Question 62098
<b>2 trains left the same city at the same time, traveling in opposite directions. the eastbound train traveles for 10 hours, and the westbound train traveles for 5 hours. They are now 1300 km. apart. the westbound train's rate is 20 km/h faster than the eastbound train. Find the rate of the speed of each train.</b>

Let {{{R[e]}}} = the rate (or speed) of the eastbound train.
Let {{{R[w]}}} = the rate (or speed) of the westbound train.
Let {{{D[e]}}} = the distance covered by the eastbound train.
Let {{{D[w]}}} = the distance covered by the westbound train.

Distance = Rate x Time.

We know that the eastbound train traveled for 10 hours.
So, {{{D[e] = R[e]*10}}}.

We know that the westbound train traveled for 5 hours.
So, {{{D[w] = R[w]*5}}}.

We know that the total distance covered = 1300 so {{{D[e]+D[w] = 1300}}}.
Or, {{{R[e]*10 + R[w]*5 =1300}}}.

We know that the westbound rate is 20 km/hour faster than the westbound rate.
So, {{{R[w]=R[e]+20}}}.

So, {{{R[e]*10 + (R[e]+20)*5) = 1300}}}.
Or, {{{R[e]*10 + R[e]*5 + 100 = 1300}}}.
Or, {{{R[e]*15 = 1200}}}.
Or, {{{R[e] = 80}}}.
So, {{{R[w] = R[e]+20 = 100}}}.

So, the eastbound train is going 80 km/hour and the westbound train is going 100 km/hour.

Verify:

In 10 hours, the eastbound train covers 10*80 km = 800 km.
In 5 hours, the westbound train covers 5*100 km = 500 km.
The total distance = 800 + 500 = 1300 km which is what the problem stated.