A train travels from Station A to Station B, a distance of 180 miles, at a constant speed … — GRE General Test

A train travels from Station A to Station B, a distance of 180 miles, at a constant speed of $v$ miles per hour. On the return trip from Station B to Station A, the train travels the first half of the distance at a constant speed that is $20\%$ less than $v$ , and the remaining half of the distance at a constant speed that is $25\%$ greater than $v$ . If the total time for the return trip is 6 minutes longer than the total time for the trip from Station A to Station B, what is the value of $v$ ?

Answer: 45 miles per hour

Answer

The value of

v

45

The outbound travel time for 180 miles at speed

v

\frac{180}{v}

hours. On the return trip, the first 90 miles at speed

0.80v

require

\frac{90}{0.80v} = \frac{112.5}{v}

hours, while the second 90 miles at speed

1.25v

require

\frac{90}{1.25v} = \frac{72}{v}

hours. The total return duration is

\frac{112.5 + 72}{v} = \frac{184.5}{v}

hours. Setting the difference between the return time and outbound time equal to 6 minutes (

0.1

hours) gives

\frac{184.5}{v} - \frac{180}{v} = 0.1

, which simplifies to

\frac{4.5}{v} = 0.1

, yielding

v = 45

Step-by-Step Solution

Write the expression for the outbound trip duration in terms of

v

T_{\text{outbound}} = \frac{180}{v}

hours.

Time is equal to total distance divided by constant speed.

Calculate the duration for each half of the return trip in terms of

v

The first 90 miles take

\frac{90}{0.80v} = \frac{112.5}{v}

hours, and the second 90 miles take

\frac{90}{1.25v} = \frac{72}{v}

hours, giving a total return duration of

\frac{184.5}{v}

hours.

The return trip consists of two 90-mile segments driven at

0.80v

and

1.25v

respectively.

Equate the difference between return and outbound times to 0.1 hours and solve for

v

\frac{184.5}{v} - \frac{180}{v} = 0.1 \implies \frac{4.5}{v} = 0.1 \implies v = 45

The time difference of 6 minutes is equal to

\frac{6}{60} = 0.1

hours.

Key Concept

Distance, Rate, and Time Modeling with Piecewise Speed Changes

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