Answer

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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.

**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\[\dfrac{{21x}}{{25}}\].

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