Given that 10% of A's income= 15% of B's income 20% of C's income. If sum of their Income is Rs. 7800, then B's income is:

Options

A. Rs. 3600
B. Rs. 3000
C. Rs. 2400
D. Rs. 1800

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Set up the equations based on the given information.
We know that:
- \( 10\% \) of A's income = \( 15\% \) of B's income = \( 20\% \) of C's income.

Let A's income be \( A \), B's income be \( B \), and C's income be \( C \). We can express the relationships as:
\[ 0.1A = 0.15B = 0.2C \]

### Step 2: Express A, B, and C in terms of a common variable.
Let \( k \) be the common value of \( 0.1A \), \( 0.15B \), and \( 0.2C \). Then we can write:
\[ A = \frac{k}{0.1} = 10k \]
\[ B = \frac{k}{0.15} = \frac{20k}{3} \]
\[ C = \frac{k}{0.2} = 5k \]

### Step 3: Write the equation for the total income.
According to the problem, the sum of their incomes is \( 7800 \):
\[ A + B + C = 7800 \]
Substituting the expressions for A, B, and C:
\[ 10k + \frac{20k}{3} + 5k = 7800 \]

### Step 4: Combine the terms.
To combine the terms, we need a common denominator. The common denominator for \( 1 \) and \( 3 \) is \( 3 \):
\[ 10k = \frac{30k}{3} \]
\[ 5k = \frac{15k}{3} \]
Now substituting back:
\[ \frac{30k}{3} + \frac{20k}{3} + \frac{15k}{3} = 7800 \]
Combining the fractions:
\[ \frac{30k + 20k + 15k}{3} = 7800 \]
\[ \frac{65k}{3} = 7800 \]

### Step 5: Solve for k.
To eliminate the fraction, multiply both sides by \( 3 \):
\[ 65k = 23400 \]
Now divide by \( 65 \):
\[ k = \frac{23400}{65} = 360 \]

### Step 6: Calculate B's income.
Now that we have \( k \), we can find B's income:
\[ B = \frac{20k}{3} = \frac{20 \times 360}{3} = 2400 \]

### Final Answer:
B's income is \( Rs. 2400 \).

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