Question

What must be subtracted from each of the numbers $23,40,57$ and $108$ so that the remainder are in proportions?

Answer 1

Hint: here in this question , the meaning of the remainder is that the numbers we get after subtracting the given number from a particular number. So that the ratio of the first two numbers will be equal to the ratio of the last two numbers. We can easily do it by cross-multiplication method.

Complete step-by-step solution:
Let the number to be subtracted from the given numbers be x.
Now, the remaining numbers are $(23 - x),(40 - x),(57 - x)$ and $(108 - x)$.
It is given that the remaining numbers are in the proportions.
So, $(23 - x):(40 - x)::(57 - x):(108 - x)$
$\dfrac{{(23 - x)}}{{(40 - x)}} = \dfrac{{(57 - x)}}{{(108 - x)}}$
Using the cross-multiplication method,
$(23 - x)(108 - x) = (57 - x)(40 - x)$
Multiply on both sides,
$23*108 - 23x - 108x + {x^2} = 40*57 - 57x - 40x + {x^2}$
$\Rightarrow 2484 - 131x + {x^2} = 2280 - 97x + {x^2}$
Cancel ${x^2}$ from both sides
$2484 - 131x = 2280 - 97x$
Now, transposing $2280$ to the left hand side and $ - 131x$ to the right hand side.
$2484 - 2280 = 131x - 97x$
$\Rightarrow 204 = 34x$
$\Rightarrow \dfrac{{204}}{{34}} = x$
$\Rightarrow x = 6$
Therefore, the required number to be subtracted is from $23,40,57$ and $108$ is $6$.

Note: Do the calculations carefully otherwise the answer would come out to be wrong. You have to subtract the required number from the given number. Don’t do the opposite of it i.e, subtract the given number from the required number.