Question

A shopkeeper purchases a sewing machine for Rs. 1200. He sells it at a discount of 10% and still makes a profit of 8%. Find (i) The selling price (ii) The marked price.

Answer 1

Hint:
Here, we are provided with the cost price and we have to find the selling price and the marked price. We are also provided with profit margins and discount margins. We will calculate the profit margin in rupees. We will obtain the value of the selling price using the formula and substituting the values in it. Similarly, we will obtain the marked price too by using the formula. For that, we will calculate the discount margin.
Formula Used: We will use the following formulas, to solve our question –
${\text{Total profit}=\text{Profit percentage}\times \text{Cost price}}$
${\text{Selling Price}} = {\text{Total Profit}} + {\text{Cost Price}}$
${\text{Total discount}} = {\text{Discount percentage}} \times {\text{Marked price}}$
${\text{Marked price}} = {\text{Selling price}} + {\text{Total discount}}$

Complete step by step solution:
We are given that the shopkeeper purchases a sewing machine for Rs. 1200. Thus, this is the cost price (C.P) given for the sewing machine.
Now we are supposed to find the selling price (S.P) of the machine.
The profit given to us is $8\% $.
Thus, the selling price can be given by –
${\text{Selling Price}} = {\text{Total Profit}} + {\text{Cost Price}}$……$\left( i \right)$
We will substitute $8\% $ for profit percentage and 1200 for cost price in the formula ${\text{Total profit}} = {\text{Profit percentage}} \times {\text{Cost price}}$.
${\text{Total profit}} = \dfrac{8}{{100}} \times 1200 = 8 \times 12$
Multiplying the terms, we get
${\text{Total profit}} = {\text{Rs}}.96$
Now, we will substitute 96 for total profit and 1200 for the cost price in equation $\left( i \right)$.
\[{\text{Selling price}} = {\text{96}} + 1{\text{2}}00 = {\text{Rs}}{\text{.1296}}\]
$\therefore $ The selling price of the sewing machine is Rs. 1296.
Since we have found the selling price, now let us find the marked price of the sewing machine.
The marked price is given by the formula –
${\text{Marked price}} = {\text{Selling price}} + {\text{Total discount}}$……$\left( {ii} \right)$
However, the discount is always found on the marked price and not on the selling price.
Thus, let us assume our marked price to be $x$. The discount percentage is given as $10\% $.
Now, we will substitute $10\% $ for discount percentage and $x$ for marked price in the formula ${\text{Total discount}} = {\text{Discount percentage}} \times {\text{Marked price}}$.
${\text{Total discount}} = \dfrac{{10}}{{100}} \times x = {\text{Rs}}{\text{.}}\dfrac{x}{{10}}$
We will substitute $x$for the marked price, 1296 for selling price, and $\dfrac{x}{{10}}$ for total discount in equation $\left( {ii} \right)$.
$x = 1296 + \dfrac{x}{{10}}$
Now as we can see that this is a linear equation in one variable, we will solve it easily now.
Firstly, we will gather all the like terms together from the equation $x = 1296 + \dfrac{x}{{10}}$.
Subtracting $\dfrac{x}{{10}}$ from both sides of the equation, we get
$x - \dfrac{x}{{10}} = 1296 + \dfrac{x}{{10}} - \dfrac{x}{{10}} \\
   \Rightarrow x - \dfrac{x}{{10}} = 1296 \\$
Now, we will simplify our equation by solving the like terms. To do that let us simplify the LHS of the equation first.
We will take the LCM of the denominators in the LHS. The denominators are 1 and 10. The LCM of 1 and 10 is 10.
We will multiply the denominators with suitable numbers to get the product 10. Subsequently, we will multiply the numerators with the same number too, to not alter the fractions.
$x - \dfrac{x}{{10}} = \dfrac{{x \times 10}}{{1 \times 10}} - \dfrac{{x \times 1}}{{10 \times 1}} \\
   = \dfrac{{10x}}{{10}} - \dfrac{x}{{10}} \\$
Now as we have obtained the similar denominators, we will simplify our numerator with the help of arithmetic operations.
$\dfrac{{10x}}{{10}} - \dfrac{x}{{10}} = \dfrac{{9x}}{{10}}$
Since we have simplified our LHS now, we will equate it to the RHS of the equation.
$\dfrac{{9x}}{{10}} = 1296$
Now, we will multiply both sides of the equation with 10 to eliminate 10 from the denominator of LHS.
$\dfrac{{9x}}{{10}} \times 10 = 1296 \times 10 \\
   \Rightarrow 9x = 12960 \\$
Now, we will divide both sides of our equation with 9 to obtain the value of $x$.
$\dfrac{{9x}}{9} = \dfrac{{12960}}{9} \\
 \Rightarrow x = 1440 \\$

$\therefore $ The marked price of our sewing machine is Rs. 1440.

Note:
The cost price of any item is the value at which the item is purchased. The selling price is the value at which the item is sold. If the selling price is greater than the cost price, then there is a profit. However, if vice versa occurs then it is a loss. Thus, the selling price is obtained after manipulating the profit or loss at the cost price. The marked price is the value that is imprinted on the item for the sale. But there can be certain discounts due to which the marked price is calculated on the selling price but manipulating the discounts.