messages listlengths 2 2 | instruction_seed stringlengths 27 7.16k | response_seed stringlengths 27 7.16k | _source stringclasses 1
value | _fasttext_score float64 1 1 | reasoning stringlengths 1.03k 110k | deepseek_solution stringlengths 7 12.4k | __original_row_idx int64 0 31.6k | final_reasoning_trace stringlengths 1.52k 112k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|---|---|
[
{
"content": "The lengths of the three sides of a triangle are $10$, $x+5$ and $3x-2$. The perimeter of the triangle is 45. What is the length of the longest side of the triangle?",
"role": "user"
},
{
"content": "We know that the perimeter of the triangle is 45, and the lengths of the three sides... | The lengths of the three sides of a triangle are $10$, $x+5$ and $3x-2$. The perimeter of the triangle is 45. What is the length of the longest side of the triangle? | The lengths of the three sides of a triangle are $10$, $x+5$ and $3x-2$. The perimeter of the triangle is 45. What is the length of the longest side of the triangle? | ai2-adapt-dev/openmath-2-math | 0.999567 | Okay, let's see. I need to find the length of the longest side of a triangle where the three sides are 10, x+5, and 3x-2. The perimeter is 45. Hmm, perimeter is the sum of all sides, right? So maybe I can set up an equation adding those three sides together and solve for x first. Once I have x, I can find the lengths o... | The lengths of the three sides of the triangle are given as 10, \(x + 5\), and \(3x - 2\). The perimeter of the triangle is 45. To find the value of \(x\), we set up the equation for the perimeter:
\[
10 + (x + 5) + (3x - 2) = 45
\]
Simplifying the left side, we combine the constant terms and the terms with \(x\):
\... | 7 | <think>
Okay, let's see. I need to find the length of the longest side of a triangle where the three sides are 10, x+5, and 3x-2. The perimeter is 45. Hmm, perimeter is the sum of all sides, right? So maybe I can set up an equation adding those three sides together and solve for x first. Once I have x, I can find the l... | [
{
"from": "human",
"value": "The lengths of the three sides of a triangle are $10$, $x+5$ and $3x-2$. The perimeter of the triangle is 45. What is the length of the longest side of the triangle?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the length of the longest s... |
[
{
"content": "Find the absolute value of the sum of the solutions of $x^2-7x-4=0$.",
"role": "user"
},
{
"content": "To find the sum of the solutions of the quadratic equation $x^2 - 7x - 4 = 0$, we can use Vieta's formulas.\n\nVieta's formulas state that for a quadratic equation $ax^2 + bx + c = 0$... | Find the absolute value of the sum of the solutions of $x^2-7x-4=0$. | Find the absolute value of the sum of the solutions of $x^2-7x-4=0$. | ai2-adapt-dev/openmath-2-math | 0.999974 | Okay, let's see. I need to find the absolute value of the sum of the solutions for the quadratic equation \(x^2 - 7x - 4 = 0\). Hmm, quadratic equations. Right, they have two solutions, and there's something about the sum of the roots. Wait, isn't there a formula or a theorem related to that? Oh yeah! Vieta's formulas.... | To find the absolute value of the sum of the solutions of the quadratic equation \(x^2 - 7x - 4 = 0\), we can use Vieta's formulas. For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is given by \(-b/a\). Here, \(a = 1\), \(b = -7\), and \(c = -4\). Therefore, the sum of the solutions is:
\[
-\frac{b... | 44 | <think>
Okay, let's see. I need to find the absolute value of the sum of the solutions for the quadratic equation \(x^2 - 7x - 4 = 0\). Hmm, quadratic equations. Right, they have two solutions, and there's something about the sum of the roots. Wait, isn't there a formula or a theorem related to that? Oh yeah! Vieta's f... | [
{
"from": "human",
"value": "Find the absolute value of the sum of the solutions of $x^2-7x-4=0$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the absolute value of the sum of the solutions for the quadratic equation \\(x^2 - 7x - 4 = 0\\). Hmm, quadratic equations. Ri... |
[
{
"content": "In a right triangle, the length of the altitude to the hypotenuse is half the length of the hypotenuse. If the hypotenuse has a length of 10 inches, what is the area of the triangle?",
"role": "user"
},
{
"content": "Since the altitude to the hypotenuse is half the length of the hypote... | In a right triangle, the length of the altitude to the hypotenuse is half the length of the hypotenuse. If the hypotenuse has a length of 10 inches, what is the area of the triangle? | In a right triangle, the length of the altitude to the hypotenuse is half the length of the hypotenuse. If the hypotenuse has a length of 10 inches, what is the area of the triangle? | ai2-adapt-dev/openmath-2-math | 0.99992 | Okay, so I need to find the area of a right triangle where the altitude to the hypotenuse is half the length of the hypotenuse, and the hypotenuse itself is 10 inches. Let me start by recalling some properties of right triangles.
First, in a right triangle, the altitude to the hypotenuse relates the two segments into ... | Given a right triangle where the altitude to the hypotenuse is half the length of the hypotenuse, and the hypotenuse itself is 10 inches, we need to find the area of the triangle.
1. The hypotenuse \( c \) is given as 10 inches.
2. The altitude \( h \) to the hypotenuse is half of the hypotenuse, so \( h = \frac{10}{2... | 53 | <think>
Okay, so I need to find the area of a right triangle where the altitude to the hypotenuse is half the length of the hypotenuse, and the hypotenuse itself is 10 inches. Let me start by recalling some properties of right triangles.
First, in a right triangle, the altitude to the hypotenuse relates the two segmen... | [
{
"from": "human",
"value": "In a right triangle, the length of the altitude to the hypotenuse is half the length of the hypotenuse. If the hypotenuse has a length of 10 inches, what is the area of the triangle?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the area of a right ... |
[
{
"content": "Find the sum of the coordinates of the midpoint of a line segment with endpoints at $(-1, 8)$ and $(7, 2)$.",
"role": "user"
},
{
"content": "To find the midpoint of a line segment with endpoints $(-1, 8)$ and $(7, 2)$, we use the midpoint formula:\n\\[ \\text{midpoint} = \\left( \\fra... | Find the sum of the coordinates of the midpoint of a line segment with endpoints at $(-1, 8)$ and $(7, 2)$. | Find the sum of the coordinates of the midpoint of a line segment with endpoints at $(-1, 8)$ and $(7, 2)$. | ai2-adapt-dev/openmath-2-math | 0.999501 | Okay, so I need to find the sum of the coordinates of the midpoint of a line segment with endpoints at (-1, 8) and (7, 2). Hmm, let me think. I remember that the midpoint formula is something like averaging the x-coordinates and averaging the y-coordinates of the two endpoints. Let me verify that.
So, if I have two p... | To find the sum of the coordinates of the midpoint of a line segment with endpoints at \((-1, 8)\) and \((7, 2)\), we use the midpoint formula. The midpoint \((M)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substit... | 58 | <think>
Okay, so I need to find the sum of the coordinates of the midpoint of a line segment with endpoints at (-1, 8) and (7, 2). Hmm, let me think. I remember that the midpoint formula is something like averaging the x-coordinates and averaging the y-coordinates of the two endpoints. Let me verify that.
So, if I ha... | [
{
"from": "human",
"value": "Find the sum of the coordinates of the midpoint of a line segment with endpoints at $(-1, 8)$ and $(7, 2)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the coordinates of the midpoint of a line segment with endpoints at (-1, 8) and (7, ... |
[
{
"content": "Given a square with side length $x$, a circle is inscribed within it, touching all four sides of the square. If the area of the circle is $4\\pi$, determine the length of the diagonal of the square.",
"role": "user"
},
{
"content": "Given that the circle is inscribed in the square, its... | Given a square with side length $x$, a circle is inscribed within it, touching all four sides of the square. If the area of the circle is $4\pi$, determine the length of the diagonal of the square. | Given a square with side length $x$, a circle is inscribed within it, touching all four sides of the square. If the area of the circle is $4\pi$, determine the length of the diagonal of the square. | ai2-adapt-dev/openmath-2-math | 0.999396 | Alright, let's see. I need to find the length of the diagonal of a square, given that a circle inscribed in it has an area of 4π. Hmm, okay. Let me start by recalling some basic concepts.
First, a square with side length x. If a circle is inscribed in the square, that means the circle touches all four sides of the squ... | Given a square with side length \( x \), a circle is inscribed within it, touching all four sides of the square. The area of the circle is given as \( 4\pi \). We need to determine the length of the diagonal of the square.
1. **Find the radius of the circle:**
The area of the circle is given by \( \pi r^2 = 4\pi \)... | 88 | <think>
Alright, let's see. I need to find the length of the diagonal of a square, given that a circle inscribed in it has an area of 4π. Hmm, okay. Let me start by recalling some basic concepts.
First, a square with side length x. If a circle is inscribed in the square, that means the circle touches all four sides of... | [
{
"from": "human",
"value": "Given a square with side length $x$, a circle is inscribed within it, touching all four sides of the square. If the area of the circle is $4\\pi$, determine the length of the diagonal of the square."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I need t... |
[
{
"content": "In a right-angled triangle, the ratio of the length of the hypotenuse to the length of the shorter leg is 2:1. If the length of the longer leg is 15 cm, find the length of the shorter leg.",
"role": "user"
},
{
"content": "Let's denote the length of the shorter leg as $x$ and the lengt... | In a right-angled triangle, the ratio of the length of the hypotenuse to the length of the shorter leg is 2:1. If the length of the longer leg is 15 cm, find the length of the shorter leg. | In a right-angled triangle, the ratio of the length of the hypotenuse to the length of the shorter leg is 2:1. If the length of the longer leg is 15 cm, find the length of the shorter leg. | ai2-adapt-dev/openmath-2-math | 0.999935 | Okay, so I need to find the length of the shorter leg in a right-angled triangle where the ratio of the hypotenuse to the shorter leg is 2:1, and the longer leg is 15 cm. Let me think about how to approach this.
First, since it's a right-angled triangle, I remember the Pythagorean theorem: in a right triangle, the squ... | Given a right-angled triangle where the ratio of the hypotenuse to the shorter leg is 2:1 and the longer leg is 15 cm, we need to find the length of the shorter leg.
1. Let the shorter leg be \( a \), the hypotenuse be \( c \), and the longer leg be \( 15 \) cm.
2. The given ratio \( \frac{c}{a} = 2 \) implies \( c = ... | 78 | <think>
Okay, so I need to find the length of the shorter leg in a right-angled triangle where the ratio of the hypotenuse to the shorter leg is 2:1, and the longer leg is 15 cm. Let me think about how to approach this.
First, since it's a right-angled triangle, I remember the Pythagorean theorem: in a right triangle,... | [
{
"from": "human",
"value": "In a right-angled triangle, the ratio of the length of the hypotenuse to the length of the shorter leg is 2:1. If the length of the longer leg is 15 cm, find the length of the shorter leg."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the length of ... |
[
{
"content": "In a school, the number of students in the school band is 2/3 of the number of students in the school choir. If the total number of students in both the band and the choir is 120, and 10 students are members of both, how many students are in the school choir?",
"role": "user"
},
{
"con... | In a school, the number of students in the school band is 2/3 of the number of students in the school choir. If the total number of students in both the band and the choir is 120, and 10 students are members of both, how many students are in the school choir? | In a school, the number of students in the school band is 2/3 of the number of students in the school choir. If the total number of students in both the band and the choir is 120, and 10 students are members of both, how many students are in the school choir? | ai2-adapt-dev/openmath-2-math | 0.999568 | Okay, so I need to figure out how many students are in the school choir. The problem says that the number of students in the band is two-thirds of the number in the choir. The total number of students in both is 120, and 10 students are in both. Hmm, let's break this down step by step.
First, let me note down what's g... | Let \( C \) be the number of students in the school choir. The number of students in the school band is \( \frac{2}{3}C \). The total number of students in both the band and the choir, considering 10 students are members of both, is given by the formula for the union of two sets:
\[
|Band \cup Choir| = |Band| + |Choir... | 13 | <think>
Okay, so I need to figure out how many students are in the school choir. The problem says that the number of students in the band is two-thirds of the number in the choir. The total number of students in both is 120, and 10 students are in both. Hmm, let's break this down step by step.
First, let me note down ... | [
{
"from": "human",
"value": "In a school, the number of students in the school band is 2/3 of the number of students in the school choir. If the total number of students in both the band and the choir is 120, and 10 students are members of both, how many students are in the school choir?"
},
{
"from... |
[
{
"content": "A water tank in the shape of a right circular cylinder has a capacity of 2000 liters. If the water is filled to a height that is 1/3 of the total height of the cylinder, what is the volume of the water in the tank?",
"role": "user"
},
{
"content": "The volume of a right circular cylind... | A water tank in the shape of a right circular cylinder has a capacity of 2000 liters. If the water is filled to a height that is 1/3 of the total height of the cylinder, what is the volume of the water in the tank? | A water tank in the shape of a right circular cylinder has a capacity of 2000 liters. If the water is filled to a height that is 1/3 of the total height of the cylinder, what is the volume of the water in the tank? | ai2-adapt-dev/openmath-2-math | 0.999619 | Okay, so I need to figure out the volume of water in this cylindrical tank when it's filled to a third of its total height. The tank's total capacity is 2000 liters. Hmm, let's see.
First, I remember that the volume of a cylinder is calculated by the formula V = πr²h, where r is the radius and h is the height. But he... | The water tank is a right circular cylinder with a total capacity of 2000 liters when completely full. The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.
Given that the tank is filled to a height that is \( \frac{1}{3} \) of the total height, t... | 97 | <think>
Okay, so I need to figure out the volume of water in this cylindrical tank when it's filled to a third of its total height. The tank's total capacity is 2000 liters. Hmm, let's see.
First, I remember that the volume of a cylinder is calculated by the formula V = πr²h, where r is the radius and h is the height... | [
{
"from": "human",
"value": "A water tank in the shape of a right circular cylinder has a capacity of 2000 liters. If the water is filled to a height that is 1/3 of the total height of the cylinder, what is the volume of the water in the tank?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I n... |
[
{
"content": "What is the sum of the solutions of the equation $-35=-x^2-2x$?\n\nNote: The original problem asks for the product of the solutions, while the new problem asks for the sum. This requires a different approach to solve, as the sum of the solutions cannot be found directly from the product of the sol... | What is the sum of the solutions of the equation $-35=-x^2-2x$?
Note: The original problem asks for the product of the solutions, while the new problem asks for the sum. This requires a different approach to solve, as the sum of the solutions cannot be found directly from the product of the solutions. | What is the sum of the solutions of the equation $-35=-x^2-2x$?
Note: The original problem asks for the product of the solutions, while the new problem asks for the sum. This requires a different approach to solve, as the sum of the solutions cannot be found directly from the product of the solutions. | ai2-adapt-dev/openmath-2-math | 0.999556 | Okay, let's see. I have this equation: -35 = -x² - 2x. The problem is asking for the sum of the solutions. Hmm, I remember that for quadratic equations, there's something called Vieta's formulas that relate the coefficients to the sum and product of the roots. Maybe I can use that?
First, I need to make sure the equat... | To find the sum of the solutions of the equation \(-35 = -x^2 - 2x\), we first rewrite the equation in standard quadratic form.
Starting with the given equation:
\[
-35 = -x^2 - 2x
\]
We add \(x^2\) and \(2x\) to both sides to get:
\[
x^2 + 2x - 35 = 0
\]
This is a quadratic equation in the form \(ax^2 + bx + c = 0... | 69 | <think>
Okay, let's see. I have this equation: -35 = -x² - 2x. The problem is asking for the sum of the solutions. Hmm, I remember that for quadratic equations, there's something called Vieta's formulas that relate the coefficients to the sum and product of the roots. Maybe I can use that?
First, I need to make sure t... | [
{
"from": "human",
"value": "What is the sum of the solutions of the equation $-35=-x^2-2x$?\n\nNote: The original problem asks for the product of the solutions, while the new problem asks for the sum. This requires a different approach to solve, as the sum of the solutions cannot be found directly from the... |
[
{
"content": "Find the number of distinct prime factors of the sum of the distinct prime factors of 315.",
"role": "user"
},
{
"content": "To find the distinct prime factors of 315, first find its prime factorization.\n\nPrime factorization of 315:\n\\[ 315 = 3^2 \\times 5 \\times 7 \\]\n\nThe disti... | Find the number of distinct prime factors of the sum of the distinct prime factors of 315. | Find the number of distinct prime factors of the sum of the distinct prime factors of 315. | ai2-adapt-dev/openmath-2-math | 0.999817 | Okay, so I need to find the number of distinct prime factors of the sum of the distinct prime factors of 315. Hmm, let's break this down step by step.
First, I need to find the distinct prime factors of 315. Then, sum those prime factors, and after that, find how many distinct prime factors that sum has. Alright, let... | To find the number of distinct prime factors of the sum of the distinct prime factors of 315, we start by determining the prime factorization of 315.
315 can be factored as follows:
\[
315 \div 5 = 63 \quad \text{(5 is a prime factor)}
\]
\[
63 \div 7 = 9 \quad \text{(7 is a prime factor)}
\]
\[
9 \div 3 = 3 \quad \t... | 151 | <think>
Okay, so I need to find the number of distinct prime factors of the sum of the distinct prime factors of 315. Hmm, let's break this down step by step.
First, I need to find the distinct prime factors of 315. Then, sum those prime factors, and after that, find how many distinct prime factors that sum has. Alri... | [
{
"from": "human",
"value": "Find the number of distinct prime factors of the sum of the distinct prime factors of 315."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the number of distinct prime factors of the sum of the distinct prime factors of 315. Hmm, let's break this down... |
[
{
"content": "Compute the number of ways to arrange the digits of the number 1293 into a 4-digit number.",
"role": "user"
},
{
"content": "To compute the number of ways to arrange the digits of the number 1293 into a 4-digit number, we first observe that there are 4 digits in total.\n\nSince the dig... | Compute the number of ways to arrange the digits of the number 1293 into a 4-digit number. | Compute the number of ways to arrange the digits of the number 1293 into a 4-digit number. | ai2-adapt-dev/openmath-2-math | 0.999969 | Alright, so I need to figure out how many different 4-digit numbers can be made by rearranging the digits of 1293. Hmm, let's see. First, I should check if there are any repeating digits because that might affect the calculation. The digits in 1293 are 1, 2, 9, and 3. Wait, 1293 has the digits 1, 2, 9, and 3. Are there... | To determine the number of ways to arrange the digits of the number 1293 into a 4-digit number, we first check if there are any repeating digits. The digits in 1293 are 1, 2, 9, and 3, all of which are unique. Since there are no repeating digits, the number of permutations is calculated using the factorial of the numbe... | 42 | <think>
Alright, so I need to figure out how many different 4-digit numbers can be made by rearranging the digits of 1293. Hmm, let's see. First, I should check if there are any repeating digits because that might affect the calculation. The digits in 1293 are 1, 2, 9, and 3. Wait, 1293 has the digits 1, 2, 9, and 3. A... | [
{
"from": "human",
"value": "Compute the number of ways to arrange the digits of the number 1293 into a 4-digit number."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to figure out how many different 4-digit numbers can be made by rearranging the digits of 1293. Hmm, let's see. First... |
[
{
"content": "In a certain quadrilateral, the lengths of the sides are in the ratio 3:4:5:6. If the length of the shortest side is 12 cm, what is the perimeter of the quadrilateral?",
"role": "user"
},
{
"content": "Let's use the ratio of the side lengths to find the lengths of the other sides.\n\nG... | In a certain quadrilateral, the lengths of the sides are in the ratio 3:4:5:6. If the length of the shortest side is 12 cm, what is the perimeter of the quadrilateral? | In a certain quadrilateral, the lengths of the sides are in the ratio 3:4:5:6. If the length of the shortest side is 12 cm, what is the perimeter of the quadrilateral? | ai2-adapt-dev/openmath-2-math | 0.999809 | Okay, so I need to find the perimeter of a quadrilateral where the sides are in the ratio 3:4:5:6, and the shortest side is 12 cm. Hmm, let's see.
First, ratios can sometimes be a bit confusing, but I remember that ratios tell us how the parts relate to each other. Here, the sides are in the ratio 3:4:5:6. That means... | Given a quadrilateral with side lengths in the ratio 3:4:5:6 and the shortest side being 12 cm, we need to find the perimeter.
1. Let the common factor be \( x \). The side lengths can be represented as \( 3x \), \( 4x \), \( 5x \), and \( 6x \).
2. The shortest side is \( 3x \), which is given as 12 cm. Solving for \... | 188 | <think>
Okay, so I need to find the perimeter of a quadrilateral where the sides are in the ratio 3:4:5:6, and the shortest side is 12 cm. Hmm, let's see.
First, ratios can sometimes be a bit confusing, but I remember that ratios tell us how the parts relate to each other. Here, the sides are in the ratio 3:4:5:6. Th... | [
{
"from": "human",
"value": "In a certain quadrilateral, the lengths of the sides are in the ratio 3:4:5:6. If the length of the shortest side is 12 cm, what is the perimeter of the quadrilateral?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the perimeter of a quadrilateral wh... |
[
{
"content": "If a company's profit is directly proportional to the number of units it sells, and inversely proportional to the square of the number of employees it has, what happens to the profit if the number of units sold doubles and the number of employees triples?",
"role": "user"
},
{
"content... | If a company's profit is directly proportional to the number of units it sells, and inversely proportional to the square of the number of employees it has, what happens to the profit if the number of units sold doubles and the number of employees triples? | If a company's profit is directly proportional to the number of units it sells, and inversely proportional to the square of the number of employees it has, what happens to the profit if the number of units sold doubles and the number of employees triples? | ai2-adapt-dev/openmath-2-math | 0.999819 | Okay, so there's this problem about a company's profit being directly proportional to the number of units sold and inversely proportional to the square of the number of employees. The question is asking what happens to the profit if the number of units sold doubles and the number of employees triples. Hmm, let me try t... | Given that a company's profit \( P \) is directly proportional to the number of units sold \( U \) and inversely proportional to the square of the number of employees \( E \), we can express this relationship as:
\[ P = k \cdot \frac{U}{E^2} \]
where \( k \) is the constant of proportionality.
Initially, the profit ... | 14 | <think>
Okay, so there's this problem about a company's profit being directly proportional to the number of units sold and inversely proportional to the square of the number of employees. The question is asking what happens to the profit if the number of units sold doubles and the number of employees triples. Hmm, let ... | [
{
"from": "human",
"value": "If a company's profit is directly proportional to the number of units it sells, and inversely proportional to the square of the number of employees it has, what happens to the profit if the number of units sold doubles and the number of employees triples?"
},
{
"from": "... |
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