Percentage, Discount, Profit and Loss

Lessons

Senior 1 · Mathematics

Percentage, Discount, Profit and Loss

5 lessons

Lesson 4 of 5

Profit and loss

25 min

Let's begin

Mutoni buys a crate of tomatoes at Kimironko market for 8,000 Frw and sells them in her neighbourhood for 10,000 Frw in total. She has made 2,000 Frw. Is that a good return? A trader who gains 2,000 Frw on goods that cost 8,000 Frw is doing far better than one who gains 2,000 Frw on goods that cost 80,000 Frw. To compare fairly, traders measure profit as a percentage of what they spent.

A vegetable stall at Kimironko market, Kigali. Every trader here buys at one price (the cost price) and sells at another (the selling price); the difference is the profit or loss. Photo: François Terrier, CC BY 2.0, via Wikimedia Commons

Definition

Cost price (CP): the total amount paid to buy or make an item, including costs such as transport.

Selling price (SP): the amount received when the item is sold.

Example: Mutoni's CP is 8,000 Frw and her SP is 10,000 Frw.

Definition

Profit: the gain when the selling price is more than the cost price, $\text{profit} = \text{SP} - \text{CP}$ .

Loss: the shortfall when the selling price is less than the cost price, $\text{loss} = \text{CP} - \text{SP}$ .

Example: Mutoni's profit is $10,000 - 8,000 = 2,000$ Frw.

Definition

Percentage profit and percentage loss: the profit or loss as a percentage of the cost price, $\text{profit}\,\% = \frac{\text{profit}}{\text{CP}} \times 100\%$ $\text{loss}\,\% = \frac{\text{loss}}{\text{CP}} \times 100\%.$

Mark-up: the amount a seller adds to the cost price to set the selling price. A mark-up of $25 %$ means $\text{SP} = 1.25 \times \text{CP}$ .

Why divide by the cost price? The cost price is the money the trader risked. Profit per cent answers the question "for every 100 Frw I spent, how much did I gain?", so the cost price is the whole. It also turns profit into a percentage change from CP to SP, so the multiplier method of the last lesson works: $\text{SP} = \left(1 + \frac{r}{100}\right) \times \text{CP}$ for a profit of $r %$ , and $\text{SP} = \left(1 - \frac{r}{100}\right) \times \text{CP}$ for a loss of $r %$ .

Item (a kiosk's price list)	CP (Frw)	SP (Frw)	Profit or loss (Frw)	Percentage
Bar of soap	500	600	profit 100	$20 %$ profit
Loaf of bread	1,000	1,200	profit 200	$20 %$ profit
Packet of milk	400	500	profit 100	$25 %$ profit
Old exercise book	250	200	loss 50	$20 %$ loss

The soap and the milk each make 100 Frw, but the milk makes the better percentage profit, because it cost less.

Worked example

(a) Mutoni's tomatoes: CP 8,000 Frw, SP 10,000 Frw. (b) A phone bought for 60,000 Frw and sold for 51,000 Frw. Find the percentage profit or loss

(a) Profit $= 2,000$ Frw; $\frac{2{,}000}{8{,}000} \times 100\% = 25\%$ profit.

(b) Loss $= 60,000 - 51,000 = 9,000$ Frw; $\frac{9{,}000}{60{,}000} \times 100\% = 15\%$ loss.

Worked example

Finding the selling price. A tailor makes a dress for 40,000 Frw and wants a $30 %$ profit. What should she charge?

\text{SP} = 1.3 \times 40{,}000 = 52{,}000 \text{ Frw}.

Worked example

Finding the cost price. (a) A bicycle sold for 27,600 Frw gave a $15 %$ profit. (b) A goat sold for 34,000 Frw gave a $15 %$ loss. Find each cost price

(a) $1.15 \times \text{CP} = 27{,}600$ , so $\text{CP} = \frac{27{,}600}{1.15} = 24{,}000$ Frw.

(b) $0.85 \times \text{CP} = 34{,}000$ , so $\text{CP} = \frac{34{,}000}{0.85} = 40{,}000$ Frw.

Check (a): $15 %$ of 24,000 is 3,600, and $24,000 + 3,600 = 27,600$ . ✓

Worked example

A cooperative of three farmers buys 2,000 kg of maize at 350 Frw per kg and pays 60,000 Frw to transport it. It sells all the maize at 420 Frw per kg. (a) Find the percentage profit. (b) The members share the profit in the ratio $2 : 3 : 3$ of their contributions. How much does each receive?

	Working	Frw
Cost of maize	$2{,}000 \times 350$	700,000
Transport		60,000
Cost price	$700,000 + 60,000$	760,000
Selling price	$2{,}000 \times 420$	840,000
Profit	$840,000 - 760,000$	80,000

(a) $\frac{80{,}000}{760{,}000} \times 100\% \approx 10.5\%$ profit. Leaving out the transport would wrongly give $20 %$ .

(b) $2 + 3 + 3 = 8$ parts, so one part is $80{,}000 \div 8 = 10{,}000$ Frw. The shares are 20,000 Frw, 30,000 Frw and 30,000 Frw.

Warning

Percentage profit is on the cost price, not the selling price. For Mutoni, $\frac{2{,}000}{10{,}000} = 20\%$ is wrong; the answer is $\frac{2{,}000}{8{,}000} = 25\%$ . In the same way, to find a cost price from a selling price, divide by the multiplier. Do not take the percentage off the selling price: $27{,}600 - 15\%$ of $27,600 = 23,460$ Frw, which is wrong.

Tip

Put every cost into the cost price: transport, market fees, packaging. A profit that forgets a cost is not a real profit, as the cooperative example shows.

SkanCity Academy: "Percentages 1 - How to Calculate Profit and Loss". Watch how the percentage is always worked out on the cost price. The prices in the video are in Ghana cedis.

✦

Practice

State whether each answer is a profit or a loss.

A pen bought for 200 Frw is sold for 250 Frw. Find the percentage profit.
A bag of rice bought for 30,000 Frw is sold for 27,000 Frw. Find the percentage loss.
A trader buys a goat for 45,000 Frw and wants a $20 %$ profit. Find the selling price.
A radio bought for 25,000 Frw is sold at a $12 %$ loss. Find the selling price.
A shop sells a mattress for 46,000 Frw at a profit of $15 %$ . Find the cost price.
A farmer sold a cow for 360,000 Frw and lost $10 %$ . What had she paid for it?
A kiosk buys 50 loaves at 900 Frw each, pays 2,000 Frw transport and sells all of them at 1,100 Frw each. Find the percentage profit, to 1 decimal place.
Gatete says: "I bought for 5,000 and sold for 6,000, so I made a $20 %$ profit; if I had bought for 6,000 and sold for 5,000, I would have made a $20 %$ loss." Is he right?

Show answersHide answers

$\frac{50}{200} \times 100\% = 25\%$ profit
$\frac{3{,}000}{30{,}000} \times 100\% = 10\%$ loss
$1.2 \times 45{,}000 = 54{,}000$ Frw
$0.88 \times 25{,}000 = 22{,}000$ Frw
$\frac{46{,}000}{1.15} = 40{,}000$ Frw
$\frac{360{,}000}{0.9} = 400{,}000$ Frw
CP $= 45,000 + 2,000 = 47,000$ ; SP $= 55,000$ ; profit $= 8,000$ ; $\frac{8{,}000}{47{,}000} \times 100\% \approx 17.0\%$ .
No. The profit is $\frac{1{,}000}{5{,}000} = 20\%$ , but the loss would be $\frac{1{,}000}{6{,}000} \approx 16.7\%$ : the cost price, the whole, is different.

Sources and credits

3.2: Solve Percent Applications (discount and mark-up), from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Kimironko Maket.jpg, François Terrier, Wikimedia Commons, licensed under CC BY 2.0. Used unchanged.
Video: Percentages 1 - How to Calculate Profit and Loss / Cost Price and Selling Price, SkanCity Academy, embedded from YouTube.

The rest of this lesson (the profit and loss definitions, the kiosk table, the tomato, phone, bicycle and cooperative examples, the bar diagram, the practice items and the quiz questions) was written by Ganzaa. The prices are models for practice, not market data.