Successive percentage increase-decrease relation
Question
The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q?
Options
p(10^4 - k^2)^2 = q × 10^8
p(10^4 - k^2)^2 = q × 10^4
p(10^4 - k^2) = q × 10^4
p(10^4 - k^2) = q × 10^8
Explanation
Applying successive percentage changes, the final price q is expressed as: q = p × (1 + k/100) × (1 - k/100) × (1 + k/100) × (1 - k/100) Using the algebraic identity (a+b)(a-b) = a² - b², we group the pairs: q = p × (1 - k²/10000) × (1 - k²/10000) ⟨MATH⟩q = p × \left(10000 - k²/10000\right)²⟨/MATH⟩ q = p × (10^4 - k²)²/(10^4)² q = p × (10^4 - k²)²/10^8 Cross-multiplying yields: q × 10^8 = p(10^4 - k²)².
Answer: (a).
Question details
Year
2025
Paper
CSAT
Question
Q30
Section
Numerical Ability
Sub-topic
Percentages
Type
Algebra & equations
Difficulty
Hard
Source hint
Percentage algebra
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