Question

What least number be subtracted from each of the numbers 12, 17, 22, 32 so that the remainders may be in proportional?

Answer 1

Then, (14 - x) : (17 - x) : : (34 - x) : (42 - x).

(14 - x) * (42 - x) = (17 - x) * (34 - x)

=> x^2 - 56x + 588 = x^2 - 51x + 578

5x = 10

x = 2

Required number = 2.
that's it.

Answer 2

Given:

Four numbers 12,17,22,32

To Find:

The least number be subtracted from each of the numbers so that the remainders may be proportional.

Solution:

Let us assume the least number is x

So, it should be subtracted from each number

Then,

12-x, 17-x,22-x,32-x

These numbers are in proportion which means the ratio of two numbers is equal to the ratio of the other two numbers.

(12-x):(17-x)::(22-x):(32-x)

Which means

(12-x)/(17-x) = (22-x)/(32-x)

By cross multiplication

(12-x)(32-x) =  (22-x)(17-x)    (comparing L.H.S and R.H.S)

384-44x-x² = 374 - 39x-x²

384-374= -39x+44x -x²+x²

10 = 5x

x = 10/5

x = 2

Now,

(12-x):(17-x)::(22-x):(32-x)  (put x=2)

10:15::20:30

Hence, the least number be subtracted from each of the numbers so that the remainders may be proportional is 2.