prompt
listlengths 1
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|---|---|---|
[
{
"content": "<EF>Accurate, clear, and methodical. Precise and thorough.</EF>Rita is reading a book with 95 pages. Each chapter has three pages more than the previous one, and the first chapter has 13 pages. How many chapters are in the book?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct. Clear and complete. Reasoning is thorough and well-structured.</EF>If vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=3$, $|\\overrightarrow{a}-\\overrightarrow{b}|=5$, $\\overrightarrow{a}\\cdot \\overrightarrow{b}=1$, then $|\\overrightarrow{b}|=$____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3\sqrt{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and conclusive, with logically sound reasoning and complete coverage.</EF>A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. Calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{12}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, well-structured, and clear. Thorough and logically sound.</EF>Find the number of different integer pairs $(x, y)$ that satisfy the equation $\\sqrt{1992} = \\sqrt{x} + \\sqrt{y}$ and the constraints $0 \\leqslant x \\leqslant y$. Express your answer as a single integer.\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, well-structured reasoning. The solution is coherent, and the final result is correct.</EF>A shopping cart contains only apples, oranges, and pears. There are 3 times as many oranges as apples, and some times as many pears as oranges. The apples are equivalent to 0.08333333333333333 fraction of the pears. How many times more pears are there than oranges?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and the final answer is accurate.</EF>A pet store sold 2 kittens and 1 puppy over the weekend. The kittens sold for a certain amount each and the puppy sold for $5. They earned $17 from selling the pets. How much did each kitten sell for?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound, well-structured, and complete.</EF>A certain number of camels cost as much as some horses, 16 horses cost as much as 4 oxen, and 6 oxen cost as much as 4 elephants. The cost of 10 elephants is Rs. 110000. If the cost of a camel is Rs. 4400, how many horses are equivalent to the cost of the camels?\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 reasonably complete; the reasoning is sound and the final value is accurate.</EF>Trevor and Joe were working together to finish a 500 piece puzzle. They put the border together first and that was 75 pieces. Trevor was able to place 105 pieces of the puzzle. Joe was able to place a certain multiple of the number of puzzle pieces as Trevor. There are 5 puzzle pieces missing. What is the ratio of the number of puzzle pieces placed by Joe to the number of puzzle pieces placed by Trevor?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3:1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is logically sound and well organized.</EF>If the symbol \"\\*\" represents a certain operation, and $a*b=a+2b^2$, then the value of $(-4)*6$ is.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
68
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct yet thorough. Clear, structured, and coherent.</EF>Given that the graph of the linear function $y=kx+\\left(2-k\\right)$ passes through the first, second, and third quadrants, the range of values for $k$ is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0 < k < 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and methodical reasoning. The solution is well-organized and convincing.</EF>In triangle \\(ABC\\), points \\(M\\) and \\(N\\) are marked on the sides \\(AB\\) and \\(BC\\) respectively, such that \\(BM = BN\\). A line is drawn through point \\(M\\) perpendicular to \\(BC\\), and another line is drawn through point \\(N\\) perpendicular to \\(AB\\). These lines intersect at point \\(O\\). The extension of segment \\(BO\\) intersects side \\(AC\\) at point \\(P\\) and divides it into segments \\(AP = 5\\) and \\(PC = 4\\). Find the length of segment \\(BP\\), given that \\(BC = 6\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent; clear and complete. Efficient and readable.</EF>Mahesh can complete a piece of work in 50 days. He works on it for 20 days and then Rajesh takes over. Along with Rajesh, Suresh also contributes to the work. Rajesh and Suresh together complete the remaining work in 30 days. If Suresh alone can complete the work in 60 days, how long would Rajesh take to complete the entire work on his own?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
300
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with logically sound reasoning and complete coverage.</EF>Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$125$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution. Thoroughly reasoned and well structured.</EF>Given the function $f(x)=2x-\\ln x$, where $x\\in(0,e)$, find the minimum value of $f(x)$. Express your answer as a single value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1+\ln 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. well-structured and logically sound.</EF>Let \\( a, b, c, \\) and \\( d \\) be the roots of the equation \\( x^{4}-15x^{2}+56=0 \\). If \\( P = a^{2} + b^{2} + c^{2} + d^{2} \\), find the value of \\( P \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and logically solid overall; clear, coherent, and fairly complete.</EF>The average of all scores on a certain algebra test was 90. The average of the 8 male students' grades was 84, and there were 24 female students who took the test. What was the average grade of the female students?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
92
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough, methodical reasoning.</EF>Officer Hopps has a goal to give out a certain number of tickets in May. The first 15 days he averages 8 tickets a day. He has to average 5 tickets each day for the rest of the month to reach his required goal. How many tickets does he have to give out in total in May?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
200
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and well-structured reasoning. The reasoning is coherent and complete.</EF>Find the coefficient of the term containing $x$ in the expansion of the binomial ($\\sqrt {x}$ - $\\frac {1}{x}$)$^5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with solid reasoning and good structure. The presentation is thorough and easy to follow.</EF>Given that the graph of the power function $f(x)$ passes through the point $A(\\frac{1}{4}, \\frac{1}{2})$, determine the equation of the tangent line to the curve $y = f(x)$ at point $A$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4x - 4y + 1 = 0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and systematic reasoning. The solution is well-structured and easy to follow.</EF>Given an arithmetic sequence $\\{a_n\\}$ and a geometric sequence $\\{b_n\\}$ with a common ratio of $q$ ($n, q \\in \\mathbb{N}^*$), let $S_n$ and $T_n$ be the sum of the first $n$ terms of $\\{a_n\\}$ and $\\{b_n\\}$, respectively. If $T_{2n}+1=S_{q^n}$, then $a_n= \\boxed{\\text{2n-1}}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2n-1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous. Clear and complete.</EF>Given the complex number $z = \\frac{2i}{1-i}$, where $i$ is the imaginary unit, identify the quadrant in which the point corresponding to $z$ is located. Express your answer as the name of the quadrant (e.g., First quadrant, Second quadrant, etc.).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
Second quadrant
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and lucid. Clear, coherent, and comprehensive.</EF>In a studio audience of 100 people, 40% of these people have an envelope taped underneath their chairs. 20% of these people will have \"You Won\" in their envelope, the rest will have empty envelopes. How many people will win a prize?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a thorough, well-structured explanation that is appropriately detailed and easy to follow.</EF>A gambler has won 40% of his poker games for the week so far. If, all of a sudden, his luck changes and he begins winning 80% of the time, he must play approximately 30 more games to end up winning 60% of all his games for the week. How many games did the gambler play initially?\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 sound and well-organized, with a correct final result.</EF>A waiter had some customers. After 5 more arrived he had 8 customers. How many customers did the waiter have initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and coherent. Thorough, well-organized, and easy to follow.</EF>Ral is three times as old as Suri. In 3 years, Suri's current age will be 16. What is Ral's current age?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
39
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and well-structured overall, with precise final numeric presentation.</EF>Given that the solution set of the inequality $x+b > 0$ is $\\{x|x > 2\\}$, find the solution set of the inequality $ \\frac {x+b}{(x-6)(x+1)} > 0$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-1,2) \cup (6,+\infty)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, well-structured, and reasonably thorough. The reasoning is sound and the final result is accurate.</EF>Given $ab=2$, $a+b=3$, find the value of $a^{2}b+ab^{2}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and logically sound, with complete case analysis.</EF>The equation of the tangent line to the function $f(x)=(x+1)\\ln x-4(x-1)$ at the point $(1,f(1))$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2x+y-2=0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and logically rigorous; well-organized and easy to follow.</EF>I apologize for providing the answer. Here is the revised problem without the answer:\n\nQueenie earns $150 a day as a part-time clerk. For overtime work, she earns an additional $5 for the first hour, and her overtime pay rate increases by $2 for each additional hour. Calculate Queenie's earnings for working 5 days with 4 hours overtime.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$782
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and logically sound, with complete case analysis.</EF>Marta works on her grandparent's farm to raise money for a new phone. She receives $10 for every hour she works and has collected $50 in tips from her grandmother. Marta has worked 19 hours on the farm so far. How much money has she collected in total?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$240
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough and accessible narration.</EF>Evaluate the definite integral $$\\int_{1}^{2} \\frac{3x+1}{x} \\, dx$$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3 + \ln 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate and coherent solution. well-organized and accessible.</EF>Given the plane vectors $a, b,$ and $c$ such that:\n$$\n\\begin{array}{l}\n|\\boldsymbol{a}|=|\\boldsymbol{b}|=|\\boldsymbol{c}|=2, \\boldsymbol{a}+\\boldsymbol{b}+\\boldsymbol{c}=\\mathbf{0}.\n\\end{array}\n$$\n\nIf $0 \\leqslant x \\leqslant \\frac{1}{2} \\leqslant y \\leqslant 1$, then find the minimum value of \n$$\n|x(\\boldsymbol{a}-\\boldsymbol{c})+y(\\boldsymbol{b}-\\boldsymbol{c})+\\boldsymbol{c}|.\n$$\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 coherent. The reasoning is solid and the final answer is accurate.</EF>What is the value of $\\frac12\\cdot\\frac41\\cdot\\frac18\\cdot\\frac{16}{1} \\dotsm \\frac{1}{512}\\cdot\\frac{1024}{1}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
32
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and well-structured reasoning. Overall, informative and convincing.</EF>A person's commute times to work on 5 different occasions (in minutes) were $x$, $y$, $10$, $11$, and $9$. It is known that the average of this set of data is $10$, and the variance is $2$. Calculate the value of $|x-y|$. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and coherent. Clear and complete.</EF>Among all six-digit natural numbers whose digits are arranged in ascending order (from left to right), we consider the numbers that contain the digit 1 and those that do not contain this digit. Which group of numbers is larger and by how many?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
28
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Efficiently organized, with thorough, accessible reasoning.</EF>There are some different movies and 21 different books in the 'crazy silly school' series. If you read 7 of the books and watched 4 of the movies, you still have to watch 4 more movies. How many different movies are there in the series?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured solution with correct final result.</EF>Let S be the set of all positive integers that, when divided by 15, have a remainder of 11. What is the 127th number in this set?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1901
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. The reasoning is sound and easy to follow.</EF>A man can do a piece of work in some days, but with the help of his son, he can do it in 4 days. The son can do it alone in 6.67 days. In how many days can the man do the work alone?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough and methodical reasoning; precise and logically sound.</EF>There are three digit numbers where the tens digit is 3 and the sum of the ones digit and the hundreds digit is 5. Each digit in this three-digit number is different, and when you add 124 to this number, all digits are equal. Find a three digit number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
431
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and the final result is verified.</EF>In digging a pond with dimensions of 20 m length, 12 m width, and some depth, the volume of the soil extracted is 1200 cubic meters. What is the depth of the pond?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Efficient and thorough.</EF>Mary is baking a cake. The recipe calls for 6 cups of flour, some cups of sugar, and 7 cups of salt. She already put in 5 cups of flour. She needs to add 1 more cup of sugar than cups of salt now. How many cups of sugar does the recipe call for?\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. well-structured and logically sound.</EF>If the volume of a cylinder is exactly equal to its surface area, calculate the minimum possible value for the volume of the cylinder. Express your answer in terms of pi.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
54\pi
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, with thorough, well-structured reasoning and readable presentation.</EF>The domain of the function $f(x)= \\sqrt {2^{x}-8}$ is __________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
[3,+\infty)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough reasoning and a tidy presentation.</EF>Adam bought 9 packages of cat food and 7 packages of dog food. Each package of cat food contained 10 cans, and each package of dog food contained 5 cans. How many more cans of cat food than dog food did Adam buy?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
55
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear. The reasoning is solid and the conclusion is well-supported.</EF>The solution set of the inequality $\\frac{{3x+5}}{{x-1}} > x$ is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\left(-\infty ,-1\right)\cup \left(1,5\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and logically coherent. Thorough and properly structured.</EF>What is the greatest prime factor of 3105?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
23
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution. Thorough and coherent reasoning overall.</EF>In the rectangular coordinate system $(xOy)$, the polar coordinate system is established with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the line $\\begin{cases} & x=-t \\ & y=\\sqrt{3}t \\ \\end{cases}$ ($t$ is the parameter) and the curve $C_{1}$: $ρ=4\\sin θ$, the point $A$ is the intersection of the line and the curve $C_{1}$ that is distinct from point $O$. Similarly, the point $B$ is the intersection of the line and the curve $C_{2}$: $ρ=2\\sin θ$ that is distinct from point $O$. Find the length $|AB|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear, logical, and well-structured.</EF>Simplify first, then evaluate: $\\left(a+1\\right)^{2}+a\\left(1-a\\right)$, where $a=\\frac{\\sqrt{3}}{3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{3}+1
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear and reasonable in reasoning and presentation.</EF>Find the value of dividing the area of a rectangle in half when the rectangle has a width of 25 centimeters (cm) and a height of 16 centimeters (cm).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
200
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete, with clear and orderly reasoning.</EF>A polygon $\\mathcal{P}$ is drawn on the 2D coordinate plane. Each side of $\\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\\mathcal{P}$ do not have to be lattice points). Given that the interior of $\\mathcal{P}$ includes the interior of the circle \\(x^{2}+y^{2}=2022\\), find the minimum possible perimeter of $\\mathcal{P}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8 \sqrt{2022}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured solution with correct final result.</EF>A vehicle is driven some miles on Monday, 18 miles on Tuesday, and 21 miles on Wednesday. The average distance traveled per day is 17 miles. How many miles were driven on Monday?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and complete coverage. The reasoning is sound and the final result is accurate.</EF>A factory is producing a batch of parts. If it produces 3 fewer parts than planned per day and reduces the production quota by 60 parts, it will take 31 days to complete the production. However, if it produces 3 more parts than planned per day and increases the production quota by 60 parts, it will take 25 days to complete the production. What is the original production quota of the parts?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
215
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and thorough and ultimately correct. Consistently organized and easy to follow.</EF>Without using any tables, find the exact value of the product:\n\\[ P = \\cos \\frac{\\pi}{15} \\cos \\frac{2\\pi}{15} \\cos \\frac{3\\pi}{15} \\cos \\frac{4\\pi}{15} \\cos \\frac{5\\pi}{15} \\cos \\frac{6\\pi}{15} \\cos \\frac{7\\pi}{15}. \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{128}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution with coherent, well-structured reasoning. The approach is sound and the final result is accurate.</EF>Vasiliy Petrov is working on an English language assignment. In this assignment, there are 10 English expressions and their translations into Russian in random order. The task is to match the correct pairs between the expressions and their translations. For each correctly matched pair, 1 point is given. Therefore, it is possible to score from 0 to 10 points. Vasya knows nothing, so he chooses the options randomly. Find the probability that he will get exactly 9 points.\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. Thorough and readable, with conventional, well-structured reasoning.</EF>Find the solution set for the inequality $\\frac {5}{x+2} \\geqslant 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(-2, 3]
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound and the result is confidently stated.</EF>a fill pipe can fill 1 / 2 of cistern in 15 minutes . in how many minutes , it can fill 1 / 2 of the cistern ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete. Clear, logical, and well-structured.</EF>Given that the base edge length of a regular square pyramid is $2$, and the side edge length is $\\sqrt{6}$, determine the volume of the pyramid.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{8}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and thorough, correctly reasoned, and easy to follow. The solution is accurate and well-structured.</EF>Determine the trajectory equation of the center of the hyperbola $x^{2}-y^{2}-6mx-4my+5m^{2}-1=0$ when $m$ takes all real numbers.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2x+3y=0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-organized; analytical and thorough.</EF>James streams on Twitch. He had 150 subscribers and then someone gifted 50 subscribers. If he gets a certain amount per month per subscriber and now makes $1800 a month, how much does he make per subscriber?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$9
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough, methodical reasoning and well-structured presentation.</EF>if x + y = 270 , x - y = 200 , for integers of x and y , y = ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
35
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Efficient and well-structured.</EF>Given the conditions:\n\n1. $|\\overrightarrow{a}|=1$\n2. $(\\overrightarrow{a}+\\overrightarrow{b})\\bot \\overrightarrow{a}$\n3. $(2\\overrightarrow{a}+\\overrightarrow{b})\\bot \\overrightarrow{b}$\n\nFind the magnitude of vector $\\overrightarrow{b}$. Express your answer as a numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Thorough and well-structured, with a solid, self-contained explanation.</EF>A coin is tossed and lands heads up 9 times in a row. What is the probability that it will land heads up on the 10th toss? Express your answer as a fraction.\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 and clear, with thorough reasoning and extra code demonstration. The solution is sound and well-presented, though the code is optional.</EF>If $\\theta$ is an angle in the second quadrant, what is the sign of the value of $\\sin(\\cos\\theta) \\cdot \\cos(\\sin 2\\theta)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
negative
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and logically rigorous; well-organized and easy to follow.</EF>Given the equation of the hyperbola $C: {y}^{2}-\\frac{{x}^{2}}{3}=1$, find the coordinates of the focus. Express your answer as coordinates (x, y), including the ± symbol if applicable.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(0, \pm 2)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and direct. Precise, coherent, and thorough.</EF>Mary starts with some eggs. She finds another 4 eggs and ends up with 31 eggs. How many eggs did Mary start with?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
27
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear solution. Thorough and coherent reasoning overall.</EF>Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$. If $a_2 + a_5 = 19$ and $S_5 = 40$, then find $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
29
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, comprehensive, and clear. Thorough, methodical, and well-structured.</EF>A census taker stands in front of Mrs. Wang's house and asks her: \"You are 40 years old. What are the ages of the three orphans you adopted?\" Mrs. Wang replies: \"The product of their ages is equal to my age, and the sum of their ages is equal to the house number.\" The census taker looks at the house number and says: \"I still can't determine their ages.\" What is Mrs. Wang's house number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough and well-organized reasoning, with an extra verification snippet that is informative but not necessary.</EF>If $\\frac{1}{{\\sqrt{x-3}}}$ is meaningful within the real number range, then the range of real number $x$ is ____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x > 3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct. Clear and thorough. The reasoning is sound and complete.</EF>The ratio of the number of students in grades 10, 11, and 12 of a certain high school is $4:4:5$. Using stratified sampling, a sample of 65 students is to be drawn from these three grades. The number of students to be sampled from grade 10 should be $\\boxed{▲}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear overall; consistently organized and convincing.</EF>Given $x(x\\ln x)' = \\ln x + 1$, and $a= \\int_{1}^{e} \\ln x \\, dx$, calculate the remainder of $a^{100}+2C_{100}^{1}a^{99}+2^2C_{100}^{2}a^{98}+\\ldots+2^{99}C_{100}^{1}a+2^{100}$ when divided by 10. Express your answer as a single digit.\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. Overall, accurate and well-structured reasoning.</EF>Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let \\( p \\) be the probability that the triangle formed by the chosen vertices is acute. Compute \\( 10p \\).\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 coherent and thorough reasoning. Efficient and well-organized.</EF>In the rectangular coordinate system, the coordinates of point $A$ are $(\\sqrt{3}, 1)$, and point $B$ is a moving point on the unit circle with the origin $O$ as the center. The maximum value of $| \\overrightarrow{OA} + \\overrightarrow{OB} |$ is _______.\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 solution; thorough and well-structured, with a solid progression and a proper check.</EF>The function $f(x)$ defined on $\\mathbb{R}$ satisfies $f(x)=3^{x-1}$ ($-3<x\\leq0$), $f(x)=f(x+3)$. Find $f(2019)$.\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. Logical, complete, and well organized.</EF>John has a toilet that uses some gallons of water per flush and his household flushed 15 times per day. He replaced it with a toilet that uses 80% less water per flush. John saved 1800 gallons of water in June. How much water did the old toilet use per flush?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, logical, and clear in its reasoning. Thorough and well-structured.</EF>The salary of a typist was at first raised by 10% and then the same was reduced by 5%. If the typist presently draws Rs. 6270, what was his original salary?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
6000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and structured presentation. Overall, precise and coherent.</EF>A reduction of 40% in the price of oil enables a housewife to obtain 8 kgs more for Rs. 2400. What is the reduced price per kg?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough. Adequately organized and easy to follow.</EF>Mrs. Hilt has 154 markers. They are divided equally into 13 packages. Mrs. Hilt wants to know how many markers are leftover?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured, with thorough and coherent reasoning.</EF>The straight line $y = a$ and the graph of the function $y = x^3 - 3x$ have three distinct intersection points. Find the range of values for $a$. Express your answer in interval notation.\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 comprehensive reasoning; overall, well-structured and convincing.</EF>Fabian went to a park to get some fresh air. He decided to walk there for some hours. Every hour he covers 5 kilometers. Fabian needed 3 more hours to walk to reach a total of 30 kilometers. How many hours did he walk initially?\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; thorough and well organized.</EF>Given an arithmetic sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, where $a_1 \\neq 0$ and $2S_n = a_n^2 + a_n$, find the sum of the first $n$ terms of the sequence ${\\frac{1}{a_n a_{n+1}}}$. Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{n}{n+1}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Precise, complete, and well-structured.</EF>Find the angle between the planes:\n\n$$\n\\begin{aligned}\n& 3x - y + 2z + 15 = 0 \\\\\n& 5x + 9y - 3z - 1 = 0\n\\end{aligned}\n$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is clear, reasoning is coherent, and the explanation is thorough and readable.</EF>If $p(t)$ and $q(t)$ are both seventh-degree polynomials in $t$, what is the degree of $p(t)\\cdot q(t)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear demonstration with thorough, explicit reasoning; well-organized and easy to follow.</EF>Find the distance between line $l_1$: $x-y-1=0$ and line $l_2$: $x-y+1=0$. Express your answer as a numerical value, including the square root symbol if necessary.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with thorough, well-structured reasoning that is logically sound and complete.</EF>Given that the radius of circle $O$ is $5$, and line $l$ is tangent to circle $O$, find the distance from point $O$ to line $l$. Express your answer as a single number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with rigorous and comprehensive reasoning. The solution is thorough, logically sound, and achieves the correct result.</EF>If $p (x) = 2-x^2$ and $q(x) = \\frac{6}{x}$, what is the value of $p (q(2))$?\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, with solid reasoning. The solution is thorough and easy to follow.</EF>A high school senior is preparing to take the independent college entrance exams and needs to choose 3 out of 6 colleges to apply to. Among these colleges, two have exams scheduled at the same time. Calculate the number of different ways the student can register for the exams. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Accessible, coherent, and thorough.</EF>A number is 7% of a third number, and another number is some percentage of the same third number. The first number is 50% of the second number. What percentage is the second number of the third number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
14\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is orderly and well-structured, yielding the right outcome.</EF>Given a regular pyramid with a square base of side length $2$ and a height of $3$, the lateral surface area of the pyramid is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4\sqrt{10}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning and a helpful verification snippet.</EF>Calculate the limit: $\\lim_{n\\to\\infty} \\frac{ C_{ n }^{ 2 }}{1+2+3+\\dots+n}=\\_\\_\\_\\_\\_\\_.$\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 rigorous solution. Thorough, coherent, and well-structured.</EF>Given that \\\\(f(x)\\\\) and \\\\(g(x)\\\\) are respectively odd and even functions defined on the domain \\\\(R\\\\), and \\\\(f(x)+g(x)=3^{x}\\\\). The value of \\\\(f(1)\\\\) is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac {4}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. Thorough, well-structured, and precise.</EF>One-third of 1206 is 300% of a certain number. What is the number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
134
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, rigorous, and clear. The reasoning is sound and the final answer is correct.</EF>At a recent small town election for mayor, a total of 979 votes were cast for the 4 candidates. The winner exceeded his opponents by 53, 79, and some votes, respectively. The candidate in fourth place received 199 votes. How many more votes did the winner receive than the candidate in fourth place?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
105
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete; logically sound and well-structured.</EF>The function $f(x)=ax^{2}+bx$ has an extreme value at $x=\\frac{1}{a}$. The value of $b$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and easy to follow, producing the expected result.</EF>Simplify: $\\sqrt{18}-\\sqrt{32}+\\sqrt{2}$ equals ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete overall.</EF>Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\\ldots$ satisfying the following conditions: $a_{1}=0$, $a_{2}=-|a_{1}+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, $\\ldots$ and so on, then the value of $a_{2023}$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-1011
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete overall.</EF>Mike picked 7 apples, Nancy picked 3 apples, and Keith picked 6 apples and 4 pears at the farm. Additionally, Jennifer picked 5 apples, 2 pears, and 3 oranges, while Tom picked 8 apples and 7 pears, and Stacy collected 4 apples, 5 oranges, and 2 plums. In total, _____ apples were picked.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
33
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear; correct result with sound reasoning and complete, though somewhat verbose, explanation.</EF>Find the value of the constant $a$ in the expansion of ${(x-\\frac{\\sqrt{a}}{x^{2}})}^{6}$ if the constant term is $60$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and complete, with clear and orderly reasoning.</EF>A train 90 meters long completely crosses a bridge of certain length in 36 seconds. The speed of the train is approximately 29 meters per second. What is the length of the bridge?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
954
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear. Efficient and direct.</EF>A rectangular wooden block has a square top and bottom, its volume is $576$ , and the surface area of its vertical sides is $384$ . Find the sum of the lengths of all twelve of the edges of the block.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
112
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct, the reasoning is coherent and complete, and the presentation is easy to follow. Overall, accurate and readable.</EF>For $x > 0$, the area of the triangle with vertices $(0, 0), (x, 2x)$, and $(x, 0)$ is 64 square units. What is the value of $x$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
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