Question

What is the fourth proportional to $72,168{\text{ and }}150$ ?
(A) $450$
(B) $300$
(C) $350$
(D) $400$

Answer 1

Hint: In the continuous proportionality, the ratio of the first two numbers will be equal to the ratio of the last two numbers. Now assume a variable for the fourth required number and use the above property to obtain a relation. This equation will contain one unknown and thus can be solved easily by multiplication and division.

Complete step-by-step answer:
Here in this problem, we are given with three numbers $72,168{\text{ and }}150$ and we are asked to find the fourth proportional number to these three. And using this information we need to check which of the four given options are correct.
Before starting with the solution to this problem we need to understand the concept of proportionality. The equality of two ratios is called a proportion. Four are said to be in proportional when the ratio of the first two is equal to the ratio of the last two numbers.
If four numbers $a,b,c{\text{ and }}d$ are in proportional, then this can be represented as:
$ \Rightarrow a:b::c:d$
The first and fourth terms $\left( {a{\text{ and }}d} \right)$ are called extreme terms. The second and third terms $\left( {b{\text{ and }}c} \right)$ are called mean terms. The product of extreme terms is equal to the product of mean terms.
The above representation of the numbers is done using proportionality; the same relation can be expressed using the fractions:
$ \Rightarrow a:b::c:d \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}{\text{ or }}\dfrac{a}{c} = \dfrac{b}{d}$
So, let us assume that the fourth unknown number is proportional to $72,168{\text{ and }}150$ be ‘m’
Therefore, from the above definition, we can represent it as:
$ \Rightarrow 72:168::150:m$
This can be further expressed in form of fractions as:
$ \Rightarrow 72:168::150:m \Rightarrow \dfrac{{72}}{{168}} = \dfrac{{150}}{m}$
Let us now transpose the unknown to one side of the equality and the constants to the other
 $ \Rightarrow \dfrac{{72}}{{168}} = \dfrac{{150}}{m} \Rightarrow m = \dfrac{{150 \times 168}}{{72}}$
Now from the above equation, we can easily find the value of the unknown
$ \Rightarrow m = \dfrac{{150 \times 168}}{{72}} \Rightarrow m = \dfrac{{25200}}{{72}}$
Therefore, we get the required value as: $m = \dfrac{{25200}}{{72}} = 350$
Thus, we get the fourth proportional number as $350$ .
Hence, the option (C) is the correct answer.

Note: The above-obtained answer can be verified by putting it again into the proportionality relation. $72:168::150:m \Rightarrow \dfrac{{72}}{{168}} = \dfrac{{150}}{{350}} = \dfrac{3}{7}$ . An alternative approach can be taken to find the fourth number in proportionality by using the property that the product of extreme terms is equal to the product of mean terms. This will give you the expression: $72 \times m = 168 \times 150$ .