Speed, Time and Distance

To cover 28 miles in 30 minutes (which is 0.5 hours), the speed required is calculated by dividing the distance by the time: speed = 28 miles ÷ 0.5 hours = 56 mph. Therefore, the correct answer is option C (56 mph). The original question had the correct calculation in the explanation but incorrectly marked option D as the correct answer. This has been corrected to option C.

After 2 hours, the first plane travels 150 mph × 2 = 300 miles north, and the second plane travels 200 mph × 2 = 400 miles west. Using the Pythagorean theorem, the distance apart is √(300² + 400²) = √(90000 + 160000) = √250000 = 500 miles.

The distance covered at 6 km/hr in 50 minutes (which is 5/6 hour) is 6 × 5/6 = 5 km. At 10 km/hr, time taken to cover 5 km is 5/10 = 0.5 hours or 30 minutes. However, since 30 minutes is option B, let's recheck calculations carefully: Distance = speed × time = 6 km/hr × (50/60) hr = 6 × 5/6 = 5 km. Time at 10 km/hr = distance/speed = 5/10 = 0.5 hr = 30 minutes. So the correct answer is B, not C. Therefore, correct_answer is B and explanation adjusted accordingly.

Each boy runs 60 km in 6 hours, so the distance covered in 1 hour is 60 km ÷ 6 hours = 10 km. However, the question asks for the distance covered by 1 boy in 1 hour, and since the question states '4th boys ran 60 km each in 6 hours,' it likely means 4 boys each ran 60 km in 6 hours. The distance per boy per hour is 60 km ÷ 6 = 10 km, so the correct answer is 10 km, which corresponds to option B. The initial calculation is correct, so the correct answer is B.

The average speed is calculated by dividing the total distance by the total time. The train travels 6 km in the first 12 minutes and 6 km in the next 8 minutes, covering 12 km in 20 minutes (1/3 hour). Thus, average speed = 12 km / (1/3) hr = 36 km/hr. However, recalculating carefully: First segment distance = (30 km/hr)*(12/60 hr) = 6 km; second segment distance = (45 km/hr)*(8/60 hr) = 6 km; total distance = 12 km; total time = 20 minutes = 1/3 hour; average speed = 12 / (1/3) = 36 km/hr. So the correct average speed is 36 km/hr, which corresponds to option B. Therefore, the original correct_answer 'B' is correct. The explanation is now added for clarity.

The average speed for a round trip is given by the formula 2 * (speed1 * speed2) / (speed1 + speed2). Here, it is 2 * (100 * 150) / (100 + 150) = 120 km/hr.

The two automobiles are moving toward each other with a combined speed of 60 km/h + 40 km/h = 100 km/h. To cover the 150 km distance between them, the time taken is distance divided by speed, which is 150 km ÷ 100 km/h = 1.5 hours. Therefore, the correct answer is 1.5 hours.

The run rate is calculated by dividing the total runs scored by the number of overs faced. In the first 10 overs, the run rate was 3.2, so runs scored = 3.2 × 10 = 32 runs. The target is 282 runs, so remaining runs needed = 282 - 32 = 250 runs. The remaining overs are 40. Therefore, the required run rate for the remaining overs = 250 ÷ 40 = 6.25 runs per over. This corresponds to option A.

The average speed for a round trip when speeds differ is given by the formula: 2 * speed1 * speed2 / (speed1 + speed2). Here, speed1 = 40 km/h and speed2 = 40 + 50% of 40 = 60 km/h. So, average speed = 2 * 40 * 60 / (40 + 60) = 4800 / 100 = 48 km/h.

Since A beats B by 50 meters or 10 seconds in a 1000 meter race, B takes 10 seconds more than A to finish. B covers 950 meters in the same time A covers 1000 meters. Thus, B's speed = 950 meters / (A's time + 10 seconds). A's speed = 1000 meters / A's time. Equating speeds: 950 / (t + 10) = 1000 / t. Solving gives t = 200 seconds for A.

A runs 1 km in 4.5 minutes, so speed of A = 1000/4.5 = 222.22 m/min. B runs 1 km in 5 minutes, so speed of B = 1000/5 = 200 m/min. To finish together, A must give B a start distance 'd' such that time taken by A to run 1000 m equals time taken by B to run (1000 - d) m. So, 1000/222.22 = (1000 - d)/200, solving gives d = 125 m.

Since A gives B a 100 m start in 1000 m, B runs 900 m when A runs 1000 m. Similarly, A gives C a 200 m start, so C runs 800 m when A runs 1000 m. Therefore, when B runs 900 m, C runs (800/1000)*900 = 720 m. Hence, B can give C a start of 900 - 720 = 180 m. However, the options do not include 180 m, so let's re-express the problem carefully: The question asks how much start B can give C in a kilometer race. Since B runs 900 m when A runs 1000 m, and C runs 800 m when A runs 1000 m, the ratio of B's speed to C's speed is 900:800 = 9:8. Therefore, in a 1000 m race, B runs 1000 m, C runs (8/9)*1000 = 888.88 m, so B can give C a start of 1000 - 888.88 = 111.12 m, which matches option A.

The effective speed downstream is 9 + 1 = 10 kmph, and upstream is 9 - 1 = 8 kmph. Let the distance be d km. Total time = d/10 + d/8 = 4.5 hours. Solving gives d = 32 km.

The speed downstream is the sum of the boat's speed in still water and the current speed: 60 + 20 = 80 kmph. The speed upstream is the difference: 60 - 20 = 40 kmph. Therefore, the correct answer is 80 kmph downstream and 40 kmph upstream.

The man rows at 7.5 km/hr in still water. It takes him twice as long to row upstream as downstream, meaning the time upstream is double the time downstream for the same distance. Let the speed of the stream be s km/hr. Then upstream speed = 7.5 - s and downstream speed = 7.5 + s. Since time is distance/speed, and time upstream is twice time downstream, we have: (distance / (7.5 - s)) = 2 × (distance / (7.5 + s)). Simplifying, (7.5 + s) = 2(7.5 - s), which leads to 7.5 + s = 15 - 2s, or 3s = 7.5, so s = 2.5 km/hr. Therefore, the rate of the stream is 2.5 km/hr, which corresponds to option B.

The man's effective speed downstream is the sum of his rowing speed in still water and the speed of the current: 15 km/h + 3 km/h = 18 km/h. Converting this speed to meters per second: (18 * 1000) / 3600 = 5 m/s. To cover 60 meters at 5 m/s, the time taken is distance/speed = 60 / 5 = 12 seconds. Therefore, the correct answer is D (12 seconds).

The effective speed downstream is the sum of the boat's speed in still water and the stream speed: 20 kmph + 6 kmph = 26 kmph. The time taken to cover 60 km downstream is distance divided by speed, which is 60/26 = 30/13 hours. This matches option D. Therefore, the correct answer is D.

Let the speed of the boat in still water be b and the speed of the current be c. Upstream speed = b - c, downstream speed = b + c. Given times and same distance d, we have d/(b-c) = 8.8 hours and d/(b+c) = 4 hours. Dividing these gives (b+c)/(b-c) = 8.8/4 = 2.2. Solving for b/c yields (b+c)/(b-c) = 2.2 => (b/c + 1)/(b/c - 1) = 2.2, which gives b/c = 3. Thus, the ratio of the speed of the boat to the speed of the current is 3:1.

Let the speed of the boat in still water be x kmph and the speed of the stream be 4 kmph. The boat's effective speed downstream is (x + 4) kmph and upstream is (x - 4) kmph. The total time for 6 km downstream and 6 km upstream is 2 hours: 6/(x+4) + 6/(x-4) = 2. Solving this equation gives x = 10 kmph.

Let the speed of the man in still water be x kmph and the speed of the stream be y kmph. Using the given data, we form two equations based on time = distance/speed: 30/(x+y) + 20/(x−y) = 4 and 45/(x+y) + 40/(x−y) = 7. Solving these simultaneously gives x = 12 kmph as the speed of the man in still water.

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.

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.

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.

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.

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.