Question

Two trains leave the station at the same time, one heading east and the other west. The eastbound train travels 10 milesper hour slower than the westbound train. If the two trains are 570 miles apart after 3 hours, what is the rate of theeastbound train?Do not do any rounding.

Answer 1

To find the rate of the eastbound train, we proceed as follows:

Step 1: We make a sketch of the two trains and their direction of travel

Step 2: We label the diagram with the speed of travel of each train

Given that the eastbound train travels 10 miles per hour slower than the westbound train, then:

Speed of eastbound train = speed of westbound train - 10 miles/hour

If we represent the speed of the westbound train as w, then:

Speed of eastbound train = w - 10 miles/hour

OR

Speed of eastbound train = w - 10

Now, see the diagram below:

Step 3: We write out the rate-distance formula, as follows:

$Dis\tan ce=Rate\text{ }\times time$

Step 4: We apply the rate-distance formula as follows:

Given that after a period of time of 3 hours , the two trains are a distance of 570 miles apart, we can say that:

Distance = 570 miles

Time = 3 hrs

Now, since the two trains are travelling in opposite directions, the overall rate of their relative motion is:

Rate = rate of eastbound train + rate of westbound train

Thus:

Rate = (w - 10) + w

Rate = (2w - 10) miles/hour

Now, substitute for rate, distance and time in the rate-distance formula;

$\begin{gathered} Dis\tan ce=Rate\text{ }\times time \\ 570=(2w-10)\times3 \end{gathered}$

Step 5: We solve the resulting equation for w, as follows:

$\begin{gathered} 570=(2w-10)\times3 \\ 570=(2w\times3)-(10\times3) \\ 570=6w-30 \\ 570+30=6w \\ 600=6w \\ \frac{600}{6}=w \\ 100=w \\ w=100\text{ miles/hour} \end{gathered}$

Thus, w = 100 miles/hour

Step 6: We now solve for the rate (speed) of the eastbound train, as follows:

Since:

Speed of eastbound train = w - 10 miles/hour

And w = 100miles/hour, we have that:

Speed of eastbound train = (100 - 10)miles/hour

Speed of eastbound train = 90 miles/hour

Therefore, the rate (or speed) of the eastbound train is 90 miles/hour