Rama gave her cousin `1/3` of her stamp collection .She gave her sister `2/5` of the remainder and had 96 stamps left . How many stamps did she have at first ?

Options

A. 240
B. 360
C. 570
D. 720

Answer

To solve the problem step by step, let's denote the total number of stamps Rama had initially as \( X \).

### Step 1: Determine the number of stamps given to her cousin
Rama gave her cousin \( \frac{1}{3} \) of her stamp collection. Therefore, the number of stamps she gave to her cousin is:
\[ \text{Stamps given to cousin} = \frac{1}{3}X \]

### Step 2: Calculate the remainder after giving stamps to her cousin
After giving away \( \frac{1}{3}X \), the number of stamps remaining with Rama is:
\[ \text{Remaining stamps} = X - \frac{1}{3}X = \frac{2}{3}X \]

### Step 3: Determine the number of stamps given to her sister
Rama then gave her sister \( \frac{2}{5} \) of the remaining stamps. The remaining stamps are \( \frac{2}{3}X \), so the number of stamps given to her sister is:
\[ \text{Stamps given to sister} = \frac{2}{5} \times \frac{2}{3}X = \frac{4}{15}X \]

### Step 4: Calculate the number of stamps left after giving to her sister
After giving stamps to her sister, the number of stamps left with Rama is:
\[ \text{Stamps left} = \frac{2}{3}X - \frac{4}{15}X \]
To perform this subtraction, we need a common denominator. The least common multiple of 3 and 15 is 15. Thus, we convert \( \frac{2}{3}X \) to have a denominator of 15:
\[ \frac{2}{3}X = \frac{10}{15}X \]
Now we can subtract:
\[ \text{Stamps left} = \frac{10}{15}X - \frac{4}{15}X = \frac{6}{15}X = \frac{2}{5}X \]

### Step 5: Set up the equation based on the number of stamps left
According to the problem, after all these transactions, Rama had 96 stamps left. Therefore, we can set up the equation:
\[ \frac{2}{5}X = 96 \]

### Step 6: Solve for \( X \)
To find \( X \), we multiply both sides by \( \frac{5}{2} \):
\[ X = 96 \times \frac{5}{2} = 96 \times 2.5 = 240 \]

### Conclusion
Thus, the total number of stamps Rama had initially is:
\[ \boxed{240} \]