Question

The sum of three numbers is 212. If the ratio of the first to the second is 13:16 and that of the second to the third is 2:3, then find the numbers.

Answer 1

Hint: First of all, we will suppose any three numbers, say x, y and z. Then, we are given that the sum x + y + z = 212. After that, we will use the ratios to calculate the values of x and z in terms of y and then we will substitute the obtained values in the sum of the three numbers. We will get the value of y from there and we can put this value of y in the equations of x and z to obtain their values.

Complete step-by-step answer:
We are given that there are any three numbers, say x, y and z, and their sum is 212 i.e., x + y + z = 212.
Now, we are told that the ratio of first and second number is 13:16.
We can write this in terms of an equation as: $\dfrac{x}{y} = \dfrac{{13}}{{16}}$
From this, we can write $x = \dfrac{{13}}{{16}}y$ equation (1)
Now, the ratio of the second and the third equation is 2:3
We can write this in an equation as: $\dfrac{y}{z} = \dfrac{2}{3}$
From this, we can say that$z = \dfrac{3}{2}y$ equation (2)
Substituting these values of x and z in the sum x + y + z = 212, we get
$
   \Rightarrow \dfrac{{13}}{{16}}y + y + \dfrac{3}{2}y = 212 \\
   \Rightarrow \dfrac{{13y + 16y + 24y}}{{16}} = 212 \\
   \Rightarrow \dfrac{{53}}{{16}}y = 212 \\
   \Rightarrow y = \dfrac{{212 \times 16}}{{53}} = 64 \\
 $
Now, we get y = 64. Substituting it in equation (1), we get
$
   \Rightarrow x = \dfrac{{13}}{{16}}y \\
   \Rightarrow x = \dfrac{{13}}{{16}} \times 64 = 13 \times 4 = 52 \\
 $
$ \Rightarrow x = 52$
Now, substituting y = 64 in equation (2), we get
$
   \Rightarrow z = \dfrac{3}{2}y \\
   \Rightarrow z = \dfrac{3}{2} \times 64 = 3 \times 32 = 96 \\
   \Rightarrow z = 96 \\
 $
Therefore, the numbers are x = 52, y = 64 and z = 96.

Note: In this question, be careful while converting the ratios in terms of the numbers x, y and z and then also with how to put their individual values in the sum of the three numbers. You can also verify that if the obtained values of x, y and z have their sum equal to 212 or not i.e., 52+64+96 = 212.