Percentage, Discount, Profit and Loss

Lessons

Senior 1 · Mathematics

Percentage, Discount, Profit and Loss

5 lessons

Lesson 3 of 5

Discount and commission

22 min

Let's begin

At the start of the school year, a shop marks a school bag at 12,000 Frw and puts up a sign: "15% off". Down the road, a saleswoman for a solar-lamp company is paid a small percentage of every lamp she sells. Both are everyday uses of percentages: one lowers a price for the buyer, the other pays the seller.

1. Discount

Definition

Marked price: the price shown on an item before any reduction.

Discount: an amount taken off the marked price. The discount rate is the discount as a percentage of the marked price.

Sale price (or net price): what the customer pays, $\text{sale price} = \text{marked price} - \text{discount}$ .

Example: 15% off a 12,000 Frw bag is a discount of 1,800 Frw, so the sale price is 10,200 Frw.

Important

With a discount rate of $r %$ : $\text{discount} = \frac{r}{100} \times \text{marked price}$ $\text{sale price} = \left(1 - \frac{r}{100}\right) \times \text{marked price}.$ $\text{discount rate} = \frac{\text{discount}}{\text{marked price}} \times 100\%.$

Why the second formula works. The marked price is $100\%$ . Taking $r %$ off leaves $(100 - r)\%$ , so a discount is a percentage decrease, and $1 - \frac{r}{100}$ is its multiplier.

Worked example

A school bag is marked 12,000 Frw, with $15 %$ off. Find the discount and the sale price

Discount: $0.15 \times 12{,}000 = 1{,}800$ Frw. Sale price: $12,000 - 1,800 = 10,200$ Frw.

In one step with the multiplier: $0.85 \times 12{,}000 = 10{,}200$ Frw.

Worked example

A shirt marked 8,000 Frw is sold for 6,800 Frw. What is the discount rate?

\text{discount} = 8{,}000 - 6{,}800 = 1{,}200 \text{ Frw}

\frac{1{,}200}{8{,}000} \times 100\% = 15\%.

Worked example

Which is the better deal for a pair of shoes: Shop A marks them 25,000 Frw with $20 %$ off; Shop B marks them 24,000 Frw with 3,500 Frw off?

Shop	Marked price	Discount	Sale price
A	25,000 Frw	$0.2 \times 25{,}000 = 5{,}000$ Frw	20,000 Frw
B	24,000 Frw	3,500 Frw	20,500 Frw

Shop A is cheaper by 500 Frw, even though its marked price is higher.

Carved wooden figures in a craft shop in Kigali. A shop that marks such items at one price and then offers a percentage discount always works the discount out from the marked price. Photo: Diego Tirira, CC BY-SA 2.0, via Wikimedia Commons

2. Commission

Definition

Commission: money paid to an agent or salesperson for making a sale, usually a percentage of the value of the sales. That percentage is the commission rate: $\text{commission} = \frac{\text{rate}}{100} \times \text{value of sales}.$

Example: at a commission rate of $3 %$ , selling a 10,000,000 Frw plot of land earns the broker 300,000 Frw.

Worked example

A saleswoman is paid a basic wage of 80,000 Frw a month plus $4 %$ commission on her sales of solar lamps. In March she sells lamps worth 1,500,000 Frw. What does she earn in March?

Commission: $0.04 \times 1{,}500{,}000 = 60{,}000$ Frw. Total: $80,000 + 60,000 = 140,000$ Frw.

Worked example

An agent earns 45,000 Frw commission at a rate of $3 %$ . What was the value of the sales?

$3 %$ of the sales is 45,000 Frw, so $1 %$ is 15,000 Frw and $100\%$ is 1,500,000 Frw. As an equation: $0.03 S = 45,000$ , so $S = \frac{45{,}000}{0.03} = 1{,}500{,}000$ Frw.

Warning

The discount rate is always a percentage of the marked price. For the shirt, $\frac{1{,}200}{6{,}800} \approx 17.6\%$ is wrong, because it divides by the sale price. Ask "a percentage of what?" and the answer is the price before the discount.

Tip

Estimate before you calculate. $15 %$ off is a bit more than one seventh off; $20 %$ off is one fifth off; $25 %$ off is one quarter off. If your sale price is not a little under what the estimate says, check the arithmetic.

Khan Academy: "Tax, discount and tip examples". Watch how each discount is found as a percentage of the original price. (A tip is a gift to a waiter; many countries add it to a bill.)

✦

Practice

Give answers in Frw.

Find the sale price of a 5,000 Frw book with $10 %$ off.
A radio marked 36,000 Frw is sold at a $25 %$ discount. Find the discount and the sale price.
A dress marked 15,000 Frw is sold for 12,000 Frw. Find the discount rate.
A broker's commission rate is $2.5 %$ . How much does she earn for selling a house for 40,000,000 Frw?
A salesman earns 50,000 Frw plus $5 %$ of his sales. Find his pay when he sells goods worth 900,000 Frw.
An agent earned 24,000 Frw commission at $4 %$ . Find the value of the sales.
After a $20 %$ discount a phone costs 96,000 Frw. What was its marked price?

Show answersHide answers

$0.9 \times 5{,}000 = 4{,}500$ Frw
Discount 9,000 Frw; sale price 27,000 Frw.
$\frac{3{,}000}{15{,}000} \times 100\% = 20\%$
$0.025 \times 40{,}000{,}000 = 1{,}000{,}000$ Frw
$50,000 + 45,000 = 95,000$ Frw
$\frac{24{,}000}{0.04} = 600{,}000$ Frw
$0.8 M = 96,000$ , so $M = 120,000$ Frw.

Sources and credits

6.3: Solve Sales Tax, Commission, and Discount Applications, from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: RW Kigali (95) (17260007071).jpg, Diego Tirira, Wikimedia Commons, licensed under CC BY-SA 2.0. Used unchanged.
Video: Tax, discount and tip examples, Khan Academy, embedded from YouTube.

The school-bag, shoe-shop, solar-lamp and broker examples, the percentage bar, the practice items and the quiz questions were written by Ganzaa. The prices and rates are models for practice.

This lesson includes CC BY-SA material and is shared under CC BY-SA 4.0.

11. Discount

22. Commission

1. Discount

2. Commission