Multiple choice

Alice and Bob decide to run a race. Alice runs the first half of the race at an average speed of 15 miles per hour and the second half at 20 miles per hour. On the other hand, Bob covers the same distance and with same speeds of 15 miles per hour and 20 miles per hour, with the ratio of the distances covered at these speeds being 3 : 4.Quantity AAlice's average speed for the entire distanceQuantity BBob's average speed for the entire distance

  1. Quantity A is greater.

  2. Quantity B is greater.

  3. The two quantities are equal.

  4. The relationship cannot be determined from the information given.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Alice's average speed = 2 * v1 * v2 / (v1 + v2) = 2 * 15 * 20 / (15 + 20) = 600 / 35 = 17.14 mph. Bob's average speed = Total Distance / Total Time. Let distance be 7d. Bob covers 3d at 15 mph and 4d at 20 mph. Time = 3d/15 + 4d/20 = d/5 + d/5 = 2d/5. Average speed = 7d / (2d/5) = 35/2 = 17.5 mph. Bob's speed is greater.

AI explanation

Because Alice travels equal distances at 15 and 20 miles per hour, her average speed is the harmonic mean 2(15)(20) / (15 + 20), which equals 120/7 or about 17.14 miles per hour. For Bob, the average speed is the weighted arithmetic mean (3(15) + 4(20)) / (3 + 4), which equals 125/7 or about 17.86 miles per hour. Comparing the two values shows that 125/7 is greater than 120/7, so Quantity B is greater.

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