solving this is mostly a matter of keeping track and simplifying as you go.
otherwise you get lost in the details.
let x = the amount of money that he starts off with.
you will need two columns.
the first column is what he has left and the second column is what he spends.
you will need to determine what he has left and then determine how much he spends of that.
the result of all your labors will result in the following:
row 1 shows what he has left and what he spends of it.
row 2 shows what he has left after he spent what was in row 1 and what he spent of that, etc.
left spends
x x/2 + 5
x/2 - 5 x/4 + 5/2
x/4 - 15/2 3x/16 - 5/8
x/16 - 55/8 which is equal to 0.
the devil is in the details.
here's the details.
he starts with x and he spends half of that plus 5.
this means he spends x/2 + 5
you need to subtract what he spends from what he has left to get what he has left after the spending.
x - (x/2 + 5) is equal to x - x/2 - 5 which is equal to 2x/2 - x/2 - 5 which is equal to x/2 - 5.
he is left with x/2 - 5 which is the second row first column in the above table.
now he spends half of this plus 5.
that means he spends 1/2 * (x/2 - 5) + 5 which means he spends x/4 - 5/2 + 5 which means he spends x/4 - 5/2 + 10/2 which means he spends x/4 + 5/2 which is the second row second column in the above table.
he is left with x/2 - 5 - (x/4 + 5/2) which means he is left with x/2 - 5 - x/4 - 5/2 which means he is left with 2x/4 - x/4 - 10/2 - 5/2 which means he is left with x/4 - 15/2 which is the third row first column in the above table.
now he spends 3/4 of this plus 5 which means he spends 3/4 * (x/4 - 15/2) + 5 which means he spends 3x/16 - 45/8 + 40/8 which means he spends 3x/16 - 5/8 which is the the third row second column in the above table.
he needs to subtract (3x/16 - 5/8) from (x/4 - 15/2) to determine what he finally has left.
(x/4 - 15/2) - (3x/16 - 5/8) becomes 4x/16 - 60/8 - 3/16 + 5/8 which becomes x/16 - 55/8 which is what he finally has left as shown in the fourth row first column.
that's what he has left.
since the remainder is 0, you set this equal to equal to 0 and solve for x.
x/16 - 55/8 = 0 is your eqution.
add 55/8 to both sides to get x/16 = 55/8
multiply both sides by 16 to get x = 55 * 16/8
simplify to get x = 55 * 2 = 110.
your solution is x = 110.
he starts with 110.
he spends half plus 5 which means he spends 60 and is left with 50
he spends half plus 5 which means he spends 30 and is left with 20
he spends three fourths plus 5 which means he spends 20 and is left with 0.
solution checks out.
answer is he starts with 110.
the table of what this looks like is shown below:
left spends
110 60
50 30
20 20
0