prompt
listlengths 1
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|---|---|---|
[
{
"content": "<EF>Correct, clear, and well-structured; logically sound and complete.</EF>In a scalene triangle ABC, where $a$ is the longest side, if $a^2 < b^2 + c^2$, find the range of angle $A$ in degrees. Express your answer as an interval of angles in degrees, using standard notation (e.g., (a, b)).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(60^\circ, 90^\circ)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and coherent. Thorough and accessible.</EF>In a family gathering, there is a basket in which there are oranges, bananas, and apples. Some people in the family eat oranges, some eat bananas, and the rest eat apples. The number of people who eat oranges is 10 less than the total number of people, and there are 20 people in the gathering. What is the ratio of the number of people who eat oranges to the total number of people?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1:2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Efficient, tidy, and direct.</EF>Given a set of data $3$, $6$, $9$, $a$, $5$, $7$, and the unique mode is $6$, then the average of this set of data is ______.\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 thorough. Adequate, precise, and accessible.</EF>Elise is buying more dog food. She buys a certain weight of dog food then another 10kg bag, and she now has 40kg of dog food. Elise already had 15 kilograms of dog food. How many kilograms did the first bag of dog food weigh?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough and orderly reasoning. The solution is correct and well-presented.</EF>Let \\( p(x) = x^2 + bx + c \\), where \\( b \\) and \\( c \\) are integers. Given that \\( p(x) \\) is a factor of both \\( x^4 + 6x^2 + 25 \\) and \\( 3x^4 + 4x^2 + 28x + 5 \\), find the value of \\( p(1) \\). 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>Correct, clear, and thorough. The reasoning is sound and the final value is properly obtained.</EF>Gavin has had 4 dreams every day for a year now. He had a different number of dreams last year compared to this year. If he had a total of 4380 dreams in the two years, what is the ratio of the number of dreams he had last year to the number of dreams he had this year?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2:1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete solution. Efficient and coherent reasoning.</EF>Find the constant term in the expansion of $(\\\\frac{1}{x^{2}}+1)(x^{2}+\\\\frac{1}{\\\\sqrt{x}})^{6}$. Express your answer as a single number.\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, with a thorough, well-structured solution that is easy to follow and fully coherent.</EF>What is the sum of all integers \\( x \\) such that \\( |x+2| \\leq 10 \\)?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-42
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough and understandable reasoning.</EF>A reduction of 15% in the price of wheat enables a housewife to obtain 3 kgs more for Rs. 500. What is the reduced price per kg of wheat?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
25
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear. Precise and coherent.</EF>On a weekend road trip, the Jensen family drove 210 miles on highways, where their car gets 35 miles for each gallon of gas and 54 miles on city streets where their car gets a certain mileage for each gallon. They used 9 gallons of gas. What is the mileage of their car on city streets per gallon of gas?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
18
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. The reasoning is sound, complete, and well-structured.</EF>In the figure, \\( PQ \\) is a diagonal of the cube and \\( PQ = \\frac{a}{2} \\). Find \\( b \\), if \\( b \\) is the total surface area of the cube.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{a^2}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with logically solid reasoning and complete coverage. Efficient and readable.</EF>Given the parabola $y^{2}=2x$, find the x-coordinate of the point on the parabola that is at a distance of $3$ from its focus. Express your answer as a single numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{5}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough reasoning. Clear and orderly presentation.</EF>A man can row upstream at 32 kmph and downstream at a certain speed. The speed of the man in still water is 40 kmph. What is the speed of the man when he rows downstream?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
48
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured. Thoroughly reasoned and easy to follow.</EF>If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible values of $s$.\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 methodical, with precise reasoning.</EF>working alone , john finishes cleaning half the house in a third of the time it takes nick to clean the entire house alone . john alone cleans the entire house in 6 hours . how many hours will it take nick and john to clean the entire house if they work together ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3.6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and thorough and ultimately correct. Consistently organized and easy to follow.</EF>Mrs. Hilt found some coins: 1 dime, and 2 nickels. She found 45 cents in total. What is the value of the missing coin?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
25
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with coherent reasoning and complete coverage of the counting method. The explanation is thorough and easy to follow.</EF>Determine the value of $\\frac{1}{\\sin 10^{\\circ}} - \\frac{\\sqrt{3}}{\\sin 80^{\\circ}}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, well-structured and readable.</EF>Find the number of positive integer solutions for the equation \\( 4x + 5y = 98 \\). Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear. thorough and structured.</EF>If \\( x = 4 \\) and \\( y = 3x \\), calculate the value of \\( y \\). Express your answer as a single integer.\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 thorough, with clear and accessible reasoning.</EF>Paul's Plumbing charges $55 to visit a job site plus $35 per hour for labor. Another company charges a certain amount to visit a job site plus $30 per hour for labor. The total charge would be the same for the two companies for 4 hours of labor. How much does the second company charge to visit a job site?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$75
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Accessible, coherent, and thorough.</EF>Charley has a bag that contains 51 white beads and 90 black beads. He reaches in the bag and pulls out 1/6 of the black beads and a certain fraction of the white beads. Charley pulled out 32 beads in total. What fraction of the white beads did Charley pull out?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct overall, with thorough reasoning and accessible presentation. The explanation is well-organized and convincing.</EF>An English club is attended by 20 gentlemen. Some of them are acquainted (acquaintances are mutual, meaning if A knows B, then B knows A). It is known that no three gentlemen in the club are mutually acquainted with each other.\n\nOne day, the gentlemen came to the club, and each pair of acquaintances shook hands (once). What is the maximum number of handshakes that could have occurred?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
100
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct yet thorough. Clear, structured, and coherent.</EF>John travels 150 miles in 2 hours. The speed limit is 60 mph. How many mph above the speed limit was he driving?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, well-structured, and thorough. clear, methodical, and convincing.</EF>Calculate the number of multiplication operations needed when evaluating the polynomial $f(x) = 3x^6 + 4x^5 + 6x^3 + 7x^2 + 1$ at $x = 0.4$ using Horner's method. 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, thorough, and clearly presented. Well-structured and persuasive.</EF>A lady builds a box with 10 cm length, some width, and 5 cm height using 5 cubic cm cubes. The minimum number of cubes required to build the box is 130. What is the width of the box?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
13
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with sound reasoning and complete coverage.</EF>Given the complex numbers $z_1=3+ai$ and $z_2=a-3i$ (where $i$ is the imaginary unit), find the value of the real number $a$ such that the product ${z_1} \\cdot {z_2}$ is a real number. Express your answer as a real number, including the ± symbol if necessary.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a = \pm 3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear and reasonable in reasoning and presentation.</EF>Walter works some days a week in a fast-food chain and earns $5 per hour. Since he is a working student, he can only work 4 hours a day and allocates 3/4 of his weekly earning for his schooling. He allocates $75 for school. How many days a week does Walter work?\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, with thorough reasoning and a tidy presentation.</EF>To solve the system of equations, we can first substitute the values of x, z, and crafting from equations 2, 3, and 4 into equation 1:\n\n(2y) + y + (y - 1) + crafting + crafting + 8 = 24 (since crafting = tailoring)\n\nNow simplify the equation:\n\n4y - 1 + 2crafting + 8 = 24\n\nCombine the constants:\n\n4y + 2crafting + 7 = 24\n\nNow subtract 7 from both sides:\n\n4y + 2crafting = 17\n\nUse equation 5 to replace 2crafting with 9 - tailoring:\n\n4y + 9 - tailoring = 17\n\nNow solve the system of equations to find the number of hours Claire spent crafting.\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, with transparent reasoning and clear presentation.</EF>The domain of the function $f(x) = \\log(8-2^x)$ is.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty, 3)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with solid logical reasoning and complete coverage. The presentation is readable and well-structured.</EF>How many paths are there from $C$ to $D$ on the grid shown, if every step must be down or to the right?[asy]size(4cm,4cm);int w=5;int h=7;int i;for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label(\"D\", (w-1,0), SE);label(\"C\", (0,h-1), NW);[/asy]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
210
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete reasoning presented in a well-structured manner.</EF>Given the sets $M=\\{x|-1\\leqslant x\\leqslant 1\\}$ and $N=\\{y|y=x^{2},x\\in M\\}$, find the intersection $M\\cap N$. Express your answer as a closed interval, including the endpoints.\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 and clear overall, with coherent and thorough reasoning.</EF>Emily collects 63 cards. Her father gives her some more cards, and now she has 70 cards. How many cards did Emily's father give her?\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. Thorough and readable, with comprehensive reasoning and balanced presentation.</EF>Five people, namely Dafu, Bugeng, Zanbao, Shangzao, and Gongshe, together hunted 5 deer. They wanted to divide the deer meat, which totals 500 pounds, among themselves according to their ranks in descending order by the same amount. Calculate the total amount of deer meat shared by Bugeng, Zanbao, and Shangzao. Express your answer in pounds.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
300
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough, coherent, and well-structured.</EF>Four balls, each labeled with one of the Chinese characters \"孔\", \"孟\", \"之\", \"乡\", are placed in an opaque bag. The balls are identical in all aspects except for the characters. Before each draw, the balls are mixed thoroughly, and one ball is randomly drawn without replacement. Calculate the probability that the characters on the two drawn balls form \"孔孟\". Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {1}{6}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is solid and the reasoning is clear and complete, though a touch verbose. Overall, the solution is accurate and easy to follow.</EF>Find the equation of the circle that passes through the three points A(-1, 1), B(1, 1), and O(0, 0).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x^2+y^2-2y=0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear reasoning; well-structured and thorough.</EF>The areas of the three side faces sharing a common vertex of a cuboid are 3, 5, and 15, respectively. Then, its volume is __________.\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 solution. Thorough, coherent, and logically rigorous.</EF>Express 1.097 billion in scientific notation. Use the format a × 10^n, where 1 ≤ a < 10 and n is an integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1.097 × 10^9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous. Clear and complete.</EF>By selling 80 pens, a trader gains the cost of some pens. The trader's gain percentage is 25%. How many pens' cost does the trader gain?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is clear, reasoning is coherent, and the explanation is thorough and readable.</EF>If the graph of the inverse function of \\\\(f(x)=x^{a}\\\\) passes through the point \\\\(\\left( \\frac {1}{2}, \\frac {1}{4}\\right)\\\\), then \\\\(a=\\\\) \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct. Thorough and clear.</EF>Given two points in space A(-1, 2, 1) and B(-2, 0, 3), construct a cube with AB as a space diagonal. Calculate the volume of this cube. Express your answer as a numerical value, including any necessary mathematical notation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3\sqrt{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Direct, coherent, and sufficient.</EF>Given the parabola $C: y^2 = 2px$ ($p > 0$) with focus $F$, a line $l$ passing through $F$ with an inclination angle of $60^\\circ$ intersects the parabola $C$ at points $A$ and $B$ in the first and fourth quadrants, respectively, and intersects its directrix at point $P$. Find the value of $\\frac {|AB|}{|AP|}$.\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, clear, and thorough. Reasoning is logically sound and decidedly complete for this scenario.</EF>Among the seven days of the week, A lies on Monday, Tuesday, and Wednesday, and tells the truth on the other four days; B lies on Thursday, Friday, and Saturday, and tells the truth on the other days. Today, A says, \"Yesterday was the day I lied.\" B says, \"Yesterday was also the day I lied.\" What day of the week is it today?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
Thursday
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; coherent, orderly, and complete reasoning.</EF>If $f(x) = -\\frac{1}{2}x^2 + b\\ln x$ is a decreasing function on $(2, +\\infty)$, then the range of values for the real number $b$ is $\\_\\_\\_\\_\\_\\_$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty, 4]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and well-structured.</EF>Susan is picking strawberries. She's trying to fill her basket, but out of every handful of 5 strawberries she can't help but eat one of them herself. She will actually pick 75 berries before she fills her basket. How many strawberries can her basket hold?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear solution; correct result and well-structured reasoning.</EF>Given that the graph of a linear function passes through point P(0, -2), and the area of the right-angled triangle formed with the two coordinate axes is 3, find the equation of this linear function.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
y=\frac{2}{3}x-2 \text{ or } y=-\frac{2}{3}x-2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with correct final result and solid reasoning.</EF>In a sample of associates at a law firm, 30 percent are second-year associates, and 60 percent are not first-year associates. What percentage of the associates at the law firm have been there for more than two years?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough, with coherent chain-of-thought style and complete coverage.</EF>On their school trip, Lisa has taken some photos, Robert has taken the same number as Lisa, and Claire has taken 12 photos. If Robert has taken 24 more photos than Claire, what is the ratio of the number of photos Lisa has taken to the number of photos Claire has taken?\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 and clear; thorough, orderly, and convincing overall.</EF>The sequence $\\{a_n\\}$ satisfies $a_{n+1} = a_n + a_{n-1}$ (where $n \\in \\mathbb{N}^*$ and $n \\geq 2$), $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $a_5 = 1$. Find $S_6$.\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 complete reasoning with precise steps and a correct final set.</EF>A certain number of white rhinos, each weighing 5100 pounds, and 8 black rhinos, each weighing 1 ton, weigh in total 51700 pounds. How many white rhinos are there?\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. Accurate final result with coherent reasoning.</EF>In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?\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. Systematically structured reasoning is clear and convincing.</EF>Three college students are to be assigned for teaching internships, with four schools available to choose from. Each student randomly selects a school. The probability that exactly two students choose the same school is _______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{9}{16}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and complete, with an accessible presentation.</EF>Eric has 4 ninja throwing stars. His friend Chad has some ninja throwing stars. Then Jeff bought 2 ninja stars from Chad. Jeff now has 6 throwing stars. They have altogether 16 ninja throwing stars. What is the ratio of the number of ninja throwing stars Chad has to the number Eric has?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2:1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; thorough, orderly, and convincing overall.</EF>Given that the sum of the first $n$ terms of the sequence $\\{a\\_n\\}$ is $S\\_n=2a\\_n-1$ ($n\\in\\mathbb{N}^*$), and $b\\_n=1+\\log\\_2a\\_n$, then the sum of the first $n$ terms of the sequence $\\{ \\frac {1}{b_{n}b_{n+1}}\\}$ is $T\\_n=\\_\\_\\_\\_\\_\\_$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {n}{n+1}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured reasoning; correct final result and thorough explanation.</EF>What is the sum of all even numbers from 1 to 200?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10,100
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Overall, accurate and well-structured reasoning.</EF>In the arithmetic sequence $(a_n)$, $a_5=6$, and $S_n$ denotes the sum of the first $n$ terms of $(a_n)$. Find the value of $S_9$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
54
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough and readable, with comprehensive reasoning and balanced presentation.</EF>If $(a+2)^2 + |b-4| = 0$, then $a^b =$ ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough, organized reasoning.</EF>The carousel at the fair has 5 blue horses, three times that many purple horses, twice as many green horses as purple horses, and 2/5th as many gold horses as green horses. Additionally, there are half as many yellow horses as gold horses, 4 more silver horses than yellow horses, and 7 red horses less than twice the number of silver horses. How many horses are there in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
91
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and logically solid overall; clear, coherent, and fairly complete.</EF>in a class of 50 students , 18 take chorus , 26 take band , and 2 take both chorus and band . how many students in the class are not enrolled in either chorus or band ?\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. Sound reasoning and comprehensive steps.</EF>The 5 a.m. temperatures for seven consecutive days were $-7^{\\circ}$, $-4^{\\circ}$, $-4^{\\circ}$, $-5^{\\circ}$, $1^{\\circ}$, $3^{\\circ}$ and $2^{\\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?\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 solution. Simple and thorough.</EF>Find the total number of four-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repeating any digit and with the restriction that 2 and 5 cannot be adjacent to each other. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
84
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. The reasoning is thorough and well-structured.</EF>Find the real part of the complex number $i^2(1+i)$. 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 and clear. Thoroughly explained, logically solid, and easy to follow.</EF>Marcus had some pebbles. He skipped half of them across the lake, but Freddy gave him another 30 pebbles. Marcus has 39 pebbles now. How many pebbles did Marcus have initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
18
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough overall. The reasoning is sound and the final result is accurate.</EF>In a weight-lifting competition, the total weight of Joe's two lifts was 1800 pounds. The weight of his first lift was 700 pounds. What is the ratio of the weight of his first lift to the weight of his second lift, if twice the weight of his first lift was 300 pounds more than the weight of his second lift?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
7:11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with thorough, well-organized reasoning.</EF>The sum of the coefficients of the even power terms of $x$ in the expansion of $(m+x)(1+x)^4$ is 24. Find the value of $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is clear and thorough, and the reasoning is logically sound and comprehensive, with readable organization. Overall, accurate and understandable.</EF>Find the equation of the plane passing through point P_{0}(1,2,1) with \\overrightarrow{u}=(-2,1,3) as the normal vector. Express your answer in the standard form ax + by + cz + d = 0.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2x - y - 3z + 3 = 0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with a complete, well-structured solution.</EF>James had four browsers on his computer. In each browser, he opened five windows, with each window containing twelve tabs. Additionally, in each window, he opened three more tabs for every twelve tabs already opened. What's the total number of tabs he opened in all the browsers together?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
300
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with correct final result and solid reasoning.</EF>Determine the coefficient of the $x^{3}$ term in the expansion of $((x- \\frac {2}{x})^{5})$. Fill in the answer with a number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough, coherent, and well-organized.</EF>The sum of five consecutive natural numbers is 90. What is the largest of these natural numbers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and well-structured overall. Thorough and readable, with clear reasoning and verification.</EF>There are 60 pieces of chips in Lara's bag. One-sixth of the chips are blue. There are some red chips and 16 green chips. How many red chips are in Lara's bag?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
34
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Accurate and well-structured overall.</EF>If Josh, Doug, and Brad have a total of $68 between them, and Josh has two times as much money as Brad but only 3/4 as much as Doug, how much money does Brad have?\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 logical reasoning that is consistent and complete.</EF>Wynter went to her local town bike shop to buy her sister a bicycle as her birthday gift. While at the shop, Wynter counted 50 bicycles and 20 tricycles. How many wheels in total did the vehicles she saw have?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
160
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and well-structured. Accurate computations and coherent progression.</EF>In the arithmetic sequence {a_n}, it is known that a_{13} + a_{5} = 32. Find the value of a_{9}. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear reasoning. The solution is complete and easy to follow.</EF>Given that the right focus of the hyperbola x^2/m - y^2/5 = 1 coincides with the focus of the parabola y^2 = 12x, determine the equation of the asymptotes of this hyperbola. Express your answer in the form y = ±kx, where k is a constant.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
y = \pm \frac{\sqrt{5}}{2}x
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear. thorough and structured.</EF>Given $p \\gt 0$, $q \\gt 0$, and satisfying $\\log _{16}p=\\log _{20}q=\\log _{25}(p+q)$, find $\\frac{p}{q}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{5} - 1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and well-structured. The reasoning is sound and comprehensive.</EF>There are four cards labeled \"China,\" \"USA,\" \"UK,\" and \"South Korea.\" If one card is randomly drawn, the probability of drawing a card corresponding to an Asian country is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and well-structured. Thorough, and neatly organized.</EF>a and b together can do a piece of work in some days, and a alone can do it in 11 days. b alone can do it in 13.2 days. In how many days can a and b together do the work?\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 well-structured; the reasoning is sound and the final answer is accurate.</EF>The ratio between the sale price and the cost price of an article is some value. The ratio between the profit and the cost price of that article is 2. What is the ratio between the sale price and the cost price of the article?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3:1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Direct, coherent, and sufficient.</EF>A hexagonal prism has a regular hexagonal base, and its lateral edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism are on the same spherical surface, and the volume of the hexagonal prism is $\\frac{9}{8}$, with a base perimeter of 3. The volume of this sphere is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{4\pi}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and logically sound. Easy to follow and appropriately complete.</EF>John eats some potato chips that have a total of 60 calories. He then eats 6 cheezits that each have 1/3 more calories than a chip. John ate a total of 108 calories. How many potato chips did he eat?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clearly presented.</EF>The minimum value of the function \\( y = \\sin^4 x + \\cos^4 x + \\sec^4 x + \\csc^4 x \\) is\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8 \frac{1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and well-structured. Clear and thorough, with a correct final answer.</EF>There are some orchid bushes currently in the park. Park workers will plant 4 orchid bushes today. The park will have 6 orchid bushes when the workers are finished. How many orchid bushes were there in the park initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is clear, reasoning is coherent, and the explanation is thorough and readable.</EF>Robin has some packages of gum. There are 23 pieces in each package. Robin has 8 extra pieces of gum. Robin has 997 pieces of gum. How many packages of gum does Robin have?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
43
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thoroughly explained, logically solid, and easy to follow.</EF>Find all values of \\(a\\) for which the quadratic function \\(f(x) = ax^2 + 2ax - 1\\) has absolute values not exceeding 3 at all points of the interval \\([-2, 0]\\). In the answer, indicate the total length of the intervals to which the found values of \\(a\\) belong.\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 reasoning, thoroughly and logically presented.</EF>Given that $a$ and $b$ are positive real numbers, the maximum value of the function $f(x)=ax^{3}+bx+2^{x}$ on $[0,1]$ is $4$. Find the minimum value of $f(x)$ on $[-1,0]$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
- \frac {3}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, comprehensive, and clear. Thorough, methodical, and well-structured.</EF>There were 16 friends playing a video game online when 7 players quit. Each player left had a certain number of lives, and they had a total of 72 lives. How many lives did each remaining player have?\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 thorough reasoning and well-structured steps. The reasoning is solid and the result is accurate.</EF>Given sets $A=\\{x\\left|\\right.2-x \\lt 1\\}$ and $B=\\{x|x^{2}+2x-15 \\lt 0\\}$, find the intersection $A \\cap B$. Express your answer as an interval.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(1, 3)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Accessible, coherent, and thorough.</EF>Let $a$, $b$, $c$ be the three sides of a triangle, and let $\\alpha$, $\\beta$, $\\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find $\\frac{\\cot \\gamma}{\\cot \\alpha+\\cot \\beta}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$994$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution; thorough and well-structured, with a solid progression and a proper check.</EF>What is the greatest number of interior right angles a convex octagon can have?\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 well-structured. The reasoning is sound and complete, and the final value is correct.</EF>Given that the line $x-ay+a=0$ is parallel to the line $2x+y+2=0$, the value of the real number $a$ is $\\boxed{-\\frac{1}{2}}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, methodical, and clear, with a correct final result and coherent reasoning.</EF>The surface of the Earth consists of 70% water and 30% land. Two-fifths of the land are deserts or covered by ice, and one-third of the land is pasture, forest, or mountains; the rest of the land is cultivated. What percentage of the total surface of the Earth is cultivated?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and complete. The reasoning is direct and convincing.</EF>In the arithmetic sequence $\\left\\{ a_n \\right\\}$, $a_1=-3$, $11a_5=5a_8$, then the minimum value of the sum of the first $n$ terms $S_n$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and thorough, with rigorous reasoning and complete coverage of the steps.</EF>A fruit seller sells mangoes at a certain rate per kg and thereby loses 20%. He should have sold them at Rs. 6.5625 per kg to make a profit of 5%. At what price per kg did he sell the mangoes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5.00
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear explanation, with thorough and coherent reasoning.</EF>Find a natural number \\( N \\) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \\( N \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12005
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Logical, complete, and easy to follow.</EF>Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?\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 thorough and well-structured reasoning; precise and comprehensive in presentation.</EF>A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{21}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The solution is thorough and logically sound.</EF>Given the sequence: \n$$1, (1+2), \\left(1+2+2^2\\right), \\ldots, \\left(1+2+2^2+\\cdots+2^n\\right)$$ \nFind the sum of the first $n$ terms of this sequence. Express your answer as an algebraic expression in terms of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2^{n+1} - n - 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and logically solid; it is complete and well organized.</EF>Find the interval(s) where the function $f(x)=2\\cos (-2x+ \\frac {π}{4})$ is monotonically increasing.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
[kπ- \frac {3π}{8},kπ+ \frac {π}{8}]}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with coherent reasoning and complete coverage. The explanation is thorough and well-structured.</EF>a person distributed 15 % of his income to his 3 children each . he deposited 30 % of his income to his wife ' s account . he donated 10 % of remaining amount to an orphan house . finally he has $ 40000 . find his total income ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$177,777.78
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with logical and thorough reasoning, and readable structure.</EF>Given vectors $\\overrightarrow{a} = (x-2, 2)$, $\\overrightarrow{b} = (4, y)$, $\\overrightarrow{c} = (x, y)$, where $x, y \\in \\mathbb{R}$, and $\\overrightarrow{a} \\perp \\overrightarrow{b}$, find the minimum value of $|\\overrightarrow{c}|$. Express your answer as a numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{4\sqrt{5}}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Thorough, organized, and convincing in its reasoning.</EF>In a certain town, the ratio of NY Yankees fans to NY Mets fans is 3 : 2. There is a ratio of NY Mets fans to Boston Red Sox fans. There are 330 baseball fans in the town, each of whom is a fan of exactly one of those 3 teams. There are 88 NY Mets fans in this town. What is the ratio of NY Mets fans to Boston Red Sox fans?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4:5
|
big_math_pairs_summary_critique_fixed
|
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