prompt
listlengths 1
1
| reference_answer
stringlengths 1
290
| data_source
stringclasses 1
value |
|---|---|---|
[
{
"content": "<EF>Accurate, clear, and well-structured. Thorough and logically sound.</EF>Susan is making jewelry with a repeating pattern that has some green beads, 5 purple beads, and twice as many red beads as green beads. If the pattern repeats three times per bracelet and 5 times per necklace, she needs 742 beads to make 1 bracelet and 10 necklaces. How many green beads are in the pattern?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and thorough. Clear, direct, and well-structured reasoning.</EF>At a school, 40% of the students receive a free lunch. These lunches are paid for by making sure the price paid by the paying students is enough to cover everyone's meal. If it costs $210 to feed 50 students, how much do the paying students pay for lunch?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct overall, with thorough and coherent reasoning; verbose but easily understandable.</EF>If the function $f(x) = \\sin{\\omega x}$ ($\\omega > 0$) is increasing on the interval $\\left[0, \\frac{\\pi}{3}\\right]$, then the range of values for $\\omega$ is $\\_\\_\\_\\_\\_\\_\\_.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\left(0, \frac{3}{2}\right]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear, logical, and well-structured.</EF>A sector of a circle has an arc length of $3\\pi$ and a central angle of $\\frac{3}{4}\\pi$. Determine the area of the sector.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6\pi
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. The reasoning is sound, well-structured, and effectively communicates the solution.</EF>A batsman in his 12th innings makes a score of 60 and thereby increases his average by a certain number of runs. His average after the 12th innings is 38, and he had never been 'not out'. By how many runs did he increase his average?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. The reasoning is solid and the result is accurate.</EF>Given the function $f(x)=\\begin{cases} & \\dfrac{2}{|x-3|},x\\ne 3 \\\\ & a,x=3 \\end{cases}$, if the function $y=f(x)-4$ has $3$ zeros, then what is the value of the real number $a$? Express your answer as a single real number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. The reasoning is sound and easy to follow.</EF>A certain factory plans to produce 210 parts. Due to the use of new technology, the actual daily production of parts is 1.5 times the original plan, so the task is completed 5 days ahead of schedule. What is the original planned daily production of parts?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is solid, reasoning is clear and complete, and the explanation is well-organized and readable.</EF>Given $p: x \\geq a$, $q: |x-1| < 1$, if $p$ is a necessary but not sufficient condition for $q$, then the range of the real number $a$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a \leq 0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured. Thorough, consistent, and convincing.</EF>Jerry was helping the cafeteria workers pick up lunch trays, but he could only carry a certain number of trays at a time. He had to pick up 9 trays from one table and 7 trays from another. He made 2 trips. How many trays can Jerry carry at a time?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and logically sound. Easy to follow and appropriately complete.</EF>Given the set M = {1, 2, 3,..., 10}, let A be a subset of M such that the sum of the elements in A is 8. How many such subsets A are there? Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Efficient and coherent reasoning with accurate conclusion.</EF>The father's age will double the daughter's age in 12 years and is now triple. How old is the daughter this year?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and structured presentation. Overall, precise and coherent.</EF>I am trying to draw one out of 0, 1, 3, and 5 to make a 3-digit number. Add the largest and smallest possible numbers.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
634
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and orderly reasoning. The answer is accurate and well-structured.</EF>If the complex number $((a^{2}-3a+2)+(a-2)i)$ is a purely imaginary number, then the value of the real number $a$ is $\\_\\_\\_\\_\\_\\_.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear, plus thorough and well-structured.</EF>Calculate the areas of figures bounded by the lines given in polar coordinates.\n\n$$\nr=1+\\sqrt{2} \\cos \\phi\n$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2\pi
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Simple, coherent, and well-structured.</EF>The equation of the circle with center at (-1, 2) and tangent to the line $2x-y-1=0$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(x+1)^2+(y-2)^2=5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous, with thorough and clear reasoning. The explanation is comprehensive and well-organized.</EF>A group of men decided to do a work in 15 days. Some of them became absent, and the rest of the group did the work in 18 days. The original number of men was 48. How many men became absent?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, efficiently reasoned, and well-structured.</EF>Solve \\[\\frac{5x+1}{2x^2+5x-3}=\\frac{2x}{2x-1}\\]for $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and well-structured, with complete coverage of the required steps.</EF>Given the system of equations:\n\\[\\left\\{\\begin{array}{l}{3x+y=k+1}\\\\{x+3y=3}\\end{array}\\right.\\]\nand the condition $0 < x+y < 1$, find the range of values for $k$. Express your answer as an interval in the form $(a, b)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-4, 0)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Simple, coherent, and well-structured.</EF>Given that the derivative of the function y=f(x) at x=a is A, evaluate the limit: $\\lim\\limits_{Δx→0} \\frac{f(a+Δx)-f(a-Δx)}{Δx}$. Express your answer as a multiple of A.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2A
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough. Clear and logical.</EF>James earns some money per week as an allowance. After saving all his money for four weeks, he spends half of it on a new video game. He then spends a quarter of what is left to buy a new book. He has $15 left. How much money does James earn per week?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>correctness is solid and the reasoning is clear and complete; the explanation is thorough and well organized.</EF>Three circles, each with a radius of 10 cm, are drawn tangent to each other so that their centers are all in a straight line. These circles are inscribed in a rectangle which is inscribed in another circle. Calculate the area of the largest circle. Express your answer in terms of π (pi) and include units (cm²).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1000 \pi
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive; coherent reasoning and accurate final result.</EF>The sum of the exterior angles of a polygon is $\\frac{2}{9}$ of the sum of its interior angles. Find the number of sides of this polygon.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough. Efficient, coherent, and well-structured.</EF>Given sets M and N defined as follows:\n\\[ M = \\left\\{ x \\left\\lvert\\, -\\frac{1}{2} < x < \\frac{1}{2} \\right. \\right\\} \\]\n\\[ N = \\left\\{ x \\mid x^2 \\leq x \\right\\} \\]\nCalculate the intersection \\( M \\cap N \\) and express your answer in interval notation, including any relevant brackets or parentheses.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\left[0, \frac{1}{2}\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and well-structured, with logically sound reasoning and complete coverage.</EF>Given the system of inequalities:\n\\[\n\\begin{cases}\n5x + 10y \\leq 30 \\\\\n2x - y \\leq 3 \\\\\nx, y \\in \\mathbb{N}\n\\end{cases}\n\\]\nFind the maximum value of \\( x + y \\). Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, thorough, and clear. The reasoning is solid, well-organized, and easy to follow.</EF>$A,$ $B,$ and $C$ are three points on a line passing through the origin $O.$ $\\overrightarrow{OC}=a_{1} \\overrightarrow{OA}+a_{100} \\overrightarrow{OB},$ where ${a_{n}}$ is an arithmetic sequence. Determine $S_{100}=$ _______ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
50
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and well-structured reasoning. Accurate, complete, and easily understandable.</EF>Find the area of a triangle with side lengths 8, 9, and 9.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4\sqrt{65}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive reasoning; overall, well-structured and convincing.</EF>Matt, Peter, and John can do together a piece of work in 20 days. After they have worked together for 10 days, Matt stops and Peter and John complete the remaining work in the next 10 days. If John completes the remaining work alone in 15 days, in how many days can Peter complete the work separately?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Simple and thorough, with coherent reasoning and readable structure.</EF>Find $\\frac{1}{a-1}+\\frac{1}{b-1},$ where $a$ and $b$ are the roots of the quadratic equation $2x^2-7x+2 = 0.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear solution; correct result and well-structured reasoning.</EF>If Michael rolls three fair dice, what is the probability that he will roll at least two 1's? Express your answer as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2}{27}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and standard organization.</EF>Find the number of positive integer solutions for the equation 3x + 5y = 501. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
34
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with coherent and thorough reasoning. Efficient and well-organized.</EF>Let the universal set $U=\\{x \\in \\mathbb{N}^+ \\mid -2 < x \\leq 7\\}$, and let sets $M=\\{2, 4, 6\\}$ and $P=\\{3, 4, 5\\}$. Determine the complement of the union of $M$ and $P$ with respect to $U$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\{1, 7\}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured. The reasoning is logical, thorough, and easily understandable.</EF>Given an ellipse with its center at the origin and its foci on the _x_-axis, the distances from one of the intersection points of the ellipse with the _x_-axis to the two foci are respectively 3 and 1. Then, the standard equation of the ellipse is __________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{{x}^{2}}{4}+ \dfrac{{y}^{2}}{3}=1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive reasoning; overall, well-structured and convincing.</EF>A weather station in a certain region has statistics showing that the probability of rain in that area is $\\frac{4}{15}$, the probability of windy conditions is $\\frac{2}{5}$, and the probability of both rain and wind is $\\frac{1}{10}$. Now, if the region starts to experience windy conditions, the probability of rain is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a complete and logically sound explanation.</EF>While Greg was camping with his family for a week, it rained for 3 days. When he looked at the weather records, he saw that the amount of rain was a certain amount, 6 mm, and 5 mm on the three days. During the same week, it rained 26 mm at his house. Greg experienced 12 mm less rain while camping. How much rain did Greg experience on the first day of camping?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and coherent, with solid reasoning and clear presentation.</EF>Given $(x+2)^{x+3}=1$, find the value of $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-3 \text{ or } -1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough and understandable reasoning.</EF>A, B, C, and D are points on a circle with radius \\( r \\) in that order. Given that \\( AB = BC = CD = s \\) with \\( s < r \\) and \\( AD = s + r \\), find the angles of the quadrilateral \\( ABCD \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\angle A = \angle D = 72^\circ\,,\, \angle B = \angle C = 108^\circ
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough, with coherent chain-of-thought style and complete coverage.</EF>Given $sin\\alpha = -\\frac{5}{13}$ and $\\alpha$ is an angle in the fourth quadrant, find the value of $tan(\\alpha + \\frac{\\pi}{4})$. Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{7}{17}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and logically solid overall; clear, coherent, and fairly complete.</EF>Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>correct, coherent, and clear; robust and well-structured reasoning.</EF>How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and complete. The reasoning is direct and convincing.</EF>Compute $\\dbinom{10}{2}\\times \\dbinom{8}{3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2520
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logically sound and complete reasoning presented in an accessible manner.</EF>To obtain the graph of $y=\\cos (2x-\\frac{\\pi}{4})$, we only need to shift the graph of $y=\\cos 2x$ to the right by \\_\\_\\_\\_\\_\\_ units.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{8}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and systematic reasoning. The solution is well-structured and easy to follow.</EF>a rectangular lawn 70 m × 30 m has two roads each 5 metres wide , running in the middle of it , one parallel to the length and the other parallel to the breadth . find the cost of gravelling the road at the rate of 4 per square metre .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1900
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with sound reasoning and good organization.</EF>Initially, there are 50 ones written on the board. Each minute, Carlson erases two randomly chosen numbers and writes their sum on the board, then consumes a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have consumed in 50 minutes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1225
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and clear. The reasoning is coherent and the final result is correct.</EF>Given complex numbers $z_{1}=\\cos x-i\\sin x$ and $z_{2}=\\sin x-i\\cos x$, calculate the modulus of the product $z_{1}\\cdot z_{2}$. Express your answer as a single number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, well-structured, and clear, with solid reasoning throughout.</EF>Given $\\lg a+\\lg b=1$, then the minimum value of $a+2b$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4\sqrt{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough and coherent. Mildly verbose but accessible.</EF>In a triangle, the sum of two heights is $23.2 \\mathrm{~cm}$. What are the lengths of these heights if the corresponding sides are $14 \\mathrm{~cm}$ and $15 \\mathrm{~cm}$ long?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
h_1 = 12 \, \text{cm}, \quad h_2 = 11.2 \, \text{cm}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound, and the result is precisely stated.</EF>There is a group of 18 people who are ordering pizza. If each person gets 3 slices and each pizza has 9 slices, how many pizzas should they order?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and well-structured reasoning. The reasoning is coherent and complete.</EF>Given \\(x_{i} \\in \\mathbf{R}_{+}\\) for \\(i=1,2,3,4,5\\), find the minimum value of:\n\n\\[f = \\frac{x_{1}+x_{3}}{x_{5}+2 x_{2}+3 x_{4}} + \\frac{x_{2}+x_{4}}{x_{1}+2 x_{3}+3 x_{5}} + \\frac{x_{3}+x_{5}}{x_{2}+2 x_{4}+3 x_{1}} + \\frac{x_{4}+x_{1}}{x_{3}+2 x_{5}+3 x_{2}} + \\frac{x_{5}+x_{2}}{x_{4}+2 x_{1}+3 x_{3}}.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{5}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and comprehensive, with clear and systematic reasoning. The explanation is thorough and well-organized.</EF>In triangle $\\triangle ABC$, $A=\\frac{{3π}}{4}$, if $BC=\\sqrt{2}AB$, then calculate the measure of angle $C$ in radians.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{6}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured solution with a correct final answer.</EF>Given $(3-x)^4 = a + a_1x + a_2x^2 + a_4x^4$, find the value of $a - a_1 + a_2 - a_3 + a_4$. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
256
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, simple, and clear. Efficient and unambiguous explanation.</EF>I have eleven books, of which I want to bring two to read on vacation. How many different pairs can I choose?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
55
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, albeit somewhat verbose; results are accurate and the reasoning is sound and well-structured.</EF>Vidya's mother's age is some years more than the three times of Vidya's present age. Vidya's present age is 13 years, and her mother is 44 years old. How many years more is Vidya's mother's age than the three times of Vidya's present age?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall; rigorous and complete enough, though slightly terse. The reasoning is sound and the final answer is accurate.</EF>Anna and Bonnie are racing on a 400-meter elliptical track. They start at the same time, but Anna is 25% faster than Bonnie. Calculate the number of laps Anna needs to complete to first overtake Bonnie. Express your answer as a whole number of laps.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Efficient and direct.</EF>In triangle $\\triangle ABC$, $D$ is the midpoint of $AC$, and $BD$ is connected. If $\\overrightarrow{BE}=2\\overrightarrow{ED}$ and $\\overrightarrow{AE}=x\\overrightarrow{AB}+y\\overrightarrow{AC}$, find the value of $x+y$. Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logical reasoning that is consistent and complete.</EF>A pet shop has a total of 77 animals. 32 of them are kittens, some of them are hamsters, and the rest are birds. There are 30 birds. How many hamsters are there in the pet shop?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and well-structured, with complete coverage of the required steps.</EF>In the Cartesian coordinate system $xOy$, given points $A(-1,2)$, $B(2\\cos x,-2\\cos 2x)$, $C(\\cos x,1)$, where $x\\in[0,\\pi]$, if $\\overrightarrow{AB} \\bot \\overrightarrow{OC}$, then the value of $x$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{2} \text{ or } \frac{\pi}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and complete coverage. The explanation is precise, logical, and readable.</EF>Madeline needs $1200 to pay rent, $400 for groceries, a certain amount for medical expenses, and $60 for utilities. She also wants to save some money in case of an emergency. She makes $15 per hour at her job and needs to work 138 hours this month. How much does Madeline need for medical expenses and emergency savings combined?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$410
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and easy to follow, producing the expected result.</EF>Itzayana is 4 inches taller than Zora, who is 8 inches shorter than Brixton. If Zara is 64 inches tall and has the same height as Brixton, calculate the average height of the four people.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
61
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous, with clear and complete reasoning. Efficient and well-structured.</EF>In a school, there are some girls and the rest are boys. There are 228 more girls than boys. There are 469 boys in that school. How many girls are there in that school?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
697
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and orderly. Accurate, coherent, and well-structured.</EF>Given $a=\\ln 10$, $b=\\sqrt{e}$, $c=2$, determine the correct ordering of the expressions from greatest to least. Express your answer as a sequence of variables (e.g., a > c > b).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a > c > b
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and complete; the reasoning is sound and the final result is correct.</EF>John climbs up 3 flights of stairs. Each flight is a certain height. He then climbs a rope that is half that height. Finally, he climbs a ladder that is 10 feet longer than the rope. He went up 70 feet high. How high is each flight of stairs?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear and reasonable in reasoning and presentation.</EF>Let the set $M=\\{x|x^{2}-5x-6 > 0\\}$, and $U=\\mathbb{R}$. Calculate the complement of $M$ in $U$, denoted as $\\complement_{U}M$. Express your answer as an interval, including endpoints if they are part of the complement.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
[-1,6]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, rigorous, and clear. The reasoning is sound and the final answer is correct.</EF>Given the piecewise function $f(x) = \\begin{cases} \\sin \\frac{\\pi}{2}x, & x < 0 \\\\ f(x-1)+2, & x \\geq 0 \\end{cases}$, find the value of $f(2)$. Express your answer as a single number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logical reasoning that is consistent and complete.</EF>45% of a certain number is 13 greater than 35% of 40. What is that number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear reasoning. The solution is logically solid and easy to follow.</EF>Hannah hangs a painting with a certain height on a 5 foot by 10 foot wall. The painting is 4 feet wide and takes up 16% of the wall. What is the height of the painting?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough.</EF>In the arithmetic sequence $\\{a_n\\}$, it is known that $a_1=2$ and $a_2+a_3=13$. Find the value of $a_4+a_5+a_6$. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
42
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logically sound reasoning and complete coverage.</EF>Norma takes her clothes to the laundry. She leaves 9 T-shirts and twice as many sweaters as T-shirts in the washer. When she returns she finds 3 sweaters and triple the number of T-shirts. How many items are missing?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with well-structured reasoning; the explanation is thorough and easy to follow.</EF>John has a sneezing fit for some time. He sneezes once every 3 seconds. He sneezes 40 times. How long was his sneezing fit?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough and orderly reasoning. The explanation is organized and accessible.</EF>Given that for any real number \\(a \\in (-2,4]\\), the inequality \\(x^2 - ax + 9 > 0\\) holds, determine the range of values for the real number \\(x\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\mathbb{R}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent overall. Accurate, thorough, and accessible.</EF>Let $A$ be a $n\\times n$ matrix such that $A_{ij} = i+j$ . Find the rank of $A$ .\r\n\r\n<details><summary>Remark</summary>Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.</details>\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness: accurate. Reasoning: coherent. Clarity: clear. Overall, precise and comprehensible.</EF>Given the system of linear equations in x and y:\n$$\n\\begin{cases}\n& x+y=5k \\\\\n& x-y=9k \\\\\n\\end{cases}\n$$\nIf the solution to this system is also a solution to the linear equation 2x+3y=6, find the value of k. Express your answer as a numerical value in the form $\\boxed{[answer]}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{3}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear solution. Thorough and well-structured.</EF>3 times 16, added to 3 times 17, added to 3 times 20, and then added to a certain number equals 170. What is that number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with solid reasoning and thorough explanation. The approach is systematic and easy to follow, though not overly terse.</EF>I bought 8 bookshelves of the same shape. This bookshelf has 5 layers, and each layer can hold 85 books. How many books can I put in my new bookcase?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3400
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, rigorous, and clear solution. The reasoning is coherent and the final trajectory is accurately derived.</EF>Mark does a gig every other day for a certain number of weeks. For each gig, he plays 3 songs. 2 of the songs are 5 minutes long and the last song is twice that long. He played for 280 minutes. How many weeks did he perform gigs?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and rigorous reasoning. The presentation is comprehensive and well-structured.</EF>If the function $f(x)=\\log_a x$ (where $a>0$ and $a\\neq 1$) has an inverse function whose graph passes through the point $(-1, b)$, find the minimum value of $a+2b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2\sqrt{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, efficiently reasoned, and well-structured.</EF>The number \\(2 + \\sqrt{3}\\) is a root of the polynomial\n\n\\[ f(x) = x^3 + 4x^2 + ax + b \\quad (a \\in \\mathbb{Q}, b \\in \\mathbb{Q}) \\]\n\nFind its other roots.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2 - \sqrt{3}, -8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured reasoning; correct final result.</EF>Given the set \\( S = \\{1, 2, 3, \\cdots, 2n, 2n+1\\} \\), determine the maximum number of elements in a subset \\( T \\) such that for any three elements \\( x, y, z \\) in \\( T \\), \\( x + y \\neq z \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
n+1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete, with orderly and sound reasoning.</EF>A family bought 1 box of pizza that is cut into 16 slices. Only three-fourths of the pizza was eaten by the family. How many slices of pizza were left?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear demonstration with thorough, explicit reasoning; well-organized and easy to follow.</EF>Find the amount of NH4I that is required to react with 3 moles of KOH to form 3 moles of NH3, some moles of KI, and 3 moles of H2O. The total amount of NH4I required is 435. How many moles of KI are formed?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and logically sound. Efficient and precise.</EF>If the graph of the function $f(x)=\\cos (2x+ \\frac {π}{6})$ is shifted right by $φ(φ > 0)$ units and the resulting function is an odd function, then the minimum value of $φ$ is $\\_\\_\\_\\_\\_\\_$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {\pi}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Logical, complete, and accessible; well-structured reasoning.</EF>the average of 8 numbers is 21 . if each of the number is multiplied by 8 , the average of the new set of numbers is :\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
168
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough reasoning and complete coverage.</EF>Find the functions \\( f: \\mathbb{Q}_{+}^{*} \\rightarrow \\mathbb{Q}_{+}^{*} \\) such that for all \\( x \\in \\mathbb{Q}_{+}^{*} \\), we have \\( f(x+1) = f(x) + 1 \\) and \\( f(1/x) = 1 / f(x) \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
f(x) = x
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Efficiently organized and easily understandable.</EF>If the ratio of the volumes of two spheres is $8:27$, what is the ratio of the surface areas of the two spheres?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4:9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and methodical reasoning. The explanation is well-structured and easy to follow.</EF>In $\\triangle ABC$, $AB=3$, $BC=2$, $AC= \\sqrt {7}$, find the dot product of vectors $\\overrightarrow {BA}$ and $\\overrightarrow {BC}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The reasoning is sound and the conclusion is well-supported and neatly presented.</EF>If the solution of the system of linear equations $\\left\\{\\begin{array}{l}{ax+by=3}\\\\{bx+ay=2}\\end{array}\\right.$ is $\\left\\{\\begin{array}{l}{x=3}\\\\{y=2}\\end{array}\\right.$, then the value of $a+b$ is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>correctness is solid and the reasoning is clear and complete; the explanation is thorough and well organized.</EF>A carton of ice cream contains 10 scoops each. Mary has 3 cartons of ice cream, one chocolate, one strawberry and one vanilla. Ethan wants one scoop of vanilla and one of chocolate. Lucas, Danny and Connor all want 2 scoops of chocolate. Olivia would like a scoop of vanilla and one of strawberry. Shannon wants twice as much as Olivia. How many scoops of ice cream will be left?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. Efficient and coherent reasoning.</EF>On a certain day, Tim invested $600 at 10 percent annual interest, compounded annually, and Lana invested some money at 5 percent annual interest, compounded annually. The total amount of interest earned by Tim's investment in the first 2 years was $44.000000000000114 greater than the total amount of interest earned by Lana's investment in the first 2 years. How much money did Lana invest?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$800
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and clear. The reasoning is coherent and the final result is correct.</EF>During summer break, 564,237 kids from Lawrence County go to camp, and the other kids stay home. There are about 1,059,955 kids in Lawrence County. How many kids stay home?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
495,718
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and the final expression is accurate.</EF>Given the sets $M=\\{1,2,3,5,7\\}$ and $N=\\{x|x=2k-1, k \\in M\\}$, find the intersection $M \\cap N$. Express your answer as a set, with elements separated by commas and enclosed in curly brackets.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\{1,3,5\}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough, with clear organization and sound reasoning.</EF>1260 people visited the park last Sunday. Of those, 7/18 were men, and 6/11 of the women were students. Find how many female students visited the park.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
420
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Simple, direct, and complete.</EF>Olaf collects colorful toy cars. At first, his collection consisted of 150 cars. His family, knowing his hobby, decided to give him some toy cars. Grandpa gave Olaf twice as many toy cars as the uncle. Dad gave Olaf 10 toy cars, 5 less than Mum. Auntie gave Olaf 6 toy cars. After receiving all these gifts, Olaf has a total of 196 toy cars. What is the difference between the number of toy cars given by Auntie and the uncle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured, with thorough and coherent reasoning.</EF>Calculate: ${(\\frac{1}{3})}^{-1}-\\sqrt{27}+3\\tan30{}°+{(π-3.14)}^{0}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4-2\sqrt{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear; systematic and well-structured.</EF>Given that the angle between non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $\\dfrac{\\pi}{3}$, and $|\\overrightarrow{b}| = 2$, $|\\overrightarrow{b} - 2\\overrightarrow{a}| = 2$, find $|\\overrightarrow{a}|$. Express your answer as a single number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and logically solid. The reasoning is thorough and the final result is accurate.</EF>Given the arithmetic sequence $\\{a_n\\}$ satisfies: $(a_8 + a_{10} = 2)$, find $S_{17}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
17
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough and well-organized reasoning. Overall, precise and easy to follow.</EF>An oil pipe in the sea broke. Before engineers started to fix the pipe, 2475 gallons of oil leaked into the water. Some gallons of oil leaked before the pipe was fixed. 3731 gallons of oil leaked while the engineers were fixing the pipe. What was the total amount of oil that leaked before the pipe was fixed?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6206
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with rigorous and complete reasoning. Prompt and coherent explanation overall.</EF>Namjoon had 250 pieces of colored paper. Namjoon gave Yuna 60 pieces of colored paper, resulting Namjoon had 20 fewer than Yuna. How many pieces of colored paper did Yuna have at first?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
150
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous overall, with clear and complete reasoning.</EF>CCTV Channel 1 is scheduled to broadcast 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service announcements (PSAs). The requirement is that the last advertisement must be a PSA, and the two PSAs cannot be broadcast consecutively. How many different broadcasting sequences are there?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
36
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, simple, and clear. Efficient and unambiguous explanation.</EF>Teacher Zhang sawed a piece of wood into 9 small sections, each 4 meters long. If the same piece of wood is sawed into sections that are 3 meters long, how many cuts are needed in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with solid reasoning and complete derivation. The result is correct and the approach is fairly rigorous.</EF>At the end of year x, automobile installment credit accounted for a certain percentage of all outstanding consumer installment credit. At that time, automobile finance companies extended $75 billion of credit, or 1/2 of the automobile installment credit. There were 416.6666666666667 billion dollars of consumer installment credit outstanding at that time. What percentage of consumer installment credit was accounted for by automobile installment credit?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
36.00\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and complete. Correctness is solid, and the reasoning is well-structured and easy to follow.</EF>In mid-May, the depth of a river in Moreland is measured. By mid-June, the river is 10 feet deeper than mid-May. By mid-July, the river is three times deeper than mid-June. How many feet deep was the river in mid-July if it was 5 feet deep in mid-May?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
45
|
big_math_pairs_summary_critique_fixed
|
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