Question

The percentage profit earned by selling an article for Rs.1920 is equal to the percentage loss incurred by selling the same article for Rs.1280. At what price should the article be sold to make \[25\% \] profit?
A. Rs.2000
B. Rs.2200
C. Rs.2400
D. Data inadequate

Answer 1

Hint: We use the formula of percentage of profit and percentage of loss and equate both the percentages. Take the given selling prices on respective sides of the equation and assume the cost price of the article as a variable. Calculate the cost price of an article. Calculate \[25\% \] of the cost price obtained and write the new selling price by adding the profit to the cost price.
* If CP is the cost price and SP is the selling price then loss and profit percentages are given by
Profit percentage \[ = \](Profit\[/\]CP)\[ \times 100\] i.e. \[ = \](SP\[ - \]CP\[/\]CP)\[ \times 100\]
Loss percentage \[ = \](Loss\[/\]CP)\[ \times 100\]i.e. \[ = \](CP\[ - \]SP\[/\]CP)\[ \times 100\]

Complete step-by-step solution:
Let the initial cost price, CP of an article be Rs.x
\[ \Rightarrow \]CP \[ = x\]................… (1)
We form two equations each for a case of profit percentage and for loss percentage.
Profit percentage:
Since we are given profit is earned by selling an article for Rs.1920
So, the selling price of article when profit is earned is Rs.1920
\[ \Rightarrow \]SP\[ = 1920\].....................… (2)
Use formula of profit percentage to write the value of percentage
\[\because {\text{Profit percentage}}\] \[ = \]\[ {\dfrac{S.P - C.P}{C.P}} \times 100\]
Substitute the value of CP from equation (1) and SP from equation (2) in the formula
\[ \Rightarrow \]Profit percentage \[ = \dfrac{{1920 - x}}{x} \times 100\] ……………..… (3)
Loss percentage:
Since we are given loss is earned by selling an article for Rs.1280
So, the selling price of article when loss is earned is Rs.1280
\[ \Rightarrow \]SP\[ = 1280\].................… (4)
Use formula of loss percentage to write the value of percentage
\[\because {\text{Loss percentage}}\] \[ = \]\[ {\dfrac{C.P - S.P}{C.P}} \times 100\]
Substitute the value of CP from equation (1) and SP from equation (4) in the formula
\[ \Rightarrow \]Loss percentage \[ = \dfrac{{x - 1280}}{x} \times 100\] ……...… (5)
Now we know the percentage of profit is equal to the percentage of loss, equate the percentages obtained from equations (3) and (5)
\[ \Rightarrow \dfrac{{1920 - x}}{x} \times 100 = \dfrac{{x - 1280}}{x} \times 100\]
Cancel same factors from both sides of the equation i.e. ‘x’ in the denominator and 100 in numerator
\[ \Rightarrow 1920 - x = x - 1280\]
Shift all constant values to LHS of the equation
\[ \Rightarrow 1920 + 1280 = x + x\]
\[ \Rightarrow 3200 = 2x\]
Divide both sides of the equation by 2
\[ \Rightarrow \dfrac{{3200}}{2} = \dfrac{{2x}}{2}\]
Cancel same factors from numerator and denominator on both sides of the equation
\[ \Rightarrow 1600 = x\]
Cost price of the article, CP is Rs.1600
Now we have to find the selling price of the article such that there is a profit of \[25\% \]
So, we calculate the \[25\% \] of the cost price and sell the article with the selling price equal to \[25\% \]more of the cost price
\[ \Rightarrow \]New selling price \[ = \]Cost Price \[ + 25\% \] Cost price
\[ \Rightarrow \]New selling price \[ = 1600 + 25\% \times 1600\]
Write percentage in fraction form
\[ \Rightarrow \]New selling price \[ = 1600 + \dfrac{{25}}{{100}} \times 1600\]
Cancel same factors from numerator and denominator in RHS
\[ \Rightarrow \]New selling price \[ = 1600 + 25 \times 16\]
\[ \Rightarrow \]New selling price \[ = 1600 + 400\]
\[ \Rightarrow \]New selling price \[ = 2000\]
\[\therefore \]Price at which article should be sold to make \[25\% \] profit is Rs.2000

\[\therefore \]Option A is correct

Note: Many students make mistakes in calculating the new selling price as they use the formula of profit percentage again which is wrong, here in the end we have to take profit percentage with respect to the cost price of the item. Also, keep in mind when converting percentage into fraction we always divide by 100 and when converting a fraction into percentage we always multiply by 100.