Question #518c1
1 Answer
Allyson is
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
#Delta_"age" = y^2 - y#
Let's say that exactly
#"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
Now, we know that when Jenny is
#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
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
#7^2 = 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