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 ?

Let the original price be p. When a price is increased by k% and then decreased by k%, the net effect is a decrease. The new price after one such cycle (increase by k%, then decrease by k%) is given by: Price' = p * (1 + k/100) * (1 - k/100) Price' = p * (1 - (k/100)^2) Price' = p * (1 - k^2/10000) This process is repeated twice. First cycle: p becomes p * (1 - k^2/10000) Second cycle: The price from the first cycle further undergoes the same transformation. So, the new price q is: q = [p * (1 - k^2/10000)] * (1 - k^2/10000) q = p * (1 - k^2/10000)^2 Now, let's simplify the term inside the parenthesis: 1 - k^2/10000 = (10000 - k^2) / 10000 Substitute this back into the equation for q: q = p * [(10000 - k^2) / 10000]^2 q = p * (10000 - k^2)^2 / (10000)^2 q = p * (10^4 - k^2)^2 / 10^8 To match the options, we rearrange the equation: q * 10^8 = p * (10^4 - k^2)^2 p * (10^4 - k^2)^2 = q * 10^8 Analyzing the options: A) p(10^4 - k^2)^2 = q * 10^8 - This matches our derived relation. B) p(10^4 - k^2)^2 = q * 10^4 - Incorrect power of 10 on the right side. C) p(10^4 - k^2) = q * 10^4 - Incorrectly missing the square on the left side. D) p(10^4 - k^2) = q * 10^8 - Incorrectly missing the square on the left side. The final answer is A