messages listlengths 2 2 | instruction_seed stringlengths 13 6.87k | response_seed stringlengths 13 6.87k | _source stringclasses 1
value | _fasttext_score float64 1 1 | reasoning stringlengths 838 96.4k | deepseek_solution stringlengths 177 3.45k | __original_row_idx int64 2 31.6k | final_reasoning_trace stringlengths 1.11k 97.7k | conversations listlengths 2 2 |
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[
{
"content": "In a right triangle, the length of the hypotenuse is 20 cm. The ratio of the lengths of the legs is 5:12. Find the area of the triangle.",
"role": "user"
},
{
"content": "Let the lengths of the legs be $5x$ and $12x$.\n\nBy the Pythagorean theorem, we have\n\\[ (5x)^2 + (12x)^2 = 20^2 ... | In a right triangle, the length of the hypotenuse is 20 cm. The ratio of the lengths of the legs is 5:12. Find the area of the triangle. | In a right triangle, the length of the hypotenuse is 20 cm. The ratio of the lengths of the legs is 5:12. Find the area of the triangle. | ai2-adapt-dev/openmath-2-math | 0.999992 | Okay, so I have this problem here about a right triangle. The hypotenuse is 20 cm, and the ratio of the legs is 5:12. I need to find the area. Hmm, let's see. Let me start by recalling some basics about right triangles. The area of a triangle is (base * height)/2, and in a right triangle, the two legs serve as the base... | Given a right triangle with a hypotenuse of 20 cm and a leg ratio of 5:12, we need to find the area of the triangle.
1. **Assign variables to the legs based on the ratio:**
Let the legs be \(5k\) and \(12k\), where \(k\) is a scaling factor.
2. **Apply the Pythagorean theorem:**
\[
(5k)^2 + (12k)^2 = 20^2
... | 20,823 | <think>
Okay, so I have this problem here about a right triangle. The hypotenuse is 20 cm, and the ratio of the legs is 5:12. I need to find the area. Hmm, let's see. Let me start by recalling some basics about right triangles. The area of a triangle is (base * height)/2, and in a right triangle, the two legs serve as ... | [
{
"from": "human",
"value": "In a right triangle, the length of the hypotenuse is 20 cm. The ratio of the lengths of the legs is 5:12. Find the area of the triangle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here about a right triangle. The hypotenuse is 20 cm, and the... |
[
{
"content": "The ratio of boys to girls in a class is 3:4. There are 24 boys. How many girls are there in the class?",
"role": "user"
},
{
"content": "The ratio of boys to girls is 3:4. Let's assume the number of boys is 3x and the number of girls is 4x.\n\nWe are given that there are 24 boys. So, ... | The ratio of boys to girls in a class is 3:4. There are 24 boys. How many girls are there in the class? | The ratio of boys to girls in a class is 3:4. There are 24 boys. How many girls are there in the class? | ai2-adapt-dev/openmath-2-math | 0.999987 | Okay, let's see. The problem says the ratio of boys to girls in a class is 3:4, and there are 24 boys. We need to find out how many girls there are. Alright, ratios can sometimes be tricky, but I think I remember that ratios compare quantities. So, if the ratio is 3:4 for boys to girls, that means for every 3 boys, the... | The ratio of boys to girls in the class is given as 3:4, and there are 24 boys. To find the number of girls, we first determine the value of each part in the ratio. Since 3 parts correspond to 24 boys, each part is calculated as follows:
\[
\text{Value of each part} = \frac{24}{3} = 8
\]
The number of girls correspon... | 26,729 | <think>
Okay, let's see. The problem says the ratio of boys to girls in a class is 3:4, and there are 24 boys. We need to find out how many girls there are. Alright, ratios can sometimes be tricky, but I think I remember that ratios compare quantities. So, if the ratio is 3:4 for boys to girls, that means for every 3 b... | [
{
"from": "human",
"value": "The ratio of boys to girls in a class is 3:4. There are 24 boys. How many girls are there in the class?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says the ratio of boys to girls in a class is 3:4, and there are 24 boys. We need to find out ... |
[
{
"content": "Simplify $2\\cdot\\frac{9}{5}\\cdot\\frac{15}{-36}$.",
"role": "user"
},
{
"content": "To simplify the expression, we can cancel out common factors between the numerator and the denominator.\n\n\\[ 2 \\cdot \\frac{9}{5} \\cdot \\frac{15}{-36} \\]\n\nFirst, notice that 9 and 36 have a c... | Simplify $2\cdot\frac{9}{5}\cdot\frac{15}{-36}$. | Simplify $2\cdot\frac{9}{5}\cdot\frac{15}{-36}$. | ai2-adapt-dev/openmath-2-math | 1.00001 | Okay, let me try to simplify this expression step by step. The problem is 2 multiplied by 9/5 multiplied by 15/-36. Hmm, so it's 2 * (9/5) * (15/-36). I need to multiply these three terms together. Let me recall how to multiply fractions and whole numbers.
First, I know that when you multiply a whole number by a frac... | To simplify \(2 \cdot \frac{9}{5} \cdot \frac{15}{-36}\), we can follow these steps:
1. Convert the whole number to a fraction: \(2 = \frac{2}{1}\).
2. Multiply the numerators together and the denominators together:
\[
\frac{2 \cdot 9 \cdot 15}{1 \cdot 5 \cdot (-36)}
\]
3. Simplify the numerator and the denom... | 3,029 | <think>
Okay, let me try to simplify this expression step by step. The problem is 2 multiplied by 9/5 multiplied by 15/-36. Hmm, so it's 2 * (9/5) * (15/-36). I need to multiply these three terms together. Let me recall how to multiply fractions and whole numbers.
First, I know that when you multiply a whole number b... | [
{
"from": "human",
"value": "Simplify $2\\cdot\\frac{9}{5}\\cdot\\frac{15}{-36}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to simplify this expression step by step. The problem is 2 multiplied by 9/5 multiplied by 15/-36. Hmm, so it's 2 * (9/5) * (15/-36). I need to multiply thes... |
[
{
"content": "Simplify $4\\cdot\\frac{23}{6}\\cdot \\frac{12}{-46}$.",
"role": "user"
},
{
"content": "Simplify the expression step-by-step:\n\\[ 4 \\cdot \\frac{23}{6} \\cdot \\frac{12}{-46} \\]\n\nFirst, simplify the numerators and denominators:\n\\[ \\frac{4 \\cdot 23 \\cdot 12}{6 \\cdot -46} \\]... | Simplify $4\cdot\frac{23}{6}\cdot \frac{12}{-46}$. | Simplify $4\cdot\frac{23}{6}\cdot \frac{12}{-46}$. | ai2-adapt-dev/openmath-2-math | 1.000009 | Okay, so I need to simplify this expression: 4 times 23 over 6 times 12 over -46. Let me write that out first to make sure I have it right. It's 4 multiplied by (23/6) multiplied by (12/-46). Hmm, okay. Let me think step by step.
First, maybe I can simplify the fractions before multiplying everything together. That mi... | To simplify \(4 \cdot \frac{23}{6} \cdot \frac{12}{-46}\):
1. Convert the whole number 4 to a fraction: \(4 = \frac{4}{1}\).
2. Multiply the numerators together and the denominators together:
\[
\frac{4}{1} \cdot \frac{23}{6} \cdot \frac{12}{-46} = \frac{4 \cdot 23 \cdot 12}{1 \cdot 6 \cdot (-46)}
\]
3. Simpl... | 11,694 | <think>
Okay, so I need to simplify this expression: 4 times 23 over 6 times 12 over -46. Let me write that out first to make sure I have it right. It's 4 multiplied by (23/6) multiplied by (12/-46). Hmm, okay. Let me think step by step.
First, maybe I can simplify the fractions before multiplying everything together.... | [
{
"from": "human",
"value": "Simplify $4\\cdot\\frac{23}{6}\\cdot \\frac{12}{-46}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify this expression: 4 times 23 over 6 times 12 over -46. Let me write that out first to make sure I have it right. It's 4 multiplied by (23/6) mult... |
[
{
"content": "Compute $\\sin 225^\\circ$.",
"role": "user"
},
{
"content": "To compute $\\sin 225^\\circ$, we can use the fact that the sine function has a period of $360^\\circ$. Therefore, we can write $225^\\circ$ as $225^\\circ - 180^\\circ = 45^\\circ$. We know that $\\sin 45^\\circ = \\frac{\\... | Compute $\sin 225^\circ$. | Compute $\sin 225^\circ$. | ai2-adapt-dev/openmath-2-math | 1.00001 | Okay, so I need to compute sin 225 degrees. Hmm, let's see. I remember that 225 degrees is in the third quadrant of the unit circle. In the third quadrant, both sine and cosine are negative because the x and y coordinates are both negative there. So whatever reference angle I get, the sine will be negative.
First, I ... | To compute \(\sin 225^\circ\), we start by noting that \(225^\circ\) is in the third quadrant, where both sine and cosine are negative. The reference angle for \(225^\circ\) is found by subtracting \(180^\circ\) from \(225^\circ\):
\[
225^\circ - 180^\circ = 45^\circ
\]
The reference angle is \(45^\circ\). In the thi... | 19,715 | <think>
Okay, so I need to compute sin 225 degrees. Hmm, let's see. I remember that 225 degrees is in the third quadrant of the unit circle. In the third quadrant, both sine and cosine are negative because the x and y coordinates are both negative there. So whatever reference angle I get, the sine will be negative.
F... | [
{
"from": "human",
"value": "Compute $\\sin 225^\\circ$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute sin 225 degrees. Hmm, let's see. I remember that 225 degrees is in the third quadrant of the unit circle. In the third quadrant, both sine and cosine are negative because t... |
[
{
"content": "There are values $A$ and $B$ such that\n\\[\\frac{Ax+20}{x^2-3x+2}=\\frac{2}{x-1}+\\frac{B}{x-2}.\\]Find $A+B$.",
"role": "user"
},
{
"content": "To solve for $A$ and $B$, we need to rewrite the equation in a way that allows us to compare coefficients.\n\nStart by finding a common deno... | There are values $A$ and $B$ such that
\[\frac{Ax+20}{x^2-3x+2}=\frac{2}{x-1}+\frac{B}{x-2}.\]Find $A+B$. | There are values $A$ and $B$ such that
\[\frac{Ax+20}{x^2-3x+2}=\frac{2}{x-1}+\frac{B}{x-2}.\]Find $A+B$. | ai2-adapt-dev/openmath-2-math | 1.00001 | Okay, so I have this problem here where I need to find values A and B such that the rational function (Ax + 20)/(x² - 3x + 2) can be expressed as 2/(x - 1) + B/(x - 2). Then, I have to find A + B. Hmm, partial fractions, right? Let me remember how that works.
First, the denominator is x² - 3x + 2. Maybe I should facto... | To find values \(A\) and \(B\) such that
\[
\frac{Ax + 20}{x^2 - 3x + 2} = \frac{2}{x - 1} + \frac{B}{x - 2},
\]
we start by factoring the denominator on the left-hand side:
\[
x^2 - 3x + 2 = (x - 1)(x - 2).
\]
Next, we express the right-hand side with a common denominator:
\[
\frac{2}{x - 1} + \frac{B}{x - 2} = \frac{... | 9,461 | <think>
Okay, so I have this problem here where I need to find values A and B such that the rational function (Ax + 20)/(x² - 3x + 2) can be expressed as 2/(x - 1) + B/(x - 2). Then, I have to find A + B. Hmm, partial fractions, right? Let me remember how that works.
First, the denominator is x² - 3x + 2. Maybe I shou... | [
{
"from": "human",
"value": "There are values $A$ and $B$ such that\n\\[\\frac{Ax+20}{x^2-3x+2}=\\frac{2}{x-1}+\\frac{B}{x-2}.\\]Find $A+B$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here where I need to find values A and B such that the rational function (Ax + 20)/(x²... |
[
{
"content": "Find the units digit of the sum, $$ 1! + 2! + 3! + \\cdots + 2006!. $$",
"role": "user"
},
{
"content": "To find the units digit of the sum, we need to look at the units digit of each term in the sum. Let's start calculating factorials:\n\n\\[ 1! = 1 \\]\n\n\\[ 2! = 1 \\times 2 = 2 \\]... | Find the units digit of the sum, $$ 1! + 2! + 3! + \cdots + 2006!. $$ | Find the units digit of the sum, $$ 1! + 2! + 3! + \cdots + 2006!. $$ | ai2-adapt-dev/openmath-2-math | 1.00001 | Okay, so I need to find the units digit of the sum \(1! + 2! + 3! + \cdots + 2006!\). Hmm, let me think about how factorials work and their units digits.
First, I remember that the units digit of a number is just the last digit of that number. For example, the units digit of 5! is 0 because 5! = 120, which ends with ... | To find the units digit of the sum \(1! + 2! + 3! + \cdots + 2006!\), we need to consider the units digits of the factorials involved.
The units digits of factorials are as follows:
- \(1! = 1\) (units digit is 1)
- \(2! = 2\) (units digit is 2)
- \(3! = 6\) (units digit is 6)
- \(4! = 24\) (units digit is 4)
- Start... | 23,525 | <think>
Okay, so I need to find the units digit of the sum \(1! + 2! + 3! + \cdots + 2006!\). Hmm, let me think about how factorials work and their units digits.
First, I remember that the units digit of a number is just the last digit of that number. For example, the units digit of 5! is 0 because 5! = 120, which en... | [
{
"from": "human",
"value": "Find the units digit of the sum, $$ 1! + 2! + 3! + \\cdots + 2006!. $$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the units digit of the sum \\(1! + 2! + 3! + \\cdots + 2006!\\). Hmm, let me think about how factorials work and their units digits.... |
[{"content":"Problem:\nIn the interval $(2,5)$, there exists a number $x_0$ such that $E(x_0)=x_0$.\(...TRUNCATED) | "Problem:\nIn the interval $(2,5)$, there exists a number $x_0$ such that $E(x_0)=x_0$.\nDetermine t(...TRUNCATED) | "Problem:\nIn the interval $(2,5)$, there exists a number $x_0$ such that $E(x_0)=x_0$.\nDetermine t(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 1.000002 | "Okay, let's see. The problem says that in the interval (2,5), there's a number x₀ where E(x₀) =(...TRUNCATED) | "The problem states that in the interval \\((2, 5)\\), there exists a number \\(x_0\\) such that \\((...TRUNCATED) | 119 | "<think>\nOkay, let's see. The problem says that in the interval (2,5), there's a number x₀ where (...TRUNCATED) | [{"from":"human","value":"Problem:\nIn the interval $(2,5)$, there exists a number $x_0$ such that $(...TRUNCATED) |
[{"content":"What day of the week will it be 75 days from Wednesday?","role":"user"},{"content":"To (...TRUNCATED) | What day of the week will it be 75 days from Wednesday? | What day of the week will it be 75 days from Wednesday? | ai2-adapt-dev/openmath-2-math | 0.999997 | "Okay, so I need to figure out what day of the week it will be 75 days from a Wednesday. Let me thin(...TRUNCATED) | "To determine what day of the week it will be 75 days from Wednesday, we start by noting that the da(...TRUNCATED) | 18,434 | "<think>\nOkay, so I need to figure out what day of the week it will be 75 days from a Wednesday. Le(...TRUNCATED) | [{"from":"human","value":"What day of the week will it be 75 days from Wednesday?"},{"from":"gpt","v(...TRUNCATED) |
[{"content":"How many positive 4-digit numbers are divisible by 9?","role":"user"},{"content":"A num(...TRUNCATED) | How many positive 4-digit numbers are divisible by 9? | How many positive 4-digit numbers are divisible by 9? | ai2-adapt-dev/openmath-2-math | 1.000009 | "Okay, so I need to figure out how many positive 4-digit numbers are divisible by 9. Hmm, let's star(...TRUNCATED) | "To determine how many positive 4-digit numbers are divisible by 9, we start by noting that the rang(...TRUNCATED) | 7,806 | "<think>\nOkay, so I need to figure out how many positive 4-digit numbers are divisible by 9. Hmm, l(...TRUNCATED) | [{"from":"human","value":"How many positive 4-digit numbers are divisible by 9?"},{"from":"gpt","val(...TRUNCATED) |
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