Solving Age Ratio Problems: A Step-by-Step Guide

Understanding and solving age ratio problems is an essential skill in algebra. This article explains a step-by-step process to solve a common age ratio problem, using the example of Sam and John. We will derive the ages of both individuals, verify the given conditions, and present the final answer.

The Problem

The ages of Sam and John are in the ratio 5:7. After 9 years this ratio will become 4:5. What is the difference between their ages?

Solution

Let's represent the current ages of Sam and John with variables. Let x be a positive integer. Then:

sam-age'>Sam's current age 5x

john-age'>John's current age 7x

After 9 Years

In 9 years, their ages will be:

Sam's age: 5x 9

John's age: 7x 9

According to the problem, after 9 years, the ratio of their ages will be 4:5. Therefore:

frac{5x 9}{7x 9} frac{4}{5}

By cross-multiplying the equation, we get:

5(5x 9) 4(7x 9)

Simplifying the left and right sides:

25x 45 28x 36

Now, we can rearrange the equation to isolate x:

25x 45 - 28x 36 - 45

-3x -9

(x frac{-9}{-3} 3)

Substituting x back into the expressions for Sam's and John's ages:

Sam's age 5x 5(3) 15

John's age 7x 7(3) 21

The difference between their ages is:

21 - 15 6 years

Conclusion

The difference in the ages of Sam and John is 6 years. This method involves using algebra to represent the current and future ages, setting up an equation based on the given conditions, and solving for the unknown integer x. It's a systematic way to tackle age ratio problems.

Additional Example

Let's consider another example for verification:

Age of Sam 5x and John 7x

After 9 years, age of Sam 5x 9 and that of John 7x 9

frac{5x 9}{7x 9} frac{4}{5}

Or, 25x 45 28x 36

Or, 28x - 25x 45 - 36

Or, 3x 9

(x frac{9}{3} 3)

Difference of their age 7x - 5x 2x 2 × 3 6 years

Therefore, the age difference is 6 years.