What least number must be subtracted from each of the numbers 23, 30, 57 and 78, so that the remainders are in proportion?

Let least number to be subtracted to numbers be x.

∴ 23 - x : 30 - x :: 57 - x : 78 - x

$\Rightarrow \dfrac{23 - x}{30 - x} = \dfrac{57 - x}{78 - x} \\[1em] \Rightarrow (23 - x)(78 - x) = (57 - x)(30 - x) \\[1em] \Rightarrow 1794 - 23x - 78x + x^2 = 1710 - 57x - 30x + x^2 \\[1em] \Rightarrow x^2 - 101x + 1794 = x^2 - 87x + 1710 \\[1em] \Rightarrow x^2 - x^2 - 101x + 87x = 1710 - 1794 \\[1em] \Rightarrow -14x = -84 \\[1em] \Rightarrow x = \dfrac{-84}{-14} \\[1em] \Rightarrow x = 6.$

Hence, least number to be subtracted to make numbers proportional is 6.