Question

Find out the third proportional to $16$ and $36$ .

Answer 1

Hint: Proportion is defined as the equality between any two ratios. We know by the definition of the third proportional of two numbers $p$ and $q$ that a number $r$ is defined to be the third proportional such that $p:q = q:r$ . So, the third proportional can be found out by using the general form $\dfrac{p}{q} = \dfrac{q}{r}$ . On simplifying we can find the third proportional using this form $r = \dfrac{{q \times q}}{p}$ .

Complete step-by-step answer:
We have two numbers $16$ and $36$ .
We need to find the third proportional to these numbers.
We know that two ratios $a:b$ and $c:d$ are said to be in proportion if $\dfrac{a}{b} = \dfrac{c}{d}$ .
Thus we can also write it as $a:b::c:d$ .
However, three numbers $a,b$ and $c$ are said to be in continued proportion if $\dfrac{a}{b} = \dfrac{b}{c}$ . Therefore we can say $c$ is the third proportional of two numbers $a$ and $b$ if $a:b::b:c$ .
Let $x$ be the third proportional of given two numbers $16$ and $36$ .
$\therefore \dfrac{{16}}{{36}} = \dfrac{{36}}{x}$
Taking $x$ on the left-hand side of the equation by cross multiplication.
$ \Rightarrow \dfrac{x}{{36}} = \dfrac{{36}}{{16}}$
Shifting everything on the right-hand side of the equation by cross multiplication except the unknown variable $x$ .
$ \Rightarrow x = \dfrac{{36 \times 36}}{{16}}$
Multiplying the terms in the numerator:
$ \Rightarrow x = \dfrac{{1296}}{{16}}$
Performing the division operation and simplifying we get:
$ \Rightarrow x = 81$
Thus, we can write $\dfrac{{16}}{{36}} = \dfrac{{36}}{{81}}$ .
Hence the third proportion to the numbers $16$ and $36$ is $81$ .

Note: So, the third proportion of two numbers is defined as the second term to the mean terms. One must be well aware of the formula of the third proportion to the two numbers. Substituting the values correctly in the equation is of utmost importance. A slight mistake would result in an incorrect answer.