In a partnership, A invests of the `1/6` capital for `1/6` of the time, B invests `1/3` of the capital for `1/3` of the time and C, the rest of the capital for whole time. Find A's share of the total profit of Rs. 2300.

Answer

To solve the problem step by step, we will calculate the investment of each partner and their respective shares of the profit.

### Step 1: Define the total capital
Let's assume the total capital is represented as \( C \).

### Step 2: Calculate A's investment
A invests \( \frac{1}{6} \) of the capital for \( \frac{1}{6} \) of the time.
- A's investment = \( \frac{1}{6}C \)
- A's time = \( \frac{1}{6} \)

### Step 3: Calculate B's investment
B invests \( \frac{1}{3} \) of the capital for \( \frac{1}{3} \) of the time.
- B's investment = \( \frac{1}{3}C \)
- B's time = \( \frac{1}{3} \)

### Step 4: Calculate C's investment
C invests the remaining capital.
- Total capital invested by A and B = \( \frac{1}{6}C + \frac{1}{3}C \)
- To combine these, we convert \( \frac{1}{3}C \) to sixths:
\[ \frac{1}{3}C = \frac{2}{6}C \]
- Therefore, total capital invested by A and B = \( \frac{1}{6}C + \frac{2}{6}C = \frac{3}{6}C = \frac{1}{2}C \)
- C's investment = Total capital - (A's investment + B's investment)
\[ C's investment = C - \frac{1}{2}C = \frac{1}{2}C \]
- C's time = 1 (whole time)

### Step 5: Calculate the effective capital contribution
Now we calculate the effective capital contribution for each partner:
- A's effective capital = \( \text{Investment} \times \text{Time} = \frac{1}{6}C \times \frac{1}{6} = \frac{1}{36}C \)
- B's effective capital = \( \text{Investment} \times \text{Time} = \frac{1}{3}C \times \frac{1}{3} = \frac{1}{9}C \)
- C's effective capital = \( \text{Investment} \times \text{Time} = \frac{1}{2}C \times 1 = \frac{1}{2}C \)

### Step 6: Convert all contributions to a common denominator
To compare the contributions, we convert all fractions to a common denominator:
- The least common multiple of 36, 9, and 2 is 36.
- A's contribution = \( \frac{1}{36}C \)
- B's contribution = \( \frac{4}{36}C \) (since \( \frac{1}{9}C = \frac{4}{36}C \))
- C's contribution = \( \frac{18}{36}C \) (since \( \frac{1}{2}C = \frac{18}{36}C \))

### Step 7: Calculate the total contributions
Total contributions = A's contribution + B's contribution + C's contribution
\[ = \frac{1}{36}C + \frac{4}{36}C + \frac{18}{36}C = \frac{23}{36}C \]

### Step 8: Find the ratio of contributions
The ratio of contributions is:
- A : B : C = \( 1 : 4 : 18 \)

### Step 9: Calculate A's share of the profit
Total profit = Rs. 2300.
A's share of the profit is calculated based on the ratio of contributions:
- Total parts = \( 1 + 4 + 18 = 23 \)
- A's share = \( \frac{1}{23} \times 2300 \)

Calculating A's share:
\[ A's \, share = \frac{2300}{23} = 100 \]

### Final Answer
A's share of the total profit is Rs. 100.

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