# Sunday Times Teaser 2906 – In Proportion

*by Andrew Skidmore*

#### Published June 3 2018 (link)

In 2000 the Sultan of Proportion told his five sons they would inherit his fortune in amounts proportionate to their ages at his death.

Accordingly, they each recently received a different whole number of tesares. Strangely, if he had lived a few more hours the five ages would have been consecutive and each son would again have received a whole number of tesares. Such a delay would have benefited just one son (by 1000 tesares).

How many tesares were distributed in total?

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The only arrangement of the son’s ages that are all different at the father’s death but which would produce consecutive ages if the father’s death had been delayed by a few hours is as follows: \[(a, a+2, a+3, a+4,a+5)\]The ages at the delayed death are then: \[(a+1, a+2, a+3,a+4,a+5)\]The inheritances are proportional to these ages so we can express the Sultan’s fortune \(f\) as: \[f = m(5 a+14) = n(5 a + 15)\] where \(m\) and \(n\) are the inheritance over age ratios for the two sets of ages. We are also told that the one son who would benefit from a delayed death would gain 1000 tesares, which gives:\[n(a + 1) – m a = 1000\] These two equations allow \(m\) and \(f\) to be determined from \(a\) as:\[m=2500\frac{(a+3)}{(2a+7)}\] \[f=2500\frac{(a + 3)(5a+14)}{(2a+7)}\]Since \(m\) is an integer, we are looking for values of \(2a+7\) that are divisors of 2500. Moreover, we know tht \(a\) is at least 17 and, say, less than 120 so the only divisors of 2500 that we need to consider are 50, 100 and 125. Of these only 125 gives an integer \(a\) value (= 59) and hence a Sultan’s fortune of 383160 tesares.

You can investigate this teaser further using Erling Torkildsen’s nice GeoGebra model here.