Question

problem 11 - part a
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 of brown
if the stylist needs to make 20 oz of dye, how much of each dye color is needed?
red tone: $\frac{5}{3}$ oz
gray tone: 10 oz
brown tone: $\frac{25}{3}$ 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 in ratio

Sum of parts: \(\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{20}{8}=\frac{5}{2}\)

Step3: Find multiplier for scaling

Total dye needed: 20 oz. Multiplier \(k = \frac{20}{\frac{5}{2}}=20\times\frac{2}{5}=8\)

Step4: Calculate each color's amount

Red: \(\frac{9}{8}\times8 = 9\)? Wait, no, wait. Wait, original ratio parts sum: \(\frac{9 + 6+5}{8}=\frac{20}{8}=\frac{5}{2}\). Wait, maybe I messed up. Wait, let's recalculate. Wait, \(\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{20}{8}=\frac{5}{2}\). Then the multiplier is \(20\div\frac{5}{2}=8\). Then red: \(\frac{9}{8}\times8 = 9\)? But the given answer was \(\frac{5}{3}\), which is wrong. Wait, maybe the original problem has a typo or I misread. Wait, no, the user's problem shows red tone as \(1\frac{1}{8}\), gray \(\frac{3}{4}\), brown \(\frac{5}{8}\). Wait, maybe the total ratio sum is \(\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{20}{8}=\frac{5}{2}\). Then each part is \(20\div\frac{5}{2}=8\). So red: \(\frac{9}{8}\times8 = 9\), gray: \(\frac{6}{8}\times8 = 6\), brown: \(\frac{5}{8}\times8 = 5\). But the given answers are wrong. Wait, maybe the original problem's brown tone was cut off? Wait, the user's image shows brown tone as \(\frac{5}{8}\) but maybe the ratio is different. Wait, maybe the initial ratio sum is \(\frac{9}{8}+\frac{3}{4}+\frac{5}{8}=\frac{9 + 6+5}{8}=\frac{20}{8}=\frac{5}{2}\). Then the multiplier is \(20\div\frac{5}{2}=8\). So red: \(9\), gray: \(6\), brown: \(5\). But the given answers in the image are wrong. However, if we follow the given answer's logic, maybe the ratio sum was miscalculated. Wait, maybe the problem was to find the amount when total is 20, but the given answers are incorrect. Wait, perhaps the user made a mistake in the problem statement. But according to correct calculation:

Red tone: \(1\frac{1}{8}=\frac{9}{8}\), gray \(\frac{3}{4}=\frac{6}{8}\), brown \(\frac{5}{8}\). Sum: \(\frac{20}{8}=\frac{5}{2}\). Multiplier: \(20\div\frac{5}{2}=8\). So red: \(9\) oz, gray: \(6\) oz, brown: \(5\) oz. But the given answers in the image are \(\frac{5}{3}\), \(10\), \(\frac{25}{3}\) which sum to \(\frac{5 + 30+25}{3}=\frac{60}{3}=20\), but the ratio is wrong. So maybe the original ratio was different. Wait, maybe the red tone was \(1\frac{1}{8}\), gray \(\frac{3}{4}\), brown \(\frac{5}{8}\) but the total ratio sum is \(\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{20}{8}=\frac{5}{2}\), but if we consider the ratio as \(\frac{9}{8}:\frac{6}{8}:\frac{5}{8}=9:6:5\). Then total parts: \(9 + 6+5 = 20\). Oh! Wait, that's the key. I converted to eighths, but the ratio is \(9:6:5\) (since \(\frac{9}{8}:\frac{6}{8}:\frac{5}{8}=9:6:5\)). So total parts: \(9 + 6+5 = 20\). Then each part is \(20\div20 = 1\). Wait, no! \(9 + 6+5 = 20\) parts. So red: \(9\times1 = 9\), gray: \(6\times1 = 6\), brown: \(5\times1 = 5\). But the given answers are wrong. However, the user's image shows red tone as \(\frac{5}{3}\), gray as \(10\), brown as \(\frac{25}{3}\). Let's check their sum: \(\frac{5}{3}+10+\frac{25}{3}=\frac{5 + 30+25}{3}=\frac{60}{3}=20\). So how? Let's see the ratio. Let red: \(x\), gray: \(y\), brown: \(z\). \(x + y+ z = 20\). If \(x=\frac{5}{3}\), \(y = 10\), \(z=\frac{25}{3}\), then the ratio is \(\frac{5}{3}:10:\frac{25}{3}=5:30:25 = 1:6:5\). But original ratio was \(1\frac{1}{8}:\frac{3}{4}:\frac{5}{8}=\frac{9}{8}:\frac{6}{8}:\frac{5}{8}=9:6:5\). So there's a discrepancy. Maybe the problem was miswritten. But according to the correct method:

1. Find the sum of the ratio parts: \(1\frac{1}{8}+\frac{3}{4}+\frac{5}{8}=\frac{9}{8}+\frac{6}{8}+\frac{5}{8}=\frac{20}{8}=\frac{5}{2}\)
2. The multiplier is \(20\div\frac{5}{2}=8\)
3. Red tone: \(1\frac{1}{8}\times8 = 9\) oz
4. Gray tone: \(\frac{3}{4}\times8 = 6\) oz
5. Brown tone: \(\frac{5}{8}\times8 = 5\) oz

But the given answers in the image are incorrect. However, if we follow the given answer's sum (20), and the values \(\frac{5}{3}\), \(10\), \(\frac{25}{3}\), their ratio is \(1:6:5\), which is different from the original ratio. So there must be a mistake in the problem or the given answers. But assuming the problem's ratio was different (e.g., red: \(\frac{1}{3}\), gray: \(2\), brown: \(\frac{5}{3}\) which sum to \(\frac{1 + 6+5}{3}=\frac{12}{3}=4\), multiplier 5 to get 20), but that's speculative.

Answer:

The correct amounts should be Red tone: \(9\) oz, Gray tone: \(6\) oz, Brown tone: \(5\) oz. (The given answers in the image are incorrect based on the provided ratio.)