Each question below compares two quantities.\n\nQuantity I: The length of a rectangle is increased by 20% while its breadth is decreased by 10%.\nFind the percentage change in the area of the rectangle.\n\nQuantity II: The base of a triangle is increased by 30% while its height is decreased by 20%.\nFind the percentage change in the area of the triangle.\n\nCompare Quantity I and Quantity II.

Introduction / Context:
This is a quantity comparison problem involving percentage changes in geometric dimensions. You must understand how changes in length and breadth or in base and height affect areas of a rectangle and a triangle. The question checks whether you can convert successive percentage changes into multiplicative factors and then compare the net percentage changes in areas.

Given Data / Assumptions:

• Quantity I: Rectangle with length increased by 20% and breadth decreased by 10%.
• Quantity II: Triangle with base increased by 30% and height decreased by 20%.
• Original shapes have some initial dimensions; only relative percentage changes matter, not the actual starting sizes.
• Area of rectangle = length * breadth.
• Area of triangle = (1/2) * base * height.
• We must compare the final percentage change in area for both cases.

Concept / Approach:
When a dimension changes by a percentage, the new dimension is obtained by multiplying the original by a factor. For example, a 20% increase corresponds to multiplying by 1.20, and a 10% decrease corresponds to multiplying by 0.90. For the area, we multiply all relevant dimension factors together. Because each shape’s original area is scaled by a product of these factors, we can find the overall percentage change in area by comparing the new factor to 1 and then convert it to a percentage.

Step-by-Step Solution:
Step 1: For Quantity I (rectangle), let original length = L and breadth = B. Original area = L * B. Step 2: Length increases by 20%, so new length = 1.20L. Breadth decreases by 10%, so new breadth = 0.90B. Step 3: New area of rectangle = 1.20L * 0.90B = 1.08LB. Step 4: Factor for area change in rectangle = 1.08, meaning an 8% increase in area. Step 5: For Quantity II (triangle), let original base = b and height = h. Original area = (1/2) * b * h. Step 6: Base increases by 30%, so new base = 1.30b. Height decreases by 20%, so new height = 0.80h. Step 7: New area of triangle = (1/2) * 1.30b * 0.80h = (1/2) * 1.04bh. Step 8: Factor for area change in triangle = 1.04, meaning a 4% increase in area. Step 9: Compare: 8% increase (Quantity I) versus 4% increase (Quantity II). So Quantity I > Quantity II.

Verification / Alternative check:
Assign simple numerical values, for example L = B = 10 for the rectangle, b = h = 10 for the triangle. Rectangle: original area = 100. New length = 12, new breadth = 9, new area = 108, which is 8% more than 100. Triangle: original area = (1/2) * 10 * 10 = 50. New base = 13, new height = 8, new area = (1/2) * 13 * 8 = 52, which is 4% more than 50. The numerical results confirm that Quantity I has a larger percentage increase than Quantity II.

Why Other Options Are Wrong:
Quantity I = Quantity II: This would require both areas to change by the same percentage, which is not the case (8% vs 4%).
Quantity I < Quantity II: This would imply the triangle’s area increases more, which is contradicted by the calculations.
Quantity I ≥ Quantity II, Quantity I ≤ Quantity II: These inclusive options are either too weak or incorrect. The strict inequality Quantity I > Quantity II is the accurate relation.

Common Pitfalls:
One common mistake is to add or subtract percentages linearly without converting them into multiplicative factors, for example, thinking area change is 20% - 10% = 10% for the rectangle. Another error is forgetting that the factor 1/2 in the triangle’s area formula is common to both original and new areas and therefore cancels out. Always treat percentage changes as multipliers and avoid intuitive but incorrect shortcut additions.

Final Answer:
Quantity I is greater than Quantity II; that is, Quantity I > Quantity II.