Question

how many cups of grape punch containing 10% fruit juice and berry punch containing 20% fruit juice must be added together to create 12 cups of punch with 18% fruit juice?

Answer 1

well, let's say we need "g" of grape punch, now how much juice is in the "g" amount? well, just 10% of it is juice or (10/100) * g, 0.1g.

let's say we need "b" of berry punch, now, is 20% juice, how much juice in "b"? well, (20/100) * b, or 0.2b.

whatever "g" and "b" are, they must add up to 12 cups, of 18%, how much juice in 12cups? well (18/100) * 12, or 2.16


`\bf \begin{array}{lccclll} &amount&concentration& \begin{array}{llll} concentrated\\ amount \end{array}\\ &-----&-----&-----\\ \textit{grape punch}&g&0.10&0.10g\\ \textit{berry punch}&b&0.20&0.20b\\ -----&-----&-----&-----\\ mixture&12&0.18&2.16 \end{array}`


`\bf \begin{cases} g+b=12\implies \boxed{b}=12-g\\ 0.1g+0.2b=2.16\\ ----------\\ 0.1g+0.2\left( \boxed{12-g} \right)=2.16 \end{cases} \\\\\\ 0.1g+2.4-0.2g=2.16\implies 2.4-2.16=0.2g-0.1g \\\\\\ 0.24=0.1g\implies \cfrac{0.24}{0.1}=g\implies \boxed{2.4=g}`

and of berry will then be 12 - g