Finding the Third Proportional to 16 and 24

The question asks us to find the third proportional to the numbers 16 and 24.

Let the two given numbers be a and b. If we consider a sequence of three numbers a, b, and x such that they are in continued proportion, it means the ratio of the first to the second is equal to the ratio of the second to the third.

Mathematically, this is expressed as:

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

In this ratio, a is the first term, b is the second term (or mean proportional), and x is the third proportional.

In our problem, the first number is 16 (so, \(a = 16\)), and the second number is 24 (so, \(b = 24\)). We need to find the third proportional, which we will call x.

Using the definition of continued proportion, we set up the equation:

\(\frac{16}{24} = \frac{24}{x}\)

To solve for x, we can cross-multiply:

\(16 \times x = 24 \times 24\)

\(16x = 576\)

Now, we isolate x by dividing both sides by 16:

\(x = \frac{576}{16}\)

Let's perform the division:

\(576 \div 16 = 36\)

So, \(x = 36\).

The third proportional to 16 and 24 is 36.

To verify, we can check the ratios:

\(\frac{16}{24} = \frac{2 \times 8}{3 \times 8} = \frac{2}{3}\)

\(\frac{24}{36} = \frac{2 \times 12}{3 \times 12} = \frac{2}{3}\)

Since \(\frac{16}{24} = \frac{24}{36}\), the numbers 16, 24, and 36 are in continued proportion, and 36 is indeed the third proportional.

The value of the third proportional is 36.