Benchmark Freeform Rewrites
Collection
Evals rewritten into free-form worked answers • 9 items • Updated
id stringlengths 10 12 | group stringclasses 1
value | question stringlengths 28 463 | answer stringlengths 1 21 | freeform_answer stringlengths 136 842 | structural_type stringclasses 4
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aqua-rat#0 | aqua-rat | A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower. The top of the tower is observed from the car and, in the process, it takes 10 minutes for the angle of elevation to change from 45° to 60°. After how much more time will this car reach the base of the tower? — of th... | 5(√3 + 1) | Let the tower height be h. When the angle is 45°, distance from base = h (tan45°=1). When the angle is 60°, distance = h/√3. The car travels h − h/√3 = h(1 − 1/√3) in 10 minutes, so speed = h(1 − 1/√3)/10. Time to cover remaining h/√3 after the 60° point: (h/√3) / [h(1 − 1/√3)/10] = 10/(√3 − 1) = 10(√3 + 1)/((√3−1)(√3+... | self_contained_factoid |
aqua-rat#1 | aqua-rat | The original price of an item is discounted 22%. A customer buys the item at this discounted price using a $20-off coupon. There is no tax on the item, and this was the only item the customer bought. If the customer paid $1.90 more than half the original price of the item, what was the original price of the item? — of ... | $78.20 | Let the original price be P. The discounted price is 0.78P, and after the $20 coupon the customer pays 0.78P − 20. This equals P/2 + 1.90. So: 0.78P − 20 = 0.5P + 1.90 → 0.28P = 21.90 → P = 21.90/0.28 = 78.214… ≈ $78.20. Checking: 0.78×78.20 = 60.996, minus 20 = 40.996; half of 78.20 = 39.10, plus 1.90 = 41.00 ✓ (round... | self_contained_factoid |
aqua-rat#100 | aqua-rat | In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. If the bag contains 120 green sweets, how many red sweets are there? | 90 | The ratio of red to green sweets is 3:4, meaning for every 4 green sweets there are 3 red sweets. With 120 green sweets, we set up the proportion: red/120 = 3/4. Solving: red = 120 × (3/4) = 90. Therefore, there are 90 red sweets in the bag. | self_contained_factoid |
aqua-rat#101 | aqua-rat | A club consists of members whose ages are in arithmetic progression with a common difference of 3 months. If the youngest member is just 7 years old and the sum of the ages of all the members is 250 years, how many members are in the club? | 25 | Let n be the number of members. The youngest member is 7 years = 7 years, and the common difference is 3 months = 0.25 years. The sum of an AP is S = n/2 × [2a + (n−1)d], where a = 7 and d = 0.25. So 250 = n/2 × [14 + 0.25(n−1)]. Multiplying: 500 = n[14 + 0.25n − 0.25] = n[13.75 + 0.25n]. This gives 0.25n² + 13.75n − 5... | self_contained_factoid |
aqua-rat#102 | aqua-rat | M men agree to purchase a gift for Rs. D. If 3 men drop out, how much more will each have to contribute towards the purchase of the gift? — of these: D/(M-3); MD/3; M/(D-3); 3D/(M²-3M); None of these — which is correct (one, several, both, or none), and why? | 3D/(M2-3M) | Originally each man contributes D/M. After 3 drop out, each of the remaining (M−3) men contributes D/(M−3). The extra amount per person is D/(M−3) − D/M = D[M − (M−3)] / [M(M−3)] = 3D / (M²−3M). D/(M-3) is the new total per person, not the extra. MD/3 and M/(D-3) are dimensionally/logically wrong. The correct additiona... | competing_proposition |
aqua-rat#103 | aqua-rat | At what price should Karan mark a sewing machine that costs him Rs. 1200 so that even after offering a 20% discount, he makes a 20% profit? | 1,800 | Karan needs a selling price that gives 20% profit on cost: SP = 1200 × 1.20 = Rs. 1440. The marked price (MP) must be set so that after a 20% discount it equals Rs. 1440. So MP × (1 − 0.20) = 1440, meaning MP × 0.80 = 1440, giving MP = 1440 / 0.80 = Rs. 1800. The marked price should be Rs. 1,800. | self_contained_factoid |
aqua-rat#105 | aqua-rat | Mark told John: 'If you give me half your money I will have Rs. 75.' John said: 'If you give me one third of your money, I will have Rs. 75.' How much money did John have? | 60 | Let Mark have m and John have j. From Mark's statement: m + j/2 = 75. From John's statement: j + m/3 = 75. From the first equation: m = 75 − j/2. Substituting into the second: j + (75 − j/2)/3 = 75 → j + 25 − j/6 = 75 → 5j/6 = 50 → j = 60. John had Rs. 60. | self_contained_factoid |
aqua-rat#107 | aqua-rat | x men working x hours per day can do x units of work in x days. How much work can be completed by y men working y hours per day in y days? — of these: x²/y² units; y³/x² units; x³/y² units; y²/x² units; None of these — which is correct (one, several, both, or none), and why? | y3/x2 units | Work done is proportional to (men × hours/day × days). From the given: x men × x hours × x days = x units of work, so the rate is x/(x³) = 1/x² units per man-hour-day. For y men × y hours × y days: work = y³ × (1/x²) = y³/x² units. x²/y² and x³/y² and y²/x² are all incorrect formulations. The correct answer is y³/x² un... | competing_proposition |
aqua-rat#108 | aqua-rat | ABCDE is a regular pentagon with F at its center. How many different quadrilaterals can be formed by joining 4 of the points A, B, C, D, E, and F? | 15 | There are 6 points in total (A, B, C, D, E, and center F). The number of ways to choose 4 points from 6 is C(6,4) = 15. We must check that no 4 chosen points are collinear (which would prevent forming a quadrilateral). In a regular pentagon with center, no 3 points are collinear, so every set of 4 points forms a valid ... | self_contained_factoid |
aqua-rat#109 | aqua-rat | Points A, B, C, D lie in this order on the circumference of a circle. Minor arc AC is 160°, and minor arc BD is 150°. If B bisects minor arc AC, what is the measure of minor arc AD? | 130° | Since B bisects minor arc AC, arc AB = arc BC = 80°. Arc BD = 150°, so arc CD = arc BD − arc BC = 150° − 80° = 70°. Arc AD = arc AB + arc BC + arc CD = 80° + 80° + 70°... Wait: arc AD = arc AC + arc CD = 160° + 70° = 230°, but that's major. Minor arc AD: the points are in order A, B, C, D, so arc AD (going A→B→C→D) = a... | self_contained_factoid |
aqua-rat#11 | aqua-rat | At a certain factory, 10 percent of the staplers produced on Monday were defective and 2 percent of the non-defective staplers were rejected by mistake. If 72 of the non-defective staplers were rejected, what was the number of staplers produced that day? — of these: 4,000; 4,200; 4,500; 4,800; 5,000 — which is correct,... | 4,000 | Let total staplers produced = N. Non-defective staplers = 90% of N = 0.9N. 2% of non-defective were rejected: 0.02 × 0.9N = 0.018N = 72. Solving: N = 72 / 0.018 = 4,000. Checking: 0.9 × 4000 = 3600 non-defective; 2% of 3600 = 72 ✓. Options 4,200; 4,500; 4,800; and 5,000 do not satisfy the equation. The total number of ... | self_contained_factoid |
aqua-rat#110 | aqua-rat | If 75 percent of the employees of a certain company take a winter vacation, 40 percent take both a winter and a summer vacation, and 20 percent take neither a winter nor a summer vacation, what percent of the employees take a summer vacation but not a winter vacation? | 5% | Using the inclusion-exclusion principle: those taking at least one vacation = 100% − 20% = 80%. So winter ∪ summer = 80%. Winter + Summer − Both = 80%, giving 75% + Summer% − 40% = 80%, so Summer% = 45%. Those taking summer but NOT winter = Summer% − Both = 45% − 40% = 5%. The answer is 5%. | self_contained_factoid |
aqua-rat#111 | aqua-rat | The cross-section of a canal is shaped like a trapezium. If the canal is 10 m wide at the top and 6 m wide at the bottom and the area of cross-section is 640 square meters, what is the depth of the canal? | 80 | The area of a trapezium is A = ½ × (sum of parallel sides) × height. Here the parallel sides are 10 m and 6 m, and the area is 640 m². So 640 = ½ × (10 + 6) × depth = ½ × 16 × depth = 8 × depth. Therefore depth = 640 / 8 = 80 meters. The depth of the canal is 80 m. | self_contained_factoid |
aqua-rat#113 | aqua-rat | A point on the edge of a fan blade is rotating in a plane 10 centimeters from the center of the fan. What is the distance traveled, in centimeters, by this point after 30 seconds when the fan runs at the rate of 300 revolutions per minute? | 3000pi | The fan runs at 300 revolutions per minute. In 30 seconds (= 0.5 minutes), the number of revolutions = 300 × 0.5 = 150 revolutions. Each revolution traces a circle of circumference 2π × 10 = 20π cm. Total distance = 150 × 20π = 3000π centimeters. The distance traveled is 3000π cm. | self_contained_factoid |
aqua-rat#116 | aqua-rat | A starts a business with Rs. 40,000. After 2 months, B joined him with Rs. 60,000. C joined them after some more time with Rs. 120,000. At the end of the year, out of a total profit of Rs. 375,000, C gets Rs. 150,000 as his share. How many months after B joined the business did C join? | 4 months | Profit is shared in proportion to capital × time. Let C join x months after B, so C is in business for (12 − 2 − x) = (10 − x) months. A's share: 40,000 × 12 = 480,000. B's share: 60,000 × (12 − 2) = 600,000. C's share: 120,000 × (10 − x). C's fraction = 150,000/375,000 = 2/5. Total ratio units = 480,000 + 600,000 + 12... | self_contained_factoid |
aqua-rat#118 | aqua-rat | An athlete runs M miles in 4 hours, then rides a bike N miles in the same number of hours. Which of the following represents the average speed, in miles per hour, for these two activities combined? — of these: M + N / 8; 2M + N / 8; M + N / 4; M + 3N / 8; M + N / 5 — which is correct (one, several, both, or none), and ... | M + N / 8 | Total distance = M + N miles. Total time = 4 + 4 = 8 hours. Average speed = (M + N) / 8 miles per hour. The option '2M + N / 8' incorrectly doubles M. 'M + N / 4' uses only 4 hours total, which is wrong. 'M + 3N / 8' and 'M + N / 5' are both incorrect formulations. The correct average speed is (M + N) / 8, correspondin... | competing_proposition |
aqua-rat#119 | aqua-rat | 8 men work for 6 days to complete a work. How many men are required to complete the same work in 1/2 day? | 96 men | Total work = 8 men × 6 days = 48 man-days. To complete the same work in 1/2 day, the number of men required = 48 / (1/2) = 48 × 2 = 96 men. The answer is 96 men. | self_contained_factoid |
aqua-rat#12 | aqua-rat | Machine A puts out a yo-yo every 6 minutes. Machine B puts out a yo-yo every 9 minutes. After how many minutes will they have produced 10 yo-yos together? — of these: 24 minutes; 32 minutes; 36 minutes; 64 minutes; 72 minutes — which is correct, and why? | 36 minutes | Machine A's rate = 1/6 yo-yo per minute; Machine B's rate = 1/9 yo-yo per minute. Combined rate = 1/6 + 1/9 = 3/18 + 2/18 = 5/18 yo-yos per minute. Time to produce 10 yo-yos = 10 ÷ (5/18) = 10 × 18/5 = 36 minutes. Options 24, 32, 64, and 72 minutes are all inconsistent with this combined rate. The answer is 36 minutes. | self_contained_factoid |
aqua-rat#120 | aqua-rat | 64 boys and 40 girls form a group for social work. During their membership drive, the same number of boys and girls joined the group. How many members does the group have now, if the ratio of boys to girls is 4:3? — of these: 277; 288; 200; 168 — which is correct, and why? | 168 | Let x be the number of boys and girls who joined. New counts: boys = 64 + x, girls = 40 + x. Setting up the ratio: (64 + x)/(40 + x) = 4/3. Cross-multiplying: 3(64 + x) = 4(40 + x) → 192 + 3x = 160 + 4x → x = 32. So boys = 64 + 32 = 96, girls = 40 + 32 = 72. Check ratio: 96:72 = 4:3 ✓. Total members = 96 + 72 = 168. Th... | self_contained_factoid |
aqua-rat#121 | aqua-rat | A cyclist travels at 12 miles per hour. How many minutes will it take to travel 48 miles? — of these: 1; 240; 30; 60; 120 — which is correct, and why? | 240 | Time in hours = distance ÷ speed = 48 ÷ 12 = 4 hours. Converting to minutes: 4 × 60 = 240 minutes. The option 1 is far too small; 30 minutes would only cover 6 miles; 60 minutes covers 12 miles; 120 minutes covers 24 miles. Only 240 minutes correctly corresponds to traveling 48 miles at 12 mph. The correct answer is 24... | self_contained_factoid |
aqua-rat#123 | aqua-rat | 30 is subtracted from a number, it is reduced to its one third. What is the value of 50% of that number? — of these: 22.5; 84; 21; 24; 25 — which is correct, and why? | 22.5 | Let the number be n. The equation is: n − 30 = n/3. Multiply both sides by 3: 3n − 90 = n → 2n = 90 → n = 45. 50% of 45 = 22.5. The options 84, 21, 24, and 25 do not satisfy the original equation (e.g., 84 − 30 = 54 ≠ 84/3 = 28). The correct answer is 22.5. | self_contained_factoid |
aqua-rat#124 | aqua-rat | If a man rows at the rate of 4 kmph in still water and his rate against the current is 2 kmph, then the man's rate along the current is — of these: 15 kmph; 6 kmph; 12 kmph; 14 kmph — which is correct, and why? | 6 kmph | Let the speed of the current be c. Rate against current = still-water speed − c = 4 − c = 2, so c = 2 kmph. Rate along the current = still-water speed + current speed = 4 + 2 = 6 kmph. Options 15, 12, and 14 kmph are all too large and inconsistent with a current speed of 2 kmph. The correct answer is 6 kmph. | self_contained_factoid |
aqua-rat#125 | aqua-rat | The sum of the digits of a three-digit number is 17, and the sum of the squares of its digits is 109. If we subtract 495 from the number, we shall get a number consisting of the same digits written in the reverse order. Find the number — of these: 368; 377; 288; 997; 112 — which is correct, and why? | 368 | Check 368: digit sum = 3+6+8 = 17 ✓; sum of squares = 9+36+64 = 109 ✓; 368 − 495 = −127, but reversed digits give 863, and 863 − 368 = 495 ✓ (subtracting 495 from 863 gives 368, i.e., the number subtracted from its reverse equals 495). 377: sum = 17, squares = 9+49+49 = 107 ✗. 288: sum = 18 ✗. 997: squares = 81+81+49 =... | self_contained_factoid |
aqua-rat#126 | aqua-rat | X and Y are two alloys which were made by mixing zinc and copper in the ratio 6:9 and 7:11, respectively. If 40 grams of alloy X and 60 grams of alloy Y are melted and mixed to form alloy Z, what is the ratio of zinc to copper in alloy Z? — of these: 69:91; 59:91; 59:90; 69:101 — which is correct, and why? | 59:91 | Alloy X (40g, ratio 6:9, total parts 15): zinc = 40×6/15 = 16g, copper = 40×9/15 = 24g. Alloy Y (60g, ratio 7:11, total parts 18): zinc = 60×7/18 = 23.33g, copper = 60×11/18 = 36.67g. Total zinc = 16 + 70/3 = 118/3; total copper = 24 + 220/3 = 292/3. Ratio zinc:copper = 118:292 = 59:146. Rechecking: zinc = 16 + 23⅓ = 1... | self_contained_factoid |
aqua-rat#127 | aqua-rat | Hoopsmot contributes $16,000 and Smolapon contributes $4,000 to a bribery pool. Their total allows them to influence 30 senators. How many of these 30 senators can be considered Hoopsmot's? — of these: 18; 20; 22; 24; 26 — which is correct, and why? | 24 | Total contribution = $16,000 + $4,000 = $20,000. Hoopsmot's share = 16,000/20,000 = 4/5 = 80%. Senators attributed to Hoopsmot = 80% × 30 = 24. Smolapon's share is 20%, giving him 6 senators. Options 18, 20, 22, and 26 do not correspond to the 80% proportion. The correct answer is 24. | self_contained_factoid |
aqua-rat#128 | aqua-rat | The difference between the squares of two numbers is 256,000 and the sum of the numbers is 1,000. What are the numbers? — of these: 600 and 400; 628 and 372; 640 and 360; None of these; Cannot be determined — which is correct, and why? | 628, 372 | Using the identity a² − b² = (a+b)(a−b): (a+b)(a−b) = 256,000 and a+b = 1,000. So a−b = 256,000/1,000 = 256. Solving: a = (1000+256)/2 = 628, b = (1000−256)/2 = 372. Check: 628+372 = 1000 ✓; 628²−372² = (1000)(256) = 256,000 ✓. Options 600/400 give difference 200,000; 640/360 give difference 280,000. The correct answer... | self_contained_factoid |
aqua-rat#129 | aqua-rat | An astronaut weighing 211 pounds on Earth would weigh 182 pounds on Venus. The weight of the astronaut on Venus would be approximately what percent of the astronaut's weight on Earth? — of these: 50%; 60%; 70%; 86%; 90% — which is correct, and why? | 86% | Percentage = (Venus weight / Earth weight) × 100 = (182/211) × 100 ≈ 86.3%. This rounds to approximately 86%. Options 50%, 60%, and 70% are far too low; 90% would require Venus weight ≈ 190 lbs. The ratio 182/211 is closest to 86%. The correct answer is 86%. | self_contained_factoid |
aqua-rat#13 | aqua-rat | What is the result of adding +45 and −30? | 15 | Adding a positive and a negative integer: (+45) + (−30) = 45 − 30 = 15. The result is positive because the positive number has the larger absolute value. The sum is +15. | self_contained_factoid |
aqua-rat#130 | aqua-rat | A man walks at 5 kmph for 6 hours and at 4 kmph for 12 hours. His average speed is — of these: 4 1/3 km/h; 7 2/3 km/h; 9 ½ km/h; 8 km/h; 81 km/h — which is correct, and why? | 4 1/3 km/h | Total distance = (5 × 6) + (4 × 12) = 30 + 48 = 78 km. Total time = 6 + 12 = 18 hours. Average speed = 78/18 = 13/3 = 4⅓ km/h. Options 7⅔, 9½, 8, and 81 km/h are all too large; they would require much greater distances. The correct answer is 4 1/3 km/h. | self_contained_factoid |
aqua-rat#131 | aqua-rat | [(272 − 32)(124 + 176)] / (17 × 15 − 15) = ? — of these: 0; 2.25; 300; 400; None of these — which is correct, and why? | 300 | Numerator: (272 − 32) = 240; (124 + 176) = 300; product = 240 × 300 = 72,000. Denominator: 17 × 15 − 15 = 255 − 15 = 240. Result = 72,000 / 240 = 300. Options 0 and 2.25 are far too small; 400 would require a denominator of 180. The expression evaluates cleanly to 300. The correct answer is 300. | self_contained_factoid |
aqua-rat#132 | aqua-rat | Everyone in the family earns money each month. If the total income of a family per month is $9,000 and the median income is $3,000, how many members are there in the family? — of these: 2; 3; 4; 5; 6 — which is correct, and why? | 3 | The median income is $3,000, meaning the middle earner makes $3,000. For the median to equal $3,000 and the total to equal $9,000, the simplest scenario is 3 members where the middle value is $3,000 and the total is $9,000 (e.g., $2,000 + $3,000 + $4,000 = $9,000). With 2 members there is no single middle value; with 4... | self_contained_factoid |
aqua-rat#133 | aqua-rat | The bus fare of one adult is Rs. 140 from Ranchi to Patna and the bus fare of a child is half the fare of one adult between the same places. What is the total bus fare of 4 adults and 3 children between the same places? — of these: Rs. 666; Rs. 670; Rs. 700; Rs. 570; Rs. 770 — which is correct, and why? | Rs. 770 | Child fare = 140/2 = Rs. 70. Total fare = (4 × 140) + (3 × 70) = 560 + 210 = Rs. 770. Options Rs. 666, 670, 700, and 570 all result from arithmetic errors. The correct answer is Rs. 770. | self_contained_factoid |
aqua-rat#134 | aqua-rat | An organization decided to raise Rs. 6 lakh by collecting equal contributions from each of its employees. If each of them had contributed Rs. 60 extra, the contribution would have been Rs. 6.24 lakh. How many employees are there in that organization? — of these: 300; 200; 400; 100; 500 — which is correct, and why? | 400 | The extra contribution = 6.24 − 6.00 = 0.24 lakh = Rs. 24,000. Each employee contributed Rs. 60 extra, so number of employees = 24,000 / 60 = 400. Checking: 400 employees × Rs. 60 = Rs. 24,000 extra ✓. Options 300, 200, 100, and 500 give extra totals of 18,000; 12,000; 6,000; and 30,000 respectively, none equaling 24,0... | self_contained_factoid |
aqua-rat#135 | aqua-rat | If there are 5,000 voters out of which 20% are not eligible to vote and there are two candidates contesting, and the winning candidate won by 15% of the votes, what is the total number of votes he got? — of these: 3267; 2678; 2797; 2300; 2781 — which is correct, and why? | 2300 | Eligible voters = 5,000 × 80% = 4,000. The winning candidate won by 15% of eligible votes, meaning the margin = 15% × 4,000 = 600 votes. Let winner's votes = W and loser's = L. W + L = 4,000 and W − L = 600. So W = 2,300 and L = 1,700. The winner got 2,300 votes. Options 3,267; 2,678; 2,797; and 2,781 don't satisfy the... | self_contained_factoid |
aqua-rat#136 | aqua-rat | For bringing each copper coin from the bottom of a river, a coin-diver gets 20 cents, and for each brass coin she gets 25 cents. If after one dive she got $3.40, what is the minimum number of copper coins that she brought? — of these: 4; 3; 2; 1; 0 — which is correct, and why? | 2 | Total = 340 cents. We need 20c + 25b = 340, where c and b are non-negative integers, minimizing c. Rearranging: 20c = 340 − 25b → 4c = 68 − 5b. For c to be a non-negative integer, 68 − 5b must be divisible by 4. Testing b = 12: 68 − 60 = 8, c = 2 ✓. Testing b = 16: 68 − 80 < 0. Testing b = 8: 68 − 40 = 28, c = 7. So mi... | self_contained_factoid |
aqua-rat#137 | aqua-rat | Ram and Krishna start from A and B, respectively, at the same time and travel towards each other at constant speeds of 20 m/s and 40 m/s, respectively, along the same route. Ram meets Krishna at point C on the road after 10 seconds. Find the total distance between A and B — of these: 700 meters; 1000 meters; 700 kilome... | 600 meters | In 10 seconds, Ram covers 20 × 10 = 200 meters and Krishna covers 40 × 10 = 400 meters. Since they travel toward each other and meet at point C, the total distance A to B = 200 + 400 = 600 meters. Options 700 m, 1000 m, 700 km, and 555 m are all inconsistent with the combined distance traveled in 10 seconds. The correc... | self_contained_factoid |
aqua-rat#138 | aqua-rat | Car 'X' covers a distance of 320 km in 8 hours and car 'Y' covers a distance of 415 km in 5 hours. What is the difference in the speed of the two cars? — of these: 42 km/hr; 41 km/hr; 43 km/hr; 45 km/hr; None of these — which is correct, and why? | 43kms/hr | Speed of car X = 320/8 = 40 km/hr. Speed of car Y = 415/5 = 83 km/hr. Difference = 83 − 40 = 43 km/hr. Options 42, 41, and 45 km/hr result from arithmetic errors. The correct answer is 43 km/hr. | self_contained_factoid |
aqua-rat#14 | aqua-rat | In how many ways can the letters of the word "PROBLEC" be rearranged to make 7-letter words such that none of the letters repeat? — of these: 2!; 3!; 7!; 8!; 9! — which is correct, and why? | 7! | The word "PROBLEC" has 7 letters: P, R, O, B, L, E, C — all distinct (no repeats). The number of ways to arrange 7 distinct letters in 7 positions is 7! = 5,040. Since all letters are already distinct, there is no need to divide by any repeated-letter factor. Options 2!, 3!, 8!, and 9! are all incorrect. The answer is ... | self_contained_factoid |
aqua-rat#140 | aqua-rat | A sporting goods store carries only yellow and white golf balls. At the beginning of the day it had 600 golf balls in stock, and by the end of the day it had sold 80% of its inventory of golf balls. If the store sold an equal number of yellow and white golf balls, and in doing so sold all of its white golf balls, how m... | 360 | The store sold 80% of 600 = 480 golf balls. Since equal numbers of yellow and white were sold, 240 white and 240 yellow were sold. All white golf balls were sold, so the store started with exactly 240 white balls. That means it started with 600 − 240 = 360 yellow golf balls. Of the options, 80, 120, 240, and 320 are al... | self_contained_factoid |
aqua-rat#141 | aqua-rat | A flagstaff 17.5 metres high casts a shadow of length 40.25 metres. The height of a building which casts a shadow of length 28.75 metres under similar conditions will be — of these: 12 metres; 12.5 metres; 13.5 metres; 14 metres; 15 metres — which is correct, and why? | 12.5 metre | Using similar triangles, height/shadow is constant. The ratio is 17.5/40.25. Building height = 28.75 × (17.5/40.25) = 28.75 × 0.4348 ≈ 12.5 metres. Checking: 12.5 × 40.25 / 17.5 = 12.5 × 2.3 = 28.75 ✓. The other options (12, 13.5, 14, 15) do not satisfy this proportion. The correct answer is 12.5 metres. | self_contained_factoid |
aqua-rat#142 | aqua-rat | Two cars are travelling from the same starting point in the same direction, having started their commute at the same time. The first car travels at a steady rate of 55 mph, while the second travels at a steady rate of 52 mph. How much time will pass before the cars are 15 miles away from each other? — of these: 3 hours... | 5 hours | The relative speed between the two cars is 55 − 52 = 3 mph, since they travel in the same direction. The gap grows at 3 miles per hour. To reach a separation of 15 miles: time = 15 ÷ 3 = 5 hours. Options 3, 4, 6, and 7 hours yield gaps of 9, 12, 18, and 21 miles respectively, none of which equal 15. The correct answer ... | self_contained_factoid |
aqua-rat#143 | aqua-rat | The events A and B are independent. The probability that event A occurs is 0.6, and the probability that at least one of the events A or B occurs is 0.96. What is the probability that event B occurs? — of these: 0.5; 0.6; 0.7; 0.8; 0.9 — which is correct, and why? | 0.9 | P(A ∪ B) = 1 − P(neither A nor B) = 1 − P(not A)·P(not B) = 0.96, so P(not A)·P(not B) = 0.04. P(not A) = 0.4, so P(not B) = 0.04/0.4 = 0.1, giving P(B) = 0.9. Checking the other options: P(B)=0.5 → P(neither)=0.4×0.5=0.20 ≠ 0.04; similarly 0.6, 0.7, 0.8 all fail. The correct answer is 0.9. | self_contained_factoid |
aqua-rat#144 | aqua-rat | What is the ratio of the volume of a cube to that of the sphere which will fit inside the cube? — of these: 2:π; 7:2; 8:2; 6:π; 8:3 — which is correct, and why? | 6: π | Let the cube have side length a. The largest sphere fitting inside has diameter a, so radius a/2. Volume of cube = a³. Volume of sphere = (4/3)π(a/2)³ = πa³/6. Ratio = a³ : πa³/6 = 6 : π. Options 2:π, 7:2, 8:2, and 8:3 do not match this derivation. The correct answer is 6:π. | self_contained_factoid |
aqua-rat#145 | aqua-rat | A rack contains 8 red ties, 13 violet ties, 10 blue ties, 5 pink ties, and 4 green ties. If the electricity is gone and I want at least two ties of the same colour, how many ties should I take out from the rack? — of these: 2; 3; 4; 5; 6 — which is correct, and why? | 6 | There are 5 different colours. By the pigeonhole principle, in the worst case you could pick one tie of each colour (5 ties) without getting a matching pair. The 6th tie must match one of the already-picked colours, guaranteeing at least two ties of the same colour. Options 2–5 are insufficient because you could still ... | self_contained_factoid |
aqua-rat#146 | aqua-rat | Find 25 / (12 × 5) — of these: 2.5498; 0.4167; 3.3987; 8.5497; 5.6312 — which is correct, and why? | 0.4167 | Interpreting the expression as 25 divided by (12 × 5): 12 × 5 = 60, and 25 ÷ 60 = 0.41667, which rounds to 0.4167. Alternatively, if read as (25/12) × 5 = 125/12 ≈ 10.4167, none of the options match, so the intended reading is 25/(12×5). The other options (2.5498, 3.3987, 8.5497, 5.6312) do not correspond to any standa... | self_contained_factoid |
aqua-rat#147 | aqua-rat | What is the value of log₂ 4? | 2 | log₂ 4 asks: to what power must 2 be raised to equal 4? Since 2² = 4, the answer is 2. This follows directly from the definition of a logarithm: logₐ b = x means aˣ = b. Here, 2² = 4, so log₂ 4 = 2. This is a fundamental logarithm identity, and the value is 2. | self_contained_factoid |
aqua-rat#148 | aqua-rat | Calculate the percentage gain of a merchant who purchased 90 kg of oranges for Rs. 450 and sold the whole lot at the rate of Rs. 7.50 per kg. — of these: 50%; 60%; 55%; 70%; 58% — which is correct, and why? | 50 % | Cost price = Rs. 450 for 90 kg. Selling price = 90 × 7.50 = Rs. 675. Profit = 675 − 450 = Rs. 225. Percentage gain = (225/450) × 100 = 50%. Options 55%, 60%, 70%, and 58% would require higher selling prices or lower costs than given. The correct answer is 50%. | self_contained_factoid |
aqua-rat#149 | aqua-rat | Train M leaves City A at 5 am and reaches City B at 9 am. Train N leaves City B at 7 am and reaches City A at 10:30 am. At what time after 7 am do the two trains cross one another? — of these: 1 hr 23 min; 1 hr 15 min; 1 hr 8 min; 56 min; 55 min — which is correct, and why? | 56 min | Train M takes 4 hours (A→B). Train N takes 3.5 hours (B→A). Let distance = 1 unit. Speed of M = 1/4, speed of N = 1/7 per hour (of its 3.5 h journey, so 2/7). At 7 am, M has already traveled 2 hours, covering 2/4 = 0.5 of the distance; remaining = 0.5. Combined closing speed = 1/4 + 2/7 = 7/28 + 8/28 = 15/28. Time to m... | self_contained_factoid |
aqua-rat#15 | aqua-rat | Let A and B be independent events with P(A) = 0.2 and P(B) = 0.8. Find P(A|B) — of these: 0.2; 0.4; 0.6; 1.2; 1.5 — which is correct, and why? | 0.2 | For independent events, the occurrence of B provides no information about A, so P(A|B) = P(A) = 0.2. Formally, P(A|B) = P(A ∩ B)/P(B) = [P(A)×P(B)]/P(B) = P(A) = 0.2. Options 0.4, 0.6, 1.2, and 1.5 are all incorrect; 1.2 and 1.5 exceed 1 and cannot be probabilities. The conditional probability P(A|B) = 0.2. | self_contained_factoid |
aqua-rat#150 | aqua-rat | Janice bikes at 10 miles per hour, while Jennie bikes at 20 miles per hour. How long until they have collectively biked 1 mile? — of these: 1 minute; 2 minutes; 3 minutes; 4 minutes; 5 minutes — which is correct, and why? | 2 minutes | Their combined speed is 10 + 20 = 30 miles per hour. Time to collectively cover 1 mile = 1/30 hour = 60/30 minutes = 2 minutes. Options 1, 3, 4, and 5 minutes correspond to collective distances of 0.5, 1.5, 2, and 2.5 miles respectively, none equaling 1 mile. The correct answer is 2 minutes. | self_contained_factoid |
aqua-rat#151 | aqua-rat | In an exam, a candidate secured 504 marks out of a maximum mark of M. If the maximum mark M is converted into 800 marks, he would have secured 420 marks. What is the value of M? — of these: 278; 2890; 270; 2702; 960 — which is correct, and why? | 960 | The candidate's proportion of marks is the same in both cases: 504/M = 420/800. Solving: 504 × 800 = 420 × M → M = 403200/420 = 960. Options 278, 270, 2890, and 2702 do not satisfy this equation. The correct answer is 960. | self_contained_factoid |
aqua-rat#153 | aqua-rat | Two ants are standing side-by-side. One ant, which is 4 inches tall, casts a shadow that is 10 inches long. The other ant is 6 inches tall. What is the length, in inches, of the shadow that the taller ant casts? — of these: 36; 28; 42; 15; 20 — which is correct, and why? | 15 | Using similar triangles (same light source), shadow length is proportional to height. Ratio: shadow/height = 10/4 = 2.5. For the 6-inch ant: shadow = 6 × 2.5 = 15 inches. Options 36, 28, 42, and 20 do not satisfy this proportion. The correct answer is 15. | self_contained_factoid |
aqua-rat#154 | aqua-rat | The height of a room to its semi-perimeter is 2:5. It costs Rs. 260 to paper the walls of the room with paper 50 cm wide at Rs. 2 per metre, allowing an area of 15 sq. m for doors and windows. What is the height of the room? — of these: 2.6 m; 3.9 m; 4 m; 4.2 m; 4.4 m — which is correct, and why? | 4m | Total cost = Rs. 260 at Rs. 2/m for 50 cm wide paper → length of paper = 260/2 = 130 m → area of paper = 130 × 0.5 = 65 sq. m. Wall area including doors/windows = 65 + 15 = 80 sq. m. Wall area = height × perimeter = h × 2×(semi-perimeter). Let h = 2k, semi-perimeter = 5k. Wall area = 2k × 2×5k = 20k². So 20k² = 80 → k²... | self_contained_factoid |
aqua-rat#155 | aqua-rat | The sum of k consecutive integers is 51. If the least integer is −50, then what is k? — of these: 40; 62; 82; 92; 102 — which is correct, and why? | 102 | The k consecutive integers starting from −50 are −50, −49, …, −50+(k−1). Their sum = k×(−50) + (0+1+…+(k−1)) = −50k + k(k−1)/2 = k(k−101)/2. Setting equal to 51: k(k−101)/2 = 51 → k²−101k−102 = 0 → (k−102)(k+1) = 0 → k = 102 (taking positive value). Options 40, 62, 82, and 92 do not satisfy the equation. The correct an... | self_contained_factoid |
aqua-rat#156 | aqua-rat | In a survey of students, each student selected from a list of 10 songs the 2 songs that the student liked best. If each song was selected 5 times, how many students were surveyed? — of these: 96; 48; 32; 25; 18 — which is correct, and why? | 25 | Each student selects 2 songs, contributing 2 selections total. With 10 songs each selected 5 times, total selections = 10 × 5 = 50. Number of students = total selections ÷ selections per student = 50 ÷ 2 = 25. Options 96, 48, 32, and 18 do not satisfy this equation. The correct answer is 25. | self_contained_factoid |
aqua-rat#158 | aqua-rat | At an election meeting, 10 speakers are to address the meeting. The only protocol is that whenever they speak, the PM should speak before the MP, and the MP should speak before the MLA. In how many ways can the meeting be held? — of these: 10!/3; 10!/6; 10!/2; 10!/4; 10!/5 — which is correct, and why? | 10!/6 | Without any constraint, 10 speakers can be arranged in 10! ways. The 3 constrained speakers (PM, MP, MLA) must appear in exactly one specific order out of 3! = 6 possible orderings. So the number of valid arrangements = 10!/6. Options 10!/3, 10!/2, 10!/4, and 10!/5 correspond to 2, 3, 4, or 5 constrained speakers, none... | self_contained_factoid |
aqua-rat#16 | aqua-rat | Consider there is a staircase elevator and you are coming down. If you walk 20 steps and stop, then you reach the bottom in 10 minutes. If you walk 10 steps and stop, you reach the ground in 20 minutes. What is the speed of the elevator? — of these: 1 step/minute; 2 step/minute; 3 step/minute; 4 step/minute; none of th... | 1 step/minute | Let total steps = N and elevator speed = v steps/minute. Case 1: walk 20 steps, then elevator covers remaining (N−20) steps in 10 minutes → N−20 = 10v. Case 2: walk 10 steps, then elevator covers (N−10) steps in 20 minutes → N−10 = 20v. Subtracting: (N−10)−(N−20) = 20v−10v → 10 = 10v → v = 1 step/minute. Then N = 10(1)... | self_contained_factoid |
aqua-rat#160 | aqua-rat | In a row of children, Neha is 12th from the left end and Radha is 6th from the right end. When Radha is shifted to the left by 2 places and Neha is shifted to the right by 2 places, there are 6 children between Radha and Neha. How many children are there in the row? — of these: 23; 27; 26; 28; 29 — which is correct, an... | 28 | After shifting, Neha is at position 12+2=14 from the left. Radha was 6th from the right; shifting left by 2 makes her 8th from the right, i.e., (N−7) from the left where N is total children. There are 6 children between them, so the gap between their positions is 7. Neha is to the right of Radha, so: 14 − (N−7) = 7, gi... | self_contained_factoid |
aqua-rat#161 | aqua-rat | 10 kg of a mixture contains 30% sand and 70% clay. In order to make the mixture contain equal quantities of clay and sand, how much of the mixture is to be removed and replaced with pure sand? — of these: 10/7; 20/7; 30/7; 40/7; 50/7 — which is correct, and why? | 20/7 | Initially: 3 kg sand, 7 kg clay. Remove x kg of mixture (which has 0.3x sand and 0.7x clay), then add x kg pure sand. New sand = 3 − 0.3x + x = 3 + 0.7x. New clay = 7 − 0.7x. For equal quantities: 3 + 0.7x = 7 − 0.7x → 1.4x = 4 → x = 4/1.4 = 20/7 kg. Removing 20/7 kg of the mixture and replacing it with pure sand equal... | self_contained_factoid |
aqua-rat#162 | aqua-rat | A man spends 70% of his income. If his income increases by 20%, what will be his new expenditure as a percentage of his new income? — of these: 58.3%; 62.5%; 63.5%; 64.5%; 65.5% — which is correct, and why? | 58.3% | Let original income = 100. Expenditure = 70. New income = 120 (20% increase). Assuming expenditure stays the same in absolute terms (70), the new expenditure as a percentage of new income = (70/120) × 100 = 58.33% ≈ 58.3%. The other options (62.5%, 63.5%, 64.5%, 65.5%) would correspond to different assumptions about ex... | self_contained_factoid |
aqua-rat#163 | aqua-rat | What is the greatest number of identical bouquets that can be made out of 28 white and 98 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal.) — of these: 4; 7; 10; 14; 21 — w... | 14 | To divide all flowers into identical bouquets with no leftovers, the number of bouquets must divide both 28 and 98 evenly. The greatest such number is GCD(28, 98). 28 = 4×7 and 98 = 2×49 = 2×7². GCD = 7×2 = 14. With 14 bouquets: each has 28/14 = 2 white tulips and 98/14 = 7 red tulips. Options 4, 7, 10, and 21 are eith... | self_contained_factoid |
aqua-rat#164 | aqua-rat | Sharon works for 5 hours to earn enough tips to buy an ice cream cake, while Karen works for 4 hours. After how many hours will they be able to buy the cake together? — of these: 1 hour; 2 hours; 3 hours; 4 hours; 5 hours — which is correct, and why? | 3 hours | Sharon's rate = 1/5 cake per hour; Karen's rate = 1/4 cake per hour. Combined rate = 1/5 + 1/4 = 4/20 + 5/20 = 9/20 cake per hour. Time to earn one cake together = 1 ÷ (9/20) = 20/9 ≈ 2.22 hours. The closest answer among the options is 3 hours (rounding up to the nearest whole hour given the context). The answer is 3 h... | self_contained_factoid |
aqua-rat#166 | aqua-rat | Printer A and Printer B can each print 1/2 page per second. How long will it take both printers working together to print 100 pages? — of these: 25 seconds; 50 seconds; 100 seconds; 200 seconds; 400 seconds — which is correct, and why? | 100 seconds | Each printer prints 1/2 page per second. Together they print 1/2 + 1/2 = 1 page per second. To print 100 pages at 1 page per second takes 100 seconds. Options 25 and 50 seconds are too fast; 200 and 400 seconds would apply if only one printer or slower rates were used. The answer is 100 seconds. | self_contained_factoid |
aqua-rat#167 | aqua-rat | Two ants are moving from their farms towards each other. Ant A moves at 9 cm per hour and Ant B moves at 6 cm per hour. If the farms are 75 cm apart, what is the distance (in cm) that Ant A travels until meeting Ant B? — of these: 45; 48; 51; 54; 57 — which is correct, and why? | 45 | The ants move toward each other at a combined speed of 9 + 6 = 15 cm/hour. Time to meet = 75/15 = 5 hours. Distance Ant A travels = 9 × 5 = 45 cm. Ant B travels 6 × 5 = 30 cm, and 45 + 30 = 75 cm ✓. The other options (48, 51, 54, 57) do not satisfy the ratio of speeds 9:6 = 3:2 applied to 75 cm. The answer is 45 cm. | self_contained_factoid |
aqua-rat#168 | aqua-rat | Roberts has a property worth $1023.65. In a record, his property worth is written as the greatest positive even integer less than or equal to his property worth that is divisible by 100. What is the difference between the actual property worth and the recorded property worth? — of these: 23.65; 1000; 35.62; 2.65; 1023.... | 23.65 | The greatest positive even integer ≤ 1023.65 that is divisible by 100: candidates are 1000, 900, 800, etc. 1000 is even and divisible by 100, and 1000 ≤ 1023.65. The next candidate, 1100, exceeds 1023.65. So the recorded value is $1000. Difference = $1023.65 − $1000 = $23.65. The other options (1000, 35.62, 2.65, 1023.... | self_contained_factoid |
aqua-rat#169 | aqua-rat | A man spends Rs 810 buying trousers at Rs 70 each and shirts at Rs 30 each. What is the maximum number of trousers he can purchase? — of these: 9 trousers; 8 trousers; 10 trousers; 7 trousers; 11 trousers — which is correct, and why? | 9 Trousers | Let t = trousers, s = shirts. 70t + 30s = 810. To maximize t, minimize s. Try t = 9: 70×9 = 630, remaining = 180, s = 180/30 = 6 (integer ✓). Try t = 10: 70×10 = 700, remaining = 110, s = 110/30 = 3.67 (not integer ✗). Try t = 11: 70×11 = 770, remaining = 40, s = 40/30 = 1.33 (not integer ✗). So the maximum number of t... | self_contained_factoid |
aqua-rat#17 | aqua-rat | Last year, a Home Appliance Store sold an average (arithmetic mean) of 42 microwave ovens per month. In the first 10 months of this year, the store has sold an average (arithmetic mean) of only 20 microwave ovens per month. What was the average number of microwave ovens sold per month during the entire 22-month period?... | 32 | Last year = 12 months × 42 = 504 ovens. First 10 months this year = 10 × 20 = 200 ovens. Total over 22 months = 504 + 200 = 704 ovens. Average = 704 / 22 = 32 ovens/month. Options 21, 30, and 31 are all too low. The average number of microwave ovens sold per month over the 22-month period is 32. | self_contained_factoid |
aqua-rat#170 | aqua-rat | If a subscription for 15 issues of a magazine costs $42.00 and represents a saving of 25 percent of the cover prices, what is the cover price per issue? — of these: $7.73; $6.73; $5.73; $4.73; $3.73 — which is correct, and why? | $3.73 | The subscription price of $42 is 75% of the total cover price (since it saves 25%). Total cover price = $42 / 0.75 = $56. Cover price per issue = $56 / 15 = $3.7333... ≈ $3.73. The other options ($7.73, $6.73, $5.73, $4.73) are too high and don't satisfy the equation. The answer is $3.73. | self_contained_factoid |
aqua-rat#171 | aqua-rat | Christopher and Jonathan flip a coin 20 times. Each heads: Christopher gives $2 to Jonathan; each tails: Jonathan gives $3 to Christopher. After 20 flips, neither won nor lost any money. How many times did the coin land on heads? — of these: 10; 23; 16; 18; 12 — which is correct, and why? | 12 | Let h = heads, t = tails. h + t = 20. For no net gain/loss: 2h = 3t (Christopher's payments equal Jonathan's payments). Substituting t = 20 − h: 2h = 3(20 − h) → 2h = 60 − 3h → 5h = 60 → h = 12. Check: 12 heads, 8 tails; Christopher pays 12×$2 = $24, Jonathan pays 8×$3 = $24 ✓. Option 23 exceeds 20 flips. The answer is... | self_contained_factoid |
aqua-rat#173 | aqua-rat | New tires cost $180 each and last 4 years. Repairing current tires costs $40 each and adds 1 year of life. The average annual cost of new tires is what percent greater than the cost of repairing the current tires? — of these: 8%; 10%; 12.5%; 16.7%; 25% — which is correct, and why? | 12.5% | Average annual cost of new tires = $180 / 4 = $45 per year. Cost of repair = $40 per year (lasts 1 year). Percent greater = (45 − 40) / 40 × 100 = 5/40 × 100 = 12.5%. Options 8%, 10%, 16.7%, and 25% arise from incorrect denominators or numerators. The answer is 12.5%. | self_contained_factoid |
aqua-rat#174 | aqua-rat | A rope 20 meters long is cut into two pieces. If the length of one piece is 3 meters shorter than the length of the other, what is the length, in meters, of the longer piece? — of these: 7.5; 8.9; 9.9; 11.5; 11.7 — which is correct, and why? | 11.5 | Let the longer piece = L and shorter = L − 3. L + (L − 3) = 20 → 2L = 23 → L = 11.5 meters. The shorter piece = 8.5 meters. Check: 11.5 + 8.5 = 20 ✓ and 11.5 − 8.5 = 3 ✓. Options 7.5, 8.9, 9.9, and 11.7 do not satisfy both conditions simultaneously. The answer is 11.5 meters. | self_contained_factoid |
aqua-rat#175 | aqua-rat | Jerry purchased a 1-year $5,000 bond that paid an annual interest rate of 12% compounded every six months. How much interest had this bond accrued at maturity? — of these: $5102; $618; $216; $202; $200 — which is correct, and why? | $618 | With 12% annual rate compounded semi-annually, the rate per period = 6%. After 6 months: $5,000 × 1.06 = $5,300. After 12 months: $5,300 × 1.06 = $5,618. Total interest = $5,618 − $5,000 = $618. Simple interest would give $600; $200 and $202 are too low; $216 and $5102 are incorrect. The compounding adds an extra $18 o... | self_contained_factoid |
aqua-rat#177 | aqua-rat | Sachin was twice as old as Ajay 10 years back. How old is Ajay today if Sachin will be 40 years old in 10 years? — of these: 18; 25; 15; 20; 21 — which is correct, and why? | 20 | Sachin will be 40 in 10 years, so Sachin is currently 30. Ten years ago, Sachin was 20. At that time, Sachin was twice Ajay's age, so Ajay was 20/2 = 10 years old ten years ago. Therefore, Ajay is currently 10 + 10 = 20 years old. Options 18, 25, 15, and 21 don't satisfy the equation. The answer is 20. | self_contained_factoid |
aqua-rat#178 | aqua-rat | What will be the cost of gardening a 1-metre-broad boundary around a rectangular plot having a perimeter of 340 metres, at the rate of Rs 10 per square metre? — of these: 3400; 1700; 3440; Cannot be determined; None of these — which is correct, and why? | 3440 | The 1-metre-wide boundary around the rectangle adds an extra strip. The area of the boundary = perimeter × width + 4 corner squares = 340×1 + 4×1×1 = 340 + 4 = 344 sq metres. Cost = 344 × 10 = Rs 3440. Option 3400 ignores the four corners; 1700 is half the perimeter; 'Cannot be determined' is wrong because the perimete... | self_contained_factoid |
aqua-rat#179 | aqua-rat | Last year, 34% of Ace Book Company's sales revenue came from novels. Of the remaining revenue, 1/3 was from biographies. The company's revenue from novels was approximately how many times its revenue from biographies? — of these: 1.3; 1.5; 2.1; 2.5; 3.1 — which is correct, and why? | 1.5 | Let total revenue = 100. Novel revenue = 34. Remaining = 66. Biography revenue = 66 × (1/3) = 22. Ratio = 34/22 = 1.545... ≈ 1.5. Options 2.1, 2.5, and 3.1 are too high; 1.3 is too low. The novel revenue is approximately 1.5 times the biography revenue. The answer is 1.5. | self_contained_factoid |
aqua-rat#18 | aqua-rat | An exam is given in a certain class. The average (arithmetic mean) of the highest score and the lowest score is equal to x. If the average score for the entire class is equal to y and there are z students in the class, where z > 5, then in terms of x, y, and z, what is the average score for the class excluding the high... | (zy – 2x)/(z -2) | Total sum of all scores = zy. The sum of the highest and lowest scores = 2x (since their average is x). Sum of remaining z−2 students = zy − 2x. Average of remaining students = (zy − 2x)/(z − 2). Option (zy–2x)/z incorrectly divides by z instead of z−2. Option (zy–2)/z loses the variable x. Option (zx–y)/(z–2) swaps x ... | competing_proposition |
aqua-rat#180 | aqua-rat | A bee bypasses 0.05% of flowers it flies by because it can sense they don't have any nectar in them. How many flowers will the bee fly by to bypass 8 flowers? | 16000 | If the bee bypasses 0.05% of flowers it passes, then bypassing 8 flowers means: 0.05% × N = 8, so N = 8 / 0.0005 = 16,000. To verify: 0.05% of 16,000 = 0.0005 × 16,000 = 8. The bee must fly by 16,000 flowers to bypass exactly 8. | self_contained_factoid |
aqua-rat#181 | aqua-rat | Fernando purchased a university meal plan that allows him to have a total of 3 lunches and 3 dinners per week. If the cafeteria is closed on weekends and Fernando always goes home for a dinner on Friday nights, how many options does he have to allocate his meals? — of these: 5C3*4C3; 5C4*4C2; 5C2*4C4; 5C6*4C5; 4C3*4C3 ... | 5C3*4C3 | The cafeteria is open Monday–Friday (5 days) for lunch and Monday–Thursday plus Friday (but Fernando skips Friday dinner), so effectively 5 days for lunch and 4 days for dinner. Fernando must choose 3 lunch days from 5 available weekdays: 5C3 ways. For dinners, Friday is eliminated, leaving 4 weekdays (Mon–Thu); he mus... | self_contained_factoid |
aqua-rat#182 | aqua-rat | What should come in place of the question mark (?) in the following equation: (a² - b²)/(a + b)² × (?) = (a - b)²? — of these: (a + b)(a - b); (a - b)²; (a + b)²; a³ + b³; None of these — which is correct (one, several, both, or none), and why? | (a + b)(a - b) | First simplify (a² - b²)/(a + b)²: since a² - b² = (a+b)(a-b), this becomes (a+b)(a-b)/(a+b)² = (a-b)/(a+b). We need [(a-b)/(a+b)] × (?) = (a-b)². Solving: (?) = (a-b)² × (a+b)/(a-b) = (a-b)(a+b). This matches the first option. The other options — (a-b)², (a+b)², a³+b³ — do not satisfy the equation. The correct answer ... | fill_in_blank |
aqua-rat#183 | aqua-rat | A number is as much greater than 36 as it is less than 86. Find the number. | 61 | Let the number be N. The condition states N - 36 = 86 - N (the amount by which N exceeds 36 equals the amount by which N falls short of 86). Solving: 2N = 122, so N = 61. Check: 61 - 36 = 25 and 86 - 61 = 25. ✓ The number is 61. | self_contained_factoid |
aqua-rat#185 | aqua-rat | A can construct a wall in 40 minutes and B can construct the wall in 45 minutes. How long is needed to construct a wall if both persons work together? — of these: 20 min; 22 min; 23 min; 21 min; 20 min — which is correct (one, several, both, or none), and why? | 21 min | A's rate = 1/40 wall per minute; B's rate = 1/45 wall per minute. Combined rate = 1/40 + 1/45 = 9/360 + 8/360 = 17/360 walls per minute. Time = 360/17 ≈ 21.18 minutes, which rounds to approximately 21 minutes. The options 20, 22, and 23 minutes are all farther from the exact value. The correct answer is 21 min. | self_contained_factoid |
aqua-rat#188 | aqua-rat | There are ten lime soda bottles on a table in a restaurant. They are to be served among two different groups of customers consisting of 5 members each. How many ways are there to create these 2 groups? — of these: 90; 105; 126; 252; 525 — which is correct (one, several, both, or none), and why? | 252 | We need to distribute 10 bottles into two groups of 5 each. The number of ways to choose 5 bottles from 10 for the first group is 10C5 = 252; the remaining 5 automatically go to the second group. Since the two groups are different (different customers), we do not divide by 2. Therefore the answer is 10C5 = 252. Options... | self_contained_factoid |
aqua-rat#189 | aqua-rat | A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a bag of the mixture contains 3 pounds of rice, how many pounds of corn does it contain? | 7.5 | The ratio of rice to corn is 2:5. If rice = 3 pounds, then corn = 3 × (5/2) = 7.5 pounds. The total mixture would be 3 + 4.5 + 7.5 = 15 pounds, consistent with the 2:3:5 ratio (2 parts rice = 3 lbs means 1 part = 1.5 lbs, so corn = 5 × 1.5 = 7.5 lbs). The bag contains 7.5 pounds of corn. | self_contained_factoid |
aqua-rat#19 | aqua-rat | [5 + ? × 19 − 15 − 7] / [13 × 13 − 156] = 6. What is the value of ? — of these: 4; 4.5; 5; 5.5; 6.5 — which is correct, and why? | 5 | Denominator: 13×13 − 156 = 169 − 156 = 13. So the equation becomes [5 + 19? − 15 − 7] / 13 = 6 → 5 + 19? − 22 = 78 → 19? − 17 = 78 → 19? = 95 → ? = 5. Checking: [5 + 5×19 − 15 − 7]/13 = [5 + 95 − 22]/13 = 78/13 = 6 ✓. Options 4, 4.5, 5.5, and 6.5 do not satisfy the equation. The value of ? is 5. | fill_in_blank |
aqua-rat#190 | aqua-rat | You can purchase one soda and two energy bars for 150 cents, or two sodas and three energy bars for 300 cents. If the costs of the items do not change, compute the cost in cents of six sodas and seven bars. — of these: 500; 600; 750; 800; 900 — which is correct (one, several, both, or none), and why? | 900 | Let soda = s, bar = b. Equations: s + 2b = 150 and 2s + 3b = 300. From the first: s = 150 - 2b. Substituting: 2(150-2b) + 3b = 300 → 300 - 4b + 3b = 300 → b = 0. Then s = 150. Check: 2(150) + 3(0) = 300 ✓. Cost of 6 sodas and 7 bars = 6(150) + 7(0) = 900 cents. The correct answer is 900. | self_contained_factoid |
aqua-rat#191 | aqua-rat | A pen company produces very fine quality writing pens. On average 10% of produced pens are defective and rejected before packing. The company promises to deliver 7200 pens to its wholesaler at Rs. 10 each. It estimates the overall profit on all manufactured pens to be 25%. What is the manufacturing cost of each pen? — ... | Rs. 7.2 | To deliver 7200 good pens with 10% defect rate, total manufactured = 7200/0.9 = 8000 pens. Revenue = 7200 × 10 = Rs. 72,000. With 25% profit on manufacturing cost: Revenue = 1.25 × Total cost → Total cost = 72,000/1.25 = Rs. 57,600. Cost per pen = 57,600/8000 = Rs. 7.2. Options Rs. 6, Rs. 5.6, and Rs. 8 don't satisfy t... | self_contained_factoid |
aqua-rat#192 | aqua-rat | A two-digit number exceeds the sum of the digits of that number by 18. If the digit at the unit's place is double the digit in the ten's place, what is the number? | 24 | Let the tens digit be t and units digit be u. Conditions: (1) 10t + u - (t + u) = 18 → 9t = 18 → t = 2. (2) u = 2t = 4. The number is 24. Verification: sum of digits = 2 + 4 = 6; 24 - 6 = 18 ✓; units digit (4) = 2 × tens digit (2) ✓. The number is 24. | self_contained_factoid |
aqua-rat#194 | aqua-rat | Three bells ring at intervals of 36 seconds, 40 seconds and 48 seconds, respectively. They start ringing together at a particular time. When will they ring together again? — of these: After 6 minutes; After 12 minutes; After 18 minutes; After 24 minutes; none — which is correct (one, several, both, or none), and why? | After 12 minutes | The bells ring together again at the LCM of 36, 40, and 48 seconds. Prime factorizations: 36 = 2²×3², 40 = 2³×5, 48 = 2⁴×3. LCM = 2⁴×3²×5 = 16×9×5 = 720 seconds = 12 minutes. Options of 6, 18, and 24 minutes correspond to 360, 1080, and 1440 seconds, none of which are divisible by all three intervals correctly. The cor... | self_contained_factoid |
aqua-rat#195 | aqua-rat | An electric pole 14 metres high casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions. | 21 | Under similar lighting conditions, height and shadow length are proportional. Setting up the proportion: 14/10 = h/15. Solving: h = 14 × 15/10 = 210/10 = 21 metres. The tree is 21 metres tall. | self_contained_factoid |
aqua-rat#196 | aqua-rat | At my favorite fruit stand, an orange costs 18 dollars, a pineapple costs 27 dollars, and a grape costs 15 dollars. Using the same logic, how much does a mango cost? — of these: 22 dollars; 15 dollars; 20 dollars; 18 dollars; 10 dollars — which is correct (one, several, both, or none), and why? | 15 dollars | The pattern is that the price equals the number of letters in the fruit's name multiplied by 3: orange = 6 letters × 3 = 18; pineapple = 9 letters × 3 = 27; grape = 5 letters × 3 = 15. Mango = 5 letters × 3 = 15 dollars. Options 22, 20, 18, and 10 do not fit this letter-count pattern. The correct answer is 15 dollars. | self_contained_factoid |
aqua-rat#197 | aqua-rat | In the coordinate plane, a triangle has vertices at (a,0), (b,0), and (x,y). If a > x > b > 0 > y, which of the following represents the area of that triangle? — of these: (ay−by)/2; (ab−ay)/2; (by−ay)/2; (ay+by)/x; (a−b)/2y — which is correct (one, several, both, or none), and why? | (by−ay)/2 | The base of the triangle lies along the x-axis from (b,0) to (a,0), with length (a−b). The height is the perpendicular distance from (x,y) to the x-axis, which is |y|. Since y < 0, |y| = −y. Area = (1/2)(a−b)(−y) = (−ay+by)/2 = (by−ay)/2. Since b < a and y < 0, by < ay (both negative but ay more negative), so by−ay > 0... | competing_proposition |
aqua-rat#200 | aqua-rat | The difference between simple interest and compound interest at the same rate for Rs.5000 for 2 years is Rs.72. The rate of interest is — of these: 10%; 12%; 6%; 8%; 4% — which is correct, and why? | 12% | For a principal P, time 2 years, and rate r, the difference between CI and SI is P·r²/100². Setting 5000·r²/10000 = 72 gives r²/2 = 72, so r² = 144, and r = 12. Checking: SI = 5000×12×2/100 = 1200; CI = 5000×(1.12²−1) = 5000×0.2544 = 1272; difference = 72. ✓ The options 10%, 6%, 8%, and 4% all yield differences of 50, ... | self_contained_factoid |
aqua-rat#201 | aqua-rat | All 250 files on Sam's hard drive are infected by either a virus or a worm or both. The number of files infected only by a worm is 2.5 times the number infected by both a virus and a worm. If 50% of the files were not infected by a virus, how many of Sam's files were NOT infected by a worm? — of these: 50; 70; 75; 100;... | 75 | 50% not infected by a virus means 125 files have no virus, so those 125 are infected only by a worm. Let both = b; then worm-only = 2.5b = 125, so b = 50. Total infected by worm = worm-only + both = 125 + 50 = 175. Files NOT infected by a worm = 250 − 175 = 75. The remaining options (50, 70, 100, 125) don't satisfy the... | self_contained_factoid |
aqua-rat#202 | aqua-rat | A father wants to divide Rs. 5100 between his two sons, Mohan (age 23) and Sohan (age 24). He divides the amount so that if their shares are invested at compound interest at 4% p.a., they will receive equal amounts on attaining age 26. Find Mohan's share — of these: 2400; 2500; 2600; 2700; None of these — which is corr... | 2500 | Mohan is 23, so his share grows for 3 years; Sohan is 24, so his share grows for 2 years. Let Mohan's share = M and Sohan's = (5100 − M). For equal final amounts: M×(1.04)³ = (5100−M)×(1.04)². Dividing both sides by (1.04)²: M×1.04 = 5100−M → 1.04M + M = 5100 → 2.04M = 5100 → M = 2500. Sohan's share = 2600. The answer ... | self_contained_factoid |
aqua-rat#203 | aqua-rat | What is 60% of 30% of 1400 grams? — of these: 450 gms; 100 gms; 252 gms; 240 gms; None of these — which is correct, and why? | 252 gms | First compute 30% of 1400: 0.30 × 1400 = 420 grams. Then compute 60% of 420: 0.60 × 420 = 252 grams. The option 450 gms would correspond to a different percentage; 100 gms and 240 gms are also incorrect computations; 'None of these' is unnecessary since 252 gms is listed. The correct answer is 252 gms. | self_contained_factoid |
aqua-rat#204 | aqua-rat | A certain liquid passes through a drain at a rate of w/25 gallons every x seconds. At that rate, how many minutes will it take y gallons to pass through the drain? — of these: y/(1200xy); 20xy/w; 5xy/(12w); w/(3xy); 3y/(wx) — which is correct, and why? | 5xy/(12w) | Rate = (w/25) gallons per x seconds = w/(25x) gallons per second. Time to drain y gallons = y ÷ [w/(25x)] = 25xy/w seconds. Convert to minutes by dividing by 60: 25xy/(60w) = 5xy/(12w). Checking the other options: 20xy/w is off by a factor; y/(1200xy) simplifies incorrectly; w/(3xy) and 3y/(wx) don't match dimensional ... | self_contained_factoid |
aqua-rat#205 | aqua-rat | A coin made of alloy of aluminum and silver measures 2 × 15 mm (2 mm thick, 15 mm diameter). The coin weighs 30 grams and the volume of aluminum equals that of silver. Silver is twice as heavy as aluminum. What will be the weight of a coin measuring 1 × 30 mm made of pure aluminum? — of these: 36 grams; 40 grams; 42 gr... | 40 grams | Volume of alloy coin: π×(7.5)²×2 = 112.5π mm³. Half is aluminum (56.25π) and half silver (56.25π). Let aluminum density = d; silver = 2d. Weight = 56.25π·d + 56.25π·2d = 56.25π·3d = 30g → d = 30/(168.75π). New coin volume: π×(15)²×1 = 225π mm³, pure aluminum. Weight = 225π·d = 225π × 30/(168.75π) = 6750/168.75 = 40 gra... | self_contained_factoid |
aqua-rat#206 | aqua-rat | If 10 is subtracted from 2/3 of a number the result is equal to the sum of 40 and 1/3 of the number. Find the number — of these: 100; 160; 150; 210; 220 — which is correct, and why? | 150 | Let the number be n. The equation is (2/3)n − 10 = 40 + (1/3)n. Subtracting (1/3)n from both sides: (1/3)n − 10 = 40. Adding 10: (1/3)n = 50, so n = 150. Verification: (2/3)×150 − 10 = 100 − 10 = 90; 40 + (1/3)×150 = 40 + 50 = 90. ✓ The other options (100, 160, 210, 220) do not satisfy the equation. The correct answer ... | self_contained_factoid |
aqua-rat#207 | aqua-rat | What is the largest integral value of k for which the quadratic equation x² − 5x + k = 0 will have two real and distinct roots? — of these: 9; 7; 3; 8; 12 — which is correct, and why? | 3 | For two real and distinct roots, the discriminant must be strictly positive: b² − 4ac > 0. Here a=1, b=−5, c=k, so 25 − 4k > 0 → k < 6.25. The largest integer less than 6.25 is 6, but 6 is not among the options. Among the given options, only k=3 satisfies k < 6.25; k=7, 8, 9, and 12 all exceed 6.25 and yield non-positi... | self_contained_factoid |
aqua-rat#208 | aqua-rat | 900 + 5 × 12 = ? — of these: 820; 202; 420; 209; 960 — which is correct, and why? | 960 | By order of operations (BODMAS/PEMDAS), multiplication is performed before addition: 5 × 12 = 60, then 900 + 60 = 960. The other options (820, 202, 420, 209) result from incorrect arithmetic or wrong order of operations. The correct answer is 960. | self_contained_factoid |
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AGIEval English Freeform Rewrites
DatologyAI is pleased to release a rewrite of AGIEval English that enables language-model evaluation through language-modeling loss on verified-correct free-form answers. Part of the DatologyAI Benchmark Freeform Rewrites collection.
Motivation
Practitioners developing language models need evaluations capable of quantifying the effects of small changes, such as adjustments to the training data mix, even when evaluating the smaller models trained during research and development.
Generative evaluation with pass/fail checks is susceptible to emergence effects when applied to development-scale models. Unconditioned language-modeling loss over held-out documents provides a denser signal, but held-out documents may not map clearly onto the capabilities of interest.
Few-shot-conditioned free-form question answering, the setup this dataset enables, sits between the two. It targets specific capabilities and measures in-context learning while providing a dense, emergence-free signal.
This data
The DatologyAI Benchmark Freeform Rewrites datasets transform existing multiple-choice and short-answer benchmarks into free-form long-answer tasks. Each row pairs a self-contained question, the original short answer, and a worked continuation verified to answer the question correctly.
A technical write-up describing the rewrite and verification process and the validation results is coming soon.
Fields
| Field | Description |
|---|---|
id |
<group>#<source id>; unique within the dataset. |
group |
Subject or subtask; equals the HF config name. |
question |
Self-contained rewritten question or problem statement. |
answer |
Short original answer from the source benchmark. |
freeform_answer |
Free-form worked answer that arrives at answer. |
structural_type |
How the source multiple-choice item was converted: self_contained_factoid, fill_in_blank, multi_statement, negative_odd_one_out, logic_symbolization or competing_proposition. |
Loading
from datasets import load_dataset
ds = load_dataset("DatologyAI/agieval-freeform-rewrites", "aqua-rat", split="train")
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