Mathematics

The principal can be found using the simple interest formula: SI = P × R × T / 100, so P = (SI × 100) / (R × T) = (50 × 100) / (5 × 2) = Rs. 500. Compound interest for 2 years at 5% is CI = P[(1 + R/100)^T - 1] = 500[(1 + 0.05)^2 - 1] = 500(1.1025 - 1) = Rs. 51.25, which matches option A.

Let the speed of the stream be x km/h. The man's speed in still water is 4.5 km/h. Downstream speed = 4.5 + x, upstream speed = 4.5 - x. Given time upstream is twice time downstream, so (distance / upstream speed) = 2 × (distance / downstream speed). Simplifying, downstream speed = 2 × upstream speed, so 4.5 + x = 2(4.5 - x). Solving gives x = 1.5 km/h.

The compound interest on Rs 16,000 for 9 months at 20% per annum compounded quarterly is calculated as follows: The quarterly interest rate is 20%/4 = 5%. The number of quarters in 9 months is 3. Using the compound interest formula A = P(1 + r)^n, the amount A = 16000 * (1 + 0.05)^3 = 16000 * 1.157625 = Rs 18,522 (approximately). The compound interest is the amount minus the principal, i.e., 18522 - 16000 = Rs 2522. This matches option C exactly, so the correct answer is C, not B. The original explanation incorrectly chose the closest option instead of the exact match.

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.

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 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%.

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.

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.

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.

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 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.

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 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.

Using the compound interest formula A = P(1 + r)^t, where A = 4913, r = 6.25% = 0.0625, and t = 3 years, we solve for P: P = 4913 / (1.0625)^3 = 4913 / 1.199951 = approximately Rs. 4096. This confirms option D is correct.

Since A beats B by 22.5 m or 6 seconds in a 300 m race, B takes 6 seconds longer than A. A's speed is 300 m divided by (B's time - 6). Using the ratio of distances and times, B's time calculates to 86 seconds.

The speed of the man in still water is the average of the upstream and downstream speeds: (25 + 35) / 2 = 30 kmph. This is because the current's effect cancels out when averaging the two speeds.

The speed of the man in still water is calculated as the average of the downstream and upstream speeds: (18 + 10) / 2 = 14 kmph. The speed of the stream is half the difference between the downstream and upstream speeds: (18 - 10) / 2 = 4 kmph. Therefore, the correct answer is 14 kmph for the man in still water and 4 kmph for the stream, which corresponds to option D.

The difference between compound interest (CI) and simple interest (SI) for 2 years at 5% per annum is given by the formula: Difference = P × (R/100)^2, where P is the principal and R is the rate of interest. Here, the difference is Rs. 1.50, and the rate R = 5%. So, 1.50 = P × (5/100)^2 = P × 0.0025. Solving for P gives P = 1.50 / 0.0025 = Rs. 600. Therefore, the sum is Rs. 600, which corresponds to option A.

A's simple interest for 2 years = (6000 × 5 × 2)/100 = Rs 600. B's compound interest for 2 years = 5000 × (1 + 8/100)^2 - 5000 = 5000 × 1.1664 - 5000 = Rs 832. The difference = 832 - 600 = Rs 232. However, since the question asks for the difference between their interest, the absolute difference is Rs 232, which matches option B. But the calculation shows the difference is Rs 232, so the correct answer is B, not A. Therefore, the original correct_answer is correct, but the explanation was missing and is now provided. The difficulty is easy, and tags are added accordingly.

The simple interest on Rs 5000 at 10% per annum for 3 years is calculated as SI = (5000 × 10 × 3) / 100 = Rs 1500. The compound interest for the same principal, rate, and time (compounded annually) is CI = 5000 × (1 + 10/100)^3 - 5000 = 5000 × 1.331 - 5000 = Rs 1655. The difference between compound interest and simple interest is Rs 1655 - Rs 1500 = Rs 155. The closest option to Rs 155 is Rs 145 (option A), which is the best available choice among the given options.

The speed of the current is half the difference between downstream and upstream speeds: (35 - 25) / 2 = 5 kmph.

The speed of the stream is calculated as half the difference between downstream and upstream speeds: (15.5 - 8.5) / 2 = 7 / 2 = 3.5 km/hr. However, option A is 3.5 km/hr, which matches this calculation, so the correct answer is A, not B. Rechecking options shows option A is 3.5 km/hr, which is correct. Therefore, correct_answer should be A.

The speed of the boat upstream is (9 - 3) = 6 km/h and downstream is (9 + 3) = 12 km/h. Let the distance between A and B be x km. The total time is given as 3 hours, so x/6 + x/12 = 3. Solving this gives x = 12 km.

The simple interest (S.I.) for 2 years is Rs. 660, so the interest for 1 year is Rs. 330. Using the formula S.I. = (P × R × T)/100, and knowing the compound interest (C.I.) for 2 years is Rs. 696.30, the rate of interest is calculated as 10%.

The speed of the man in still water is the average of his speed against the stream and with the stream. Given: Distance = 0.75 km, Time against stream = 11 ¼ minutes = 11.25/60 = 0.1875 hours, Time with stream = 7 ½ minutes = 7.5/60 = 0.125 hours. Speed against stream = 0.75 / 0.1875 = 4 km/hr. Speed with stream = 0.75 / 0.125 = 6 km/hr. Speed in still water = (4 + 6) / 2 = 5 km/hr. Therefore, the correct answer is D (5 km/hr).