Interests and Amounts 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? 
 A: 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 $4\cdot
7777\cdot \frac{1}{12}=\frac{7777}{3}$.
A: Let $x$ be the amount of money the girl started with.
Then we can derive this equation.
$x-\frac{1}{2}x-(\frac{1}{2}x(\frac{1}{3}))-(\frac{1}{2}x(\frac{2}{3}(\frac{1}{4})))=7777$  
Multiply out the fractions and factor out x, you get:
$x(1-\frac{1}{2}-\frac{1}{6}-\frac{1}{12})=7777$    
So $x(\frac{12}{12}-\frac{6}{12}-\frac{2}{12}-\frac{1}{12})=7777$  
Thus :
$\frac{1}{4}x=7777$  
Multiply both sides by $4$ to get:
$x=31108$  
Divide this by 12, for each month,
$31108/12\approx2592.33$    
So the girl gets an allowance of $2592.33$$/month  
