Profit, loss and discount

A shop­keeper buys goods at one price and sells them at another. If he sells for more than he paid, he makes a profit. If he sells for less, he makes a loss. Many shops also print a price on the item and then give a dis­count on it. In this les­son you will learn the words and for­mu­las for all of this. Every prob­lem in the next three lessons uses them.

Three prices#

Cost price (C.P.). The price at which an arti­cle is bought.

Sell­ing price (S.P.). The price at which an arti­cle is sold.

Marked price (M.P.). The price that is printed or writ­ten on the arti­cle. It is also called the list price, printed price, cat­a­logue price, quo­ta­tion price or invoice price.

Profit#

If the sell­ing price is more than the cost price, there is a profit (also called gain).

$$\text{profit} = \text{S.P.} - \text{C.P.}$$

$$\displaystyle \text{profit}\% = \left(\frac{\text{profit}}{\text{C.P.}} \times 100\right)\%$$

Profit per cent is always worked out on the cost price.

Sup­pose a pen is bought for ₹10 and sold for ₹12. The profit is $12 - 10 = 2$ rupees. On ₹10 the profit is ₹2, so on ₹100 it would be $\displaystyle \frac{2}{10} \times 100 = 20$ rupees. The profit is 20%.

Sell­ing price from the profit per cent#

The sell­ing price is the cost price plus the profit:

$$\displaystyle \begin{aligned} \text{S.P.} &= \text{C.P.} + \text{profit} \\ &= \text{C.P.} + \frac{\text{profit}\%}{100} \times \text{C.P.} \\ &= \left(1 + \frac{\text{profit}\%}{100}\right)\text{C.P.} \end{aligned}$$

$$\displaystyle \text{S.P.} = \left(\frac{100 + \text{profit}\%}{100}\right)\text{C.P.} \qquad \text{C.P.} = \left(\frac{100}{100 + \text{profit}\%}\right)\text{S.P.}$$

Loss#

If the sell­ing price is less than the cost price, there is a loss.

$$\text{loss} = \text{C.P.} - \text{S.P.}$$

$$\displaystyle \text{loss}\% = \left(\frac{\text{loss}}{\text{C.P.}} \times 100\right)\%$$

$$\displaystyle \text{S.P.} = \left(\frac{100 - \text{loss}\%}{100}\right)\text{C.P.} \qquad \text{C.P.} = \left(\frac{100}{100 - \text{loss}\%}\right)\text{S.P.}$$

Sup­pose the pen bought for ₹10 is sold for ₹8. The loss is $10 - 8 = 2$ rupees, and the loss is $\displaystyle \frac{2}{10} \times 100 = 20$ per cent.

Dis­count#

A dis­count is money taken off the marked price.

$$\text{discount} = \text{M.P.} - \text{S.P.}$$

$$\displaystyle \text{discount}\% = \left(\frac{\text{discount}}{\text{M.P.}} \times 100\right)\%$$

Dis­count per cent is always worked out on the marked price, not on the cost price.

$$\displaystyle \text{S.P.} = \text{M.P.} - \text{discount} = \text{M.P.} - \frac{\text{discount}\%}{100} \times \text{M.P.}$$

$$\displaystyle \text{S.P.} = \left(\frac{100 - \text{discount}\%}{100}\right)\text{M.P.}$$

Putting the three prices together#

Say a pen has a marked price of ₹15. The shop sells it for ₹12, and it cost the shop ₹10.

• Dis­count $= 15 - 12 = 3$ rupees. Dis­count per cent $\displaystyle = \frac{3}{15} \times 100 = 20$.
• Profit $= 12 - 10 = 2$ rupees.

Going down from the marked price: take off the dis­count to reach the sell­ing price, then take off the profit to reach the cost price.

If there is a profit:

$$\text{C.P.} = \text{S.P.} - \text{profit} = \text{M.P.} - \text{discount} - \text{profit}$$

For the pen: $15 - 3 - 2 = 10$. That is the cost price.

If there is a loss: the sell­ing price is below the cost price, so this time we add the loss back.

$$\text{C.P.} = \text{S.P.} + \text{loss} = \text{M.P.} - \text{discount} + \text{loss}$$

For exam­ple, a bag is marked ₹500, sold with a dis­count of ₹50, and the shop loses ₹30. Then the sell­ing price is $500 - 50 = 450$ rupees, and the cost price is $450 + 30 = 480$ rupees.

Worked exam­ples#

Exam­ple 1#

A cycle is bought for ₹2400 and sold for ₹2700. Find the profit per cent.

S.P. is more than C.P., so there is a profit. Profit $= 2700 - 2400 = 300$ rupees.

$$\displaystyle \text{profit}\% = \left(\frac{300}{2400} \times 100\right)\% = 12.5\%$$

Exam­ple 2#

A fan bought for ₹1500 is sold at a loss of 8%. Find the sell­ing price.

$$\displaystyle \text{S.P.} = \frac{100 - 8}{100} \times 1500 = \frac{92}{100} \times 1500 = 1380$$

The sell­ing price is ₹1380.

Exam­ple 3#

By sell­ing a clock for ₹1150, a shop­keeper gains 15%. Find the cost price.

$$\displaystyle \text{C.P.} = \frac{100}{100 + 15} \times 1150 = \frac{100}{115} \times 1150 = 1000$$

The cost price is ₹1000.

Exam­ple 4#

A shirt is marked ₹800. The shop gives a dis­count of 15%. Find the sell­ing price.

$$\displaystyle \text{S.P.} = \frac{100 - 15}{100} \times 800 = \frac{85}{100} \times 800 = 680$$

The sell­ing price is ₹680.

Prac­tice#

1. C.P. ₹450, S.P. ₹540. Find the profit and the profit per cent.
2. C.P. ₹1200, S.P. ₹1020. Find the loss and the loss per cent.
3. An arti­cle costs ₹800 and is sold at a profit of 12%. Find the sell­ing price.
4. An arti­cle costs ₹650 and is sold at a loss of 6%. Find the sell­ing price.
5. An arti­cle is sold for ₹1035 at a profit of 15%. Find the cost price.
6. An arti­cle is sold for ₹1140 at a loss of 5%. Find the cost price.
7. A watch is marked ₹2500 and sold at a dis­count of 12%. Find the sell­ing price.
8. A toy is marked ₹600 and sold for ₹510. Find the dis­count per cent.
9. An arti­cle is marked ₹1000. The shop gives a dis­count of ₹100 and still makes a profit of ₹150. Find the cost price.
10. A lamp is marked ₹750. The shop gives a dis­count of ₹75 and makes a loss of ₹25. Find the cost price.

Answers#