Granny's age
Weekly Problem 10 - 2015
Granny is four times as old as I am. Five years ago she was five times as old as I was. What is the sum of our ages?
Granny is four times as old as I am. Five years ago she was five times as old as I was. What is the sum of our ages?
Age
14 to 16
Challenge level
Problem
Granny has calculated that she is four times as old as I am now, but remembers that 5 years ago she was five times as old as I was at that time.
What is the sum of our ages now?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $100$
One unknown
Let my age now be $x$. So Granny's age is $4x$.
Five years ago, I was $x-5$ years old and Granny's age was $4x-5$.
At that time, she was five times as old as I was and so $$4x-5=5(x-5)\\
4x-5=5x-25\\
4x+20=5x\\
x=20$$
So Granny is $80$ and I am $20$, which means that the sum of our ages now is $100$.
Two unknowns
My age now: $m$
Granny's age now: $g$
$\therefore g=4m$
My age five years ago: $m-5$
Granny's age five years ago: $g-5$
$\therefore g-5=5(m-5)=5m-25\\
\Rightarrow g=5m-20$
If $g=4m$ and $g=5m-20$, then $4m=5m-20$ so $m=20$
$\therefore g=80$ and $m+g=20+80=100$
\Rightarrow g=5m-20$
If $g=4m$ and $g=5m-20$, then $4m=5m-20$ so $m=20$
$\therefore g=80$ and $m+g=20+80=100$