José is making fruit-filled tart shells for a party. He has 72 raspberries and 108 blueberries. He wants the same number of each type of fruit in each tart. If he puts the greatest possible number of fruits in each tart, how many tarts can he make?
If he puts the greatest possible number of fruits in each tart, José can make 5 tarts?
It's not clear just what you mean by "the greatest number" If you put 72 of each berry into each tart, you can make only 1.
If you put 36 of each kind of berry into each tart, then you can make only 2 tarts, with 36 blueberries left over.
If you put 3 raspberries and 4 blueberries in each tart, you can make exactly 24 tarts.
If you put 8 raspberries and 12 blueberries in each tart, you can make exactly 9 tarts.
Since there are 2/3 as many raspberries as blueberries, to use up all the berries, you will need 2/3 as many in each tart.
The GCF for both 72 and 108 is 36. So 36 berries will be in each tart.
To find out the number of tarts José can make, we need to determine the highest common factor (HCF) of the number of raspberries and blueberries he has.
First, let's list the factors of 72:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Next, let's list the factors of 108:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
From these lists, we can see that the HCF of 72 and 108 is 36.
This means that José can make 36 tarts with the same number of raspberries and blueberries in each tart.