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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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.