Henry Ernest Dudeney/Puzzles and Curious Problems/286 - Unlucky Breakdowns/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $286$
- Unlucky Breakdowns
- On a day of great festivities, a large crowd gathered for a day's outing and pleasure.
- They all agreed to pile into a bunch of wagons, each of which was to carry the same number of people.
- But ten of the wagons broke down half way, so each of the other wagons then had to carry one more person than had been planned.
- As they were about to start back, it was discovered that $15$ more of these wagons had become unserviceable,
- and so there were three more people in each working wagon on the way back than started out.
- How many people were there in the party?
Solution
There were $900$ people, all travelling in $100$ wagons, $9$ to a wagon.
Then they were down to $90$ wagons, in which they travelled $10$ to a wagon.
Then they were down to $75$ wagons, in which they travelled $12$ to a wagon.
Proof
Let $P$ be the number of people going on the outing.
Let $W$ be the number of wagons.
Let $n$ be the number of people per wagon who started out.
We have:
\(\ds P\) | \(=\) | \(\ds W n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {W - 10} \paren {n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {W - 25} \paren {n + 3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds W n\) | \(=\) | \(\ds W n - 10 n + W - 10\) | multiplying out | ||||||||||
\(\ds \) | \(=\) | \(\ds W n - 25 n + 3 W - 75\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds W - 10 n\) | \(=\) | \(\ds 10\) | |||||||||||
\(\ds 3 W - 25 n\) | \(=\) | \(\ds 75\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 W - 30 n\) | \(=\) | \(\ds 30\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 5 n\) | \(=\) | \(\ds 45\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n\) | \(=\) | \(\ds 9\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds W\) | \(=\) | \(\ds 100\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds P\) | \(=\) | \(\ds 900\) |
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $286$. -- Unlucky Breakdowns
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $247$. Unlucky Breakdowns