# Gold coins and the Girls

Question: A group of fewer than 10 girls found a number of gold-coins which they were able to divide equally among them.
After this division had been done, Lalita - one of the girls, suggested that it would be more equitable to divide the gold-coins by families rather than by individuals. Among the them, there were two groups with two sisters, of course Lalita was not in either group. The rest of the girls were unrelated to each other. A re-division by families would have meant that the gold-coins per family were 5 more than the gold-coins per girl.

The girls argued among themselves over this way of dividing the gold-coins. Before a final decision is made, Ash - one of the girls, decided that she did not want any gold-coins. Her
share was equally divided (without breaking/cutting any gold-coin) among the other girls.

Finally, Lalita decided to withdraw her suggestion of dividing the gold-coins by families.

How many girls were there and how many gold-coins did each girl end up with?

There were total 6 girls. Each end up with 12 gold-coins.
The number of gold-coins is evenly divisible by the number of girls as well as the number of families.
Let's assume that N is the number of gold-coins each girl received initially and G is the total number of girls.
Then, total number of gold-coins = NG
If the gold-coins had been divided by families rather than by individuals, the number of recipients would be (G - 2) and each share would be (N + 5).
Again, total number of gold-coins = (N + 5)(G - 2)
But, the total number of the gold-coins is the same.
NG = (N + 5)(G - 2)
NG = NG -2N +5G -10
2N = 5G -10
N = (5/2)G - 5

Now, N and G are the positive integers and also total number of gold-coins must be divisible by G, (G-1) and (G-2). This is because initially there were G girls; then since it was divided family wise, the total number of units, the coins were to be divided would be (G-2) (as two groups had two sisters, so two girls got combined as one group, one per group); and after Ash backed out, there were (G-1) girls.

Now trying different EVEN values for G, starting with 2; there were total 6 girls and 60 gold-coins. The gold-coins are divided among 5 girls and hence each girl ends up with 12 gold-coins.