Henry Ernest Dudeney/Puzzles and Curious Problems/31 - A Queer Settling Up/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $31$

Person 1: "Here is my purse, give me just as much money as you find in it."

Person 2, having done that: "If you give me as much as I have left of my own, we shall be square."

After Person 2 has done that, Person 1 find his purse contains three shillings and sixpence,

while Person 2 has three shillings.

How much did each possess at the start?

Person $1$ started with $2 \shillings 6 \oldpence$

Person $2$ started with $4 \shillings$

Before the second transaction:

person $2$ has half of three shillings, that is, $1 \shillings 6 \oldpence$

person $1$ has ${3 \shillings 6 \oldpence} + {1 \shillings 6 \oldpence}$, that is, $5 \shillings$

So before the first transaction:

person $1$ has half of $5 \shillings$, that is, $2 \shillings 6 \oldpence$

person $2$ has ${1 \shillings 6 \oldpence} + {2 \shillings 6 \oldpence}$, that is, $4 \shillings$

Hence the result.

$\blacksquare$