Difficulty: Easy
Correct Answer: 6 : 5
Explanation:
Introduction / Context:
This is a ratio based mensuration problem that involves percentage increase in one dimension and comparison of areas of two shapes. You are given a relationship between the length and breadth of a rectangle and asked to compare its area with that of a square whose side equals the breadth of the rectangle. This is a direct application of area formulas and percentage interpretation.
Given Data / Assumptions:
Concept / Approach:
Use algebraic expressions in terms of b. Express both areas in terms of b so that b cancels when forming the ratio. The percentage increase of 20 percent translates into a factor of 1.2. This allows easy comparison between rectangle area and square area and simplifies to a ratio of simple integers.
Step-by-Step Solution:
Let breadth = b
Length = 20 percent more than b = b + 0.2b = 1.2b
Area of rectangle A_rect = length * breadth = 1.2b * b = 1.2b^2
Area of square A_sq = side^2 = b^2
Ratio of areas A_rect : A_sq = 1.2b^2 : b^2
Cancel b^2 to get 1.2 : 1
Convert 1.2 to fraction: 1.2 = 6/5
So ratio = 6/5 : 1 = 6 : 5
Verification / Alternative check:
You can choose a simple value for b, for example b = 5 units. Then length = 1.2 * 5 = 6 units. Rectangle area = 6 * 5 = 30. Square area = 5^2 = 25. The ratio 30 : 25 simplifies to 6 : 5, confirming the algebraic result.
Why Other Options Are Wrong:
2 : 1: This would imply rectangle area is double the square area, which is not correct for a 20 percent increase.
5 : 6: This inverts the correct ratio.
Data is inadequate: All necessary information is provided.
4 : 3: Does not match any consistent calculation using 20 percent increase.
Common Pitfalls:
Misinterpreting 20 percent more as just 0.2 instead of 1.2 factor.
Using breadth plus 20 units instead of 20 percent of breadth.
Incorrect simplification of decimal ratios.
Final Answer:
The required area ratio is 6 : 5
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