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If the length of a rectangle is increased by 40%, by what percentage should the breadth be reduced to ensure that the area remains unchanged (round up to two decimal places)?
A) 20.50%
B) 22.22%
C) 30.51%
D) 28.57%
28.57%
Given: Length of the rectangle is increased by 40% (x = 40).
Length of the rectangle is increased by 40% (x = 40).
PercentageReduction=(100+xx×100)
Use the fraction method: 40% increase is +2/5. To keep area constant, we need a -2/(5+2) reduction. Since 2/7 is 28.57%, this is the answer.
Many students mistakenly apply a 40% reduction, failing to realize that a percentage increase on one dimension requires a different percentage decrease on the other to maintain constant product.
Define Relationship
If length increases by x%, breadth must be reduced by y% such that (1+x/100)×(1−y/100)=1.
y=100+xx×100
Substitute Values
Given x=40, substitute this into the formula.
y=100+4040×100
Final Calculation
Calculate the resulting fraction and percentage.
y=14040×100=72×100≈28.57%
D is correct because applying the reduction formula gives 28.57%, which maintains the constant area.
This concept of inverse proportional change is identical to calculations in Price-Consumption problems (e.g., if price increases, how much consumption must drop) and Speed-Time relations.