# What number must be added to each of the numbers 5, 11, 19 and 37 so that they are in proportion

## Mathematics

Let the number to be added be x. So, new numbers are 5 + x, 11 + x, 19 + x, 37 + x

Since, these numbers are in proportion,

∴5+x11+x=19+x37+x⇒(5+x)(37+x)=(19+x)(11+x)(185+5x+37x+x2)=209+19x+11x+x2⇒x2−x2+42x−30x+185−209=0⇒12x−24=0⇒12x=24⇒x=2412=2.\therefore \dfrac{5 + x}{11 + x} = \dfrac{19 + x}{37 + x} \\[0.5em] \Rightarrow (5 + x)(37 + x) = (19 + x)(11 + x) \\[0.5em] (185 + 5x + 37x + x^2) = 209 + 19x + 11x + x^2 \\[0.5em] \Rightarrow x^2 - x^2 + 42x - 30x + 185 - 209 = 0 \\[0.5em] \Rightarrow 12x - 24 = 0 \\[0.5em] \Rightarrow 12x = 24 \\[0.5em] \Rightarrow x = \dfrac{24}{12} = 2.

Hence, the number that must be added to make the numbers in proportion is 2.