If `10%` of A's Income `=15%` of B's Income `=30%` of C's Income and sum of income of A, B and C is `Rs.7800` then find Income of B

Options

A. `2600`
B. `3000`
C. `2400`
D. `1800`

Answer

To solve the problem, let's denote the incomes of A, B, and C as \( A \), \( B \), and \( C \) respectively.

### Step 1: Set up the equations based on the given information

From the problem, we know:
- \( 10\% \) of A's income = \( 15\% \) of B's income = \( 30\% \) of C's income.

Let \( k \) be the common value of these percentages. Therefore, we can write:

\[ 0.1A = k \quad (1) \]
\[ 0.15B = k \quad (2) \]
\[ 0.3C = k \quad (3) \]

### Step 2: Express A, B, and C in terms of k

From equation (1):
\[ A = \frac{k}{0.1} = 10k \]

From equation (2):
\[ B = \frac{k}{0.15} = \frac{20k}{3} \]

From equation (3):
\[ C = \frac{k}{0.3} = \frac{10k}{3} \]

### Step 3: Write the equation for the sum of incomes

According to the problem, the sum of the incomes of A, B, and C is Rs. 7800:

\[ A + B + C = 7800 \]

Substituting the expressions for A, B, and C:

\[ 10k + \frac{20k}{3} + \frac{10k}{3} = 7800 \]

### Step 4: Combine the terms

To combine the terms, we need a common denominator. The common denominator for the fractions is 3:

\[ 10k + \frac{20k}{3} + \frac{10k}{3} = 10k + \frac{30k}{3} = 10k + 10k = 20k \]

So, we have:

\[ 20k = 7800 \]

### Step 5: Solve for k

Now, divide both sides by 20:

\[ k = \frac{7800}{20} = 390 \]

### Step 6: Find the income of B

Now that we have \( k \), we can find B's income using the expression we derived earlier:

\[ B = \frac{20k}{3} = \frac{20 \times 390}{3} = \frac{7800}{3} = 2600 \]

### Final Answer

Thus, the income of B is Rs. 2600.

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