Directions () : Study the following information carefully and answer the given questions
. Aman prints visiting cards of four standard sizes - `(L xx 2)` sq cm, `(6 xx 3)` sq cm, `(5 xx 5)` sq cm and `(2 xx 2)` sq cm. On Monday he printed cards of two sizes `(L xx 2)` sq cm and `(6 xx 3)` sq cm for a client. The respective ratio between the number of `(L xx 2)` sq cm cards and the number of `(6 xx 3)` sq cm cards printed on Monday was 2:3. The client had to pay a total of rs 1600 for the `(L xx 2)` sq cm card and a total of rs 5400 for `(6 xx 3)` sq cm card, (Both at the rate of 2rs per sq. cm).
What is the value of L?

Options

A. 3
B. 8
C. 5
D. 4

Answer

To solve the problem, we need to find the value of \( L \) given the information about the visiting cards printed by Aman.

### Step-by-Step Solution:

1. **Understand the Sizes and Ratios**:
- The sizes of the cards are:
- \( (L \times 2) \) sq cm
- \( (6 \times 3) \) sq cm
- The ratio of the number of \( (L \times 2) \) sq cm cards to \( (6 \times 3) \) sq cm cards printed is \( 2:3 \).

2. **Assume Variables**:
- Let the number of \( (L \times 2) \) cards printed be \( 2x \).
- Let the number of \( (6 \times 3) \) cards printed be \( 3x \).

3. **Calculate the Area of Each Card**:
- Area of \( (L \times 2) \) card = \( 2L \) sq cm.
- Area of \( (6 \times 3) \) card = \( 18 \) sq cm (since \( 6 \times 3 = 18 \)).

4. **Calculate the Cost for Each Card**:
- Cost of one \( (L \times 2) \) card = \( 2 \times (2L) = 4L \) Rs.
- Cost of one \( (6 \times 3) \) card = \( 2 \times 18 = 36 \) Rs.

5. **Total Cost for Each Type of Card**:
- Total cost for \( (L \times 2) \) cards = \( 2x \times 4L = 8xL \).
- Total cost for \( (6 \times 3) \) cards = \( 3x \times 36 = 108x \).

6. **Set Up the Equations**:
- According to the problem, the total cost for \( (L \times 2) \) cards is Rs 1600:
\[ 8xL = 1600 \]
- The total cost for \( (6 \times 3) \) cards is Rs 5400:
\[ 108x = 5400 \]

7. **Solve for \( x \)**:
- From \( 108x = 5400 \):
\[ x = \frac{5400}{108} = 50 \]

8. **Substitute \( x \) Back to Find \( L \)**:
- Substitute \( x = 50 \) into the equation \( 8xL = 1600 \):
\[ 8 \times 50 \times L = 1600 \]
\[ 400L = 1600 \]
\[ L = \frac{1600}{400} = 4 \]

### Final Answer:
The value of \( L \) is **4**.