Answer: If 34% of the mixed fruit juice is apple juice, then 66% of it is a combination of other fruit juices. We can assume that the ratio of apple juice to other fruit juices remains the same in the ice cream and mixed fruit juice mixture.
Let the volume of ice cream in each batch be 3x litres and the volume of mixed fruit juice be 2x litres. Then the total volume of mixture in each batch is 5x litres.
If 34% of the mixed fruit juice is apple juice, then the volume of apple juice in each batch is 0.34 × 2x = 0.68x litres. The volume of other fruit juices in each batch is 2x − 0.68x = 1.32x litres.
The ratio of ice cream to mixed fruit juice is 3 : 2, so we have:
3x : 2x = ice cream : mixed fruit juice
Simplifying, we get:
ice cream = 1.5 × mixed fruit juice
Substituting the values we obtained earlier, we have:
3x = 1.5 × 2x
x = 0.75
Therefore, the volume of ice cream in each batch is 3x = 2.25 litres, and the volume of mixed fruit juice is 2x = 1.5 litres.
To make 1.5 litres of mixed fruit juice, we need 0.68 litres of apple juice and 0.82 litres of other fruit juices.
To make 12.5 litres of smoothies, we need 2.25 litres of ice cream and 10.25 litres of mixed fruit juice. Of this, 0.68 litres is apple juice and 9.57 litres is other fruit juices.
Therefore, to make 12.5 litres of smoothies, we need 0.68/1.5 × 12.5 = 5.67 litres of apple juice.
To make 51 litres of apple juice, we can make a maximum of 51/5.67 ≈ 9 batches of smoothies.