An order was placed for a carpet whose breadth was 6 metres and whose length was 1.44 times its breadth. What will be the cost of a second carpet whose length and breadth are 40% more and 25% more respectively than the first carpet, if the rate of the carpet is Rs. 45 per square metre?

Introduction:
This question combines geometry of rectangles with percentage increase and cost calculation. The original carpet has given dimensions based on a relationship between length and breadth. The new carpet has both dimensions increased by certain percentages, and the cost is based on area at a fixed rate per square metre. The problem checks a student's ability to handle successive percentage increases, compute the area of a rectangle correctly, and then apply a unit rate to find the total cost.

Given Data / Assumptions:

• Breadth of the first carpet = 6 m. • Length of the first carpet = 1.44 times its breadth. • Length and breadth of the second carpet are increased by 40% and 25% respectively compared to the first carpet. • Rate of carpet = Rs. 45 per square metre. • We need the cost of the second carpet.

Concept / Approach:
First, compute the dimensions of the original carpet. Then apply the given percentage increases to find the new dimensions for the second carpet. Use the area formula for rectangles, area = length * breadth, to find the area of the second carpet. Once the area is known, multiply by the cost per square metre to find the total cost. Careful handling of percentages and decimal multiplication is essential.

Step-by-Step Solution:
Step 1: Compute length of the first carpet. Breadth of first carpet = 6 m. Length of first carpet = 1.44 * 6 = 8.64 m. Step 2: Compute area of the first carpet (for reference). Area first = 6 * 8.64 = 51.84 sq m. Step 3: Compute new length of second carpet. Increase in length = 40% of 8.64 = 0.40 * 8.64 = 3.456 m. New length = 8.64 + 3.456 = 12.096 m. Step 4: Compute new breadth of second carpet. Increase in breadth = 25% of 6 = 0.25 * 6 = 1.5 m. New breadth = 6 + 1.5 = 7.5 m. Step 5: Compute area of the second carpet. Area second = 12.096 * 7.5 = 90.72 sq m. Step 6: Compute cost at Rs. 45 per sq m. Cost = 90.72 * 45. Cost = 4082.4, that is Rs. 4082.40.

Verification / Alternative check:
We can also reason proportionally: The area of a rectangle scales by the product of the scale factors applied to each dimension. Here, length is multiplied by 1.40 and breadth is multiplied by 1.25. So area is multiplied by 1.40 * 1.25 = 1.75. The first carpet area is 51.84 sq m, so the second carpet area should be 51.84 * 1.75 = 90.72 sq m, matching our previous result. Multiplying by the rate 45 gives 51.84 * 1.75 * 45 = 4082.4, reinforcing the answer.

Why Other Options Are Wrong:
Option Rs. 1024.21: This is far too low and would correspond to a much smaller carpet area. Option Rs. 2810.60: This underestimates the scaled area and cost, as it corresponds to reducing one of the multipliers incorrectly. Option Rs. 3214.00: This might result from an error in applying only one percentage change correctly or miscalculating the rate. Option Rs. 3500.00: This is an approximate round number but does not come from correct dimensional increases or exact multiplication.

Common Pitfalls:
Common mistakes include applying the percentage increases to the area directly instead of to each dimension, or adding 40% and 25% and assuming a 65% increase in area, which is incorrect. Another error is misplacing decimals while multiplying, especially with non-integer dimensions like 8.64 and 12.096. A step-by-step approach with clear intermediate results avoids most calculation errors.

Final Answer:
The cost of the second carpet is Rs. 4082.40.