prompt
listlengths 1
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|---|---|---|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and well-structured, yielding the expected result.</EF>a man buys a cycle for rs . 2300 and sells it at a loss of 30 % . what is the selling price of the cycle ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1610
|
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>Given an arithmetic sequence $ \\{a_n\\} $ whose sum of the first n terms is $ S_n = -n^2 + 4n $, find the common difference $ d $.\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 thorough, methodical reasoning and well-structured presentation.</EF>Given the matrix $$M= \\begin{bmatrix} 1 & a \\\\ 3 & b\\end{bmatrix}$$ has an eigenvalue $\\lambda_1 = -1$ and the corresponding eigenvector $$\\overrightarrow {e}= \\begin{bmatrix} 1 \\\\ -1 \\end{bmatrix}$$, find the inverse matrix of $M$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\begin{bmatrix} - \frac {1}{2} & \frac {1}{2} \\ \frac {3}{4} & - \frac {1}{4}\end{bmatrix}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough reasoning presented in a clear, logical, and comprehensive manner.</EF>John is very unfit and decides to work up to doing a push-up. He trains 5 days a week for them and starts with wall push-ups. He adds 1 rep a day and once he gets to 15 reps he will start training high elevation push-ups, and then low elevation push-ups, and finally floor push-ups. How many weeks will it take him to get to floor push-ups?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and comprehensive. Clear, organized, and convincing.</EF>The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
666
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>correctness is solid and the reasoning is clear and complete; the explanation is thorough and well organized.</EF>The average age of an adult class is some years. 2 new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 2. What was the original average age of the class?\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 and orderly reasoning. Overall, precise and reliable.</EF>Find a perfect number of the form \\( p \\times q \\), where \\( p \\) and \\( q \\) are prime numbers.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and well-structured overall, with precise final numeric presentation.</EF>When you divide a number by 11, the result is 2. What is a number multiplied by 6?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
132
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with rigorous and complete reasoning. Prompt and coherent explanation overall.</EF>Billy has some horses. Each one eats 4 pounds of oats, twice a day. He needs 96 pounds of oats to feed his horses for 3 days. How many horses does Billy have?\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 thorough. Clear, direct, and well-structured reasoning.</EF>there were 35 students in a hostel . if the number of students increases by 7 , the expenses of the mess increase by rs . 42 per day while the average expenditure per head diminishes by re 1 . find the original expenditure of the mess .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
420
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. The reasoning is logical and complete, with a well-organized presentation.</EF>Two adjacent sides of a rectangle are in the ratio 3:7. What is the area of the rectangle if its perimeter is 40 cm?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
84 \text{ cm}^2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and coherent solution. well-organized and accessible.</EF>Matt gets a $1,300 commission on a big sale. This commission alone raises his average commission by a certain amount. Matt's new average commission is $400, and he has made 6 sales. By how much did the big sale raise his average commission?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$180
|
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>Find $\\left(\\frac{1}{2}\\right)^{8} \\cdot \\left(\\frac{3}{4}\\right)^{-3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{108}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, efficiently stated, and logically sound.</EF>Find the maximum value of the function y = x - sin(x) on the interval [π/2, 3π/2]. Express your answer as a numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3\pi}{2}+1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with solid reasoning and accessible structure. The approach is thorough and easy to follow.</EF>A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
60
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution. Thorough and well-organized, with a transparent, linear reasoning process.</EF>If the sequence $\\{a_n\\}$ is an arithmetic sequence and the sum of the first $n$ terms is $S_n$, given that $$\\frac {a_{5}}{a_{3}}= \\frac {5}{9},$$ find the value of $$\\frac {S_{9}}{S_{5}}.$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, simple, and clear. Efficient and unambiguous explanation.</EF>Lilah's family gallery has 400 photos. On a two-day trip to the Grand Canyon, they took half as many photos they have in the family's gallery on the first day and some more photos on the second day. After adding all these photos to the family gallery, the total number of photos in the gallery is 920. How many more photos did they take on the second day compared to the first day?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent; clear and complete. Efficient and readable.</EF>Find the smallest positive integer \\( k \\) such that the equation \\( 2xy - 3x - 5y = k \\) has an odd number of positive integral solutions.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear; well-organized and easy to follow.</EF>A certain candy manufacturer reduced the weight of candy bar M by 25 percent but left the price unchanged. What was the resulting percent increase in the price per ounce of candy bar M?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
33.33\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough reasoning and a tidy presentation.</EF>Given $a,b \\in \\mathbb{R}$ and $i$ is the imaginary unit, if $a+i=2-bi$, then find the value of $(a+bi)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3-4i
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough steps and a correct final result.</EF>There were 1,250 bookshelves. Each bookshelf had 45 books. There were _____ books on the shelves.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
56,250
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and complete coverage. The reasoning is sound and the final result is accurate.</EF>Given the equations of two circles: x^2 + y^2 - 4 = 0 and x^2 + y^2 - 4x + 4y - 12 = 0, find the length of their common chord.\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 and coherent, with clear organization and complete coverage. Overall, the reasoning is solid and the result is reliable.</EF>Find the distance from the point on the parabola $y^2 = -4x$ with an x-coordinate of -6 to its focus. Express your answer as a whole number.\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 overall, with thorough reasoning and a tidy presentation.</EF>Determine the smallest positive period of the function $y=1-\\cos 2x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
π
|
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>In a tug-of-war competition, class 3, class 4, and class 5 were victorious in the preliminary rounds. Three judges made predictions about the champion: Judge A said: \"The champion will not be class 3, nor class 4\"; Judge B said: \"The champion will not be class 3, it must be class 5\"; Judge C said: \"The champion will not be class 5, it will be class 3\". After the result of the match came out, one of them had both predictions right, one had both wrong, and another had one correct and one incorrect prediction. Thus, the champion is class ＿.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, rigorous, complete, and clear.</EF>cara leaves her home and walks toward don ' s house . two hours later , don leaves his home and walks toward cara ' s house . the distance between their homes is 45 kilometers , cara ' s walking speed is 6 km / h , and don ' s walking speed is 5 km / h . how many kilometers will cara walk before she meets don ?\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. The reasoning is coherent and well-structured for the given problem.</EF>In $\\triangle ABC$, if $\\frac {\\sin C}{\\sin A}=3$ and $b^{2}-a^{2}= \\frac {5}{2}ac$, calculate the value of $\\cos B$. Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and well-structured. The reasoning is sound and the result is properly derived.</EF>On Thursday, Mabel handled some transactions. Anthony handled 10% more transactions than Mabel, Cal handled 2/3rds of the transactions that Anthony handled, and Jade handled 15 more transactions than Cal. Jade handled 81 transactions. How many transactions did Mabel handle?\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 coherent solution; thorough and clear, with solid reasoning and complete coverage.</EF>Printer A and Printer B can finish a task together in some minutes. Printer A alone would finish the task in 60 minutes. The task contains 480 pages, and Printer B prints 4 pages a minute more than Printer A. How many minutes would it take for Printer A and Printer B to finish the task together?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
24
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logical reasoning that is consistent and complete.</EF>Two circles are given: C1: $x^2 + y^2 = 1$ and C2: $(x - 3)^2 + (y - 4)^2 = 16$. Find the number of common tangents to both circles. Express your answer as a whole number.\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 well-structured reasoning. Overall, informative and convincing.</EF>Given that the function $y=f(x)$ is an even function on $\\mathbb{R}$, and when $x \\geq 0$, $f(x) = 2^x + 1$, then when $x < 0$, $f(x) = \\ $.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2^{-x} + 1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and methodical reasoning. The solution is well-organized and convincing.</EF>Among the real numbers $\\frac{1}{3}$, $\\pi$, $0$, $\\sqrt{5}$, the irrational numbers are ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\pi \text{ and } \sqrt{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and the final value is properly obtained.</EF>In an election, candidate Douglas won 54 percent of the total vote in counties X and Y. He won 62 percent of the vote in county X. If the ratio of people who voted in county X to county Y is 2:1, what percent of the vote did candidate Douglas win in county Y?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
38
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and coherent. Thorough, well-structured, and convincing.</EF>Given the hyperbola equation $\\dfrac{x^{2}}{1-2k}-\\dfrac{y^{2}}{k-2}=1$ with foci on the $y$-axis, find the range of the real number $k$. Express your answer as an interval in standard notation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(\dfrac{1}{2},2)
|
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>Let the set \\( M = \\{1, 2, 3, \\cdots, 2002\\} \\). For any non-empty subset \\( A \\) of \\( M \\), define \\( x_{A} \\) as the sum of the maximum and minimum numbers in \\( A \\). Find the arithmetic mean of all such \\( x_{A} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2003
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with solid reasoning and complete derivation. The result is correct and the approach is fairly rigorous.</EF>Evaluate 22 plus the square root of (-4 + 6 * 4 * 3). What is the result?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30.246211251
|
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 the hyperbola $$C： \\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$, one of its asymptotes is intercepted by the circle $(x-2)^2+y^2=4$ to form a chord of length 2. The eccentricity of the hyperbola C is $\\boxed{\\text{answer}}$.\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 with thorough, well-organized reasoning.</EF>In the sequence\n\n$$\n1,4,7,10,13,16,19, \\ldots\n$$\n\neach term is 3 more than the previous term. Find the 1000th term of the sequence.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2998
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, coherent, and clear.</EF>In a right triangle with legs in the ratio 1:2, a perpendicular is drawn from the vertex of the right angle to the hypotenuse. Find the ratio of the corresponding segments of the hypotenuse. Express your answer as a ratio, using a colon (:) to separate the parts of the ratio.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1:4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough reasoning presented in a clear, logical, and comprehensive manner.</EF>James joins a football team and becomes the star. He scores 4 touchdowns per game and each touchdown is worth some points. There are 15 games in the season. He also manages to score 2 point conversions 6 times during the season. James beat the old record of 300 points during the season by 72 points. How many points is each touchdown worth?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, methodical, and clear, with a correct final result and coherent reasoning.</EF>Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation: $x = \\sqrt{19} + \\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{x}}}}}}}}}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$383$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough and methodical reasoning; precise and logically sound.</EF>The equation of a circle is given as $x^2+y^2-4x-2y-5=0$. Find the coordinates of the center of the circle. Express your answer as a coordinate pair (x, y).\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 thorough. The reasoning is sound and complete.</EF>If the solution set of the inequality $|ax + 2| < 6$ with respect to $x$ is $(-1, 2)$, then calculate the value of the real number $a$. 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 logically solid. The reasoning is thorough and the final result is accurate.</EF>The sum of three natural numbers \\(A\\), \\(B\\), and \\(C\\) is 111. It is known that the average of \\(A\\) and \\(B\\) is 31, and the average of \\(A\\) and \\(C\\) is 37. Calculate the average of \\(B\\) and \\(C\\). Express your answer as a whole number.\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 uncluttered, with clear and logical reasoning and thorough coverage.</EF>At 6 o'clock, a clock ticks 6 times. The time between the first and last ticks is a certain number of seconds. At 12 o'clock, it ticks for 55 seconds. How long is the time between the first and last ticks at 6 o'clock?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
25
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct. Clear and thorough. The reasoning is sound and complete.</EF>In a class, it is observed that no two boys were born on the same day of the week and no two girls were born in the same month. If the addition of another child would cause one of these conditions to no longer hold true, calculate the total number of children currently in the class.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
19
|
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>For an integer $k$ let $T_k$ denote the number of $k$ -tuples of integers $(x_1,x_2,...x_k)$ with $0\\le x_i < 73$ for each $i$ , such that $73|x_1^2+x_2^2+...+x_k^2-1$ . Compute the remainder when $T_1+T_2+...+T_{2017}$ is divided by $2017$ .\n\n[i]Proposed by Vincent Huang\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
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>John buys 2 packs of index cards for all his students. He has 6 classes and a certain number of students in each class. He bought 360 packs. How many students are in each class?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, simple, and clear. Efficient and unambiguous explanation.</EF>Let $I$ be the set of all points in the plane such that both their x-coordinate and y-coordinate are irrational numbers, and let $R$ be the set of points whose both coordinates are rational. What is the maximum number of points from $R$ that can lie on a circle with an irrational radius and a center that belongs to $I$?\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 coherent solution. Clear, thorough, and systematically organized.</EF>Given that $\\tan (α+80^{\\circ})=4\\sin 420^{\\circ}$, find the value of $\\tan (α+20^{\\circ})$. Express your answer as a fraction or decimal, including the square root symbol if necessary.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{\sqrt{3}}{7}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; thorough and well-structured, with precise computations and final result.</EF>An engineer undertakes a project to build a road 15 km long in 300 days and employs 50 men for the purpose. After a certain number of days, he finds only 2.5 km of the road has been completed. He must employ approximately 75 extra men to finish the work in time. How many days had passed when he realized the progress?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
100
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and complete. The reasoning is direct and convincing.</EF>In a school, a total of 100 students from the third, fourth, and fifth grades participated in a tree planting activity, where they planted a total of 566 trees. It is known that the number of third and fourth grade students are equal. On average, a third grade student planted 4 trees, a fourth grade student planted 5 trees, and a fifth grade student planted 6.5 trees. How many trees did the third grade students plant in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
84
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear and thorough. Efficient and well-structured.</EF>The result of combining like terms $3a^{2}b-4ba^{2}$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-a^{2}b
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with solid reasoning. The solution is thorough and easy to follow.</EF>Given the identity: $(x^2-x+1)^6 = a_0 + a_1x + a_2x^2 + a_3x^3 + \\ldots + a_{10}x^{10} + a_{11}x^{11} + a_{12}x^{12}$, then $(a_0 + a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12})^2 - (a_1 + a_3 + a_5 + a_7 + a_9 + a_{11})^2 = \\ $\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
729
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, clear, and well-structured solution. The reasoning is solid and the final answer is correct.</EF>Mike buys some rose bushes at 75 dollars each, 2 of them are for his friend and the rest are for him. He also buys 2 tiger tooth aloes for 100 dollars each. He spent 500 dollars on plants for himself. How many rose bushes did he buy in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear. Logical, thorough, and well-structured reasoning.</EF>A list of $8$ numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown: \\[\\text{\\underline{\\hspace{3 mm}?\\hspace{3 mm}}\\hspace{1 mm},\\hspace{1 mm} \\underline{\\hspace{7 mm}}\\hspace{1 mm},\\hspace{1 mm} \\underline{\\hspace{7 mm}}\\hspace{1 mm},\\hspace{1 mm} \\underline{\\hspace{7 mm}}\\hspace{1 mm},\\hspace{1 mm} \\underline{\\hspace{7 mm}}\\hspace{1 mm},\\hspace{1 mm}\\underline{\\hspace{2 mm}16\\hspace{2 mm}}\\hspace{1 mm},\\hspace{1 mm}\\underline{\\hspace{2 mm}64\\hspace{2 mm}}\\hspace{1 mm},\\hspace{1 mm}\\underline{\\hspace{1 mm}1024\\hspace{1 mm}}}\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$\frac{1}{4}$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; logically rigorous and complete. The reasoning is well-structured and easy to follow, yielding the expected result.</EF>Given that the ratio of the three sides of a triangle is 3:5:7, find the measure of the largest interior angle of this triangle. Express your answer in degrees as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>correct, coherent, and clear; robust and well-structured reasoning.</EF>Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's *vicious hobgoblin +1 of viciousness,* Hugo rolls $25$ $20$ -sided dice, obtaining a sum of (alas!) only $70$ . Trying to console him, Fidel notes that, given that sum, the product of the numbers was as large as possible. How many $2$ s did Hugo roll?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and coherent. Thorough, well-structured, and convincing.</EF>A young sprinter runs 90 meters in 10 seconds with the wind. Under the same wind speed, he runs 70 meters against the wind, also in 10 seconds. How many seconds will it take for him to run 100 meters in no wind condition? Express your answer in seconds as a decimal number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12.5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and neatly presented.</EF>A can do a piece of work in 4 hours. A and C together can do it in just 2 hours, while B and C together need some hours to finish the same work. B can complete the work in 12 hours. In how many hours can B and C together complete the work?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clearly presented.</EF>If $\\log_6 (4x)=2$, find $\\log_x 27$. Express your answer in simplest fractional form.\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, clear, and complete reasoning with precise steps and a correct final set.</EF>Given that sin(alpha) = 3/4 and alpha is an angle in the second quadrant, find the value of sin(2alpha + pi/6). Express your answer as a simplified fraction or decimal, including any negative signs and square roots if applicable.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{3\sqrt{21}+1}{16}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and verification. The reasoning is coherent and well-organized.</EF>Logan watched television for 5 hours with breaks of 10, 15, 20, and 25 minutes between the shows. How many minutes did he spend watching television, excluding the time spent during breaks?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
230
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, well-structured, and clear enough. The reasoning is sound and the final answer is appropriate.</EF>Given the complex number $z = \\frac{(2-i)}{i}$, determine the quadrant in which the corresponding point lies in the complex plane. Express your answer as the name of the quadrant (e.g. first, second, third, fourth).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
third
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. Thoroughly organized and precise.</EF>Given line $l_1$: $y=2x+3$, and line $l_2$ is symmetric to $l_1$ with respect to the x-axis. The slope of line $l_2$ is \\_\\_\\_\\_\\_\\_.\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. Correct final result with sound reasoning and good organization.</EF>A man sitting in a train which is traveling at 60 kmph observes that a goods train, traveling in the opposite direction, takes 12 seconds to pass him. If the goods train is 300 m long, what is the speed of the goods train?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
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>Factorize: $2x^{2}y-8y=$____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2y(x+2)(x-2)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. The reasoning is sound, well-structured, and effectively communicates the solution.</EF>Grandpa is 65 years old this year, and Mingming is 5 years old. Next year, Grandpa's age will be times Mingming's age.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and clear. The reasoning is coherent and the final result is correct.</EF>What is the probability that when tossing a fair coin four times, heads do not appear consecutively?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
B. \frac{1}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The reasoning is sound and the conclusion is well-supported and neatly presented.</EF>Given vectors $\\overrightarrow{a}=(\\sin x, \\frac {3}{2})$ and $\\overrightarrow{b}=(\\cos x,-1)$, when $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, find the value of $\\frac {2\\sin x-\\cos x}{4\\sin x+3\\cos x}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {4}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and clear. The reasoning is sound, thorough, and easy to follow.</EF>If the graph of the function $f(x) = \\sin 2x + \\cos 2x$ is shifted to the left by $\\varphi$ units, and the resulting graph is symmetric about the y-axis, then the minimum positive value of $\\varphi$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{8}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Accurate result with coherent, rigorous reasoning.</EF>The area of a sector with a radius of 2cm and a central angle of 120° is to be calculated. Find the area of the sector. Express your answer in square centimeters (cm²) and include π in your answer if necessary.\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>Accurate and comprehensive. Well-organized and convincing, with correct conclusion.</EF>The minimum value of \\\\(f(x)=x^{2}+2x+1\\\\), where \\\\(x \\in [-2,2]\\\\), is \\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough and slightly verbose reasoning presented in two equivalent ways; the explanation is easy to follow and logically sound.</EF>Given sets $M = \\{x | x(x-a-1) < 0, x \\in \\mathbb{R}\\}$, $N = \\{x | x^2 - 2x - 3 \\leq 0\\}$, if $M \\cup N = N$, find the range of the real number $a$.\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 logically sound. Easy to follow and appropriately complete.</EF>A baker already had seventy-eight cakes but made some extra. The baker had a total of 87 cakes. How many extra cakes did the baker make?\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 coherent. Efficient and thorough.</EF>Given a quadratic function $y=f(x)$ with a maximum value of 13, and $f(3) = f(-1) = 5$, find the explicit formula for $f(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
f(x) = -2x^2 + 4x + 11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough. Well-structured and easily understandable.</EF>Given the function $f(x)=\\sqrt{a{x}^{2}+bx+c}(a＜b)$ with the domain of $R$, find the maximum value of $\\frac{b-a}{a+2b+4c}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{8}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, rigorous, and clear. The reasoning is sound and the final answer is correct.</EF>At the fair, the 'Twirly Tea Cups' ride can hold nine people per teacup. The ride has a certain number of tea cups, and a total of 63 people can ride at a time. How many tea cups are there on the ride?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with complete reasoning and a correct final answer.</EF>Given a rectangle ABCD with a perimeter of 20 inches, find the minimum value of the diagonal AC. Express your answer in inches, possibly using a square root.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{50}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and clear solution; correct result and well-structured reasoning.</EF>Given that $\\{a_n\\}$ is a geometric sequence with all positive terms, where $S_5=3$, $S_{15}=21$, find the value of $S_{20}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
45
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and complete. The reasoning is direct and convincing.</EF>What least number should be subtracted from 2590 so that the remainder when divided by 9, 11, and 13 will leave in each case the same remainder 6?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, thorough, and clear. The reasoning is solid and the final answer is correct.</EF>When x is even, [x] = x / 2 + 1, and when x is odd, [x] = 2x + 1. If [6] * [x] equals 28, what is the value of x?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent overall. Accurate, thorough, and accessible.</EF>Lorie has 2 pieces of $100 bills. He requested to change one piece of the $100 bills into $50 bills. Half of the remaining $100 bill is changed to $10 bills while the rest is changed to $5 bills. How many pieces of bills will she have?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
17
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and well-structured overall, with precise final numeric presentation.</EF>For the past n days, the average daily production at a company was 50 units. If today's production of 60 units raises the average to a certain number of units per day, and the value of n is 1, what is the new average daily production?\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 coherent solution. Clear, thorough, and systematically organized.</EF>At a teachers' gathering, there were 12 more female teachers than male teachers. A teacher is randomly selected to perform a show. If the probability of selecting a male teacher is $\\frac{9}{20}$, then the total number of teachers attending the gathering is __________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear explanation, with thorough and coherent reasoning.</EF>Find $\\begin{pmatrix} 1 & 5 \\\\ -2 & 4 \\end{pmatrix} + \\begin{pmatrix} 0 & -3 \\\\ 8 & -5 \\end{pmatrix}.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\begin{pmatrix} 1 & 2 \\ 6 & -1 \end{pmatrix}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Systematic, precise, and well-structured.</EF>Fill in the blank with $\\in$ or $\\notin$: $0$ \\_\\_\\_\\_\\_\\_ $\\emptyset$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\notin
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, clear, and thorough. The reasoning is coherent and logically sound, yielding the correct result.</EF>What is the molecular weight of some moles of K2Cr2O7 if the molecular weight of 4 moles is 1184?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
296
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Logical, complete, and accessible; well-structured reasoning.</EF>A train leaves Mumbai at 9 am at a speed of 40 kmph. After one hour, another train leaves Mumbai in the same direction as that of the first train at a certain speed. The two trains meet at a distance of 120 km from Mumbai. What is the speed of the second train?\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 thorough. The reasoning is sound, and the result is precisely stated.</EF>along a yard 414 metres long , 24 trees are palnted at equal distances , one tree being at each end of the yard . what is the distance between two consecutive trees\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 reasoning. The solution is complete and easy to follow.</EF>Calculate the limit of the function:\n\n\\[ \\lim_{x \\to 10} \\frac{\\lg x - 1}{\\sqrt{x - 9} - 1} \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{5 \ln 10}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough steps and a correct final result.</EF>How many natural numbers less than 1000 have exactly three distinct positive integer divisors?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Overall, precise and well-structured reasoning.</EF>We ordered 7 pizzas. Each pizza has the same number of slices. If there were a total of 14 slices, how many slices are there in each pizza?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, logically sound and well organized.</EF>a can do a work in 15 days b in 14 days and c in 16 days . if they work on it together then in how many days required to complete the 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 reasoning; well-structured and thorough.</EF>Joe drives 360 miles at 60 miles per hour, and then he drives the next 120 miles at 40 miles per hour. What is his average speed for the entire trip in miles per hour?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
53.33
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, logical, and clear in its reasoning. Thorough and well-structured.</EF>Matt has 8 baseball cards worth $6 each. He trades two of them to Jane in exchange for 3 $2 cards and 1 $9 card. After that, he trades one of the $2 cards and one of his original $6 cards to Sam for two $5 cards and one $8 card. Next, he trades one of the $5 cards and one $9 card to Alex for three $3 cards, one $10 card, and one $1 card. Calculate Matt's overall profit after these three transactions.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$19
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Precise, coherent, and complete.</EF>Suppose there is a bag containing one red, one yellow, and one blue ball. Each time a ball is drawn from the bag (the probability of drawing any particular color ball is the same), the color is noted, and then the ball is replaced into the bag. This process continues until two consecutive red balls are drawn. Let $\\xi$ denote the number of draws at that point. What is the expected value of $\\xi$?\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 reasoning; well-structured and thorough.</EF>Beth has 72 marbles of three colors. Each color has the same amount of marbles. If Beth loses 5 red marbles, twice as many blue marbles, and some yellow marbles, she has 42 marbles left. What is the ratio of the number of yellow marbles lost to the number of red marbles lost?\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 overall, with a coherent and methodical explanation; the reasoning is sound and complete.</EF>On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent sides. Calculate the probability that the area of this rectangle is greater than 60cm². Express your answer as a simplified fraction in the form \\boxed{\\frac{a}{b}}.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{4}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with a thorough but slightly verbose derivation. The reasoning is sound and effectively leads to the correct result.</EF>On a trip covering 350 km, a bicyclist travels the first 200 km at 20 km per hour and the remainder of the distance at a certain speed. The average speed for the entire trip is 17.5 km per hour. What was the speed of the bicyclist during the remainder of the distance?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
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