A Day's Walk - Solving a Fractional Distance Challenge

Understanding Distances

Jim walked 24 miles in one day, breaking down his journey into two parts: the morning and the afternoon. This problem involves basic algebra and understanding of fractions, making it a great exercise for students and anyone looking to improve their problem-solving skills. Let's break it down step by step.

Breaking Down the Walk

Let's denote the distance Jim walked in the morning as x miles. According to the problem, he walked twice as far in the morning as he did in the afternoon. Thus, the distance he walked in the afternoon would be x/2 miles.

Solving for Morning Distance

The total distance Jim walked is the sum of the distances walked in the morning and the afternoon:

x x/2 24

To solve for x, we first combine the terms:

2x/2 x/2 24

This simplifies to:

3x/2 24

NExt, multiply both sides by 2 to eliminate the fraction:

3x 48

Then, divide both sides by 3:

x 16

This means Jim walked 16 miles in the morning.

Calculating the Afternoon Distance

To find the distance Jim walked in the afternoon, we use the relationship x/2:

Distance in the afternoon x/2 16/2 8 miles

So, Jim walked 8 miles in the afternoon.

Visual Representation

To make this process easier to understand, you can use a visual approach. Draw a rectangle and mark it as the distance walked in the afternoon. This distance is 8 miles. Now, for the morning walk, draw another rectangle that is twice as wide as the first one since the morning walk is twice the afternoon walk. This second rectangle will be 16 miles wide.

Adding these sections together gives you the total distance of 24 miles, proving the solution is correct.

Conclusion

In summary, Jim walked 16 miles in the morning and 8 miles in the afternoon, making a total of 24 miles. This problem not only tests algebraic skills but also reinforces the concept of fractions and their practical applications in everyday scenarios.

Frequently Asked Questions

Q: How do you check if the solution is correct?

A: You can double-check by adding the distances: 16 miles (morning) 8 miles (afternoon) 24 miles (total).

Q: Can you solve this problem using a different method?

A: Yes, you can also set up a system of equations:

M A 24

M 2A

Replace M in the first equation with 2A:

2A A 24

3A 24

A 8 (afternoon distance)

M 2 * 8 16 (morning distance)