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 sq. cm)..
On Tuesday, cards of only (xxy) sq cm size were printed. Number of cards printed on Tuesday was `20%` more than the total number of cards printed on Monday and were printed at the cost of 4 rs per sq cm. If the client paid a total of rs 76800, what is the size of each card printed on Tuesday?

Options

A. 64 sq cm
B. 36 sq cm
C. None of those given as options
D. 16 sq cm

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the number of cards printed on Monday
- The ratio of `(L x 2)` sq cm cards to `(6 x 3)` sq cm cards is given as 2:3.
- Let the number of `(L x 2)` sq cm cards be `2x` and the number of `(6 x 3)` sq cm cards be `3x`.

### Step 2: Calculate the cost of cards printed on Monday
- The cost per square cm is Rs 2.
- The area of `(L x 2)` sq cm card = L * 2 = 2L sq cm.
- The area of `(6 x 3)` sq cm card = 6 * 3 = 18 sq cm.
- The cost of one `(L x 2)` card = 2L * 2 = 4L Rs.
- The cost of one `(6 x 3)` card = 18 * 2 = 36 Rs.

### Step 3: Set up the equations for total costs
- Total cost for `(L x 2)` cards = 1600 Rs.
- Therefore, \( 4L \cdot (2x) = 1600 \)
- This simplifies to \( 8Lx = 1600 \) → \( Lx = 200 \) → \( x = \frac{200}{L} \)

- Total cost for `(6 x 3)` cards = 5400 Rs.
- Therefore, \( 36 \cdot (3x) = 5400 \)
- This simplifies to \( 108x = 5400 \) → \( x = 50 \)

### Step 4: Calculate the value of L
- From \( Lx = 200 \) and \( x = 50 \):
- \( L \cdot 50 = 200 \) → \( L = \frac{200}{50} = 4 \)

### Step 5: Find the total number of cards printed on Monday
- Total number of cards printed on Monday = \( 2x + 3x = 5x = 5 \cdot 50 = 250 \)

### Step 6: Calculate the number of cards printed on Tuesday
- The number of cards printed on Tuesday is 20% more than Monday.
- Therefore, the number of cards printed on Tuesday = \( 250 + 0.2 \cdot 250 = 250 + 50 = 300 \)

### Step 7: Determine the cost per card printed on Tuesday
- Total cost for Tuesday = Rs 76800.
- Cost per card = \( \frac{76800}{300} = 256 \) Rs.

### Step 8: Calculate the size of each card printed on Tuesday
- The cost per square cm on Tuesday is Rs 4.
- Therefore, the area of one card = \( \frac{256}{4} = 64 \) sq cm.

### Conclusion
- The size of each card printed on Tuesday is 64 sq cm.

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