Percentages Toolkit

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Lesson notes

Basic Percentages

Percentage means 'out of 100' (e.g., 50% = 50/100).
Convert a fraction to a percentage by finding an equivalent fraction with denominator 100, or by using a calculator: e.g., 234/650 = 0.36 = 36%.
Key building blocks: 50% (halve), 25% (halve twice), 10% (divide by 10), 1% (divide by 100).
Combine blocks: e.g., 20% = 10% × 2; 5% = 10% ÷ 2; 12% = 10% + 1% + 1%.
With a calculator, use a multiplier (decimal equivalent): e.g., 12% of 650 = 0.12 × 650 = 78.
To express one number as a percentage of another, write as a fraction then convert to decimal and multiply by 100.
Worked example: shade 35% of a 40-square grid — 10% = 4, so 30% = 12 and 5% = 2, total 14 squares shaded.

35% of a 40-square grid

Percentage Increases & Decreases

To increase by a percentage, find the percentage of the original and add it on, or find (100 + x)% and use multiplier (1 + x/100).
To decrease by a percentage, find the percentage and subtract, or find (100 - x)% and use multiplier (1 - x/100).
For repeated percentage changes, multiply all multipliers together: e.g., increase by 10% (×1.10) then discount 15% (×0.85) gives combined multiplier 1.10 × 0.85 = 0.935.
Percentage change = (after - before) / before × 100; positive = increase, negative = decrease.
Percentage profit/loss = (selling price - cost price) / cost price × 100; positive = profit, negative = loss.

Reverse Percentages

A reverse percentage question gives the value after a percentage change and asks for the original (before) value.
Identify the multiplier p: for an increase of x%, p = 1 + x/100; for a decrease of x%, p = 1 - x/100.
Use the equation: before × p = after, so before = after / p.
Common mistake: a 10% increase followed by a 10% decrease does NOT return to the original (e.g., 100 → 110 → 99).

Slides

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Practice questions

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1.What is 12% of 650?
Easy
- A78
- B65
- C72
- D80
2.Write $1.92 as a percentage of $1.60.
Medium
- A120%
- B83.3%
- C80%
- D125%
3.In a box of 80 glasses, 3 are broken. Work out the percentage of broken glasses.
Easy
- A3.75%
- B2.5%
- C4%
- D3%
4.Work out 5% of $25.
Easy
- A$1.25
- B$1.00
- C$1.50
- D$0.75
5.Work out $1.20 as a percentage of $16.
Medium
- A7.5%
- B8%
- C6%
- D9%
6.67 of 645 parcels are damaged. Calculate the percentage of parcels that are damaged.
Medium
- A10.4%
- B9.8%
- C11.2%
- D10.8%
7.Increase 42 by 16%.
Medium
- A48.72
- B46.2
- C49.2
- D47.8
8.The surface area of France is 6.41 × 10⁵ km² and China is 9.60 × 10⁶ km². Find France's area as a percentage of China's.
Medium
- A6.68%
- B7.5%
- C5.9%
- D8.1%

More questions

9.The Earth's population in 2017 was 7.53 × 10⁹ and in 2000 was 6.02 × 10⁹. Calculate the percentage increase.
Medium
- A25.1%
- B20.0%
- C22.5%
- D18.9%
10.Manjeet uses 220 litres of water daily. She reduces usage by 15%. How many litres does she now use?
Medium
- A187
- B190
- C185
- D180
11.Marianne sells photos at $6 each, making a profit of $2.25 each. Calculate her profit as a percentage of the selling price.
Medium
- A37.5%
- B40%
- C35%
- D30%
12.The selling price of a shirt is $26.50 including 6% tax. Calculate the price before tax.
Medium
- A$25.00
- B$24.91
- C$25.50
- D$24.80
13.Adil gets a 7% pay decrease to $427.80. Brian gets a 7% increase from the same original wage. Work out Brian's new wage.
Hard
- A$492.20
- B$480.00
- C$500.00
- D$475.80
14.Shona buys a chair for $435.60 in a 12% reduction sale. Calculate the original price.
Medium
- A$495.00
- B$488.00
- C$500.00
- D$475.00
15.Jerry bought a house in 2003. In 2007 he sold it to Mia at a 20% profit. In 2012 Mia sold it for £162,000 at a 10% loss. How much did Jerry pay in 2003?
Hard
- A£150,000
- B£145,000
- C£155,000
- D£140,000
16.Marie invests £8000 for one year. Interest is added and she pays 20% tax on the interest, totalling £28.80. What is the percentage interest rate?
Hard
- A1.8%
- B2.0%
- C1.5%
- D2.2%
17.The Earth has a surface area of approximately 510,100,000 km². Water covers 70.8% of the Earth's surface. Work out the area of the Earth's surface covered by water.
Medium
- A361,150,800 km²
- B361,150,000 km²
- C361,151,000 km²
- D361,000,000 km²
18.The Muller family have $1962 New Zealand dollars. They spend 15% of their dollars on a tour. Calculate the number of dollars they have left.
Medium
- A$1667.70
- B$294.30
- C$1667.00
- D$196.20
19.Work out $1.45 as a percentage of $72.50.
Easy
- A2%
- B0.02%
- C20%
- D0.2%
20.Bill's weight decreases from 64.8 kg to 59.3 kg. Calculate the percentage decrease in Bill's weight. Give your answer correct to 3 significant figures.
Medium
- A8.49%
- B9.27%
- C8.5%
21.Tom is a ski jumper. The maximum length of skis he can use is 145% of his height. Tom's height is 1.80 m. Work out the maximum length of skis Tom can use.
Medium
- A2.61 m
- B2.6 m
- C2.16 m
22.Emily buys a pack of 12 bottles of water. The pack costs £5.64. Emily sells all 12 bottles for 50p each. Work out Emily's percentage profit. Give your answer correct to 1 decimal place.
Medium
- A6.4%
- B6.5%
- C6.3%
23.One month, Gretal spent a total of $360 on presents. She spent 1/5 of this total on presents for her parents. She spent 2/3 of the remaining money on presents for her friends. She spent the rest on presents for her sisters. Calculate the percentage of the $360 that she spent on presents for her sisters.
Hard
- A26.7%
- B26.6%
- C26.8%
24.Dhanu has a model railway. He has a train that consists of a locomotive and 4 coaches. The mass of the locomotive is 87 g and the mass of each coach is 52 g. Work out the mass of the locomotive as a percentage of the total mass of the train.
Medium
- A29.5%
- B29.6%
- C29.4%
25.Increase $22 by 15%.
Easy
- A$25.30
- B$25.20
- C$25.00
26.The cost of a Premium ticket is reduced from $114 to $96.90. Calculate the percentage reduction in the cost of a ticket.
Medium
- A15%
- B17.7%
- C15.5%
27.Arjun earned $36,515 in 2019. This was an increase of 9% on his earnings in 2018. Work out his earnings in 2018.
Medium
- A$33,500
- B$33,600
- C$33,400
28.In a sale the original prices are reduced by 15%. Calculate the original price of a jacket that has a sale price of $38.25.
Medium
- A$45.00
- B$44.00
- C$45.50
29.In September the total income at a campsite was $37,054. This was a decrease of 4.5% on the total income in August. Calculate the total income in August.
Medium
- A$38,800
- B$38,700
- C$38,900
30.The cost of a railway magazine increases by 12.5% to $2.70. Calculate the cost of the magazine before this increase.
Medium
- A$2.40
- B$2.50
- C$2.45
31.A shop increases the price of a coat from $80 by 10%. A month later, the shop discounts the coat by 15%. What is the final price of the coat?
Medium
- A$74.80
- B$75.20
- C$76.00
- D$79.20
32.A loaf of bread loses 12% of its weight when baked. Claire wants the baked loaf to weigh 1.1 kg. What weight of dough should she use before baking?
Medium
- A1.25 kg
- B1.232 kg
- C1.212 kg
- D1.1 kg
33.A force of 70 newtons acts on an area of 20 cm². The force is increased by 10 newtons and the area by 10 cm². Helen says 'The pressure decreases by less than 20%'. Is Helen correct?
Hard
- AYes, the pressure decreases by 15%
- BNo, the pressure decreases by 25%
- CYes, the pressure decreases by 18%
- DNo, the pressure decreases by 22%
34.George wants to watch all 23 home games. A season ticket costs £425, or 23 separate tickets at £24 each. What percentage does he save by buying a season ticket?
Medium
- A23.0%
- B22.8%
- C23.2%
- D22.5%
35.What is the decimal equivalent of 25%?
Easy
- A0.25
- B2.5
- C0.025
- D25.0
36.To find 50% of a number, you multiply the number by 0.5.
Easy
True or false?
37.A shop has a sale with 20% off all items. What is the sale price of a jacket originally priced at $85?
Easy
- A$68
- B$65
- C$80
- D$17
38.The number of visitors to a museum increased from 2400 to 3000. What is the percentage increase?
Medium
- A25%
- B20%
- C30%
- D12.5%
39.Which of the following multipliers represent a percentage decrease? (Select all that apply.)
Medium
- A0.92
- B1.05
- C0.75
- D1.00
- E1.20
40.Arrange the following in order of increasing size: 0.3, 30%, 0.03, 3%.
Easy
- 0.3
- 30%
- 0.03
- 3%
41.What is 1% of 500?
Easy
- A5
- B0.5
- C50
- D0.05
42.Match each percentage change to its correct multiplier.
Medium
- Increase by 8%
- Decrease by 15%
- Increase by 150%
- Decrease by 2.5%
- 0.975
- 1.08
- 2.5
- 0.85
43.Put the following in order from smallest to largest: 0.08, 8%, 0.8, 80%, 0.088.
Medium
- 0.08
- 8%
- 0.8
- 80%
- 0.088
44.To find 30% of a number, you can multiply the number by 0.3.
Easy
True or false?
45.Find 10% of 350.
Easy
46.Complete the sentence.
Easy
A percentage increase of 8% corresponds to a multiplier of ____.
47.Explain what a reverse percentage is in one sentence.
Easy
48.After a 15% increase, a worker's salary is $36,800. What was the original salary?
Medium
49.The price of a laptop is increased by 12% and then decreased by 10%. The final price is $756. Find the original price before any changes.
Hard
50.A shop increases the price of a coat from $80 by 10%. A month later, the shop discounts the coat by 15%. What is the final price? Show your working.
Medium
51.Find 10% of 350.
Easy
- A31
- B43
- C35
- D39
52.After a 15% increase, a worker's salary is $36,800. What was the original salary?
Medium
- A32000 $
- B28800 $
- C35200 $
- D38400 $
53.The price of a laptop is increased by 12% and then decreased by 10%. The final price is $756. Find the original price before any changes.
Hard
- A675 $
- B900 $
- C825 $
- D750 $

Revision notes

## Percentages Toolkit ### Basic Percentages - **Percentage** means "out of 100". - To find a percentage of an amount without a calculator: use 50%, 25%, 10%, 1% as building blocks. - With a calculator: use a **multiplier** (decimal equivalent). E.g., 12% = 0.12; 126% = 1.26. - To express one number as a percentage of another: write as a fraction, convert to decimal, multiply by 100. ### Percentage Increases & Decreases - **Increase by x%**: new amount = original × (1 + x/100). Multiplier > 1. - **Decrease by x%**: new amount = original × (1 - x/100). Multiplier < 1. - **Percentage change**: - Multiplier method: m = after / before. (m > 1 = increase, m < 1 = decrease) - Formula: % change = (after - before) / before × 100. (positive = increase, negative = decrease) - **Percentage profit/loss**: - Multiplier: m = selling price / cost price. (m > 1 = profit, m < 1 = loss) - Formula: % profit = (selling price - cost price) / cost price × 100. - For repeated percentage changes, multiply all multipliers together first. ### Reverse Percentages - Given the final amount after a percentage change, find the original amount. - Let multiplier p = 1 ± (percentage/100). - Use: before × p = after → before = after / p. - Common mistake: a 10% increase followed by a 10% decrease does NOT return to the original. ### Key Formulas - **Multiplier for increase**: 1 + (percentage/100) - **Multiplier for decrease**: 1 - (percentage/100) - **Percentage change**: (new - original) / original × 100 - **Reverse percentage**: original = new / multiplier

Percentage: A number or ratio expressed as a fraction of 100, denoted by the symbol '%'.
Multiplier: The decimal equivalent of a percentage used to increase or decrease an amount. For an increase of x%, multiplier = 1 + x/100; for a decrease of x%, multiplier = 1 - x/100.
Reverse percentage: Finding the original amount before a percentage change, given the final amount. Formula: original = final / multiplier.
Percentage change: The relative change between an old and new value, expressed as a percentage. Formula: (new - old) / old × 100.
Percentage profit/loss: Profit or loss expressed as a percentage of the cost price. Formula: (selling price - cost price) / cost price × 100.

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