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A shopkeeper allows a discount of $10\% $ on the marked price. How much above the cost price must he mark his goods to gain $8\% $ ?
(A) $20\% $
(B) $100\% $
(C) $80\% $
(D) None of these

Answer
VerifiedVerified
570.9k+ views
Hint: As we know that the selling price can be represented as ${\text{Selling Price}} = \left( {1 - \dfrac{{Discount\% }}{{100}}} \right) \times {\text{ Marked Price}}$ and then we can represent the profit percentage using the equation ${\text{Profit % }} = \dfrac{{{\text{Selling Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} \times 100$. Using the first equation expresses selling price in terms of marked price and substitutes the known values in the second equation. Now transform the equation to get the required percentage.

Complete step-by-step answer:
Here in this problem, a shopkeeper is offering a discount of $10\% $ on the marked price of his goods. Now he wants to earn a profit of $8\% $ on his goods. And we need to find out how much more than the cost price he must mark the goods.
Before starting with the solution to this problem we should understand the concepts of percentage. In mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, , although the abbreviations ‘pct.’, ‘pct’ and sometimes ‘pc’ are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement.
The selling of the goods will be the amount paid by the customer after taking away the discount from the marked price. The profit is defined as the difference in selling price and cost price, where the selling price should always be greater than the cost price.
$ \Rightarrow {\text{Selling Price}} = \left( {1 - \dfrac{{Discount\% }}{{100}}} \right) \times {\text{ Marked Price}}$
And also
$ \Rightarrow {\text{Profit}} = {\text{Selling Price}} - {\text{Cost Price}}$ and ${\text{Profit % }} = \dfrac{{{\text{Selling Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} \times 100$
From the above equation, using the given value of the discount, we get:
$ \Rightarrow {\text{Selling Price}} = \left( {1 - \dfrac{{10}}{{100}}} \right) \times {\text{ Marked Price}} = \dfrac{9}{{10}} \times {\text{Marked Price}} = 0.9 \times {\text{Marked Price}}$
Now let’s use this value in the above equation of the profit percentage to obtain a relation between marked price and cost price.
$ \Rightarrow 8 = \dfrac{{{\text{Selling Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} \times 100 = \left( {\dfrac{{SP}}{{CP}} - 1} \right) \times 100 = \left( {\dfrac{{0.9 \times {\text{Marked Price}}}}{{{\text{Cost Price}}}} - 1} \right) \times 100$
Now, this can be further solved as:
$ \Rightarrow 8 = \left( {\dfrac{{0.9 \times {\text{Marked Price}}}}{{{\text{Cost Price}}}} - 1} \right) \times 100 \Rightarrow \dfrac{8}{{100}} + 1 = \dfrac{{0.9 \times {\text{Marked Price}}}}{{{\text{Cost Price}}}}$
This can be simplified and rewritten as:
$ \Rightarrow \dfrac{8}{{100}} + 1 = \dfrac{{0.9 \times {\text{Marked Price}}}}{{{\text{Cost Price}}}} \Rightarrow \dfrac{{0.9 \times {\text{Marked Price}}}}{{{\text{Cost Price}}}} = \dfrac{{108}}{{100}}$
Now we can express the ratio of marked price and cost price as:
$ \Rightarrow \dfrac{{{\text{Marked Price}}}}{{{\text{Cost Price}}}} = \dfrac{{108}}{{100}} \times \dfrac{1}{{0.9}} = \dfrac{{12}}{{10}} = 1.2$
Now let’s subtract one from both sides of this equation, we will get:
$ \Rightarrow \dfrac{{{\text{Marked Price}}}}{{{\text{Cost Price}}}} - 1 = 1.2 - 1 \Rightarrow \dfrac{{{\text{Marked Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} = 0.2$
If we multiply both sides with $100$ then we will get the percentage of how more is marked with respect to the cost price. Therefore, on multiplying $100$ both sides, we get:
$ \Rightarrow \dfrac{{{\text{Marked Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} \times 100 = 0.2 \times 100 = 20\% $
Therefore, we get that the market price of $20\% $ should be more than the cost price for the shopkeeper to make a profit of $8\% $.
Hence, the option (A) is the correct answer.

Note: In questions like this, the use of fundamental concepts of profit and loss plays a crucial role in the solution. An alternate approach can be to obtain an expression for what the question requires, i.e. $\dfrac{{{\text{Marked Price}} - {\text{Cost Price}}}}{{{\text{Cost Price}}}} \times 100$ , which can be expressed as $\left( {\dfrac{{{\text{Marked Price}}}}{{{\text{Cost Price}}}} - 1} \right) \times 100$. Now for evaluating this required expression you just need the ratio of marked price and cost price.
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