d) The 3 Graces were carrying baskets of apples, and in each was the same number. The 9 Muses met them and asked each for apples and they gave the same number to each Muse and the 9 and the 3 each had the same number. Tell me how many they gave and how they all had the same number. (This problem is indeterminate. Find the smallest permissible solution.)
Let x represent the number of apples in one of the Graces’ baskets, then 3x represents the total number of apples.
The minimum number of apples that each Grace could give to each Muse is 1, therefore each Muse now has 3 apples and each Grace has x – 9 apples.
Since the Muses and the Graces must end up with the same number of apples, the Graces must also have 3 apples in their baskets. Then,
x – 9 = 3
x = 12
Therefore each Grace began with a minimum of 12 apples
If each Grace gave away 2 apples, then
x – 9(2) = 3(2)
x – 18 = 6
x = 24
and each Grace began with 24 apples.
Therefore, 3x = 12n, where n is the number of apples each Grace gave away and x is the number of apples each Grace started with.