Understanding Ratios Through a Reading Challenge: Michelle and John's Book Problem

In this educational article, we dive into a fascinating math problem involving Michelle and John's love for reading. By exploring the relationship between the number of books they have, we can enhance our understanding of ratios and fractions in a practical and engaging context.

The Problem

Michelle and John both enjoy reading books. There is an interesting fact about their collection of books: 3/8 of Michelle's books is equal to 1/9 of John's books. Our task is to find the ratio of Michelle's books to John's books in their simplest form.

Mathematical Representation

To make the problem more digestible, let us denote the number of books Michelle and John have as x and y respectively.

Given that 3/8 of Michelle's books is equal to 1/9 of John's books, we can express this relationship as:

3/8x1/9y.

Solving the Equation

First, we isolate the ratio x/y by rearranging the equation:

xy1/93/8

When we simplify the right-hand side, we get:

xy1/9×8/3 xy827
x/y 8/27

Therefore, the ratio of Michelle's books to John's books is 8/27, which is already in its simplest form.

Interpreting the Ratio

A ratio of 8:27 means that for every 8 books Michelle has, John has 27 books. This can help us understand the relative sizes of their book collections.

Verification Through Alternative Methods

Let us verify this calculation using a different approach:

1. Cross-Multiplication Method

We can cross-multiply the fraction to find the equivalent ratio:

3/8x 1/9y

Cross-multiplying gives:

3 × y 8 × (1/9)x

Further simplification yields:

x/y 8/27

2. Common Denominator Method

Another way to express the ratio is by finding a common denominator:

Mary's books 3/8x
John's books 1/9y

Expressing both as fractions with a common denominator (72) of the equivalent books gives:

Michelle's books 3/8x 27/72x
John's books 1/9y 8/72y

Hence, the ratio is:

27/72 : 8/72 27 : 8

Conclusion

This problem not only tests our understanding of ratios and fractions but also highlights the importance of problem-solving techniques in mathematics. The ratio of 8:27 serves as a valuable insight into the relative sizes of Michelle's and John's book collections.

Additional Resources

For readers interested in enhancing their knowledge of ratios and fractions, we recommend the following resources:

MathIsFun: Ratio Khan Academy: Solving Ratio Problems GreenMath: Ratios