Question

A shopkeeper marks his goods at 40% above the cost price and allows a discount of 40% on the marked price. His loss or gain is:

Answer 1

Hint: The shopkeeper first marks his goods at 40% above the cost price so first we will find the Marked price then he allows a discount of 40% on the marked price and we need to find his loss or gain after that.

Complete step-by-step answer:
It is given that the shopkeeper marks his goods at 40% above the cost price.
Let, cost price (CP) of goods =Rs. \[x\]
Since the market price (MP) would give him 40% profit.
Thus \[MP = CP{\rm{ }} + {\rm{ }}Profit\% \times CP\]
\[MP = {\rm{ }}CP + 40\% \times CP\]
By substituting the cost price in the above equation we get,
The marked price of the goods\[ = x + 40\% \times x\]
Here by using the percentage formula we get,
\[x + 40\% \times x\]\[ = x + \dfrac{{40}}{{100}} \times x\]
On solving the above terms we get,
Marked price\[ = x + \dfrac{{2x}}{5}\]\[ = \dfrac{{7x}}{5}\]
Hence the marked price is \[\dfrac{{7x}}{5}\]
Also given that, he allows a discount of 40% on the marked price.
So, the Selling price = marked price - 40% ×marked price
\[SP = MP - 40\% of{\rm{ }}MP\]
Here by substituting the marked price we get,
The selling price of the goods\[ = \dfrac{{7x}}{5} - \dfrac{{40}}{{100}} \times \dfrac{{7x}}{5}\]
Let us solve the above equation,
\[ = \dfrac{{7x}}{5} - \dfrac{2}{5} \times \dfrac{{7x}}{5}\]
\[ = \dfrac{{35x - 14x}}{{25}}\]
The selling price of the goods\[ = \dfrac{{21x}}{{25}}\]
Since \[SP = \dfrac{{21x}}{{25}} < x\], the cost price is greater than selling price
Thus he is getting loss after discounting 40%,
Now we have to find the percentage of loss,
We know that, \[Loss = cost{\text{ }}price{\text{ }}-{\text{ }}selling{\text{ }}price\]
Let us substitute the cost and selling price in the above equation we get,
Loss\[ = x - \dfrac{{21x}}{{25}}\]
On solving the above equation we get,
Loss \[ = \dfrac{{25x - 21x}}{{25}}\]\[ = \dfrac{{4x}}{{25}}\]
The formula to find the percentage of loss is
\[Loss\% {\rm{ }} = \dfrac{{Loss}}{{CP}} \times 100\]
Percentage of loss\[ = \dfrac{{\dfrac{{4x}}{{25}}}}{x} \times 100\]
\[ = \dfrac{{4x}}{{25}} \times \dfrac{1}{x} \times 100\]
On solving the above equation we get,
The percentage of loss \[ = 4 \times 4 = 16\% \]
Hence, the shopkeeper has a loss of 16%

Note: The loss percentage is given by, \[loss\% = \dfrac{{SP - CP}}{{CP}} \times 100\]
We have used the fact that if the cost price is less than the selling price then there is a profit. If the cost price is greater than the selling price then there is a loss. In the problem we have lost because the cost price x is less than the selling price\[\dfrac{{21x}}{{25}}\].