# Sunday Times Teaser 2909 – Pensionable Service

### by Angela Newing

#### Published June 24 2018 (link)

A few years ago my father was lucky enough to have been able to retire and take his pension at sixty. I explained to my young son — not yet a teenager — that this was not now usual, and that neither I nor he would be so lucky in the future.

When I am as old as my father is now, I shall be six times my son’s present age. By then, my son will be nine years older than I am now. (These statements all refer to whole years and not fractions of years.)

How old am I now?

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1. Let the son’s, the mother’s and the (grand)father’s ages be $$s$$, $$m$$ and $$g$$ respectively and let $$x$$ years elapse before the mother reaches her father’s age now. We have two equations:
\begin{align}g=m+x&=6s\\s +x&=m+9\end{align} which allow $$x$$ to be eliminated giving:$2m=7s-9$ Multiplying this by 4 and rewriting it$$\pmod 7$$ now gives: $m = 6\pmod 7$and hence:$m=7k+6$ where $$k$$ is an integer. We can now substitute this into the earlier equations to show that:\begin{align}s&=2k+3\\ x&=5k+12\\g&=12k+18\end{align}
We know that $$g$$ is greater than 60, which means that $$k >= 4$$. We also know that $$s$$ is less than 13 so $$k < 5$$. So $$k=4$$, which gives the mother's age now as 34. This makes the son's age 11 and the grandfather's age 66.