Mathematics

Given the problem: P, Q, and R together earn Rs.1620 in 9 days, so their combined daily earning is Rs.1620/9 = Rs.180 per day. P and R together earn Rs.600 in 5 days, so their combined daily earning is Rs.600/5 = Rs.120 per day. Q and R together earn Rs.910 in 7 days, so their combined daily earning is Rs.910/7 = Rs.130 per day. From these, we have three equations: 1) p + q + r = 180 2) p + r = 120 3) q + r = 130 Subtracting equation (2) from (1) gives q = 60. Subtracting equation (3) from (1) gives p = 50. Using p + r = 120 and p = 50, we get r = 70. Therefore, R earns Rs.70 per day, which corresponds to option B.

The total work can be considered as 28 men × 15 days = 420 man-days. Since the amount of work is constant, the time taken is inversely proportional to the number of men. Therefore, if 21 men are working, the time taken will be 420 man-days ÷ 21 men = 20 days. Hence, the correct answer is 20 days, which corresponds to option D.

First, calculate total bull-hours needed to plough 20 fields: 10 bulls × 3 days × 10 hours = 300 bull-hours. So, 1 field requires 300/20 = 15 bull-hours. For 32 fields, total bull-hours = 32 × 15 = 480. With 30 bulls working 8 hours a day, daily bull-hours = 30 × 8 = 240. Days required = 480 / 240 = 2 days. However, this conflicts with the options, so rechecking: 10 bulls × 3 days × 10 hours = 300 bull-hours for 20 fields, so 1 field = 15 bull-hours. For 32 fields, total bull-hours = 480. With 30 bulls × 8 hours = 240 bull-hours/day, days = 480/240 = 2 days. Option A is 2, so correct answer is A, not B. Correction made accordingly.

The two taps fill the cistern at rates of 1/12 and 1/15 cisterns per minute, respectively. Combined, they fill at (1/12 + 1/15) = 9/60 = 3/20 cisterns per minute. When the outlet pipe is also open, the cistern fills in 20 minutes, so the net filling rate is 1/20 cisterns per minute. Let the outlet pipe's emptying rate be 1/x. Then, (3/20) - (1/x) = 1/20, solving gives x = 40 minutes.

Given the edges of the cuboid are in the ratio 1:2:3, let the edges be x, 2x, and 3x. The surface area is 2(x*2x + 2x*3x + 3x*x) = 2(2x^2 + 6x^2 + 3x^2) = 2(11x^2) = 22x^2. Given surface area = 88 cm², so 22x² = 88, hence x² = 4 and x = 2. The volume = x * 2x * 3x = 6x³ = 6 * 8 = 48 cm³. However, this contradicts the calculation; rechecking: volume = x * 2x * 3x = 6x³ = 6 * (2)³ = 6 * 8 = 48 cm³. So the volume is 48 cm³, which matches option B. The initial explanation was incorrect; the correct volume is 48 cm³, so the correct answer is B.

Jameel works from 9:00 am to 2:00 pm, which is 5 hours. His work rate is 1/15 of the task per hour, so in 5 hours he completes 5/15 = 1/3 of the task. The remaining task is 2/3. Nasir starts working at 4:00 pm after a 2-hour break. Nasir's work rate is 1/12 of the task per hour. To complete the remaining 2/3 of the task, Nasir needs (2/3) / (1/12) = 8 hours. Starting at 4:00 pm and working for 8 hours means the task finishes at 12:00 am (midnight). Therefore, the correct answer is C (12:00 am).

Since Kamal completes the work in 20 days, his work rate is 1/20 of the work per day. Suresh is 25% more efficient than Kamal, meaning Suresh's work rate is 1.25 times Kamal's rate. Therefore, Suresh's work rate = 1.25 × (1/20) = 1/16 of the work per day. Hence, Suresh will complete the work in 16 days (1 ÷ (1/16) = 16).

Mansoor-Ul-Haque types at a rate of 32 pages / 6 hours = 16/3 pages per hour. Aaqib types at a rate of 40 pages / 5 hours = 8 pages per hour. Together, their combined rate is (16/3) + 8 = (16/3) + (24/3) = 40/3 pages per hour. To type 110 pages, time needed = 110 / (40/3) = 110 * (3/40) = 8.25 hours, which is 8 hours 15 minutes. However, this contradicts the initial correct answer C. Rechecking calculations: Mansoor-Ul-Haque rate = 32/6 ≈ 5.33 pages/hour, Aaqib rate = 40/5 = 8 pages/hour, combined rate = 5.33 + 8 = 13.33 pages/hour. Time = 110 / 13.33 ≈ 8.25 hours = 8 hours 15 minutes. So the original correct answer C (8 hour 15 minutes) is correct. Therefore, the initial calculation was incorrect. The correct answer is C.

The tap fills the tank at a rate of 1/32 per minute, and the other empties it at 1/16 per minute. The net rate when both taps are open is (1/32) - (1/16) = -1/32 tank per minute, meaning the tank is emptying at a rate of 1/32 of the tank per minute. Since the tank is initially half full, it will take (1/2) ÷ (1/32) = 16 minutes to empty the tank completely. Therefore, the correct answer is 16 minutes, option C.

If the three adjacent faces of a rectangular box have areas A = xy, B = yz, and C = zx, where x, y, and z are the edges meeting at a point, then the product ABC = (xy)(yz)(zx) = (xyz)^2, which is the square of the volume of the box.