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Difficulty: MediumRate, Time, and Distance Problems

A cyclist travels from City A to City B at a constant speed of 2020 miles per hour. On the return trip along the exact same route, the cyclist increases their speed by 25%25\%. If the total time taken for the round trip is 99 hours, what is the distance, in miles, between City A and City B?

Answer: 100 miles

Answer

The distance between City A and City B is 100 miles.
To find the one-way distance dd, determine the return speed as 20×1.25=2520 \times 1.25 = 25 miles per hour. The outbound time is d20\frac{d}{20} hours and the return time is d25\frac{d}{25} hours. Adding these gives d20+d25=9\frac{d}{20} + \frac{d}{25} = 9, which simplifies to 9d100=9\frac{9d}{100} = 9, so d=100d = 100 miles.

Step-by-Step Solution

1
Calculate the speed on the return trip.
The return speed is 20×1.25=2520 \times 1.25 = 25 miles per hour.
The return speed increases by 25%25\% over the outbound speed of 2020 miles per hour.
2
Express outbound and inbound travel times using distance dd.
Outbound time t1=d20t_1 = \frac{d}{20} hours, Inbound time t2=d25t_2 = \frac{d}{25} hours.
Using the rate-time-distance formula t=drt = \frac{d}{r}.
3
Set up and solve the total time equation.
d20+d25=9  ⟹  9d100=9  ⟹  d=100\frac{d}{20} + \frac{d}{25} = 9 \implies \frac{9d}{100} = 9 \implies d = 100 miles.
The total duration of the round trip is given as 99 hours.

Key Concept

Rate, Time, and Distance Relationship in Round Trips
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Topics

Rate, Time, and Distance Problems

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