Train X departs from a station traveling due east at a constant speed of 50 miles per hour… — GRE General Test

Train $X$ departs from a station traveling due east at a constant speed of $50$ miles per hour. Exactly $1$ hour later, Express Train $Y$ departs from the same station along the same track, traveling due east at a constant speed of $75$ miles per hour. How many hours after Express Train $Y$ departs will it catch up to Train $X$ ?

A
$1$
B
$1.5$
$2$ Answer
D
$2.5$
E
$3$

Answer

Express Train

Y

will catch up to Train

X

exactly

2

hours after Express Train

Y

departs.

Let

t

represent the number of hours Express Train

Y

travels. Because Train

X

departed

1

hour earlier, it has been traveling for

t + 1

hours. For Express Train

Y

to catch up to Train

X

, both trains must cover the exact same distance from the starting station. Setting up the distance equation

75t = 50(t + 1)

gives

75t = 50t + 50

, which simplifies to

25t = 50

, yielding

t = 2

hours.

Step-by-Step Solution

Define variables for the time traveled by each train.

Let

t

be the time in hours that Express Train

Y

travels. Since Train

X

departed

1

hour earlier, Train

X

travels for

t + 1

hours.

Train

X

has a

1

-hour head start.

Express the distance traveled by each train using

\text{Distance} = \text{Rate} \times \text{Time}

Distance of Train

X = 50(t + 1)

miles; Distance of Express Train

Y = 75t

miles.

Both trains travel at constant rates along the same path.

Equate the two distance expressions to solve for

t

75t = 50(t + 1) \implies 75t = 50t + 50 \implies 25t = 50 \implies t = 2

Express Train

Y

catches Train

X

when both have covered the exact same distance.

Key Concept

Distance, Rate, and Time Modeling for Catch-up Scenarios

Question

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