Henry Ernest Dudeney/Modern Puzzles/6 - Generous Gifts

by : $6$

 * 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: