How many pirates were there, and how much was the booty?

Question

A pirate crew at the end of the day split the booty. The first pirate got 100 gold pieces and 1/6 of the remaining booty. The second one got 200 gold pieces and 1/6 of the remaining booty. The third one got 300 gold pieces and 1/6 of the remaining booty, etc. The last one only got what was left of the booty.

At the end, every pirate had the same amount of gold pieces (from the booty).

How many pirates were there, and how much was the booty?

Answer 1

Call the initial booty $x$, so the first pirate gets $100 + \frac{x - 100}{6} = \frac{x+500}{6}$ gold pieces. The second one gets $200 + \frac{x - \frac{x+500}{6} - 200}{6} = \frac{x - \frac{x+500}{6} + 1000}{6}$ gold pieces, which is the same as the first one. So
$$x + 500 = x - \frac{x+500}{6} + 1000$$ $$0=- \frac{x+500}{6}+500$$ $$x+500 = 3000$$ $$x= 2500$$ So the first pirate gets $100 + \frac{2400}{6} = 500$ gold pieces, and there must be 5 pirates.

One should check all pirates get the same amount:

After the first pirate took his/her share, there are $2000$ gold pieces left. The second one gets $200 + \frac{1800}{6} = 500$ gold pieces. The third one $300 + \frac{1200}{6} = 500$; the fourth one $400 + \frac{600}{6} = 500$ and there are $500$ gold pieces left for the final pirate.

Answer 2

Here's another approach:

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Let $x$ be each pirate's share and $B$ the total booty.

For the first pirate,

$x = 100 + \frac{B-100}{6}$, which implies $6x = B + 500$.

For the second pirate,

$x = 200 + \frac{B-200-x}{6}$, which implies $7x = B + 1000$.

For the third pirate,

$x = 300 + \frac{B-300-2x}{6}$, which implies $8x = B + 1500$.

(We only need the first two, but the pattern is more apparent with three.)

When you add one $x$ the value on the right increases by 500.

More formally,

We can solve the system using substitution: $7x = B + 1000 = (B + 500) + 500 = 6x + 500$, which implies $x = 500$.

And the result is that each pirate

got 500 gold pieces and, using any of the three equations to solve for $B$, the total booty is 2500 gold pieces. Dividing the total by the booty for each pirate gives the number of pirates: $B/x = 5$.