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?

Mathematics Questions

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?

Short Answer

The process begins by setting up a proportion to compare distances traveled across plains and mountains, denoting the mountain distance as x. After cross-multiplying and simplifying the equation, it is solved to find that the distance through the mountains is approximately 211.76 miles.

Step-by-Step Solution

Step 1: Set Up the Proportion

Begin by identifying the speeds and distances involved in the trip. You need to compare the distance traveled across the plains and through the mountains. Let’s denote the distance through the mountains as x. Set up a proportion like this:

x / 90 = (300 – x) / 37.5

Step 2: Cross-Multiply and Simplify

Next, perform cross-multiplication to eliminate the fraction and set up a solvable equation. This will allow you to express everything in terms of x. The equation will look like this:

37.5x = 90(300 – x)

Now simplify this equation to:

37.5x = 27000 – 90x

Step 3: Solve for Distance

Combine like terms to isolate x on one side of the equation. Bring together all x terms by adding 90x to both sides:

127.5x = 27000

Finally, solve for x to find the distance through the mountains:

x ‚Äöaa 211.76 miles

This means that approximately 211.76 miles of the trip were through the mountains.

A train averages a speed of 90 miles per hour …