Henry Ernest Dudeney/Modern Puzzles/6 - Generous Gifts/Solution

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Modern Puzzles by Henry Ernest Dudeney: $6$

Generous Gifts
A generous man set aside a certain sum of money for equal distribution weekly to the needy of his acquaintance.
One day he remarked:
"If there are five fewer applicants next week, you will each receive $2$ shillings more."
Unfortunately, instead of there being fewer there were actually four more persons applying for the gift.
"This means," he pointed out, "that you will each receive one shilling less."
Now, how much did each person receive at that last distribution?


Solution

$20$ people each received a dole of $6$ shillings.

There is a total available weekly amount of $240$ shillings.


With $5$ fewer people, we have $120$ shillings divided between $15$ people, that is $8$ shillings each.

However, with $4$ more people, we have $120$ shillings divided between $24$ people, that is $5$ shillings each.


Proof

Let $N$ denote the total number of shillings to be disbursed.

Let $m$ denote the number of shillings disbursed per person today.

Let $p$ denote the total number of people who are receiving a donation today.

Then we have:

\(\text {(1)}: \quad\) \(\ds N\) \(=\) \(\ds p m\)
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \paren {p - 5} \paren {m + 2}\)
\(\text {(3)}: \quad\) \(\ds \) \(=\) \(\ds \paren {p + 4} \paren {m - 1}\)
\(\ds \leadsto \ \ \) \(\ds p m - 5 m + 2 p - 10\) \(=\) \(\ds p m + 4 m - p - 4\) eliminating $N$ between $(2)$ and $(3)$
\(\ds \leadsto \ \ \) \(\ds 3 p\) \(=\) \(\ds 9 m + 6\) simplifying
\(\ds \leadsto \ \ \) \(\ds p\) \(=\) \(\ds 3 m + 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds m \paren {3 m + 2}\) \(=\) \(\ds \paren {\paren {3 m + 2} - 5} \paren {m + 2}\) substituting for $p$ in $(1)$ and $(2)$
\(\ds \leadsto \ \ \) \(\ds 3 m^2 + 2 m\) \(=\) \(\ds \paren {3 m - 3} \paren {m + 2}\) simplifying
\(\ds \) \(=\) \(\ds 3 m^2 + 3 m - 6\) more simplifying
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds 6\) more simplifying
\(\ds \leadsto \ \ \) \(\ds p\) \(=\) \(\ds 20\) from $p = 3 m + 2$

$\blacksquare$


Sources

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