A man spends a `(1)/(4)` th of his in come on food `(2)/(3)` rd of it on house rent and the remaining income which is Rs 630 on other commodities. Find his house rent.

Options

A. Rs 5040
B. Rs 3520
C. Rs 4890
D. Rs 4458

Answer

To solve the problem step by step, we will follow the information given in the question and use algebra to find the man's house rent.

### Step 1: Define the variables
Let the man's total income be \( x \).

### Step 2: Calculate the amount spent on food
The man spends \( \frac{1}{4} \) of his income on food. Therefore, the amount spent on food is:
\[ \text{Amount spent on food} = \frac{1}{4} x \]

### Step 3: Calculate the amount spent on house rent
The man spends \( \frac{2}{3} \) of his income on house rent. Therefore, the amount spent on house rent is:
\[ \text{Amount spent on house rent} = \frac{2}{3} x \]

### Step 4: Set up the equation for remaining income
According to the problem, the remaining income after spending on food and house rent is Rs 630. Therefore, we can write the equation:
\[ x - \left(\frac{1}{4} x + \frac{2}{3} x\right) = 630 \]

### Step 5: Simplify the equation
To simplify the left side, we need to find a common denominator for the fractions. The least common multiple of 4 and 3 is 12. Thus, we can rewrite the fractions:
\[ \frac{1}{4} x = \frac{3}{12} x \quad \text{and} \quad \frac{2}{3} x = \frac{8}{12} x \]
Now substituting these into the equation gives:
\[ x - \left(\frac{3}{12} x + \frac{8}{12} x\right) = 630 \]
This simplifies to:
\[ x - \frac{11}{12} x = 630 \]

### Step 6: Combine like terms
Combining the terms on the left side:
\[ \frac{1}{12} x = 630 \]

### Step 7: Solve for \( x \)
To find \( x \), multiply both sides by 12:
\[ x = 630 \times 12 = 7560 \]

### Step 8: Calculate the house rent
Now that we have the total income, we can find the house rent:
\[ \text{House rent} = \frac{2}{3} x = \frac{2}{3} \times 7560 \]
Calculating this gives:
\[ \text{House rent} = 5040 \]

### Final Answer
The man's house rent is Rs 5040.