Question

the compositions of three powdered drink mixes are shown in the table below. drink mix composition mix a mix b mix c citric acid 0.10 0.15 0.20 flavoring 0.10 0.05 0.10 sugar 0.80 0.80 0.70 xavier needs to create 100 grams of a drink mix containing 18% citric acid, 8% flavoring, and 74% sugar. how much of mix c should he use? 0 grams 20 grams 40 grams 60 grams

Explanation:

Step1: Let the amount of Mix A be $x$ grams, Mix B be $y$ grams and Mix C be $z$ grams.

We know that $x + y+z=100$ (total - mass equation).
The amount of citric - acid equation: $0.1x + 0.15y+0.2z=0.18\times100 = 18$.
The amount of flavoring equation: $0.1x + 0.05y+0.1z=0.08\times100 = 8$.
The amount of sugar equation: $0.8x + 0.8y+0.7z=0.74\times100 = 74$.
Since $x + y+z = 100$, then $x=100 - y - z$.

Step2: Substitute $x = 100 - y - z$ into the citric - acid and flavoring equations.

Citric - acid: $0.1(100 - y - z)+0.15y + 0.2z=18$.
Expand: $10-0.1y-0.1z + 0.15y+0.2z=18$.
Combine like - terms: $0.05y + 0.1z=8$.
Flavoring: $0.1(100 - y - z)+0.05y+0.1z=8$.
Expand: $10-0.1y-0.1z + 0.05y+0.1z=8$.
Combine like - terms: $-0.05y=-2$, so $y = 40$.

Step3: Substitute $y = 40$ into $0.05y + 0.1z=8$.

$0.05\times40+0.1z=8$.
$2 + 0.1z=8$.
Subtract 2 from both sides: $0.1z=6$.
Divide both sides by 0.1: $z = 60$.

Answer:

60 grams