source stringclasses 9
values | problem stringlengths 10 1.42k | solution stringlengths 17 4.99k | messages listlengths 2 2 | answer stringlengths 1 15 | id int64 0 273k | pass@32 int64 28 32 |
|---|---|---|---|---|---|---|
orca_math | if jake loses 8 pounds , he will weigh twice as much as his sister . together they now weigh 278 pounds . what is jake ’ s present weight , in pounds ? | Let's call Jake's present weight J and his sister's weight S.
According to the problem, if Jake loses 8 pounds, he will weigh twice as much as his sister. So we can write the equation:
J - 8 = 2S (1)
We also know that together they now weigh 278 pounds, so we can write another equation:
J + S = 278 (2)
Now we ha... | [
{
"content": "if jake loses 8 pounds , he will weigh twice as much as his sister . together they now weigh 278 pounds . what is jake ’ s present weight , in pounds ?",
"role": "user"
},
{
"content": "Let's call Jake's present weight J and his sister's weight S.\n\nAccording to the problem, if Jake l... | 188 | 162,986 | 31 |
orca_math | During an 8-hour Relay for Life event, I walked on a route with varying elevations. I started walking at a pace of 2 miles per hour but gradually decreased my pace by 0.2 miles per hour after every 2 hours to account for rest breaks and shifts in energy levels. If the route consisted of climbing uphill and then descend... | To calculate the total distance covered during the event, we need to consider the pace at which you walked during each 2-hour segment and the elevation changes.
For the first 2 hours, you walked at a pace of 2 miles per hour. So in 2 hours, you would have covered:
2 hours * 2 miles/hour = 4 miles
After every 2 hours,... | [
{
"content": "During an 8-hour Relay for Life event, I walked on a route with varying elevations. I started walking at a pace of 2 miles per hour but gradually decreased my pace by 0.2 miles per hour after every 2 hours to account for rest breaks and shifts in energy levels. If the route consisted of climbing u... | 13.6 | 171,708 | 31 |
synthetic_math | How many integers \(n\) satisfy \((n+5)(n-9) \le 0\)? | The inequality \((n+5)(n-9) \le 0\) suggests that the product of the two factors \((n+5)\) and \((n-9)\) is less than or equal to zero. This occurs when one factor is non-negative and the other is non-positive.
1. **Set each factor to zero and solve for \(n\)**:
- \(n+5 = 0 \Rightarrow n = -5\)
- \(n-9 = 0 \Righ... | [
{
"content": "How many integers \\(n\\) satisfy \\((n+5)(n-9) \\le 0\\)?",
"role": "user"
},
{
"content": "The inequality \\((n+5)(n-9) \\le 0\\) suggests that the product of the two factors \\((n+5)\\) and \\((n-9)\\) is less than or equal to zero. This occurs when one factor is non-negative and th... | 15 | 128,908 | 32 |
orca_math | Brandon sold 86 geckos last year. He sold a certain number of geckos the year before. Brandon has sold 258 geckos in the last two years. What is the ratio of the number of geckos sold the year before to the number of geckos sold last year? | Let's call the number of geckos Brandon sold the year before x.
According to the information given, the total number of geckos sold in the last two years is 258. We know that Brandon sold 86 geckos last year, so the equation to find the number of geckos sold the year before is:
x (geckos sold the year before) + 86 (g... | [
{
"content": "Brandon sold 86 geckos last year. He sold a certain number of geckos the year before. Brandon has sold 258 geckos in the last two years. What is the ratio of the number of geckos sold the year before to the number of geckos sold last year?",
"role": "user"
},
{
"content": "Let's call t... | 2:1 | 182,316 | 31 |
orca_math | In a sports club with 30 members, 17 play badminton and 19 play tennis. There are 9 members who play both badminton and tennis. How many members do not play either badminton or tennis? | To find out how many members do not play either badminton or tennis, we can use the principle of inclusion-exclusion.
First, we add the number of badminton players and tennis players:
17 (badminton players) + 19 (tennis players) = 36
However, this sum counts the members who play both badminton and tennis twice, so we... | [
{
"content": "In a sports club with 30 members, 17 play badminton and 19 play tennis. There are 9 members who play both badminton and tennis. How many members do not play either badminton or tennis?",
"role": "user"
},
{
"content": "To find out how many members do not play either badminton or tennis... | 3 | 239,658 | 31 |
orca_math | Before the recent housing boom, there were 1426 houses in Lawrence County. Developers built 574 houses during the housing boom. How many houses are there in Lawrence County now? | To find the total number of houses in Lawrence County now, you simply need to add the number of houses built during the housing boom to the number of houses that were there before the boom.
Before the boom: 1426 houses
During the boom: +574 houses
---------------------------
Total now: 1426 + 574 = 2000 houses
There ... | [
{
"content": "Before the recent housing boom, there were 1426 houses in Lawrence County. Developers built 574 houses during the housing boom. How many houses are there in Lawrence County now?",
"role": "user"
},
{
"content": "To find the total number of houses in Lawrence County now, you simply need... | 2000 | 67,258 | 32 |
orca_math | the moon revolves around the earth at a speed of approximately 1.02 kilometers per second . this approximate speed is how many kilometers per hour ? | To convert the speed from kilometers per second to kilometers per hour, you multiply by the number of seconds in an hour. There are 60 seconds in a minute and 60 minutes in an hour, so there are 60 * 60 = 3600 seconds in an hour.
So, if the moon revolves around the Earth at approximately 1.02 kilometers per second, th... | [
{
"content": "the moon revolves around the earth at a speed of approximately 1.02 kilometers per second . this approximate speed is how many kilometers per hour ?",
"role": "user"
},
{
"content": "To convert the speed from kilometers per second to kilometers per hour, you multiply by the number of s... | 3672 | 75,053 | 32 |
orca_math | A certain bag of apples costs 3.50 dollars. At the same rate, the cost of 7 pounds of apples would be 4.9 dollars. How many pounds of apples are in the bag? | Let's denote the cost per pound of apples as \( c \) dollars per pound. We are given that 7 pounds of apples cost 4.9 dollars, so we can write the following equation:
\[ 7 \text{ pounds} \times c = 4.9 \text{ dollars} \]
To find the cost per pound (\( c \)), we divide both sides of the equation by 7 pounds:
\[ c = \... | [
{
"content": "A certain bag of apples costs 3.50 dollars. At the same rate, the cost of 7 pounds of apples would be 4.9 dollars. How many pounds of apples are in the bag?",
"role": "user"
},
{
"content": "Let's denote the cost per pound of apples as \\( c \\) dollars per pound. We are given that 7 p... | 5 | 117,806 | 32 |
cn_k12 | Simplify first, then evaluate: $(\frac{x}{x-2}-\frac{x}{x+2})÷\frac{4x}{x-2}$, where $x=3$. | To simplify and then evaluate the expression $(\frac{x}{x-2}-\frac{x}{x+2})÷\frac{4x}{x-2}$ for $x=3$, we follow these steps:
1. Combine the fractions in the numerator and simplify the entire expression:
\begin{align*}
Original\ expression &= \left[\frac{x(x+2)}{(x-2)(x+2)}-\frac{x(x-2)}{(x-2)(x+2)}\right] \cdot \frac... | [
{
"content": "Simplify first, then evaluate: $(\\frac{x}{x-2}-\\frac{x}{x+2})÷\\frac{4x}{x-2}$, where $x=3$.",
"role": "user"
},
{
"content": "To simplify and then evaluate the expression $(\\frac{x}{x-2}-\\frac{x}{x+2})÷\\frac{4x}{x-2}$ for $x=3$, we follow these steps:\n\n1. Combine the fractions ... | \frac{1}{5} | 1,120 | 32 |
math | Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$. | To find the sum of the solutions of the given quadratic equation $-32x^2 + 84x + 135=0$, we can use the formula for the sum of the roots of a quadratic equation, which is given by $-\frac{b}{a}$, where $a$ is the coefficient of $x^2$ and $b$ is the coefficient of $x$.
Given the equation $-32x^2 + 84x + 135=0$, we iden... | [
{
"content": "Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$.",
"role": "user"
},
{
"content": "To find the sum of the solutions of the given quadratic equation $-32x^2 + 84x + 135=0$, we can use the formula for the sum of the roots of a quadratic equation, which is given by $-... | \frac{21}{8} | 2,620 | 32 |
gsm8k | Melanie baked 10 batches of brownies, with 20 brownies in each batch. She set aside 3/4 of the brownies in each batch for a bake sale, put 3/5 of the remaining in a container, and gave out the rest. How many brownies are given out? | To solve this problem, let's break it down step by step, following the given solution closely:
1. **Brownies Set Aside for Bake Sale**: Melanie baked 10 batches of brownies, with 20 brownies in each batch. She set aside 3/4 of the brownies in each batch for a bake sale. The calculation for the brownies set aside per b... | [
{
"content": "Melanie baked 10 batches of brownies, with 20 brownies in each batch. She set aside 3/4 of the brownies in each batch for a bake sale, put 3/5 of the remaining in a container, and gave out the rest. How many brownies are given out?",
"role": "user"
},
{
"content": "To solve this proble... | 20 | 218,316 | 32 |
orca_math | Dan picked 9 limes and Sara gave him 4 limes. How many limes does Dan have now? | Dan originally picked 9 limes, and then Sara gave him 4 more. To find out how many limes Dan has now, you add the two amounts together:
9 (limes Dan picked) + 4 (limes Sara gave) = 13 limes
Dan now has $\boxed{13}$ limes. | [
{
"content": "Dan picked 9 limes and Sara gave him 4 limes. How many limes does Dan have now?",
"role": "user"
},
{
"content": "Dan originally picked 9 limes, and then Sara gave him 4 more. To find out how many limes Dan has now, you add the two amounts together:\n\n9 (limes Dan picked) + 4 (limes S... | 13 | 260,126 | 32 |
synthetic_math | Evaluate \(\frac{5 + x(5 + x) - 5^2}{x - 5 + x^2}\) for \( x = -3 \). | We simplify the expression \(\frac{5 + x(5 + x) - 5^2}{x - 5 + x^2}\) under the given \( x = -3 \):
\[
5 + x(5 + x) = 5 + (-3)(5 + (-3)) = 5 + (-3)(2) = 5 - 6 = -1
\]
\[
5^2 = 25
\]
Thus, the numerator becomes:
\[
-1 - 25 = -26
\]
Now the denominator:
\[
x - 5 + x^2 = -3 - 5 + (-3)^2 = -3 - 5 + 9 = 1
\]
Plugging the... | [
{
"content": "Evaluate \\(\\frac{5 + x(5 + x) - 5^2}{x - 5 + x^2}\\) for \\( x = -3 \\).",
"role": "user"
},
{
"content": "We simplify the expression \\(\\frac{5 + x(5 + x) - 5^2}{x - 5 + x^2}\\) under the given \\( x = -3 \\):\n\n\\[\n5 + x(5 + x) = 5 + (-3)(5 + (-3)) = 5 + (-3)(2) = 5 - 6 = -1\n\\... | -26 | 1,875 | 32 |
cn_k12 | A two-digit number, where the sum of the digits in the units and tens place is $8$. If the digits in the units and tens place are swapped, the resulting two-digit number is $18$ greater than the original two-digit number. Find the original two-digit number. | Let's break down the solution into detailed steps:
1. **Define Variables:**
Let's denote the tens digit of the two-digit number as $x$ and the units digit as $y$.
2. **Set Up Equations:**
According to the problem, we can set up two equations based on the given information:
- The sum of the digits is $8$, whi... | [
{
"content": "A two-digit number, where the sum of the digits in the units and tens place is $8$. If the digits in the units and tens place are swapped, the resulting two-digit number is $18$ greater than the original two-digit number. Find the original two-digit number.",
"role": "user"
},
{
"conte... | 35 | 33,186 | 32 |
synthetic_math | Given that one root of the equation $3x^2 + kx = 5$ is 3, what is the other root? | Rearranging the equation provided, we obtain:
\[3x^2 + kx - 5 = 0\]
The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$, and the product of the roots is given by $\frac{c}{a}$. For the given equation $3x^2 + kx - 5 = 0$, where $a = 3$, $b = k$, and $c = -5$, the product of the ro... | [
{
"content": "Given that one root of the equation $3x^2 + kx = 5$ is 3, what is the other root?",
"role": "user"
},
{
"content": "Rearranging the equation provided, we obtain:\n\\[3x^2 + kx - 5 = 0\\]\nThe sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\\frac{b}{a}$, and ... | -\frac{5}{9} | 55,683 | 29 |
synthetic_math | The number 340! ends in how many zeros when expressed as an integer? | To find the number of terminal zeros in 340!, we need to determine how many times the factor 5 appears in the factorial, as there will be more factors of 2 than factors of 5.
Using the formula for the number of times a prime divides a factorial:
- $\left\lfloor \frac{340}{5} \right\rfloor = 68$
- $\left\lfloor \frac{3... | [
{
"content": "The number 340! ends in how many zeros when expressed as an integer?",
"role": "user"
},
{
"content": "To find the number of terminal zeros in 340!, we need to determine how many times the factor 5 appears in the factorial, as there will be more factors of 2 than factors of 5.\n\nUsing... | 83 | 88,306 | 32 |
orca_math | Cheryl starts with 8 Skittles. Kathryn gives Cheryl some more Skittles. Cheryl ends with 97 Skittles. How many Skittles did Kathryn give Cheryl? | Cheryl started with 8 Skittles and ended with 97 Skittles. To find out how many Skittles Kathryn gave her, we subtract the number of Skittles Cheryl started with from the number she ended with:
97 Skittles (end amount) - 8 Skittles (start amount) = 89 Skittles
Kathryn gave Cheryl $\boxed{89}$ Skittles. | [
{
"content": "Cheryl starts with 8 Skittles. Kathryn gives Cheryl some more Skittles. Cheryl ends with 97 Skittles. How many Skittles did Kathryn give Cheryl?",
"role": "user"
},
{
"content": "Cheryl started with 8 Skittles and ended with 97 Skittles. To find out how many Skittles Kathryn gave her, ... | 89 | 22,204 | 32 |
orca_math | Tina can do a certain job in some hours. Working independently, Ann can do the same job in 9 hours. Tina worked independently at the job for 8 hours and then Ann worked independently. It took Ann 3 hours to complete the remainder of the jobs. How many hours does it take for Tina to do the job independently? | Let's denote the time it takes for Tina to do the job independently as T hours.
Tina worked for 8 hours, so she completed \( \frac{8}{T} \) of the job.
Ann can complete the job in 9 hours, so her work rate is \( \frac{1}{9} \) of the job per hour.
Ann worked for 3 hours to complete the job, so she completed \( 3 \ti... | [
{
"content": "Tina can do a certain job in some hours. Working independently, Ann can do the same job in 9 hours. Tina worked independently at the job for 8 hours and then Ann worked independently. It took Ann 3 hours to complete the remainder of the jobs. How many hours does it take for Tina to do the job inde... | 12 | 145,903 | 32 |
orca_math | A number is doubled and 5 is added. If the resultant is trebled, it becomes a certain value. The original number is 20. What is the value of the trebled resultant? | Let's calculate step by step:
1. The original number is 20.
2. The number is doubled: 20 * 2 = 40.
3. Then 5 is added: 40 + 5 = 45.
4. Finally, the resultant is trebled: 45 * 3 = 135.
So, the value of the trebled resultant is $\boxed{135}$ . | [
{
"content": "A number is doubled and 5 is added. If the resultant is trebled, it becomes a certain value. The original number is 20. What is the value of the trebled resultant?",
"role": "user"
},
{
"content": "Let's calculate step by step:\n\n1. The original number is 20.\n2. The number is doubled... | 135 | 211,914 | 32 |
cn_k12 | Given $\frac{1}{m}-\frac{1}{n}=6$, then the value of $\frac{mn}{m-n}$ is ____. | Given the equation $\frac{1}{m}-\frac{1}{n}=6$, we aim to find the value of $\frac{mn}{m-n}$.
Starting from the given equation:
\begin{align*}
\frac{1}{m}-\frac{1}{n} &= 6 \\
\Rightarrow \frac{n-m}{mn} &= 6 & \text{(Finding a common denominator)} \\
\Rightarrow n-m &= 6mn & \text{(Cross multiplying)} \\
\Rightarrow m-... | [
{
"content": "Given $\\frac{1}{m}-\\frac{1}{n}=6$, then the value of $\\frac{mn}{m-n}$ is ____.",
"role": "user"
},
{
"content": "Given the equation $\\frac{1}{m}-\\frac{1}{n}=6$, we aim to find the value of $\\frac{mn}{m-n}$.\n\nStarting from the given equation:\n\\begin{align*}\n\\frac{1}{m}-\\fra... | -\frac{1}{6} | 181,828 | 32 |
orca_math | In a bus, the left side has 15 seats available, and there are 3 fewer seats on the right side because of the rear exit door. Each seat can hold 3 people. In addition, there is a seat at the back that can sit a certain number of people all together. The bus can hold 90 people. How many people can sit at the back seat? | Let's calculate the total number of seats on the right side first. Since there are 3 fewer seats on the right side compared to the left side, which has 15 seats, the right side has:
15 seats (left side) - 3 seats = 12 seats (right side)
Now, let's calculate the total number of people that can be seated on the left an... | [
{
"content": "In a bus, the left side has 15 seats available, and there are 3 fewer seats on the right side because of the rear exit door. Each seat can hold 3 people. In addition, there is a seat at the back that can sit a certain number of people all together. The bus can hold 90 people. How many people can s... | 9 | 256,249 | 32 |
orca_math | If the given two numbers are respectively 6% and 30% of a third number, what percentage is the first number of the second number? | Let's denote the third number as X.
According to the problem, the first number is 6% of X and the second number is 30% of X.
So, the first number = (6/100) * X
And the second number = (30/100) * X
Now, we want to find out what percentage the first number is of the second number. To do this, we divide the first numbe... | [
{
"content": "If the given two numbers are respectively 6% and 30% of a third number, what percentage is the first number of the second number?",
"role": "user"
},
{
"content": "Let's denote the third number as X.\n\nAccording to the problem, the first number is 6% of X and the second number is 30% ... | 20\% | 22,820 | 31 |
orca_math | Hajun went to school 5 times and it took him 1 hour and 36 minutes. How many minutes does it take to go once? | To find out how many minutes it takes for Hajun to go to school once, we need to first convert the total time into minutes and then divide by the number of times he went to school.
1 hour and 36 minutes is the same as 60 minutes (for the hour) plus 36 minutes, which equals 96 minutes in total.
Now, we divide the tota... | [
{
"content": "Hajun went to school 5 times and it took him 1 hour and 36 minutes. How many minutes does it take to go once?",
"role": "user"
},
{
"content": "To find out how many minutes it takes for Hajun to go to school once, we need to first convert the total time into minutes and then divide by ... | 19.2 | 61,150 | 31 |
olympiads | If $\frac{1}{2} \sin ^{2} x + C = - \frac{1}{4} \cos 2x$ is an identity, what is the value of $C$? | We are given the identity:
\[ \frac{1}{2} \sin^2 x + C = -\frac{1}{4} \cos 2x \]
Our goal is to find the value of \( C \) so that this equation holds for all \( x \).
1. **Use a Trigonometric Identity:**
Recall the double-angle identity for cosine:
\[ \cos 2x = 1 - 2 \sin^2 x \]
Substituting \(\cos 2x \) in... | [
{
"content": "If $\\frac{1}{2} \\sin ^{2} x + C = - \\frac{1}{4} \\cos 2x$ is an identity, what is the value of $C$?",
"role": "user"
},
{
"content": "We are given the identity:\n\n\\[ \\frac{1}{2} \\sin^2 x + C = -\\frac{1}{4} \\cos 2x \\]\n\nOur goal is to find the value of \\( C \\) so that this ... | -\frac{1}{4} | 38,249 | 32 |
orca_math | If the number is decreased by 5 and divided by 7, the result is 7. What would be the result if the number is subtracted by 34 and divided by 10? | Let's call the number we're looking for "x".
According to the first condition, if the number is decreased by 5 and then divided by 7, the result is 7. We can write this as an equation:
(x - 5) / 7 = 7
To find x, we solve for it:
x - 5 = 7 * 7
x - 5 = 49
x = 49 + 5
x = 54
Now that we know x is 54, we can find out w... | [
{
"content": "If the number is decreased by 5 and divided by 7, the result is 7. What would be the result if the number is subtracted by 34 and divided by 10?",
"role": "user"
},
{
"content": "Let's call the number we're looking for \"x\".\n\nAccording to the first condition, if the number is decrea... | 2 | 166,278 | 32 |
synthetic_math | Determine the value of $\left(1 - \frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \dotsm \left(1-\frac{1}{100}\right)$. Express your answer as a common fraction. | First, note that each term in the product can be rewritten as $\left(1 - \frac{1}{n}\right) = \frac{n-1}{n}$. Therefore, the entire product becomes:
$$
\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times \cdots\times\frac{99}{100}.
$$
This sequence simplifies as each numerator cancels out the preceding denominator, lea... | [
{
"content": "Determine the value of $\\left(1 - \\frac{1}{2}\\right)\\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{4}\\right)\\left(1-\\frac{1}{5}\\right) \\dotsm \\left(1-\\frac{1}{100}\\right)$. Express your answer as a common fraction.",
"role": "user"
},
{
"content": "First, note that each ter... | \frac{1}{100} | 63,860 | 32 |
synthetic_math | What is the value of $650^2 - 350^2$? | To solve for $650^2 - 350^2$, we use the difference of squares formula, which states:
\[ a^2 - b^2 = (a+b)(a-b) \]
Using \( a = 650 \) and \( b = 350 \):
\[ 650^2 - 350^2 = (650 + 350)(650 - 350) \]
\[ = 1000 \cdot 300 \]
\[ = 300,000 \]
Thus, the solution is \( \boxed{300000} \). | [
{
"content": "What is the value of $650^2 - 350^2$?",
"role": "user"
},
{
"content": "To solve for $650^2 - 350^2$, we use the difference of squares formula, which states:\n\\[ a^2 - b^2 = (a+b)(a-b) \\]\n\nUsing \\( a = 650 \\) and \\( b = 350 \\):\n\\[ 650^2 - 350^2 = (650 + 350)(650 - 350) \\]\n\... | 300000 | 127,490 | 32 |
orca_math | If the average (arithmetic mean) of a and b is 115, and the average of b and c is 160, what is the value of a - c? | The average of a and b is 115, which means that (a + b)/2 = 115. Multiplying both sides by 2, we get:
a + b = 230
Similarly, the average of b and c is 160, which means that (b + c)/2 = 160. Multiplying both sides by 2, we get:
b + c = 320
Now, we want to find the value of a - c. To do this, we can subtract the seco... | [
{
"content": "If the average (arithmetic mean) of a and b is 115, and the average of b and c is 160, what is the value of a - c?",
"role": "user"
},
{
"content": "The average of a and b is 115, which means that (a + b)/2 = 115. Multiplying both sides by 2, we get:\n\na + b = 230\n\nSimilarly, the av... | -90 | 249,472 | 32 |
orca_math | Maynard's dog dug some holes in the lawn. Maynard filled in 75% of the holes with dirt, and 2 holes remain unfilled. How many holes did Maynard's dog dig in total? | If 75% of the holes have been filled and 2 holes remain unfilled, that means the 2 holes represent the remaining 25% that have not been filled.
To find the total number of holes, we can set up a proportion where 25% (the unfilled portion) is to 2 holes as 100% (the total) is to the total number of holes (which we'll c... | [
{
"content": "Maynard's dog dug some holes in the lawn. Maynard filled in 75% of the holes with dirt, and 2 holes remain unfilled. How many holes did Maynard's dog dig in total?",
"role": "user"
},
{
"content": "If 75% of the holes have been filled and 2 holes remain unfilled, that means the 2 holes... | 8 | 175,422 | 32 |
synthetic_math | You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you and your friend each eat a quarter of the pizza. On each successive trip, both of you eat half of the previous amount each of you ate. After four trips to the refrigerator, what fractional part of the pizza have you together eaten? | - On the first trip, each person eats $\frac{1}{4}$ of the pizza, so together they eat $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ of the pizza.
- On the second trip, each eats half of what they ate before, so each eats $\frac{1}{8}$ of the pizza, and together, they eat $\frac{1}{8} + \frac{1}{8} = \frac{1}{4}$.
- On the... | [
{
"content": "You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you and your friend each eat a quarter of the pizza. On each successive trip, both of you eat half of the previous amount each of you ate. After four trips to the refrigerator, what fractional part of the pizza hav... | \frac{15}{16} | 270,800 | 32 |
synthetic_math | Dr. Van Helsing opens a savings account that has an annual interest rate of 8 percent but it compounds quarterly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.) | The bank account compounds quarterly at an interest rate of $8/4 = 2$ percent per quarter. Therefore, the equivalent annual interest rate, compounded annually, must be calculated as follows:
- The interest factor for each quarter is: $1 + 0.02 = 1.02$.
- Therefore, over one year (four quarters), the compounded interest... | [
{
"content": "Dr. Van Helsing opens a savings account that has an annual interest rate of 8 percent but it compounds quarterly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.)",
"role": "user"
},
{
... | 8.24\% | 16,686 | 31 |
orca_math | Hoseok jumped the rope during this weekday. If he jumped 88 times, 75 times, 62 times, 91 times, and 80 times from Monday to Friday, find how many times he did more comparing the days he jumped the most and the least. | To find out how many more times Hoseok jumped on the day he jumped the most compared to the day he jumped the least, we first need to identify the maximum and minimum numbers of jumps he made during the weekdays.
From the given data:
- Monday: 88 times
- Tuesday: 75 times
- Wednesday: 62 times
- Thursday: 91 times
- F... | [
{
"content": "Hoseok jumped the rope during this weekday. If he jumped 88 times, 75 times, 62 times, 91 times, and 80 times from Monday to Friday, find how many times he did more comparing the days he jumped the most and the least.",
"role": "user"
},
{
"content": "To find out how many more times Ho... | 29 | 168,258 | 32 |
synthetic_math | In how many ways can I choose 4 captains from a team of 15 people? | The process involves selecting a subset (4 captains) from a larger set (15 players). Since the order in which the captains are chosen does not matter, we use combinations to find the number of ways to choose the captains. The formula for combinations is given by:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
For this proble... | [
{
"content": "In how many ways can I choose 4 captains from a team of 15 people?",
"role": "user"
},
{
"content": "The process involves selecting a subset (4 captains) from a larger set (15 players). Since the order in which the captains are chosen does not matter, we use combinations to find the nu... | 1365 | 50,108 | 32 |
orca_math | In a cricket team of 11 members, the captain is 26 years old and the wicket keeper is 5 years older. If the ages of these two are excluded, the average age of the remaining players is a certain amount less than the average age of the whole team, which is 24. What is the difference between the average age of the remaini... | Let's denote the average age of the remaining players (excluding the captain and the wicket keeper) as A.
The captain is 26 years old, and the wicket keeper is 5 years older, so the wicket keeper is 26 + 5 = 31 years old.
The total age of the team (including the captain and the wicket keeper) is the average age of th... | [
{
"content": "In a cricket team of 11 members, the captain is 26 years old and the wicket keeper is 5 years older. If the ages of these two are excluded, the average age of the remaining players is a certain amount less than the average age of the whole team, which is 24. What is the difference between the aver... | 1 | 190,851 | 32 |
olympiads | There were 80 students who brought water and 70 students who brought fruit on the spring excursion. There were 6 students who did not bring either. If the number of students who brought both water and fruit is half of the total number of participants, how many students participated in the spring excursion? | 1. **Define the Variables**: Let the number of students who brought both water and fruit be \( x \).
2. **Express Total Participants**: Since we know that these students who brought both items make up half of the total participants, thus the total number of participants is \( 2x \).
3. **Set up the Equation with Give... | [
{
"content": "There were 80 students who brought water and 70 students who brought fruit on the spring excursion. There were 6 students who did not bring either. If the number of students who brought both water and fruit is half of the total number of participants, how many students participated in the spring e... | 104 | 242,887 | 32 |
cn_k12 | The variables $x$ and $y$ satisfy the constraint conditions $\begin{cases}\begin{matrix}x+y-2\geq0 \\\\ x-y-2\leq0 \\\\ y\geq1\\end{matrix} \end{cases}$. The minimum value of the objective function $z=x+3y$ is ( ). | First, let's simplify the given inequalities:
1. $x+y-2\geq0 \Rightarrow x+y\geq2$
2. $x-y-2\leq0 \Rightarrow x-y\leq2$
3. $y\geq1$
The feasible region determined by these constraints is a polygon in the $xy$-plane. To find the minimum value of $z$, we need to evaluate it at each vertex of the feasible region.
The ve... | [
{
"content": "The variables $x$ and $y$ satisfy the constraint conditions $\\begin{cases}\\begin{matrix}x+y-2\\geq0 \\\\\\\\ x-y-2\\leq0 \\\\\\\\ y\\geq1\\\\end{matrix} \\end{cases}$. The minimum value of the objective function $z=x+3y$ is ( ).",
"role": "user"
},
{
"content": "First, let's simplify... | 4 | 190,728 | 28 |
orca_math | A salesman's terms were changed from a flat commission of 5% on all his sales to a fixed salary of Rs. 1300 plus 2.5% commission on all sales exceeding Rs. 4,000. His sales were worth Rs. 12,000. How much more was his remuneration as per the new scheme compared to the previous scheme? | First, let's calculate the salesman's remuneration under the old scheme, which was a flat commission of 5% on all his sales.
Old scheme remuneration = 5% of Rs. 12,000
= 0.05 * 12,000
= Rs. 600
Now, let's calculate the salesman's remuneration under the new scheme, which is a fixed salary of Rs. 1300 plus 2.5% commiss... | [
{
"content": "A salesman's terms were changed from a flat commission of 5% on all his sales to a fixed salary of Rs. 1300 plus 2.5% commission on all sales exceeding Rs. 4,000. His sales were worth Rs. 12,000. How much more was his remuneration as per the new scheme compared to the previous scheme?",
"role"... | 900 | 190,484 | 32 |
cn_k12 | Given a sequence $\{a_n\}$, the sum of the first $n$ terms $S_n=2n(n+1)$, find the value of $a_5$. | The sum of the first $n$ terms of the sequence is given by $S_n=2n(n+1)$. To find $a_5$, the fifth term of the sequence, we need to use the following relation for the $n$th term of a sequence expressed in terms of the sum of the first $n$ terms:
$$a_n = S_n - S_{n-1}$$
For $a_5$, this becomes:
$$a_5 = S_5 - S_4$$
W... | [
{
"content": "Given a sequence $\\{a_n\\}$, the sum of the first $n$ terms $S_n=2n(n+1)$, find the value of $a_5$.",
"role": "user"
},
{
"content": "The sum of the first $n$ terms of the sequence is given by $S_n=2n(n+1)$. To find $a_5$, the fifth term of the sequence, we need to use the following r... | 20 | 65,911 | 32 |
orca_math | Ryan builds model mustang cars with different scales. A full-size mustang is 240 inches long. Ryan creates four different scaled models:
1. The mid-size model is 1/10th the size of the full-size mustang.
2. The small model is 1/4 the size of the mid-size model.
3. The mini model is 1/3 the size of the small model. ... | Let's calculate the length of each model step by step:
1. Mid-size model:
The mid-size model is 1/10th the size of the full-size Mustang. So, we calculate:
\( \frac{1}{10} \times 240 \) inches = 24 inches
2. Small model:
The small model is 1/4 the size of the mid-size model. So, we calculate:
\( \frac{1}{4} \times 24... | [
{
"content": "Ryan builds model mustang cars with different scales. A full-size mustang is 240 inches long. Ryan creates four different scaled models: \n\n1. The mid-size model is 1/10th the size of the full-size mustang. \n2. The small model is 1/4 the size of the mid-size model. \n3. The mini model is 1/3 the... | 32.4 | 74,151 | 31 |
orca_math | There is 1 green bead, 2 brown beads, and 3 red beads in a container. Tom took some beads out of the container and left a certain number of beads in. Tom took out 2 beads. How many beads were left in the container? | Originally, there were 1 green bead + 2 brown beads + 3 red beads = 6 beads in total.
Tom took out 2 beads, so the number of beads left in the container would be 6 beads - 2 beads = $\boxed{4}$ beads. | [
{
"content": "There is 1 green bead, 2 brown beads, and 3 red beads in a container. Tom took some beads out of the container and left a certain number of beads in. Tom took out 2 beads. How many beads were left in the container?",
"role": "user"
},
{
"content": "Originally, there were 1 green bead +... | 4 | 199,551 | 32 |
cn_k12 | In a class of 50 students, 32 participated in a school computer drawing competition, and 24 participated in a computer typesetting competition. If there are 3 students who did not participate in either competition, how many students participated in both competitions? | Let's denote the total number of students in the class as $N = 50$.
Given that 3 students did not participate in either competition, the number of students who participated in at least one competition is $N - 3 = 50 - 3 = 47$.
Let's define set $A$ as the students who participated in the computer drawing competition... | [
{
"content": "In a class of 50 students, 32 participated in a school computer drawing competition, and 24 participated in a computer typesetting competition. If there are 3 students who did not participate in either competition, how many students participated in both competitions?",
"role": "user"
},
{
... | 9 | 216,495 | 32 |
cn_k12 | Calculate: $(-0.91) \div (-0.13) =$ . | **Solution**: $(-0.91) \div (-0.13) = 7$.
Therefore, the answer is $\boxed{7}$. | [
{
"content": "Calculate: $(-0.91) \\div (-0.13) =$ .",
"role": "user"
},
{
"content": "**Solution**: $(-0.91) \\div (-0.13) = 7$.\n\nTherefore, the answer is $\\boxed{7}$.",
"role": "assistant"
}
] | 7 | 83,030 | 31 |
olympiads | 1 kilogram of soybeans can be made into 3 kilograms of tofu. To produce 1 kilogram of soybean oil, 6 kilograms of soybeans are required. Tofu costs 3 yuan per kilogram, and soybean oil costs 15 yuan per kilogram. A batch of soybeans weighs 460 kilograms, and the total sales revenue from the tofu and soybean oil produce... |
1. Determine the revenue from 1000 kg of soybeans when converted to tofu and soy oil.
- Revenue from tofu:
\[
\text{Revenue per 1000 kg of tofu} = 3 \, \text{yuan/kg}
\]
\[
\text{Tofu yield from 1000 kg of soybeans} = 3 \times 1000 = 3000 \, \text{kg}
\]
\[
\text{Total reve... | [
{
"content": "1 kilogram of soybeans can be made into 3 kilograms of tofu. To produce 1 kilogram of soybean oil, 6 kilograms of soybeans are required. Tofu costs 3 yuan per kilogram, and soybean oil costs 15 yuan per kilogram. A batch of soybeans weighs 460 kilograms, and the total sales revenue from the tofu a... | 360 | 34,436 | 31 |
orca_math | My dog had 493 bones. Then, he dug up 367 bones. How many bones does he have now? | If your dog had 493 bones and then dug up 367 more, you would add the two amounts together to find the total number of bones he has now:
493 (original bones) + 367 (dug up bones) = 860 bones
So, your dog now has $\boxed{860}$ bones. | [
{
"content": "My dog had 493 bones. Then, he dug up 367 bones. How many bones does he have now?",
"role": "user"
},
{
"content": "If your dog had 493 bones and then dug up 367 more, you would add the two amounts together to find the total number of bones he has now:\n\n493 (original bones) + 367 (du... | 860 | 71,437 | 32 |
gsm8k | Jim is baking loaves of bread. He has 200g of flour in the cupboard, 100g of flour on the kitchen counter, and 100g in the pantry. If one loaf of bread requires 200g of flour, how many loaves can Jim bake? | To solve this problem, we first need to calculate the total amount of flour Jim has. He has flour in three different places:
1. In the cupboard: $200g$
2. On the kitchen counter: $100g$
3. In the pantry: $100g$
Adding these amounts together gives us the total amount of flour:
\[200g + 100g + 100g = 400g\]
Now, sinc... | [
{
"content": "Jim is baking loaves of bread. He has 200g of flour in the cupboard, 100g of flour on the kitchen counter, and 100g in the pantry. If one loaf of bread requires 200g of flour, how many loaves can Jim bake?",
"role": "user"
},
{
"content": "To solve this problem, we first need to calcul... | 2 | 70,547 | 32 |
orca_math | Two taps can fill a cistern in some minutes and 18 minutes respectively. They are turned on at the same time. If the first tap is turned off after 4 minutes, tap B takes 8 minutes to fill the rest of the cistern. How long does it take for tap A to fill the cistern alone? | Let's denote the time it takes for tap A to fill the cistern alone as \( t \) minutes.
Since tap B takes 18 minutes to fill the cistern alone, its filling rate is \( \frac{1}{18} \) of the cistern per minute.
When both taps are open together for 4 minutes, they fill a certain portion of the cistern. The combined rate... | [
{
"content": "Two taps can fill a cistern in some minutes and 18 minutes respectively. They are turned on at the same time. If the first tap is turned off after 4 minutes, tap B takes 8 minutes to fill the rest of the cistern. How long does it take for tap A to fill the cistern alone?",
"role": "user"
},
... | 12 | 267,931 | 31 |
cn_k12 | Simplify and then evaluate: $\left(1+\dfrac{1}{a-1}\right)\div \dfrac{2a}{a^{2}-1}$, where $a=-2$. | To simplify and then evaluate the given expression $\left(1+\dfrac{1}{a-1}\right)\div \dfrac{2a}{a^{2}-1}$ with $a=-2$, we follow these steps:
1. **Simplify the Expression:**
First, we notice that $a^2 - 1 = (a+1)(a-1)$, which is a difference of squares. This allows us to rewrite the denominator of the division as $2... | [
{
"content": "Simplify and then evaluate: $\\left(1+\\dfrac{1}{a-1}\\right)\\div \\dfrac{2a}{a^{2}-1}$, where $a=-2$.",
"role": "user"
},
{
"content": "To simplify and then evaluate the given expression $\\left(1+\\dfrac{1}{a-1}\\right)\\div \\dfrac{2a}{a^{2}-1}$ with $a=-2$, we follow these steps:\... | -\frac{1}{2} | 28,943 | 32 |
orca_math | A teacher grades students’ tests by subtracting twice the number of incorrect responses from the number of correct responses. If Student A answers each of the 100 questions on her test and receives a score of 64, how many questions did Student A answer correctly? | Let's denote the number of correct answers as \( C \) and the number of incorrect answers as \( I \). According to the grading system, the score is calculated as follows:
\[ \text{Score} = C - 2I \]
We know that Student A answered all 100 questions, so the sum of correct and incorrect answers is 100:
\[ C + I = 100 ... | [
{
"content": "A teacher grades students’ tests by subtracting twice the number of incorrect responses from the number of correct responses. If Student A answers each of the 100 questions on her test and receives a score of 64, how many questions did Student A answer correctly?",
"role": "user"
},
{
... | 88 | 68,653 | 32 |
cn_k12 | If ( ) - (-2) = 3, then the number in the parentheses is ____. | **Analysis:** According to the rule that the minuend equals the difference plus the subtrahend, we can set up an equation and then calculate it using the addition rule.
Let's denote the number in the parentheses as $x$. According to the problem, we have:
\[x - (-2) = 3\]
Using the addition rule, we simplify the left... | [
{
"content": "If ( ) - (-2) = 3, then the number in the parentheses is ____.",
"role": "user"
},
{
"content": "**Analysis:** According to the rule that the minuend equals the difference plus the subtrahend, we can set up an equation and then calculate it using the addition rule.\n\nLet's denote the... | 1 | 181,986 | 32 |
synthetic_math | For how many values of the digit $B$ is it true that $45$ is divisible by $B$ and $451{,}B7$ is divisible by $3$? | 1. **Divisibility by $B$:** The divisors of $45$ are $1, 3, 5, 9, 15, 45$. However, $B$ must be a single digit, so the possible values for $B$ are $1, 3, 5, 9$.
2. **Divisibility by $3$:** The number $451{,}B7$ must be divisible by $3$. To check this, sum the digits of the number and see if the sum is divisible by $3$... | [
{
"content": "For how many values of the digit $B$ is it true that $45$ is divisible by $B$ and $451{,}B7$ is divisible by $3$?",
"role": "user"
},
{
"content": "1. **Divisibility by $B$:** The divisors of $45$ are $1, 3, 5, 9, 15, 45$. However, $B$ must be a single digit, so the possible values for... | 1 | 158,756 | 29 |
gsm8k | A construction company is building 2 apartment buildings with 12 floors each. The apartments are almost ready to sell but all of them need doors to be completed. Each floor has 6 apartments, and each apartment needs 7 doors in total. How many doors does the company need to buy? | To calculate the total number of doors needed by the construction company, we follow these steps:
1. Determine the total number of floors across both buildings:
\[12 \, \text{floors/building} \times 2 \, \text{buildings} = 24 \, \text{floors}\]
2. Calculate the total number of apartments based on the number of floors... | [
{
"content": "A construction company is building 2 apartment buildings with 12 floors each. The apartments are almost ready to sell but all of them need doors to be completed. Each floor has 6 apartments, and each apartment needs 7 doors in total. How many doors does the company need to buy?",
"role": "user... | 1008 | 115,795 | 32 |
orca_math | In 1 week , Mitch 's family drank 0.5 carton of regular milk and 0.1 carton of soy milk . How much milk did they drink in all ? | To find out how much milk Mitch's family drank in all, we need to add the amount of regular milk they drank to the amount of soy milk they drank.
Regular milk: 0.5 carton
Soy milk: 0.1 carton
Total milk consumed: 0.5 carton + 0.1 carton = 0.6 cartons
Mitch's family drank a total of $\boxed{0.6}$ cartons of milk in ... | [
{
"content": "In 1 week , Mitch 's family drank 0.5 carton of regular milk and 0.1 carton of soy milk . How much milk did they drink in all ? ",
"role": "user"
},
{
"content": "To find out how much milk Mitch's family drank in all, we need to add the amount of regular milk they drank to the amount o... | 0.6 | 188,767 | 32 |
orca_math | There are 4 wolves out hunting for the pack. The pack contains 16 additional wolves. Each wolf needs to eat 8 pounds of meat a day and the wolves won't hunt again for five days. Each deer contains 200 pounds of meat. How many deer does each wolf need to kill? | First, let's calculate the total amount of meat needed for the entire pack for five days.
The pack has a total of 4 (hunting wolves) + 16 (additional wolves) = 20 wolves.
Each wolf needs 8 pounds of meat per day, so for five days, each wolf needs 8 * 5 = 40 pounds of meat.
Therefore, the entire pack needs 20 wolves ... | [
{
"content": "There are 4 wolves out hunting for the pack. The pack contains 16 additional wolves. Each wolf needs to eat 8 pounds of meat a day and the wolves won't hunt again for five days. Each deer contains 200 pounds of meat. How many deer does each wolf need to kill?",
"role": "user"
},
{
"con... | 1 | 271,182 | 29 |
synthetic_math | What is the largest divisor of 540 that is also a factor of 315? | First, we will find the factors of 540 and 315.
The factors of 540 are:
$$540 = 2^2 \times 3^3 \times 5 $$
The factors are 1, 2, 4, 3, 6, 12, 9, 18, 36, 5, 10, 20, 15, 30, 45, 60, 90, 135, 180, 270, 540.
The factors of 315 are:
$$315 = 3^2 \times 5 \times 7$$
The factors are 1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 31... | [
{
"content": "What is the largest divisor of 540 that is also a factor of 315?",
"role": "user"
},
{
"content": "First, we will find the factors of 540 and 315.\n\nThe factors of 540 are:\n$$540 = 2^2 \\times 3^3 \\times 5 $$\nThe factors are 1, 2, 4, 3, 6, 12, 9, 18, 36, 5, 10, 20, 15, 30, 45, 60, ... | 45 | 138,186 | 32 |
orca_math | A trader sells 60 meters of cloth for Rs. 8400. The cost price of one meter of cloth is Rs. 128. What is the profit per meter of cloth? | To calculate the profit per meter of cloth, we first need to determine the total cost price for 60 meters of cloth and then the total profit made on the sale. After that, we can find the profit per meter by dividing the total profit by the number of meters sold.
Total cost price for 60 meters of cloth:
Cost price per ... | [
{
"content": "A trader sells 60 meters of cloth for Rs. 8400. The cost price of one meter of cloth is Rs. 128. What is the profit per meter of cloth?",
"role": "user"
},
{
"content": "To calculate the profit per meter of cloth, we first need to determine the total cost price for 60 meters of cloth a... | 12 | 157,023 | 32 |
orca_math | A train of a certain length crosses a platform in 48 seconds while it crosses a signal pole in 18 seconds. The length of the platform is 500 meters. What is the length of the train? | Let's denote the length of the train as \( L \) meters.
When the train crosses a signal pole, it only needs to cover its own length. Given that it takes 18 seconds to pass the pole, we can express the speed of the train as \( \frac{L}{18} \) meters per second.
When the train crosses the platform, it needs to cover th... | [
{
"content": "A train of a certain length crosses a platform in 48 seconds while it crosses a signal pole in 18 seconds. The length of the platform is 500 meters. What is the length of the train?",
"role": "user"
},
{
"content": "Let's denote the length of the train as \\( L \\) meters.\n\nWhen the ... | 300 | 226,631 | 31 |
cn_k12 | Given $x^{a}=2$ and $x^{b}=9$, find $x^{3a-b}$. | Given $x^{a}=2$ and $x^{b}=9$, we aim to find $x^{3a-b}$.
Starting with the given equations:
1. $x^{a}=2$
2. $x^{b}=9$
We can manipulate these equations to find $x^{3a-b}$ as follows:
- First, we raise both sides of the first equation to the power of 3, which gives us $(x^{a})^{3} = 2^{3}$.
- This simplifies to $x^{3... | [
{
"content": "Given $x^{a}=2$ and $x^{b}=9$, find $x^{3a-b}$.",
"role": "user"
},
{
"content": "Given $x^{a}=2$ and $x^{b}=9$, we aim to find $x^{3a-b}$.\n\nStarting with the given equations:\n1. $x^{a}=2$\n2. $x^{b}=9$\n\nWe can manipulate these equations to find $x^{3a-b}$ as follows:\n- First, we... | \frac{8}{9} | 260,608 | 32 |
cn_k12 | Simplify first, then evaluate: $x\left(x+2y\right)-\left(x+1\right)^{2}+2x$, where $x=\frac{1}{25}$ and $y=-25$. | To simplify the given expression $x(x+2y)-(x+1)^2+2x$ and then evaluate it with $x=\frac{1}{25}$ and $y=-25$, we follow these steps:
1. Expand and simplify the expression:
\begin{align*}
x(x+2y)-(x+1)^2+2x &= x^2 + 2xy - (x^2 + 2x + 1) + 2x \\
&= x^2 + 2xy - x^2 - 2x - 1 + 2x \\
&= 2xy - 1
\end{align*}
2. Substitute ... | [
{
"content": "Simplify first, then evaluate: $x\\left(x+2y\\right)-\\left(x+1\\right)^{2}+2x$, where $x=\\frac{1}{25}$ and $y=-25$.",
"role": "user"
},
{
"content": "To simplify the given expression $x(x+2y)-(x+1)^2+2x$ and then evaluate it with $x=\\frac{1}{25}$ and $y=-25$, we follow these steps:\... | -3 | 83,022 | 32 |
orca_math | In a school, there are 542 girls and 387 boys. So, there are _____ pupils in that school. | To find the total number of pupils in the school, you need to add the number of girls to the number of boys:
542 (girls) + 387 (boys) = 929 pupils
So, there are $\boxed{929}$ pupils in that school. | [
{
"content": "In a school, there are 542 girls and 387 boys. So, there are _____ pupils in that school.",
"role": "user"
},
{
"content": "To find the total number of pupils in the school, you need to add the number of girls to the number of boys:\n\n542 (girls) + 387 (boys) = 929 pupils\n\nSo, there... | 929 | 94,425 | 32 |
orca_math | There are some players on a football team, 31 are throwers. The rest of the team is divided so one third are left-handed and the rest are right-handed. Assuming that all throwers are right-handed, there are 57 right-handed players in total. How many players are there on the football team? | Let's denote the total number of players on the football team as \( P \).
We know that there are 31 throwers, and all of them are right-handed. So, the number of right-handed players that are not throwers is \( 57 - 31 = 26 \).
The problem states that the rest of the team, excluding the throwers, is divided into one ... | [
{
"content": "There are some players on a football team, 31 are throwers. The rest of the team is divided so one third are left-handed and the rest are right-handed. Assuming that all throwers are right-handed, there are 57 right-handed players in total. How many players are there on the football team?",
"r... | 70 | 98,487 | 31 |
olympiads | If the real number \( x \) satisfies \( \log_{2} x = \log_{4}(2x) + \log_{8}(4x) \), then \( x = \) ? |
To solve the equation involving logarithms, \(\log_{2} x = \log_{4}(2x) + \log_{8}(4 x)\), we will first convert all logarithms to have the same base, base 2.
1. **Convert the logarithms to base 2:**
\[
\log_{4}(2x) = \frac{\log_{2}(2x)}{\log_{2}(4)} = \frac{\log_{2}(2x)}{2}
\]
\[
\log_{8}(4x) =... | [
{
"content": "If the real number \\( x \\) satisfies \\( \\log_{2} x = \\log_{4}(2x) + \\log_{8}(4x) \\), then \\( x = \\) ?",
"role": "user"
},
{
"content": "\nTo solve the equation involving logarithms, \\(\\log_{2} x = \\log_{4}(2x) + \\log_{8}(4 x)\\), we will first convert all logarithms to hav... | 128 | 93,261 | 32 |
synthetic_math | Given that \(\|\mathbf{u}\| = 7,\) find \(\mathbf{u} \cdot \mathbf{u}.\) | Given the magnitude of vector \(\mathbf{u}\), we know that:
\[
\mathbf{u} \cdot \mathbf{u} = \|\mathbf{u}\|^2
\]
Since \(\|\mathbf{u}\| = 7,\) we calculate:
\[
\|\mathbf{u}\|^2 = 7^2
\]
\[
\|\mathbf{u}\|^2 = 49
\]
Thus, \(\mathbf{u} \cdot \mathbf{u} = \boxed{49}.\)
Conclusion:
The calculation verifies that the dot pro... | [
{
"content": "Given that \\(\\|\\mathbf{u}\\| = 7,\\) find \\(\\mathbf{u} \\cdot \\mathbf{u}.\\)",
"role": "user"
},
{
"content": "Given the magnitude of vector \\(\\mathbf{u}\\), we know that:\n\\[\n\\mathbf{u} \\cdot \\mathbf{u} = \\|\\mathbf{u}\\|^2\n\\]\nSince \\(\\|\\mathbf{u}\\| = 7,\\) we cal... | 49 | 130,401 | 32 |
orca_math | If 90 percent of 600 is 50 percent of x, what is the value of x? | To find the value of x, we can set up the following equation:
90% of 600 = 50% of x
First, we convert the percentages to decimals:
0.90 * 600 = 0.50 * x
Now, we can solve for x:
540 = 0.50 * x
To find x, we divide both sides of the equation by 0.50:
x = 540 / 0.50
x = 1080
Therefore, the value of x is $\boxed{1... | [
{
"content": "If 90 percent of 600 is 50 percent of x, what is the value of x?",
"role": "user"
},
{
"content": "To find the value of x, we can set up the following equation:\n\n90% of 600 = 50% of x\n\nFirst, we convert the percentages to decimals:\n\n0.90 * 600 = 0.50 * x\n\nNow, we can solve for ... | 1080 | 7,453 | 32 |
cn_k12 | Determine the minimum value of the function $y=\cos{2x} - 6\cos{x} + 6$. | Given the function $y = \cos{2x} - 6\cos{x} + 6$, we can utilize the trigonometric identity $\cos{2x} = 2\cos^2{x} - 1$ to rewrite the function as:
$$
y = (2\cos^2{x} - 1) - 6\cos{x} + 6.
$$
Now, we can simplify this further to get a quadratic expression in terms of $\cos{x}$:
\begin{align*}
y & = 2\cos^2{x} - 6\cos{x... | [
{
"content": "Determine the minimum value of the function $y=\\cos{2x} - 6\\cos{x} + 6$.",
"role": "user"
},
{
"content": "Given the function $y = \\cos{2x} - 6\\cos{x} + 6$, we can utilize the trigonometric identity $\\cos{2x} = 2\\cos^2{x} - 1$ to rewrite the function as:\n$$\ny = (2\\cos^2{x} - 1... | 1 | 37,152 | 31 |
orca_math | A man took a loan at a certain rate per annum simple interest. After 3 years, he had to pay $9,000 interest. The principal amount borrowed by him was $25,000. What was the interest rate per annum? | To find the interest rate per annum, we can use the formula for simple interest:
Simple Interest (SI) = Principal (P) * Rate (R) * Time (T) / 100
We know the Simple Interest (SI) is $9,000, the Principal (P) is $25,000, and the Time (T) is 3 years. We need to find the Rate (R).
Plugging in the values we have:
$9,00... | [
{
"content": "A man took a loan at a certain rate per annum simple interest. After 3 years, he had to pay $9,000 interest. The principal amount borrowed by him was $25,000. What was the interest rate per annum?",
"role": "user"
},
{
"content": "To find the interest rate per annum, we can use the for... | 12\% | 126,899 | 29 |
orca_math | If the horse can run 10 miles/hour, it would take 1 day for a horse to run a certain distance. What is the distance from New York City to Washington D.C.? | To calculate the distance from New York City to Washington D.C. based on the information provided, we need to know how many hours are in a day and then multiply that by the horse's speed.
There are 24 hours in a day. If the horse can run at a speed of 10 miles per hour, then in one day (24 hours), the horse would cove... | [
{
"content": "If the horse can run 10 miles/hour, it would take 1 day for a horse to run a certain distance. What is the distance from New York City to Washington D.C.?",
"role": "user"
},
{
"content": "To calculate the distance from New York City to Washington D.C. based on the information provided... | 240 | 228,660 | 32 |
cn_k12 | The coefficient of $x^{2}$ in the expansion of $\left(1+ \frac {1}{x^{2}}\right)(1+x)^{6}$ is __________. | To solve this, when choosing $1$ from $\left(1+ \frac {1}{x^{2}}\right)$, the term involving $x^{2}$ from the expansion of $(1+x)^{6}$ is $C_{6}^{2}x^{2}$. When choosing $\frac {1}{x^{2}}$ from $\left(1+ \frac {1}{x^{2}}\right)$, the term from the expansion of $(1+x)^{6}$ to be selected is $C_{6}^{4}x^{4}$.
Therefore,... | [
{
"content": "The coefficient of $x^{2}$ in the expansion of $\\left(1+ \\frac {1}{x^{2}}\\right)(1+x)^{6}$ is __________.",
"role": "user"
},
{
"content": "To solve this, when choosing $1$ from $\\left(1+ \\frac {1}{x^{2}}\\right)$, the term involving $x^{2}$ from the expansion of $(1+x)^{6}$ is $C... | 30 | 67,889 | 30 |
orca_math | A father told his son "I was as old as you are at present, at the time of your birth". If the son's age 5 years ago was 14 years, how old is the father now? | Let's denote the son's current age as S and the father's current age as F.
According to the information given, the son's age 5 years ago was 14 years. Therefore, the son's current age (S) is 14 + 5 = 19 years.
The father said that he was as old as the son is now at the time of the son's birth. This means that when th... | [
{
"content": "A father told his son \"I was as old as you are at present, at the time of your birth\". If the son's age 5 years ago was 14 years, how old is the father now?",
"role": "user"
},
{
"content": "Let's denote the son's current age as S and the father's current age as F.\n\nAccording to th... | 38 | 85,802 | 32 |
olympiads | Solve the equation \(\frac{1}{\sqrt{\log _{5}(5 x)}+\sqrt{\log _{5} x}}+\sqrt{\log _{5} x}=2\). | 1. **Rewrite the given equation step-by-step:**
\[
\frac{1}{\sqrt{\log_{5}(5x)} + \sqrt{\log_{5} x}} + \sqrt{\log_{5} x} = 2
\]
To simplify, use the properties of logarithms. Note that:
\[
\log_{5}(5x) = \log_{5}5 + \log_{5}x = 1 + \log_{5} x
\]
Substituting this back into the equation, we get:
... | [
{
"content": "Solve the equation \\(\\frac{1}{\\sqrt{\\log _{5}(5 x)}+\\sqrt{\\log _{5} x}}+\\sqrt{\\log _{5} x}=2\\).",
"role": "user"
},
{
"content": "1. **Rewrite the given equation step-by-step:**\n \\[\n \\frac{1}{\\sqrt{\\log_{5}(5x)} + \\sqrt{\\log_{5} x}} + \\sqrt{\\log_{5} x} = 2\n \\... | 125 | 165,603 | 32 |
olympiads | Vanya decided to give Masha a bouquet of an odd number of flowers, consisting of yellow and red tulips, so that the count of one color differs from the count of the other by exactly one. Yellow tulips cost 50 rubles each, and red tulips cost 31 rubles each. What is the maximum number of tulips he can buy for Masha's bi... |
**Step 1: Define Variables**
Let:
- $r$ be the number of red tulips,
- $y$ be the number of yellow tulips.
**Step 2: Set Up Equations**
We are given two conditions:
1. The bouquet contains an odd number of flowers.
2. The number of flowers of one color is exactly one more than the number of flowers of the other colo... | [
{
"content": "Vanya decided to give Masha a bouquet of an odd number of flowers, consisting of yellow and red tulips, so that the count of one color differs from the count of the other by exactly one. Yellow tulips cost 50 rubles each, and red tulips cost 31 rubles each. What is the maximum number of tulips he ... | 15 | 17,343 | 28 |
orca_math | Rosa had 67.0 flowers. Andre gave her some more flowers, and now Rosa has a total of 157 flowers. How many flowers did Andre give her? | To find out how many flowers Andre gave Rosa, we need to subtract the original number of flowers Rosa had from the total number of flowers she has now.
So, we calculate:
157 (total flowers now) - 67.0 (original flowers) = 90 flowers
Andre gave Rosa $\boxed{90}$ flowers. | [
{
"content": "Rosa had 67.0 flowers. Andre gave her some more flowers, and now Rosa has a total of 157 flowers. How many flowers did Andre give her?",
"role": "user"
},
{
"content": "To find out how many flowers Andre gave Rosa, we need to subtract the original number of flowers Rosa had from the to... | 90 | 117,339 | 32 |
synthetic_math | The value of $\log_{10}{46328}$ is between which consecutive integers $a$ and $b$? Find the sum $a+b$. | - Observe that $\log_{10}10000 = 4$ and $\log_{10}100000 = 5$.
- Since $\log_{10}x$ increases as $x$ increases, it follows that $\log_{10}10000 < \log_{10}46328 < \log_{10}100000$.
- Thus, $\log_{10}{46328}$ lies between $4$ and $5$. Therefore, the sum $a+b = 4+5$.
- Final answer: $\boxed{9}$. | [
{
"content": "The value of $\\log_{10}{46328}$ is between which consecutive integers $a$ and $b$? Find the sum $a+b$.",
"role": "user"
},
{
"content": "- Observe that $\\log_{10}10000 = 4$ and $\\log_{10}100000 = 5$.\n- Since $\\log_{10}x$ increases as $x$ increases, it follows that $\\log_{10}10000... | 9 | 59,803 | 32 |
orca_math | John is building a tree house to serve as his hideout. For the pillars, he needs two sets of wood with different lengths. The first set of wood is 4 feet long, and the second set is 5 times longer than the first set. How long is the second set of wood? | The second set of wood is 5 times longer than the first set, which is 4 feet long. To find the length of the second set, you multiply the length of the first set by 5:
Length of second set = 4 feet * 5 = 20 feet
So, the second set of wood is $\boxed{20}$ feet long. | [
{
"content": "John is building a tree house to serve as his hideout. For the pillars, he needs two sets of wood with different lengths. The first set of wood is 4 feet long, and the second set is 5 times longer than the first set. How long is the second set of wood?",
"role": "user"
},
{
"content": ... | 20 | 128,672 | 32 |
orca_math | 10 % people of a village in sri lanka died by bombardment , 15 % of the remainder left the village on account of fear . if now the population is reduced to 3213 , how much was it in the beginning ? | Let's denote the initial population of the village as P.
According to the problem, 10% of the population died due to bombardment. So, the population after the bombardment is:
90% of P = P - (10% of P)
90% of P = 0.9 * P
Then, 15% of the remaining population left the village due to fear. So, the population after thes... | [
{
"content": "10 % people of a village in sri lanka died by bombardment , 15 % of the remainder left the village on account of fear . if now the population is reduced to 3213 , how much was it in the beginning ?",
"role": "user"
},
{
"content": "Let's denote the initial population of the village as ... | 4200 | 47,144 | 32 |
orca_math | The average monthly salary of 24 employees in an organisation is a certain amount. If the manager's salary is added, then the average salary increases by Rs. 100. The manager's monthly salary is Rs. 4900. What is the average monthly salary of the employees excluding the manager? | Let's denote the average monthly salary of the 24 employees as A. The total salary of these employees would then be 24 * A.
When the manager's salary is added, the total number of employees becomes 25, and the average salary increases by Rs. 100. So the new average salary is A + 100.
The total salary with the manager... | [
{
"content": "The average monthly salary of 24 employees in an organisation is a certain amount. If the manager's salary is added, then the average salary increases by Rs. 100. The manager's monthly salary is Rs. 4900. What is the average monthly salary of the employees excluding the manager?",
"role": "use... | 2400 | 180,165 | 32 |
cn_k12 | A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$. | The work done by a force $F$ on an object as it moves from point $A$ to point $B$ is given by the integral of $F$ with respect to $x$ from $A$ to $B$. In this case, the work $W$ done by the force $F$ on the mass $m$ as it moves from $x = 1$ to $x = 10$ is:
\[W = \int_{1}^{10} (x^2 + 1) dx\]
To solve this integral, we... | [
{
"content": "A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.",
"role": "user"
},
{
"content... | 342 | 150,410 | 32 |
orca_math | 0.20 of a number decreased by 4 is equal to a certain value. The number is 50. What is the value? | First, we find 0.20 of the number 50:
0.20 * 50 = 10
Then, we decrease this result by 4:
10 - 4 = 6
So, the value is $\boxed{6}$ . | [
{
"content": "0.20 of a number decreased by 4 is equal to a certain value. The number is 50. What is the value?",
"role": "user"
},
{
"content": "First, we find 0.20 of the number 50:\n\n0.20 * 50 = 10\n\nThen, we decrease this result by 4:\n\n10 - 4 = 6\n\nSo, the value is $\\boxed{6}$ .",
"rol... | 6 | 187,305 | 32 |
cn_k12 | Zhang Hua's brother works in Shanghai. On the eve of "Father's Day" this year, she wanted her brother to buy some automotive technology books and send them to their father, who loves automotive technology, as a gift. Therefore, she sent a text message to her brother, but she couldn't remember the order of the last thre... | Since the possible outcomes are: 028, 082, 208, 280, 802, 820,
Therefore, the probability of Zhang Hua successfully sending a text message in one attempt is:
$$\frac{1}{6}$$
Thus, the answer is:
$$\boxed{\frac{1}{6}}$$ | [
{
"content": "Zhang Hua's brother works in Shanghai. On the eve of \"Father's Day\" this year, she wanted her brother to buy some automotive technology books and send them to their father, who loves automotive technology, as a gift. Therefore, she sent a text message to her brother, but she couldn't remember th... | \frac{1}{6} | 3,903 | 31 |
cn_k12 | If $|a+2|$ and $(b-3)^2$ are opposites of each other, then $a^b+3(a-b)=$ ? | According to the problem, we have:
- For $a+2=0$, we get $a=-2$.
- For $b-3=0$, we get $b=3$.
Therefore, the original expression becomes $-8+3(-2-3)=-23$.
Hence, the answer is $\boxed{-23}$. | [
{
"content": "If $|a+2|$ and $(b-3)^2$ are opposites of each other, then $a^b+3(a-b)=$ ?",
"role": "user"
},
{
"content": "According to the problem, we have:\n\n- For $a+2=0$, we get $a=-2$.\n- For $b-3=0$, we get $b=3$.\n\nTherefore, the original expression becomes $-8+3(-2-3)=-23$.\n\nHence, the a... | -23 | 217,872 | 32 |
orca_math | There were a total of 323 football games in the season. The season is played for 17 months. How many football games were played each month, if each month has the same number of games? | To find out how many football games were played each month, we need to divide the total number of games by the number of months in the season.
Total number of games = 323
Number of months = 17
Games per month = Total number of games / Number of months
Games per month = 323 / 17
Games per month = 19
So, $\boxed{19}$ ... | [
{
"content": "There were a total of 323 football games in the season. The season is played for 17 months. How many football games were played each month, if each month has the same number of games?",
"role": "user"
},
{
"content": "To find out how many football games were played each month, we need ... | 19 | 95,578 | 32 |
synthetic_math | Compute
$$\frac{1625^2 - 1618^2}{1632^2 - 1611^2}.$$ | Firstly, factor the numerator using the difference of squares formula:
$$ 1625^2 - 1618^2 = (1625 - 1618)(1625 + 1618) = 7 \cdot (1625 + 1618) = 7 \cdot 3243. $$
Next, factor the denominator:
$$ 1632^2 - 1611^2 = (1632 - 1611)(1632 + 1611) = 21 \cdot (1632 + 1611) = 21 \cdot 3243. $$
Thus, the fraction simplifies to:... | [
{
"content": "Compute\n\n$$\\frac{1625^2 - 1618^2}{1632^2 - 1611^2}.$$",
"role": "user"
},
{
"content": "Firstly, factor the numerator using the difference of squares formula:\n$$ 1625^2 - 1618^2 = (1625 - 1618)(1625 + 1618) = 7 \\cdot (1625 + 1618) = 7 \\cdot 3243. $$\n\nNext, factor the denominato... | \frac{1}{3} | 78,499 | 32 |
orca_math | If 63 percent of a class answered the first question on a certain test correctly, 50 percent answered the second question on the test correctly, and a certain percentage answered neither of the questions correctly. What percent answered neither of the questions correctly if 33 percent answered both correctly? | Let's denote the following:
- Let A be the event that a student answered the first question correctly.
- Let B be the event that a student answered the second question correctly.
- Let A ∩ B be the event that a student answered both questions correctly.
We are given the following probabilities:
- P(A) = 63% (the pro... | [
{
"content": "If 63 percent of a class answered the first question on a certain test correctly, 50 percent answered the second question on the test correctly, and a certain percentage answered neither of the questions correctly. What percent answered neither of the questions correctly if 33 percent answered bot... | 20 | 97,490 | 31 |
orca_math | Gus eats an eggs-omelet for breakfast made with some eggs. He has an egg salad sandwich made with 3 eggs for lunch. He then has an egg drop soup made with 1 egg for dinner. Gus ate 6 eggs altogether. How many eggs were in the omelet for breakfast? | Gus ate a total of 6 eggs throughout the day. For lunch, he had an egg salad sandwich made with 3 eggs, and for dinner, he had egg drop soup made with 1 egg.
To find out how many eggs were in the omelet for breakfast, we subtract the eggs he ate for lunch and dinner from the total number of eggs he ate:
6 eggs (tota... | [
{
"content": "Gus eats an eggs-omelet for breakfast made with some eggs. He has an egg salad sandwich made with 3 eggs for lunch. He then has an egg drop soup made with 1 egg for dinner. Gus ate 6 eggs altogether. How many eggs were in the omelet for breakfast?",
"role": "user"
},
{
"content": "Gus ... | 2 | 129,527 | 32 |
synthetic_math | Given that $a-b=3$ and $a^2+b^2=27$, find $a^3-b^3$. | First, use the formula $(a-b)^2 = a^2 - 2ab + b^2$.
- Plugging in the given values: \(3^2 = 27 - 2ab\), we get \(9 = 27 - 2ab\).
- Solving for \(ab\), we calculate \(18 = 2ab\) so \(ab = 9\).
Now, use the identity \(a^3-b^3=(a-b)(a^2+ab+b^2)\).
- Plugging in our known values: \(a^3-b^3 = 3(27 + 9) = 3 \cdot 36 = 108\... | [
{
"content": "Given that $a-b=3$ and $a^2+b^2=27$, find $a^3-b^3$.",
"role": "user"
},
{
"content": "First, use the formula $(a-b)^2 = a^2 - 2ab + b^2$. \n- Plugging in the given values: \\(3^2 = 27 - 2ab\\), we get \\(9 = 27 - 2ab\\).\n- Solving for \\(ab\\), we calculate \\(18 = 2ab\\) so \\(ab = ... | 108 | 176,880 | 29 |
synthetic_math | Find $2537 + 240 \times 3 \div 60 - 347$. | Start by simplifying the expression following the order of operations (PEMDAS/BODMAS rules):
1. **Multiplication and Division**:
\[
240 \times 3 = 720, \quad 720 \div 60 = 12.
\]
2. **Addition and Subtraction**:
\[
2537 + 12 - 347.
\]
Simplify the addition and subtraction:
\[
2537 + 12 = 2549... | [
{
"content": "Find $2537 + 240 \\times 3 \\div 60 - 347$.",
"role": "user"
},
{
"content": "Start by simplifying the expression following the order of operations (PEMDAS/BODMAS rules):\n1. **Multiplication and Division**:\n \\[\n 240 \\times 3 = 720, \\quad 720 \\div 60 = 12.\n \\]\n2. **Addit... | 2202 | 60,952 | 32 |
orca_math | The sum of ages of 5 children born 2 years different each is 40 years. What is the age of the elder child? | Let's denote the age of the youngest child as x. Since each child is born 2 years apart, the ages of the children can be represented as follows:
Youngest child: x
2nd child: x + 2
3rd child: x + 4
4th child: x + 6
Eldest child: x + 8
The sum of their ages is given as 40 years. Therefore, we can write the equation:
x... | [
{
"content": "The sum of ages of 5 children born 2 years different each is 40 years. What is the age of the elder child?",
"role": "user"
},
{
"content": "Let's denote the age of the youngest child as x. Since each child is born 2 years apart, the ages of the children can be represented as follows:\... | 12 | 177,188 | 32 |
synthetic_math | Calculate $2 + 3 \cdot 4 - 5 + 6 \cdot (2 - 1)$. | Follow the order of operations: first handle operations inside parentheses, then multiplication and division, and finally addition and subtraction.
1. Evaluate the parentheses:
$$
2 - 1 = 1
$$
2. Substitute and simplify using multiplication:
$$
2 + 3 \cdot 4 - 5 + 6 \cdot 1
$$
$$
2 + 12 - 5 + ... | [
{
"content": "Calculate $2 + 3 \\cdot 4 - 5 + 6 \\cdot (2 - 1)$.",
"role": "user"
},
{
"content": "Follow the order of operations: first handle operations inside parentheses, then multiplication and division, and finally addition and subtraction.\n\n1. Evaluate the parentheses:\n $$\n 2 - 1 = 1\... | 15 | 200,913 | 32 |
synthetic_math | Jenny places a total of 30 red Easter eggs in several purple baskets and a total of 42 blue Easter eggs in some yellow baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket? | The number of eggs in each basket is a common divisor of 30 and 42 that is at least 5. To solve this, determine the greatest common divisor (GCD) of 30 and 42 that meets this criterion.
1. First, find all divisors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
2. Find all divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
3. The common di... | [
{
"content": "Jenny places a total of 30 red Easter eggs in several purple baskets and a total of 42 blue Easter eggs in some yellow baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket?",
"role": "user"
},
{
"... | 6 | 174,283 | 32 |
gsm8k | On a particular week, Fatima's restaurant served 25 people food and drinks, and 3/5 of that number came to buy coffee. How many people did not buy coffee in the restaurant? | To solve this problem, we start by calculating the number of customers who bought coffee in the restaurant. Since 3/5 of the customers came to buy coffee, we multiply the total number of customers by 3/5:
\[ \frac{3}{5} \times 25 = 15 \]
This means that 15 customers bought coffee. To find out how many customers did n... | [
{
"content": "On a particular week, Fatima's restaurant served 25 people food and drinks, and 3/5 of that number came to buy coffee. How many people did not buy coffee in the restaurant?",
"role": "user"
},
{
"content": "To solve this problem, we start by calculating the number of customers who boug... | 10 | 249,419 | 30 |
orca_math | A man rows his boat 120 km downstream and some distance upstream, taking 2 hours each time. The speed of the stream is 15 km/h. How far did he row upstream? | Let's denote the speed of the boat in still water as \(V_b\) and the speed of the stream as \(V_s\). We are given that \(V_s = 15\) km/h.
When the man is rowing downstream, the effective speed of the boat is the sum of the speed of the boat in still water and the speed of the stream, so the downstream speed is \(V_b +... | [
{
"content": "A man rows his boat 120 km downstream and some distance upstream, taking 2 hours each time. The speed of the stream is 15 km/h. How far did he row upstream?",
"role": "user"
},
{
"content": "Let's denote the speed of the boat in still water as \\(V_b\\) and the speed of the stream as \... | 60 | 132,826 | 32 |
orca_math | Lucy put sixty-eight shells in her bucket. She found some more shells and now she has eighty-nine shells. How many more shells did she find? | Lucy originally had 68 shells and now has 89 shells. To find out how many more shells she found, subtract the original number of shells from the new total:
89 (total shells now) - 68 (original shells) = 21 shells
Lucy found $\boxed{21}$ more shells. | [
{
"content": "Lucy put sixty-eight shells in her bucket. She found some more shells and now she has eighty-nine shells. How many more shells did she find?",
"role": "user"
},
{
"content": "Lucy originally had 68 shells and now has 89 shells. To find out how many more shells she found, subtract the o... | 21 | 143,896 | 32 |
orca_math | two tains are running in opposite directions with the same speed . if the length of each train is 120 metres and they cross each other in 12 seconds , then the speed of each train ( in km / hr ) is : | When two trains are running in opposite directions and cross each other, the relative speed is the sum of their individual speeds. Since both trains have the same speed, let's denote the speed of each train as 'v'.
The total distance covered when the trains cross each other is the sum of their lengths. So, the total d... | [
{
"content": "two tains are running in opposite directions with the same speed . if the length of each train is 120 metres and they cross each other in 12 seconds , then the speed of each train ( in km / hr ) is :",
"role": "user"
},
{
"content": "When two trains are running in opposite directions a... | 36 | 162,769 | 31 |
synthetic_math | Let $a, b, c, d$ be nonzero real numbers. Find the number of real roots of the equation
\[
\begin{vmatrix} x & -c & b \\ c & x & -d \\ -b & d & x \end{vmatrix} = 0.
\] | We can expand the determinant as follows:
\begin{align*}
\begin{vmatrix} x & -c & b \\ c & x & -d \\ -b & d & x \end{vmatrix} &= x \begin{vmatrix} x & -d \\ d & x \end{vmatrix} + c \begin{vmatrix} c & -d \\ -b & x \end{vmatrix} + b \begin{vmatrix} c & x \\ -b & d \end{vmatrix} \\
&= x(x^2 + d^2) + c(cx + bd) + b(-bc + ... | [
{
"content": "Let $a, b, c, d$ be nonzero real numbers. Find the number of real roots of the equation\n\\[\n\\begin{vmatrix} x & -c & b \\\\ c & x & -d \\\\ -b & d & x \\end{vmatrix} = 0.\n\\]",
"role": "user"
},
{
"content": "We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatr... | 1 | 149,015 | 32 |
orca_math | There are 40 students in a class. If 1/10 are absent, a certain fraction of the students who are present are in the classroom, and the rest are in the canteen. There are 9 students in the canteen. What fraction of the students who are present are in the classroom? | First, let's find out how many students are absent. Since 1/10 of the students are absent, we can calculate this as:
Absent students = 1/10 * 40 = 4 students
Now, let's find out how many students are present:
Present students = Total students - Absent students
Present students = 40 - 4 = 36 students
We know that th... | [
{
"content": "There are 40 students in a class. If 1/10 are absent, a certain fraction of the students who are present are in the classroom, and the rest are in the canteen. There are 9 students in the canteen. What fraction of the students who are present are in the classroom?",
"role": "user"
},
{
... | 3/4 | 102,025 | 32 |
synthetic_math | If $\frac{1}{8}$ of $2^{36}$ is $8^x$, what is the value of $x$? | We start by simplifying $\frac{1}{8} \cdot 2^{36}$:
\[
\frac{1}{8} \cdot 2^{36} = \frac{2^{36}}{2^3} = 2^{36-3} = 2^{33}
\]
Next, we express $8^x$ as a power of 2:
\[
8^x = (2^3)^x = 2^{3x}
\]
Setting these equal gives:
\[
2^{33} = 2^{3x}
\]
Equating the exponents leads to:
\[
3x = 33 \implies x = \frac{33}{3} = \boxed... | [
{
"content": "If $\\frac{1}{8}$ of $2^{36}$ is $8^x$, what is the value of $x$?",
"role": "user"
},
{
"content": "We start by simplifying $\\frac{1}{8} \\cdot 2^{36}$:\n\\[\n\\frac{1}{8} \\cdot 2^{36} = \\frac{2^{36}}{2^3} = 2^{36-3} = 2^{33}\n\\]\nNext, we express $8^x$ as a power of 2:\n\\[\n8^x =... | 11 | 108,224 | 32 |
orca_math | Paul had 78.0 strawberries in his basket. He ate 42.0 of them. How many strawberries does he now have. | Paul had 78.0 strawberries and ate 42.0 of them. To find out how many strawberries he has now, we subtract the number of strawberries he ate from the number he had originally:
78.0 - 42.0 = 36.0
Paul now has $\boxed{36.0}$ strawberries left in his basket. | [
{
"content": "Paul had 78.0 strawberries in his basket. He ate 42.0 of them. How many strawberries does he now have.",
"role": "user"
},
{
"content": "Paul had 78.0 strawberries and ate 42.0 of them. To find out how many strawberries he has now, we subtract the number of strawberries he ate from the... | 36.0 | 56,637 | 32 |
orca_math | Paige was helping her mom plant flowers in 8 flowerbeds. They planted 32 seeds altogether. If they put same number of seeds in each flower bed. How many seeds did they plant in each flowerbed? | To find out how many seeds they planted in each flowerbed, you divide the total number of seeds by the number of flowerbeds.
Total seeds = 32
Number of flowerbeds = 8
Seeds per flowerbed = Total seeds / Number of flowerbeds
Seeds per flowerbed = 32 / 8
Seeds per flowerbed = 4
So, they planted $\boxed{4}$ seeds in e... | [
{
"content": "Paige was helping her mom plant flowers in 8 flowerbeds. They planted 32 seeds altogether. If they put same number of seeds in each flower bed. How many seeds did they plant in each flowerbed?",
"role": "user"
},
{
"content": "To find out how many seeds they planted in each flowerbed, ... | 4 | 36,640 | 32 |
orca_math | One fourth of one third of two fifth of a number is 16. What is the value of one third of two fifth of that number? | Let's call the number we're looking for "N".
According to the problem, one fourth of one third of two fifth of N is 16. We can write this as an equation:
(1/4) * (1/3) * (2/5) * N = 16
To find N, we need to solve for it. First, let's simplify the left side of the equation by multiplying the fractions:
(1/4) * (1/3)... | [
{
"content": "One fourth of one third of two fifth of a number is 16. What is the value of one third of two fifth of that number?",
"role": "user"
},
{
"content": "Let's call the number we're looking for \"N\".\n\nAccording to the problem, one fourth of one third of two fifth of N is 16. We can writ... | 64 | 197,755 | 31 |
orca_math | If ( a + b ) = 11 , ( b + c ) = 9 and ( c + d ) = 3 , what is the value of ( a + d )? | To find the value of (a + d), we can use the given equations:
(a + b) = 11
(b + c) = 9
(c + d) = 3
We want to eliminate b and c to find a relationship between a and d. To do this, we can subtract the second equation from the first and the third equation from the second:
(a + b) - (b + c) = 11 - 9
(a - c) = 2
(b + c... | [
{
"content": "If ( a + b ) = 11 , ( b + c ) = 9 and ( c + d ) = 3 , what is the value of ( a + d )?",
"role": "user"
},
{
"content": "To find the value of (a + d), we can use the given equations:\n\n(a + b) = 11\n(b + c) = 9\n(c + d) = 3\n\nWe want to eliminate b and c to find a relationship between... | 5 | 219,322 | 30 |
orca_math | A sum was put at simple interest at a certain rate for 9 years. Had it been put at a higher rate, it would have fetched $1,350 more. The sum is $3,000. What is the difference in percentage between the original rate and the higher rate? | Let's denote the original rate as R% and the higher rate as H%. We know that the sum of money is $3,000 and it was put at simple interest for 9 years.
The simple interest formula is:
Interest (I) = Principal (P) * Rate (R) * Time (T) / 100
Using the original rate, the interest earned would be:
I_original = $3,000 * R... | [
{
"content": "A sum was put at simple interest at a certain rate for 9 years. Had it been put at a higher rate, it would have fetched $1,350 more. The sum is $3,000. What is the difference in percentage between the original rate and the higher rate?",
"role": "user"
},
{
"content": "Let's denote the... | 5\% | 38,643 | 29 |
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