Question

The ratio of two numbers is `2/3`. If 2 is subtracted from the first and 8 from the second, the ratio becomes the reciprocal of the original ratio. Find the numbers.

Answer 1

Let the two numbers be x and y.
According to the question,
`x/y = 2/3`
3x - 2y = 0                        .....(1)

Also,`[ x - 2 ]/[ y - 8 ] = 3/2`

2x - 3y = - 20                   .....(2)

Multiplying equations no. (1) by 2 and (2) by 3
6x - 4y = 0                       .....(3)
6x - 9y = - 60                  ......(4)
Subtracting equation (4) from (3)
      6x - 4y = 0
-    6x - 9y = - 60   
      -     +     +        
             5y = 60
               y = 12

From (1), we get
3x - 2(12) = 0
x = `24/3`
x = 8
Thus, the numbers are 8 and 12.