Finding the Third Proportional to 9 and 45

The question asks us to find the third proportional to the numbers 9 and 45.

Let the two given numbers be \(a\) and \(b\). If \(a, b,\) and \(c\) are in continued proportion, then \(a:b :: b:c\), and \(c\) is called the third proportional to \(a\) and \(b\).

This relationship can be written as a fraction:

\( \frac{a}{b} = \frac{b}{c} \)

In this problem, the first number \(a = 9\) and the second number \(b = 45\). Let the third proportional be \(c\).

According to the definition of continued proportion:

\( \frac{9}{45} = \frac{45}{c} \)

To find the value of \(c\), we can cross-multiply:

\( 9 \times c = 45 \times 45 \)

\( 9c = 45 \times 45 \)

Now, we need to calculate the value of \(45 \times 45\).

\( 45 \times 45 = 2025 \)

So, the equation becomes:

\( 9c = 2025 \)

To find \(c\), divide both sides by 9:

\( c = \frac{2025}{9} \)

Performing the division:

\( c = 225 \)

Thus, the third proportional to 9 and 45 is 225.

We can check this by setting up the proportion: \( 9:45 :: 45:225 \).

The ratio \( 9:45 \) simplifies to \( \frac{9}{45} = \frac{1}{5} \).

The ratio \( 45:225 \) simplifies to \( \frac{45}{225} = \frac{45 \div 45}{225 \div 45} = \frac{1}{5} \).

Since both ratios are equal \( \left( \frac{1}{5} \right) \), the numbers are in continued proportion, and 225 is indeed the third proportional.

The final answer is 225.