+10=22>>22.
Judy and Angel scored a total of 8 + 11 = <<8+11=19>>19.
Thus, the first team scores 22 - 19 = <<22-19=3>>3 more points than the second team.
#### 3 | [
"<bos><start_of_turn>user\nIn the first team, Beth scored 12 while Jan scored 10. In the second team, Judy scored 8 while Angel scored 11. How many more points did the first team get than the second team?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```python\nfirst_team_score = ... | [
"Thus the first team scored \\boxed{3} more points than the second team."
] | [
"3"
] | [
null
] | ||
53 | If $a = 10^{5} \cdot 10^{-3}$, $b = 10^{3} \cdot 10^{-1}$, and $c = 10^{2} \cdot 10^{-2}$, calculate the value of $7a + 5b + 3c \pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive. | gpt-3.5-turbo | To calculate $7a+5b+3c$, we can start by simplifying each term.
We have $a=10^5 \cdot 10^{-3} = 10^{5-3} = 10^2 = 100$.
We have $b=10^3 \cdot 10^{-1} = 10^{3-1} = 10^2 = 100$.
We have $c=10^2 \cdot 10^{-2} = 10^{2-2} = 10^0 = 1$.
Now, we substitute these values into $7a+5b+3c$ and calculate modulo $1001$.
$7a + 5b +... | [
"<bos><start_of_turn>user\nIf $a = 10^{5} \\cdot 10^{-3}$, $b = 10^{3} \\cdot 10^{-1}$, and $c = 10^{2} \\cdot 10^{-2}$, calculate the value of $7a + 5b + 3c \\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive.<end_of_turn>\n<start_of_turn>model\nWe can compute the value of $7a + 5b + 3c$ ... | [
"So the answer is $\\boxed{202}$."
] | [
"202"
] | [
null
] | ||
54 | If Nathan purchased a large box of bananas consisting of six bunches with eight bananas in each bunch and five bunches with seven bananas in each bunch, what is the total number of bananas that Nathan has? | GSM_Rephrased | Nathan has 6 bunches x 8 bananas = 48 bananas from the first set of bunches.
He also has 5 bunches x 7 bananas = 35 bananas from the second set of bunches.
In total, Nathan has 48 bananas + 35 bananas = 83 bananas.
#### 83
The answer is: 83 | [
"<bos><start_of_turn>user\nIf Nathan purchased a large box of bananas consisting of six bunches with eight bananas in each bunch and five bunches with seven bananas in each bunch, what is the total number of bananas that Nathan has?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n``... | [
"Thus Nathan has a total of \\boxed{83} bananas."
] | [
"83"
] | [
null
] | ||
55 | The Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least X girl?
If we know the answer to the above question is \frac{295}{323}, what is the value of unknown variable X? | MATH_FOBAR | We want to find the value of $X$ in the given situation.
The probability of choosing a committee with at least 1 boy and at least $X$ girls can be found by considering the total number of possible committees and the number of committees that satisfy the given condition.
The total number of possible committees is given... | [
"<bos><start_of_turn>user\nThe Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least X girl?\nIf we know the answer to the above question is \\frac{295}{323}, what is the value of unknown variable X?<... | [
"We can now see that if X = 9, we get the desired probability.\nHence, the answer is \\boxed{9}."
] | [
"9"
] | [
null
] | ||
56 | Amanda charges $20.00 per hour to help clean out and organize a person's home. She has 5 x hours appointments on Monday, a 3-hours appointment on Tuesday and 2 2-hours appointments on Thursday. On Saturday, she will spend 6 hours at one client's house. How much money will she make this week?
If we know the answer to... | GSM_FOBAR | On Monday, Amanda has 5 x hours appointments, so she will work a total of 5x hours.
On Tuesday, she has a 3-hour appointment.
On Thursday, she has 2 2-hour appointments, so she will work a total of 2 * 2 = 4 hours.
On Saturday, she will spend 6 hours at one client's house.
The total number of hours she will work this w... | [
"<bos><start_of_turn>user\nAmanda charges $20.00 per hour to help clean out and organize a person's home. She has 5 x hours appointments on Monday, a 3-hours appointment on Tuesday and 2 2-hours appointments on Thursday. On Saturday, she will spend 6 hours at one client's house. How much money will she make this... | [
"Thus Amanda will work for \\boxed{3} hours on Monday."
] | [
"3"
] | [
null
] | ||
57 | If you multiply the skew-symmetric matrix
\[\begin{pmatrix} 0 & m & -l \\ -m & 0 & k \\ l & -k & 0 \end{pmatrix}\]
with the symmetric matrix
\[\begin{pmatrix} k^2 & kl & km \\ kl & l^2 & lm \\ km & lm & m^2 \end{pmatrix},\]
what is the resulting matrix? | gpt-3.5-turbo | To find the product of two matrices, we need to multiply each element of the first matrix by the corresponding element in the second matrix and sum the results.
Let's denote the skew-symmetric matrix as $A$ and the symmetric matrix as $B$.
$A = \begin{pmatrix} 0 & m & -l \\ -m & 0 & k \\ l & -k & 0 \end{pmatrix}$
... | [
"<bos><start_of_turn>user\nIf you multiply the skew-symmetric matrix\n\\[\\begin{pmatrix} 0 & m & -l \\\\ -m & 0 & k \\\\ l & -k & 0 \\end{pmatrix}\\]\nwith the symmetric matrix\n\\[\\begin{pmatrix} k^2 & kl & km \\\\ kl & l^2 & lm \\\\ km & lm & m^2 \\end{pmatrix},\\]\nwhat is the resulting matrix?<end_of_turn>\n<... | [
"Thus the answer is $\\boxed{\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}}$."
] | [
"\\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\\end{pmatrix}"
] | [
null
] | ||
58 | Given $z = -24 + 7i$ and $w = 4 - 3i$, calculate the magnitude of the vector from $z$ to $w$, i.e., find $|w - z|$. | gpt-3.5-turbo | The magnitude of the vector from $z$ to $w$ can be found by subtracting the components of $z$ from the components of $w$ and then taking the absolute value.
We have:
$w - z = (4 - 3i) - (-24 + 7i)$
$= 4 - 3i + 24 - 7i$
$= 28 - 10i$
To find the magnitude of a complex number $a + bi$, we use the formula:
$|a + bi| = \s... | [
"<bos><start_of_turn>user\nGiven $z = -24 + 7i$ and $w = 4 - 3i$, calculate the magnitude of the vector from $z$ to $w$, i.e., find $|w - z|$.<end_of_turn>\n<start_of_turn>model\nWe can compute the magnitude of the vector by finding the complex number representation of $w-z$.\n```python\nfrom sympy import I, symbol... | [
"So the magnitude of the vector from $z$ to $w$ is $\\boxed{2\\sqrt{221}}$."
] | [
"2\\sqrt{221}"
] | [
null
] | ||
59 | Stella wanted to buy a new dress for the upcoming dance. At the store she found out that the dress she wanted was x. The store was offering 30% off of everything in the store. What was the final cost of the dress?
If we know the answer to the above question is 35, what is the value of unknown variable x? | GSM_FOBAR | The original cost of the dress is x.
The store is offering 30% off, which means the final cost of the dress is 70% of the original cost.
To calculate the final cost, we can multiply the original cost by 70%: x * 0.7 = 0.7x.
We are given that the final cost of the dress is 35, so we can write: 0.7x = 35.
Dividing both s... | [
"<bos><start_of_turn>user\nStella wanted to buy a new dress for the upcoming dance. At the store she found out that the dress she wanted was x. The store was offering 30% off of everything in the store. What was the final cost of the dress?\nIf we know the answer to the above question is 35, what is the value of... | [
"Thus the dress costs \\boxed{50} dollars."
] | [
"50"
] | [
null
] | ||
60 | Compute $\cos 210^\circ$. | math | Let $P$ be the point on the unit circle that is $210^\circ$ counterclockwise from $(1,0)$, and let $D$ be the foot of the altitude from $P$ to the $x$-axis, as shown below.
[asy]
pair A,C,P,O,D;
draw((0,-1.2)--(0,1.2),p=black+1.2bp,Arrows(0.15cm));
draw((-1.2,0)--(1.2,0),p=black+1.2bp,Arrows(0.15cm));
A = (1,0);
... | [
"<bos><start_of_turn>user\nCompute $\\cos 210^\\circ$.<end_of_turn>\n<start_of_turn>model\nWe can compute the value of $\\cos 210^\\circ$ using sympy as follows.\n```python\nfrom sympy import cos, pi\n\nangle = 210\nresult = cos(angle * pi/180)\nprint(result)\n```<end_of_turn>\n<start_of_turn>user\n```output\n-sqrt... | [
"So $\\cos 210^\\circ$ is $\\boxed{-\\frac{\\sqrt{3}}{2}}$."
] | [
"-\\frac{\\sqrt{3}}{2}"
] | [
null
] | ||
61 | The solution of $8x+1\equiv 5 \pmod{12}$ is $x\equiv a\pmod{m}$ for some positive integers $m\geq X and $a<m$. Find $a+m$. The answer is 5. What is the value of unknown variable X? | MATH_SV | To solve this problem, we need to find the value of x that satisfies the given congruence equation.
The congruence equation is:
$8x + 1 \equiv 5 \pmod{12}$
To simplify the equation, we subtract 1 from both sides:
$8x \equiv 4 \pmod{12}$
Since we want to find the value of x in terms of modular arithmetic, we can divide ... | [
"<bos><start_of_turn>user\nThe solution of $8x+1\\equiv 5 \\pmod{12}$ is $x\\equiv a\\pmod{m}$ for some positive integers $m\\geq X and $a<m$. Find $a+m$. The answer is 5. What is the value of unknown variable X?<end_of_turn>\n<start_of_turn>model\nWe can compute the value of X using sympy as follows.\n```python\nf... | [
"The smallest value of X is $\\boxed{2}$."
] | [
"2"
] | [
null
] | ||
62 | The circumference of a wheel is 30 cm. What is the area of the wheel in square centimeters? Write your answer as a fraction in terms of $\pi$. | gpt-3.5-turbo | To find the area of the wheel, we need to know the radius of the wheel. The circumference of a circle is given by the formula $C = 2\pi r$, where $C$ is the circumference and $r$ is the radius.
In this problem, we are given that the circumference is 30 cm. So, we can set up the equation $30 = 2\pi r$ and solve for $r... | [
"<bos><start_of_turn>user\nThe circumference of a wheel is 30 cm. What is the area of the wheel in square centimeters? Write your answer as a fraction in terms of $\\pi$.<end_of_turn>\n<start_of_turn>model\nWe can compute the radius of the wheel and then use it to compute the area.\n```python\nimport sympy as sp\n\... | [
"So the answer is $\\boxed{225/\\pi}$."
] | [
"225/\\pi"
] | [
null
] | ||
63 | James buys a jar of hot sauce. Each serving is .5 ounces. He uses x servings every day. The container will last for 20 days if it is 2 ounces less than 1 quart. What is the value of unknown variable x? | GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of servings of hot sauce James uses every day.
Let's break down the information given:
Size of each serving: 0.5 ounces
Number of servings used per day: x
Number of days the container will last: 20
Size of 1 quart: 32 ounces
Size of... | [
"<bos><start_of_turn>user\nJames buys a jar of hot sauce. Each serving is .5 ounces. He uses x servings every day. The container will last for 20 days if it is 2 ounces less than 1 quart. What is the value of unknown variable x?<end_of_turn>\n<start_of_turn>model\nLet's solve this problem using Python code.\n```... | [
"Thus James uses \\boxed{3} servings of hot sauce per day."
] | [
"3"
] | [
null
] | ||
64 | What single digit replaces $\star$ in the equation $\star4_7 = \star1_{8}$ to make it true? | gpt-3.5-turbo | To solve this problem, we need to convert the numbers from base 7 to base 10.
Let's first convert $\star4_{7}$ to base 10. In base 7, the digit $\star$ could be any digit from 0 to 6. So we have:
$\star4_{7} = \star \cdot 7^1 + 4 \cdot 7^0$
To convert this to base 10, we simply evaluate the expression:
$\star4_{7} ... | [
"<bos><start_of_turn>user\nWhat single digit replaces $\\star$ in the equation $\\star4_7 = \\star1_{8}$ to make it true?<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import Poly\n\ndef get_value_for_digit(number, base, digit_position):\n poly = Poly(number, base)\n co... | [
"We can see that 74 in base-7 is [1, 7, 4] and 81 in base-8 is [1, 8, 1]. So if we set $\\star=6$, it will hold that [1, 7, 4] = [1, 6, 1].\nThus answer is $\\boxed{6}$."
] | [
"6"
] | [
null
] | ||
65 | What is the modulo $4$ remainder of the sum $2+4+6+8+10+ \ldots + 244+246+248?$ | gpt-3.5-turbo | We can observe that each term in the sum is obtained by multiplying $2$ by consecutive even numbers. Let's denote the first term of the sum as $a$, the common difference between consecutive terms as $d$, and the number of terms as $n$.
We can use the formula for the sum of an arithmetic series to find the sum of this ... | [
"<bos><start_of_turn>user\nWhat is the modulo $4$ remainder of the sum $2+4+6+8+10+ \\ldots + 244+246+248?$<end_of_turn>\n<start_of_turn>model\nWe can compute the sum of the given numbers manually and find the modulo 4 remainder.\n```python\ntotal_sum = 0\n\nfor i in range(2, 249, 2):\n total_sum += i\n\n# Compu... | [
"So the answer is $\\boxed{0}$."
] | [
"0"
] | [
null
] | ||
66 | How can you express the binomial $t^2-49$ as a multiplication of two linear binomials? | gpt-3.5-turbo | To express the binomial $t^2-49$ as a product of two linear binomials, we need to find two binomials of the form $(t+a)(t+b)$ whose product is equal to $t^2-49$.
We can start by factoring out the perfect square from $t^2-49$. The expression $t^2-49$ can be rewritten as $(t^2-7^2)$.
Now, we have a difference of squa... | [
"<bos><start_of_turn>user\nHow can you express the binomial $t^2-49$ as a multiplication of two linear binomials?<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import Poly, pprint\n\n# create polynomial\np = Poly(\"t**2 - 49\")\n\n# factor\nfactors = p.all_factors()\n\n# prin... | [
"And now `factor` method returns an expression which has desired form:\n$\\boxed{(t-7)(t+7)}$."
] | [
"(t-7)(t+7)"
] | [
null
] | ||
67 | If $x = \frac{9 \pi}{4} + \pi n$ and $y = \frac{5 \pi}{4} + \pi m$ for any integers $n$ and $m$, find the value of $\tan(x) \cdot \tan(y) + \tan^2(x)$. | gpt-3.5-turbo | To find the value of $\tan(x) \cdot \tan(y) + \tan^2(x)$, we first express $\tan(x)$ and $\tan(y)$ in terms of $\sin$ and $\cos$.
Using the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$, we have:
$\tan(x) = \frac{\sin\left(\frac{9 \pi}{4} + \pi n\right)}{\cos\left(\frac{9 \pi}{4} + \pi n\right)}$
Using the angle sum... | [
"<bos><start_of_turn>user\nIf $x = \\frac{9 \\pi}{4} + \\pi n$ and $y = \\frac{5 \\pi}{4} + \\pi m$ for any integers $n$ and $m$, find the value of $\\tan(x) \\cdot \\tan(y) + \\tan^2(x)$.<end_of_turn>\n<start_of_turn>model\nWe can compute the tangent of an angle in terms of trigonometric functions as $\\tan(\\thet... | [
"We can now see that $\\tan(x) \\cdot \\tan(y) + \\tan^2(x)$ is equal to\n$$\n\\tan(x) \\cdot \\tan(y) + \\tan^2(x) = \\boxed{\\tan(\\frac{9\\pi}{4} + \\pi n) \\cdot \\tan(\\frac{5\\pi}{4} + \\pi m) + \\tan^2(\\frac{9\\pi}{4} + \\pi n)}\n$$"
] | [
"\\tan(\\frac{9\\pi}{4}+\\pin)\\tan(\\frac{5\\pi}{4}+\\pim)+\\tan^2(\\frac{9\\pi}{4}+\\pin)"
] | [
null
] | ||
68 | Determine the diameter of the circle that is inscribed in triangle ABC, given that AB = 11, AC = 6, and BC = 7. Express your answer in simplest radical form. | MATH_Rephrased | We can use Heron's formula to find the area of triangle ABC. Let $s$ be the semiperimeter of the triangle, which is defined as $s=\frac{11+6+7}{2}=12$.
Then, the area of triangle ABC is $\sqrt{s(s-11)(s-6)(s-7)}=\sqrt{12(12-11)(12-6)(12-7)}=\sqrt{12\cdot1\cdot6\cdot5}=6\sqrt{10}$.
The area of a triangle can also be exp... | [
"<bos><start_of_turn>user\nDetermine the diameter of the circle that is inscribed in triangle ABC, given that AB = 11, AC = 6, and BC = 7. Express your answer in simplest radical form.<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this.\nFirst, we need to find the area of the triangle by Heron's formula:... | [
"So the diameter of the inscribed circle is $\\boxed{3.1622}$."
] | [
"3.1622"
] | [
null
] | ||
69 | Let $a, b, c, d$ be complex numbers, with $a, d$ real, such that \[a+b+c+d=ab+bc+cd+da=abc+bcd+cda+dab=abcd=4.\]Find $a+d$. | gpt-3.5-turbo | We are given that $a+b+c+d=4$. Let's express $ab+bc+cd+da$ in terms of $a+b+c+d$. We have \begin{align*}
ab+bc+cd+da&= (a+c)(b+d)\\
&=(4-b-d)(b+d)\\
&=4(b+d)-b^2-d^2.
\end{align*}
Since we know that $ab+bc+cd+da=4$, we get \[4(b+d)-b^2-d^2=4,\] and rearranging the terms gives \[b^2+d^2-4(b+d)+4=0.\]
Next, we are gi... | [
"<bos><start_of_turn>user\nLet $a, b, c, d$ be complex numbers, with $a, d$ real, such that \\[a+b+c+d=ab+bc+cd+da=abc+bcd+cda+dab=abcd=4.\\]Find $a+d$.<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import Poly, symbols\n\n# Define symbols\na, b, c, d = symbols('a b c d')\n\n... | [
"We can see that all values of $a+d$ are the same:\n```python\nresult = [4, 4, 4, 4]\n```\nSo the answer is $\\boxed{4}$."
] | [
"4"
] | [
null
] | ||
70 | Given matrices $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$, and $C = \begin{pmatrix} j & k \\ l & m \end{pmatrix}$ with $\det(A) = 4$, $\det(B) = 5$, and $\det(C) = 6$, find the determinant of the matrix $M = \begin{pmatrix} A & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & C ... | gpt-3.5-turbo | To find the determinant of the matrix $M$, we can use the property that the determinant of a block diagonal matrix is equal to the product of the determinants of the diagonal blocks.
We have the matrix $M = \begin{pmatrix} A & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & C \end{pmatrix}$, where the diagonal blocks are $A$, $B$, and ... | [
"<bos><start_of_turn>user\nGiven matrices $A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$, $B = \\begin{pmatrix} e & f \\\\ g & h \\end{pmatrix}$, and $C = \\begin{pmatrix} j & k \\\\ l & m \\end{pmatrix}$ with $\\det(A) = 4$, $\\det(B) = 5$, and $\\det(C) = 6$, find the determinant of the matrix $M = \\begi... | [
"Unfortunately, we are stuck again. Let's look at the expression for the determinant and see if we can figure out a solution.\nThe determinant of M is \\[\\det(M) = \\det(A) \\det(B) \\det(C) = (ad - bc)M.\\]\nNow, let's go back to sympy and look at the expressions that it produced for $A$, $B$, and $C$. We can see... | [
"120"
] | [
null
] | ||
71 | Let $r$ be the positive real solution to $x^3 + \frac{2}{5} x - 1 = 0.$ Find the exact numerical value of
\[r^2 + 2r^5 + 3r^8 + 4r^{11} + \dotsb.\] | math | Let $S = r^2 + 2r^5 + 3r^8 + 4r^{11} + \dotsb.$ Then
\[r^3 S = r^5 + 2r^8 + 3r^{11} + 4r^{13} + \dotsb.\]Subtracting this equation from $S = r^2 + 2r^5 + 3r^8 + 4r^{11} + \dotsb,$ we get
\[S (1 - r^3) = r^2 + r^5 + r^8 + r^{11} + \dotsb = \frac{r^2}{1 - r^3}.\]Hence,
\[S = \frac{r^2}{(1 - r^3)^2}.\]Since $r^3 + \frac{... | [
"<bos><start_of_turn>user\nLet $r$ be the positive real solution to $x^3 + \\frac{2}{5} x - 1 = 0.$ Find the exact numerical value of\n\\[r^2 + 2r^5 + 3r^8 + 4r^{11} + \\dotsb.\\]<end_of_turn>\n<start_of_turn>model\nWe can use sympy for this:\n```python\nfrom sympy import Poly, symbols\n\n# create symbols\nx, a = ... | [
"We can use sympy for this:\n```python\nfrom sympy import Poly, symbols\n\n# create symbols\nx, a = symbols('x a')\n\n# create polynomial\np = x**3 + (2/5) * x - 1\n\n# find all the roots of the polynomial\nroots = Poly(p).all_roots()\n\n# compute sum of powers for only real positive root\nsum_value = 0\nfor root i... | [
"1000"
] | [
null
] | ||
72 | Find the least integer whose square is 25 more than four times the integer. | gpt-3.5-turbo | Let's assume the integer we want to find is $x$.
According to the given problem, we know that the square of $x$ is 25 more than four times $x$. Mathematically, this can be expressed as:
$x^2 = 4x + 25$
To find the least integer that satisfies this equation, we can solve it by rearranging terms and solving for $x$.
... | [
"<bos><start_of_turn>user\nFind the least integer whose square is 25 more than four times the integer.<end_of_turn>\n<start_of_turn>model\nWe can formulate the following equation:\n$n^2 = 4n + 25$\nThis is a quadratic equation and we can solve it using sympy, so\n```python\nimport sympy as sp\nn = sp.symbols('n')\n... | [
"We can now examine the options and find which one has a square which is exactly 25 more than 4 times the number, and that is \\boxed{5}."
] | [
"5"
] | [
null
] | ||
73 | Stella’s antique shop has 3 dolls, x clocks and 5 glasses for sale. She sells the dolls for $5 each. The clocks are priced at $15 each. The glasses are priced at $4 each. If she spent $40 to buy everything and she sells all of her merchandise, She will make 25 profit. What is the value of unknown variable x? | GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of clocks Stella has for sale.
Let's break down the information given:
Number of dolls: 3
Number of clocks: x
Number of glasses: 5
Price per doll: $5
Price per clock: $15
Price per glass: $4
Total cost to buy everything: $40
Total p... | [
"<bos><start_of_turn>user\nStella’s antique shop has 3 dolls, x clocks and 5 glasses for sale. She sells the dolls for $5 each. The clocks are priced at $15 each. The glasses are priced at $4 each. If she spent $40 to buy everything and she sells all of her merchandise, She will make 25 profit. What is the value of... | [
"Thus the number of clocks is \\boxed{2}."
] | [
"2"
] | [
null
] | ||
74 | Don is holding a bottle containing $\frac{1}{5}$ gallon of milk. His friend Rachel sips $\frac{2}{3}$ of the milk from the bottle. What is the volume of milk that Rachel consumed? | gpt-3.5-turbo | To find the volume of milk that Rachel consumed, we need to find $\frac{2}{3}$ of $\frac{1}{5}$ gallon.
To do this, we can multiply $\frac{1}{5}$ gallon by $\frac{2}{3}$.
\[
\frac{1}{5} \times \frac{2}{3} = \frac{2}{15}
\]
So, Rachel consumed $\frac{2}{15}$ gallon of milk.
Since we want the volume in gallon, we can... | [
"<bos><start_of_turn>user\nDon is holding a bottle containing $\\frac{1}{5}$ gallon of milk. His friend Rachel sips $\\frac{2}{3}$ of the milk from the bottle. What is the volume of milk that Rachel consumed?<end_of_turn>\n<start_of_turn>model\nWe can compute the volume of milk that Rachel consumed by subtracting t... | [
"So the answer is $\\boxed{\\frac{2}{15}}$ gallon."
] | [
"\\frac{2}{15}"
] | [
null
] | ||
75 | Consider Rural School USA, with an enrollment of 105 students - 60 of them boys and the remaining 45 girls. If on a particular school day, $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent, how much of the school population does not show up, when expressed as a percentage? | gpt-3.5-turbo | The number of boys absent is $\frac{1}{10}$ of the total number of boys, which can be calculated as $\frac{1}{10} \times 60 = 6$ boys.
The number of girls absent is $\frac{1}{3}$ of the total number of girls, which can be calculated as $\frac{1}{3} \times 45 = 15$ girls.
The total number of students absent is the sum... | [
"<bos><start_of_turn>user\nConsider Rural School USA, with an enrollment of 105 students - 60 of them boys and the remaining 45 girls. If on a particular school day, $\\frac{1}{10}$ of the boys and $\\frac{1}{3}$ of the girls are absent, how much of the school population does not show up, when expressed as a percen... | [
"So $\\boxed{20\\%}$ of the school population was absent."
] | [
"20"
] | [
null
] | ||
76 | Linda owns a store. She sells jeans at x dollars each and tees at 8 dollars each. If she sold 7 tees and 4 jeans in a day, how much money, in dollars, did she have at the end of the day?
If we know the answer to the above question is 100, what is the value of unknown variable x? | GSM_FOBAR | Linda sold 7 tees at 8 dollars each, which means she earned 7 * 8 = 56 dollars from the tees.
She also sold 4 jeans at x dollars each, which means she earned 4x dollars from the jeans.
The total amount of money she had at the end of the day is the sum of the money earned from the tees and the money earned from the jean... | [
"<bos><start_of_turn>user\nLinda owns a store. She sells jeans at x dollars each and tees at 8 dollars each. If she sold 7 tees and 4 jeans in a day, how much money, in dollars, did she have at the end of the day?\nIf we know the answer to the above question is 100, what is the value of unknown variable x?<end_of_t... | [
"We can now just try different values of x and see which one makes the equation equal to 100.\nIt's easy to see that as soon as x is set to 16, the total_revenue surpasses 100, so the answer is \\boxed{16}."
] | [
"16"
] | [
null
] | ||
77 | Two numbers add up to $50$ and the larger number is $16$ more than the smaller number. What is the value of the smaller number? | gpt-3.5-turbo | Let's assume the smaller number as $x$. According to the problem, the larger number is $16$ more than the smaller number, so the larger number can be represented as $x+16$.
We are given that the sum of the two numbers is $50$, so we can write the equation as:
\[ x + (x+16) = 50 \]
Simplifying this equation, we have:... | [
"<bos><start_of_turn>user\nTwo numbers add up to $50$ and the larger number is $16$ more than the smaller number. What is the value of the smaller number?<end_of_turn>\n<start_of_turn>model\nWe can formulate the following system of equations and solve them:\n\n$$\\begin{align*} x + y &= 50 \\\\ x &= y + 16 \\end{al... | [
"So the answer is $\\boxed{17}$."
] | [
"17"
] | [
null
] | ||
78 | An open-air party is scheduled for tomorrow, and Sheila has been invited; the event will proceed irrespective of the weather. Sheila is considering whether to attend, with a $20\%$ chance of her going if it is raining and an $80\%$ chance if it is sunny. With a forecast indicating a $40\%$ chance of rain for tomorrow, ... | gpt-3.5-turbo | To find the probability of Sheila's attendance at the party, we need to consider both the probability of rain and the conditional probability of her attendance given the weather.
Let $R$ represent the event that it is raining, and let $A$ represent the event of Sheila attending the party.
We are given:
$P(R) = 40\% =... | [
"<bos><start_of_turn>user\nAn open-air party is scheduled for tomorrow, and Sheila has been invited; the event will proceed irrespective of the weather. Sheila is considering whether to attend, with a $20\\%$ chance of her going if it is raining and an $80\\%$ chance if it is sunny. With a forecast indicating a $40... | [
"The probability of Sheila's attendance at the party is \\boxed{56\\%}"
] | [
"56"
] | [
null
] |
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