A man first sold `(2)/(3)rd` of his total quantity of rice. Again he sold `(1)/(2)` of the remaining quantity. If the total remaining quantity of the stock is 150 kg. Then, what was the original stock of rice ?

Options

A. 2100 kg
B. 900 kg
C. 2400 kg
D. 2000 kg

Answer

To solve the problem step by step, we will denote the original stock of rice as \( X \).

### Step 1: Calculate the quantity sold in the first transaction
The man sold \( \frac{2}{3} \) of his total quantity of rice. Therefore, the quantity sold in the first transaction is:
\[ \text{First Selling} = \frac{2}{3} X \]

### Step 2: Calculate the remaining quantity after the first sale
To find the remaining quantity after the first sale, we subtract the quantity sold from the original stock:
\[ \text{Remaining after first sale} = X - \frac{2}{3} X \]
To simplify this, we can express \( X \) as \( \frac{3}{3} X \):
\[ \text{Remaining after first sale} = \frac{3}{3} X - \frac{2}{3} X = \frac{1}{3} X \]

### Step 3: Calculate the quantity sold in the second transaction
The man then sold \( \frac{1}{2} \) of the remaining quantity. The remaining quantity after the first sale is \( \frac{1}{3} X \), so the quantity sold in the second transaction is:
\[ \text{Second Selling} = \frac{1}{2} \left( \frac{1}{3} X \right) = \frac{1}{6} X \]

### Step 4: Calculate the remaining quantity after the second sale
Now, we find the remaining quantity after the second sale:
\[ \text{Remaining after second sale} = \frac{1}{3} X - \frac{1}{6} X \]
To subtract these fractions, we need a common denominator. The common denominator for 3 and 6 is 6:
\[ \text{Remaining after second sale} = \frac{2}{6} X - \frac{1}{6} X = \frac{1}{6} X \]

### Step 5: Set up the equation based on the remaining quantity
We know from the problem that the remaining quantity is 150 kg:
\[ \frac{1}{6} X = 150 \]

### Step 6: Solve for \( X \)
To find \( X \), we multiply both sides of the equation by 6:
\[ X = 150 \times 6 = 900 \]

### Conclusion
The original stock of rice was:
\[ \text{Original Stock} = 900 \text{ kg} \]