Mathematics

The speed of the boat downstream is the speed in still water plus the speed of the stream: 9 km/h + 1.5 km/h = 10.5 km/h. The speed upstream is the speed in still water minus the speed of the stream: 9 km/h - 1.5 km/h = 7.5 km/h. The time taken to travel 105 km downstream is 105 / 10.5 = 10 hours. The time taken to travel 105 km upstream is 105 / 7.5 = 14 hours. Therefore, the total time for the round trip is 10 + 14 = 24 hours.

The swimmer's speed in still water is 4 km/h, and the speed of the water current is 2 km/h. When swimming against the current, the effective speed is the swimmer's speed minus the current speed, i.e., 4 - 2 = 2 km/h. To cover 6 km against the current, the time taken is distance divided by effective speed, which is 6 / 2 = 3 hours. Therefore, the correct answer is 3 hours, which corresponds to option A.

Using the compound interest formula CI = P(1 + r)^t - P, where P = 30000, r = 0.07, and CI = 4347, solving for t gives approximately 3 years. Hence, the correct answer is 3 years.

The amount grows from Rs. 800 in 3 years to Rs. 840 in 4 years, so the interest for one year on Rs. 800 is Rs. 40. The rate of interest per annum is (40/800)*100 = 5%. However, since this is compound interest, the increase from Rs. 800 to Rs. 840 represents one year's compound interest, so the rate r satisfies 800*(1 + r) = 840, giving r = 40/800 = 0.05 or 5%. But the question states compound interest, so the rate should be calculated as (840/800) - 1 = 0.05 or 5%. Therefore, the correct answer is 5%. However, the options given are fractions and percentages; option A is 2 1/2 %, option B is 4%, option C is 5%, and option D is 6 2/3 %. The correct answer is 5%, which corresponds to option C. So the original correct_answer 'C' is correct. The explanation was missing and is now provided.

The man's rate (speed in still water) is the average of the speeds with and against the stream: (6 + 4) / 2 = 5 km/h.

From the first race, A beats B by 100 m in 1000 m, so B runs 900 m when A runs 1000 m. From the second race, B beats C by 100 m in 800 m, so C runs 700 m when B runs 800 m. Combining these ratios, when A runs 600 m, B runs 540 m (600 * 900/1000), and C runs 472.5 m (540 * 700/800). Therefore, A beats C by 600 - 472.5 = 127.5 m.

When A beats B by 25 m in a 100 m race, B runs 75 m when A runs 100 m, so the speed ratio A:B = 100:75 = 4:3. When B beats C by 4 m in a 100 m race, C runs 96 m when B runs 100 m, so the speed ratio B:C = 100:96 = 25:24. Combining these ratios, A:C = (4/3) * (25/24) = 100:72. Therefore, when A runs 100 m, C runs 72 m, meaning A beats C by 28 m. Hence, the correct answer is option C (28 m). The original explanation's calculation is correct, but the choice of 29 m due to rounding is not justified mathematically.

X runs 48 meters in the same time Y runs 42 meters, so X is faster. The ratio of their speeds is 48:42 = 8:7. In a 1000 meter race, when X runs 1000 meters, Y runs (7/8) × 1000 = 875 meters. Therefore, X beats Y by 1000 - 875 = 125 meters. Hence, the correct answer is 125 meters, option B.

Using the compound interest formula A = P(1 + r/100)^t, where P = 7500, r = 4%, and t = 2 years, the amount is 7500 × (1.04)^2 = 7500 × 1.0816 = Rs. 8112. Hence, option D is correct.

Let the speed of the man in still water be x kmph and the speed of the stream be y kmph. Downstream speed = x + y, upstream speed = x - y. Given downstream distance 30 km in 3 hours, so downstream speed = 10 kmph. Upstream distance 18 km in 3 hours, so upstream speed = 6 kmph. Solving: x + y = 10 and x - y = 6, adding gives 2x = 16, so x = 8 kmph. However, the options do not include 8 kmph as the correct answer. Rechecking the problem, the man swims downstream 30 km in 3 hours (speed 10 kmph) and upstream 18 km in 3 hours (speed 6 kmph). The average speed in still water is (downstream speed + upstream speed)/2 = (10 + 6)/2 = 8 kmph. Therefore, the correct answer is B (8 kmph). The original correct_answer was B, so it is correct. The explanation was missing and is now provided.