Question

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

Answer 1

Let the required number to be added be $$x$$.

Then, $$(5 + x), (11 + x), (19 + x), (37 + x)$$ are in proportion.

Therefore, $$(5 + x)(37 + x) = (11 + x)(19 + x)$$

$$185 + 5x + 37x + x^2 = 209 + 11x + 19x + x^2$$

$$x^2 + 42x + 185 = x^2 + 30x + 209$$

Subtracting $$x^2$$ from both sides:

$$42x - 30x = 209 - 185$$

$$12x = 24$$

$$x = 2$$

Hence, the least number to be added is 2.