# Question #518c1

Feb 28, 2017

Allyson is $7$ years old now.

#### Explanation:

The key to this problem is the fact that the difference between their ages remains constant as they get older.

So, you know that at the moment, Jenny is ${y}^{2}$ years old and Allyson is $y$ years old. The difference between their ages, let's say ${\Delta}_{\text{age}}$, can be written as

${\Delta}_{\text{age}} = {y}^{2} - y$

Let's say that exactly $1$ year passes. Jenny's age will now be

$\text{Jenny: } {y}^{2} + 1$

and Allyson's age will now be

$\text{Allyson: } y + 1$

But the difference between their ages remains the same, since

${\Delta}_{\text{age}} = {y}^{2} + 1 - \left(y + 1\right)$

${\Delta}_{\text{age}} = {y}^{2} + \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}} - y - \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}}$

${\Delta}_{\text{age}} = {y}^{2} - y$

This means that regardless of how many years pass, the difference between the ages of the two girls will always be equal to ${y}^{2} - y$.

Now, we know that when Jenny is $13 y$ years old, Allyson will be ${y}^{2}$ years old. The difference between their ages will be

${\Delta}_{\text{age now}} = 13 y - {y}^{2}$

But this must be equal to

${\Delta}_{\text{age}} = {y}^{2} - y$

You can thus say that

${y}^{2} - y = 13 y - {y}^{2}$

This is equivalent to

$2 {y}^{2} - 14 y = 0$

which simplifies to

$2 y \left(y - 7\right) = 0$

You now have two possibilities here

$2 y = 0 \text{ }$ or $\text{ } y - 7 = 0$

Notice that

$2 y = 0 \implies y = 0$

is not really a suitable solution here because Jenny and her daughter cannot be $0$ years old.

This means that the only suitable solution will be

$y - 7 = 0 \implies \textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{y = 7}}}$

Therefore, you can say that at the moment, Allyson is $7$ years old and her mother is

${7}^{2} = 49$

years old. Notice that when Jenny is

$13 \cdot 7 = 91$

years old, her daughter will be

$91 - {\overbrace{\left(49 - 7\right)}}^{\textcolor{b l u e}{\text{the difference between their ages}}} = 49$

years old, which is equal to ${7}^{2}$.