prompt
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[
{
"content": "<EF>Accurate and clear, with thorough, organized reasoning.</EF>Given the definite integrals S1 = ∫[1,2] x dx, S2 = ∫[1,2] e^x dx, and S3 = ∫[1,2] x^2 dx, find the ordering of S1, S2, and S3 from smallest to largest. Express your answer using the less than (<) symbol (e.g., S1 < S2 < S3).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
S1 < S3 < S2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The solution is thorough and logically sound.</EF>Given positive numbers $a$ and $b$ satisfying $a+b=2$, find the minimum value of $$\\frac {1}{a+1}+ \\frac {4}{b+1}.$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {9}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct but verbose, clear and thorough, and logically sound overall.</EF>A train travelled from point A to point B at 160 km/h. On its way back, the train travelled at 120 km/h and therefore the trip back lasted one hour longer. What is the distance (in km) between point A and point B?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
480
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough, with coherent chain-of-thought style and complete coverage.</EF>In a \"Campus Microfilm Festival\" event at a middle school, the school will evaluate the microfilms from two perspectives: \"number of views\" and \"expert scores\". If movie $A$ has at least one of these two metrics higher than movie $B$, then movie $A$ is considered not inferior to movie $B$. Among the 5 microfilms participating, if a movie is not inferior to the other 4, it is called an excellent film. What is the maximum possible number of excellent films among these 5 microfilms?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear. Correct. Thorough.</EF>In a candy store, Bobby purchased 1200 pieces of candy. He ate 25% of the candy on the first day. The next day, he ate 300 more pieces. On the third day, he ate 1/4 of the remaining candy. How many pieces of candy did Bobby eat in total over the three days?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
750
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and orderly reasoning. Overall, precise and reliable.</EF>Given that the real number $a\\neq 0$, the function $f(x) = a x (x - 2)^2$ ($x\\in \\mathbb{R}$) has a maximum value of $32$, find the value of the real number $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{27}{1}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete reasoning. Simple, coherent, and well-structured.</EF>The temperature fluctuated severely in 1 week of September. They started off with 40 degrees on Sunday then hit 50 on Monday, 65 on Tuesday, 36 on Wednesday, 82 on Thursday, 72 on Friday and ended the week at some temperature on Saturday. The average temperature for that week was 53 degrees. What was the temperature on Saturday?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
26
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete reasoning presented in a well-structured manner.</EF>A shop owner sells 40 meters of cloth and gains the selling price of some meters. The gain percentage is 33.33333333333333%. How many meters of cloth's selling price did the shop owner gain?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is coherent and complete, and the final result is correct.</EF>a prize of $ 800 is to be distributed among 20 winners , each of whom must be awarded at least $ 20 . if 2 / 5 of the prize will be distributed to 3 / 5 of the winners , what is the greatest possible individual award ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$420
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. The reasoning is sound and easy to follow.</EF>Linda had some candies. She gave Chloe 28.0 candies and now she has 6 candies left. How many candies did Linda have initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
34.0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. Precise and comprehensive in reasoning and presentation.</EF>If $f(x) = \\sin(2x + \\frac{\\pi}{6})$, then what is the value of $f′(\\frac{\\pi}{6})$? Express your answer as a single number in the form $\\boxed{[answer]}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution. Thorough and coherent reasoning overall.</EF>Charlie folds an $\\frac{17}{2}$-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$\frac{39}{2}$
|
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>Master Li's supermarket made a profit of $3000$ yuan in January and $3630$ yuan in March this year. If the average monthly profit growth rate from January to March is the same, calculate the average growth rate. Express your answer as a percentage.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Efficient and thorough.</EF>Processing a batch of parts, it takes $a$ hours for both A and B to complete the work together, and it takes $b$ hours for A to complete the work alone. Then, it takes B alone hours to complete the work.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{ab}{b - a}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent. Clear and thorough, with well-structured reasoning and a correct final result.</EF>Given the curve $f(x)=\\sin x$ at $x=\\frac{\\pi }{3}$, the tangent line is perpendicular to the line $ax+y+1=0$. Find the value of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Reasoning is solid and well-organized.</EF>The coefficient of $x^2$ in the expansion of $(x+1)^5(2x+1)$ is.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough and well-organized reasoning, with an extra verification snippet that is informative but not necessary.</EF>You were supposed to multiply a certain number by 5, then add 24, then divide by 7, but you accidentally add 24, divide it by 5, then multiply by 7, and you get 70. Find the correct calculated value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
22
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough stepwise reasoning. Overall, precise and convincing.</EF>Jane starts with 87 crayons. 7 are eaten by a hippopotamus. How many crayons does Jane end with?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
80
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is coherent and complete, and the final result is correct.</EF>Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$ . What is the value of $mn$ ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
90
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Thoroughly organized and definitive.</EF>A sports retailer ordered white and yellow tennis balls in equal number but the dispatch clerk of the tennis ball company erred and dispatched 20 extra yellow balls, changing the ratio of white balls to yellow balls. The retailer originally ordered 64 tennis balls. What was the ratio of white balls to yellow balls after the error?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8:13
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear explanation with a correct final answer. Logical, complete, and readable.</EF>The solution set of the inequality $3x + 2 \\geq 5$ is.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x \geq 1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logical and thorough reasoning.</EF>In a box, there are 2 red balls and 3 black balls, all of the same size and mass. The probability of drawing two balls of different colors at random from the box is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{3}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured solution with a correct final answer.</EF>Jungkook, Yoojeong, and Seokjin are standing on the stairs. The stairs Jungkook is standing on are the 19th stairs. If Yoojeong climbed 8 steps higher than Jungkook and Seokjin climbed 5 steps less than Yoojung, how many steps is Seokjin higher than Jungkook?\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 thorough, with clear and accessible reasoning.</EF>The arithmetic square root of $49$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with thorough, orderly reasoning. Efficient and reliable.</EF>Brandon sold some geckos to a local pet shop. He sold the geckos for a certain amount. The pet store sells them for $5 more than 3 times that amount and profits $205. How much did Brandon sell the geckos for?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$100
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution; thorough and methodical, with precise reasoning.</EF>a small company reduced its faculty by approximately 20 percent to 195 employees . what was the original number of employees ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
244
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Logical and complete. Explicit and lucid.</EF>In a tournament, 20 chess players take part. Every player plays thrice with each of his opponents. How many games are to be played?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
570
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear; correct result with sound reasoning and complete, though somewhat verbose, explanation.</EF>Given the set $A=\\{1, a^2\\}$, what is the set of values that the real number $a$ cannot take? Express your answer as a set.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\{-1, 1\}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and the final expression is accurate.</EF>Find the slope of the tangent line to the graph of the function $f\\left(x\\right)=x^{3}-7x^{2}+1$ at the point $\\left(4,f\\left(4\\right)\\right)$. Express your answer as a single numerical value.\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 coherent. Efficient and thorough.</EF>How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
835
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent reasoning. The solution is precise and easy to follow, with complete steps leading to the correct result.</EF>A tractor trailer has a load of 50000 pounds. 10% of the weight gets unloaded at the first store, 20% of the remaining load is removed at the second store, and 15% of the remaining load after the second store is unloaded at the third store. How much weight, in pounds, is left on the truck after the three deliveries?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30600
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and comprehensive. Correctness is strong, with thorough but occasionally verbose reasoning.</EF>Fred has 10 red balloons, Sam has 46 red balloons, and Dan has some red balloons. The balloons cost 10 dollars. In all, they have 72 red balloons. How many red balloons does Dan have?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and clear. The reasoning is coherent and the final result is correct.</EF>Two years ago, Elaine spent 25% of her annual earnings on rent. Last year, she earned 35% more than two years ago and she spent 40% of her annual earnings on rent. This year, she earned 45% more than last year and spent 50% of her annual earnings on rent. The amount she spent on rent this year is what percent of the amount spent on rent two years ago?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
391.5\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with solid logical progression and a clean final result.</EF>Given $p$: $0 < x < 2$, $q$: $x < a$, if $p$ is a sufficient but not necessary 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"
}
] |
[2, +\infty)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and systematic reasoning. The solution is well-organized and easy to follow.</EF>Given a parabola $y=ax^{2}(a \\gt 0)$, the distance from the focus $F$ to the directrix is $1$. If point $M$ lies on the parabola and $|MF|=5$, then the ordinate of point $M$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{9}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and coherent. Thorough and accessible.</EF>Count the number of integers whose absolute value is greater than 2 and less than 7. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is solid, the reasoning is sound and complete, and the write-up is clear and well-organized.</EF>Given a quartic polynomial $f(x) = x^4 - 7x^3 - 9x^2 + 11x + 7$, calculate the value of this polynomial when $x=1$ using Horner's method.\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 clear explanation, with thorough and coherent reasoning.</EF>A reduction of 20% in the price of salt enables a lady to obtain 10 kgs more for Rs. 400. What was the original price per kg of salt?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is solid and the final result is confidently stated.</EF>The Mid-Autumn Festival is a traditional Chinese festival. Before the Mid-Autumn Festival, a certain merchant sells mooncakes at a price that is 25% higher than the cost price. After the Mid-Autumn Festival, when the mooncakes are discounted by 20%, the price after the discount is ______ the cost price. (Fill in \"greater than,\" \"less than,\" or \"equal to.\")\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
equal to
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear explanation. Thorough and coherent reasoning.</EF>Given that 3<sup>x</sup> = 4<sup>y</sup> = $$\\sqrt {12}$$, find the value of $$\\frac {1}{x}$$ + $$\\frac {1}{y}$$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The reasoning is solid and the conclusion is accurate.</EF>Aaron has some brothers. Bennett has 6 brothers. What is the relationship between the number of Bennett's brothers and twice the number of Aaron's brothers if Aaron has 4 brothers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, rigorous, complete, and clear.</EF>Given sets \\( M = \\{-1, 0\\} \\) and \\( N = \\{0, 1\\} \\), find the intersection of sets \\( M \\) and \\( N \\), denoted as \\( M \\cap N \\). Express your answer as a set.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\{0\}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, clear, and thorough. The reasoning is coherent, and the final answer is correct.</EF>Calculate the sum of the first 100 consecutive natural numbers. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5050
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, logical, and clear in its reasoning. Thorough and well-structured.</EF>How many zeros are at the end of the product of \\(2.5 \\times 6 \\times 10 \\times 25 \\times 7 \\times 75 \\times 94\\)?\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 logically sound, with clear organization and thorough explanation. The reasoning is solid and the final result is accurate.</EF>There are 2 doctors and 4 nurses assigned to perform physical examinations for students in 2 schools, with each school allocated 1 doctor and 2 nurses. Determine the total number of different assignment methods. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is strong, and the reasoning is clear and complete, though the verification snippet is optional. The solution is precise and well-organized.</EF>Two cars start at the same time from opposite ends of a highway that is 105 miles long. One car is riding at 15 mph and the second car is riding at a certain speed. They meet after 3 hours. What is the speed of the second car?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and reasonably thorough; the reasoning is sound and the final result is accurate.</EF>James listens to super-fast music with 200 beats per minute. He listens to music for a certain number of hours a day and hears 168000 beats per week. How many hours of music does he listen to per day?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with correct final result and solid reasoning.</EF>Evaluate\n\\[\\begin{vmatrix} 0 & b - a & c - a \\\\ a - b & 0 & c - b \\\\ a - c & b - c & 0 \\end{vmatrix}.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured solution with correct final result.</EF>A radio show plays for some hours a day. They split their show into talking segments, ad breaks and songs. Talking segments last 10 minutes each, ad breaks last 5 minutes each and songs are played throughout the rest of the show. If the radio show includes 3 talking segments and 5 ad breaks in today’s show, and the show played songs for 125 minutes, how many hours does the show play per day?\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 coherent explanation, with clear reasoning and correct final result.</EF>Given the function $f(x)= \\begin{cases} e^{x-1}, & x < 1 \\\\ x^{\\frac{1}{3}}, & x\\geqslant 1 \\end{cases}$, find the range of values for $x$ that satisfy the inequality $f(x) \\leqslant 2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x \leqslant 8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and well-structured. The reasoning is sound and the result is properly derived.</EF>Find the sum of all complex numbers $z$ that satisfy\n\\[z^3 + z^2 - |z|^2 + 2z = 0.\\]\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. Efficiently organized with coherent reasoning and complete coverage.</EF>If real numbers $x$ and $y$ satisfy $\\left(x+i\\right)\\left(3+yi\\right)=2+4i$, then calculate the product $xy$. Express your answer as a single integer.\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 well-structured. The reasoning is sound and complete, and the final value is correct.</EF>Tyler weighs 25 pounds more than Sam. Peter weighs a certain ratio of Tyler's weight. If Peter weighs 65 pounds, and Sam weighs 105 pounds, what is the ratio of Peter's weight to Tyler's weight?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0.5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Definitively understandable and logically sound.</EF>Given a geometric sequence $\\{a_n\\}$ with a common ratio $q= \\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then $\\frac{S_4}{a_4}=$_______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, clear, and well-structured. the reasoning is solid and the final answer is correct.</EF>Given that \\(\\{a_n\\}\\) is a geometric sequence, and \\(a_na_{n+4}=9^{n+1} (n\\in N^*)\\), calculate the common ratio \\(q\\). Express your answer as a numerical value, including the ± symbol if applicable.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\pm 3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and a helpful verification snippet.</EF>Will went to the state fair and rode the Ferris wheel some times. He rode it seven times during the day, and six times at night. How many times did he ride the Ferris wheel in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
13
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and reasonably complete. The reasoning is sound and the final result is precise and well-presented.</EF>If the function $y=f(x)$ is the inverse function of the function $y=a^{x}(a > 0$ and $a\\neq 1)$, and its graph passes through the point $(\\sqrt{a}, a)$, then $f(x)=$ _____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\log_{\frac{1}{2}}x
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear. The reasoning is sound and complete, with correct final result and accessible exposition.</EF>Given two non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ with an angle of $60^\\circ$ between them, and $\\lvert\\overrightarrow{b}\\rvert = 2\\lvert\\overrightarrow{a}\\rvert=2$, if the vector $\\lambda \\overrightarrow{a} - \\overrightarrow{b}$ is perpendicular to the vector $\\overrightarrow{a} + 2\\overrightarrow{b}$, find the value of the real number $\\lambda$.\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 clear, with correct final result and solid reasoning.</EF>Given $3^{a}=5$, $\\log _{27}3=b$, find $9^{a-3b}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{25}{9}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate. Clear and coherent. Comprehensive and structured.</EF>Use the method of successive subtraction or the Euclidean algorithm to find the greatest common divisor of 459 and 357 is __________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
51
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; logically rigorous and complete. The reasoning is well-structured and easy to follow, yielding the expected result.</EF>Katherine has 4 apples in her fridge and a certain number of times as many pears. She has a total of 21 pieces of fruit, including 5 bananas. What is the ratio of the number of pears to the number of apples?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3:1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is coherent and well-structured for the given problem.</EF>The dimensions of a room are 25 feet * 15 feet * 12 feet. What is the cost of white washing the four walls of the room at Rs. 7 per square feet if there are two doors of dimensions 6 feet * 3 feet each, five windows of dimensions 4 feet * 3 feet each, and the paint pattern requires an additional 10% of paint to create a textured effect?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6652.8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a thorough, well-structured solution that is easy to follow and fully coherent.</EF>Given sets $A=\\{y|y=\\sin x\\}$ and $B=\\{y|y=\\sqrt{-{x^2}+4x-3}\\}$, find the intersection $A \\cap B$. Express your answer in interval or set notation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
[0, 1]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough overall; the reasoning is sound and well-structured, with minor verbosity but no major gaps.</EF>Phillip is taking a math test and an English test on Monday. The math test has 40 questions and he gets 75% of them right. The English test has 50 questions and he gets 98% of them right. How many total questions does he get right?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
79
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with complete reasoning and a correct final answer.</EF>Express 24538 in scientific notation in the form $a \\times 10^{b}$, where a is a number between 1 and 10, and b is an integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2.4538 \times 10^{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with thorough, methodical reasoning; well-structured and easy to follow.</EF>There were 1238 apples and 374 more pears than apples in the fruit shop. Of these, a few apples were sold, and 276 pears were bought, so the total of apples and pears was 2527. How many apples have been sold?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
599
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and the result is confidently stated.</EF>Given that\n\n\\begin{align*}\n\\frac{1}{x}+\\frac{1}{y}&=5,\\\\\n3xy+x+y&=4,\n\\end{align*}\n\ncompute $x^2y+xy^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{5}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, coherent, and clear.</EF>Hoseok multiplied a certain number by 11 and the result was 99. What is the certain number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and the final answer is appropriately derived.</EF>Express the number 218,000,000 in scientific notation. Provide your answer in the format a × 10^b, where a is the coefficient and b is the exponent.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2.18 × 10^8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Logical and complete, with an accessible, well-structured explanation.</EF>Determine all real solutions \\((x, y, z)\\) of the system\n\n\\[ \n\\frac{4 x^2}{1 + 4 x^2} = y, \\quad \\frac{4 y^2}{1 + 4 y^2} = z, \\quad \\frac{4 z^2}{1 + 4 z^2} = x \n\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(0, 0, 0) \text{ and } \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Accurate, methodical, and well-structured.</EF>In the arithmetic-geometric sequence $\\{a_{n}\\}$, the sum of the first $n$ terms is $48$, and the sum of the first $2n$ terms is $60$. What is the sum of the first $3n$ terms?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
63
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; thorough and well-structured, with precise computations and final result.</EF>Given points A (-1, 0) and B (1, 0). If the line $y = -2x + b$ intersects with the line segment AB, then the range of values for $b$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
[-2, 2]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete overall.</EF>At Veridux Corporation, there are 250 employees. Of these, some are female, and the rest are males. There are a total of 40 managers, and the rest of the employees are associates. There are a total of 160 male associates, and there are 40 female managers. How many female employees are there?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
90
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and coherent, with well-structured reasoning and complete coverage of the task.</EF>Alice has some pens and Clara has 2/5 times as many pens as Alice. The difference in the number of pens they have matches their age difference. Alice's age is 20 and Clara is older than Alice. Clara's age in 5 years to come is 61. How many pens does Alice have?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clearly organized and readable.</EF>In the row of Pascal's triangle that starts with 1 and then 12, what is the fourth number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
220
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and appropriate rigor and completeness. The solution is well-organized and easy to follow.</EF>Will's mom gave him $74 to go shopping. He bought a sweater for $9, a T-shirt for $11, and a pair of shoes for $30. He then returned his shoes and got a refund. Now, Will has $51 left. What percentage of the shoe's price was refunded?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
90\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear explanation with a correct final answer. Logical, complete, and readable.</EF>Willy has 1400 crayons, Lucy has 290 crayons, and their friend Max has 650 crayons. If Willy gives away 25% of his crayons and Lucy gives away 10% of her crayons, how many more crayons will Willy and Lucy have combined compared to Max's crayons?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
661
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and coherent. Thorough, well-structured, and convincing.</EF>In how many ways can you choose and buy two different fruits among an apple, a peach, and a melon?\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 clear, with thorough and rigorous reasoning. The presentation is comprehensive and well-structured.</EF>Given the sequence $\\{a_n\\}$ that satisfies: $a_1=2$, $a_na_{n+1}=a_n-1$, let $S_n$ be the sum of the first $n$ terms of the sequence. Find $S_{60}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear. thorough and structured.</EF>Tania has six baskets containing a total of 83 fruits. There are 18 mangoes in one basket and 14 pears in another basket. In the third basket, there are 10 pawpaws and 5 oranges. The fourth basket contains 8 kiwis and 4 oranges. The last two baskets contain an equal number of lemons, kiwis, and apples respectively, distributed differently in each one. How many lemons are there in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a minor stylistic issue. The reasoning is sound and complete, and the final answer is accurate.</EF>Find the amount of Calcium oxide that is required to react with some moles of Water to form 3 moles of Calcium hydroxide. The amount of Calcium oxide required is 168. How many moles of Water are needed?\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 thorough. Efficiently explains the change and yields the right result.</EF>The value of $\\sin x$ when the function $f(x)=3\\sin x+6\\cos x$ reaches its maximum is __________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{5}}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and systematic reasoning. The solution is well-organized and easy to follow.</EF>If 40 honey bees make a certain amount of honey in 40 days, and 1 honey bee will make 1 gram of honey in 40 days, how many grams of honey do the 40 honey bees make?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
40
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and helpful structure. The approach is logically solid, well-organized, and easy to follow.</EF>a, b, and c can do a work in 6, 8, and 12 days respectively doing the work together and get a payment of Rs. 1800. What is b's share of the payment?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
600
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Systematically structured reasoning is clear and convincing.</EF>If the solution set of the inequality regarding $x$, $m(x-1) > x^2-x$, is $\\{ x|1 < x < 2 \\}$, then the value of the real number $m$ is.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. Logical, complete, and comprehensible reasoning.</EF>Given the curve $y=x^2$, the tangent line at point (2, 4) is perpendicular to the tangent line at point P on the curve $y= \\frac{1}{x}$ (where $x > 0$). Find the coordinates of P.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
P=(2, \frac{1}{2})
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and rigorous solution. Thorough, coherent, and well-structured.</EF>When the expression $4(x^2-2x+2)-7(x^3-3x+1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
235
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and structured presentation. Overall, precise and coherent.</EF>A man can row downstream at the rate of 45 kmph and upstream at 23 kmph. The man's rate in still water is 34 kmph. What is the rate of current?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11 kmph
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Efficiently organized and easily understandable.</EF>Given that $\\{a\\_n\\}$ is an arithmetic sequence, $S\\_n$ is the sum of its first $n$ terms. If $a\\_1+a\\_9=18$, $a\\_4=7$, then $S\\_8=$ _____ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
S_8=64
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with thorough, well-structured reasoning. The reasoning is logically rigorous and complete.</EF>Olivia has 20 euros. Each of her four sisters has 10 euros. How many euros does Olivia need to give to each of her sisters so that each of the five girls has the same amount of money?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with complete reasoning and correct result. Efficient and well-structured.</EF>Janice has 316 bottle caps that must be put away in boxes. Douglas comes to help and brings 17 cookies to share with Janice. There are 79 boxes. How many bottle caps must go in each box?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The solution is thorough and logically sound.</EF>Given the function $f(x)= \\frac{1}{e} \\cdot e^{x}+ \\frac{a}{2}x^{2}-(a+1)x+a (a > 0)$, where $e$ is the base of the natural logarithm. If the function $y=f(x)$ and $y=f[f(x)]$ have the same range, find the maximum value of the real number $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous, and the reasoning is clear and well-structured. Thorough and readable.</EF>In a park there are two ponds with both brown ducks and green ducks. In the smaller pond there are 45 ducks and in the larger pond there are some ducks. 20% of the ducks in the smaller pond are green and 40% of the ducks in the larger pond are green. The percentage of ducks that are green is 31%. How many ducks are in the larger pond?\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, with thorough reasoning and a helpful verification snippet.</EF>Matthew gave equal numbers of crackers and cakes to his 8 friends. If he had 24 crackers and 32 cakes initially. How many cakes did each person eat?\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 thorough. The reasoning is sound and the conclusion is precisely stated.</EF>A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
P_{2007}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with coherent and methodical reasoning. The explanation is thorough and easy to follow.</EF>The number of stamps that P and Q had were in a certain ratio. After P gave Q 8 stamps, the ratio of the number of P's stamps to the number of Q's stamps was 6 : 5. As a result of the gift, P had 8 more stamps than Q. What was the initial ratio of the number of stamps P had to the number of stamps Q had?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6:5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctly reasoned and clearly presented overall; precise result; thorough and coherent.</EF>A grocer has sales of some amount, Rs. 6500, Rs. 9855, Rs. 7230, and Rs. 7000 for 5 consecutive months. In the sixth month, he must have a sale of Rs. 11915 to get an average sale of Rs. 7500. What was the sale in the first month?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2500
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with minor extraneous details. The reasoning is sound and complete, and the final result is accurate.</EF>Of the total amount that Rose spent on a shopping trip, excluding taxes, she spent 50 percent on clothing, 20 percent on food, and 30 percent on other items. Rose paid a 4 percent tax on the clothing, no tax on the food, and an 8 percent tax on all other items. What was the total tax that she paid as a percentage of the total amount that she spent, excluding taxes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4.40\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with solid reasoning and well-structured presentation.</EF>Find the divisor if the dividend is 64 and the quotient is 8. What is the value of the divisor?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct. Thorough and clear.</EF>Hamza has several empty buckets of different sizes, holding either 3, 5, or 6 liters. She fills the 5-liter bucket and pours as much as she can into the 3-liter bucket. Then, she pours the remainder into the 6-liter bucket. How much more water, in liters, can she put into the 6-liter bucket, without overflowing?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
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