What least number must be added to each of the numbers 5, 11, 19 and 37, so that the resulting numbers are proportional.

Let least number to be added to numbers be x.

∴ 5 + x : 11 + x :: 19 + x : 37 + x

$\Rightarrow \dfrac{5 + x}{11 + x} = \dfrac{19 + x}{37 + x} \\[1em] \Rightarrow (5 + x)(37 + x) = (19 + x)(11 + x) \\[1em] \Rightarrow 185 + 5x + 37x + x^2 = 209 + 19x + 11x + x^2 \\[1em] \Rightarrow x^2 + 42x + 185 = x^2 + 30x + 209 \\[1em] \Rightarrow x^2 - x^2 + 42x - 30x = 209 - 185 \\[1em] \Rightarrow 12x = 24 \\[1em] \Rightarrow x = 2.$

Hence, least number to be added to make numbers proportional is 2.