Question

What is the number to be added to each of $6,14,18,38$ to make the new numbers proportional?

Answer 1

We want to find a such that $6+a:14+a::18+a:38+a$. This implies that
$(6+a)(38+a)=(14+a)(18+a)$.
This simplifies to
$228+44a+a^2=252+32a+a^2$.
Hence, $12a=24$ or $a=2$. Therefore, if we add $2$ to all the terms, the new numbers are proportional.
We can also do this as follows. Using
$\frac{6+a}{14+a}=\frac{18+a}{38+a}$,
we may obtain
$\frac{6+a}{14+a}=\frac{18+a}{38+a}=\frac{(18+a)-(6+a)}{(38+a)-(14+a)}=\frac{12}{24}=\frac{1}{2}$.
Hence, 2(6 + a) = 14 + a. This reduces to $a=14-12=2$.