Profit and Loss

Every shopkeeper in your bazaar is doing this chapter in their head all day: buy at one price, sell at another, and hope the difference covers the rent. The mathematics is short — four formulas — but the words matter, because profit percent is always measured against the cost while discount percent is always measured against the marked price. Mixing those two up is the classic mistake of this unit.

5.1 Cost price, selling price, profit and loss

• Cost price (CP) is the price at which an article is bought, including any expense of bringing it.
• Selling price (SP) is the price at which it is sold.
• If SP > CP, the difference is profit: \text{profit}=\text{SP}-\text{CP}.
• If SP < CP, the difference is loss: \text{loss}=\text{CP}-\text{SP}.

Percentages are always measured against the cost price:

Rearranged, these let you find a price from a percentage:

5.2 Marked price and discount

Shops do not usually sell at the price printed on the label. The printed price is the marked price (MP), also called the list price, and the reduction allowed is the discount.

A discount is measured on the marked price, while profit and loss are measured on the cost price. A sale can therefore show a big discount and still be profitable, which is exactly what shopkeepers rely on.

5.3 Commission

A commission is a payment to an agent for selling something on someone else's behalf, and it is a percentage of the sale value.

5.4 Marked price, discount and VAT in order

The order of operations is where most marks are lost: discount comes off the marked price, VAT goes on to what is left.

Quantity	Formula	Always on
Profit	S.P. - C.P.	—
Profit per cent	\frac{\text{profit}}{\text{C.P.}}\times100	cost price
Loss per cent	\frac{\text{loss}}{\text{C.P.}}\times100	cost price
Discount	\frac{d}{100}\times \text{M.P.}	marked price
VAT	\frac{v}{100}\times \text{(M.P. after discount)}	discounted price

Chapter summary

• \text{profit}=\text{SP}-\text{CP} when SP exceeds CP; \text{loss}=\text{CP}-\text{SP} when CP exceeds SP.
• Profit percent and loss percent are measured against the cost price.
• \text{SP}=\text{CP}(1\pm\frac{\%}{100}); \text{CP}=\frac{100\times\text{SP}}{100\pm\%}.
• \text{discount}=\text{MP}-\text{SP}, and discount percent is measured against the marked price.
• \text{SP}=\text{MP}(1-\frac{\text{discount}\%}{100}).
• A sale can give a discount and a profit at the same time, because the two percentages are measured on different bases.
• Commission is a percentage of the sale value and is paid to the agent.

Practice questions

Group A — Objective (1 mark each)

1. Profit percent is calculated on the [1]aselling pricebmarked priceccost priceddiscount
2. Discount percent is calculated on the [1]aselling pricebmarked priceccost pricedprofit
3. An article bought for Rs 500 is sold for Rs 450. The loss percent is [1]a10%b11%c12.5%d50%
4. If CP is Rs 800 and profit is 25 %, the SP is [1]aRs 825bRs 1000cRs 1025dRs 1200
5. MP Rs 1,000 with a 30 % discount gives an SP of [1]aRs 970bRs 700cRs 300dRs 1300
6. A 5 % commission on a sale of Rs 40,000 is [1]aRs 200bRs 2000cRs 500dRs 8000

Answers to Group A

1. (c) — cost price.

2. (b) — marked price.

3. (a) — \frac{50}{500}\times100=10\%.

4. (b) — 800\times1.25=1000.

5. (b) — 1000\times0.7=700.

6. (b) — \frac{5}{100}\times40\,000=2000.

Group B — Short answer (2 marks each)

1. Find the profit percent when an article bought for Rs 2,500 is sold for Rs 2,800. [2]
2. The selling price of an article is Rs 1,440 at a loss of 10 %. Find its cost price. [2]
3. Find the selling price after a discount of 12 % on a marked price of Rs 4,000. [2]
4. An agent gets a commission of Rs 2,250 on a sale of Rs 45,000. Find the rate of commission. [2]

Answers to Group B

1. Profit =300; profit percent =\frac{300}{2500}\times100=12\%.

2. \text{CP}=\frac{100\times1440}{90}=\text{Rs }1600.

3. Discount =480, so SP =4000-480=\text{Rs }3520.

4. Rate =\frac{2250}{45000}\times100=5\%.

Group C — Long answer (4 marks each)

1. A shopkeeper marks his goods 40 % above the cost price and then allows a discount of 20 % on the marked price. Find his profit percent, and explain why the profit is not 20 %. [4]
2. A dealer sold two mobile phones for Rs 9,000 each. On one he made a profit of 20 % and on the other a loss of 20 %. Find his total profit or loss, and explain the result. [4]

Answers to Group C

1. Take CP = Rs 100. Then \text{MP}=100\times1.40=\text{Rs }140, and the discount is 20 % of the marked price: 0.20\times140=\text{Rs }28, so \text{SP}=140-28=\text{Rs }112. Profit =112-100=\text{Rs }12, that is 12 % on the cost. Why not 20 %. The 40 % is added to the cost price while the 20 % discount comes off the marked price, which is the larger number: 0.40\times100=40, but 0.20\times140=28, so only 12 remains. Percentages added and subtracted in this way only cancel when they are measured on the same base.

2. First phone (20 % profit). \text{CP}=\frac{100\times9000}{120}=\text{Rs }7500; profit =1500. Second phone (20 % loss). \text{CP}=\frac{100\times9000}{80}=\text{Rs }11\,250; loss =2250. Total. \text{CP}=7500+11\,250=\text{Rs }18\,750, and \text{SP}=9000+9000=\text{Rs }18\,000. So there is a net loss of Rs 750, which is \frac{750}{18\,750}\times100=4\% of the total cost. Explanation. The loss is worked on a larger cost price than the profit, so the two do not cancel. Equal percentage profit and loss on equal selling prices always gives a net loss of \frac{x^2}{100}\% — here \frac{20^2}{100}=4\% ✓.

Extra practice — textbook-style questions

1. A sweater is sold for Rs 1,075 after a 14 % discount. Find its marked price.

2. A torch bought for Rs 1,400 is marked 40 % above the cost price and sold at a 20 % discount. Find the MP, the discount, the SP and the profit.

3. Rahaman marks a suitcase at Rs 6,000 and allows a 15 % discount; he still makes a profit of Rs 500. Find the SP, the CP and the profit percent.

4. VAT of Rs 338 was charged at 13 %. Find the price before VAT.

Extra practice — answers

1. SP =86\% of MP, so MP =\frac{1075\times100}{86}= Rs 1,250.

2. MP =1400+560= Rs 1,960; discount =20\% of 1960= Rs 392; SP =1960-392= Rs 1,568; profit =1568-1400= Rs 168 (12 %).

3. Discount =15\% of 6000=900; SP = Rs 5,100; CP =5100-500= Rs 4,600; profit percent =\frac{500}{4600}\times100\approx10.87\%.

4. Price = \frac{338\times100}{13} = Rs 2,600.

Objective practice

Choose the correct answer. [5×1=5]

1. An article bought for Rs 600 and sold for Rs 690 gives a profit of (a) 10 % (b) 15 % (c) 20 % (d) 9 %

2. Profit per cent is calculated on the (a) selling price (b) marked price (c) cost price (d) discount

3. VAT is charged on the (a) marked price (b) price after discount (c) cost price (d) profit

4. A watch marked Rs 2,000 sold at 20 % discount costs (a) Rs 1,600 (b) Rs 1,800 (c) Rs 2,400 (d) Rs 400

5. If the cost price is Rs 800 and the loss is 15 %, the selling price is (a) Rs 680 (b) Rs 720 (c) Rs 920 (d) Rs 640

Objective practice — answers

1. (b) — profit 90; \frac{90}{600}\times100=15 %.

2. (c) — on the cost price.

3. (b) — on the discounted price.

4. (a) — 2000-400=1600.

5. (a) — 800-120=680.