Question

By selling a bicycle for Rs. $ 1638 $, Ishita loses $ 9\% $. At what price should she sell it to make a profit of $ 15\% ? $

Answer 1

Hint: Profit and loss percentages are always calculated according to cost price. 

If the selling price is more that the cost price, it will be considered as Profit. $\text{Profit = Selling price (S.P) - Cost price(C.P)}$

Similarly, If the selling price is less than the cost price that will be considered as Loss. $\text{Loss = Cost price(C.P) - Selling price(S.P)}$

In the given question, Ishita encountered a $9\%$ loss which indicates that she sold the bicycle for $9\%$ less than the cost price. Using this relation we will find the cost price of the bicycle. 

And we use this cost price to calculate the selling price where she makes a profit of $15\%$.

Formula used:

$\text{Loss%}=\dfrac{\text{Cost price - Selling price}}{\text{Cost price}} \times 100$

$\text{Profit%}=\dfrac{\text{Selling price - Cost price}}{\text{Cost price}} \times 100$

Complete step-by-step answer:
Given that, Selling Price, $ S.P. = 1638 $ Rs.
Loss $ = 9\% . $
Let us assume that the cost price of the bicycle be “x”
 $ loss\% = \dfrac{{C.P. - S.P.}}{{C.P.}} \times 100 $
Place the given values in the above expressions –
\[9 = \dfrac{{x - 1638}}{{C.P.}} \times 100\]
Simplify the above expression –
\[9 = \dfrac{{100x - 163800}}{x}\]
Perform Cross-multiplication in the above expression where the numerator is multiplied with the denominator of the opposite side and vice versa.
\[9x = 100x - 163800\]
Make like terms together. When you move any term from one side to another the sign of the term also changes. Positive terms become negative and the negative becomes positive.
\[100x - 9x = 163800\]
Simplify the above expression finding the subtraction of the term on the left hand side –
\[91x = 163800\]
Term multiplicative on one side of the equation moves to the opposite side then it goes in the denominator.
\[x = \dfrac{{163800}}{{91}}\]
Simplify the above expression –
$ \Rightarrow x = 1820 $ Rs.
Now, let us assume that the new selling price be “y”
Profit $ \% = 15\% $
Profit ${{\% = }}\dfrac{{SP - CP}}{{CP}} \times 100 $
Place the known values in the above equation –
$ {\text{15 = }}\dfrac{{y - 1820}}{{1820}} \times 100 $
Cross multiply the above expression –
$ 15(1820) = (y - 1820)100 $
Simplify the above expression by finding the product of the terms on both the sides of the equation –
$ 27300 = 100y - 182000 $
Make like terms together –
$ 27300 + 182000 = 100y $
Simplify the terms by adding –
$ 100y = 209300 $
Make required term the subject –
$ y = \dfrac{{209300}}{{100}} $
Simplify the above expression –
$ y = 2093 $ Rs.
Therefore, to make a profit of $ 15\% $, the selling price should be $Rs. 2093 $.

Note: Remember that cost price is always greater than the selling price when it is loss and when it is profit, selling price is always greater than the cost price. There will be one more price that is Marked price(M.P.) which is marked as MRP on the object. It is different from Selling price if the discount is given. The Discount is calculated w.r.t. Marked price.