Aglass holds 6 ounces of 60-proof rum (30 alcohol). how much fruit juice must be added to the rum so that the mixture is diluted to 20% alcohol

We need an equation that will relate the concentrated mixture and the diluted one. To solve this we use the equation, 

M1 V1 = M2 V2

where M1 is the concentration of the stock solution, V1 is the volume of the stock solution, M2 is the concentration of the new solution and V2 is its volume.

M1V1 = M2V2

30 % x 6 oz = 20 %  x V2

V2 = 9 oz 

The volume of the diluted mixture would be 9 oz. Therefore, you will need to add 9 oz - 6 oz = 3 oz of fruit juice to dilute the 30 percent alcohol to 20 percent alcohol.