While measuring the sides of a rectangle, one side is measured 5% more than its actual length and the other side is measured 4% less than its actual length. Using these incorrect measurements, what is the percentage error in the calculated area of the rectangle?

Introduction:
This problem is about percentage error in the area of a rectangle when both dimensions are not measured exactly. In aptitude tests, it is common to see questions where one side is measured in excess and the other in deficit. The goal is to understand how these small percentage errors combine and affect the final computed area. Instead of recomputing exact numeric values every time, there is a neat percentage error shortcut that helps you quickly reach the answer.

Given Data / Assumptions:

• One side of the rectangle is measured 5% more than the true length (that is, +5% error). • The other side is measured 4% less than the true length (that is, -4% error). • Area of a rectangle = length * breadth. • Errors are small, so percentage error combination rules can be applied accurately.

Concept / Approach:
When two quantities with small percentage errors a% and b% are multiplied, the approximate percentage error in the product is given by a + b + (a*b)/100. Here, the product is the area (length * breadth). The positive error increases the area, while the negative error reduces it. The interaction term (a*b)/100 accounts for the combined effect of both errors happening simultaneously, and it is very important for accurate results.

Step-by-Step Solution:
Step 1: Let the true length be L and the true breadth be B. Step 2: Measured length = L * (1 + 5/100) = 1.05L. Step 3: Measured breadth = B * (1 - 4/100) = 0.96B. Step 4: Measured area = 1.05L * 0.96B = 1.008L * B. Step 5: This means the measured area is 1.008 times the true area, so the percentage error in area is 0.8%. Step 6: Using the shortcut: a = +5, b = -4, error = a + b + (a*b)/100 = 5 - 4 + (5 * -4)/100 = 1 - 0.2 = 0.8%.

Verification / Alternative check:
Assume a simple rectangle where the true length and breadth are both 100 units. Then the true area = 100 * 100 = 10000 square units. With errors, measured length = 105 and measured breadth = 96. The measured area = 105 * 96 = 10080 square units. The increase in area = 10080 - 10000 = 80. Percentage error in area = (80 / 10000) * 100 = 0.8%, which exactly matches the error obtained using the shortcut formula.

Why Other Options Are Wrong:
Option 0.7%: This slightly underestimates the interaction term and does not match exact calculation. Option 0.9%: This overestimates the combined effect of the percentage errors. Option 0.3%: This assumes almost complete cancellation of the errors and ignores the multiplicative effect. Option 1.2%: This would arise if both sides had positive errors, not when one is in excess and the other is in deficit.

Common Pitfalls:
Candidates often simply add 5% and -4% and conclude that the error is 1%, without considering the product term (a*b)/100. Another common mistake is to ignore the sign of the deficit error or to apply the formula to sums or differences instead of products. Remember that the shortcut formula applies only when we are multiplying quantities, such as when calculating area, volume, or combined value.

Final Answer:
The percentage error in the calculated area is 0.8%.