## Teaser 2906: In proportion

**From The Sunday Times, 3rd June 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?

[teaser2906]

## Jim Randell 11:06 am

on7 June 2019 Permalink |Assuming ages are expressed as a whole number of years.

In the actual situation the five ages are all different, and in the hypothetical situation (which is a few hours later) the ages are consecutive, lets say:

The proportion going to each son would be:

In the actual case one of the sons is one year younger, and as the ages are all different it can only be the youngest son, so that actual ages are:

and the actual proportions are:

So the youngest son is better off by 1000 tesares in the hypothetical situation, so if the total number of tesares to be distributed is

T, we have:As the 5 sons were around in 2000 and the puzzle was set in 2018, we can suppose that each son is older than 18 (otherwise we can find a second solution).

This Python program runs in 68ms.

Run:[ @repl.it ]Solution:In total 383,160 tesares were distributed.The actual ages of the sons at the time of the Sultan’s death were: 59, 61, 62, 63, 64. The youngest son turning 60 shortly after.

If we allow ages less than 18 then there is a further solution where the actual ages of the sons are: 9, 11, 12, 13, 14, and 70,800 tesares are distributed amongst them.

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