How many originally… Counting Backwards

Three pirates were stranded on a desert island just outside of Pittsburgh. They came upon a treasure chest, which they opened and found to be full of oranges. Unfortunately they couldn't decide how to split them up, so they put all the oranges back in the chest. On the first afternoon, one pirate snuck away from the others and went to the chest. He separated the oranges into three equal piles. There was 1 extra so he gave it to a passing monkey. Then he ate one of the three piles and put the rest back in the chest. A short time later, the second pirate did the same thing. He separated the oranges into four equal piles. There were two extra oranges, which he gave to a passing monkey. He then ate one of the four piles and put the rest back. Later, the third pirate came to do the same thing. He split the oranges into five equal groups. He had 3 left which he gave to a passing money and ate 1 of the 5 equal piles and put the rest back. The next day the three pirates split the remaining. Each of the three pirates got 20 each.. How many oranges were in the chest originally? and how many did each pirate receive?

I listed out of the cases

.Treasure of oranges . Divide into 3 groups. . Gave away 1 . Ate 1/3 of oranges

Divide into 4 groups. Gave away 2 Ate 1/4 of oranges

Divide into 5. Gave away 3. Ate 1/5 oranges.

Split rest into 3 groups. Each got 20 oranges.

TO find how many I started with I started by working backwards.

They end up with 20 oranges each. Multiply by 3 = 60 oranges. One guy ate 1/5 so 75 oranges He gave away 3 so + 3 = 78

I get confused when I get to split into five groups. Do I count that again? Or do I go past the divide into 5 groups and straight to ate 1/4 of piles..

Basically, which answer would be correct for next?

Divide into 5 groups which would be : 78$\times$5= 390 oranges

OR.....

Ate 1/4 of oranges which would be 1/4 of 78 which would = 104

-
It's a trick question! Pittsburgh is inland, so there can't be any desert islands just outside it. – Henning Makholm Apr 25 '14 at 1:45

Working from the end to the beginning, you have 3 piles of 20 oranges. So 60. No extra are mentioned so I'm assuming it's exactly 60.

On the third day, a pirate splits the piles into 5, gives away 3, then eats one pile. 60 oranges remain. Oranges at start of steal 3 = $x$. $$(x - 3)\cdot \frac{4}{5} = 60 \rightarrow x = 78$$

On the second day, a pirate splits the piles into 4, gives away 2, then eats one pile. 78 oranges remain. Oranges at start of steal 2 = $x$. $$(x - 2)\cdot \frac{3}{4} = 78 \rightarrow x = 106$$

On the third day, a pirate splits the piles into 3, gives away 1, then eats one pile. 106 oranges remain. Oranges at start of steal 3 = $x$. $$(x - 1)\cdot \frac{2}{3} = 106 \rightarrow x = 160$$

As for what they got, you find what they stole and add the 20 oranges they each got when they split them up at the end. Pirate 1 had $$\frac{160-1}{3} = 53 \rightarrow 53+20=73$$

Pirate 2 had $$\frac{106-2}{4} = 26 \rightarrow 26+20=46$$

Pirate 3 had $$\frac{78-3}{5} = 15 \rightarrow 15+20=35$$

And the monkey got 6.

-