The ratio of incomes of C and D is `3 : 2`. Ratio of income of D and E is `5 : 4`. If one-third of C’s income is Rs 4000 more than the half of E’s income. then what is the D’s income (in Rs)?

Options

A. 40000
B. 43000
C. 50000
D. 60000

Answer

To solve the problem step by step, we will first set up the relationships based on the ratios provided and then use the information given to find D's income.

### Step 1: Set up the ratios
We know:
- The ratio of incomes of C and D is \(3:2\).
- The ratio of incomes of D and E is \(5:4\).

Let’s denote:
- C's income = \(3x\)
- D's income = \(2x\)

For D and E:
- D's income = \(5y\)
- E's income = \(4y\)

### Step 2: Equate D's income
Since D's income is represented in both ratios, we can equate them:
\[ 2x = 5y \]
From this, we can express \(y\) in terms of \(x\):
\[ y = \frac{2x}{5} \]

### Step 3: Substitute E's income
Now we can express E's income in terms of \(x\):
\[ E's\ income = 4y = 4 \left(\frac{2x}{5}\right) = \frac{8x}{5} \]

### Step 4: Set up the equation from the problem statement
According to the problem, one-third of C’s income is Rs 4000 more than half of E’s income. We can write this as:
\[ \frac{1}{3} \times 3x = \frac{1}{2} \times \frac{8x}{5} + 4000 \]

### Step 5: Simplify the equation
Now simplify the left and right sides:
\[ x = \frac{4x}{5} + 4000 \]

### Step 6: Eliminate fractions
To eliminate the fraction, multiply the entire equation by 5:
\[ 5x = 4x + 20000 \]

### Step 7: Solve for \(x\)
Subtract \(4x\) from both sides:
\[ 5x - 4x = 20000 \]
\[ x = 20000 \]

### Step 8: Find D's income
Now we can find D's income:
\[ D's\ income = 2x = 2 \times 20000 = 40000 \]

### Final Answer
Thus, D's income is Rs 40,000.

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