Explain in Your Own Words: Improving Reasoning via Token-Selective Dual Knowledge Distillation
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user_prompt stringlengths 155 4.45k | question stringlengths 155 4.45k | output stringlengths 130 4.59k | answer stringlengths 1 159 | step_char_ends listlengths 3 161 | id int64 0 79.7k |
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Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\%$ of the diameter. How many cubic feet of s... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\%$ of the diameter. How many cubic feet of s... | To solve this problem, we will follow these steps:
1. **Identify the given values and formulas:**
- The diameter of the conical pile of sand is $8$ feet.
- The height of the conical pile is $75\%$ of the diameter.
- The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$, where $r$ is the radius an... | 32\pi | [
52,
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158,
221,
342,
383,
428,
505,
546,
593,
672,
726,
777,
823,
855,
889,
937,
1009,
1037,
1089,
1166
] | 0 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? | To solve the problem, we need to use the properties of an ellipse and its foci. An ellipse is defined as the set of all points \(P\) such that the sum of the distances from \(P\) to the two foci \(F_1\) and \(F_2\) is constant. Let's denote this constant by \(2a\).
Given:
- Foci \(F_1 = (0,2)\) and \(F_2 = (3,0)\)
- T... | \frac{15}{4} | [
267,
274,
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485,
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615,
618,
628,
631,
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882,
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1333,
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1366,
1369,
1378,
13... | 1 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
i bought two books ; for rs . 540 . i sold one at a loss of 15 % and other at a gain of 19 % and then i found each book was sold at the same price . find the cost of the book sold a... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
i bought two books ; for rs . 540 . i sold one at a loss of 15 % and other at a gain of 19 % and then i found each book was sold at the same price . find the cost of the book sold a... | Let's denote the cost of the first book (sold at a loss) as \(C_1\) and the cost of the second book (sold at a gain) as \(C_2\). We know that the total cost of both books is Rs. 540, so we can write:
\[C_1 + C_2 = 540\]
We are also told that the first book was sold at a 15% loss, and the second book was sold at a 19%... | 315 | [
201,
222,
432,
497,
530,
596,
629,
729,
752,
794,
835,
896,
920,
938,
971,
998,
1030,
1096
] | 3 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The farmers reported that they harvest 45 sacks of apples from each of the 8 sections of the orchard daily. _____ sacks are harvested every day. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The farmers reported that they harvest 45 sacks of apples from each of the 8 sections of the orchard daily. _____ sacks are harvested every day. | To find out how many sacks of apples are harvested every day from the entire orchard, you simply need to multiply the number of sacks harvested from each section by the total number of sections in the orchard.
Given:
- Each section of the orchard yields 45 sacks of apples daily.
- There are 8 sections in the orchard.
... | 360 | [
211,
218,
281,
321,
401,
479,
555
] | 12 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a , band c can do a piece of work in 11 days , 5 days and 55 days respectively , working alone . how soon can the work be done if a is assisted by band c on alternate days ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a , band c can do a piece of work in 11 days , 5 days and 55 days respectively , working alone . how soon can the work be done if a is assisted by band c on alternate days ? | To solve this problem, we'll first determine the work rates of a, b, and c, and then calculate the total work done in two days when they work in the described pattern (a assisted by b on one day and a assisted by c on the next day).
1. **Determine the work rates:**
- \(a\) can do the work in 11 days, so \(a\)’s wo... | 5 | [
234,
268,
369,
468,
570,
629,
792,
960,
1011,
1129,
1208,
1393,
1506,
1509,
1519,
1521
] | 15 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Define $f(x)=3x-8$. If $f^{-1}$ is the inverse of $f$, find the value(s) of $x$ for which $f(x)=f^{-1}(x)$. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Define $f(x)=3x-8$. If $f^{-1}$ is the inverse of $f$, find the value(s) of $x$ for which $f(x)=f^{-1}(x)$. | To solve the problem, we need to find the value(s) of $x$ for which $f(x) = f^{-1}(x)$, given that $f(x) = 3x - 8$.
### Step 1: Find the Inverse Function $f^{-1}(x)$
First, we need to find the inverse of $f(x) = 3x - 8$.
1. Start by setting $y = f(x) = 3x - 8$.
2. To find $f^{-1}(x)$, we need to solve for $x$ in term... | 4 | [
117,
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264,
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1027,
1033,
1058,
1064,
1075,
... | 17 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
riya and priya set on a journey . riya moves eastward at a speed of 18 kmph and priya moves westward at a speed of 24 kmph . how far will be priya from riya after 15 minutes | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
riya and priya set on a journey . riya moves eastward at a speed of 18 kmph and priya moves westward at a speed of 24 kmph . how far will be priya from riya after 15 minutes | To find out how far Priya will be from Riya after 15 minutes, we need to calculate the distance each of them travels in that time and then sum these distances, since they are moving in opposite directions.
1. **Calculate the distance Riya travels in 15 minutes:**
Riya's speed is 18 km/h. First, convert 15 minutes ... | 10.5 | [
207,
266,
332,
338,
411,
418,
499,
505,
598,
605,
665,
726,
732,
824,
831,
858,
1003,
1009,
1084,
1091,
1173
] | 18 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a and b began business with rs . 2000 and rs . 4000 after 8 months , a withdraws rs . 1000 and b advances rs . 1000 more . at the end of the year , their profits amounted to rs . 63... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a and b began business with rs . 2000 and rs . 4000 after 8 months , a withdraws rs . 1000 and b advances rs . 1000 more . at the end of the year , their profits amounted to rs . 63... | To solve this problem, we need to calculate the share of profit for B based on the amount of money each partner invested and the duration of their investments. Here's the step-by-step process:
1. **Identify Initial Investments and Duration:**
- A initially invested Rs. 2000.
- B initially invested Rs. 4000.
-... | 455 | [
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280,
316,
403,
454,
512,
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915,
1034,
1080,
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1311,
1441,
1520,
1551,
1670,
1724
] | 23 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a rectangular farm has to be fenced one long side , one short side and the diagonal . if the cost of fencing is rs . 11 per meter . the area of farm is 1200 m 2 and the short side i... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a rectangular farm has to be fenced one long side , one short side and the diagonal . if the cost of fencing is rs . 11 per meter . the area of farm is 1200 m 2 and the short side i... | To solve this problem, we need to find the total cost of fencing a rectangular farm given the cost per meter and the dimensions of the farm. Let's break it down step-by-step.
1. **Identify the dimensions of the farm:**
- The area of the farm is given as 1200 m².
- The short side of the farm is given as 30 m.
... | 1320 | [
176,
220,
267,
317,
321,
367,
421,
511,
519,
543,
551,
575,
583,
628,
637,
682,
765,
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1022,
1093,
1101,
1174,
1183,
1231,
1286,
1321,
1329,
1387,
1396,
1448
] | 28 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Selena got a tip today that amounted to $99. She pampered herself by eating at a 5-star hotel. She indulged herself with 2 steak meals that cost $24 each plate. She also ordered 2 t... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Selena got a tip today that amounted to $99. She pampered herself by eating at a 5-star hotel. She indulged herself with 2 steak meals that cost $24 each plate. She also ordered 2 t... | To find out how much money Selena will be left with, we need to calculate the total cost of her meal and then subtract that from the $99 tip she received.
1. **Calculate the cost of the steak meals**: Selena ordered 2 steak meals at $24 each.
\[ 2 \times 24 = 48 \]
2. **Calculate the cost of the burgers**: She ordere... | 38 | [
156,
244,
268,
361,
385,
481,
503,
548,
571,
641,
661,
735
] | 33 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
at company x , senior sales representatives visit the home office once every 15 days , and junior sales representatives visit the home office once every 10 days . the number of visi... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
at company x , senior sales representatives visit the home office once every 15 days , and junior sales representatives visit the home office once every 10 days . the number of visi... | To solve this problem, we first need to calculate the number of visits each type of sales representative makes in a 2-year period and then find the percentage increase from the senior representative's visits to the junior representative's visits.
**Step 1: Calculate the number of visits for each type of representative... | 52.08 | [
248,
344,
444,
522,
584,
658,
816,
894,
956,
1021,
1107,
1170,
1262,
1482
] | 35 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
d... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
d... | To solve this problem, we need to use the properties of trapezoids and triangles, as well as the given area of the trapezoid and the relationship between the lengths of the parallel sides.
1. **Identify the given information:**
- The trapezoid \(PQRS\) has an area of 12.
- \(RS\) is twice the length of \(PQ\).
... | 4 | [
190,
229,
276,
320,
368,
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648,
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1338,
1344,
1419,
1425,
1431,
1487,
1493,
1499,
1536,
... | 36 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Isabella’s hair is 18 inches long. If her hair grows 4 more inches, it will be _____ inches long . | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Isabella’s hair is 18 inches long. If her hair grows 4 more inches, it will be _____ inches long . | To solve this problem, follow these steps:
1. **Identify the initial length of Isabella’s hair**: The problem states that Isabella’s hair is initially 18 inches long.
2. **Identify the amount of growth**: The problem also states that her hair will grow 4 more inches.
3. **Calculate the new length of her hair**: Add th... | 22 | [
44,
168,
269,
388,
452,
528
] | 45 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
An garage has 20 bike wheels. How many bikes he can assemble using 20 wheels? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
An garage has 20 bike wheels. How many bikes he can assemble using 20 wheels? | To solve the problem, we need to determine how many bicycles can be assembled with 20 wheels, given that each bicycle requires 2 wheels.
Step 1: Identify the number of wheels required for one bicycle.
Each bicycle requires 2 wheels.
Step 2: Calculate the number of bicycles that can be assembled with 20 wheels.
To fin... | 10 | [
138,
202,
235,
314,
411,
509,
556,
594,
684
] | 46 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the charge for a single room at hotel p is 40 percent less than the charge for a single room at hotel r and 10 percent less than the charge for a single room at hotel g . the charge... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the charge for a single room at hotel p is 40 percent less than the charge for a single room at hotel r and 10 percent less than the charge for a single room at hotel g . the charge... | Let's denote the charge for a single room at Hotel P as \(P\), at Hotel R as \(R\), and at Hotel G as \(G\).
From the problem, we have the following relationships:
1. \(P\) is 40% less than \(R\). This can be expressed as:
\[P = R - 0.4R = 0.6R\]
2. \(P\) is 10% less than \(G\). This can be expressed as:
\[P = G - 0... | 50 | [
110,
166,
225,
250,
309,
334,
373,
386,
400,
480,
497,
589,
637,
732,
862
] | 52 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if the diameter of circle r is 50 % of the diameter of circle s , the area of circle r is what percent of the area of circle s ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if the diameter of circle r is 50 % of the diameter of circle s , the area of circle r is what percent of the area of circle s ? | To solve this problem, let's start by defining the diameters and radii of the circles involved and then use the area formula for circles to find the required percentage.
Let's denote:
- The diameter of circle \(R\) as \(d_R\).
- The diameter of circle \(S\) as \(d_S\).
Given that the diameter of circle \(R\) is 50% o... | 25 | [
171,
185,
228,
272,
387,
415,
467,
493,
520,
589,
672,
718,
766,
811,
909,
952,
1057,
1148,
1270,
1323,
1353,
1438
] | 53 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Lidia has a collection of books. Her collection is four times bigger than the collection which her friend Susan has. In total Susan and Lidia, both have 3000 books. How many books d... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Lidia has a collection of books. Her collection is four times bigger than the collection which her friend Susan has. In total Susan and Lidia, both have 3000 books. How many books d... | To solve this problem, let's define the number of books Susan has as \(S\). According to the problem, Lidia's collection is four times bigger than Susan's, so Lidia has \(4S\) books.
The total number of books that Susan and Lidia have together is given as 3000. Therefore, we can write the equation:
\[ S + 4S = 3000 \]... | 600 | [
184,
301,
322,
365,
382,
440,
472,
526
] | 54 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Adult tickets for a play cost 11 dollars and the child tickets cost 10 dollars. If there were 23 people at a performance and the theater collected 246 dollars from tickets sales , h... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Adult tickets for a play cost 11 dollars and the child tickets cost 10 dollars. If there were 23 people at a performance and the theater collected 246 dollars from tickets sales , h... | To solve this problem, we can set up a system of equations based on the given information. Let's define:
- \(A\) as the number of adult tickets sold.
- \(C\) as the number of child tickets sold.
From the problem, we have two pieces of information that can be translated into equations:
1. The total number of people (... | 7 | [
106,
151,
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289,
369,
385,
543,
564,
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624,
638,
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666,
670,
761,
806,
822,
877,
905,
943,
969,
988,
1008,
1026,
1037,
1103
] | 57 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
In a big box, there are marbles in 4 colors. Inside, there are 20 red marbles. There are three times more green marbles than red. Yellow marbles make up 20% of the green marbles. ... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
In a big box, there are marbles in 4 colors. Inside, there are 20 red marbles. There are three times more green marbles than red. Yellow marbles make up 20% of the green marbles. ... | Let's break down the problem step by step.
1. **Identify the number of red marbles:**
- There are 20 red marbles.
2. **Determine the number of green marbles:**
- There are three times more green marbles than red marbles.
- Therefore, the number of green marbles = 3 * 20 = 60.
3. **Determine the number of ye... | 88 | [
44,
87,
119,
165,
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336,
390,
466,
523,
610,
667,
728,
795,
828,
863,
899,
987,
1101,
1163,
1230
] | 59 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$? | To find the smallest possible value of the expression \((a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\), we will use the method of inequalities and symmetry. Let's denote the expression by \(E\):
\[E = (a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\]
First, we will use the Cauchy-Schwar... | \frac{9}{2} | [
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] | 60 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
joe needs to paint all the airplane hangars at the airport , so he buys 360 gallons of paint to do the job . during the first week , he uses 1 / 9 of all the paint . during the seco... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
joe needs to paint all the airplane hangars at the airport , so he buys 360 gallons of paint to do the job . during the first week , he uses 1 / 9 of all the paint . during the seco... | To solve the problem step-by-step, let's first determine how much paint Joe uses in the first week, then how much he uses in the second week, and finally add the two amounts together to find the total amount of paint used.
1. **First Week:**
- Joe starts with 360 gallons of paint.
- He uses \(\frac{1}{9}\) of al... | 104 | [
224,
243,
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351,
414,
422,
471,
480,
521,
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636,
645,
665,
744,
808,
816,
865,
874,
899,
1003,
1011,
1046,
1055,
1124
] | 64 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A van is delivering 180 bottles of drinks to a neighborhood, and each bottle contains either cider or beer or a mixture of two. Out of the 180 bottles, 40 contain only cider, 80 c... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A van is delivering 180 bottles of drinks to a neighborhood, and each bottle contains either cider or beer or a mixture of two. Out of the 180 bottles, 40 contain only cider, 80 c... | To solve this problem, we need to follow these steps:
1. Determine the number of bottles that contain a mixture of cider and beer.
2. Calculate half the number of each type of bottle (only cider, only beer, and the mixture) that the delivery man gives to the first house.
3. Sum these amounts to find the total number o... | 90 | [
55,
132,
273,
357,
439,
591,
617,
700,
769,
837,
906,
994,
1019,
1074
] | 67 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
James buys 10 boxes of Capri-sun. Each box has 6 pouches in it. If he paid $12 how many cents does each pouch cost? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
James buys 10 boxes of Capri-sun. Each box has 6 pouches in it. If he paid $12 how many cents does each pouch cost? | To solve the problem, we will follow these steps:
1. Calculate the total number of Capri-sun pouches James bought.
2. Convert the total cost from dollars to cents.
3. Calculate the cost per pouch in cents.
**Step 1: Calculate the total number of Capri-sun pouches.**
- James buys 10 boxes.
- Each box contains 6 pouche... | 20 | [
51,
116,
165,
208,
269,
292,
323,
398,
456,
474,
553,
604,
641,
674,
790,
831
] | 73 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The class has to guess how many jellybeans are in a jar the teacher brought in. He lets 4 students make their guesses. The first thinks it contains 100 jellybeans. The second says t... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The class has to guess how many jellybeans are in a jar the teacher brought in. He lets 4 students make their guesses. The first thinks it contains 100 jellybeans. The second says t... | To solve the problem, we'll follow the steps methodically, starting with the first student's guess and then moving on to the subsequent guesses as described.
1. **First Student's Guess**: The first student guesses 100 jellybeans.
\[
\text{First guess} = 100
\]
2. **Second Student's Guess**: The second studen... | 525 | [
159,
231,
237,
265,
272,
394,
400,
444,
451,
549,
555,
595,
602,
722,
785,
791,
934,
941,
994,
1000,
1040,
1047,
1112
] | 77 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a watch was sold at a loss of 9 % . if it was sold for rs . 220 more , there would have been a gain of 4 % . what is the cost price ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a watch was sold at a loss of 9 % . if it was sold for rs . 220 more , there would have been a gain of 4 % . what is the cost price ? | To solve this problem, let's denote the cost price of the watch as \(CP\).
Given:
- The watch was sold at a loss of 9%.
- If the selling price was Rs. 220 more, there would have been a gain of 4%.
First, let's express the selling price (SP) when the watch was sold at a loss of 9%:
\[SP_1 = CP - 0.09 \times CP = CP \t... | 1692.31 | [
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83,
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434,
506,
589,
612,
675,
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761,
795,
821,
841,
867,
885,
947
] | 84 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
there are 437 doctors and nurses in a hospital . if the ratio of the doctors to the nurses is 8 : 11 , then how many nurses are there in the hospital ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
there are 437 doctors and nurses in a hospital . if the ratio of the doctors to the nurses is 8 : 11 , then how many nurses are there in the hospital ? | To solve this problem, let's break it down step-by-step.
The ratio of doctors to nurses is given as 8:11. This means for every 8 doctors, there are 11 nurses.
1. **Identify the total parts of the ratio**: The total parts of the ratio (8 + 11) is 19. This means the total number of people (doctors and nurses) can be di... | 253 | [
58,
161,
342,
429,
656,
662,
720,
727,
897,
903,
951,
958,
1025
] | 91 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Larry and Barry want to pick apples out of the tree, but neither is tall enough to reach the apples. Barry can reach apples that are 5 feet high. Larry is 5 feet tall, but his sho... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Larry and Barry want to pick apples out of the tree, but neither is tall enough to reach the apples. Barry can reach apples that are 5 feet high. Larry is 5 feet tall, but his sho... | To find out how high Larry and Barry can reach when Barry stands on Larry's shoulders, we need to first determine how high Larry's shoulders are and then add Barry's reach to that height.
1. **Calculate Larry's Shoulder Height:**
- Larry's full height is 5 feet.
- His shoulder height is 20% less than his full he... | 9 | [
189,
231,
267,
326,
410,
540,
598,
660,
754,
848,
878,
937,
982,
1091
] | 92 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the number of moles of H2O formed on combining 2 moles of CH4 and 4 moles of O2 | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the number of moles of H2O formed on combining 2 moles of CH4 and 4 moles of O2 | To solve this problem, we need to consider the chemical reaction between methane (CH4) and oxygen (O2) to form water (H2O) and carbon dioxide (CO2). The balanced chemical equation for the combustion of methane is:
\[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \]
From the balanced equation... | 4 | [
215,
294,
424,
431,
448,
465,
505,
581,
657,
816,
871,
940,
1017,
1074
] | 93 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Mary just held tryouts for the high school band. 80% of the 20 flutes got in, half the 30 clarinets got in, 1/3 of the 60 trumpets got in, and 1/10th of the 20 pianists got in. How ... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Mary just held tryouts for the high school band. 80% of the 20 flutes got in, half the 30 clarinets got in, 1/3 of the 60 trumpets got in, and 1/10th of the 20 pianists got in. How ... | To find out how many people are in the band total, we will calculate the number of each type of instrument player that got in based on the given percentages and fractions, and then sum these numbers.
1. **Flutes**: 80% of the 20 flutes got in.
\[
0.80 \times 20 = 16 \text{ flutes}
\]
2. **Clarinets**: Half o... | 53 | [
201,
245,
251,
289,
296,
347,
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401,
408,
456,
462,
509,
516,
567,
573,
620,
627,
746,
749,
846,
850,
918
] | 95 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
There are 78 pieces of fruit in a crate. One-third of the box contains kiwi. The rest are strawberries. How many strawberries are there? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
There are 78 pieces of fruit in a crate. One-third of the box contains kiwi. The rest are strawberries. How many strawberries are there? | To find the number of strawberries in the crate, we first need to determine how many kiwis are in the crate. Given that one-third of the crate contains kiwis, we can calculate the number of kiwis as follows:
1. Calculate the number of kiwis:
\[ \text{Number of kiwis} = \frac{1}{3} \times 78 = 26 \]
2. Subtract the nu... | 52 | [
209,
243,
302,
402,
454,
510
] | 98 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A boat carrying 20 sheep, 10 cows and 14 dogs capsized. 3 of the sheep drowned. Twice as many cows drowned as did sheep. All of the dogs made it to shore. How many total animals... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A boat carrying 20 sheep, 10 cows and 14 dogs capsized. 3 of the sheep drowned. Twice as many cows drowned as did sheep. All of the dogs made it to shore. How many total animals... | To solve this problem, we need to calculate how many of each type of animal made it to the shore and then sum these numbers to find the total number of animals that survived.
1. **Sheep**: Initially, there were 20 sheep. 3 of the sheep drowned. Therefore, the number of sheep that made it to the shore is \(20 - 3 = 17\... | 35 | [
176,
324,
557,
605,
635,
647,
657,
669,
749,
790
] | 99 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Paddy's Confidential has 600 cans of stew required to feed 40 people. How many cans would be needed to feed 30% fewer people? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Paddy's Confidential has 600 cans of stew required to feed 40 people. How many cans would be needed to feed 30% fewer people? | To solve this problem, we'll follow these steps:
1. **Determine the number of people after a 30% reduction:**
- First, calculate 30% of 40 people: \( 40 \times \frac{30}{100} = 40 \times 0.30 = 12 \).
- Subtract this from the original number of people: \( 40 - 12 = 28 \).
2. **Find the ratio of cans to people f... | 420 | [
50,
111,
205,
281,
348,
402,
518,
576,
692,
727,
759,
852
] | 101 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
right triangle abc is the base of the prism in the figure above . if ab = ac = â ˆ š 10 and the height of the prism is 4 , what is the volume of the prism ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
right triangle abc is the base of the prism in the figure above . if ab = ac = â ˆ š 10 and the height of the prism is 4 , what is the volume of the prism ? | To solve for the volume of the prism, we first need to calculate the area of the base, which is a right triangle, and then multiply it by the height of the prism.
Given:
- \(AB = AC = \sqrt{10}\)
- The height of the prism is 4 units.
Since \(AB = AC\) and the triangle is a right triangle, \(AB\) and \(AC\) are the le... | 20 | [
164,
171,
197,
236,
458,
533,
587,
716,
822,
908,
961
] | 107 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
15 men take 21 days of 8 hrs . each to do a piece of work . how many days of 2 hrs . each would it take for 21 women if 3 women do as much work as 2 men ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
15 men take 21 days of 8 hrs . each to do a piece of work . how many days of 2 hrs . each would it take for 21 women if 3 women do as much work as 2 men ? | To solve this problem, we need to break it down into several steps, converting the given information into a form that allows us to compare the work done by men and women, and then calculate the required number of days.
### Step 1: Determine the total work in terms of man-hours.
Given:
- 15 men take 21 days of 8 hours ... | 90 | [
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1514,
1529,
1559,
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1580,
1591,
1593
] | 108 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
If $2^{x-3}=4^2$, find $x$. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
If $2^{x-3}=4^2$, find $x$. | To solve the equation \(2^{x-3} = 4^2\), we will follow these steps:
1. **Express 4 as a power of 2**:
Since \(4 = 2^2\), we can rewrite the equation as:
\[
2^{x-3} = (2^2)^2
\]
2. **Simplify the right-hand side**:
Using the property of exponents \((a^m)^n = a^{mn}\), we get:
\[
(2^2)^2 = 2^{2 \c... | 7 | [
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531,
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601,
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650,
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676,
686,
688
] | 109 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\%$-$69\%$ range? \begin{tabular}{|c|c|}
Test S... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\%$-$69\%$ range? \begin{tabular}{|c|c|}
Test S... | To find the percentage of the class that received a score in the $60\%$-$69\%$ range, we need to follow these steps:
1. **Count the total number of students in the class.**
2. **Count the number of students who scored in the $60\%$-$69\%$ range.**
3. **Calculate the percentage of students who scored in the $60\%$-$69\... | 20 | [
118,
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625,
665,
704,
739,
772,
800,
916,
1049,
1052,
1220,
1224,
1325
] | 111 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
One of Robi's new year's resolutions is to start saving. He started to save $2 in January. This is followed by $4 in February and $8 in March. If he continues this saving pattern, h... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
One of Robi's new year's resolutions is to start saving. He started to save $2 in January. This is followed by $4 in February and $8 in March. If he continues this saving pattern, h... | To find out Robi's total savings after 6 months following the given saving pattern, we need to first understand the pattern and then sum up the savings for each month.
From the problem, we can see that Robi's savings form a geometric sequence where the first term \(a_1 = 2\) (dollars in January) and the common ratio \... | 126 | [
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545,
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847,
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924,
976,
1045,
1077,
1117,
1133,
1150,
1169,
1188,
1209,
1255,
1272
] | 113 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $P= \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \dotsm \left( 1 - \frac{1}{n} \right)$. What is the value of $P$ if $n=2007$? E... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $P= \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \dotsm \left( 1 - \frac{1}{n} \right)$. What is the value of $P$ if $n=2007$? E... | To solve the given problem, we start by simplifying the expression for $P$:
\[P= \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \dotsm \left( 1 - \frac{1}{n} \right)\]
First, simplify each term inside the product:
\[1 - \frac{1}{k} = \frac{k-1}{k}\]
for any integer $k \geq... | \frac{1}{2007} | [
76,
215,
261,
297,
363,
481,
610,
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1127,
1153
] | 114 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the number of moles of Water formed on combining 3 moles of Methane and 6 moles of Oxygen | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the number of moles of Water formed on combining 3 moles of Methane and 6 moles of Oxygen | To find the number of moles of water formed when methane (CH₄) reacts with oxygen (O₂), we need to start with the balanced chemical equation for the combustion of methane:
\[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \]
This equation tells us that 1 mole of methane reacts with 2 moles of... | 6 | [
173,
252,
387,
394,
421,
447,
488,
603,
709,
813,
860,
982,
1104,
1164
] | 118 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units? | To find the measure of the hypotenuse of a right triangle with leg lengths of 75 and 100 units, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematical... | 125 | [
347,
368,
492,
516,
540,
557,
639,
661,
674,
740
] | 121 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Kay spends 250 minutes per week exercising. Her ratio of time spent on aerobics to time spent on weight training is 3 to 2. How many minutes per week does she spend on aerobics? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Kay spends 250 minutes per week exercising. Her ratio of time spent on aerobics to time spent on weight training is 3 to 2. How many minutes per week does she spend on aerobics? | To solve the problem, let's start by understanding the ratio and the total time spent exercising.
The ratio of time spent on aerobics to time spent on weight training is 3 to 2. This means for every 3 parts of time spent on aerobics, 2 parts are spent on weight training. The total parts of time spent exercising can be... | 150 | [
99,
390,
572,
679,
855,
950,
1017
] | 124 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the side of a square is increased by 25 % then how much % does its area increases ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the side of a square is increased by 25 % then how much % does its area increases ? | To solve this problem, we will follow these steps:
1. **Define the original side length of the square**: Let the original side length of the square be \( s \).
2. **Calculate the original area of the square**: The area of the square is \( s^2 \).
3. **Determine the new side length after the increase**: If the side len... | 56.25 | [
52,
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481,
531,
537,
562,
568,
664,
670,
701,
707,
863,
869,
935,
942,
1008
] | 129 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a company that ships boxes to a total of 13 distribution centers uses color coding to identify each center . if either a single color or a pair of two different colors is chosen to ... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a company that ships boxes to a total of 13 distribution centers uses color coding to identify each center . if either a single color or a pair of two different colors is chosen to ... | To solve this problem, we need to determine the minimum number of colors needed such that each distribution center is uniquely represented by either a single color or a pair of two different colors. We will start by calculating the number of unique representations possible with a given number of colors and see when thi... | 5 | [
351,
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1043,
1048,
1087,
1092,
1153,
1174,
1179,
1219,
1224,
1277,
1397,
1476
] | 131 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Alex had 36 ounces of jelly beans. He ate 6 ounces. Then he divided the rest equally into 3 piles. How much does each pile weigh? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Alex had 36 ounces of jelly beans. He ate 6 ounces. Then he divided the rest equally into 3 piles. How much does each pile weigh? | To solve this problem, we can break it down into a few simple steps:
1. **Find out how many ounces of jelly beans Alex had left after eating 6 ounces.**
- Alex started with 36 ounces.
- He ate 6 ounces.
- Therefore, the remaining amount is \(36 - 6 = 30\) ounces.
2. **Divide the remaining jelly beans equally... | 10 | [
70,
154,
188,
210,
275,
337,
367,
409,
518,
553,
556,
567,
569
] | 133 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
What is the molecular weight of 3 moles of Ascorbic acid | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
What is the molecular weight of 3 moles of Ascorbic acid | To solve this problem, we first need to find the molecular weight of ascorbic acid (vitamin C), and then we will multiply this by 3 to find the molecular weight of 3 moles of ascorbic acid.
Step 1: Determine the molecular formula of ascorbic acid.
The molecular formula of ascorbic acid is \( C_6H_8O_6 \).
Step 2: Cal... | 528.372 | [
191,
249,
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366,
430,
496,
561,
606,
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854,
900,
975,
1043,
1084,
1163
] | 138 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$? | To solve the problem of finding the length of the segment from $A$ to $A'$ after reflecting triangle $ABC$ over the $y$-axis, we can follow these steps:
1. **Identify the coordinates of $A'$ after reflection:**
- The reflection of a point $(x, y)$ over the $y$-axis results in the point $(-x, y)$.
- For point $A(... | 2 | [
154,
212,
302,
386,
438,
565,
630,
638,
736,
745,
809
] | 145 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if w is 40 percent less than e , e is 40 percent less than y , and z is 46 percent less than y , then z is greater than w by what percent of w ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if w is 40 percent less than e , e is 40 percent less than y , and z is 46 percent less than y , then z is greater than w by what percent of w ? | To solve the problem, we need to express all the given quantities in terms of one variable and then find the relationship between \(z\) and \(w\).
1. **Expressing \(w\) in terms of \(e\):**
Given \(w\) is 40 percent less than \(e\), we can write:
\[
w = e - 0.4e = 0.6e
\]
Therefore,
\[
e = \frac{w... | 50 | [
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1362,
1364
] | 151 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A classroom of 15 students turned in essays on Friday morning. The first 5 students each turned in essays with 2 pages. The next 5 students each turned in essays with 3 pages. The l... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A classroom of 15 students turned in essays on Friday morning. The first 5 students each turned in essays with 2 pages. The next 5 students each turned in essays with 3 pages. The l... | To find the average page count per essay for the 15 students, we need to follow these steps:
1. Calculate the total number of pages turned in by the students.
2. Divide the total number of pages by the number of students to find the average page count per essay.
First, let's calculate the total number of pages:
- Th... | 2 | [
94,
160,
265,
316,
446,
575,
703,
778,
896,
899,
1020,
1024,
1105
] | 153 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
indu gave bindu rs . 2500 on compound interest for 2 years at 4 % per annum . how much loss would indu has suffered had she given it to bindu for 2 years at 2 % per annum simple int... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
indu gave bindu rs . 2500 on compound interest for 2 years at 4 % per annum . how much loss would indu has suffered had she given it to bindu for 2 years at 2 % per annum simple int... | To solve this problem, we need to calculate the amount that would be received under two different interest schemes and then find the difference between them, which represents the loss Indu would have suffered.
### Step 1: Calculate the Compound Interest Amount
The formula for the amount with compound interest is:
\[ A... | 104 | [
211,
262,
316,
361,
368,
414,
458,
518,
542,
590,
629,
664,
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1117,
1182,
1238,
1285,
1306,
1322,
1387,
1418,
1569,
1601,
1626,
1685
] | 154 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Tom turned 32 years old 5 years ago. 7 years ago Jim was 5 years older than half Tom's age. How old is Jim in 2 years? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Tom turned 32 years old 5 years ago. 7 years ago Jim was 5 years older than half Tom's age. How old is Jim in 2 years? | To solve this problem, let's break it down step-by-step.
1. **Determine Tom's current age:**
- If Tom turned 32 years old 5 years ago, then Tom's current age is \(32 + 5 = 37\) years old.
2. **Determine Tom's age 7 years ago:**
- If Tom is currently 37 years old, then 7 years ago, Tom was \(37 - 7 = 30\) years ... | 29 | [
58,
94,
193,
233,
326,
366,
467,
534,
598,
634,
730,
769,
865,
906
] | 159 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
5/8 of shoppers at All Goods Available store prefer to avoid the check-out line on weekends and instead go through the express lane. If the number of shoppers in the store is 480, c... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
5/8 of shoppers at All Goods Available store prefer to avoid the check-out line on weekends and instead go through the express lane. If the number of shoppers in the store is 480, c... | To solve this problem, we first need to determine the fraction of shoppers who prefer the check-out line rather than the express lane. Since \( \frac{5}{8} \) of shoppers prefer to use the express lane, the remaining fraction of shoppers who prefer the check-out lane is:
\[ 1 - \frac{5}{8} = \frac{8}{8} - \frac{5}{8} ... | 180 | [
273,
338,
416,
663,
693,
744,
774,
801,
824,
859,
884,
953,
956,
968,
970
] | 161 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
there are 3 workers in the factory a , b , c . the age of a is 4 more than the age of b and the age of b is 4 more than the age of c . if the sum of the square of the ages is 1202 .... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
there are 3 workers in the factory a , b , c . the age of a is 4 more than the age of b and the age of b is 4 more than the age of c . if the sum of the square of the ages is 1202 .... | Let's denote the age of worker \(C\) as \(x\).
According to the problem:
- The age of \(B\) is 4 more than the age of \(C\), so the age of \(B\) is \(x + 4\).
- The age of \(A\) is 4 more than the age of \(B\), so the age of \(A\) is \((x + 4) + 4 = x + 8\).
We're also given that the sum of the squares of their ages ... | 57 | [
48,
74,
160,
261,
329,
332,
367,
371,
418,
421,
469,
473,
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496,
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629,
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982,
985,
988,
1026,
1029,
1032,
1065,
1068,
1071,
1104,
1107,
1110,
1... | 162 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | To solve for $n$ in the equation $8^n \cdot 8^n \cdot 8^n = 64^3$, we can start by simplifying the equation using the properties of exponents and expressing all terms with the same base.
First, note that $8$ and $64$ can be expressed as powers of $2$:
- $8 = 2^3$
- $64 = 2^6$
Therefore, we can rewrite the equation as... | 2 | [
188,
253,
265,
279,
331,
334,
409,
413,
447,
450,
465,
469,
498,
501,
522,
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611,
614,
630,
634,
711,
714,
722,
726,
744,
747,
768,
772,
796,
799,
809,
811
] | 166 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
John buys 20 hours of music a month. The average length of a song is 3 minutes. He buys each song for $.50. How much does he pay for music a year? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
John buys 20 hours of music a month. The average length of a song is 3 minutes. He buys each song for $.50. How much does he pay for music a year? | To find out how much John pays for music a year, we'll follow these steps:
1. **Calculate the number of songs John buys each month:**
- There are 60 minutes in an hour.
- John buys 20 hours of music a month.
- Therefore, the total number of minutes of music he buys each month is \(20 \times 60 = 1200\) minute... | 2400 | [
76,
135,
173,
215,
323,
454,
498,
525,
597,
631,
667,
737,
805
] | 170 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Freeport McMoran projects that in 2010 the world grain supply will be 1800000 metric tons and the supply will be only 0.75 of the world grain demand. What will the world grain deman... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Freeport McMoran projects that in 2010 the world grain supply will be 1800000 metric tons and the supply will be only 0.75 of the world grain demand. What will the world grain deman... | To find the world grain demand in 2010 based on the given information, we can set up a simple equation. Let's denote the world grain demand in 2010 as \(D\).
According to the problem, the supply in 2010 is projected to be 1,800,000 metric tons, and this supply is only 0.75 (or 75%) of the demand. This can be written a... | 2400000 | [
159,
324,
357,
448,
482,
503,
580,
595
] | 171 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a computer store offers employees a 10 % discount off the retail price . if the store purchased a computer from the manufacturer for $ 1200 dollars and marked up the price 10 % to t... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a computer store offers employees a 10 % discount off the retail price . if the store purchased a computer from the manufacturer for $ 1200 dollars and marked up the price 10 % to t... | To solve this problem, let's break it down into steps:
1. **Determine the final retail price**:
- The store marked up the price by 10% from the purchase price of $1200.
- The markup amount is \( 1200 \times \frac{10}{100} = 1200 \times 0.10 = 120 \) dollars.
- Therefore, the final retail price is \( 1200 + 12... | 132 | [
56,
97,
173,
266,
342,
388,
454,
549,
630,
681,
785,
844,
994
] | 172 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
an aeroplane covers a certain distance of 420 kmph in 6 hours . to cover the same distance in 4 2 / 3 hours , it must travel at a speed of ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
an aeroplane covers a certain distance of 420 kmph in 6 hours . to cover the same distance in 4 2 / 3 hours , it must travel at a speed of ? | To solve this problem, we will first determine the distance covered by the aeroplane and then calculate the required speed to cover this distance in the given time.
Step 1: Calculate the distance covered by the aeroplane.
The problem states that the aeroplane covers a certain distance at a speed of 420 km/h in 6 hours... | 540 | [
166,
223,
419,
508,
567,
657,
709,
792,
887,
963,
1009,
1092,
1221
] | 176 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} + \mathbf{b}\| = \sqrt{3},$ and
\[\mathbf{c} - \mathbf{a} ... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} + \mathbf{b}\| = \sqrt{3},$ and
\[\mathbf{c} - \mathbf{a} ... | To find $\mathbf{b} \cdot \mathbf{c}$, we need to carefully analyze the given information and work through the problem step-by-step.
Given:
1. $\|\mathbf{a}\| = 1$ and $\|\mathbf{b}\| = 1$.
2. $\|\mathbf{a} + \mathbf{b}\| = \sqrt{3}$.
3. $\mathbf{c} - \mathbf{a} - 2 \mathbf{b} = 3 (\mathbf{a} \times \mathbf{b})$.
###... | \frac{5}{2} | [
134,
141,
191,
236,
317,
356,
443,
446,
607,
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1904,
2045,
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2105,
2108,
2114,
2117,
... | 178 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a circle graph shows how the megatech corporation allocates its research and development budget : 14 % microphotonics ; 25 % home electronics ; 15 % food additives ; 29 % geneticall... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
a circle graph shows how the megatech corporation allocates its research and development budget : 14 % microphotonics ; 25 % home electronics ; 15 % food additives ; 29 % geneticall... | To find out how many degrees of the circle are used to represent basic astrophysics research, we first need to calculate the percentage of the budget allocated to basic astrophysics. We can do this by subtracting the sum of the percentages allocated to the other sectors from 100%.
The percentages given are:
- Micropho... | 32.4 | [
283,
310,
332,
356,
378,
421,
450,
483,
529,
618,
644,
825,
912,
941,
1041,
1124,
1127,
1140,
1142
] | 184 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
donald duck can can swim his pool downstream ( with the pool current helping time ) in exact 40 seconds and upstream ( against the pool current ) in a pool in exact 60 seconds . the... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
donald duck can can swim his pool downstream ( with the pool current helping time ) in exact 40 seconds and upstream ( against the pool current ) in a pool in exact 60 seconds . the... | To solve the problem, we need to determine Donald Duck's swimming speed in a still pool. We are given the times it takes for Donald to swim downstream and upstream, as well as the length of the pool. Let's denote Donald's swimming speed in still water as \( v \) (in kilometers per second) and the speed of the pool's cu... | 48 | [
366,
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415,
475,
537,
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677,
746,
760,
865,
899,
918,
940,
987,
1027,
1045,
1096,
1192,
1294,
1340,
1460
] | 187 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A soccer team had 24 players prepared for a soccer game. The first half saw 11 players start the game with 2 substitutions made. In the second half, the team made twice as many subs... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A soccer team had 24 players prepared for a soccer game. The first half saw 11 players start the game with 2 substitutions made. In the second half, the team made twice as many subs... | To solve this problem, we need to carefully track the number of players who played in the game and subtract this from the total number of players on the team to find out how many did not play.
1. **Total Players:** The soccer team had 24 players prepared for the game.
2. **First Half Substitutions:** In the first half... | 7 | [
194,
270,
557,
782,
1252,
1316,
1398,
1540,
1874,
2101,
2169,
2195,
2267,
2348,
2469,
2488,
2571
] | 188 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Solve the equation $27 = 3(9)^{x-1}$ for $x.$ | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Solve the equation $27 = 3(9)^{x-1}$ for $x.$ | To solve the equation \(27 = 3(9)^{x-1}\) for \(x\), we'll follow these steps:
1. **Express all terms with the same base:**
Note that \(27 = 3^3\) and \(9 = 3^2\). Therefore, we can rewrite the equation using base 3.
2. **Rewrite the equation using base 3:**
\[
3^3 = 3 \cdot (3^2)^{x-1}
\]
Simplify the... | 2 | [
80,
125,
222,
264,
270,
299,
305,
349,
355,
383,
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417,
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624,
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722,
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923,
947,
971,
977,
991,
997,
1003,
1013,
1019,
1046,
1052,
1061,
106... | 190 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers? | To solve the problem, we start by analyzing the function \( f(n) = n^4 - 360n^2 + 400 \). Our goal is to determine when \( f(n) \) is a prime number and then sum all such values.
First, let's rewrite the function \( f(n) \) in a potentially more manageable form. Notice that the expression resembles a quadratic in term... | 41 | [
180,
337,
370,
417,
448,
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1725,
1741,
1807,
1837,
1853,
1919,
1949,
2068,
2143,
2146,
2157,
2159
] | 191 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A theater sold 900 tickets to a play. Floor seats cost 12 dollars each and balcony seats 10 dollars each. Total receipts were 9,780 dollars. How many balcony seats were sold? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A theater sold 900 tickets to a play. Floor seats cost 12 dollars each and balcony seats 10 dollars each. Total receipts were 9,780 dollars. How many balcony seats were sold? | To solve this problem, we can use a system of equations based on the information provided.
Let's define:
- \(x\) as the number of floor seats sold.
- \(y\) as the number of balcony seats sold.
We know two things:
1. The total number of tickets sold was 900.
2. The total receipts from ticket sales were $9,780.
From t... | 510 | [
92,
106,
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195,
215,
260,
314,
355,
374,
399,
486,
505,
584,
616,
646,
685,
710,
768,
793,
812,
837,
864,
879,
937
] | 194 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Angelina is 4 years older than Justin is now. In 5 years, Angelina will be 40 years old. Calculate the age of Justin currently. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Angelina is 4 years older than Justin is now. In 5 years, Angelina will be 40 years old. Calculate the age of Justin currently. | To find Justin's current age, we can start by setting up equations based on the information given.
Let's denote Justin's current age as \( J \).
According to the problem:
1. Angelina is 4 years older than Justin is now. So, Angelina's current age can be represented as \( J + 4 \).
2. In 5 years, Angelina will be 40 y... | 31 | [
100,
147,
173,
284,
389,
453,
477,
500,
521,
539,
603,
620,
634,
686
] | 195 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $f(x)=\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $f(x)=\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. | To solve the problem, we need to evaluate the expression $f(g(x)) + g(f(x))$ at $x=1$. Let's break this down step-by-step.
### Step 1: Evaluate $g(x)$ at $x=1$
Given $g(x) = x - 1$, we substitute $x = 1$:
\[
g(1) = 1 - 1 = 0
\]
### Step 2: Evaluate $f(x)$ at $x=1$
Given $f(x) = \frac{3x^2 + 5x + 8}{x^2 - x + 4}$, we ... | 5 | [
124,
161,
206,
209,
226,
230,
267,
340,
343,
441,
445,
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711,
714,
731,
734,
763,
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837,
840,
870,
874,
906,
909,
919,
921
] | 197 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Andy walks 50 meters from his house to school. After school, he comes back to the house and goes to the market. If he walks 140 meters in total, how many meters is the distance betw... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Andy walks 50 meters from his house to school. After school, he comes back to the house and goes to the market. If he walks 140 meters in total, how many meters is the distance betw... | To find the distance between Andy's house and the market, let's break down his journey step by step.
1. Andy walks from his house to school, which is a distance of 50 meters.
2. After school, he walks back to his house, covering another 50 meters.
So, the distance covered just to go to school and back home is \(50 + ... | 40 | [
102,
176,
250,
340,
485,
582,
649,
734
] | 198 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if a = 105 and a ^ 3 = 21 × 25 × 45 × b , what is the value of b ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
if a = 105 and a ^ 3 = 21 × 25 × 45 × b , what is the value of b ? | To solve the given problem, let's first express the equation and then simplify it step-by-step.
Given:
\[ a = 105 \]
\[ a^3 = 21 \times 25 \times 45 \times b \]
First, we calculate \( a^3 \):
\[ a^3 = 105^3 \]
To simplify the process, we factorize each number on the right-hand side of the equation.
1. Factorize 21:... | 49 | [
97,
104,
118,
163,
194,
213,
304,
321,
344,
361,
377,
394,
419,
457,
529,
573,
621,
641,
673,
679,
746,
781,
849,
887,
911,
932,
968,
999,
1015
] | 200 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in a certain archery competition , points were awarded as follows : the first place winner receives 11 points , the second place winner receives 7 points , the third place winner re... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in a certain archery competition , points were awarded as follows : the first place winner receives 11 points , the second place winner receives 7 points , the third place winner re... | To solve this problem, we need to determine how many times John participated in the archery competition given the product of the points he received is 15400. Let's break down the solution step-by-step.
First, let's factorize 15400 to understand its prime factorization:
\[ 15400 = 154 \times 100 = (14 \times 11) \times... | 7 | [
203,
271,
417,
466,
491,
516,
540,
566,
632,
681,
693,
737,
781,
824,
869,
903,
920,
936,
953,
977,
994,
1028,
1063,
1098,
1141,
1208,
1233,
1318
] | 202 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $p$ and $q$ be positive integers such that\[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\]and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qq... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Let $p$ and $q$ be positive integers such that\[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\]and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qq... | To solve the problem, we need to find the smallest positive integer \( q \) such that there exists a positive integer \( p \) satisfying the inequality \(\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\).
### Step 1: Understanding the Inequality
We start by converting the given fractions to decimal form for easier comparison:... | 7 | [
198,
239,
321,
324,
401,
404,
461,
464,
494,
498,
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999,
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1336,
1377,
1380,
1407,
1410,
1460,
1463,
1466,
1498,
1501,
1504,
1524,
... | 208 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Bridge and Sarah have $3 between them. If Bridget has 50 cents more than Sarah, how many cents does Sarah have? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Bridge and Sarah have $3 between them. If Bridget has 50 cents more than Sarah, how many cents does Sarah have? | Let's denote the amount of money Sarah has as \(S\) dollars, and the amount of money Bridget has as \(B\) dollars. According to the problem, we have two key pieces of information:
1. The total amount of money they have together is $3. This can be expressed as:
\[B + S = 3\]
2. Bridget has 50 cents more than Sarah, wh... | 125 | [
181,
262,
277,
345,
363,
514,
536,
556,
575,
637,
652,
695,
709,
833,
860,
898
] | 209 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.75 km and 1.10 km respectively . the time taken by the slower train to cross the fast... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.75 km and 1.10 km respectively . the time taken by the slower train to cross the fast... | To solve this problem, we need to find out how long it takes for the two trains to completely pass each other when moving in opposite directions. Here's a step-by-step approach:
1. **Calculate the total length the trains need to cover to completely pass each other:**
- Length of the first train: 1.75 km
- Length... | 68 | [
179,
269,
309,
350,
399,
458,
518,
573,
613,
654,
771,
826,
907,
971,
1057,
1110,
1176,
1250,
1333,
1426
] | 214 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Mara and Riley went to a carnival, Mara rode the bumper car two times, Riley rode the space shuttle four times, and they both rode the Ferris wheel three times. If a ticket for a bu... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Mara and Riley went to a carnival, Mara rode the bumper car two times, Riley rode the space shuttle four times, and they both rode the Ferris wheel three times. If a ticket for a bu... | To solve this problem, let's break down the cost of each ride and then calculate the total cost based on how many times Mara and Riley rode each attraction.
1. **Calculate the cost for the bumper car rides:**
- Mara rode the bumper car 2 times.
- The cost for one bumper car ride is $2.
- Therefore, the total ... | 50 | [
158,
210,
249,
294,
385,
440,
483,
531,
627,
681,
742,
789,
882,
976,
1080,
1124,
1265,
1330,
1425
] | 224 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Melanie is selling 4 gumballs for 8 cents each. Melanie get _____ cents from selling the gumballs. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Melanie is selling 4 gumballs for 8 cents each. Melanie get _____ cents from selling the gumballs. | To solve this problem, follow these steps:
1. Identify the cost per gumball: Each gumball costs 8 cents.
2. Determine the number of gumballs: Melanie is selling 4 gumballs.
3. Multiply the number of gumballs by the cost per gumball to find the total amount of money:
\[
4 \text{ gumballs} \times 8 \text{ cents per gum... | 32 | [
44,
106,
174,
269,
272,
345,
349,
413
] | 228 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The product of a number $M$ and six less than $M$ is $-5$. What is the sum of all possible values of $M$? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The product of a number $M$ and six less than $M$ is $-5$. What is the sum of all possible values of $M$? | To solve the given problem, we start by translating the given statement into a mathematical equation. We are told that the product of a number \( M \) and six less than \( M \) is \(-5\). This can be expressed as:
\[
M \times (M - 6) = -5
\]
Expanding and rearranging the equation, we get:
\[
M^2 - 6M = -5
\]
To for... | 6 | [
215,
218,
240,
244,
293,
296,
310,
314,
378,
381,
398,
402,
566,
605,
608,
662,
666,
765,
768,
830,
834,
864,
867,
906,
910,
913,
951,
955,
1063,
1066,
1076,
1080,
1133,
1136,
1146,
1148
] | 229 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A girl saves 1-dollar coins and 2-dollar coins. She takes it to a bank , she has 180 coins in total that equals 260 dollars. How many 1-dollar coins are there? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A girl saves 1-dollar coins and 2-dollar coins. She takes it to a bank , she has 180 coins in total that equals 260 dollars. How many 1-dollar coins are there? | To solve this problem, we can set up a system of equations based on the information provided.
Let's define:
- \(x\) as the number of 1-dollar coins.
- \(y\) as the number of 2-dollar coins.
We are given two pieces of information:
1. The total number of coins is 180.
2. The total value of the coins is 260 dollars.
Fr... | 100 | [
95,
109,
150,
192,
232,
269,
318,
382,
399,
464,
482,
493,
511,
548,
551,
565,
580,
593,
605,
609,
713,
716,
747,
750,
753,
773,
776,
779,
786,
790,
892,
895,
908,
911,
914,
927,
930,
933,
941,
945,
999
] | 230 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Given the four digits 2, 4, 6, and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Given the four digits 2, 4, 6, and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? | To solve this problem, we need to determine how many different two-digit integers can be formed using the digits 2, 4, 6, and 7 without repeating any digit.
Step 1: Identify the total number of digits available.
We have 4 digits: 2, 4, 6, and 7.
Step 2: Determine the number of choices for the tens place.
Since we can... | 12 | [
158,
213,
248,
308,
378,
439,
538,
606,
761,
851,
952
] | 234 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in a can , there is a mixture of milk and water in the ratio 5 : 3 . if the can is filled with an additional 20 liters of milk , the can would be full and the ratio of milk and wate... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in a can , there is a mixture of milk and water in the ratio 5 : 3 . if the can is filled with an additional 20 liters of milk , the can would be full and the ratio of milk and wate... | Let's denote the initial amount of milk in the can as \(5x\) liters and the initial amount of water as \(3x\) liters. According to the problem, the initial ratio of milk to water is \(5:3\).
After adding 20 liters of milk, the amount of milk becomes \(5x + 20\) liters, and the amount of water remains \(3x\) liters. Th... | 60 | [
192,
367,
426,
429,
462,
466,
504,
507,
520,
524,
567,
570,
583,
586,
589,
597,
600,
603,
609,
613,
696,
699,
743,
746,
749,
794,
798,
887,
890,
926,
930,
1011,
1014,
1089,
1093,
1127,
1130,
1141,
1143
] | 236 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the average age of applicants for a new job is 10 , with a standard deviation of 8 . the hiring manager is only willing to accept applicants whose age is within one standard deviati... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
the average age of applicants for a new job is 10 , with a standard deviation of 8 . the hiring manager is only willing to accept applicants whose age is within one standard deviati... | To solve the problem, we need to determine the range of ages that the hiring manager is willing to accept and then count the number of integer ages within that range.
1. **Identify the average age and standard deviation:**
- Average age (\(\mu\)) = 10
- Standard deviation (\(\sigma\)) = 8
2. **Determine the ran... | 17 | [
168,
224,
256,
298,
345,
464,
537,
595,
644,
695,
761,
823,
888,
896,
991,
1000,
1084
] | 237 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
When the repeating decimal $0.\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
When the repeating decimal $0.\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator? | To solve the problem of expressing the repeating decimal $0.\overline{12}$ as a common fraction in lowest terms and then finding the sum of its numerator and denominator, we follow these steps:
Step 1: Let $x = 0.\overline{12}$.
Step 2: Multiply both sides of the equation by 100 (since the repeating block "12" is two... | 37 | [
195,
231,
387,
416,
474,
524,
538,
590,
613,
745,
809,
916,
963,
980,
1050
] | 243 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\sqrt{c} - a$ cm. What is the value of $a... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\sqrt{c} - a$ cm. What is the value of $a... | To solve this problem, we will find the radius of the sphere inscribed in a right cone with a base radius of $12$ cm and height of $24$ cm. The key to solving this problem is to use the geometric relationship between the cone and the sphere.
First, consider the right triangle formed by the height of the cone, the radi... | 11 | [
243,
453,
547,
686,
738,
800,
906,
957,
1114,
1300,
1322,
1336
] | 248 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
what is the value of n if the sum of the consecutive odd intergers q from 1 to n equals 169 ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
what is the value of n if the sum of the consecutive odd intergers q from 1 to n equals 169 ? | To find the value of \( n \) such that the sum of the consecutive odd integers from 1 to \( n \) equals 169, we can follow these steps:
1. **Understand the Sum of Consecutive Odd Integers:**
The sum of the first \( k \) odd integers is given by the formula:
\[
S = k^2
\]
where \( k \) is the number of t... | 25 | [
137,
192,
262,
268,
279,
285,
376,
404,
457,
463,
476,
483,
509,
548,
554,
577,
584,
624,
666,
672,
686,
692,
720,
726,
763,
770,
820
] | 249 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Round $3.45$ to the nearest tenth. | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Round $3.45$ to the nearest tenth. | To round the number $3.45$ to the nearest tenth, follow these steps:
1. Identify the tenths and hundredths place in the number. In $3.45$, the digit $4$ is in the tenths place, and the digit $5$ is in the hundredths place.
2. Determine whether to round up or down based on the hundredths place. The rule is that if the ... | 3.5 | [
70,
224,
454,
539,
654,
712,
723
] | 250 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue or green. If no two adjacent strips can be the same color, how many distinct flag... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue or green. If no two adjacent strips can be the same color, how many distinct flag... | To solve this problem, we need to determine the number of distinct flags that can be made under the condition that no two adjacent strips can be the same color. The flag consists of three horizontal strips, and each strip can be one of four colors: red, white, blue, or green.
Let's break down the problem step by step:... | 36 | [
278,
322,
506,
673,
842,
954,
957,
1119,
1123,
1196
] | 258 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
How many positive three-digit integers with each digit greater than 4 are divisible by 6? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
How many positive three-digit integers with each digit greater than 4 are divisible by 6? | To solve this problem, we'll first identify the range of digits that can be used, then determine the conditions for divisibility by 6, and finally, count the numbers that satisfy these conditions.
### Step 1: Identify the Range of Digits
Digits greater than 4 can be 5, 6, 7, 8, or 9. This gives us 5 options for each d... | 32 | [
198,
239,
351,
396,
472,
583,
669,
704,
775,
804,
927,
1000,
1041,
1082,
1119,
1160,
1201,
1242,
1279,
1320,
1361,
1398,
1435,
1476,
1517,
1554,
1595,
1636,
1677,
1714,
1755,
1796,
1837,
1874,
1915,
1956,
1994,
2108,
2201,
2230,
2353,
... | 266 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
John jogs at a speed of 4 miles per hour when he runs alone, but runs at 6 miles per hour when he is being dragged by his 100-pound German Shepherd dog. If John and his dog go on a ... | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
John jogs at a speed of 4 miles per hour when he runs alone, but runs at 6 miles per hour when he is being dragged by his 100-pound German Shepherd dog. If John and his dog go on a ... | To solve the problem, we need to break down John's total distance traveled into two parts: the distance he covers when running with his dog and the distance he covers when running alone.
1. **Distance when running with the dog:**
- John's speed when running with his dog is 6 miles per hour.
- They run together f... | 5 | [
188,
231,
296,
355,
423,
459,
517,
577,
645,
677,
766,
843
] | 267 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Queenie earns $150 a day as a part-time clerk. She earns an additional $5 per hour as overtime pay. How much will Queenie receive for working 5 days with 4 hours overtime? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Queenie earns $150 a day as a part-time clerk. She earns an additional $5 per hour as overtime pay. How much will Queenie receive for working 5 days with 4 hours overtime? | To solve the problem, let's break it down into steps.
1. **Calculate Queenie's regular pay for 5 days:**
Queenie earns $150 a day. For 5 days, her regular pay is:
\[
150 \text{ dollars/day} \times 5 \text{ days} = 750 \text{ dollars}
\]
2. **Calculate Queenie's overtime pay for 4 hours:**
Queenie earns... | 770 | [
55,
106,
167,
173,
244,
251,
304,
407,
413,
483,
490,
522,
617,
623,
689,
696,
808
] | 269 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in one hour , a boat goes 21 km / hr along the stream and 11 km / hr against the stream . the speed of the boat in still water ( in km / hr ) is : | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
in one hour , a boat goes 21 km / hr along the stream and 11 km / hr against the stream . the speed of the boat in still water ( in km / hr ) is : | To find the speed of the boat in still water, we can use the information given about the boat's speed downstream and upstream.
Let's denote:
- \(x\) as the speed of the boat in still water (in km/hr),
- \(y\) as the speed of the stream (in km/hr).
When the boat is moving downstream, its effective speed is the sum of ... | 16 | [
128,
142,
202,
250,
386,
402,
548,
564,
611,
629,
648,
723,
755,
767,
779,
846
] | 279 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
what will be the cost of building a fence around a square plot with area equal to 144 sq ft , if the price per foot of building the fence is rs . 58 ? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
what will be the cost of building a fence around a square plot with area equal to 144 sq ft , if the price per foot of building the fence is rs . 58 ? | To find the cost of building a fence around a square plot with an area of 144 square feet, given that the price per foot of building the fence is Rs. 58, we can follow these steps:
### Step 1: Determine the Side Length of the Square Plot
- The area of a square is given by \( A = s^2 \), where \( s \) is the length of ... | 2784 | [
182,
239,
344,
408,
413,
425,
430,
435,
452,
457,
462,
482,
488,
543,
605,
653,
658,
692,
698,
757,
801,
895,
900,
919,
924,
957,
962,
984,
990,
1052
] | 287 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The infinite sequence $S=\{s_1,s_2,s_3,\ldots\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$? | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
The infinite sequence $S=\{s_1,s_2,s_3,\ldots\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$? | To solve the problem of finding the remainder when $s_{100}$ is divided by 5, we need to understand the behavior of the sequence modulo 5. Given the sequence $S=\{s_1,s_2,s_3,\ldots\}$ with $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$, we can start by examining the powers of 7 modulo 5.
First, note that $7 \eq... | 3 | [
297,
410,
438,
484,
538,
550,
656,
684,
712,
749,
787,
816,
1000,
1052,
1088,
1127,
1179,
1303,
1352,
1436,
1509
] | 289 |
Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the point of tangency of the parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593.$ | Solve the following math problem step-by-step.
Simplify your answer as much as possible. Present your final answer as \boxed{Your Answer}.
Find the point of tangency of the parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593.$ | To find the point of tangency of the given parabolas, we need to find a common point \((x, y)\) that satisfies both equations, and also ensure that the slopes of the tangents at this point are equal.
First, let's denote the equations of the parabolas as follows:
\[ y = x^2 + 15x + 32 \]
\[ x = y^2 + 49y + 593 \]
We n... | (-7, -24) | [
201,
264,
289,
316,
449,
480,
513,
707,
738,
812,
843,
977,
1031,
1060,
1088,
1117,
1174,
1248,
1305,
1441,
1463,
1517,
1571,
1630,
1743,
1791,
1841,
1871,
1901,
2175,
2207,
2230
] | 294 |
End of preview. Expand in Data Studio
UltraInteract (math only), responses of Qwen2.5-14B-Instruct
The math chain-of-thought part of the UltraInteract prompts, with the responses that the TSD-KD authors generated with
the teacher Qwen2.5-14B-Instruct (paper: https://arxiv.org/abs/2603.13260; source release
Minsang/TSD-KD-Qwen2.5-14B-Instruct-Gen,
79,751 rows). The authors' repository (https://github.com/kmswin1/TSD-KD) describes it as the teacher-generated data for
off-policy distillation; the dataset card of the release itself is a copy of the 1.5B card and calls the responses
student-generated, which is a mistake. The license follows the source release (MIT).
Filter: the instruction starts with Solve the following math problem step-by-step. (22,582 rows) and the response
ends in a \boxed{} answer (39 rows dropped) -> 22,543 rows.
Fields of tsdkd-clean.jsonl:
id: row number in the original release file (the two releases share the same prompts in the same order)user_prompt/question: the release's own user turn (no extra instruction added)output: theresponseof the release rowanswer: content of the last\boxed{}of the responsestep_char_ends: end offsets of the response lines (one reasoning step per non-empty line)
The student-generated counterpart (Qwen2.5-1.5B-Instruct) is
phuocsang/UltraInteract-math-Qwen2.5-1.5B-Instruct-Gen.
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