Henry Ernest Dudeney/Puzzles and Curious Problems/6 - A New Partner/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $6$

A New Partner
Two partners named Smugg and Williamson have decided to take a Mr. Rogers into partnership.
Smugg has one and a half times as much capital invested in the business as Williamson
and Rogers has to pay down $\pounds 2500$, which sum shall be divided between Smugg and Williamson,
so that the three partners shall have an equal interest in the business.
How shall that sum be divided?


Solution

Rogers pays $\pounds 500$ to Smugg and $\pounds 2000$ to Williamson.


Proof

The assumption that must be made is that the contribution of Rogers is one third of the total business value before he entered.

Let $T$ pounds be the total investment in the business.

Let $S$ and $W$ pounds be the amounts of capital invested by Smugg and Williamson respectively.

We have:

\(\ds T\) \(=\) \(\ds S + W\)
\(\ds T\) \(=\) \(\ds 3 \times 2500\)
\(\ds \) \(=\) \(\ds 7500\)
\(\ds S\) \(=\) \(\ds \dfrac 3 2 W\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 3 2 W + W\) \(=\) \(\ds 7500\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 5 2 W\) \(=\) \(\ds 7500\)
\(\ds \leadsto \ \ \) \(\ds W\) \(=\) \(\ds 3000\)
\(\ds \leadsto \ \ \) \(\ds S\) \(=\) \(\ds 4500\)

So Rogers pays $\pounds 500$ to Smugg and $\pounds 2000$ to Williamson.

$\blacksquare$


Sources

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