# Your age and my age [closed]

This is not a very difficult problem, but it seems that it hasn't been asked here. It's one of my favorites, so here we go:

I am twice as old as you were when I was as old as you are.
When you will be as old as I am, the sum of our ages will be 81.
How old are we?

Edit: I totally misremembered the second sentence, it's fixed now.

I am 36 years old, you are 27 years old.

Proof:

Let's say I am $X$ years old and you are $Y$ years old. I am older than you by $(X-Y)$ years. Then when I was $Y$ you were $Y-(X-Y)$ years old. Therefore:

$X = 2 \cdot (Y-(X-Y)) = 4 \cdot Y-2 \cdot X$

or

$X = 4/3 \cdot Y$ (1)

When you will be X years old I will be $X+(X-Y)$ years old. Therefore:

$X+(X+(X-Y)) = 81$

or

$Y = 3 \cdot X-81$ (2)

From (1) and (2) we get:

$X = 4/3 \cdot (3 \cdot X-81) = 4 \cdot X-108$

So $X = 36$. $Y = 3/4 \cdot X = 27$.

Solution from the perspective of the asker.

The difference $x$.

The first sentence gives us two equations: $$\text{I}~~~~ inow=2 \cdot youthen$$ $$\begin{pmatrix} {\scriptsize ithen\overset{\text{as old as}}= younow} \\ {\scriptsize younow=inow-x}\\ {\scriptsize \implies ithen=inow-x} \\ {\scriptsize \implies ithen-x=inow-2x}\\ {\scriptsize \text{and since}~~~~ youthen = ithen -x} \\ \end{pmatrix}$$

$$\text{II}~~~~ youthen = inow - 2x$$ $$\Rightarrow inow=2 \cdot (inow-2x) \\ \Rightarrow inow=4x$$

When you become as old as I am now, I will be x years older (2nd sentence):

$$2 \cdot inow+x=81$$ $$inow=4x$$ $$\Rightarrow x=9$$

Our ages.

Now we know that $$inow=36$$ $$inow-younow=9$$ $$\Rightarrow younow =27$$