In measuring the sides of a rectangle, one side is taken 20% in excess of its actual length and the other side is taken 10% less than its actual length; what is the percentage error in the calculated area compared with the true area?

Introduction / Context:
This question deals with the effect of errors in measurements on the calculated area of a rectangle. Small percentage errors in each dimension can combine to produce a net percentage error in the area that is not simply the sum or difference of those percentages. Such problems commonly appear in the error and approximation section of quantitative aptitude and help build an understanding of how multiplication of dimensions affects area estimates.

Given Data / Assumptions:

• Let the true length of one side be L.
• Let the true breadth of the rectangle be B.
• Area of the rectangle is L * B.
• In measurement, length is overestimated by 20%, so measured length is 1.2 * L.
• In measurement, breadth is underestimated by 10%, so measured breadth is 0.9 * B.
• We must find the percentage error in the area calculated using these wrong measurements.

Concept / Approach:
The true area is L * B. The wrong measured area is (1.2 * L) * (0.9 * B). The ratio of wrong area to true area gives the factor by which the area is overestimated or underestimated. Percentage error can be found by subtracting 1 from this ratio and multiplying by 100. If the ratio is greater than 1, the error is excess; if the ratio is less than 1, the error is deficit.

Step-by-Step Solution:
True area A is L * B.Measured length is 1.2 * L due to 20% excess.Measured breadth is 0.9 * B due to 10% deficit.Measured area Am is (1.2 * L) * (0.9 * B) = 1.2 * 0.9 * L * B.Compute multiplier: 1.2 * 0.9 = 1.08.So measured area Am = 1.08 * L * B = 1.08 * A.Thus the measured area is 1.08 times the true area.Percentage error = (Am - A) / A * 100 = (1.08A - A) / A * 100 = 0.08 * 100 = 8%.Since the factor is greater than 1, the error is 8% in excess.

Verification / Alternative check:
Take an easy numeric example. Suppose true length L is 10 units and true breadth B is 10 units. True area is 10 * 10 = 100 square units. Measured length would be 20% greater, so 12 units. Measured breadth would be 10% less, so 9 units. Measured area becomes 12 * 9 = 108 square units. The difference is 108 - 100 = 8 square units. Percentage error is 8 / 100 * 100 = 8%, and since the measured area is larger, this is an 8% excess. This simple example confirms the general calculation.

Why Other Options Are Wrong:
9% deficit and 11% deficit both assume the measured area is smaller than the true area, which contradicts the computed factor of 1.08.

12% excess would require the multiplier to be 1.12 rather than 1.08, which is not the result of multiplying 1.2 and 0.9.

Common Pitfalls:
A common error is to simply add the percentage changes, for example 20% minus 10%, and conclude that the net error is 10%, which is wrong. Another mistake is to treat the wrong measurements separately and not realize that area is the product of both length and breadth. The correct method is always to convert percentage changes into multiplicative factors, multiply them, and then interpret the final factor relative to 1 to determine both the magnitude and direction of the error.

Final Answer:
The calculated area is in 8% excess compared with the true area.