A, B and C are partners. A receives 9/10 of the profit and B and C share the remaining profit equally. A's income is increased by 270 when the profit rises from 12 to `15%`. Find the capital invested by B and C each

Options

A. `' 5000`
B. `' 1000`
C. `'500`
D. `'1500`

Answer

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

### Step 1: Understand the profit distribution
A receives \( \frac{9}{10} \) of the total profit, while B and C share the remaining profit equally. This means that B and C together receive \( \frac{1}{10} \) of the total profit, and each of them receives \( \frac{1}{20} \) of the total profit.

### Step 2: Define the total investment
Let the total investment of A, B, and C be \( 10x \). Therefore:
- A's share of the profit = \( \frac{9}{10} \) of total profit
- B's share of the profit = \( \frac{1}{20} \) of total profit
- C's share of the profit = \( \frac{1}{20} \) of total profit

### Step 3: Calculate A's profit at different rates
When the profit rate is 12%, A's profit is:
\[ A's\ profit\ at\ 12\% = \frac{9}{10} \times 12\% \times 10x = \frac{9 \times 12x}{100} = \frac{108x}{100} \]

When the profit rate is 15%, A's profit is:
\[ A's\ profit\ at\ 15\% = \frac{9}{10} \times 15\% \times 10x = \frac{9 \times 15x}{100} = \frac{135x}{100} \]

### Step 4: Determine the increase in A's profit
The increase in A's profit when the profit rate changes from 12% to 15% is:
\[ Increase\ in\ A's\ profit = A's\ profit\ at\ 15\% - A's\ profit\ at\ 12\% \]
Substituting the values:
\[ Increase = \frac{135x}{100} - \frac{108x}{100} = \frac{27x}{100} \]
According to the problem, this increase is equal to 270:
\[ \frac{27x}{100} = 270 \]

### Step 5: Solve for \( x \)
To find \( x \), multiply both sides by 100:
\[ 27x = 27000 \]
Now, divide by 27:
\[ x = \frac{27000}{27} = 1000 \]

### Step 6: Calculate total investment
Now that we have \( x \), we can find the total investment:
\[ Total\ investment = 10x = 10 \times 1000 = 10000 \]

### Step 7: Find the capital invested by B and C
Since B and C share the remaining profit equally, their individual investments are:
\[ B's\ investment = C's\ investment = \frac{Total\ investment\ by\ B\ and\ C}{2} = \frac{10000 - A's\ investment}{2} \]
A's investment is:
\[ A's\ investment = \frac{9}{10} \times 10000 = 9000 \]
Thus, the total investment by B and C is:
\[ Total\ investment\ by\ B\ and\ C = 10000 - 9000 = 1000 \]
Therefore, each of their investments is:
\[ B's\ investment = C's\ investment = \frac{1000}{2} = 500 \]

### Final Answer
The capital invested by B and C each is **500**.

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