Question

Mandy and Diana baked some muffins. Mandy gave 4/7 of her muffins to Diana. Diana then gave 1/9 of her muffins to Mandy. As a result, Mandy had 1/2 as many muffins as Diana. If Mandy gave Diana 78 more muffins than what Diana gave to Mandy, how many muffins did Diana have at first?

Answer 1

to start, they have m and d muffins.
After the first swap, they have
Mandy: 3/7 m
Diana: d + 4/7 m
After the 2nd swap, they have
Mandy: 3/7 m + 1/9 (d + 4/7 m)
Diana : 8/9 (d + 4/7 m)
so at the end,
3/7 m + 1/9 (d + 4/7 m) = 1/2 * 8/9 (d + 4/7 m)
4/7 m = 78 + 1/9 (d + 4/7 m)
whew. Simplifying things a bit, we have
5m = 7d
32m = 7d + 4914
so
5m = 32m - 4914
m = 182
so d = 130
check:
Mandy had 182, and gave 104 to Diana, who then had 234
Diana then gave 26 to Mandy, who then had 104. Diana had 208.
Mandy then had 1/2 as many as Diana
Note that Mandy gave 104=78+26 to Diana.