Solving a Distance Travelled Problem Using Mathematical Reasoning

Statement of the Problem

In the first hour, the coach traveled 1/3 of the total distance. In the second hour, it traveled 1/3 of what was left. In the third hour, it traveled 1/4 of the remainder. The coach now has 25 miles left to its destination. How many miles has the coach traveled?

Solution

Let's denote the total distance up to the destination as D miles. Following the problem, we can break down the journey as follows:

First Hour:

Distance covered in the first hour D/3 Distance remaining after the first hour 2D/3

Second Hour:

Distance covered in the second hour 1/3 × 2D/3 2D/9 Distance remaining after the second hour 2D/3 - 2D/9 4D/9

Third Hour:

Distance covered in the third hour 1/4 × 4D/9 D/9 Distance remaining after the third hour 4D/9 - D/9 3D/9 D/3

According to the problem statement, the coach has 25 miles left, so:

D/3 25

Therefore, the total distance D is:

D 25 × 3 75 miles

Distance Traveled in 3 Hours

The distance traveled in the first 3 hours is:

3/9D D/9 4D/9

Therefore, the coach has traveled:

4D/9 4 × 75/9 300/9 50 miles

Verification

To verify our solution, let's retrace the steps with the calculated total distance of 75 miles:

First Hour:

Distance covered in the first hour 75 × 1/3 25 miles

Distance remaining after the first hour 75 - 25 50 miles

Second Hour:

Distance covered in the second hour 50 × 1/3 50/3 miles

Distance remaining after the second hour 50 - 50/3 125/3 miles

Third Hour:

Distance covered in the third hour (125/3) × 1/4 125/12 miles

Distance remaining after the third hour 125/3 - 125/12 500/12 - 125/12 375/12 25 miles

The coach has 25 miles left after 3 hours, which matches the problem statement.