Question

Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, then the ration becomes 4 : 5. Find the two numbers.

Answer 1

Hint: Assume a variable x that will represent the first number and assume a variable y that will represent the second variable. It is given in the question that the ratio of x and y is 5 : 6. Also, if we subtract 8 from the two numbers, we get x – 8 and y – 8. It is given in the question that the ratio of these two numbers is 4 : 5. Hence, we get two equations which can be solved with each other to obtain the value of x and y.

Complete step-by-step answer:
Let us assume a variable x which represents the first number and let us assume a variable y that which represents the second number.
In this question, we are given that the two numbers are in the ratio 5 : 6. So, we can say,
$\dfrac{x}{y}=\dfrac{5}{6}$
$\Rightarrow y=\dfrac{6x}{5}$ . . . . . . . . . . . . . . . . (1)
Also, in the question, it is given that if we subtract 8 from both the numbers then the ratio of the two numbers becomes 4 : 5. So, we can say,
$\begin{align}
  & \dfrac{x-8}{y-8}=\dfrac{4}{5} \\
 & \Rightarrow 5\left( x-8 \right)=4\left( y-8 \right) \\
 & \Rightarrow 5x-40=4y-32 \\
 & \Rightarrow 5x=4y+8 \\
\end{align}$
Substituting y = $\dfrac{6x}{5}$ from equation (1) in the above equation, we get,
$\begin{align}
  & 5x=4\left( \dfrac{6x}{5} \right)+8 \\
 & \Rightarrow 5x-\dfrac{24x}{5}=8 \\
 & \Rightarrow \dfrac{25x-24x}{5}=8 \\
 & \Rightarrow \dfrac{x}{5}=8 \\
 & \Rightarrow x=40 \\
\end{align}$
Substituting this value of x in equation (1), we get,
$\begin{align}
  & y=\dfrac{6\left( 40 \right)}{5} \\
 & \Rightarrow y=48 \\
\end{align}$
Hence, x = 40 and y = 48.


Note: There is a faster and direct way to solve this question. One can directly take the two numbers as 5x and 6x since it is given that the two numbers are in the ratio 5 : 6. If we directly take the two numbers as 5x and 6x, we have to write one equation less and we can solve the question in a lesser time.