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Every month, a girl gets An allowance. Assume last year she had no money, and kept all the money she has earned up to now. Then she spends $\frac{1}{2}$ of her money on clothes, then $\frac{1}{3}$ of the remaining money on games, and then $\frac{1}{4}$ 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?

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work your way backwards. how much money did she have before spending on toys? before spending on games? before spending on clothes? now what is her monthly allowance? – Yuval Filmus Feb 9 '11 at 7:14
why "Interests" in the title? – Américo Tavares Feb 10 '11 at 23:09

Let $x$ be the amount of money the girl started with.

Then we can derive this equation.


Multiply out the fractions and factor out x, you get:

So $x(\frac{12}{12}-\frac{6}{12}-\frac{2}{12}-\frac{1}{12})=7777$

Thus :

Multiply both sides by $4$ to get:

Divide this by 12, for each month,

So the girl gets an allowance of $2592.33$$/month

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Please check $-(\frac{1}{2}x(\frac{1}{3}(\frac{1}{4})))$ in your equation. – Américo Tavares Feb 9 '11 at 20:51
... I've got $-\frac{1}{12}x$ instead. And $x=31108$, $x/12=2592.3$. – Américo Tavares Feb 9 '11 at 20:55
You are correct, thanks. – fdart17 Feb 9 '11 at 21:47

Solution without equations, only fractions. If we represent by $1$ the total money earned by the girl before spending on cloths, games and toys, we can split this unit according to the fractions she spent:

$$1=\overset{3/4}{\overbrace{\underset{2/3}{\underbrace{\frac{1}{2}+\frac{1}{3} \cdot \frac{1}{2}}}+\frac{1}{4}\cdot \frac{1}{3}}}+\frac{1}{4}$$

The fraction $\frac{1}{4}$ represents the money left, which we know is $7777 $. So the money she earned in a year is $4\cdot 7777$. An in a month $4\cdot 7777\cdot \frac{1}{12}=\frac{7777}{3}$.

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