Question #518c1

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#.