Question

Question #518c1

Answer 1

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_"age"`, can be written as

`Delta_"age" = y^2 - y`

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

`"Jenny: " y^2 + 1`

and Allyson's age will now be

`"Allyson: " y + 1`

But the difference between their ages remains the same, since

`Delta_"age" = y^2 + 1 - (y + 1)`

`Delta_"age" = y^2 + color(red)(cancel(color(black)(1))) - y - color(red)(cancel(color(black)(1)))`

`Delta_"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 `13y` years old, Allyson will be `y^2` years old. The difference between their ages will be

`Delta_"age now" = 13y - y^2`

But this must be equal to

`Delta_"age"= y^2 - y`

You can thus say that

`y^2 - y = 13y - y^2`

This is equivalent to

`2y^2 - 14y = 0`

which simplifies to

`2y(y - 7) = 0`

You now have two possibilities here

`2y = 0" "` or `" "y-7=0`

Notice that

`2y = 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 color(darkgreen)(ul(color(black)(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 * 7 = 91`

years old, her daughter will be

`91 - overbrace((49 - 7))^(color(blue)("the difference between their ages")) = 49`

years old, which is equal to `7^2`.