If the third proportional of 16 and 56 be x, then what is the value of x?

The correct answer is 196.

To find the third proportional of two numbers, let's recall the definition of proportion.

If three numbers a, b, and c are in continued proportion, then the ratio of the first two numbers is equal to the ratio of the second and third numbers:

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

Here, c is called the third proportional to a and b.

Cross-multiplying gives the formula for the third proportional c:

$a\times c=b^2\Rightarrow c=\frac{b^2}{a}$

In this question, we are given:
a = 16
b = 56
The third proportional is given as x (so c = x).

Substitute the given values into the formula:

$x=\frac{{56}^2}{16}=\frac{56\times 56}{16}$

Simplifying the fraction step-by-step:
Divide 56 and 16 by their common factor 8:

$x=\frac{7\times 56}{2}$

Now, divide 56 by 2:

$x=7\times 28$

Multiplying 7 by 28:

$x=196$

Thus, the value of x is 196.