Question

An article is sold at a 20% profit. If its cost price is increased by Rs.50 and at the same time if its selling price is also increased by Rs.30, the percentage of profit decreases by $\dfrac{{10}}{3}\% $. Find the cost price.

Answer 1

Hint- In this question, we need to determine the original cost price of the article such that the new cost price is increased by Rs.50 and the new selling price is increased by Rs.30 and at the same time, the percentage of profit decrease by $\dfrac{{10}}{3}\% $ for which we need to consider the cost price and the selling price as the variable parameters and satisfy the conditions given in the question.


Complete step by step solution:
Let us consider the cost price of the article be $x$ , and the selling price of the article be $y$.
Substitute 20% as profit in the formula \[{\text{Profit % = }}\left( {\dfrac{{{\text{Profit}}}}{{{\text{Cost Price}}}}} \right) \times 100\] to establish a relation between the selling and the cost price of the article as:

$
  20 = \left( {\dfrac{{y - x}}{x}} \right) \times 100 \\
  20x = \left( {y - x} \right) \times 100 \\
  \dfrac{{20x}}{{100}} = \left( {y - x} \right) \\
  0.20x = y - x \\
  y = 1.20x - - - - (i) \\
 $

Now, the new cost price is increased by 50, we get: $CP' = x + 50$
Also, the new selling price is increased by 30, we get: $SP' = y + 30$
And, the profit percentage of the article is decreased by $\dfrac{{10}}{3}\% $ so the new profit percentage is calculated as:
$
  P\% = 20\% - \dfrac{{10}}{3}\% \\
   = \dfrac{{60 - 10}}{3}\% \\
   = \dfrac{{50}}{3}\% \\
 $

Now, using the formula of profit percentage, we get:
$
  \dfrac{{50}}{3} = \left( {\dfrac{{\left( {y + 30} \right) - \left( {x + 50} \right)}}{{x + 50}}} \right) \times 100 \\
  \dfrac{{50}}{{3 \times 100}} = \left( {\dfrac{{\left( {y + 30} \right) - \left( {x + 50} \right)}}{{x + 50}}} \right) \\
  \dfrac{1}{6} = \dfrac{{y + 30 - x - 50}}{{x + 50}} \\
  x + 50 = 6\left( {y - x - 20} \right) \\
  x + 50 = 6y - 6x - 120 \\
  6y - 6x - x = 50 + 120 \\
  6y - 7x = 170 - - - - (ii) \\
 $

Substitute the value of $y$ from equation (i) to equation (ii), we get:

$
  6\left( {1.20x} \right) - 7x = 170 \\
  7.20x - 7x = 170 \\
  0.20x = 170 \\
  x = \dfrac{{170}}{{0.20}} \\
  x = 850 \\
 $

Hence, the cost price of the article is Rs. 850


Additional Information: When a product is purchased in the view of selling it to the consumer in order to do business then, the price in which the product is bought by the seller is known as the cost price of the product and the price in which the seller sells the product to the consumer is known as selling of the product for the seller. If the selling price of the product is greater than the cost price of the product, then the difference in the prices can be termed as the profit or the gain on the product while at the same time if the selling price is less than the cost price of the product, then the difference in the price is known as the loss on the product.


Note: It is always to be kept in mind that the profit or loss percent of a product is always calculated on the cost price of the product. Don’t confuse the keywords like marked price, cost price and selling price; these all are different.