{
  "summary": {
    "method": "CoT",
    "model": "qwen2.5:32b",
    "task": "math",
    "samples": 10,
    "correct": 9,
    "accuracy": 0.9,
    "avg_time_s": 36.46
  },
  "details": [
    {
      "question": "Find the value of $x$ such that $\\sqrt{4x + 5} = 7$.",
      "expected": "11",
      "predicted": "11",
      "full_response": "To solve for \\( x \\) in the equation \\( \\sqrt{4x + 5} = 7 \\), we will follow these steps:\n\n1. **Isolate the square root**: The given equation is already isolated with the square root on one side.\n   \\[\n   \\sqrt{4x + 5} = 7\n   \\]\n\n2. **Square both sides** to eliminate the square root:\n   \\[\n   (\\sqrt{4x + 5})^2 = 7^2\n   \\]\n   Simplifying this, we get:\n   \\[\n   4x + 5 = 49\n   \\]\n\n3. **Solve for \\( x \\)** by isolating the variable:\n   - Subtract 5 from both sides of the equation:\n     \\[\n     4x + 5 - 5 = 49 - 5\n     \\]\n     Simplifying this, we get:\n     \\[\n     4x = 44\n     \\]\n\n   - Divide both sides by 4 to solve for \\( x \\):\n     \\[\n     x = \\frac{44}{4}\n     \\]\n     Simplifying this, we get:\n     \\[\n     x = 11\n     \\]\n\nTherefore, the value of \\( x \\) is \\(\\boxed{11}\\).",
      "correct": true,
      "time_s": 29.56,
      "tokens": 261
    },
    {
      "question": "A box contains 5 red balls and 7 blue balls. If two balls are drawn at random without replacement, what is the probability that both balls are red? Express your answer as a fraction in simplest form.",
      "expected": "5/33",
      "predicted": "\\frac{5}{33}",
      "full_response": "To solve this problem, we need to calculate the probability of drawing two red balls from a box containing 5 red and 7 blue balls. We will do this step-by-step:\n\n1. **Determine the total number of balls**:\n   - There are 5 red balls and 7 blue balls.\n   - Total number of balls = \\(5 + 7 = 12\\).\n\n2. **Calculate the probability of drawing a red ball on the first draw**:\n   - The number of favorable outcomes (drawing a red ball) is 5.\n   - The total possible outcomes for the first draw are 12.\n   - Probability of drawing a red ball on the first draw = \\(\\frac{5}{12}\\).\n\n3. **Calculate the probability of drawing a second red ball given that the first one was red**:\n   - After drawing one red ball, there are now \\(5-1=4\\) red balls left.\n   - The total number of balls remaining is \\(12-1=11\\).\n   - Probability of drawing a red ball on the second draw = \\(\\frac{4}{11}\\).\n\n4. **Calculate the combined probability**:\n   - Since these are dependent events (the outcome of the first event affects the second), we multiply the probabilities.\n   - Combined probability = \\(\\left( \\frac{5}{12} \\right) \\times \\left( \\frac{4}{11} \\right)\\).\n\n5. **Simplify the combined probability**:\n   - Multiply the numerators: \\(5 \\times 4 = 20\\).\n   - Multiply the denominators: \\(12 \\times 11 = 132\\).\n   - Combined probability = \\(\\frac{20}{132}\\).\n\n6. **Simplify the fraction**:\n   - Find the greatest common divisor (GCD) of 20 and 132, which is 4.\n   - Divide both numerator and denominator by their GCD: \n     \\[\n     \\frac{20 \\div 4}{132 \\div 4} = \\frac{5}{33}\n     \\]\n\nTherefore, the probability that both balls drawn are red is:\n\n\\boxed{\\frac{5}{33}}",
      "correct": false,
      "time_s": 50.01,
      "tokens": 549
    },
    {
      "question": "What is the sum of the first 20 positive integers?",
      "expected": "210",
      "predicted": "210",
      "full_response": "To find the sum of the first 20 positive integers, we can use the formula for the sum of an arithmetic series. The general formula for the sum \\( S \\) of the first \\( n \\) terms of an arithmetic sequence is:\n\n\\[ S = \\frac{n}{2} (a + l) \\]\n\nwhere:\n- \\( n \\) is the number of terms,\n- \\( a \\) is the first term, and\n- \\( l \\) is the last term.\n\nFor this problem:\n- The number of terms \\( n \\) is 20.\n- The first term \\( a \\) is 1 (the first positive integer).\n- The last term \\( l \\) is 20 (since we are summing up to the 20th positive integer).\n\nNow, substitute these values into the formula:\n\n\\[ S = \\frac{20}{2} (1 + 20) \\]\n\nFirst, calculate the value inside the parentheses:\n\n\\[ 1 + 20 = 21 \\]\n\nNext, multiply by \\( \\frac{20}{2} \\):\n\n\\[ S = 10 \\times 21 \\]\n\nFinally, perform the multiplication:\n\n\\[ S = 210 \\]\n\nTherefore, the sum of the first 20 positive integers is:\n\n\\boxed{210}",
      "correct": true,
      "time_s": 29.51,
      "tokens": 295
    },
    {
      "question": "If $f(x) = 2x^2 + 3x - 5$, find $f(-2)$.",
      "expected": "-3",
      "predicted": "-3",
      "full_response": "To solve for \\( f(-2) \\), we need to substitute \\( x = -2 \\) into the function \\( f(x) = 2x^2 + 3x - 5 \\).\n\nStep-by-step solution:\n\n1. Substitute \\( x = -2 \\) into the function:\n   \\[\n   f(-2) = 2(-2)^2 + 3(-2) - 5\n   \\]\n\n2. Calculate \\( (-2)^2 \\):\n   \\[\n   (-2)^2 = 4\n   \\]\n   So, we have:\n   \\[\n   f(-2) = 2(4) + 3(-2) - 5\n   \\]\n\n3. Multiply the terms:\n   \\[\n   2(4) = 8\n   \\]\n   and\n   \\[\n   3(-2) = -6\n   \\]\n   So, we have:\n   \\[\n   f(-2) = 8 - 6 - 5\n   \\]\n\n4. Combine the terms:\n   \\[\n   8 - 6 = 2\n   \\]\n   and then,\n   \\[\n   2 - 5 = -3\n   \\]\n\nTherefore, the value of \\( f(-2) \\) is:\n\\[\n\\boxed{-3}\n\\]",
      "correct": true,
      "time_s": 29.33,
      "tokens": 202
    },
    {
      "question": "The sum of two numbers is 20 and their difference is 4. What is the product of the two numbers?",
      "expected": "96",
      "predicted": "96",
      "full_response": "To solve this problem, we will use a system of equations based on the information given:\n\n1. Let \\( x \\) be one number.\n2. Let \\( y \\) be the other number.\n\nFrom the problem statement, we have:\n- The sum of the two numbers is 20: \n\\[ x + y = 20 \\]\n- Their difference is 4:\n\\[ x - y = 4 \\]\n\nWe now have a system of linear equations:\n1. \\( x + y = 20 \\)\n2. \\( x - y = 4 \\)\n\nTo solve for \\( x \\) and \\( y \\), we can add these two equations together to eliminate \\( y \\):\n\n\\[ (x + y) + (x - y) = 20 + 4 \\]\n\\[ 2x = 24 \\]\n\nSolving for \\( x \\):\n\\[ x = \\frac{24}{2} \\]\n\\[ x = 12 \\]\n\nNow that we have \\( x \\), substitute it back into one of the original equations to solve for \\( y \\). We use the first equation:\n\\[ x + y = 20 \\]\n\\[ 12 + y = 20 \\]\n\\[ y = 20 - 12 \\]\n\\[ y = 8 \\]\n\nSo, the two numbers are \\( x = 12 \\) and \\( y = 8 \\).\n\nNext, we need to find their product:\n\\[ xy = 12 \\times 8 \\]\n\\[ xy = 96 \\]\n\nTherefore, the product of the two numbers is:\n\n\\boxed{96}",
      "correct": true,
      "time_s": 35.69,
      "tokens": 320
    },
    {
      "question": "How many distinguishable permutations can be made from the letters in the word 'MISSISSIPPI'?",
      "expected": "34650",
      "predicted": "34650",
      "full_response": "To solve this problem, we need to calculate the number of distinguishable permutations of the letters in the word \"MISSISSIPPI\". \n\nStep 1: Identify the total number of letters.\nThe word \"MISSISSIPPI\" has a total of 11 letters.\n\nStep 2: Count the frequency of each distinct letter.\n- M appears once (1 time)\n- I appears four times (4 times)\n- S appears four times (4 times)\n- P appears two times (2 times)\n\nStep 3: Use the formula for permutations with repetitions.\nThe number of distinguishable permutations is given by:\n\\[ \\frac{n!}{n_1! \\cdot n_2! \\cdot ... \\cdot n_k!} \\]\nwhere \\( n \\) is the total number of items to arrange, and \\( n_1, n_2, ..., n_k \\) are the frequencies of each distinct item.\n\nFor \"MISSISSIPPI\":\n\\[ n = 11 \\]\n\\[ n_M = 1, \\quad n_I = 4, \\quad n_S = 4, \\quad n_P = 2 \\]\n\nStep 4: Plug in these values into the formula.\n\\[ \\frac{11!}{1! \\cdot 4! \\cdot 4! \\cdot 2!} \\]\n\nStep 5: Calculate the factorials.\n- \\( 11! = 39916800 \\)\n- \\( 1! = 1 \\)\n- \\( 4! = 24 \\)\n- \\( 2! = 2 \\)\n\nStep 6: Substitute these factorial values into the formula and simplify.\n\\[ \\frac{39916800}{1 \\cdot 24 \\cdot 24 \\cdot 2} \\]\n\\[ \\frac{39916800}{1152} = 34650 \\]\n\nTherefore, the number of distinguishable permutations is:\n\\boxed{34650}",
      "correct": true,
      "time_s": 44.3,
      "tokens": 410
    },
    {
      "question": "Find the area of a circle with circumference $8\\pi$.",
      "expected": "16\\pi",
      "predicted": "16\\pi",
      "full_response": "To find the area of a circle given its circumference, we can follow these steps:\n\n1. **Recall the formula for the circumference of a circle**:\n   \\[\n   C = 2\\pi r\n   \\]\n   where \\(C\\) is the circumference and \\(r\\) is the radius.\n\n2. **Substitute the given value into the formula**:\n   Given that the circumference \\(C = 8\\pi\\), we can set up the equation:\n   \\[\n   8\\pi = 2\\pi r\n   \\]\n\n3. **Solve for the radius \\(r\\)**:\n   Divide both sides of the equation by \\(2\\pi\\) to isolate \\(r\\):\n   \\[\n   r = \\frac{8\\pi}{2\\pi} = 4\n   \\]\n   So, the radius \\(r\\) is 4.\n\n4. **Recall the formula for the area of a circle**:\n   \\[\n   A = \\pi r^2\n   \\]\n\n5. **Substitute the value of the radius into the area formula**:\n   Using \\(r = 4\\):\n   \\[\n   A = \\pi (4)^2 = \\pi \\cdot 16 = 16\\pi\n   \\]\n\nTherefore, the area of the circle is:\n\n\\boxed{16\\pi}",
      "correct": true,
      "time_s": 30.27,
      "tokens": 278
    },
    {
      "question": "If $\\log_2(x) + \\log_2(x-2) = 3$, find $x$.",
      "expected": "4",
      "predicted": "4",
      "full_response": "To solve the equation \\(\\log_2(x) + \\log_2(x-2) = 3\\), we can follow these steps:\n\n1. **Combine Logarithms**: Use the logarithm property that states \\(\\log_b(a) + \\log_b(c) = \\log_b(ac)\\). Applying this to our equation, we get:\n   \\[\n   \\log_2(x) + \\log_2(x-2) = \\log_2(x(x-2))\n   \\]\n   So the equation becomes:\n   \\[\n   \\log_2(x(x-2)) = 3\n   \\]\n\n2. **Exponentiate Both Sides**: To remove the logarithm, we can exponentiate both sides of the equation with base 2:\n   \\[\n   x(x-2) = 2^3\n   \\]\n   Simplifying \\(2^3\\), we get:\n   \\[\n   x(x-2) = 8\n   \\]\n\n3. **Form a Quadratic Equation**: Expand and rearrange the equation to form a standard quadratic equation:\n   \\[\n   x^2 - 2x = 8\n   \\]\n   Subtract 8 from both sides:\n   \\[\n   x^2 - 2x - 8 = 0\n   \\]\n\n4. **Solve the Quadratic Equation**: Factorize the quadratic equation \\(x^2 - 2x - 8\\):\n   \\[\n   (x-4)(x+2) = 0\n   \\]\n   Setting each factor to zero gives us:\n   \\[\n   x - 4 = 0 \\quad \\text{or} \\quad x + 2 = 0\n   \\]\n   Solving these, we get:\n   \\[\n   x = 4 \\quad \\text{or} \\quad x = -2\n   \\]\n\n5. **Check for Valid Solutions**: Since \\(x\\) must be positive (as logarithms are only defined for positive arguments), we discard \\(x = -2\\). Thus, the valid solution is:\n   \\[\n   x = 4\n   \\]\n\n6. **Verify the Solution**: Substitute \\(x = 4\\) back into the original equation to ensure it satisfies the conditions:\n   \\[\n   \\log_2(4) + \\log_2(4-2) = \\log_2(4) + \\log_2(2)\n   \\]\n   Simplifying, we get:\n   \\[\n   2 + 1 = 3\n   \\]\n   This is true.\n\nTherefore, the solution to the equation is:\n\n\\boxed{4}",
      "correct": true,
      "time_s": 58.51,
      "tokens": 514
    },
    {
      "question": "A geometric sequence has first term 3 and common ratio 2. What is the 8th term?",
      "expected": "384",
      "predicted": "384",
      "full_response": "To find the 8th term of a geometric sequence with the given parameters, we can use the formula for the nth term of a geometric sequence:\n\n\\[a_n = a_1 \\cdot r^{(n-1)}\\]\n\nwhere:\n- \\(a_n\\) is the nth term,\n- \\(a_1\\) is the first term,\n- \\(r\\) is the common ratio,\n- \\(n\\) is the term number.\n\nGiven in the problem:\n- The first term, \\(a_1 = 3\\),\n- The common ratio, \\(r = 2\\),\n- We need to find the 8th term, so \\(n = 8\\).\n\nStep-by-step solution:\n\n1. Substitute the given values into the formula for the nth term of a geometric sequence:\n   \\[a_8 = 3 \\cdot 2^{(8-1)}\\]\n\n2. Simplify the exponent:\n   \\[a_8 = 3 \\cdot 2^7\\]\n\n3. Calculate \\(2^7\\):\n   \\[2^7 = 128\\]\n\n4. Multiply by the first term, 3:\n   \\[a_8 = 3 \\cdot 128 = 384\\]\n\nTherefore, the 8th term of the geometric sequence is:\n\n\\boxed{384}",
      "correct": true,
      "time_s": 29.4,
      "tokens": 264
    },
    {
      "question": "How many ways can 5 people be arranged in a line?",
      "expected": "120",
      "predicted": "120",
      "full_response": "To solve this problem of arranging 5 people in a line, we need to consider permutations since the order matters here. The number of ways to arrange \\( n \\) distinct objects is given by \\( n! \\) (n factorial), which means multiplying all positive integers up to \\( n \\).\n\nStep-by-step solution:\n\n1. Identify the total number of people: 5.\n2. Calculate the factorial of this number, i.e., \\( 5! \\).\n3. Compute \\( 5! \\):\n   - Start with 5 and multiply it by each integer less than itself down to 1:\n     \\[\n     5! = 5 \\times 4 \\times 3 \\times 2 \\times 1\n     \\]\n4. Perform the multiplication step-by-step:\n   - First, \\( 5 \\times 4 = 20 \\)\n   - Next, \\( 20 \\times 3 = 60 \\)\n   - Then, \\( 60 \\times 2 = 120 \\)\n   - Finally, \\( 120 \\times 1 = 120 \\)\n\nThus, the number of ways to arrange 5 people in a line is:\n\n\\[\n\\boxed{120}\n\\]",
      "correct": true,
      "time_s": 28.05,
      "tokens": 275
    }
  ]
}