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# A freight train and an express train leave towns $390$ miles apart at the same time, traveling toward one another. The freight train travels $30 \text{ mph}$ slower than the express train. They pass one another $3$ hours later. What are their speeds?

Solution

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Let $s_1$ be the speed of freight train, and let $s_2$ be the speed of express train.

We know that the speed $s$ is:

$\boxed{s=\dfrac{d}{t},~~(1)}$

where $d$ is distance and $t$ is time.

From (1) we get $d=st$. So the freight train in 3 hours passed $d_1=3s_1$ and express train passed $d_2=3s_2$

• both trains travel 3 hours and together they covered 390 miles, so we get

$d_1+d_2=390\Rightarrow 3s_1+3s_2=390$

• we also know that freight train travels 30 mph slower than express train, so we get:

$s_1=s_2-30$

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