Fractions, Decimals and Percentages
These are three ways of writing the same thing. Questions test whether you can move between them fluently and calculate with fractions without a calculator — which is exactly where the marks are lost.
1. Converting between the three forms
| To do this | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide top by bottom | 3/8 = 3 ÷ 8 = 0.375 |
| Decimal → Fraction | Place value, then simplify | 0.35 = 35/100 = 7/20 |
| Fraction → Percentage | × 100 | 13/20 = 0.65 = 65% |
| Percentage → Fraction | over 100, then simplify | 45% = 45/100 = 9/20 |
| Decimal → Percentage | × 100 | 0.7 = 70% |
| Percentage → Decimal | ÷ 100 | 8% = 0.08 |
A percentage sign and a decimal are not interchangeable. 0.65 and 65% are equal, but “0.65%” means something completely different (it is 0.0065). Adding a % sign to a decimal answer was a recorded error.
Worth memorising:
| Fraction | Decimal | % | Fraction | Decimal | % | |
|---|---|---|---|---|---|---|
| 1/2 | 0.5 | 50% | 1/5 | 0.2 | 20% | |
| 1/4 | 0.25 | 25% | 2/5 | 0.4 | 40% | |
| 3/4 | 0.75 | 75% | 1/8 | 0.125 | 12.5% | |
| 1/3 | 0.333… | 33.3% | 1/10 | 0.1 | 10% |
2. Mixed numbers and improper fractions
Mixed → Improper: multiply the whole number by the denominator, add the numerator, keep the denominator.
3¼ = (3 × 4 + 1)/4 = 13/4
Improper → Mixed: divide, and the remainder becomes the numerator.
17/5 = 3 remainder 2 = 3⅖
Convert mixed numbers to improper fractions BEFORE multiplying or dividing. Trying to multiply 2½ × 1⅓ in mixed form is the standard route to a wrong answer. Convert first, every time.
3. Simplifying
Divide the numerator and denominator by their HCF.
18/24 → both divide by 6 → 3/4
Simplify fully in one step if you can. Dividing 18/24 by 2 to get 9/12 is correct but unfinished; the mark is usually for the fully simplified form.
4. The four operations
Adding and subtracting — common denominator needed
You must have the same denominator before adding or subtracting.
2/3 + 1/4
- LCM of 3 and 4 is 12
- 2/3 = 8/12, 1/4 = 3/12
- = 11/12
Never add the denominators. 2/3 + 1/4 is not 3/7. Adding without a common denominator was one of the most frequent errors recorded.
The LCM is the SMALLEST common multiple, not the product. For 4 and 6 it is 12, not 24. Multiplying the denominators always works but leaves more simplifying to do — and taking the product as the LCM was a documented error.
With mixed numbers, either convert to improper fractions, or deal with whole numbers and fractions separately — but be careful when the fraction part needs borrowing.
Multiplying — the easy one
Multiply the numerators, multiply the denominators. No common denominator needed.
2/3 × 4/5 = 8/15
Cancel before multiplying where possible: 3/4 × 8/9 → cancel 3 with 9 and 4 with 8 → 1/1 × 2/3 = 2/3
“Of” means multiply. 5/8 of 24 = 5/8 × 24 = 15.
Dividing — flip and multiply
Keep, Change, Flip — keep the first fraction, change ÷ to ×, flip the second.
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Flip the SECOND fraction only. The reciprocal of 4/5 is 5/4 — swapping the wrong one, or flipping both, gives a wrong answer that looks tidy.
5. Recurring decimals to fractions (Extended)
Convert 0.4̇5̇ (0.454545…) to a fraction.
- Let x = 0.454545…
- Two digits repeat, so multiply by 100: 100x = 45.454545…
- Subtract: 100x − x = 45 → 99x = 45
- x = 45/99 = 5/11
Multiply by 10 for one repeating digit, 100 for two, 1000 for three — enough to line the recurring parts up so they cancel on subtraction.
Quick check: 0.3̇ = 3/9 = 1/3, and 0.6̇ = 6/9 = 2/3.
6. Ordering
Convert everything to decimals first, then compare.
Order 0.65, 3/5, 62%:
- 0.65, 3/5 = 0.6, 62% = 0.62
- Ascending: 3/5, 62%, 0.65
Give the answer in the original form if the question asks for it — convert to compare, then convert back.
Read whether the question wants ascending or descending order.
7. Calculators
Tutors were explicit that some papers are non-calculator, and this topic is where that matters most.
Check which of your papers allow a calculator. For a non-calculator paper you need the fraction methods above written out step by step. Advice heard in lessons on this point was inconsistent, so confirm against your own exam timetable and syllabus version.
Show every step in fraction working. Marks are available for the common denominator and the unsimplified answer, even if the final simplification slips.
8. Mistakes that cost marks
Adding denominators instead of finding a common one.
Using the product of the denominators as the LCM.
Multiplying mixed numbers without converting first.
Flipping the wrong fraction when dividing.
Not simplifying fully.
Adding a % sign to a decimal answer.
Dividing by 100 when you should multiply, converting between decimals and percentages.
Converting a fraction to a percentage wrongly — 13/20 is 65%, not 70%.
Comparing without converting to a common form.
Ordering in the wrong direction.
Frequently asked questions
How do I convert a fraction to a decimal? Divide the numerator by the denominator.
How do I convert a fraction to a percentage? Convert to a decimal, then multiply by 100.
How do I add fractions? Find a common denominator, convert both, then add the numerators only.
Do I need a common denominator to multiply? No — multiply the tops and the bottoms.
How do I divide fractions? Keep, Change, Flip — flip the second fraction and multiply.
How do I convert a mixed number to an improper fraction? Whole × denominator, add the numerator, keep the denominator.
How do I turn a recurring decimal into a fraction? Multiply by 10, 100 or 1000 so the recurring parts align, subtract, and solve.
What does “of” mean? Multiply.
How do I order fractions, decimals and percentages? Convert them all to decimals, compare, then answer in the form requested.
Quick revision checklist
- I can convert freely between fractions, decimals and percentages
- I know the common conversions by heart
- I never write a % sign on a decimal answer
- I can convert between mixed numbers and improper fractions
- I convert mixed numbers before multiplying or dividing
- I simplify fully using the HCF
- I find a common denominator before adding or subtracting
- I use the lowest common multiple
- I multiply fractions straight across, cancelling first
- I know “of” means multiply
- I flip the second fraction when dividing
- I can convert a recurring decimal to a fraction
- I convert to decimals to order, and answer in the required form
- I know which of my papers allow a calculator
These notes cover fractions, decimals and percentages in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus, formula list and calculator rules for your own exam series.
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