IGCSE Maths · Number · Core & Extended

Fractions, Decimals and Percentages: Moving Between Forms

Fractions, decimals and percentages are three different costumes for the same underlying number. Being able to switch between them instantly — without a calculator — is one of the most exam-rewarding skills in the whole Number topic, because it shows up inside almost every other question, from percentages to probability to statistics.

Why this matters A shop advertises “25% off,” a recipe calls for “¾ of a cup,” and a weather app says there’s a “0.3 probability of rain.” Three completely different-looking numbers — a percentage, a fraction, a decimal — and yet you compare them constantly without a second thought. That fluency is exactly what this note is building on purpose.

1. Types of Fractions

Proper fraction — numerator smaller than denominator, e.g. 3/5
Improper fraction — numerator larger than (or equal to) denominator, e.g. 9/4
Mixed number — a whole number and a proper fraction together, e.g. 2 1/4
Simplest form — numerator and denominator share no common factor except 1
Worked example — simplifying (using HCF) Simplify 18/24 to its simplest form.
HCF of 18 and 24 is 6. Divide both by 6: 18÷6 = 3, 24÷6 = 4.
Answer: 3/4
Worked example — converting improper ↔ mixed Write 17/5 as a mixed number.
17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5

Write 4 3/7 as an improper fraction.
(4 × 7) + 3 = 31, so 4 3/7 = 31/7
Try it — Fraction Basics
  1. Simplify 20/28 to its simplest form.
  2. Simplify 45/60 to its simplest form.
  3. Write 23/6 as a mixed number.
  4. Write 5 2/9 as an improper fraction.
  5. Is 11/4 a proper fraction, an improper fraction, or a mixed number?
Show answers
  1. 5/7
  2. 3/4
  3. 3 5/6
  4. 47/9
  5. Improper fraction (numerator bigger than denominator).

2. Converting Fractions to Decimals

A fraction is really a division waiting to happen — a/b means “a divided by b.” Use short or long division to convert.

Worked example Convert 3/8 to a decimal.
3 ÷ 8 = 0.375
Worked example Convert 7/20 to a decimal.
7 ÷ 20 = 0.35
Try it — Fractions to Decimals
  1. Convert 1/4 to a decimal.
  2. Convert 9/40 to a decimal.
  3. Convert 5/8 to a decimal.
  4. Convert 11/25 to a decimal.
Show answers
  1. 0.25
  2. 0.225
  3. 0.625
  4. 0.44

3. Converting Decimals to Fractions

Use place value: the digits after the decimal point become the numerator, and the denominator is 10, 100, 1000, and so on, depending on how many decimal places there are. Then simplify.

Worked example Convert 0.65 to a fraction in simplest form.
0.65 = 65/100. HCF of 65 and 100 is 5.
65 ÷ 5 = 13, 100 ÷ 5 = 20 → 13/20
Worked example Convert 0.125 to a fraction in simplest form.
0.125 = 125/1000. HCF of 125 and 1000 is 125.
125 ÷ 125 = 1, 1000 ÷ 125 = 8 → 1/8
Try it — Decimals to Fractions
  1. Convert 0.4 to a fraction in simplest form.
  2. Convert 0.75 to a fraction in simplest form.
  3. Convert 0.32 to a fraction in simplest form.
  4. Convert 0.008 to a fraction in simplest form.
Show answers
  1. 2/5
  2. 3/4
  3. 8/25
  4. 1/125

4. Converting To and From Percentages

“Percent” literally means “per hundred” — so a percentage is just a fraction with denominator 100.

Fraction/decimal → percentage: multiply by 100
Percentage → fraction/decimal: divide by 100
Worked examples Write 3/4 as a percentage: 3/4 × 100 = 75%
Write 0.6 as a percentage: 0.6 × 100 = 60%
Write 45% as a fraction in simplest form: 45/100 = 9/20
Write 8% as a decimal: 8 ÷ 100 = 0.08
Try it — Percentages
  1. Write 2/5 as a percentage.
  2. Write 0.34 as a percentage.
  3. Write 70% as a fraction in simplest form.
  4. Write 125% as a decimal.
  5. Write 5% as a fraction in simplest form.
Show answers
  1. 40%
  2. 34%
  3. 7/10
  4. 1.25
  5. 1/20

5. Common Equivalents Worth Memorising

Knowing these instantly — without working them out — saves real time in the exam.

FractionDecimalPercentage
1/20.550%
1/40.2525%
3/40.7575%
1/50.220%
1/100.110%
1/80.12512.5%
1/30.333…33.3%
2/30.666…66.7%

Notice 1/3 and 2/3 never terminate — this is a perfect bridge into the next section.

Try it — Mixed Conversions
  1. Put these in order from smallest to largest: 0.4, 3/8, 45%, 2/5.
  2. A test score is 17/20. Write this as a percentage.
  3. A shirt is reduced by 30%. Write 30% as a fraction in simplest form.
Show answers
  1. 3/8 (0.375), 0.4, 2/5 (0.4 — equal to the decimal 0.4), 45% → correctly: 3/8, 0.4, 2/5, 45% (0.4 and 2/5 are equal, so either order between them is fine)
  2. 85%
  3. 3/10

6. Recurring Decimals Extended only

Some fractions never terminate when converted to decimals — instead, a block of digits repeats forever. These are called recurring decimals. In exams, a dot is placed above the first and last digit of the repeating block (a single dot if only one digit repeats). Here, we’ll also show the repeating block with a line over it, which means the same thing and is often easier to read.

Reading recurring decimal notation 0.17 means the 7 repeats forever: 0.1777…
0.123 means “23” repeats forever: 0.1232323…
0.123 means the whole block “123” repeats forever: 0.123123123…
Try it — Reading Recurring Decimals Extended
  1. Write out 0.6 as a decimal to 6 decimal places.
  2. Write out 0.245 as a decimal to 7 decimal places.
  3. Convert 1/3 to a decimal and describe the repeating pattern.
Show answers
  1. 0.666666…
  2. 0.2454545…
  3. 1/3 = 0.333… — the digit 3 repeats forever.

7. Converting Recurring Decimals to Fractions Extended only

This is a genuinely elegant algebra trick: multiply by a power of 10 that shifts the repeating block exactly one cycle, then subtract to make the repeating part cancel out completely.

Worked example — single repeating digit Write 0.16 as a fraction.
Let x = 0.1666…
10x = 1.666…
100x = 16.666…
Subtract: 100x − 10x = 16.666… − 1.666… = 15
90x = 15 → x = 15/90 = 1/6
Worked example — two-digit repeating block Write 0.47 as a fraction.
Let x = 0.474747…
100x = 47.474747…
Subtract: 100x − x = 47.474747… − 0.474747… = 47
99x = 47 → x = 47/99
Worked example — three-digit repeating block Write 0.123 as a fraction.
Let x = 0.123123123…
1000x = 123.123123…
Subtract: 1000x − x = 123
999x = 123 → x = 123/999, which simplifies (divide by 3) to 41/333
Try it — Recurring Decimals to Fractions Extended
  1. Write 0.4 as a fraction.
  2. Write 0.18 as a fraction.
  3. Write 0.25 as a fraction (careful — the 2 does not repeat).
  4. Write 0.371 as a fraction in simplest form.
Show answers
  1. Let x = 0.4444…; 10x = 4.444…; 10x − x = 4 → 9x = 4 → x = 4/9
  2. Let x = 0.181818…; 100x = 18.1818…; 100x − x = 18 → 99x = 18 → x = 18/99 = 2/11
  3. Let x = 0.2555…; 10x = 2.555…; 100x = 25.555…; 100x − 10x = 23 → 90x = 23 → x = 23/90
  4. Let x = 0.371371…; 1000x = 371.371…; 1000x − x = 371 → 999x = 371 → x = 371/999 (this is already in simplest form, since 371 = 7 × 53 and 999 = 3³ × 37, sharing no common factor)
Common mistakes to watch for
  • Simplifying a fraction using the wrong common factor — always find the HCF, don’t just guess a number that “looks like” it divides evenly.
  • Forgetting to convert a mixed number to an improper fraction before further calculations.
  • Multiplying a fraction or decimal by 100 to get a percentage, then forgetting the % sign — 0.6 → 60, not 60%, unless the symbol is written.
  • For recurring decimals, choosing the wrong power of 10 to multiply by — it must shift the decimal point by exactly one full repeating cycle, not more, not less.
  • Treating every non-terminating decimal as irrational — recurring decimals are rational (that’s the whole point of this section); only non-repeating, non-terminating decimals like π are irrational.

8. Quick-Fire Challenge Round

True or False?
  1. Every terminating decimal can be written as a fraction.
  2. 0.5 and 1/2 represent the same number.
  3. A recurring decimal is always irrational.
  4. 75% is bigger than 3/4.
  5. To convert a fraction to a percentage, you divide by 100.
Show answers
  1. True.
  2. True.
  3. False — recurring decimals are rational; they can always be written as a fraction. Ext.
  4. False — they’re exactly equal (75% = 3/4).
  5. False — you multiply by 100 to convert to a percentage, not divide.
Spot the Error
  1. “0.4 as a fraction is 4/10, fully simplified.”
  2. “To write 3/5 as a percentage, divide 3 by 5 to get 0.6, then the percentage is 0.6%.”
  3. “0.3333… is irrational because it never ends.”
Show answers
  1. 4/10 hasn’t been simplified fully — the HCF of 4 and 10 is 2, so it should be 2/5.
  2. The decimal must be multiplied by 100, not left as-is: 0.6 × 100 = 60%, not 0.6%.
  3. Not ending isn’t enough — 0.3333… repeats forever, which makes it rational (= 1/3). Only decimals that never end and never repeat, like π, are irrational.

9. Mixed Practice — Bringing It All Together

Core & Extended
  1. Simplify 32/40 to its simplest form.
  2. Convert 9/16 to a decimal.
  3. Convert 0.28 to a fraction in simplest form.
  4. Write 5/8 as a percentage.
  5. Order from smallest to largest: 0.6, 3/5, 58%, 5/8.
Show answers
  1. 4/5
  2. 0.5625
  3. 7/25
  4. 62.5%
  5. 58%, 0.6 and 3/5 (equal), 5/8 → 58%, 3/5, 0.6, 5/8
Extended-level mixed set Extended only
  1. Write 0.8 as a fraction.
  2. Write 0.36 as a fraction.
  3. Convert 5/6 to a decimal. Is it terminating or recurring?
  4. Write 0.16 as a fraction in simplest form.
Show answers
  1. x = 0.888…; 10x = 8.888…; 9x = 8 → x = 8/9
  2. x = 0.363636…; 100x = 36.3636…; 99x = 36 → x = 36/99 = 4/11
  3. 5/6 = 0.8333… — recurring (the 3 repeats).
  4. x = 0.1666…; 10x = 1.666…; 100x = 16.666…; 90x = 15 → x = 15/90 = 1/6

Practice at Home

A large set again, in stages — no need to finish it all in one sitting.

Part A — Fraction basics
  1. Simplify 36/48 to its simplest form.
  2. Simplify 21/35 to its simplest form.
  3. Write 29/8 as a mixed number.
  4. Write 6 4/5 as an improper fraction.
Part B — Fractions and decimals
  1. Convert 7/8 to a decimal.
  2. Convert 3/40 to a decimal.
  3. Convert 0.85 to a fraction in simplest form.
  4. Convert 0.056 to a fraction in simplest form.
Part C — Percentages
  1. Write 4/5 as a percentage.
  2. Write 0.09 as a percentage.
  3. Write 65% as a fraction in simplest form.
  4. Write 110% as a decimal.
Part D — Mixed ordering
  1. Order from smallest to largest: 0.55, 5/9, 56%, 11/20.
  2. Order from smallest to largest: 2/3, 0.65, 68%, 0.7.
Part E — Recurring decimals Extended
  1. Write out 0.54 to 8 decimal places.
  2. Convert 2/9 to a decimal and describe the pattern.
  3. Convert 5/11 to a decimal (this one has a two-digit repeating block).
Part F — Converting recurring decimals to fractions Extended
  1. Write 0.7 as a fraction.
  2. Write 0.27 as a fraction in simplest form.
  3. Write 0.36 as a fraction (careful with the non-repeating part).
  4. Write 0.142857 as a fraction in simplest form (hint: this is a famous one — check what fraction it comes from once you’re done).
Part G — Applied / challenge
  1. A student scores 34/40 on a test. Write this as a percentage, then as a decimal.
  2. Two shops offer the same item: Shop A discounts it by 2/5, Shop B discounts it by 38%. Which discount is bigger?
  3. Show algebraically that 0.9 (i.e. 0.999…) is exactly equal to 1 — not just “very close to” 1.

Key Vocabulary Recap

Proper fraction — numerator smaller than denominator
Improper fraction — numerator equal to or bigger than denominator
Mixed number — whole number + proper fraction
Terminating decimal — a decimal that ends
Recurring decimal — a decimal with a repeating block of digits Ext.
Percent — “per hundred”; a fraction with denominator 100

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C1.4 (Core subject content) and section E1.4 (Extended subject content) — Fractions, decimals and percentages. Core candidates are not expected to use recurring decimal notation or convert recurring decimals to and from fractions; both are Extended-only, marked above.

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