Mathematics

The boatman's speed against the current is 2 km/h, and along the current is 6 km/h. Using these, the speed of the stream is (6 - 2)/2 = 2 km/h, and the boat's speed in still water is (6 + 2)/2 = 4 km/h. To cover 5 km in still water, time = distance/speed = 5/4 = 1.25 hours = 75 minutes. However, the options do not include 75 minutes; rechecking calculations: speed against current = 2 km/h, speed along current = 6 km/h, so speed of boat in still water = (2 + 6)/2 = 4 km/h. Time to cover 5 km = 5/4 = 1.25 hours = 75 minutes (1 hr 15 min). Option C is 1 hr 15 min, which matches the calculation. Therefore, correct answer is C.

The boat covers 24 km downstream in 10 hours, so downstream speed = 24/10 = 2.4 km/hr. It takes 2 hours more to cover the same distance upstream, so upstream time = 12 hours, and upstream speed = 24/12 = 2 km/hr. The speed of the boat in still water is the average of downstream and upstream speeds: (2.4 + 2)/2 = 2.2 km/hr. Since 2.2 km/hr is not an option, the closest given option is 2.8 km/hr (option B). The original correct answer (C) 4 km/hr is incorrect based on these calculations. Therefore, the correct answer should be B (2.8 km/hr) as the closest approximation.

The compound interest is calculated using the formula A = P(1 + r/n)^(nt). Here, P = 15625, r = 0.16, n = 4 (quarterly), and t = 0.75 years (9 months). Calculating gives A ≈ Rs. 17466, so compound interest = A - P ≈ Rs. 1941, which corresponds to option B.

If a sum doubles itself in 5 years at compound interest, it means the amount multiplies by 2 every 5 years. To become eight times itself, the amount must double three times (since 8 = 2^3). Therefore, the time required is 3 times the doubling period: 5 years × 3 = 15 years. This matches option C. This is consistent with the compound interest formula where the time to multiply by a factor is proportional to the logarithm of that factor, so the time to multiply by 8 is three times the time to multiply by 2.

The man's effective speed downstream is 6 + 1.2 = 7.2 km/h, and upstream is 6 - 1.2 = 4.8 km/h. The total time for the round trip is 1 hour, so if the distance one way is d km, then d/7.2 + d/4.8 = 1. Solving for d gives d = 4.32 km. Therefore, the total distance traveled is 2 × 4.32 = 8.64 km. However, none of the options match 8.64 km, so rechecking calculations: The sum of times is 1 hour, so d/7.2 + d/4.8 = 1. The least common denominator is 14.4, so (2d + 3d)/14.4 = 1 → 5d = 14.4 → d = 2.88 km one way. Total distance = 5.76 km, which matches option C. Therefore, option C is correct. The initial explanation was incorrect; the correct total distance is 5.76 km.

Given the man's speed in still water is 9 1/3 km/hr (which is 28/3 km/hr), and it takes thrice as much time to row upstream as downstream for the same distance, we set the speed of the current as x km/hr. Upstream speed = (28/3 - x), downstream speed = (28/3 + x). Since time is inversely proportional to speed, the time upstream is thrice the time downstream, so: 1/(28/3 - x) = 3/(28/3 + x). Solving this equation gives x = 14/3 km/hr, which is 4 2/3 km/hr. Therefore, the speed of the current is 4 2/3 km/hr, corresponding to option D.

Let the speed of the boat in still water be b kmph and the speed of the stream be s kmph. Downstream speed = b + s, upstream speed = b - s. Given time downstream = time upstream = 2.5 hours. So, 85/(b+s) = 2.5 and 45/(b-s) = 2.5. From these, b + s = 34 and b - s = 18. Adding, 2b = 52, so b = 26 kmph. Subtracting, 2s = 16, so s = 8 kmph. However, the question asks for the speed of the stream, which is 8 kmph. But the options have 8 kmph as D, so the correct answer is D. The initial calculation shows the stream speed is 8 kmph, matching option D.

The difference between the amounts for 3 years and 2 years is Rs. 10,648 - Rs. 9,680 = Rs. 968, which represents the interest earned in the third year. Since the amount at 2 years is Rs. 9,680, the interest rate is (968 / 9680) × 100 = 10%. This calculation assumes yearly compounding, and the rate of interest is 10%.

If the time taken against the stream is twice the time taken with the stream, then the speed against the stream is half the speed with the stream. Let the speed of the boat in still water be b and the speed of the stream be s. Then, (b - s) = 1/2 (b + s). Solving this gives the ratio b : s = 3 : 2.

The boatman's speed against the current is 2 km/h, and along the current is 6 km/h (1 km in 10 minutes). Using these, the speed of the stream is (6 - 2)/2 = 2 km/h, and the boat's speed in still water is (6 + 2)/2 = 4 km/h. To cover 5 km in still water at 4 km/h, time = 5/4 = 1.25 hours or 1 hour 15 minutes.