Aman 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 Rs 400 left with him, his income is

Options

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

Answer

To find Aman's income based on his expenditures and savings, we can follow these steps:

### Step 1: Define Aman's Income
Let Aman's income be represented as \( I \).

### Step 2: Calculate Expenditures
Aman spends:
- \( \frac{1}{3} \) of his income on food
- \( \frac{2}{5} \) of his income on house rent
- \( \frac{1}{5} \) of his income on clothes

### Step 3: Find a Common Denominator
To simplify the calculations, we need a common denominator for the fractions. The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.

### Step 4: Express Income in Units
Let’s express Aman's income in terms of units. We can assume:
- \( I = 15 \) units

### Step 5: Calculate Each Expenditure
Now, we can calculate the amount spent on each category:
1. **Food**:
\[ \text{Food expenditure} = \frac{1}{3} \times 15 = 5 \text{ units} \]
2. **House Rent**:
\[ \text{House rent expenditure} = \frac{2}{5} \times 15 = 6 \text{ units} \]
3. **Clothes**:
\[ \text{Clothes expenditure} = \frac{1}{5} \times 15 = 3 \text{ units} \]

### Step 6: Calculate Total Expenditure
Now, we can find the total expenditure:
\[ \text{Total Expenditure} = 5 + 6 + 3 = 14 \text{ units} \]

### Step 7: Calculate Savings
Aman's savings can be calculated as:
\[ \text{Savings} = \text{Income} - \text{Total Expenditure} = 15 - 14 = 1 \text{ unit} \]

### Step 8: Relate Savings to Actual Amount
We know that Aman's savings amount to Rs 400:
\[ 1 \text{ unit} = 400 \text{ Rs} \]

### Step 9: Calculate Total Income
Now we can find Aman's total income:
\[ \text{Income} = 15 \text{ units} \times 400 \text{ Rs/unit} = 6000 \text{ Rs} \]

### Final Answer
Aman's income is Rs 6000.

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