Question

a fruit punch that contains 80% of fruit juice is combined with 100% fruit juice. how many ounces of each should be used to make a 36 oz of a mixture that is 95% fruit juice?

Answer 1

Let the amount of 80% fruit juice needed be "x".

Let the amount of 100% fruit juice needed be "y".

We need 36 ounces in total, so we can write:

`x+y=36`

Also,

80% of x and 100% of y would be needed to form 95% of x and y, thus we can write:

`\begin{gathered} 0.8x+1y=0.95(x+y) \\ 0.8x+y=0.95x+0.95y \end{gathered}`

The first equation can be solved for y:

`\begin{gathered} x+y=36 \\ y=36-x \end{gathered}`

Substituting this into the second equation, we can solve for x. Shown below:

`\begin{gathered} 0.8x+y=0.95x+0.95y \\ 0.8x+(36-x)=0.95x+0.95(36-x) \\ 0.8x+36-x=0.95x+34.2-0.95x \\ -0.2x+36=34.2 \\ 0.2x=1.8 \\ x=(1.8)/(0.2) \\ x=9 \end{gathered}`

Now, we can easily solve for y:

`\begin{gathered} y=36-x \\ y=36-9 \\ y=27 \end{gathered}`

Thus, we need to combine 9 oz of 80% fruit juice with 27 oz of 100% fruit juice to get 36 oz of 95% fruit juice.