Finding the Third Proportional of 32 and 24

The third proportional of two numbers $a$ and $b$ is a number $x$ such that the ratio $a:b$ is equal to the ratio $b:x$. This can be written as:

$ \frac{a}{b} = \frac{b}{x} $

Setting up the Proportion

In this problem, we are given $a = 32$ and $b = 24$. We need to find the third proportional, $x$. The proportion is:

$ 32 : 24 :: 24 : x $

Which translates to the equation:

$ \frac{32}{24} = \frac{24}{x} $

Calculating the Value of x

To find $x$, we can cross-multiply:

$ 32 \times x = 24 \times 24 $

Now, solve for $x$:

$ x = \frac{24 \times 24}{32} $

Calculate the product in the numerator:

$ x = \frac{576}{32} $

Finally, perform the division:

$ x = 18 $

Conclusion

The value of $x$, which is the third proportional of 32 and 24, is 18.