prompt
listlengths 1
1
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|---|---|---|
[
{
"content": "<EF>Accurate and clear. Reasoning is solid, and the final result is correct.</EF>Find the sum of the largest and third largest three-digit numbers with different digits that can be made with the natural numbers 8, 1, and 6.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1542
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Precise and well-structured reasoning with a correct conclusion.</EF>Given the complex number $z = \\frac{1}{2} + \\frac{5}{2}i$, calculate the modulus $|z|$. Express your answer as a single value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{26}}{2}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear; direct and thorough.</EF>Among the positive integers from 1 to 2014, how many integers x satisfy the condition x - [sqrt(x)]^2 = 9? Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
40
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct but verbose, clear and thorough, and logically sound overall.</EF>The minimum positive period of the function $y=\\sin \\pi x$ 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 thorough. Efficiently presented with accurate result and sound reasoning.</EF>Find the mass percentage of Cl in a mixture containing 3 moles of NaClO and 2 moles of NaClO2.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
43.85\%
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct result, clear and logical, and complete in scope.</EF>Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?\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 clear. Methodical and comprehensive.</EF>A polygon has an interior angle sum that is 180° more than 4 times its exterior angle sum. Find the number of edges of this polygon.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
n=11
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear overall. Minor verbosity in the step-by-step doubling, but logically sound and complete.</EF>Misha and Grisha are writing numbers on the board. Misha writes threes, and Grisha writes fives. In total, they wrote 20 numbers. How many fives were written if the sum of all the numbers is 94?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
17
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, thorough, and clear reasoning. The solution is complete and easy to follow.</EF>There are 1000 doors $D_1, D_2, . . . , D_{1000}$ and 1000 persons $P_1, P_2, . . . , P_{1000}$ .\nInitially all the doors were closed. Person $P_1$ goes and opens all the doors.\nThen person $P_2$ closes door $D_2, D_4, . . . , D_{1000}$ and leaves the odd numbered doors open. Next $P_3$ changes the state of every third door, that\nis, $D_3, D_6, . . . , D_{999}$ . (For instance, $P_3$ closes the open door $D_3$ and opens\nthe closed door D6, and so on). Similarly, $P_m$ changes the state of the\nthe doors $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$ while leaving the other doors untouched. Finally, $P_{1000}$ opens $D_{1000}$ if it was closed or closes it if it were\nopen. At the end, how many doors will remain open?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
31
|
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>If 5x = 6y and xy ≠ 0, what is the ratio of 1/3 * x to 1/5 * 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 and clear, with direct reasoning and complete coverage. Thoroughly organized and easy to follow.</EF>A rectangular pool is $6\\text{ m}$ wide, $12\\text{ m}$ long and $4\\text{ m}$ deep. If the pool is half full of water, what is the volume of water in the pool?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
144 \mbox{ m}^3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. The reasoning is sound and the final answer is accurate.</EF>A cube is said to have a side length of 8 centimeters (cm). What is the surface area of this cube in square centimeters (cm2)?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
384
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Measured, coherent, and reasonable in length.</EF>Given that the first term of an arithmetic sequence is 24 and it starts to become negative from the 10th term, find the range of the common difference d. Express your answer as an inequality using standard notation.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-3 \leq d < -\frac{8}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with solid logical progression and a clean final result.</EF>Given the fraction $\\frac{x-4}{x^2}=0$, find the value of $x$.\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 comprehensive. Well-organized and convincing, with correct conclusion.</EF>The coordinates of the point $M(2,-3)$ with respect to the origin are ______.\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 and clear reasoning with a thorough, well-structured presentation.</EF>A rectangle-shaped notebook and a square-shaped notebook have the same area. If the lengths of the two sides of the notebook are 9 centimeters (cm) and 16 centimeters (cm), respectively. How many centimeters (cm) are the length of one side of the notebook?\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. Thoroughly organized and definitive.</EF>20 buckets of water fill a tank when the capacity of each bucket is a certain amount of litres. If 30 buckets are required to fill the same tank with 9-litre buckets, what is the capacity of each of the initial 20 buckets in litres?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
13.5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and coherent, with a clear and complete explanation. Precise and comprehensible.</EF>Given $\\dfrac{1}{\\log_{2}a} + \\dfrac{1}{\\log_{4}a} = 3$, then $a=$ ________.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. The reasoning is logically solid and the final result is accurate.</EF>If the value of the linear function $y=(2a-4)x+3$ increases as the value of $x$ increases, then the range of real number $a$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a > 2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness: accurate. Reasoning: coherent. Clarity: clear. Overall, precise and comprehensible.</EF>Susan had a bouquet of some dozen roses. She gave half to her daughter, and then placed the rest in a vase. The next day, one-third of the flowers in the vase were wilted. After removing the wilted flowers, 12 flowers remained in the vase. How many dozens of roses were in the bouquet initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough and orderly reasoning; completely correct.</EF>Three circles, each with radius \\( r \\), are tangent to each other. Find the area of the figure located outside the circles and bounded by the arcs between the points of tangency.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
r^{2}\left( \sqrt{3} - \frac{\pi}{2}\right)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Overall, the reasoning is sound and accessible.</EF>A city has a total of 1500 stores, including large, medium, and small stores, with a ratio of 1:5:9. If 30 stores are to be sampled using stratified sampling, calculate the number of medium-sized stores that should be sampled. Express your answer as a whole number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with a logically sound flow and complete progression. The reasoning is thorough and well-organized.</EF>There are some items that are members of set U. Of these items, 41 are members of set B, 59 are not members of either set A or set B, and 23 are members of both sets A and B. There are 116 members of set U that are members of set A. How many items are members of set U?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
193
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. The reasoning is solid and the final answer is accurate.</EF>Given that $$sin( \\frac {π}{6}-α)= \\frac {4}{5}$$, calculate the value of $$cos( \\frac {π}{3}+α)$$. Express your answer as a fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{4}{5}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thoroughly explained and well-structured.</EF>In the Cartesian coordinate system $xOy$, angles $\\alpha$ and $\\beta$ both start from $Ox$, and their terminal sides are symmetric about the $y$-axis. If $\\sin \\alpha= \\frac {1}{3}$, then $\\sin \\beta=$ ______.\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, thorough, and clear. Logical, complete, and comprehensible reasoning.</EF>There are 34 bottles of juice in the market, and there are 3/2 times as many bottles of water as juice plus 3 bottles of water. How many bottles of bottled water and juice are there in total in the market?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
88
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and methodical. Definitive and consistent.</EF>Given a triangle \\( ABC \\) with the condition:\n\n\\[ \\cos (\\angle A - \\angle B) + \\sin (\\angle A + \\angle B) = 2 \\]\n\nFind the side \\( BC \\) if \\( AB = 4 \\).\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>Accurate and coherent explanation, with clear reasoning and correct final result.</EF>Alexandra bought 8 magazines at the bookstore on Friday. On Saturday, she went back and bought 12 more. Sunday morning, she saw that the store was having a sale and she bought four times the number of magazines she did on Friday. Later that day she discovered that her dog had chewed up 4 of the magazines. How many magazines does she have now?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
48
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Accurate and well-structured overall.</EF>A shopkeeper sells 600 metres of cloth for a certain amount at a loss of Rs. 10 per metre. His cost price for one metre of cloth is Rs. 70. What was the total amount he sold the cloth for?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
36,000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and thorough. Clear, direct, and well-structured reasoning.</EF>A post office has two types of irregular envelopes. Type A has the following dimensions: the bottom width is 5 inches, the top width is 7 inches, and the height is 6 inches. Type B has dimensions as follows: the bottom width is 4 inches, the top width is 6 inches, and the height is 5 inches.\n\nYou have 3 envelopes of type A and 2 envelopes of type B. What is the combined area of all these envelopes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
158
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and complete. Correctness is solid, and the reasoning is well-structured and easy to follow.</EF>Mahesh can do a piece of work in 45 days. He works at it for 20 days and then Rajesh finished it in a certain number of days. It takes 54 days to complete the work. How many days did it take Rajesh to finish the work after Mahesh stopped working on it?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
34
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness is solid, reasoning is coherent and complete, and the explanation is clear and accessible.</EF>Let $a,$ $b,$ $c,$ $d$ be nonnegative real numbers such that $a + b + c + d = 1.$ Find the minimum value of\n\\[a^2 + b^2 + c^2 + d^2.\\]\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, with thorough reasoning. The explanation is comprehensive and easy to follow.</EF>How many kings can be placed on an $8 \\times 8$ chessboard without putting each other in check?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
16
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>accurate, thorough, and clear. The reasoning is solid, well-organized, and easy to follow.</EF>Given $f(x)=ax^{3}+bx-2$, if $f(2017)=7$, find the value of $f(-2017)$.\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 complete overall; the reasoning is sound and the answer is definite.</EF>Given the function $f\\left(x\\right)=\\sin x+\\cos x+2\\sin x\\cos x+2$, find the maximum value of $f\\left(x\\right)$.\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, clear, and thorough. The reasoning is sound and the conclusion is precisely stated.</EF>3 times a certain number, added to 3 times another number, added to 3 times a third number, and then added to 11 equals 143. What is the second number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
44
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear explanation with a thorough, well-organized reasoning process.</EF>The sum of the first $n$ terms of a geometric sequence with positive terms is $3$, and the sum of the first $3n$ terms is $21$. Find the sum of the first $2n$ terms. Express your answer as a single numerical value.\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. Thoroughly organized and logically sound, with a complete daily breakdown and accurate final total.</EF>Approximate the product of $2.46\\times 8.163\\times (5.17+4.829)$ to the nearest whole number.\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 calculation. Simple, direct, and unambiguous.</EF>There are 6 candies in a pile on the desk. Each candy comes in a package of 15. Some candies are added to the pile, making a total of 10 candies in the pile. How many candies were added to the pile?\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 clear, with thorough reasoning. The explanation is comprehensive and easy to follow.</EF>James has barrels that he uses to store water. They store 3 gallons more than twice as much as a large cask. If he has 4 barrels and a cask stores 20 gallons, how much water can he store?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
172
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear and thorough. Efficient and well-structured.</EF>Determine the coefficient of the term containing $x^3$ in the expansion of $x(1 + x)^6$. Express your answer as a single integer.\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. Efficient, clear, and logically sound.</EF>A and B started a partnership business. A's investment was thrice the investment of B and the period of his investment was two times the period of investments of B. If B received Rs 3000 as profit, what is their total profit?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
21000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and well-structured. Clear and thorough, with a correct final answer.</EF>If p and q are prime numbers, the product of p raised to a certain power and q^5 has 18 divisors. What is the power of p in the product?\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 in reasoning and presentation; thorough and well-structured.</EF>Steven is preparing a shipment of boxes to deliver to a customer for his chemical supply business. He has three trucks that can be sent on each delivery. Each truck can carry a certain load of cargo. Some of the boxes weigh 10 pounds after being packed, and some of the boxes weigh 40 pounds when packed. Steven’s customer has ordered equal quantities of both the lighter and heavier products. Steven can ship 240 boxes of products to his customer in each delivery. What is the maximum load each truck can carry?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Thorough, coherent, and well-structured.</EF>Chris wanted to buy a new video game and an assortment of candy that costs $5. To earn the money, he agreed to babysit his little sister for $8 per hour. He worked 9 hours and had $7 left over after making his purchases. How much does the video game cost?\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. Accurate conclusion with well-structured reasoning.</EF>There are four cups, each with the mouth facing up. Each time, three cups are flipped, and cups that have been flipped before are allowed to be flipped again. After $n$ flips, all the cups have the mouth facing down. What is the smallest value of the positive integer $n$?\n\nNote: Flipping a cup means turning it from mouth up to mouth down or from mouth down to mouth up.\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 well-structured. Thoroughly understandable and sufficiently complete.</EF>Given that vectors $a=(2,4)$ and $b=(x,6)$ are collinear, find the value of the real number $x$. Express your answer as a single real number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and thorough. Precise, direct, and well-structured.</EF>Given the equation $x^{4}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+a_{3}(x-2)^{3}+a_{4}(x-2)^{4}$, calculate the value of $a_{2}$ using the binomial theorem. Express your answer as a single integer.\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 rigorous and comprehensive reasoning. The solution is thorough, logically sound, and achieves the correct result.</EF>A cricket player has played 10 innings with an average of 33 runs. He needs to make 77 runs in his next innings to increase his average of runs by a certain amount. By how much does he want to increase his average of runs per innings?\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 reasonably rigorous. The reasoning is solid and the final result matches the expected answer.</EF>Tim drops a ball off the roof of a 96-foot tall building. The ball bounces to half the height from which it fell on each bounce. How high will it bounce on the fifth bounce?\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 a thorough and orderly reasoning process. Overall, the solution is accurate and well-organized.</EF>The base-10 numbers 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and comprehensive. Well-organized and convincing, with correct conclusion.</EF>Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C$: $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1(a>0,b>0)$. The line $l$ passing through the left focus $F_{1}$ intersects $C$ at a point $P$ in the first quadrant. If $|F_{1}P|=|F_{1}F_{2}|$ and the cosine value of the slope of line $l$ is $\\frac{7}{8}$, then the eccentricity of $C$ 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 and clear, with coherent and thorough reasoning; informative and readable.</EF>Mira jogs every morning at 5 miles per hour. If she jogs for a certain number of hours every morning, she can jog 50 miles in five days. How many hours does Mira jog every morning?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Persistent, coherent, and precise.</EF>A profit of Rs. 800 is divided between X and Y in a certain ratio. The difference between their profit shares is Rs. 160. What is the ratio of Y's share to the total profit?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2:5
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound, and the result is precisely stated.</EF>A line passing through the intersection point of the medians of triangle \\(ABC\\) intersects the sides \\(BA\\) and \\(BC\\) at points \\(A^{\\prime}\\) and \\(C_1\\) respectively. Given that:\n\n\\(BA^{\\prime} < BA = 3\\), \\(BC = 2\\), and \\(BA^{\\prime} \\cdot BC_1 = 3\\). Find \\(BA^{\\prime}\\).\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, methodical reasoning; well-structured and easy to follow.</EF>In the game of Dubblefud, red balls are worth 2 points, blue balls are worth 4 points, and green balls are worth 5 points. In a certain selection of balls, the product of the point values of the balls is 16000. The number of blue balls in this selection equals the number of green balls. How many red balls are in the selection?\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, well-structured and easy to follow. The reasoning is thorough and convincing.</EF>A square piece of paper with side length 10 cm is folded twice as shown in the figure. After cutting off one corner along the line connecting the midpoints of two sides, the remaining unfolded single-layer paper has an area of $\\qquad$ square centimeters.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
75
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and coherent reasoning. The approach is solid and the final result is accurate.</EF>A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units?\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, albeit somewhat verbose; results are accurate and the reasoning is sound and well-structured.</EF>Given $\\frac{1}{a}-\\frac{1}{b}=4$, find the value of $\\frac{a-2ab-b}{2a-2b+7ab}$.\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. Accurate and orderly.</EF>On day 1, 215 liters (l) of water was added to the pool, and on day 2, 76 liters (l) of water was added than on day 1. On the third day, I put 53 liters (l) less than on the second day. How many liters (l) of water do you drink on average each day?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
248
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough. Reliable, methodical, and readable.</EF>A train of some length running at 90 km/hr takes 9 seconds to pass a pole. How long is the train?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
225
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and complete. Efficient, logical, and well-structured.</EF>Given the function f(x) = 1/2*x^2 - 2x - 3*ln(x), find the interval where f(x) is monotonically decreasing. Express your answer as an interval in the format (a, b).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(0, 3)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and thorough. Clear, orderly, and convincing.</EF>2 toucans are sitting on a tree limb. 1 more toucan joins them. In all , there are _____ toucans .\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. Methodical and comprehensive.</EF>Find the LCM of 360, 450, 560, 900, and 1176.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
176400
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and well-structured. The reasoning is sound and complete, and the final value is correct.</EF>Solve the equation $\\lg(4^x+3) = \\lg2^x + \\lg4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
0 \text{ or } \log_{2}3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. Efficiently presented with accurate result and sound reasoning.</EF>The surface area of a sphere is some value. If its radius is doubled, then the surface area of the new sphere will be 9856 cm². What is the surface area of the original sphere?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2464
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and thorough. Clear and organized, with solid reasoning and complete coverage.</EF>Given the vectors $\\overrightarrow{a}=(-1,2), \\overrightarrow{b}=(1,-1)$, calculate the dot product $(\\overrightarrow{a}-\\overrightarrow{b})\\cdot \\overrightarrow{a}=$_______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and coherent. Thorough, methodical, and well-structured.</EF>Select four people from four boys and three girls to participate in the city's high school knowledge competition. The selection must include both boys and girls. How many different selection methods are there?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
34
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear and correct. Logical, complete, and easy to follow.</EF>A sweater is sold for 120 yuan, which is 40% cheaper than its original price. What is the original price? (Solve by setting up an equation)\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. The reasoning is logical and complete, with a well-organized presentation.</EF>The distance between places A and B is 100 kilometers. A bus carrying 20 passengers travels from A to B at a speed of 60 kilometers per hour. Half an hour after departure, the bus breaks down. The passengers have to be picked up by the next bus, which already has 22 passengers and departs from A at a speed of 50 kilometers per hour. After picking up the passengers from the first bus, the bus continues towards B. Along the way, for every additional 2 passengers, the bus's speed decreases by 1 kilometer per hour. The question is: How much time does the second bus take to travel from A to B? (The maximum capacity of the bus is 42 people)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2\text{ hours and }21\text{ minutes}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough and coherent reasoning. The explanation is comprehensive and well organized.</EF>It takes Avery 4 hours to build a brick wall, while Tom takes a certain amount of time to do it. If the two start working together and after an hour Avery leaves, it takes Tom 30 more minutes to complete the wall on his own. How long does it take Tom to build the wall by himself?\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 reasoning and complete steps. The approach is methodical and easy to follow.</EF>A man's speed with the current is 15 km/hr and the speed of the current is some km/hr. The man's speed against the current is 9.4 km/hr. What is the speed of the current?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
2.8
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough reasoning. Accurate and complete.</EF>Given non-collinear vectors $\\overrightarrow{e_1}$ and $\\overrightarrow{e_2}$, vector $\\overrightarrow{a} = 3\\overrightarrow{e_1} - 2\\overrightarrow{e_2}$, and vector $\\overrightarrow{b} = \\overrightarrow{e_1} + \\lambda\\overrightarrow{e_2}$. If $\\overrightarrow{a}$ is parallel to $\\overrightarrow{b}$, find the value of $\\lambda$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{2}{3}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, simple, and clear. Efficient and unambiguous explanation.</EF>There are 285 sweets in the bowl. If 49 of the sweets are red and 59 of the sweets are green, how many of the sweets are neither red nor green?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
177
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, albeit somewhat verbose. The reasoning is thorough and logically solid.</EF>Given a right prism with a regular triangular base and volume V, calculate the side length of the base when the surface area is minimized. Express your answer as a function of V.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt[3]{4V}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough steps and a correct final result.</EF>If Harold split 15 apples between 3 people in her class and kept the left overs, how many apples did each classmate get?\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. The reasoning is logically solid and the final result is accurate.</EF>Let the set \\( X = \\{1, 2, \\cdots, 20\\} \\). The set \\( A \\) is a subset of \\( X \\) with at least 2 elements, and all elements of \\( A \\) can be arranged as consecutive positive integers. Find the number of such subsets \\( A \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
190
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is sound, well-structured, and comprehensive.</EF>Given that \\( p \\) is a prime number and the set \\( \\mathrm{F}=\\{0,1,2, \\ldots, p-1\\} \\), if a subset \\( \\mathrm{A} \\) of \\( \\mathrm{F} \\) satisfies the condition that for any \\( a, b \\in A \\), \\( ab+1 \\pmod{p} \\in A \\), then \\( A \\) is called a good set. Find the possible number of elements in a good set.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
1 \text{ or } p
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and direct. Precise, coherent, and thorough.</EF>Given the complex number $2-bi$ (where $b \\in \\mathbb{R}$), if the sum of its real and imaginary parts is zero, find the value of $b$. Express your answer as a single real number.\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 fairly complete; the reasoning is sound and the final result is correct.</EF>sakshi can do a piece of work in 25 days . tanya is 25 % more efficient than sakshi . the number of days taken by tanya to do the same piece of work :\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and methodical. The reasoning is solid and well-structured.</EF>Given that \\( x_{0}^{2} + x_{0} - 1 = 0 \\). If \\( m = x_{0}^{3} + 2 x_{0}^{2} + 2 \\), find the value of \\( m \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and thorough. The reasoning is coherent and well-structured, with complete coverage of the necessary steps.</EF>Find the intersection point of the line $x+y=5$ and the line $x-y=1$. Express your answer as an ordered pair (x, y).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
(3,2)
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and rigorous. Clear and thorough.</EF>In the interval $[0,\\pi]$, a number $x$ is randomly selected. The probability that $\\sin x$ falls between $0$ and $\\frac{1}{2}$ 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>Clear and correct. Logical and thorough.</EF>John buys a vacuum cleaner for $250 and a dishwasher for a certain price. He has a $75 off coupon and spent $625 in total. How much did the dishwasher cost?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$450
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate, clear, and well-structured, with thorough and coherent reasoning.</EF>Express the repeating decimal 2.5252525... as a fraction in lowest terms and calculate the sum of the numerator and denominator.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
349
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, clear, and comprehensive. The reasoning is sound and easy to follow.</EF>A chocolate chip cookie recipe calls for 2 cups of chocolate chips. You want to make 23 recipes for a bake sale. How many cups of chocolate chips will be needed to make all the cookie recipes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
46
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct, coherent, and comprehensive. Clear, organized, and convincing.</EF>Polly and Gerald went for a fun afternoon riding mini race cars at the munchkin track, which is a one-quarter mile circular track. Polly managed to circle the track a certain number of times in one half hour, but Gerald's car was malfunctioning, and he only moved at an average speed half of what Polly did. Gerald's car averaged a speed of 3 miles per hour. How many times did Polly circle the track?\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 comprehensive, with clear and systematic reasoning. The explanation is thorough and well-organized.</EF>Matt gets a certain commission on a big sale. This commission alone raises his average commission by $150. Matt's new average commission is $250, and he has made 6 sales. What was the commission amount for the big sale?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\$1000
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correctness: accurate. Reasoning: coherent. Clarity: clear. Overall, precise and comprehensible.</EF>The $\\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \\[|x_1 - x_2| + |y_1 - y_2|.\\] For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
$\, 841$
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear overall, with coherent and thorough reasoning.</EF>Suppose f is a polynomial function such that for all real values of x, f(x^2 + 1) = x^4 + 5x^2 + 3. Find f(x^2 - 1). Express your answer as a polynomial.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
x^4 + x^2 - 3
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>The answer is correct, clear, and well-structured; it presents a sound and complete explanation.</EF>Tom catches crabs for a living. He has 8 crab buckets that he checks every day. Each crab bucket gets 12 crabs. He sells each of these crabs for a certain amount. He makes $3360 per week. How much does he sell each crab for?\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 coherent, with clear and thorough reasoning. The explanation is comprehensive and easy to follow.</EF>The minimum positive period of the function $y=\\tan \\left( \\frac{\\pi}{2}x- \\frac{\\pi}{3}\\right)$ 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 well-structured solution. Clear, logical, and complete.</EF>A railway freight station is organizing and dispatching 8 coal freight trains into two groups, each containing 4 trains. The conditions are that trains A and B cannot be in the same group, train A departs first, and train B departs last. If the group containing train A departs first, find the total number of different departure sequences for these 8 trains. Express your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
720
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and uncluttered, with clear and logical reasoning and thorough coverage.</EF>Given set A={x|x=2k-1, k∈Z}, B={x|x=2k, k∈Z}, find A∩B.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\emptyset
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct and clear, with thorough, well-organized reasoning. The approach is solid and the final answer is accurate.</EF>Tom had some violet balloons, he gave Fred 16 of the balloons. Tom now has 14 violet balloons. How many violet balloons did Tom have initially?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear. Correct. Thorough.</EF>Starting in town A, Biker Bob rides his bike 45 miles west encountering a 10% uphill slope, then he goes 25 miles northwest along a winding path with a 5% downhill slope. After that, he rides 35 miles south through a rocky terrain, and finally, he covers 50 miles east with a consistent 7% uphill slope to reach town B. Taking into account the effect of slopes and terrains on his speed, determine the total distance covered by Biker Bob between town A and town B. (Ignore the curvature of the Earth.)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
155
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Accurate and clear, with thorough, methodical reasoning and well-structured presentation.</EF>Find the length of the chord intercepted by the circle $x^{2}+y^{2}=4$ on the line $\\begin{cases} x=2-\\frac{1}{2}t \\\\ y=-1+\\frac{1}{2}t \\end{cases} (t \\text{ is a parameter})$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{14}
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Correct final result, logically solid, and clearly presented.</EF>There are four letters to be put into 3 different mailboxes. How many different ways are there to do this?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
81
|
big_math_pairs_summary_critique_fixed
|
[
{
"content": "<EF>Clear, correct, and thorough reasoning; well-organized and readable.</EF>There are 6 married couples at a party. At the start of the party, every person shakes hands once with every other person except his or her spouse. How many handshakes are there?\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 overall, with a coherent and methodical explanation; the reasoning is sound and complete.</EF>Factorize: $a^3 - 16a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] |
a(a + 4)(a - 4)
|
big_math_pairs_summary_critique_fixed
|
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