Module 2 of 3Lesson 1 of 11 · Unit 12 · Percentage increase, decrease and change
Mathematics Form 2 · Term 2 · Unit 12

🔢 Percentage increase, decrease and change

NumberIGCSE path ✓JC ✓7Nf.058Nf.05
💡
Why this matters

Prices go up, data bundles get bigger, sale signs shout “30% OFF!” and every till slip adds VAT. Percentages tell you how big a change really is. Once you know the multiplier trick, you can work out any of them in ONE step, and spot when a “great deal” is not so great.

🎯 By the end of this unit

  • I can write a percentage increase or decrease as a multiplier, such as 1.15 or 0.85.
  • I can increase or decrease an amount by a percentage in one step.
  • I can work out the percentage change between an old value and a new value.
  • I can use percentages greater than 100%.
  • I can calculate VAT, discounts and percentage profit or loss.
📖 Key words

multiplier
the number you multiply by to make a change, such as 1.15 for a 15% increase
percentage change
the change as a percentage of the original value
absolute change
the actual amount of the change, such as P12
VAT
Value Added Tax: a percentage added to the price of most goods and services
discount
an amount taken off a price, often given as a percentage
profit and loss
selling for more than you paid is a profit; selling for less is a loss

1 Learn it

Percentages as multipliers

The original amount is always 100% of itself.

An increase of 15% gives \(100\% + 15\% = 115\%\) of the original. As a decimal, 115% is 1.15, so you multiply by 1.15.

A decrease of 15% leaves \(100\% – 15\% = 85\%\). As a decimal that is 0.85, so you multiply by 0.85.

ChangePercentage of originalMultiplier
increase 15%115%1.15
increase 4%104%1.04
decrease 15%85%0.85
decrease 40%60%0.6
100% (the original)+15%115%Increase by 15%: multiply by 1.1585% is left−15%Decrease by 15%: multiply by 0.85
The coral piece is the 15% that is added or taken away.
⭐ Key ideaMultiplier for an increase: bigger than 1. Multiplier for a decrease: smaller than 1.

Increase or decrease in one step

You could find 15% and then add it on. The multiplier does both jobs in one calculation, which means fewer slips.

Example: increase P240 by 15%.

  1. \(100\% + 15\% = 115\%\)
  2. So the multiplier is 1.15
  3. \(240 \times 1.15 = 276\)
  4. The new amount is P276.00
⭐ Key ideaIncrease: × (1 + rate). Decrease: × (1 − rate). Write the rate as a decimal first.

Percentages greater than 100%

A percentage can be bigger than 100%. 200% of something is double it, and 250% is two and a half times it.

So an increase of 100% doubles an amount (multiplier 2), and an increase of 150% multiplies it by 2.5.

Percentages below 1% exist too: 0.5% of P400 is P2.

  1. 250% as a decimal is 2.5
  2. \(40 \times 2.5 = 100\)
100% of 40= 40200% of 40= 80250% of 40= 100
200% of 40 is 80; 250% of 40 is 100.
⭐ Key ideaOver 100% means more than the whole. Divide the percentage by 100 to get the multiplier.

Finding the percentage change

The absolute change is the difference: new value − original value.

The percentage change compares that change with the original value, because the original is the 100% we started from.

Example: a price rises from P80 to P92.

  1. \(92 – 80 = 12\) (change)
  2. Percentage change: \(\dfrac{12}{80} \times 100\)
  3. \(= 15\%\) increase
P70P75P80P85P90P95P100old pricenew pricechange = P12Compare the change with the ORIGINAL price: 12 out of 80
The change is P12. We compare it with the original P80.
⭐ Key ideaPercentage change \(= \dfrac{\text{change}}{\text{original}} \times 100\). Always divide by the ORIGINAL.

Profit and loss as a percentage

A trader who buys for a cost price and sells for a selling price makes a profit (or a loss).

Percentage profit or loss is just a percentage change, using the cost price as the original.

  1. Neo buys a goat for P800 and sells it for P1 000
  2. \(1000 – 800 = 200\) (profit)
  3. Percentage profit: \(\dfrac{200}{800} \times 100\)
  4. \(= 25\%\)
⭐ Key ideaPercentage profit \(= \dfrac{\text{profit}}{\text{cost price}} \times 100\).

VAT and discounts

VAT (Value Added Tax) is added to the price of most things. In Botswana the standard rate is currently 14%. Always use the rate the question gives you.

Adding 14% VAT means multiplying by 1.14. A 20% discount means you pay 80%, so multiply by 0.8.

When there is a discount and VAT, do them one after the other: two multipliers.

Adding VAT at 14%price before VAT 100%114%: ×1.14Taking 20% off in a saleyou pay 80%×0.8
VAT makes the bar longer; a discount makes it shorter.
⭐ Key ideaVAT at r%: × (1 + r ÷ 100). Discount of d%: × (1 − d ÷ 100).

Why +10% then −10% does not cancel

Increase P100 by 10% and you get P110. Now decrease P110 by 10%: you take 10% of 110, which is P11, not P10.

You end at P99. The second percentage is of a different, bigger amount. Multipliers show this at once: \(1.1 \times 0.9 = 0.99\).

P100P110P99×1.1×0.9+10% then −10% does NOT get you back to P100
Each percentage is of the amount at that moment.
⭐ Key ideaFor several changes, multiply the multipliers together. Never just add or subtract the percentages.

2 Watch it work

EXAMPLE 1

Increase with a multiplier

Kabelo’s monthly data bundle is 2.5 GB. His provider increases it by 40%. How big is the new bundle?
  1. Increase of 40%: \(100\% + 40\% = 140\%\)
  2. Multiplier: 1.4
  3. \(2.5 \times 1.4 = 3.5\)
Answer 3.5 GB
EXAMPLE 2

Decrease with a multiplier

A laptop costs P6 800. In one year it loses 15% of its value. What is it worth after one year?
  1. Decrease of 15%: \(100\% – 15\% = 85\%\)
  2. Multiplier: 0.85
  3. \(6800 \times 0.85 = 5780\)
Answer P5 780.00
EXAMPLE 3

Percentage increase

The price of a loaf of bread rises from P8.00 to P9.20. Find the percentage increase.
  1. \(9.20 – 8.00 = 1.20\) (change)
  2. \(\dfrac{1.20}{8.00} \times 100\)
  3. \(= 15\%\)
Answer 15% increase
EXAMPLE 4

Percentage loss

Lesego bought a bicycle for P1 250 and later sold it for P1 000. Find her percentage loss.
  1. \(1250 – 1000 = 250\) (loss)
  2. Compare with the cost price: \(\dfrac{250}{1250} \times 100\)
  3. \(= 20\%\)
Answer 20% loss
EXAMPLE 5

Discount, then VAT

A fridge is marked P1 500 before VAT. The shop gives a 20% discount, then adds VAT at 14%. How much does the customer pay?
  1. \(1500 \times 0.8 = 1200\) (after the discount)
  2. \(1200 \times 1.14 = 1368\) (add VAT)
Answer P1 368.00
EXAMPLE 6

A change bigger than 100%

A baby elephant in the Okavango has a mass of 120 kg. A year later its mass is 300 kg. Find the percentage increase.
  1. \(300 – 120 = 180\) (change)
  2. \(\dfrac{180}{120} \times 100\)
  3. \(= 150\%\)
Answer 150% increase (its mass is now 250% of what it was)

3 Your turn

Try each question in your book first. Then tap the button to check every step.

1🟢 Warm-up
Write down the multiplier for:
(a) an increase of 12% (b) a decrease of 30% (c) an increase of 5%
👀 Show the answer
Step by step
  1. (a) \(100\% + 12\% = 112\%\), which is 1.12
  2. (b) \(100\% – 30\% = 70\%\), which is 0.7
  3. (c) \(100\% + 5\% = 105\%\), which is 1.05
Answer (a) 1.12 (b) 0.7 (c) 1.05
2🟢 Warm-up
Increase 50 by 20%.
👀 Show the answer
Step by step
  1. Multiplier: 1.2
  2. \(50 \times 1.2 = 60\)
Answer 60
3🟢 Warm-up
Decrease P80 by 25%.
👀 Show the answer
Step by step
  1. \(100\% – 25\% = 75\%\)
  2. So the multiplier is 0.75
  3. \(80 \times 0.75 = 60\)
Answer P60.00
4🟢 Warm-up
Find 200% of 35 and 150% of 60.
👀 Show the answer
Step by step
  1. 200% is 2: \(35 \times 2 = 70\)
  2. 150% is 1.5: \(60 \times 1.5 = 90\)
Answer 70 and 90
5🔵 Core
A combi fare rises from P5.50 to P6.60. Find the percentage increase.
👀 Show the answer
Step by step
  1. \(6.60 – 5.50 = 1.10\) (change)
  2. \(\dfrac{1.10}{5.50} \times 100\)
  3. \(= 20\%\)
Answer 20%
6🔵 Core
A pair of trainers costs P349.00. In a sale there is 15% off. Find the sale price.
👀 Show the answer
Step by step
  1. \(100\% – 15\% = 85\%\)
  2. So the multiplier is 0.85
  3. \(349 \times 0.85 = 296.65\)
Answer P296.65
7🔵 Core
A radio costs P450.00 before VAT. VAT is charged at 14%. Find the price including VAT.
👀 Show the answer
Step by step
  1. Multiplier: 1.14
  2. \(450 \times 1.14 = 513\)
Answer P513.00
8🔵 Core
The population of a village near Kanye grew from 1 250 to 1 600. Find the percentage increase.
👀 Show the answer
Step by step
  1. \(1600 – 1250 = 350\) (change)
  2. \(\dfrac{350}{1250} \times 100\)
  3. \(= 28\%\)
Answer 28%
9🔵 Core
A farmer had 48 cattle. After a dry season he had 42. Find the percentage decrease.
👀 Show the answer
Step by step
  1. \(48 – 42 = 6\) (change)
  2. \(\dfrac{6}{48} \times 100\)
  3. \(= 12.5\%\)
Answer 12.5%
10🔵 Core
A morula seedling grew from 1.2 m to 3 m tall. Find the percentage increase.
👀 Show the answer
Step by step
  1. \(3 – 1.2 = 1.8\) (change)
  2. \(\dfrac{1.8}{1.2} \times 100\)
  3. \(= 150\%\)
Answer 150% (a change can be more than 100%)
11🔵 Core
Naledi buys sweets for P60 and sells them for P75. Find her percentage profit.
👀 Show the answer
Step by step
  1. \(75 – 60 = 15\) (profit)
  2. \(\dfrac{15}{60} \times 100\)
  3. \(= 25\%\)
Answer 25% profit
12🟠 Stretch
A price of P100 is increased by 10%. The new price is then decreased by 10%. Thato says the price is back to P100. Is she right?
👀 Show the answer
Step by step
  1. \(100 \times 1.1 = 110\) (after the increase)
  2. \(110 \times 0.9 = 99\) (after the decrease)
  3. The 10% decrease is 10% of 110, which is P11
Answer No. The final price is P99.00, a 1% decrease overall.
13🟠 Stretch
Shop A gives 30% off. Shop B gives 20% off and then another 10% off the reduced price. Both start with the same price. Which is cheaper?
👀 Show the answer
Step by step
  1. Shop A multiplier: 0.7
  2. Shop B multipliers: \(0.8 \times 0.9 = 0.72\)
  3. 0.7 is smaller than 0.72, so Shop A leaves you paying less
Answer Shop A (you pay 70% of the price, not 72%).
14🟠 Stretch
Mpho’s electricity bill was P320.00 before VAT. Next month she used 15% more electricity at the same rates. VAT is 14%. How much was next month’s bill including VAT?
👀 Show the answer
Step by step
  1. \(320 \times 1.15 = 368\) (next month, before VAT)
  2. \(368 \times 1.14 = 419.52\) (add VAT)
Answer P419.52

4 Watch out!

✗ WrongDecrease P200 by 15%: \(200 \times 0.15 = 30\), so the answer is P30.
✓ Right\(200 \times 0.85 = 170\), so the answer is P170.00.

0.15 finds the 15% that is taken away, not what is left. What is left is 85%, so the multiplier is 0.85.

✗ WrongA price goes from P80 to P100. Percentage increase: \(\dfrac{20}{100} \times 100 = 20\%\).
✓ Right\(\dfrac{20}{80} \times 100 = 25\%\)

Always divide by the ORIGINAL value, because the original is the 100% you started from.

✗ WrongIncrease by 5%: multiply by 1.5.
✓ RightIncrease by 5%: multiply by 1.05.

5% is 0.05 as a decimal, so 105% is 1.05. Multiplying by 1.5 would be a 50% increase.

✗ Wrong20% off and then 14% VAT is the same as 6% off.
✓ Right\(0.8 \times 1.14 = 0.912\), so the final price is 91.2% of the marked price.

Percentages of different amounts cannot be added or subtracted. Multiply the multipliers.

5 In your exams

🇧🇼 Junior Certificate (JCE)

Percentage change is tested often in the JCE. In Paper 1 (no calculator) you may need a multiplier, a simple percentage increase or decrease, or a percentage of an amount with easy numbers. In Paper 2, Section B usually sets it in real life: discounts, VAT on bills and invoices, profit and loss, or a price that changes. Show the change and what you divided by.

🎓 Later in IGCSE

In IGCSE 0580 multipliers are used everywhere: percentage increase and decrease, percentage profit and loss, then reverse percentages (finding the original price), compound interest and depreciation, where the same multiplier is used again and again.

🏆 Challenge
After a 20% increase, a pair of school shoes costs P96. What was the price before the increase?
👀 Reveal the solution
Step by step
  1. The new price is 120% of the old price
  2. So old price \(\times 1.2 = 96\)
  3. To undo \(\times 1.2\), divide by 1.2
  4. \(96 \div 1.2 = 80\)
  5. Check: \(80 \times 1.2 = 96\) ✓
Answer P80.00 (finding 20% of P96 and taking it off gives P76.80, which is wrong!)
📝 Remember

  • The original is 100%. Increase by r%: multiply by \(1 + \dfrac{r}{100}\). Decrease by r%: multiply by \(1 – \dfrac{r}{100}\).
  • Percentage change \(= \dfrac{\text{change}}{\text{original}} \times 100\).
  • Percentages can be more than 100%: 200% means double.
  • VAT at 14%: × 1.14. A 20% discount: × 0.8. Profit and loss % use the cost price as the original.
  • For several changes, multiply the multipliers: +10% then −10% gives × 0.99, not × 1.

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