After dinner, the
five boys of a household
happened to find a parcel of sugar-plums.

It was quite unexpected loot,
and an exciting scramble ensued, the full details of which I will
recount with accuracy, as it forms an interesting puzzle.

You see, Andrew managed to get possession of just
two-thirds of the parcel of sugar-plums.

Bob at once grabbed
three-eighths of these, and Charlie managed to seize three-tenths
also.

Then young David dashed upon the scene, and captured all that Andrew
had left, except one-seventh, which Edgar artfully secured for himself
by a cunning trick.

Now the fun began in real earnest, for Andrew and
Charlie jointly set upon Bob, who stumbled against the fender and
dropped half of all that he had, which were equally picked up by David
and Edgar, who had crawled under a table and were waiting.

Next, Bob
sprang on Charlie from a chair, and upset all the latter's collection
on to the floor.

Of this prize Andrew got just a quarter, Bob gathered
up one-third, David got two-sevenths, while Charlie and Edgar divided
equally what was left of that stock.

They were just
thinking the fray was over when
David suddenly struck out in two directions at once, upsetting
three-quarters of what Bob and Andrew had last acquired.

The two
latter, with the greatest difficulty, recovered five-eighths of it in
equal shares, but the three others each carried off one-fifth of the
same.

Every sugar-plum was now accounted for, and they called a truce,
and divided equally amongst them the remainder of the parcel.

What is
the smallest number of sugar-plums there could have been at the start,
and what proportion did each boy obtain?