Ratio Problem: Solving for (x - y) : (x + y)

This problem involves working with ratios and algebraic expressions. We are given the ratio between two variables, \(x\) and \(y\), and asked to find a new ratio involving expressions of \(x\) and \(y\). Let's break down the steps to find the solution.

Understanding the Given Ratio

We are given that the ratio \(x : y\) is equal to \(8:3\). This can be written mathematically as:

\( \frac{x}{y} = \frac{8}{3} \)

This means that for some non-zero constant \(k\), we can express \(x\) and \(y\) as:

• \(x = 8k\)
• \(y = 3k\)

Using this representation helps us work with the variables in the subsequent steps.

Calculating the New Expressions

The question asks for the ratio \((x - y) : (x + y)\). We need to find the values of the expressions \((x - y)\) and \((x + y)\) using our representations of \(x\) and \(y\).

• Calculating \((x - y)\): Substitute \(x = 8k\) and \(y = 3k\) into the expression \((x - y)\): \((x - y) = 8k - 3k = 5k\)
• Calculating \((x + y)\): Substitute \(x = 8k\) and \(y = 3k\) into the expression \((x + y)\): \((x + y) = 8k + 3k = 11k\)

Finding the Final Ratio

Now we have the values for both parts of the required ratio:

• \((x - y) = 5k\)
• \((x + y) = 11k\)

The ratio \((x - y) : (x + y)\) can be written as:

\( (x - y) : (x + y) = 5k : 11k \)

To simplify this ratio, we can divide both terms by the common factor \(k\) (assuming \(k \neq 0\)):

\( \frac{5k}{11k} = \frac{5}{11} \)

Therefore, the ratio \((x - y) : (x + y)\) simplifies to \(5:11\).

Conclusion

Based on the calculations, when \(x : y = 8:3\), the ratio \((x - y) : (x + y)\) is found to be \(5:11\).