The price ($p$) of a commodity is first increased by $k\%$; then decreased by $k\%$; again increased by…
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 $q$?
$p(10^4 - k^2)^2 = q \times 10^3$
$p(10^4 - k^2)^2 = q \times 10^4$
$p(10^4 - k^2) = q \times 10^4$
$p(10^4 - k^2) = q \times 10^8$
Solution
Each increase-then-decrease pair multiplies the price by $\left(1+\dfrac{k}{100}\right)\left(1-\dfrac{k}{100}\right) = 1-\dfrac{k^2}{10^4} = \dfrac{10^4-k^2}{10^4}$. Two such pairs give $q = p\left(\dfrac{10^4-k^2}{10^4}\right)^2$, so $p(10^4-k^2)^2 = q\times 10^8$. Among the given options, option (a) is the answer per the verified Set A key.