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Difficulty: EasyRate, Time, and Distance Problems

An athlete swims across a lake at a constant rate of 1010 miles per hour and swims back to the starting point along the exact same path at a constant rate of 1515 miles per hour. What is the athlete's average speed, in miles per hour, for the entire trip?

  1. A
    11.511.5
  2. 1212Answer
  3. C
    12.512.5
  4. D
    1313
  5. E
    2525

Answer

1212 miles per hour
To find the average speed for a round trip, divide the total distance by the total time. If the distance of one leg is dd, the time for the first leg is d10\frac{d}{10} and for the second leg is d15\frac{d}{15}. Total distance is 2d2d, and total time is d10+d15=d6\frac{d}{10} + \frac{d}{15} = \frac{d}{6}. Dividing total distance by total time yields 2dd6=12\frac{2d}{\frac{d}{6}} = 12 miles per hour.

Step-by-Step Solution

1
Assign a convenient distance for one leg of the trip.
Let the distance across the lake be d=30d = 30 miles (the least common multiple of 1010 and 1515).
Choosing a concrete distance simplifies calculating travel times.
2
Calculate the time spent on each leg.
Outbound time t1=3010=3t_1 = \frac{30}{10} = 3 hours; return time t2=3015=2t_2 = \frac{30}{15} = 2 hours.
Time equals distance divided by rate.
3
Calculate the total distance and total time.
Total distance =30+30=60= 30 + 30 = 60 miles; total time =3+2=5= 3 + 2 = 5 hours.
The round trip consists of two equal legs.
4
Compute the average speed.
Average speed =605=12= \frac{60}{5} = 12 miles per hour.
Average speed is defined as total distance divided by total time.

Key Concept

Average speed for a multi-leg trip is equal to the total distance divided by the total time, not the simple arithmetic mean of the individual speeds.
Estimated Time:1m 0s
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Topics

Rate, Time, and Distance Problems

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