A man spends `(1)/(3)` of his income on food, `(2)/(5)` of his income on house rent and `(1)/(5)` of his income on clothes. If he still has Rs400 left with him, his income is

Options

A. ₹ 4000
B. ₹ 5000
C. ₹ 6000
D. ₹ 7000

Answer

To find the man's income based on his spending and remaining amount, we can follow these steps:

### Step 1: Define the Income
Let the man's income be Rs. \( X \).

### Step 2: Calculate the Amount Spent on Each Category
- Amount spent on food: \( \frac{1}{3}X \)
- Amount spent on house rent: \( \frac{2}{5}X \)
- Amount spent on clothes: \( \frac{1}{5}X \)

### Step 3: Calculate Total Amount Spent
Now, we need to find the total amount spent:
\[ \text{Total spent} = \frac{1}{3}X + \frac{2}{5}X + \frac{1}{5}X \]

### Step 4: Find a Common Denominator
The common denominator for \( 3 \) and \( 5 \) is \( 15 \). We convert each fraction:
- \( \frac{1}{3}X = \frac{5}{15}X \)
- \( \frac{2}{5}X = \frac{6}{15}X \)
- \( \frac{1}{5}X = \frac{3}{15}X \)

Now, we can add these fractions:
\[ \text{Total spent} = \frac{5}{15}X + \frac{6}{15}X + \frac{3}{15}X = \frac{14}{15}X \]

### Step 5: Calculate Remaining Income
The remaining income after spending is given as Rs. 400. Therefore, we can write:
\[ X - \frac{14}{15}X = 400 \]

### Step 6: Simplify the Equation
This simplifies to:
\[ \frac{1}{15}X = 400 \]

### Step 7: Solve for \( X \)
To find \( X \), multiply both sides by \( 15 \):
\[ X = 400 \times 15 \]
\[ X = 6000 \]

### Conclusion
The man's income is Rs. 6000.

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