Tim and Sally were playing marbles. In the first game, Tim lost 3/5 of what he had and another 7 more marbles to Sally. In the second game, Sally lost 1/4 of what she had and another 4 marbles to Tim. In the third game, Tim lost 2/3 of what he had to Sally. At the end of three games, Tim had 61 marbles and Sally had 352 marbles. How many marbles did Sally have at first?
to start,
Tim had x
Sally had y
after game 1,
Tim had 2/5 x - 7
Sally had y + 3/5 x + 7
after game 2,
Sally had 3/4 (y + 3/5 x + 7) - 4 = 3/4 y + 9/20 x + 5/4
Tim had 2/5 x - 7 + 1/4 (y + 3/5 x + 7) + 4 = 11/20 x + 1/4 y - 5/4
after game 3,
Tim had 1/3 (11/20 x + 1/4 y - 5/4) = 11/60 x + 1/12 y - 5/12
Sally had (3/4 y + 9/20 x + 5/4) + 2/3 (11/20 x + 1/4 y - 5/4) = 11/12 y + 49/60 x + 5/12
so we are left with
11x + 5y = 3685
49x + 55y = 21095
solve that to get
x = 270
y = 142 <--- Sally