Question

The price of an article was increased by r%. Later the new price was decreased by r%. If the latest price was Rs.1, then the original price was
A) Rs.1
B) Rs.$ \left( {\dfrac{{1 - {r^2}}}{{100}}} \right) $
C) Rs.$ \dfrac{{\sqrt {1 - {r^2}} }}{{100}} $
D) Rs.$ \left( {\dfrac{{10000}}{{10000 - {r^2}}}} \right) $

Answer 1

Hint: This is a simple percentage question. To solve the above question, first we will obtain the increased price of the article assuming the original price of the article as x. Then we will derive the new decreased price of the article in terms of x. As the new price was 1, we will get an equation. Solving that we will get the original price.

Complete step-by-step answer:
Let the original price of article was = Rs. x
According to the question, the price of an article was increased by r%.
Hence, total price after r% increase $ = (100 + r)\% \times x $
I.e. the total price of article after r% increase $ = \dfrac{{(100 + r)}}{{100}} \times x $
Again it is given in the question that the new price was decreased by r%.
Hence, new price after r% decrease $ = (100 - r)\% \times \left[ {\dfrac{{(100 + r)}}{{100}}x} \right] $
I.e. the total price of article after r% increase $ = \dfrac{{(100 - r)}}{{100}} \times \dfrac{{(100 + r)}}{{100}} \times x $
It is given in the question that the latest price was Rs.1.
 $ \therefore $ $ \dfrac{{(100 - r)}}{{100}} \times \dfrac{{(100 + r)}}{{100}} \times x = 1 $
Simplifying the above equation using the formula $ (a + b)(a - b) = {a^2} - {b^2} $ ,
 $ \dfrac{{{{(100)}^2} - {{(r)}^2}}}{{10000}} \times x = 1 $
Taking denominator of left hand side to the right hand side we get,
 $ {(100)^2} - {(r)^2} \times x = 10000 $
Simplifying the above equation we get the value of x,
 $ x = \dfrac{{10000}}{{{{(100)}^2} - {{(r)}^2}}} $
 $ \therefore $ $ x = \dfrac{{10000}}{{10000 - {r^2}}} $
 $ \therefore $ The original price of the article was Rs. $ \dfrac{{10000}}{{10000 - {r^2}}} $ .

Option D is the correct answer.

Note: Many times students make the mistake of not writing the percentage with respect to the price, keep in mind in these types of questions we take percentage with reference to one price, so we add or subtract accordingly. Also, add the value of percentage whenever it is stated that there was an increased percentage and subtract the percentage whenever it is stated that there was decreased percentage.