Question

A man marks his goods at a price that would give him a 20% profit. He sells $\dfrac{3}{5}$ of the goods at the marked price and sells the remaining at 20% discount. Find his gain percent on the whole transaction.

Answer 1

Hint: Selling Price = Cost price + Profit

Selling price = cost price + profit%×cost price

Profit% =$\dfrac{{profit}}{{CP}} \times 100$ 

Selling price =Marked price- Discount%×marked price

The man first marked the price at 20 % profit so first, we will find the marked price then $\dfrac{3}{5}$ of the goods are sold at the marked price and we get that by multiplying $\dfrac{3}{5}$ and the marked price.

Then we will find the discounted marked price and multiply $\left(1-\dfrac{3}{5}\right)$ with it.

We will get the Selling price by adding the two selling amounts. Then by the above formula, we can calculate the gain percentage.

Complete step by step solution:

Let, cost price (CP) of goods be Rs. x

Since the marked price (MP) would give him 20% profit.

Thus $MP = CP{\rm{ }} + {\rm{ }}Profit\%  \times CP$ 

$MP = {\rm{ }}CP + 20\%  \times CP$ 

That is 

$MP = x + \dfrac{{20}}{{100}} \times x$ 

By solving the above equation we get,

$MP = x + \dfrac{x}{5} = \dfrac{{5x + x}}{5}$

$Mp = \dfrac{{6x}}{5}$

Given that, he sells $\dfrac{3}{5}$ of the goods at the market price.

So Selling price of $\dfrac{3}{5}$ of the goods=$\dfrac{3}{5} \times MP$

$ = \dfrac{3}{5} \times \dfrac{{6x}}{5}$

Selling price of $\dfrac{3}{5}$ of the goods $ = \dfrac{{18x}}{{25}}$ 

Remaining $\left( {1 - \dfrac{3}{5}} \right)$=$\dfrac{2}{5}$ of the goods are sold at 20% discount.

So the selling price of $\dfrac{2}{5}$ of the goods is found the following formula,

Selling price of remaining goods is =$\dfrac{2}{5}\times \text{MP} -20\%$ of $(\dfrac{2}{5} \times \text{MP})$

$ = \dfrac{2}{5} \times \dfrac{{6x}}{5} - \left\{\dfrac{{20}}{{100}} \times \left(\dfrac{2}{5} \times \dfrac{{6x}}{5}\right)\right\}$

On solving this we get,

$ = \dfrac{{12x}}{{25}} - \dfrac{{12x}}{{125}}$

$ = \dfrac{{60x - 12x}}{{125}}$ 

The selling price of remaining goods$ = \dfrac{{48x}}{{125}}$

So the total selling price ($SP$) is equal to the sum of both the selling prices,

That is Selling price$ = \dfrac{{18x}}{{25}} + \dfrac{{48x}}{{125}}$

$ = \dfrac{{90x + 48x}}{{125}}$

Selling price $ = \dfrac{{138x}}{{125}}$ 

 We know that, $\text{Profit = selling price – cost price} =SP – CP$

Hence, profit$ = \dfrac{{138x}}{{125}} - x$ 

On solving the above equation we get,

Profit $ = \dfrac{{138x - 125x}}{{125}}$

That is Profit = $\dfrac{{13x}}{{125}}$.

We know that $\{\rm\{Profit\%  = \}\}\dfrac{{SP - CP}}{{CP}} \times 100 = \dfrac{{{\rm{profit}}}}{{CP}} \times 100$ as we know that $\left(\text{profit} =  SP -  CP \right)$

Hence profit percentage$ = \dfrac{{\dfrac{{13x}}{{125}}}}{x} \times 100$

On solving the above we get,

$ = \dfrac{{13x}}{{125}} \times \dfrac{1}{x} \times 100$

Profit percentage$ = 10.4\% $ 

$\therefore$ The gain percent on the whole transaction is 10.4%

Note:

We should be careful while calculating the value of the goods sold with a discount to subtract the percentage of discount from the marked price.

Also here while finding the profit percentage we use the fact that division of two fractions is equal to the multiplication of the numerator fraction with the reciprocal of the denominator fraction. That is $\dfrac{{\dfrac{a}{b}}}{{\dfrac{c}{d}}} = \dfrac{a}{b} \times \dfrac{d}{c}$.