Question

Find the fourth proportion to numbers 30, 32, 42.

Answer 1

Hint: If four numbers $a,{\text{ }}b,{\text{ }}c$ and $d$ are in proportion then the ratio of the first two must be equal to the ratio of the next. This can be represented as $a:b = c:d$ or $a:b::c:d$. In other words, we can say that:
$ \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}$
Use this equation to determine the fourth proportion of the above numbers.

Complete step-by-step answer:
According to the question, the three given numbers are 30, 32 and 42. A fourth number is required such that the four numbers are in proportion.
We know that if four numbers $a,{\text{ }}b,{\text{ }}c$ and $d$ are in proportion then the ratio of the first two must be equal to the ratio of the next i.e. $a:b = c:d$ or $a:b::c:d$.
From this we can write:
$ \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}$
 Here $d$ is the fourth proportion.
So if we put $a = 30$, $b = 32$ and $c = 42$ in the above equation, we’ll get the value of $d$ i.e. fourth proportion.
Using this, we’ll get:
$
   \Rightarrow \dfrac{{30}}{{32}} = \dfrac{{42}}{d} \\
   \Rightarrow d = \dfrac{{32}}{{30}} \times 42 \\
   \Rightarrow d = \dfrac{{1344}}{{30}} = 44.8
 $

Thus the required fourth proportion to the given numbers is 44.8.

Note: If three numbers $a,{\text{ }}b$ and $c$ are in proportion then again the ratio of first two numbers must be equal to the ratio of the last two. This is shown as:
$ \Rightarrow \dfrac{a}{b} = \dfrac{b}{d}$
If we simplify it further, we can conclude that the numbers are in Geometric Progression. So we have:
\[ \Rightarrow {b^2} = ac\]
From this we can say that $a,{\text{ }}b$ and $c$ are in Geometric Progression such that $b$ is the Geometric Mean of $a$ and $c$. The value of the proportion ratios is the common ratio of Geometric Progression:
$ \Rightarrow \dfrac{a}{b} = \dfrac{b}{d} = r$