Question

June bought three books. For the first book, she paid one half of her allowance plus $1 more. For the second book, she paid one half of her left-over money plus $2 more. For the third book, she paid one half of her left-over money plus $3 more. At this point, she had no money left. How much was her allowance?

Answer 1

$34

Explanation:

I'd work backwards.

For the third book, she paid all her remaining money. The problem says she paid "1/2 her leftover money + $3". This means that: (let m = money used to buy book 3)

m = 1/2m + 3

1/2m = 3

m = 6

For the second book: (let n = money before book 2)

n - m (money left after book 2) = 1/2n + 2

1/2n +2 is money used up for book, which is the same as n-m.

n = 1/2n + 2 + m

1/2n = 2 + m = 2 + 6

1/2n =8

n = 16

For the first book, she spent 1/2 her money + 1. If o = money before book 1 (or the whole allowance):

o - n = 1/2o + 1

o = 1/2o + n + 1 = 1/2o + 16 + 1

1/2o = 17

o = 34

Check!

Spent 17 (half 34) + 1 on book 1

16 left

Spent 8 (half 16) +2 on book 2

6 left

Spent 3 (half 6) + 3 on book 3

0 left