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# A train averages a speed of 90

## Question

A train averages a speed of 90 miles per hour across the plains and 37.5 miles per hour through the mountains. If a trip of 300 miles took 3 hours and 48 minutes, how much of it was through the mountains?

The final answer to the question is 0.8 hours or 48 minutes.

### Explanation

Data: Information from the question

Across the plains, the average speed (sp) = 90 miles per hour

Through the mountains, the average speed (sm) = 37.5 miles per hour

Distance = 300 miles

We also have time across the plains, tp = ?

And time through the mountains, tm = ?

Total time, t = 3 hours 48 minutes = (19/5) = 3.8 hours

Formula

Since we are looking for a time through the mountains, we will use the following equations:

First step: sp = xp / tp

Second step: sm = xm / tm

Third step: xp + xm = 300

Fourth step: t = tp + tm

#### Solution

First step:

We will make xp subject of the formula from the equation sp = xp / tp

Therefore, xp = sp tp = 90tp

Second step:

We will find xm from the equation sm = xm / tm

xm = sm tm = 37.5tm

Third Step:

Applying our final answer from the first and second steps into the equation xp + xm = 300, we will now obtain

90tp + 37.5tm = 300

And since we are looking for tm, we can now say:

tm = (300 – 90tp) / 37.5

Fourth step:

t = tp + tm

3.8 = tp + tm

Therefore, tp = 3.8 – tm

Applying the above equation into our final answer in the third step, we will have:

tm = (300 – 90(3.8 – tm)) / 37.5

After simplifying the above equation, we will now have

tm = (300 – 342 + 90tm) / 37.5

Which will now give us

37.5tm – 90tm = – 42

– 52.5tm = – 42

And tm = 42/52.5 = 0.8 hours = 48 minutes

Therefore, much of it that was through the mountains is 0.8 hours or 48 minutes.

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