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:

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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.

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$\dfrac{{21x}}{{25}}$.