The ratio of number of children, women and men in colony is 6 : 4 : 3 respectively and the colony has at least 200 members. What could be the minimum number of women in the colony?

Options

A. 75
B. 80
C. 64
D. 20

Answer

To solve the problem step by step, we will follow the given information about the ratio of children, women, and men in the colony, and calculate the minimum number of women based on the total population.

### Step 1: Understand the Ratio
The ratio of the number of children, women, and men is given as:
- Children : Women : Men = 6 : 4 : 3

### Step 2: Calculate the Total Parts in the Ratio
To find the total parts in the ratio, we add the parts together:
- Total parts = 6 + 4 + 3 = 13 parts

### Step 3: Set Up the Equation for Total Members
Let the total number of members in the colony be represented by \( x \). According to the problem, the colony has at least 200 members:
- \( x \geq 200 \)

### Step 4: Find the Minimum Value of \( x \) that is Divisible by 13
To find the minimum number of members that is also a multiple of 13, we need to find the smallest multiple of 13 that is greater than or equal to 200.
- The multiples of 13 are: 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, ...
- The smallest multiple of 13 that is greater than or equal to 200 is 208.

### Step 5: Calculate the Value of One Part
Now that we have determined that \( x = 208 \), we can find the value of one part:
- Value of one part = \( \frac{x}{\text{Total parts}} = \frac{208}{13} = 16 \)

### Step 6: Calculate the Number of Women
Since the number of women corresponds to 4 parts in the ratio, we can calculate the number of women:
- Number of women = \( 4 \times \text{Value of one part} = 4 \times 16 = 64 \)

### Conclusion
Thus, the minimum number of women in the colony is **64**.

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