Ram left `(1)/(3)` of his property to his widow and `(3)/(5)` of the remainder to his daughter. He gave the rest to his son who received Rs. 6,400. How much was his original property worth?

Options

A. Rs 16,000
B. Rs 32,000
C. Rs 24,000
D. Rs 1,600

Answer

To solve the problem step by step, we will denote the total property of Ram as \( P \).

### Step 1: Determine the amount left to the widow
Ram left \( \frac{1}{3} \) of his property to his widow. Therefore, the amount left to the widow is:
\[ \text{Amount to widow} = \frac{1}{3} P \]

### Step 2: Calculate the remainder after giving to the widow
After giving \( \frac{1}{3} \) of his property to his widow, the remaining property is:
\[ \text{Remainder} = P - \frac{1}{3} P = \frac{2}{3} P \]

### Step 3: Determine the amount left to the daughter
Ram then left \( \frac{3}{5} \) of the remainder to his daughter. The amount given to the daughter is:
\[ \text{Amount to daughter} = \frac{3}{5} \times \frac{2}{3} P = \frac{3 \times 2}{5 \times 3} P = \frac{2}{5} P \]

### Step 4: Calculate the remaining amount after giving to the daughter
After giving \( \frac{3}{5} \) of the remainder to his daughter, the remaining property is:
\[ \text{Remaining amount} = \frac{2}{3} P - \frac{2}{5} P \]

To subtract these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15. Thus, we convert the fractions:
\[ \frac{2}{3} P = \frac{10}{15} P \quad \text{and} \quad \frac{2}{5} P = \frac{6}{15} P \]

Now we can subtract:
\[ \text{Remaining amount} = \frac{10}{15} P - \frac{6}{15} P = \frac{4}{15} P \]

### Step 5: Set the remaining amount equal to the amount received by the son
According to the problem, the son received Rs. 6,400. Therefore, we can set up the equation:
\[ \frac{4}{15} P = 6400 \]

### Step 6: Solve for \( P \)
To find \( P \), we will multiply both sides of the equation by \( \frac{15}{4} \):
\[ P = 6400 \times \frac{15}{4} \]

Calculating the right side:
\[ P = 6400 \times 3.75 = 24000 \]

Thus, the original property worth is:
\[ \boxed{24000} \]