Question

Three numbers are in the ratio \[\dfrac{1}{2}:\dfrac{2}{3}:\dfrac{3}{4}\] . The difference between the greatest and the smallest number is 36. The numbers are
(A) 72, 84, 108
(B) 60, 72, 96
(C) 72, 84, 96
(D) 72, 96, 108

Answer 1

Hint: First of all, let us assume that the given ratio is in x. The modified ratio is \[\dfrac{1}{2}x:\dfrac{2}{3}x:\dfrac{3}{4}x\] . The greatest number and the smallest number are \[\dfrac{3}{4}x\] and \[\dfrac{1}{2}x\] respectively. Also, it is given that the difference between the greatest and the smallest number is 36. Now, proceed further and get the value of x. Then, put the value of x in the ratio \[\dfrac{1}{2}x:\dfrac{2}{3}x:\dfrac{3}{4}x\] and get the numbers.

Complete step-by-step answer:
According to the question, we are given that three numbers are in the ratio \[\dfrac{1}{2}:\dfrac{2}{3}:\dfrac{3}{4}\] and the difference between the greatest and the smallest number is 36.
First of all, let us assume that the ratio is in x.
Now, on modifying the given ratio, we get
\[=\dfrac{1}{2}x:\dfrac{2}{3}x:\dfrac{3}{4}x\] ……………………………………………..(1)
After observing the ratio, we can say that
The greatest number among the three given numbers = \[\dfrac{3}{4}x\] …………………………………………………(2)
The smallest number among the three given numbers = \[\dfrac{1}{2}x\] ………………………………………………….(3)
Since it is given that the difference between the greatest number and the smallest number is 36 so, the difference between the numbers \[\dfrac{3}{4}x\] and \[\dfrac{1}{2}x\] is 36 ……………………………………………(4)
Now, on solving equation (4) further, we get
\[\begin{align}
  & \Rightarrow \dfrac{3}{4}x-\dfrac{1}{2}x=36 \\
 & \Rightarrow \dfrac{3x-2x}{4}=36 \\
 & \Rightarrow \dfrac{x}{4}=36 \\
 & \Rightarrow x=36\times 4 \\
\end{align}\]
\[\Rightarrow x=144\] …………………………………………….(5)
On putting the value of x from equation (5) in equation (1), we get
\[\begin{align}
  & =\dfrac{1}{2}\times 144:\dfrac{2}{3}\times 144:\dfrac{3}{4}\times 144 \\
 & =72:2\times 48:3\times 36 \\
\end{align}\]
\[=72:96:108\] ……………………………………….(6)
Now, from equation (6), we have the numbers.
Therefore, the three numbers are 72, 96, and 108.

So, the correct answer is “Option D”.

Note: We can also solve this question by making the fractional ratio into an integral ratio.
The given ratio is \[\dfrac{1}{2}:\dfrac{2}{3}:\dfrac{3}{4}\] ……………………………………..(1)
First of all, take the LCM of the denominators of the given ratio.
On taking the ratio of the denominators that are 2, 3, and 4, we get
LCM of 2, 3, and 4 = 12 …………………………………………(2)
Now, on multiplying by 12 in equation (1), we get
\[=\dfrac{1}{2}\times 12:\dfrac{2}{3}\times 12:\dfrac{3}{4}\times 12\]
\[=6:8:9\] …………………………………………………….(3)
Now, let us assume the ratio in x.
Now, on modifying the given ratio, we get
\[=6x:8x:9x\] ………………………………………..(4)
From the above ratio,
The greatest number = \[9x\] ………………………………………………(5)
The smallest number = \[6x\] ……………………………………………….(6)
It is given that the difference between the greatest number and the smallest number is 36. So,
\[\begin{align}
  & \Rightarrow 9x-6x=36 \\
 & \Rightarrow 3x=36 \\
\end{align}\]
\[\Rightarrow x=12\] …………………………………(7)
Now, on putting the value of x in equation (4), we get
\[\begin{align}
  & =6\times 12:8\times 12:9\times 12 \\
 & =72:96:108 \\
\end{align}\]
Therefore, the three numbers are 72, 96, and 108.
Hence, the correct option is (D).