A train averages a speed of 90 miles per hour across the plains and 37.5 miles per

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.

You may also like to read:

A high-speed drill reaches 2000 rpm in 0.50 s