Certain number of people work in three farms. A , B and C . Three are only type farms and each person works in one or more farms. `72%` of people were in farm and people working in only farms C was `(1)/(36)`the of people working in farm A. Number of people working in both B and C but not in Farm A are 55. people working in only farms B are 65 more thn the people working in only farm C.
Number of people working in only farm A is `(160)/(3)%` more than number of people working in farm B.
What is the number of people who working in farm, A but not only in farm ?

Options

A. 225
B. 245
C. 115
D. 105

Explanation

To solve the problem step by step, we will define variables for the number of people working in each farm and then use the information provided in the question to set up equations.

### Step 1: Define Variables
Let:
- \( x \) = Total number of people working in the farms.
- \( A \) = Number of people working in only Farm A.
- \( B \) = Number of people working in only Farm B.
- \( C \) = Number of people working in only Farm C.
- \( AB \) = Number of people working in both Farms A and B but not in Farm C.
- \( AC \) = Number of people working in both Farms A and C but not in Farm B.
- \( BC \) = Number of people working in both Farms B and C but not in Farm A (given as 55).
- \( ABC \) = Number of people working in all three farms.

### Step 2: Use Given Information
1. **72% of people work in Farm A**:
\[ A + AB + AC + ABC = 0.72x \]

2. **People working in only Farm C is \( \frac{1}{36} \) of people working in Farm A**:
\[ C = \frac{1}{36} A \]

3. **People working in only Farm B are 65 more than people working in only Farm C**:
\[ B = C + 65 \]

4. **Number of people working in only Farm A is \( \frac{160}{3}\% \) more than number of people working in Farm B**:
\[ A = B + \frac{160}{3} \cdot \frac{1}{100} B = B \left(1 + \frac{160}{300}\right) = B \cdot \frac{460}{300} = \frac{23}{15}B \]

### Step 3: Substitute and Solve
From the equations derived, we can substitute \( C \) into the equation for \( B \):
\[ B = \frac{1}{36} A + 65 \]

Now substituting \( B \) in the equation for \( A \):
\[ A = \frac{23}{15} \left(\frac{1}{36} A + 65\right) \]

Expanding and simplifying:
\[ A = \frac{23}{540} A + \frac{23 \cdot 65}{15} \]
\[ A - \frac{23}{540} A = \frac{1495}{15} \]
\[ \left(1 - \frac{23}{540}\right) A = \frac{1495}{15} \]
\[ \frac{517}{540} A = \frac{1495}{15} \]
\[ A = \frac{1495 \cdot 540}{15 \cdot 517} \]
Calculating \( A \):
\[ A = \frac{1495 \cdot 36}{517} = 105 \]

### Step 4: Find the Number of People Working in Farm A but Not Only in Farm A
To find the number of people working in Farm A but not only in Farm A, we need to consider:
\[ AB + AC + ABC = 0.72x - A \]
Using the total number of people \( x \) and the values we have, we can calculate the total number of people working in Farm A but not only in Farm A.

### Conclusion
The number of people working in Farm A but not only in Farm A is:
\[ AB + AC + ABC = 105 + 55 + 0 = 160 \]