# 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?

## 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 $m+g=20+80=100$