Question

problem 11 - part b
to make a specific hair dye, a hair stylist uses a ratio of $1\frac{1}{8}$ oz of red tone, $\frac{3}{4}$ oz of gray tone, and $\frac{5}{8}$ oz
if the stylist needs to make 100 oz of dye, how much of each dye color is needed?
red tone: \boxed{} oz
gray tone: \boxed{} oz
brown tone: \boxed{} oz

Explanation:

Step1: Convert mixed number to improper fraction

Red tone: \(1\frac{1}{8}=\frac{9}{8}\) oz. Gray tone: \(\frac{3}{4}=\frac{6}{8}\) oz. Brown tone: \(\frac{5}{8}\) oz.

Step2: Find total parts

Sum of fractions: \(\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{9 + 6+5}{8}=\frac{20}{8}=\frac{5}{2}\) oz (per ratio unit).

Step3: Find multiplier

Total dye needed: 100 oz. Multiplier \(k=\frac{100}{\frac{5}{2}} = 100\times\frac{2}{5}=40\).

Step4: Calculate each tone

Red: \(\frac{9}{8}\times40=\frac{360}{8} = 45\) oz. Gray: \(\frac{6}{8}\times40=\frac{240}{8}=30\) oz. Brown: \(\frac{5}{8}\times40=\frac{200}{8}=25\) oz.

Answer:

Red tone: 45 oz
Gray tone: 30 oz
Brown tone: 25 oz