Question

1. A fruit juice cocktail is prepared by mixing apple juice concentrate (30% w/w sugar), cranberry concentrate (17% w/w sugar) and grape juice concentrate (25% w/w sugar) in 1:0.5:0.5 proportions ( 1 part of apple concentrate for 0.5 part of cranberry concentrate and 0.5 part of grape concentrate) respectively. Water is added to this mix such that the sugar content in the final juice product is 10% (w/w). Calculate the amount of each juice concentrate necessary to prepare 1000 kg of fruit juice cocktail. Also calculate the amount of water needed.

Answer 1

Apple juice concentrate: 196 kg

Cranberry concentrate: 98 kg

Grape juice concentrate: 98 kg

Water: 608 kg

Explanation:

We can start by calculating how much sugar we have per kg of concentrate, in the proportion 1:0.5:0.5.

Apple juice has 30% w/w

Cranberry has 17% w/w

Grape has 25% w/w

Then,

`(A*0.3+C*0.17+G*0.25)/(A+C+G)\\\\(1*0.3+0.5*0.17+0.5*0.25)/(1+0.5+0.5)\\\\0.51/2=0.255`

The concentrate has 25.5% w/w of sugar.

Assuming water (W) has no sugar, we have to calculate the proportions of water and concentrate (Co) to meet the final condition of 10% w/w of sugar in the cocktail:

`W+Co=1000\\\\W*0+Co*0.255=1000*0.10\\\\0.255Co=100\\\\Co=100/0.255=392\\\\\\W=1000-Co=1000-392=608`

Now we know that per 1000 kg of fruit juice cocktail, 608 kg are water and 392 kg of the contentrate mix.

Each concentrate has a proportion of 1 in 2 for the apple or 0.5 in 2 for the cranberry and grape.

The proportion of each concentrate is then:

`A=392*(1/2)=196\\\\ C =392*(0.5/2)=98\\\\G=392*(0.5/2)=98`