11. The teenagers Sam and Jo notice the following facts about their ages:
The difference between the squares of their ages is four times the sum of their ages.
The sum of their ages is eight times the difference between their ages.
What is the age of the older of the two?
A 15
B 16
C 17
D 18
E 19
Solution
D
Suppose that the ages of the teenagers are $a$ and $b$, with $a>b$.
Because the difference between the squares of their ages is four times the sum of their ages
$$
a^{2}-b^{2}=4(a+b) \text {. }
$$

By factorizing its left hand side, we may rewrite this last equation as
$$
(a-b)(a+b)=4(a+b)
$$

Because $a+b \neq 0$, we may divide both sides of this last equation by $a+b$ to give
$$
a-b=4 .
$$

Because the sum of their ages is eight times their difference
$$
a+b=8(a-b)
$$

Hence, by (1)
$$
a+b=32 \text {. }
$$

By adding equations (1) and (2), we obtain
$$
2 a=36
$$
and hence
$$
a=18 \text {. }
$$

Therefore the age of the older of the two teenagers is 18 .