Percentage, Discount, Profit and Loss

Lessons

Senior 1 · Mathematics

Percentage, Discount, Profit and Loss

5 lessons

Lesson 2 of 5

Percentage change and multipliers

25 min

Let's begin

Suppose a bag of beans that cost 800 Frw per kg last season now costs 1,000 Frw. The price went up by 200 Frw. Is that a big rise? For beans, yes, but 200 Frw on a 150,000 Frw phone would hardly be noticed. To judge a change fairly, compare it with what you started with. That is a percentage change.

Definition

Percentage change: a change written as a percentage of the original value, $\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100\%.$ It is a percentage increase if the value went up and a percentage decrease if it went down.

Example: 800 Frw to 1,000 Frw is a change of 200 Frw, and $\frac{200}{800} \times 100\% = 25\%$ , a 25% increase.

Definition

Multiplier: the single number you multiply by to make a percentage change in one step.

Increase by $r %$ : multiplier $= 1 + \frac{r}{100}$ . Example: $+ 18 %$ gives $1.18$ .
Decrease by $r %$ : multiplier $= 1 - \frac{r}{100}$ . Example: $- 20 %$ gives $0.8$ .

Why a multiplier works

The original is $100\%$ . After an increase of $20 %$ , you have $100\% + 20\% = 120\%$ of the original, and $120\% = 1.2$ . After a decrease of $20 %$ , you have $100\% - 20\% = 80\% = 0.8$ of it. So "add $20 %$ " and "multiply by 1.2" are the same thing, done in one step instead of two.

Change	New value as a % of the original	Multiplier
increase by $5 %$	$105\%$	$1.05$
increase by $18 %$	$118\%$	$1.18$
increase by $100\%$	$200\%$	$2$
decrease by $15 %$	$85 %$	$0.85$
decrease by $2.5 %$	$97.5\%$	$0.975$

Worked example

(a) A worker earning 150,000 Frw a month gets a $12 %$ rise. What is the new wage? (b) A 60,000 Frw radio loses $15 %$ of its value in a year. What is it worth now?

(a) Multiplier $1.12$ : $150{,}000 \times 1.12 = 168{,}000$ Frw.

(b) Multiplier $0.85$ : $60{,}000 \times 0.85 = 51{,}000$ Frw.

Worked example

The price of beans rose from 800 to 1,000 Frw per kg, then later fell back from 1,000 to 800. Find both percentage changes

Rise: $\frac{200}{800} \times 100\% = 25\%$ increase. Fall: $\frac{200}{1{,}000} \times 100\% = 20\%$ decrease.

The change is 200 Frw both times, but the original is different, so the percentages differ.

Worked example

Reverse percentage. A phone costs 144,000 Frw after a $20 %$ price rise. What did it cost before?

The new price is $120\%$ of the old one: $1.2 \times \text{old} = 144{,}000$ , so $\text{old} = \frac{144{,}000}{1.2} = 120{,}000 \text{ Frw}.$ Check: $120{,}000 \times 1.2 = 144{,}000$ . ✓ Taking $20 %$ off 144,000 gives 115,200, which is wrong: the $20 %$ was a percentage of the old price, not the new one.

Worked example

Successive changes. A shop raises a price by $20 %$ , then cuts the new price by $20 %$ . Is the price back where it started?

1.2 \times 0.8 = 0.96.

No. The final price is $96 %$ of the original, a $4 %$ decrease overall. The cut was $20 %$ of a bigger number than the rise was.

Percent or percentage points?

When the quantity is itself a percentage, be careful. The chart shows the share of people surveyed in Rwanda who said they had used a mobile-money account in the past year: about $18 %$ in 2014 and about $31 %$ in 2017.

The share rose by $31 - 18 = 13$ percentage points. As a percentage change it rose by $\frac{13}{18} \times 100\% \approx 72\%$ : the number of users grew by almost three quarters. Both statements are true, and they answer different questions.

Warning

Divide by the original, not the new value. The rise from 800 to 1,000 Frw is $\frac{200}{800} = 25\%$ , not $\frac{200}{1{,}000} = 20\%$ . The base of a percentage change is always the value before the change. In a reverse-percentage problem you do not know that value yet, so divide by the multiplier instead of taking a percentage off.

Tip

A multiplier bigger than 1 means an increase, and smaller than 1 means a decrease. After two changes, multiply the multipliers: $+ 10 %$ then $+ 10 %$ is $1.1 \times 1.1 = 1.21$ , a $21 %$ increase, not $20 %$ .

Cognito: "Percentage Increase and Decrease (Multiplier Method)". Watch how each percentage is turned into a multiplier before any calculating is done.

✦

Practice

Use a multiplier where you can.

Write the multiplier for (a) an increase of $7 %$ ; (b) a decrease of $30 %$ ; (c) an increase of $150\%$ .
Increase 45,000 Frw by $8 %$ .
Decrease 2,500 kg by $12 %$ .
A school's enrolment rose from 640 to 720. Find the percentage increase.
A bus fare fell from 500 Frw to 450 Frw. Find the percentage decrease.
After a $25 %$ bonus, Mugisha's data bundle is 5 GB. How big was it before the bonus?
A price rises by $10 %$ and then falls by $10 %$ . Find the overall percentage change.
A town's population was 20,000. It grew by $5 %$ in one year and $5 %$ the next. What was it after two years?

Show answersHide answers

(a) $1.07$ ; (b) $0.7$ ; (c) $2.5$
$45{,}000 \times 1.08 = 48{,}600$ Frw
$2{,}500 \times 0.88 = 2{,}200$ kg
$\frac{80}{640} \times 100\% = 12.5\%$
$\frac{50}{500} \times 100\% = 10\%$
$\frac{5}{1.25} = 4$ GB
$1.1 \times 0.9 = 0.99$ : a $1 %$ decrease.
$20{,}000 \times 1.05 \times 1.05 = 22{,}050$

Sources and credits

3.2: Solve Percent Applications, 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.
Grade 7, Unit 4: Proportional Relationships and Percentages, IM 6–8 Math by Illustrative Mathematics, originally developed by Open Up Resources (© 2017–2019 Open Up Resources; adaptations © 2019 Illustrative Mathematics), provided by Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Chart: Share of people using mobile money accounts, Rwanda, Our World in Data (data: World Bank, 2021), via Wikimedia Commons, licensed under CC BY 4.0. Used unchanged.
Video: GCSE Maths - Percentage Increase and Decrease (Multiplier Method), Cognito, embedded from YouTube.

The bean-price, wage, phone and data-bundle examples, the bar diagram, the practice items and the quiz questions were written by Ganzaa. The prices are models for practice.

1Why a multiplier works

2Percent or percentage points?

Why a multiplier works

Percent or percentage points?