Module 3 of 3Lesson 1 of 14 · Unit 25 · Fractions, decimals and percentages
Mathematics Form 1 · Term 3 · Unit 25

🔢 Fractions, decimals and percentages

NumberIGCSE path ✓JC ✓7Nf.017Nf.057Nf.06
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Why this matters

“50% off!”, “battery 7%”, “data bundle 80% used”. Percentages are everywhere because they make comparing easy: everything is turned into “out of 100”. In this unit you learn to switch between fractions, decimals and percentages, and to work out percentages of real amounts of money.

🎯 By the end of this unit

  • I can explain that per cent means “out of 100”.
  • I can convert between fractions, decimals and percentages.
  • I can understand percentages less than 1% and more than 100%.
  • I can find a percentage of an amount, with and without a calculator.
  • I can write one quantity as a percentage of another.
📖 Key words

per cent (%)
out of 100: 37% means 37 out of every 100
percentage
an amount written as a number out of 100
equivalent
the same value written in a different form, such as \(\dfrac{1}{2}\), 0.5 and 50%
decimal
a number using place value after the decimal point, such as 0.37
multiplier
the decimal you multiply by to find a percentage, such as 0.17 for 17%

1 Learn it

Per cent means “out of 100”

Per cent comes from the Latin per centum: “for every hundred”. The sign is %.

In the hundred square, 37 of the 100 small squares are shaded. That is 37%.

37 out of 100 can also be written as a fraction and as a decimal. All three describe exactly the same amount.

37 out of 100= 37%= 0.3737100
37 squares out of 100: three ways to write it.
⭐ Key ideaPercentage = how many out of 100. So 37% = \(\dfrac{37}{100}\) = 0.37.

Changing between the three forms

Percentage → decimal: divide by 100 (move the digits two places to the right). 37% = 0.37 and 8% = 0.08.

Decimal → percentage: multiply by 100. 0.6 = 60%.

Fraction → decimal: divide the top by the bottom, or make the denominator 100.

Percentage → fraction: write it over 100, then simplify.

  1. \(\dfrac{3}{20} = \dfrac{15}{100}\)
  2. \(\dfrac{15}{100} = 0.15\)
  3. 0.15 = 15%
FractionDecimalPercentagetop ÷ bottom× 100or place value÷ 100 backthree ways to write the same amount
Moving between fractions, decimals and percentages.
⭐ Key ideaDecimal × 100 = percentage. Percentage ÷ 100 = decimal.

Equivalents worth knowing by heart

These come up all the time. Learn them like your times tables, and use the number line to picture them.

FractionDecimalPercentage
\(\dfrac{1}{2}\)0.550%
\(\dfrac{1}{4}\)0.2525%
\(\dfrac{3}{4}\)0.7575%
\(\dfrac{1}{5}\)0.220%
\(\dfrac{1}{10}\)0.110%
\(\dfrac{1}{8}\)0.12512.5%
\(\dfrac{1}{3}\)0.333…33.3…%
000%140.2525%120.550%340.7575%11100%
Fractions, decimals and percentages on one number line.
⭐ Key ideaTo compare or order a mix of fractions, decimals and percentages, change them all to the same form (decimals are usually easiest).

Percentages less than 1%

1% is one square of the hundred square. Half of one square is 0.5%.

0.5% is less than one hundredth. As a decimal: \(0.5 \div 100 = 0.005\).

Bank charges and interest rates often use percentages like 0.5% or 0.25%.

  1. 0.5% = 0.005
  2. \(0.005 = \dfrac{5}{1000}\)
  3. \(\dfrac{5}{1000} = \dfrac{1}{200}\)
half of one square= 0.5%= 0.005
Zoom into one square: half of it is 0.5%.
⭐ Key ideaA percentage less than 1% is less than one hundredth of the whole: 0.5% = 0.005.

Percentages more than 100%

100% is the whole amount. More than 100% means more than the whole: one full hundred square and some more.

135% is one whole square and 35 more small squares. As a decimal it is 1.35.

If a school had 400 learners last year and 135% as many this year, it has more learners now: \(1.35 \times 400 = 540\).

100%35%135%= 1.35
135% is one whole and 35 hundredths more.
⭐ Key ideaOver 100% means bigger than the original. 200% is double; 150% is one and a half times.

Percentage of an amount: no calculator

Build any percentage from easy pieces. 10% is one tenth, so divide by 10. 50% is a half. 25% is a quarter. 1% is one hundredth.

To find 35% of P240: 10% is P24, so 30% is P72. 5% is half of 10%, which is P12. Add them: P84.

  1. 10% of 240: 240 ÷ 10 = 24
  2. 30% of 240: 24 × 3 = 72
  3. 5% of 240: 24 ÷ 2 = 12
  4. 35% of 240: 72 + 12 = 84
2424242424242424241210% = 24 each35% = 84whole bar = P240
Ten pieces of 10%; three and a half pieces make 35%.
⭐ Key idea10% = ÷ 10. 5% = half of 10%. 1% = ÷ 100. Build the rest from these.

Percentage of an amount: with a calculator

Change the percentage into a decimal (the multiplier), then multiply.

17% of P86: 17% = 0.17, and \(0.17 \times 86 = 14.62\). So the answer is P14.62.

This works because 17% means \(\dfrac{17}{100}\), and “of” means multiply.

⭐ Key ideaPercentage of an amount = decimal × amount. Give money answers to 2 decimal places.

One quantity as a percentage of another

Write the first quantity as a fraction of the second, then multiply by 100.

Kabelo got 18 out of 24 correct. As a fraction that is \(\dfrac{18}{24}\), which simplifies to \(\dfrac{3}{4}\). Three quarters is 75%.

Both quantities must be in the same units first: 45 cm out of 2 m means 45 cm out of 200 cm.

  1. \(\dfrac{18}{24} \times 100 = 75\)
  2. So 18 out of 24 is 75%
18 out of 24 correct1824=34= 75%
18 out of 24 is the same as 75 out of 100.
⭐ Key ideaPercentage = \(\dfrac{\text{part}}{\text{whole}} \times 100\). Same units first!

2 Watch it work

EXAMPLE 1

Fraction to decimal and percentage

Write \(\dfrac{3}{8}\) as a decimal and as a percentage.
  1. Divide the top by the bottom: 3 ÷ 8 = 0.375
  2. Multiply by 100: 0.375 × 100 = 37.5
Answer 0.375 and 37.5%
EXAMPLE 2

Very small and very large percentages

Write 0.6% and 145% as decimals and as fractions in their simplest form.
  1. 0.6%: 0.6 ÷ 100 = 0.006
  2. \(0.006 = \dfrac{6}{1000}\)
  3. \(\dfrac{6}{1000} = \dfrac{3}{500}\)
  4. 145%: 145 ÷ 100 = 1.45
  5. \(1.45 = \dfrac{145}{100}\)
  6. \(\dfrac{145}{100} = \dfrac{29}{20}\)
  7. \(\dfrac{29}{20} = 1\dfrac{9}{20}\)
Answer 0.6% = 0.006 = \(\dfrac{3}{500}\); 145% = 1.45 = \(1\dfrac{9}{20}\)
EXAMPLE 3

Percentage of an amount without a calculator

Find 35% of P240.
  1. 10% of P240 = P24
  2. 30%: 3 × P24 = P72
  3. 5%: half of P24 = P12
  4. 35%: P72 + P12 = P84
Answer P84
EXAMPLE 4

Percentage of an amount with a calculator

A farmer near Serowe harvests 2 450 kg of sorghum. 18% is kept as seed. How much is kept?
  1. 18% = 0.18
  2. 0.18 × 2450 = 441
Answer 441 kg
EXAMPLE 5

One quantity as a percentage of another

Lesego scored 42 out of 56 in a test. What is this as a percentage?
  1. \(\dfrac{42}{56} = \dfrac{3}{4}\)
  2. \(\dfrac{3}{4} \times 100 = 75\)
Answer 75%
EXAMPLE 6

Same units first

Write 300 g as a percentage of 2 kg.
  1. Change to the same units: 2 kg = 2000 g
  2. \(\dfrac{300}{2000} \times 100 = 15\)
  3. So 300 g is 15% of 2 kg
Answer 15%

3 Your turn

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

1🟢 Warm-up
Write 23% as a decimal and as a fraction.
👀 Show the answer
Step by step
  1. As a decimal: 23 ÷ 100 = 0.23
  2. As a fraction: \(\dfrac{23}{100}\)
Answer 0.23 and \(\dfrac{23}{100}\)
2🟢 Warm-up
Write 0.7 as a percentage.
👀 Show the answer
Step by step
  1. 0.7 × 100 = 70
Answer 70%
3🟢 Warm-up
Write \(\dfrac{3}{5}\) as a percentage.
👀 Show the answer
Step by step
  1. \(\dfrac{3}{5} = \dfrac{60}{100}\)
  2. 60 out of 100 is 60%
Answer 60%
4🟢 Warm-up
Find 10% of P350.
👀 Show the answer
Step by step
  1. 10% means divide by 10
  2. 350 ÷ 10 = 35
Answer P35
5🔵 Core
Write in order, smallest first: 0.45, \(\dfrac{2}{5}\), 42%
👀 Show the answer
Step by step
  1. Change all to decimals
  2. \(\dfrac{2}{5} = 0.4\)
  3. 42% = 0.42
  4. Order: 0.4, 0.42, 0.45
Answer \(\dfrac{2}{5}\), 42%, 0.45
6🔵 Core
Write 0.4% as a decimal and as a fraction in its simplest form.
👀 Show the answer
Step by step
  1. 0.4 ÷ 100 = 0.004
  2. \(0.004 = \dfrac{4}{1000}\)
  3. \(\dfrac{4}{1000} = \dfrac{1}{250}\)
Answer 0.004 and \(\dfrac{1}{250}\)
7🔵 Core
Write 250% as a decimal and as a mixed number.
👀 Show the answer
Step by step
  1. 250 ÷ 100 = 2.5
  2. \(2.5 = 2\dfrac{1}{2}\)
Answer 2.5 and \(2\dfrac{1}{2}\)
8🔵 Core
Without a calculator, find 15% of 80.
👀 Show the answer
Step by step
  1. 10% of 80 = 8
  2. 5% of 80 = 4
  3. 15% of 80: 8 + 4 = 12
Answer 12
9🔵 Core
Without a calculator, find 12.5% of 64 kg.
👀 Show the answer
Step by step
  1. 12.5% = \(\dfrac{1}{8}\)
  2. 64 ÷ 8 = 8
Answer 8 kg
10🔵 Core
VAT in Botswana is 14%. Find the VAT on a fridge that costs P1 250 before VAT.
👀 Show the answer
Step by step
  1. 14% = 0.14
  2. 0.14 × 1250 = 175
Answer P175.00
11🔵 Core
Mpho got 27 out of 45 in a test. What percentage is this?
👀 Show the answer
Step by step
  1. \(\dfrac{27}{45} = \dfrac{3}{5}\)
  2. \(\dfrac{3}{5} \times 100 = 60\)
Answer 60%
12🔵 Core
Write 45 minutes as a percentage of 2 hours.
👀 Show the answer
Step by step
  1. Same units: 2 hours = 120 minutes
  2. \(\dfrac{45}{120} \times 100 = 37.5\)
  3. So 45 minutes is 37.5% of 2 hours
Answer 37.5%
13🟠 Stretch
The population of a village grew from 8 000 to 10 000. Write the new population as a percentage of the old population.
👀 Show the answer
Step by step
  1. \(\dfrac{10000}{8000} = \dfrac{5}{4}\)
  2. \(\dfrac{5}{4} = 1.25\)
  3. 1.25 × 100 = 125
Answer 125% (more than 100%, because the village grew)
14🟠 Stretch
A bank charges P0.60 to send P150. What percentage of the amount sent is the charge?
👀 Show the answer
Step by step
  1. \(\dfrac{0.60}{150} = 0.004\)
  2. 0.004 × 100 = 0.4
Answer 0.4% (less than 1%)
15🟠 Stretch
Thato scored 34 out of 40 in Science and 42 out of 50 in Maths. In which subject did she do better?
👀 Show the answer
Step by step
  1. Science: \(\dfrac{34}{40} \times 100 = 85\)%
  2. Maths: \(\dfrac{42}{50} \times 100 = 84\)%
  3. 85% is more than 84%
Answer Science (85% against 84%)

4 Watch out!

✗ Wrong0.5 = 5%
✓ Right0.5 = 50%

To change a decimal to a percentage, multiply by 100: 0.5 × 100 = 50. Half is 50%, not 5%.

✗ Wrong3% = 0.3
✓ Right3% = 0.03

3% means 3 out of 100, which is 3 hundredths. 0.3 is 3 tenths, which is 30%.

✗ WrongTo find 20%, divide by 20.
✓ Right20% = \(\dfrac{1}{5}\), so divide by 5 (or find 10% and double it).

Only 10% means “divide by 10”. 20% is twice 10%, which is one fifth.

✗ Wrong50 cm as a percentage of 2 m: \(\dfrac{50}{2} \times 100\) = 2500%
✓ Right2 m = 200 cm, so \(\dfrac{50}{200} \times 100\) = 25%

Both quantities must be in the same units before you divide.

5 In your exams

🇧🇼 Junior Certificate (JCE)

Converting between fractions, decimals and percentages is a common JCE Paper 1 multiple-choice question, often as “which of these is the largest?”. Finding a percentage of an amount and writing one quantity as a percentage of another usually appear in Paper 2 money problems, such as VAT, discount, marks and bills. Show your fraction or multiplier as a working step.

🎓 Later in IGCSE

IGCSE 0580 builds straight on this: percentages of amounts, percentage increase and decrease using multipliers, reverse percentages and simple and compound interest. Converting between fractions, decimals and percentages, and ordering them, are frequent non-calculator Paper 1 questions.

🏆 Challenge
Neo spends 35% of his money on airtime and \(\dfrac{2}{5}\) of it on transport. He has P75 left. How much money did he have at the start?
👀 Reveal the solution
Step by step
  1. \(\dfrac{2}{5}\) = 40%
  2. Spent: 35% + 40% = 75%
  3. Left: 100% − 75% = 25%
  4. 25% is P75, so 100% is P75 × 4
  5. P75 × 4 = P300
Answer P300
📝 Remember

  • Per cent means out of 100: 37% = \(\dfrac{37}{100}\) = 0.37.
  • Decimal × 100 = percentage; percentage ÷ 100 = decimal; fraction: divide top by bottom.
  • Less than 1% is less than one hundredth (0.5% = 0.005); more than 100% is more than the whole (135% = 1.35).
  • Percentage of an amount: build from 10%, 5% and 1%, or multiply by the decimal.
  • One quantity as a percentage of another: \(\dfrac{\text{part}}{\text{whole}} \times 100\), with the same units.

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