`(1)/(8)` of the audience were children, `(2)/(5)`th of the audience were men and the rest women. If the number of women audience is 380, find the total number of audience in the program?

Options

A. 600
B. 500
C. 400
D. 800

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the total audience
Let the total number of audience be represented by \( x \).

### Step 2: Calculate the number of children
According to the problem, \( \frac{1}{8} \) of the audience were children. Therefore, the number of children can be calculated as:
\[ \text{Number of children} = \frac{1}{8} x \]

### Step 3: Calculate the number of men
Next, \( \frac{2}{5} \) of the audience were men. Thus, the number of men is:
\[ \text{Number of men} = \frac{2}{5} x \]

### Step 4: Calculate the number of women
The rest of the audience consists of women. To find the number of women, we can subtract the number of children and men from the total audience:
\[ \text{Number of women} = x - \left(\frac{1}{8} x + \frac{2}{5} x\right) \]

### Step 5: Find a common denominator
To simplify the expression for the number of women, we need a common denominator for \( \frac{1}{8} \) and \( \frac{2}{5} \). The least common multiple of 8 and 5 is 40. Therefore, we convert the fractions:
\[ \frac{1}{8} = \frac{5}{40}, \quad \frac{2}{5} = \frac{16}{40} \]
Now we can rewrite the number of women:
\[ \text{Number of women} = x - \left(\frac{5}{40} x + \frac{16}{40} x\right) = x - \frac{21}{40} x \]
This simplifies to:
\[ \text{Number of women} = \frac{40}{40} x - \frac{21}{40} x = \frac{19}{40} x \]

### Step 6: Set up the equation
We know from the problem that the number of women is 380. Therefore, we can set up the equation:
\[ \frac{19}{40} x = 380 \]

### Step 7: Solve for \( x \)
To find \( x \), we multiply both sides by \( \frac{40}{19} \):
\[ x = 380 \times \frac{40}{19} \]
Calculating the right side:
\[ x = 380 \times \frac{40}{19} = 380 \times 2.105263 = 800 \]

### Conclusion
The total number of audience in the program is \( 800 \).