The royal fruit company produces two types of fruit drinks the first type is 70% pure fruit juice and the second type is 95% pure fruit juice. the company is attempting to produce a fruit drink that contains 75% pure fruit juice how many pints of each of the two existing types of drinks must be used to make 60 pints of a make sure that is 75% pure fruit juice ?

This is a typical mixture problem where using a table is the best thing.
                 pints juice        % juice     =      total % juice
juice A            x                  .70          =        .70x
juice B        60-x                 .95          =    .95(60-x)
total              60                  .75          =    60(.75)

The last row gives you a total that you need to use, so do that multiplication to get 45. You now to need to add the last columns together for rows 1 and 2 and set them equal to 45: .7x +.95(60-x) = 45. When you do that math, you'll get that x = 48. This means that you need 48 pints of juice A and 60-48 pints of juice B, which is 12 pints.

Iam unsure about what the proof is, therefore as much as wish to i can not.