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 2 rs per sq. cm).
Number of cards of `(2xx 2)` sq cm size Aman printed on Wednesday, was 120 more than the number of `(L xx 2)` sq. om cards he printed on Monday. If he charged 8 rs per card on Wednesday to a client, how much did the client pay for all the `(2 xx 2)` sq cm cards printed?

Options

A. Rs... 1720
B. Rs... 1820
C. Rs... 1760
D. Rs... 1480

Answer

To solve the problem step by step, we will follow the information provided in the question.

### Step 1: Understand the sizes and ratios of the cards printed on Monday
Aman printed two sizes of cards on Monday:
- Size 1: \( L \times 2 \) sq cm
- Size 2: \( 6 \times 3 \) sq cm

The ratio of the number of \( L \times 2 \) cards to \( 6 \times 3 \) cards is given as 2:3.

Let the number of \( L \times 2 \) cards be \( 2x \) and the number of \( 6 \times 3 \) cards be \( 3x \).

### Step 2: Calculate the area of each card and the total cost for each size
1. **Area of \( L \times 2 \) card**:
- Area = \( L \times 2 \) sq cm
- Cost per sq cm = Rs 2
- Total cost for \( 2x \) cards = \( 2x \times (L \times 2) \times 2 = 8xL \)

2. **Area of \( 6 \times 3 \) card**:
- Area = \( 6 \times 3 = 18 \) sq cm
- Total cost for \( 3x \) cards = \( 3x \times 18 \times 2 = 108x \)

### Step 3: Set up equations based on total costs
From the problem, we know:
- Total cost for \( L \times 2 \) cards = Rs 1600
- Total cost for \( 6 \times 3 \) cards = Rs 5400

Setting up the equations:
1. \( 8xL = 1600 \)
2. \( 108x = 5400 \)

### Step 4: Solve for \( x \)
From the second equation:
\[ x = \frac{5400}{108} = 50 \]

### Step 5: Substitute \( x \) back to find \( L \)
Using \( x = 50 \) in the first equation:
\[ 8 \times 50 \times L = 1600 \]
\[ 400L = 1600 \implies L = \frac{1600}{400} = 4 \]

### Step 6: Find the number of cards printed on Monday
- Number of \( L \times 2 \) cards = \( 2x = 2 \times 50 = 100 \)
- Number of \( 6 \times 3 \) cards = \( 3x = 3 \times 50 = 150 \)

### Step 7: Calculate the number of \( 2 \times 2 \) cards printed on Wednesday
Aman printed 120 more \( 2 \times 2 \) cards than \( L \times 2 \) cards:
\[ \text{Number of } 2 \times 2 \text{ cards} = 100 + 120 = 220 \]

### Step 8: Calculate the total cost for \( 2 \times 2 \) cards
Cost per \( 2 \times 2 \) card = Rs 8
Total cost for \( 220 \) cards:
\[ \text{Total cost} = 220 \times 8 = 1760 \]

### Final Answer
The client paid Rs 1760 for all the \( 2 \times 2 \) cards printed.

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