To make a specific hair dye, a hair stylist uses a ratio of [tex]$1 \frac{1}{8}$[/tex] oz of red tone, [tex]$\frac{3}{4}$[/tex] oz of gray tone, and [tex]$\frac{5}{8}$[/tex] oz of brown tone.

a. If the stylist needs to make 20 oz of dye, how much of each dye color is needed?

- Red tone:
[tex]\[
1 \frac{1}{8} \times 20 = \frac{9}{8} \times \frac{20}{1} = \frac{180}{8} = 22 \frac{1}{2} \text{ oz}
\][/tex]

- Gray tone:
[tex]\[
\frac{3}{4} \times 20 = \frac{60}{4} = 15 \text{ oz}
\][/tex]

- Brown tone:
[tex]\[
\frac{5}{8} \times 20 = \frac{100}{8} = 12 \frac{1}{2} \text{ oz}
\][/tex]

b. If the stylist needs to make 100 oz of dye, how much of each dye color is needed?

Sure! Let's break down the problem step-by-step.

The hair stylist uses the following ratios for the hair dye:
- Red tone: [tex]$1 \frac{1}{8}$[/tex] oz
- Gray tone: [tex]$\frac{3}{4}$[/tex] oz
- Brown tone: [tex]$\frac{5}{8}$[/tex] oz

These amounts are for a base mix. We need to use these to find out how much of each color is necessary to make a larger total amount of dye.

a. If the stylist needs to make 20 oz of dye:

1. First, you convert [tex]$1 \frac{1}{8}$[/tex] oz to an improper fraction. So, [tex]$1 \frac{1}{8} = \frac{9}{8}$[/tex].

2. Calculate how much of each tone is needed for 20 oz:
- Red tone: [tex]$\frac{9}{8} \times 20 = 22.5$[/tex] oz
- Gray tone: [tex]$\frac{3}{4} \times 20 = 15$[/tex] oz
- Brown tone: [tex]$\frac{5}{8} \times 20 = 12.5$[/tex] oz

b. If the stylist needs to make 100 oz of dye:

1. Again, using the ratio fractions:

2. Calculate how much of each tone is needed for 100 oz:
- Red tone: [tex]$\frac{9}{8} \times 100 = 112.5$[/tex] oz
- Gray tone: [tex]$\frac{3}{4} \times 100 = 75$[/tex] oz
- Brown tone: [tex]$\frac{5}{8} \times 100 = 62.5$[/tex] oz

In summary, to make:
- 20 oz of dye, you'll need 22.5 oz of red, 15 oz of gray, and 12.5 oz of brown.
- 100 oz of dye, you'll need 112.5 oz of red, 75 oz of gray, and 62.5 oz of brown.