Henry Ernest Dudeney/Puzzles and Curious Problems/18 - The Red Death League/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $18$

The Red Death League
In a story too tedious to relate, we are given to find the number of members and cost of membership when the total subscription is $\pounds 323, 5 \shillings 4 \tfrac 1 4 \oldpence$
We are also given that the number of members is under $500$.


Solution

$311$ members, each paying dues of $\pounds 1, 0 \shillings 1 \tfrac 3 4 \oldpence$


Proof

We convert $\pounds 313, 5 \shillings 4 \tfrac 1 4 \oldpence$ into farthings:

\(\ds \pounds 313, 5 \shillings 4 \tfrac 1 4 \oldpence\) \(=\) \(\ds 313 \times {20 \shillings} + {5 \shillings 4 \tfrac 1 4 \oldpence}\)
\(\ds \) \(=\) \(\ds 6265 \shillings 4 \tfrac 1 4 \oldpence\)
\(\ds \) \(=\) \(\ds {6265 \times 12 \oldpence} + 4 \tfrac 1 4 \oldpence\)
\(\ds \) \(=\) \(\ds 75 \, 184 \tfrac 1 4 \oldpence\)
\(\ds \) \(=\) \(\ds \dfrac {75 \, 184 \times 4 + 1} 4 \oldpence\)
\(\ds \) \(=\) \(\ds 300 \, 737 \text { farthings}\)
\(\ds \) \(=\) \(\ds 311 \times 967 \text { farthings}\)

Thus there were $311$ members, and the dues were $967$ farthings.

Hence the result, after conversion back into pounds, shillings and pence.

$\blacksquare$


Sources

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