If `x/5 = y/8`, then find `(x+5):(y+8)`.

Answer

To solve the problem, we start with the equation given:

\[ \frac{x}{5} = \frac{y}{8} \]

**Step 1: Cross-multiply to express x in terms of y.**

Cross-multiplying gives us:

\[ 8x = 5y \]

**Step 2: Rearranging the equation to find the ratio x:y.**

From the equation \(8x = 5y\), we can express the ratio \(x:y\) as:

\[ \frac{x}{y} = \frac{5}{8} \]

**Step 3: Express x in terms of y.**

From the ratio, we can express \(x\) in terms of \(y\):

\[ x = \frac{5}{8}y \]

**Step 4: Substitute x into the expression \(x + 5\) and \(y + 8\).**

Now we need to find \(x + 5\) and \(y + 8\):

\[ x + 5 = \frac{5}{8}y + 5 \]

To combine these, we need a common denominator:

\[ x + 5 = \frac{5y}{8} + \frac{40}{8} = \frac{5y + 40}{8} \]

And for \(y + 8\):

\[ y + 8 = y + 8 \]

**Step 5: Write the ratio \((x + 5):(y + 8)\).**

Now we can write the ratio:

\[ (x + 5):(y + 8) = \frac{5y + 40}{8} : (y + 8) \]

**Step 6: Simplify the ratio.**

To simplify the ratio, we can express it as:

\[ \frac{5y + 40}{8} : (y + 8) \]

This can be rewritten as:

\[ \frac{5y + 40}{8(y + 8)} \]

**Step 7: Factor out common terms.**

Now we can simplify:

\[ = \frac{5(y + 8)}{8(y + 8)} \]

Since \(y + 8\) is common in both the numerator and denominator, we can cancel it out (assuming \(y + 8 \neq 0\)):

\[ = \frac{5}{8} \]

Thus, the final answer is:

\[ (x + 5):(y + 8) = 5:8 \]

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