Mathematics

A completes 2 days of work at a rate of 1/8 per day, so completes 2/8 = 1/4 of the work. Remaining work is 3/4. B completes two-thirds of the remaining work, which is (2/3)*(3/4) = 1/2 of the total work. B's rate is 1/14 per day, so time taken by B is (1/2)/(1/14) = 7 days. Remaining work after B is 1/4. C finishes this remaining 1/4 work at a rate of 1/16 per day, so time taken by C is (1/4)/(1/16) = 4 days. Total time = 2 + 7 + 4 = 13 days. This matches option A, so the correct answer is A (13 days).

Given that 12 men can complete the work in 20 days, the total work can be expressed as 12 men × 20 days = 240 man-days. Similarly, 20 women can complete the same work in 16 days, so total work = 20 women × 16 days = 320 woman-days. Since the total work is the same, 240 man-days = 320 woman-days. The capacity of one man per day is 1/240 of the work, and the capacity of one woman per day is 1/320 of the work. Therefore, the ratio of the capacity of a man to a woman is (1/240) : (1/320) = 320 : 240 = 4 : 3. Hence, the correct ratio is 4:3, which corresponds to option C.

The first pipe fills the tank in 20 minutes, so its rate is 1/20 tank per minute. The second pipe fills the tank in 60 minutes, so its rate is 1/60 tank per minute. Both pipes open for 10 minutes fill (1/20 + 1/60) * 10 = (3/60 + 1/60)*10 = (4/60)*10 = 2/3 of the tank. The remaining 1/3 of the tank is filled by the second pipe alone at 1/60 tank per minute, which takes 20 minutes. Total time = 10 + 20 = 30 minutes.

Given: 10 men can complete the work in 7 days, so the work done by 10 men in 1 day is 1/7. Therefore, work done by 1 man in 1 day is (1/7)/10 = 1/70. Similarly, 10 women can complete the work in 14 days, so work done by 10 women in 1 day is 1/14. Therefore, work done by 1 woman in 1 day is (1/14)/10 = 1/140. Now, work done by 5 men in 1 day = 5 × (1/70) = 1/14. Work done by 10 women in 1 day = 10 × (1/140) = 1/14. Total work done by 5 men and 10 women in 1 day = 1/14 + 1/14 = 1/7. Hence, the total number of days required to complete the work = 1 / (1/7) = 7 days. Therefore, the correct answer is 7 days, which corresponds to option A.

Let the work done by P, Q, and R per day be p, q, and r respectively. From the given: p + q = 1/30, q + r = 1/24, r + p = 1/20. Adding all three: 2(p + q + r) = 1/30 + 1/24 + 1/20 = (4 + 5 + 6)/120 = 15/120 = 1/8, so p + q + r = 1/16. After 10 days working together, work done = 10 × 1/16 = 10/16 = 5/8. Remaining work = 3/8. Since Q and R left, only P works now at rate p = (p + q + r) - (q + r) = 1/16 - 1/24 = (3 - 2)/48 = 1/48 per day. Time for P to finish remaining work = (3/8) ÷ (1/48) = 18 days. However, the calculation shows 18 days, but the options have 8 as B and 18 as C. Rechecking: p = (p + q + r) - (q + r) = 1/16 - 1/24 = (3/48 - 2/48) = 1/48. Remaining work = 3/8. Time = (3/8) / (1/48) = (3/8) × 48 = 18 days. So correct answer is C (18). The original answer was correct. The explanation was missing and difficulty was not set; difficulty is medium. Tags added for clarity.

The total work done by 6 men in 2 days is 6 × 2 = 12 man-days, which produces 10 sofas. Therefore, the man-days required to make one sofa is 12 ÷ 10 = 1.2 man-days. For 8 sofas, the total man-days required is 8 × 1.2 = 9.6 man-days. With 8 men working, the number of days required is 9.6 ÷ 8 = 1.2 days. Hence, the correct answer is 1.2 days, which corresponds to option C.

P's work rate is 1/16 per day, Q's is 1/12 per day. Together with R, they complete the job in 4 days, so combined rate is 1/4 per day. R's rate = (1/4) - (1/16 + 1/12) = 1/4 - (3/48 + 4/48) = 1/4 - 7/48 = 12/48 - 7/48 = 5/48. Therefore, R alone takes 48/5 = 9.6 days, which is 9(3/5) days.

The total work can be calculated as 12 men × 8 hours/day × 10 days = 960 man-hours. To complete the same work in 5 days working 12 hours a day, the number of men required is 960 ÷ (12 hours/day × 5 days) = 16 men.

First, find the total work in man-days for the first wall: 15 men × 30 days = 450 man-days to build 500 m. The work per meter is 450 man-days / 500 m = 0.9 man-days per meter. For a 1.2 km (1200 m) wall, total work = 1200 m × 0.9 man-days/m = 1080 man-days. With 30 men working, days required = 1080 man-days / 30 men = 36 days. However, this contradicts the initial calculation, so let's re-check: 15 men × 30 days = 450 man-days for 500 m, so 1 m requires 450/500 = 0.9 man-days. For 1200 m, total man-days = 1200 × 0.9 = 1080. With 30 men, days = 1080/30 = 36 days. So the correct answer is 36 days, which corresponds to option D. The initial correction was wrong; the original correct_answer D is correct.

C takes 6 days to complete half the work, so C's 1 day work = 1/12 of the work. In 4 days, C completes 4 × 1/12 = 1/3 of the work. The remaining work is 2/3. D takes 5 days to complete one-third of the work, so D's 1 day work = 1/15 of the work. To complete the remaining 2/3 of the work, D needs (2/3) ÷ (1/15) = (2/3) × 15 = 10 days. Therefore, D will complete the rest of the work in 10 days, which corresponds to option A.

The volume of the cylindrical tank is given as 246.4 litres, which converts to 0.2464 cubic meters (since 1 litre = 0.001 cubic meters). The height of the tank is 4 meters. Using the formula for the volume of a cylinder V = πr²h, we substitute the known values: 0.2464 = πr² × 4. Solving for r² gives r² = 0.2464 / (4π) ≈ 0.0196, so r ≈ 0.14 meters. The diameter is twice the radius, so diameter ≈ 0.28 meters. This matches option B. The explanation in the original question had confusing statements about decimal placement and incorrect mentions of 2.8 m, which are not relevant. The correct diameter is 0.28 m, which is option B.

The car covers 42 km in 1 hour 40 minutes 48 seconds at 5/7 of its actual speed. Converting the time to hours: 1 hr + 40/60 hr + 48/3600 hr = 1.68 hours. Speed at 5/7 actual speed = distance/time = 42/1.68 = 25 km/hr. Therefore, actual speed = (7/5) * 25 = 35 km/hr. However, this calculation shows 35 km/hr, matching option B. Rechecking: 1 hr 40 min 48 sec = 1 + 40/60 + 48/3600 = 1 + 0.6667 + 0.0133 = 1.68 hr. Speed = 42/1.68 = 25 km/hr (5/7 actual speed). Actual speed = (7/5)*25 = 35 km/hr. So the correct answer is B, not D. Explanation corrected accordingly.

The problem states that Farjan and Kashif together complete the task in 30 days, so their combined work rate is 1/30 of the task per day. They work together for 20 days, completing 20/30 = 2/3 of the task. The remaining 1/3 of the task is completed by Kashif alone in 25 days, so Kashif's work rate is (1/3)/25 = 1/75 of the task per day. Farjan's work rate is the combined rate minus Kashif's rate: (1/30) - (1/75) = (5/150) - (2/150) = 3/150 = 1/50 of the task per day. Therefore, Farjan alone takes 50 days to complete the entire task. This matches option A, confirming that the correct answer is A (50).

The total work is the same in both cases. Work done by 20 women in 16 days equals work done by 16 men in 15 days. Calculating daily work rates and comparing, the ratio of a man's capacity to a woman's capacity is 2:1.

P, Q, and R's rates of work per day are 1/24, 1/6, and 1/12 respectively. Adding these gives (1/24 + 1/6 + 1/12) = (1/24 + 4/24 + 2/24) = 7/24 work per day. Therefore, the total time to complete the work together is 24/7 days, which is approximately 3 3⁄7 days.

Each child takes twice the time a man takes to finish the work, so each child works at half the rate of a man. Therefore, 8 children have the combined work rate of 4 men (since 8 × 0.5 = 4). Together, 8 children and 12 men have the combined work rate of 16 men (12 + 4 = 16). They complete the work in 9 days, so the total work is equivalent to 16 men × 9 days = 144 man-days. To find how many days 12 men will take to finish the same work, divide the total work by 12 men: 144 man-days ÷ 12 men = 12 days. Hence, 12 men will finish the work in 12 days.

P is 30% more efficient than Q, so the efficiency ratio of P to Q is 130:100 or 13:10. P completes the work in 23 days, so P's work rate is 1/23 work per day. Q's work rate is (10/13) times P's rate, which is 10/(13 × 23). Together, their combined work rate is 1/23 + 10/(13 × 23) = (13 + 10)/(13 × 23) = 23/(13 × 23) = 1/13 work per day. Therefore, working together, they complete the work in 13 days. Hence, the correct answer is option D (13 days).

First, calculate the total distance covered. Distance = Speed × Time = 240 × 5 = 1200 km. Since the same distance must be covered in 5/3 hours, use the formula Speed = Distance ÷ Time. Therefore, Speed = 1200 ÷ (5/3) = 1200 × (3/5) = 720 km/hr. Hence, the correct answer is Option D (720 km/hr). The provided answer key marking Option A (660 km/hr) is incorrect, as the calculation clearly gives 720 km/hr.

Let the work done by A, B, and C in one day be a, b, and c respectively. Given: a + b = 1/20, b + c = 1/30, c + a = 1/40. Adding all three: 2(a + b + c) = 1/20 + 1/30 + 1/40 = (6 + 4 + 3)/120 = 13/120. So, a + b + c = 13/240. Subtracting b + c = 1/30 from this, we get a = 13/240 - 1/30 = (13/240) - (8/240) = 5/240 = 1/48. Hence, A alone finishes the work in 48 days.

If 3 men or 4 women can complete the work in 43 days, then the work done by 1 man in 1 day is 1/(3×43) and by 1 woman is 1/(4×43). The combined daily work of 7 men and 5 women is 7/(3×43) + 5/(4×43) = (28/12 + 15/12)/43 = 43/12/43 = 1/(12/43) = 43/12. Therefore, total days = 1 / (combined daily work) = 12/43 days, which is approximately 14 days.

Let the work done by one man in one day be M and by one woman be W. From the first condition: 2M + 12W = 1/4 (since the job is finished in 4 days). From the second: 4M + 6W = 1/5. Solving these equations gives M = 1/60 and W = 1/20. Therefore, one man and one woman together do (1/60 + 1/20) = 1/15 of the job per day, so they complete the job in 15 days. However, since 15 is not an option, rechecking the calculation: 2M + 12W = 1/4 => 2M + 12W = 0.25; 4M + 6W = 1/5 => 4M + 6W = 0.2. Multiply the first equation by 2: 4M + 24W = 0.5; subtract the second equation: (4M + 24W) - (4M + 6W) = 0.5 - 0.2 => 18W = 0.3 => W = 0.3/18 = 1/60. Substitute W into first equation: 2M + 12*(1/60) = 0.25 => 2M + 0.2 = 0.25 => 2M = 0.05 => M = 0.025 = 1/40. So, one man and one woman together do (1/40 + 1/60) = (3/120 + 2/120) = 5/120 = 1/24 of the job per day, so they complete the job in 24 days. Therefore, the correct answer is C (24 days).

Let the work done by one man in one day be M and by one woman be W. From the first condition: 3M + 7W = 1/10 (work per day). From the second: 4M + 6W = 1/8. Solving these equations gives M = 1/40 and W = 1/50. Therefore, 10 women will complete the work in 1/(10*W) = 50 days.

P's work rate is 1/15 per day and Q's is 1/20 per day. Together, they complete (1/15 + 1/20) = 7/60 of the work per day. In 4 days, they complete 4 × 7/60 = 28/60 = 7/15 of the work. Therefore, the fraction of work left is 1 - 7/15 = 8/15. However, since the question asks for the fraction of work left, the correct answer is 8/15, which corresponds to option A. On rechecking, the initial calculation was correct, so the correct answer is A.

P's work rate is 1/20 per day, Q's is 1/30 per day, and R's is 1/60 per day. For two days, only P works, completing 2*(1/20) = 1/10 of the work. On every third day, P, Q, and R work together, completing (1/20 + 1/30 + 1/60) = 1/10 of the work. Thus, every 3 days, they complete 1/10 + 1/10 = 1/5 of the work. To complete the entire work, it takes 5 cycles of 3 days each, totaling 15 days. Therefore, the correct answer is 15 days (option C).

A and B together complete 1/30 of the work per day. In 20 days, they complete 20/30 = 2/3 of the work. The remaining 1/3 is done by A alone in 20 days, so A's one day work is (1/3)/20 = 1/60. Therefore, A alone can finish the work in 60 days.