Understanding the Ratio and Expression

The problem provides a ratio involving two variables, \(x\) and \(y\). We are given that the ratio \(\frac{x}{y}\) is equal to \(\frac{5}{3}\). Our goal is to find the value of a different expression, \(\frac{x + y}{x - y}\), using the information from the given ratio.

Using the Given Ratio \(\frac{x}{y} = \frac{5}{3}\)

When we are given a ratio like \(\frac{x}{y} = \frac{5}{3}\), it means that \(x\) and \(y\) are in the proportion 5 is to 3. We can represent \(x\) and \(y\) using a common multiplier, say \(k\), where \(k\) is a non-zero constant. So, we can write:

• \(x = 5k\)
• \(y = 3k\)

This representation satisfies the given ratio because \(\frac{x}{y} = \frac{5k}{3k} = \frac{5}{3}\) (assuming \(k \neq 0\)).

Evaluating the Expression \(\frac{x + y}{x - y}\)

Now we substitute the expressions for \(x\) and \(y\) (\(x = 5k\) and \(y = 3k\)) into the expression \(\frac{x + y}{x - y}\):

First, let's find the numerator, \(x + y\):

\(x + y = 5k + 3k = (5 + 3)k = 8k\)

Next, let's find the denominator, \(x - y\):

\(x - y = 5k - 3k = (5 - 3)k = 2k\)

Now, we can put these back into the expression \(\frac{x + y}{x - y}\):

\(\frac{x + y}{x - y} = \frac{8k}{2k}\)

Since we established that \(k\) is a non-zero constant, we can cancel \(k\) from the numerator and the denominator:

\(\frac{8k}{2k} = \frac{8}{2}\)

Finally, we simplify the fraction:

\(\frac{8}{2} = 4\)

So, the value of the expression \(\frac{x + y}{x - y}\) is 4.

Conclusion

Given the ratio \(\frac{x}{y} = \frac{5}{3}\), by representing \(x\) as \(5k\) and \(y\) as \(3k\), we can substitute these values into the expression \(\frac{x + y}{x - y}\). The calculation shows that the expression simplifies to \(\frac{8k}{2k}\), which further reduces to 4. Therefore, \(\frac{x + y}{x - y}\) is equal to 4.