There are `3/5` as many men as women in the hall. `2/3` of the men and `1/5` of the women wear formal dress. What is the ratio of the people in the hall who do not wear formal dress to the total number of people in the hall?

Options

A. `8:5`
B. `3:5`
C. `5:8`
D. `4:5 `

Answer

To solve the problem step by step, let's break it down clearly.

### Step 1: Define Variables
Let the total number of women in the hall be \( W \). According to the problem, the number of men \( M \) is given as \( \frac{3}{5} \) of the number of women:
\[ M = \frac{3}{5}W \]

### Step 2: Calculate Total People
The total number of people in the hall is the sum of men and women:
\[ \text{Total} = M + W = \frac{3}{5}W + W = \frac{3}{5}W + \frac{5}{5}W = \frac{8}{5}W \]

### Step 3: Calculate the Number of Men and Women
From the total number of people, we can express \( W \) in terms of the total:
\[ W = \frac{5}{8} \times \text{Total} \]
\[ M = \frac{3}{5}W = \frac{3}{5} \times \frac{5}{8} \times \text{Total} = \frac{3}{8} \times \text{Total} \]

### Step 4: Calculate the Number of People Wearing Formal Dress
Next, we calculate how many men and women wear formal dress:
- The number of men wearing formal dress is \( \frac{2}{3}M \):
\[ \text{Men wearing formal dress} = \frac{2}{3} \times \frac{3}{8} \times \text{Total} = \frac{2}{8} \times \text{Total} = \frac{1}{4} \times \text{Total} \]
- The number of women wearing formal dress is \( \frac{1}{5}W \):
\[ \text{Women wearing formal dress} = \frac{1}{5} \times \frac{5}{8} \times \text{Total} = \frac{1}{8} \times \text{Total} \]

### Step 5: Calculate Total Wearing Formal Dress
Now, we find the total number of people wearing formal dress:
\[ \text{Total wearing formal dress} = \frac{1}{4} \times \text{Total} + \frac{1}{8} \times \text{Total} \]
To add these fractions, we need a common denominator:
\[ \frac{1}{4} = \frac{2}{8} \]
Thus,
\[ \text{Total wearing formal dress} = \frac{2}{8} \times \text{Total} + \frac{1}{8} \times \text{Total} = \frac{3}{8} \times \text{Total} \]

### Step 6: Calculate People Not Wearing Formal Dress
To find the number of people not wearing formal dress, we subtract the number wearing formal dress from the total:
\[ \text{People not wearing formal dress} = \text{Total} - \text{Total wearing formal dress} = \text{Total} - \frac{3}{8} \times \text{Total} = \frac{5}{8} \times \text{Total} \]

### Step 7: Find the Ratio
Finally, we need the ratio of people not wearing formal dress to the total number of people:
\[ \text{Ratio} = \frac{\text{People not wearing formal dress}}{\text{Total}} = \frac{\frac{5}{8} \times \text{Total}}{\text{Total}} = \frac{5}{8} \]

### Conclusion
The ratio of the people in the hall who do not wear formal dress to the total number of people in the hall is:
\[ \frac{5}{8} \]