Question

A cloth merchant sold half of his cloth at 20% profit, half of the remaining at 20% loss and the rest was sold at the cost price. In the total transaction, his gain or loss will be:
(a) Neither loss nor gain
(b) 5% loss
(c) 5% gain
(d) 10% gain

Answer 1

Hint: First, we will take a variable for the cost price of the full piece of cloth. Then we will find the cost price of half the cloth and find a selling price at 20% gain. Then we will find the cost price of half of the remaining half of the cloth and find the selling price of this piece at 20% loss. Then we will find the cost price of the remaining cloth. The selling price of this piece will be the same as the cost price of this piece. Then we will add the selling prices to find the selling price and compare it with the cost price of the complete piece to know whether the merchant sold the cloth in profit or loss. Then we will find the change in percentage by the relation $\dfrac{\text{sp}-\text{cp}}{\text{cp}}\times 100$, where sp is the selling price of the full piece of cloth and cp is cost price of full piece of cloth.

Complete step by step answer:
Let x be the cost price of all the cloth with the cloth merchant.
The cost price of the half of his cloth will be $\dfrac{\text{x}}{\text{2}}$= 0.5x. It is given that he sold half the cloth at 20% profit. Thus, 20% of 0.5x will be $\dfrac{20}{100}\times $0.5x = 0.1x
Therefore, the selling price of half of the cloth sold at 20% is (cost price + 20% of cost price) = (0.5x + 0.1x) = 0.6x……………………….…(1)
The cost price of half of the remaining cloth will be $\dfrac{0.5\text{x}}{2}$ = 0.25x. It is given that he sells this piece at 20% loss. Thus, 20% of 0.25x will be $\dfrac{20}{100}\times $0.25x = 0.05x
Therefore, the selling price of half of the remaining half cloth sold at 20% loss is (cost price – 20% of cost price) = (0.25x – 0.05x) = 0.2x………………………………………………...……(2)
Then, he sold the remaining cloth at cost price. The cost price of the remaining cloth will be (cost price of the total cloth – (cost price of cloth sold at 20% profit + cost price of cloth sold at 20% loss))
Therefore, the cost price of remaining cloth = (x – (0.5x + 0.25x)) = (x – 0.75x) = 0.25x
Since, he sold this part of the cloth at cost price, selling price = cost price = 0.25x………….…(3)
The total selling price of the complete cloth = selling price of cloth sold at 20% profit + selling price of cloth sold at 20% loss + selling price of cloth sold at cost price.
Therefore, selling price of complete cloth = 0.6x + 0.2x + 0.25x = 1.05x. The selling price is more than cost price and thus the merchant is in profit.
We can find the percentage change by the relation pp = $\dfrac{\text{sp}-\text{cp}}{\text{cp}}\times 100$, where pp is the percentage profit, sp is selling price and cp is the cost price of the cloth.
pp = $\dfrac{1.05\text{x}-\text{x}}{\text{x}}\times 100=5%$.
Therefore, the merchant earned a profit of 5%.
Hence, option (c) is the correct option.

Note: Other word for profit is gain. Profit and loss questions make major use of percentage and thus students are advised to have a good hold on concepts of percentage. By comparing the quantities we can solve the given problem.