Aryaraj gives `1//3` of his salary to his mother, `1//7` of remaining salary to his sister and `2//7` of rest salary to his father. If the salary `Rs.42000` left with him, then find his monthly salary ?

Options

A. Rs. `102900`
B. Rs. `132900`
C. Rs. `102000`
D. Rs. `162900`

Answer

To find Aryaraj's monthly salary, we can break down the problem step by step.

### Step 1: Define the total salary
Let Aryaraj's monthly salary be represented as \( S \).

### Step 2: Calculate the amount given to his mother
Aryaraj gives \( \frac{1}{3} \) of his salary to his mother. Therefore, the amount given to his mother is:
\[ \text{Amount to mother} = \frac{1}{3} S \]

### Step 3: Calculate the remaining salary after giving to his mother
After giving \( \frac{1}{3} S \) to his mother, the remaining salary is:
\[ \text{Remaining salary} = S - \frac{1}{3} S = \frac{2}{3} S \]

### Step 4: Calculate the amount given to his sister
Aryaraj then gives \( \frac{1}{7} \) of the remaining salary to his sister. The remaining salary after giving to his mother is \( \frac{2}{3} S \), so the amount given to his sister is:
\[ \text{Amount to sister} = \frac{1}{7} \left( \frac{2}{3} S \right) = \frac{2}{21} S \]

### Step 5: Calculate the remaining salary after giving to his sister
The remaining salary after giving to his sister is:
\[ \text{Remaining salary} = \frac{2}{3} S - \frac{2}{21} S \]
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 21 is 21. Thus:
\[ \frac{2}{3} S = \frac{14}{21} S \]
Now we can subtract:
\[ \text{Remaining salary} = \frac{14}{21} S - \frac{2}{21} S = \frac{12}{21} S = \frac{4}{7} S \]

### Step 6: Calculate the amount given to his father
Aryaraj gives \( \frac{2}{7} \) of the remaining salary to his father. The remaining salary after giving to his sister is \( \frac{4}{7} S \), so the amount given to his father is:
\[ \text{Amount to father} = \frac{2}{7} \left( \frac{4}{7} S \right) = \frac{8}{49} S \]

### Step 7: Calculate the remaining salary after giving to his father
The remaining salary after giving to his father is:
\[ \text{Remaining salary} = \frac{4}{7} S - \frac{8}{49} S \]
Again, we need a common denominator. The least common multiple of 7 and 49 is 49. Thus:
\[ \frac{4}{7} S = \frac{28}{49} S \]
Now we can subtract:
\[ \text{Remaining salary} = \frac{28}{49} S - \frac{8}{49} S = \frac{20}{49} S \]

### Step 8: Set up the equation with the remaining salary
According to the problem, the remaining salary is \( Rs. 42000 \):
\[ \frac{20}{49} S = 42000 \]

### Step 9: Solve for \( S \)
To find \( S \), we can multiply both sides by \( \frac{49}{20} \):
\[ S = 42000 \times \frac{49}{20} \]
Calculating this gives:
\[ S = 42000 \times 2.45 = 102900 \]

### Final Answer
Aryaraj's monthly salary is \( Rs. 102900 \).

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