YankeeJim: Where x=total income (assumes 12 months allowance saved up.)
[...]
x (( 1 - .5/3 - (.5 - .5/3)/4)) = $7777
x ( 1 - .167 - .083) = $7777
.25x = $7777
x = $31108
I don't know how you solved the last bracket into .25 ;)
but FWIW, the solution is even simpler:
Let M1, M2, M3 be the money before the actions.
The toys are 1/4 of the money M3, 3/4 = 7777 $ are left => M3 = 7777*4/3
The games are 1/3 of M2, => M2 = M3 * 3/2
The clothes are 1/2 of M1 => M1 = M2 * 2
Replace M2 and M3 => M1 = 4/3 * 3/2 * 2 * 7777,
eliminating identical dividends and divisors (2 and 3), M1 = 4 * 7777 = 31108
Q.E.D!