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?
Answers
oobleck
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.
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.