Question

Every month, a girl gets allowance. Assume last year she had no money, and kept it up to now. Then she spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After she bought all of that, she had $7777 left. Assuming she only gets money by allowance, how much money does she earn every month?

Answer 1

Let A = monthly allowance, M = 12A (allowance total for one year)

She starts with M, spends 1/2M on clothes and is left with:
M - 1/2M = 1/2M = M/2

Next, she spends 1/3 of her remaining money (M/2) on games, leaving her with:

M/2 - (1/3)(M/2) = M/2 - M/6 = 2M/6 = M/3

Next she spends 1/4 of her remaining money (M/3) on toys, leaving her with:

M/3 - (1/4)(M/3) = M/3 - M/12 = 4M/12 - M/12 = 3M/12 = M/4

This is her remaining money, M/4, which is 7777. So,
M/4 = 7777
M = 7777(4)
M = 31,108

But we want to know A:
M = 12A
A = M/12
A = 31,108/12

A = 2592.33, her monthly allowance

Let's check:
(2592.33)(12) = 31108 Starting money

Spend 1/2 on clothes:
31108 - (1/2)(31108) = 15554

Spend 1/3 of this on games:
15554 - (1/3)(15554) = 15554 - 5184.67 = 10369.33

Spend 1/4 of this on toys:
10369.33 - (1/4)(10369.33) = 10369.33 - 2492.33 = 7777

Answer 2

$2222

Answer 3

check again

Answer 4

nope, $2222 doesn't work :-( the right answer is 2592.33