## 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?

## Answer

**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 (s_{p}) = 90 miles per hour

Through the mountains, the average speed (s_{m}) = 37.5 miles per hour

Distance = 300 miles

We also have time across the plains, t_{p} = ?

And time through the mountains, t_{m} = ?

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: s_{p} = x_{p} / t_{p}

Second step: s_{m} = x_{m} / t_{m}

Third step: x_{p} + x_{m} = 300

Fourth step: t = t_{p} + t_{m}

**Solution**

**First step:**

We will make x_{p} subject of the formula from the equation s_{p} = x_{p} / t_{p}

Therefore, x_{p} = s_{p} t_{p} = 90t_{p}

**Second step:**

We will find x_{m} from the equation s_{m} = x_{m} / t_{m}

x_{m} = s_{m} t_{m} = 37.5t_{m}

**Third Step:**

Applying our final answer from the first and second steps into the equation x_{p} + x_{m} = 300, we will now obtain

90t_{p} + 37.5t_{m} = 300

And since we are looking for t_{m}, we can now say:

t_{m} = (300 – 90t_{p}) / 37.5

**Fourth step:**

We already have:

t = t_{p} + t_{m}

3.8 = t_{p} + t_{m}

Therefore, t_{p} = 3.8 – t_{m}

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

**t _{m} = (300 – 90(3.8 – t_{m})) / 37.5**

After simplifying the above equation, we will now have

t_{m} = (300 – 342 + 90t_{m}) / 37.5

Which will now give us

37.5t_{m} – 90t_{m} = – 42

– 52.5t_{m} = – 42

And t_{m} = 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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