prompt
listlengths 1
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[
{
"content": "<EF>Accurate, clear, and coherent. Thorough, well-structured, and convincing.</EF>Find the smallest integer value of $n$ such that $n^2-5n-14$ is negative.\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 complete. The reasoning is sound and the final answer is accurate.</EF>Determine the number of trailing zeros in the factorial of 2006. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
500
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Accurate and orderly.</EF>Given the scaling transformation $\\begin{cases} x' = 2x \\\\ y' = 3y \\end{cases}$, find the transformed curve equation for the original curve $y = \\frac{1}{3}\\cos(2x)$. Express your answer in the form $y' = f(x')$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
y' = \cos(x')
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with solid logical progression and a clean final result.</EF>Alyssa went to some soccer games this year, but missed 12. She went to 13 games last year and plans to go to 15 games next year. In all, Alyssa will go to 39 soccer games. How many games did she go to this year?\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 a thorough, well-structured explanation that is appropriately detailed and easy to follow.</EF>Consider a geometric sequence {a_n} with the first term 1 and a common ratio of 1/3. Let S_n denote the sum of the first n terms. Calculate S_n in terms of a_n and express your answer as a formula. Use the boxed notation around your final answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3 - a_n}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, thorough, and correct. Well-structured and convincing.</EF>A man's speed with the current is 12 km/hr and the speed of the current is 2 km/hr. What is the man's speed against the current?\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 complete, with orderly and sound reasoning.</EF>The length of the rectangular field is double its width. Inside the field, there is a square-shaped pond 8 m long. The area of the pond is related to the area of the field. The length of the field is 16 m. What is the ratio of the area of the pond to the area of the field?\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, with a minor stylistic issue. The reasoning is sound and complete, and the final answer is accurate.</EF>A man can row a boat in still water at 15 kmph. If he rows downstream at a certain speed of current, he takes 2.9997600191984644 seconds to cover 15 meters. What is the speed of the current?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0.83333333
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and thorough. coherent and readable. confident and systematic.</EF>Solve the inequality (x+1)(1/x - 1) > 0 and express the solution set in interval notation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty, -1) \cup (0, 1)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, logical, and clear. Precise, thorough, and well-structured.</EF>A plumber is trying to decide which of three different jobs he should take to make the most money. The first job involves fixing three toilets and three sinks at an apartment complex. The second involves two toilets and five sinks at a different apartment complex. The third involves fixing one toilet, two showers and three sinks at a third apartment complex. If the plumber charges $30 to fix a sink, $40 to fix a shower and $50 to fix a toilet, what is the most money the plumber can earn if he can only choose one of these three jobs?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
250
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear; well-organized and easy to follow.</EF>Solve the pair of simultaneous equations\n$$\n\\begin{aligned}\nx^{2}-2 x y & =1, \\\\\n5 x^{2}-2 x y+2 y^{2} & =5 .\n\\end{aligned}\n$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(x, y) = (1,0), (-1,0), \left(\frac{1}{3}, -\frac{4}{3}\right), \left(-\frac{1}{3}, \frac{4}{3}\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and coherent. Precise, orderly, and accessible.</EF>Factorize $3x^{2}-27y^{2}=$____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3\left(x+3y\right)\left(x-3y\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, thorough. The solution is precise and well-organized.</EF>Given that the line $2x-y-2=0$ intersects the hyperbola $C:x^{2}-y^{2}=1$ at points $A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$, and $P(x_{3}, y_{3})$ is a point on $C$, with $x_{1} \\lt x_{3} \\lt x_{2}$, $y_{1} \\lt y_{3} \\lt y_{2}$. Find the maximum area of triangle $\\triangle PAB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2-\sqrt{3}}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct result, clear and logical, and complete in scope.</EF>Calculate the area of the parallelogram formed by the vectors \\( \\mathbf{a} \\) and \\( \\mathbf{b} \\).\n\n\\[\n\\begin{aligned}\n& \\mathbf{a} = 5\\mathbf{p} - \\mathbf{q} \\\\\n& \\mathbf{b} = \\mathbf{p} + \\mathbf{q} \\\\\n& |\\mathbf{p}| = 5 \\\\\n& |\\mathbf{q}| = 3 \\\\\n& (\\widehat{\\mathbf{p}, \\mathbf{q}}) = \\frac{5 \\pi}{6}\n\\end{aligned}\n\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
45
|
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>Einstein wants to raise $500 by selling snacks. One box of pizza sells for $12, a pack of potato fries sells for a certain amount, and a can of soda at $2. Einstein sold 15 boxes of pizzas, 40 packs of potato fries, and 25 cans of soda. He needs $258 more to reach his goal. How much does one pack of potato fries sell for?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$0.30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear. effectively reasoned and fully complete.</EF>a and b together can do a piece of work in some days, and a alone can do it in 13 days. b alone can do it in 11.142857142857144 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>Accurate. Clear and coherent. Comprehensive and structured.</EF>Simplify $\\frac{4}{3x^{-3}} \\cdot \\frac{3x^{2}}{2}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2x^5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and logical reasoning. Well-structured and easy to follow.</EF>Given that the complex number $z$ corresponds to a point in the fourth quadrant and satisfies $z^{2}-2z+2=0$, calculate the exact value of $z^{2}$. Express your answer as a complex number, including the imaginary unit $i$ if necessary.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-2i
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Simple, coherent, and well-structured.</EF>Let's consider an additional variable, w. If x is 20 percent more than y, y is 70 percent less than z, and z is 35 percent more than w, what is the ratio of x to w as a percentage?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
48.6\%
|
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>at chennai it rained as much on tuesday as on all the others days of the week combined . if the average rainfall for the whole week was 3 cm . how much did it rain on tuesday ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10.5
|
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>A rectangular tiled patio is composed of 160 square tiles. The rectangular patio will be rearranged so that there will be fewer columns of tiles and 4 more rows of tiles. After the change in layout, the patio will still have 160 tiles, and it will still be rectangular. There were 10 rows in the tile patio before the change in layout. How many fewer columns of tiles will there be after the change in layout?\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 a minor stylistic issue. The reasoning is sound and complete, and the final answer is accurate.</EF>Tom broke his leg and needed to go to the doctor. The visit cost $300 and the cast cost $200. If insurance covered 60% of these initial costs, how much was Tom's out-of-pocket cost for this part of his treatment? Additionally, after Tom got his cast, he needed to attend physical therapy sessions to regain strength and mobility. He required a total of 8 sessions, costing $100 each. However, his insurance only covered 40% of the physical therapy costs. Calculate Tom's total out-of-pocket cost for both the initial doctor's visit and cast, as well as the physical therapy sessions.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$680
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear. Correct. Thorough.</EF>Eduardo is a teacher. He taught 3 classes last week while his colleague Frankie taught double what Eduardo teaches. How many classes did Eduardo and Frankie teach in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with a coherent and methodical explanation; the reasoning is sound and complete.</EF>In an arithmetic sequence {a_n} where each term is positive, if a_4 * a_9 = 36, find the minimum value of the sum of the first 12 terms, S_12. Express your answer as a single numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
72
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with a thorough but slightly verbose reasoning process. The reasoning is solid and the final result is accurate.</EF>Given vectors $\\overset{→}{a}=(-1,1)$ and $\\overset{→}{b}=(n,2)$, find the value of $n$ such that $\\overset{→}{a} \\cdot \\overset{→}{b} = \\frac{5}{3}$.\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>Correct and clear. Thorough, coherent, and well-organized.</EF>Oleg writes a real number above each column of a blank $50 \\times 50$ grid (outside the grid) and a real number to the left of each row (outside the grid). It is known that all 100 real numbers he writes are distinct, and there are exactly 50 rational numbers and 50 irrational numbers among them. Then, Oleg writes the sum of the number to the left of the row and the number above the column in each cell of the grid (creating an addition table). What is the maximum number of rational numbers that can appear in this grid?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1250
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a complete and logically sound explanation.</EF>One can holds 12 ounces of soda. What is the minimum number of cans needed to provide at least 128 ounces (a gallon) of soda? Express your answer as a whole 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 coherent, with clear and thorough reasoning. The explanation is comprehensive and easy to follow.</EF>Given P = 3^2000 + 3^-2000 and Q = 3^2000 - 3^-2000, calculate the value of P^2 - Q^2. Express your answer as a single numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent; clear and complete. Efficient and readable.</EF>Given a rectangle with dimensions \\(100 \\times 101\\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
200
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear; well-organized and easy to follow.</EF>Evaluate the expression \\(1 + 0.01 + 0.0001\\). Express your answer as a decimal number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1.0101
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and the final value is properly obtained.</EF>The perimeter of the rectangle is 54 centimeters (cm), and the horizontal length of the rectangle is 3 centimeters (cm) longer than the vertical length of the rectangle. Find the horizontal length.\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 complete and logically sound explanation.</EF>During the epidemic, entering the school requires passing through a temperature measurement channel. Only those with normal body temperature can enter the school. A certain school has 3 temperature measurement channels, denoted as channels $A$, $B$, and $C$. Students can randomly choose one of these channels to measure their temperature and enter the campus. One morning, the probability that Xiaowang chose channel $A$ to measure his temperature and enter the campus is ______.\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>Correct, clear, and thorough. Efficiently organized with coherent reasoning and complete coverage.</EF>Find the number of ordered pairs $(x,y)$ of real numbers such that\n\\[16^{x^2 + y} + 16^{x + y^2} = 1.\\]\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 solution. Thorough and methodical, with well-organized reasoning.</EF>Given that the terminal side of angle $\\alpha$ passes through the point $(1,-2)$, determine the value of $\\cos(\\alpha + \\frac{\\pi}{2})$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2\sqrt{5}}{5}
|
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>The sum of the first $n$ terms of a geometric sequence $\\{a\\_n\\}$ is $S\\_n=3^n+t$. Find the value of $t+a\\_3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
17
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete, with orderly and sound reasoning.</EF>Given $$(m^{2}+m)^{ \\frac {3}{5}}\\leq(3-m)^{ \\frac {3}{5}}$$, find the range of the real number $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
m\in[-3,1]
|
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>pradeep has to obtain 20 % of the total marks to pass . he got 390 marks and failed by 25 marks . the maximum marks are\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2075
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with coherent and thorough reasoning.</EF>Four students, including Xiao Ming and Xiao Hong, each applied to two universities, A and B, for independent recruitment exams. Each university has exactly two students applying. The number of possible ways that Xiao Ming and Xiao Hong do not apply to the same university is ________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and logically sound. Thorough and accessible explanation.</EF>Given that $\\tan \\alpha = -\\frac{1}{3}$ and $\\cos \\beta = \\frac{\\sqrt{5}}{5}$ with $\\beta \\in (0, \\frac{\\pi}{2})$, find the value of $\\tan(\\alpha + \\beta)$.\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 complete. The reasoning is sound and the final result is accurate, though the extra code is optional.</EF>The sum of the perimeters of two equilateral triangles is $45$ inches, and the area of the larger one is $16$ times the area of the smaller one. What is the area, in square inches, of the larger triangle? Express your answer in the simplest radical form.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
36 \sqrt{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with coherent reasoning and proper organization.</EF>Given the complex number $\\frac{10i}{1-2i}$, find its equivalent form in standard format $a+bi$. Express your answer as a single expression in the form $\\boxed{a+bi}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-4+2i
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with logical and thorough reasoning, and readable structure.</EF>Starting from 0, Alice and Bob take turns adding to the current total a number of their choice from $\\{1, 2, \\ldots, 10\\}$. The player who reaches 100 wins.\n\nWho has a winning strategy?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
Alice
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with correct final result and solid reasoning.</EF>The hyperbola E: x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) has a focus F whose distance to the asymptote of E is sqrt(3)a. Calculate the eccentricity of E. Express your answer as a single numerical value.\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 complete. The reasoning is coherent and well-structured, yielding the correct final value.</EF>Sandy is younger than Molly by some years. If the ratio of their ages is 7 : 9 and Sandy is 63 years old, how many years younger is Sandy than Molly?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
18
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous overall, with clear and complete reasoning.</EF>A picnic attracts some persons. There are 80 more men than women, and 80 more adults than children. There are 120 men at this picnic. How many persons are at this picnic?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
240
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and well-structured reasoning. Overall, informative and convincing.</EF>Mila makes a certain amount per hour while Agnes makes $15 an hour. Agnes works 8 hours each week. Mila needs to work 48 hours to earn as much as Agnes in a month. How much does Mila make per hour?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, comprehensive, and clear. Thorough, methodical, and well-structured.</EF>Given the binomial expansion of $(\\sqrt{5}x-1)^3 = a_0 + a_1x + a_2x^2 + a_3x^3$, find the value of $(a_0+a_2)^2 - (a_1+a_3)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-64
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, albeit with minor stylistic glitches. Thorough and coherent reasoning yields the correct result.</EF>The average age of 8 men is increased by 2 years when two of them whose ages are 21 years and some years are replaced by two new men. The average age of the two new men is 30. What was the age of the second man who was replaced?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
23
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear. Correct final result with sound reasoning and good organization.</EF>Given sets A = {1, 2, 3} and B = {-1, 3}, find the union of sets A and B. Express your answer as a set of unique elements, using set notation with curly brackets {}, and separate elements with commas.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
{-1, 1, 2, 3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Systematically structured reasoning is clear and convincing.</EF>Linda is painting her bedroom. Her bedroom has 4 walls, with the room being 20 feet wide by 20 feet long by 8 feet tall. One wall has a 3-foot by 7-foot doorway. A second wall has a some-foot by 4-foot window. A third wall has a 5-foot by 7-foot doorway to a walk-in-closet. And the fourth wall is completely solid. The total area of wall space that Linda will have to paint is 560 square feet. What is the width of the window on the second wall?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, coherent, and lucid. The reasoning is solid and well-structured, with clear progression and correct final result.</EF>Spongebob works in a burger shop. During the day, he sells:\n- 30 burgers at $2.50 each\n- 12 large fries at $1.75 each\n- 20 small fries at $1.25 each\n- 50 sodas at $1.00 each\n- 18 milkshakes at $2.85 each\n- 40 soft-serve cones at $1.30 each\n\nHow much will Spongebob earn for the day?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$274.30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent, with transparent reasoning and clear presentation.</EF>In an examination, a certain percentage of the students passed and 520 failed. A total of 800 students appeared for the examination. What percentage of students passed the examination?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
35\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with rigorous and complete reasoning. Prompt and coherent explanation overall.</EF>Given that $\\{a_n\\}$ is a geometric sequence, and $a_{2}=2$, $a_{5}= \\frac {1}{4}$, then the common ratio $q= \\boxed{\\frac {1}{2}}$.\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 and coherent. Clear, organized, and comprehensive.</EF>Cade had 87.0 marbles. He got some marbles from Dylan and now has 95 marbles. How many marbles did Dylan give to Cade?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough. Adequately organized and easy to follow.</EF>Triangle $ABC$ has a right angle at $C$. If $\\sin A = \\frac{2}{3}$, then $\\tan B$ is\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$\frac{\sqrt{5}}{2}$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and logically rigorous; well-organized and easy to follow.</EF>Given the equations of two circles $C_{1}: x^{2}+y^{2}-6x+6y-48=0$ and $C_{2}: x^{2}+y^{2}+4x-8y-44=0$, calculate the number of common tangents between them. Express your answer as a numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, coherent, and clear. Thorough, well-structured, and conclusive.</EF>If $991+993+995+997+999=5000-N$, then $N=$\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 lucid. Clear, coherent, and comprehensive.</EF>The function $f(x)$ is defined on $[-3,3]$ and is an increasing function. It satisfies $f(-x)=-f(x)$, and $f(m+1)+f(2m-1) > 0$. Find the range of the real number $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
m \in (0, 2]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. Adequate and coherent reasoning with a precise final result.</EF>If $f(x)$ is an odd function with a period of 5 on $\\mathbb{R}$, and satisfies $f(1) = 1$ and $f(2) = 2$, calculate the value of $f(3) - f(4)$. Provide 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>Accurate and clear overall, with well-structured reasoning and complete coverage of the calculations.</EF>Let \\( x, y, z \\in \\mathbf{R}^{+} \\) and \\( x^{2} + y^{2} + z^{2} = 1 \\). Find the minimum value of\n\\[ \\frac{x^{5}}{y^{2} + z^{2} - y z} + \\frac{y^{5}}{z^{2} + x^{2} - z x} + \\frac{z^{5}}{x^{2} + y^{2} - x y} .\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{3}}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear. effectively reasoned and fully complete.</EF>Given the operations Ä and Å defined as: (a,b) Ä (c,d) = (ac+bd, bc-ad) and (a,b) Å (c,d) = (a + c, b + d), and given that (3,4) Ä (x, y) = (11, -2), calculate the value of (3,4) Å (x, y). Express your answer as a pair of real numbers (a, b).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(4,6)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and complete. Clear and methodical.</EF>Martin went to a store to buy a new television set. He decided to spend a certain amount, and the sales clerk offered him the TV for $100 less and an additional 20% off. The price of the television was $280 lower than the amount Martin decided to spend. How much did Martin decide to spend initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$1000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with coherent and thorough reasoning.</EF>Given the hyperbola $\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1\\left(a \\gt 0,b \\gt 0\\right)$ with eccentricity $e=2$, the equation of the asymptotes of the hyperbola is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
y = \pm\sqrt{3}x
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a complete and logically sound explanation.</EF>In triangle \\( \\triangle ABC \\), \\( BD \\) is a median, point \\( P \\) lies on \\( BD \\) with the ratio \\( |BP| : |PD| = 3:1 \\), and \\( AP \\) intersects \\( BC \\) at \\( K \\). Find the ratio of the area of \\( \\triangle ABK \\) to the area of \\( \\triangle ACK \\).\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 and clear solution with thorough, well-organized reasoning.</EF>Given the point $(\\cos \\theta, \\sin \\theta)$, the distance from this point to the line $x \\sin \\theta + y \\cos \\theta - 1 = 0$ is $\\frac{1}{2}$, where $0 \\leqslant \\theta \\leqslant \\frac{\\pi}{2}$. Find the value(s) of $\\theta$ in radians.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{12} \text{ or } \frac{5 \pi}{12}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with complete reasoning and correct result. Efficient and well-structured.</EF>If a parabola has the same direction of opening and shape as $y=-5x^{2}+2$, and its vertex has coordinates $\\left(4,-2\\right)$, then its analytical expression is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
y = -5(x-4)^2 - 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and conclusive, with logically sound reasoning and complete coverage.</EF>Vann is a veterinarian. Today he is going to be doing dental cleanings only. Dogs have 42 teeth, cats have a certain number of teeth and pigs have 28 teeth. He is to do 5 dogs, 10 cats and 7 pigs, and will clean 706 teeth today. How many teeth do cats have?\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 comprehensive. Efficient and coherent reasoning.</EF>Given the line $l\\_n$: $y=x- \\sqrt {2n}$ and the circle $C\\_n$: $x^{2}+y^{2}=2a\\_n+n$ intersect at two distinct points $A\\_n$, $B\\_n$, where $n∈N^{+}$, and the sequence ${a_n}$ satisfies: $a_1=1$, $a_{n+1}= \\frac {1}{4}|A\\_nB\\_n|^2$, find the general term formula for the sequence ${a_n}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a_n = 2^{n-1}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and coherent reasoning. Overall, precise and complete reasoning.</EF>An individual deposited 20% of 25% of 30% of what he had into his account with a bank. If he deposited Rs. 750, how much was he having initially with him before depositing?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
50000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and coherent reasoning.</EF>The difference between a two-digit number and the number obtained by interchanging the digits is some value. The difference between the sum and the difference of the digits of the number is 8 when the ratio between the digits of the number is 1:2. What is the value of the difference between the original two-digit number and the number obtained by interchanging the digits?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
36
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and methodical reasoning. The solution is well-organized and convincing.</EF>Given $$\\sin\\alpha + \\cos\\alpha = \\frac{1}{5}$$, and $$- \\frac{\\pi}{2} \\leq \\alpha \\leq \\frac{\\pi}{2}$$, find the value of $\\tan\\alpha$. Express your answer as a fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{3}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and well-structured; the reasoning is sound and the final answer is accurate.</EF>Given angle $A$ is an interior angle of $\\triangle ABC$, if $\\sin A + \\cos A = \\frac{3}{5}$, then $\\sin A - \\cos A$ equals \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{41}}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is clear and the reasoning is coherent but with minor room for tighter explanation; the work is accessible and well-structured.</EF>Given an angle θ with its vertex at the origin of a Cartesian coordinate system, the initial side being the non-negative half of the x-axis, and the terminal side passing through the point P(2,4), find the value of tan(θ + π/4). Express your answer as a single number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and well-structured.</EF>Nina bought one pair of shoes for a certain price, and the second one was 50% more expensive. She paid $55 for both pairs of shoes. How much did she pay for the first pair of shoes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$22
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear, plus thorough and well-structured.</EF>The average age of a group of 20 persons was decreased by 4 years when one person, whose age was 60 years, was replaced by a new person. Find the age of the new person.\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 overall, with solid logical progression and a clean final result.</EF>Given the quadratic equation x^2 + 12x + c = 0, where x = 18 is one of the solutions, find the value of the other solution. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear reasoning. The solution is logically solid and easy to follow.</EF>Given $f(\\tan x) = \\sin^2x - \\sin x \\cdot \\cos x$, find the value of $f(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough reasoning and well-structured steps.</EF>There is a 5,200 sq. ft. house and a 7,300 sq. ft. house next to each other. The smaller house is being expanded by 3,500 sq. ft. What is the new total square footage of both houses after the expansion?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16,000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive reasoning; overall, well-structured and convincing.</EF>Solve $\\cos 3x = 1$ for $0 \\le x \\le 2 \\pi.$ Enter all the solutions, separated by commas.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0, \frac{2 \pi}{3}, \frac{4 \pi}{3}, 2 \pi
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and reasonably thorough. The reasoning is sound and the conclusion is accurate.</EF>Solve the equation $\\lg x + \\lg (x - 2) = \\lg 3 + \\lg (x + 2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x = 6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete overall; the reasoning is sound and the answer is definite.</EF>Find the equation of the circle that is externally tangent to the circle $x^{2}+y^{2}+2x-4y+1=0$ with center at $(2,-2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(x-2)^{2}+(y+2)^{2}=9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and thorough. Clear, direct, and well-structured reasoning.</EF>In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\\rho=2\\sin\\theta$, then the Cartesian equation of curve $C$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x^2+(y-1)^2=1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Accurate and well-explained.</EF>If the graph of the function $f\\left(x\\right)=x-a\\ln x$ has a tangent line at the point $\\left(1,1\\right)$ with the equation $y=3x-2$, then 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 clear, with a thorough but somewhat lengthy explanation. The reasoning is solid and well-structured.</EF>Given that $\\overrightarrow{a}$ and $\\vec{b}$ are unit vectors, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 0$. If vector $\\overrightarrow{c}$ satisfies $| \\overrightarrow{c}-( \\overrightarrow{a}+\\overrightarrow{b} )| = | \\overrightarrow{a}-\\overrightarrow{b} |$, then the maximum value of $| \\overrightarrow{c} |$ is.\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>Correct, rigorous, and clear. Thorough, coherent, and approachable.</EF>There are 64 tokens numbered 1, 2, ..., 64. The tokens are arranged in a circle around 1996 lamps which are all turned off. Each minute, the tokens are all moved: token number \\( n \\) is moved \\( n \\) places clockwise. More than one token is allowed to occupy the same place. After each move, we count the number of tokens which occupy the same place as token 1 and turn on that number of lamps. Where is token 1 when the last lamp is turned on?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
32
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and coherent. Precise, orderly, and accessible.</EF>If the derivative of the function $f(x)$ is $f'(x) = 2x - 4$, then the interval of decrease for the function $f(x-1)$ 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, clear, and thorough. The reasoning is coherent and complete, and the final result is correct.</EF>If Karen sold 36 boxes of Tagalongs, how many cases of 12 boxes does Karen pickup from the cookie mom?\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 reasoning. Clear and orderly presentation.</EF>In a renowned city, the average birth rate is 7 people every two seconds and the death rate is some people every two seconds. The estimated size of the population net increase that occurs in one day is 259200. How many people die every two seconds?\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 sound, though there is a minor typographical quirk in the final arithmetic.</EF>For a set of positive integers, the mean is 6 greater than the median. The integers are x, x + 2, x + 4, x + c, and x + 37. If the mean is 6 greater than the median, what is the value of c?\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 explanation; thorough and well-structured.</EF>Billy was making ice using ice trays. Each tray held nine ice cubes. If he had eight trays, how many cubes could he make?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
72
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear solution; correct result and well-structured reasoning.</EF>A broker invested her own money in the stock market. During the first year, she increased her stock market wealth by some percent. In the second year, largely as a result of a slump in the stock market, she suffered a 30 percent decrease in the value of her stock investments. The net increase on her overall stock investment wealth by the end of the second year was 11.99999999999999 percent. What was the percentage increase in the first year?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. Thoroughly organized and precise.</EF>Express the number 2,600,000 in scientific notation in the standard form of a number between 1 and 10 multiplied by a power of 10.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2.6 \times 10^{6}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and well-organized, with approachable explanations and a correct final result.</EF>Given that the function f(x) = 2 / (2^x + 1) + m is an odd function, use the property of odd functions f(0) = 0 to find the value of the real number m. Express your answer as a single real number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and complete. Well-organized reasoning and accurate final result.</EF>Given the function f(x) = (x-1)/x, find the root(s) of the equation f(4x) = x. Express your answer as a single value, exact if possible.\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 reasonably thorough. The reasoning is logically sound and the final result is accurate.</EF>You had to add 48 to a number, but you mistakenly subtract it, and the result was 52. Find the result of the correct calculation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
148
|
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 two circles $C_1$: $x^2 + y^2 + D_1x + E_1y + 3 = 0$ and $C_2$: $x^2 + y^2 + D_2x + E_2y + 3 = 0$ both pass through the point $A(1, 1)$, find the equation of the straight line passing through the points $(D_1, E_1)$ and $(D_2, E_2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x + y + 5 = 0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logical and thorough reasoning.</EF>Given the line equation 3y - 2x - m = 0 and the points (2,1) and (5,-1) on opposite sides of the line, determine the range of values for m. Express your answer as an inequality in terms of m.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-13 < m < -1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and reasonably complete; coherent and logically sound overall.</EF>Given the ellipse $\\dfrac {x^{2}}{49}+ \\dfrac {y^{2}}{24}=1$, if a point $P$ on the ellipse has lines connecting it to the two foci $F_{1}$ and $F_{2}$ that are perpendicular to each other, then the area of $\\triangle PF_{1}F_{2}$ is _______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
24
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Efficient and thorough.</EF>Laurie has some more marbles than Kurt. Kurt has 45 marbles less than Dennis. Dennis has 70 marbles. Laurie has 37 marbles. How many more marbles does Laurie have than Kurt?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, thorough, and correct. Well-structured and convincing.</EF>Given the function $f(x) = -x^2 + 2x + 3$, find its minimum value on the interval $[0,3]$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
big_math_pairs_summary_critique_fixed
|
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