Mathematics

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 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.

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).

Let the cost of keeping 1 cow for 1 day be C and 1 goat for 1 day be G. Given 5G = C, so G = C/5. Total cost for 20 cows and 40 goats for 10 days is 460, so 10*(20C + 40G) = 460. Substitute G = C/5: 10*(20C + 40*(C/5)) = 460 => 10*(20C + 8C) = 460 => 10*28C = 460 => 280C = 460 => C = 460/280 = 23/14. Now, cost for 50 cows and 30 goats for 12 days is 12*(50C + 30G) = 12*(50C + 30*(C/5)) = 12*(50C + 6C) = 12*56C = 672C = 672*(23/14) = 672*1.642857 = 1104. Hence, option A is correct.

At 50 kmph, mileage is 20 km/l. At 70 kmph, mileage decreases by 10%, so new mileage = 20 - 10% of 20 = 18 km/l. With 12 liters, distance = 18 × 12 = 216 km, which corresponds to option A.

To find the distance covered, first convert speed from km/h to m/s: 108 km/h = (108 × 1000) / 3600 = 30 m/s. Then, distance = speed × time = 30 m/s × 15 s = 450 metres.

The side length of the largest cube that can exactly divide the cuboid is the greatest common divisor (GCD) of 6, 9, and 12, which is 3 cm. The volume of the cuboid is 6×9×12 = 648 cm³, and the volume of each cube is 3×3×3 = 27 cm³. Therefore, the number of cubes is 648 ÷ 27 = 24.

A's work rate is 1/16 per day, B's is 1/12 per day. Together with C, they finish in 4 days, so combined rate is 1/4 per day. C's rate = 1/4 - (1/16 + 1/12) = 1/4 - (3/48 + 4/48) = 1/4 - 7/48 = 12/48 - 7/48 = 5/48 per day. Thus, C alone takes 48/5 = 9.6 days, which corresponds to option B. However, the calculation shows 9.6 days, so option B is correct, but the original correct_answer was B and options are correct. The explanation was missing and difficulty was missing, so I added them. The original answer B is correct.

Given that the total wages for 40 women for 30 days is Rs.21600, the daily wage of one woman is Rs.21600 / (40 * 30) = Rs.18. Since a man's daily wage is double that of a woman, a man's daily wage is Rs.36. To find how many men should work for 25 days to earn Rs.14400, use the equation: number of men * 25 days * Rs.36 = Rs.14400. Solving for the number of men gives 14400 / (25 * 36) = 16 men. Therefore, the correct answer is option C (16).

Pipe P fills the cistern in 12 minutes, so its rate is 1/12 per minute. Pipe Q fills it in 16 minutes, so its rate is 1/16 per minute. Both pipes open together for t minutes, then only P runs for the remaining (9 - t) minutes. The total work done is (1/12 + 1/16) * t + (1/12) * (9 - t) = 1. Solving this gives t = 3 minutes, so Q should be closed after 3 minutes.