Cambridge IGCSE Mathematics 0580: Fractions, decimals and percentages, ordering and the four operations – Study Guide

This guide teaches three subtopics of Topic 1 Number in the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027: 1.4 Fractions, decimals and percentages, 1.5 Ordering and 1.6 The four operations. It covers both tiers. Core candidates (Papers 1 and 3) need everything except the parts marked Extended only; Extended candidates (Papers 2 and 4) need all of it. The only Extended-only content in these three subtopics is recurring decimal notation and converting between recurring decimals and fractions (E1.4).

Calculators are not allowed on Papers 1 and 2, so every method here is shown by hand.

Course links: Cambridge IGCSE Mathematics hub and the printable 0580 checklist. To find your gaps first, take the Core diagnostic or the Extended diagnostic. For the whole of Topic 1, see the Number study guide and Number revision notes; the examples below are different from theirs.

What this unit covers

Syllabus section	What you must be able to do	Tier
C1.4 / E1.4 item 1	Use the language and notation of proper fractions, improper fractions, mixed numbers, decimals and percentages; write fractions in their simplest form	Core and Extended
C1.4 / E1.4 item 2	Recognise equivalence and convert between these forms	Core and Extended
E1.4	Use recurring decimal notation; convert recurring decimals to fractions and fractions to recurring decimals	Extended only
C1.5 / E1.5	Order quantities by magnitude; use the symbols =, ≠, >, <, ⩾ and ⩽	Core and Extended
C1.6 / E1.6	Use the four operations with integers, fractions and decimals, with the correct order of operations and brackets – including negative numbers, improper fractions, mixed numbers and practical situations such as temperature changes	Core and Extended

1.4 Fractions, decimals and percentages

The language

• A proper fraction has a numerator smaller than its denominator, such as 5/9.
• An improper fraction has a numerator larger than its denominator, such as 29/6.
• A mixed number is a whole number plus a proper fraction, such as 4⅚.
• A decimal uses place value after the decimal point: in 0.425 the 4 is tenths, the 2 hundredths and the 5 thousandths.
• A percentage is a number of hundredths: 42.5% means 42.5/100.

Simplest form means the numerator and denominator have no common factor other than 1. Divide both by their highest common factor. For 30/72, the HCF is 6, so 30/72 = 5/12.

Converting between the forms

From	To	Method
Fraction	Decimal	Divide numerator by denominator
Decimal	Fraction	Write over 10, 100, 1000 … by place value, then simplify
Decimal	Percentage	Multiply by 100
Percentage	Decimal	Divide by 100
Percentage	Fraction	Write over 100, then simplify
Mixed number	Improper fraction	Whole number × denominator, add numerator, keep the denominator
Improper fraction	Mixed number	Divide; the remainder goes over the same denominator

Worked example 1 (non-calculator). Write 7/16 as a decimal.

Short division of 7 by 16, adding zeros after the decimal point:

7.0000 ÷ 16
70 ÷ 16 = 4 remainder 6     ->  0.4
60 ÷ 16 = 3 remainder 12    ->  0.43
120 ÷ 16 = 7 remainder 8    ->  0.437
80 ÷ 16 = 5 remainder 0     ->  0.4375

So 7/16 = 0.4375, which is 43.75%.

Worked example 2. Write 1.15 as (a) a fraction in its simplest form, (b) a mixed number, (c) a percentage.

(a) 1.15 = 115/100     HCF of 115 and 100 is 5    ->  23/20
(b) 23 ÷ 20 = 1 remainder 3                         ->  1 3/20
(c) 1.15 × 100                                      ->  115%

A percentage over 100% means more than one whole.

Core: what to do with a fraction that never ends. Some fractions give a decimal that goes on for ever, such as 2/3 = 0.666… At Core you do not need recurring notation. Write enough digits to show the pattern, or round to the accuracy the question asks for: 2/3 = 0.667 to 3 decimal places.

Recurring decimals (Extended only)

A recurring decimal has a digit or block of digits that repeats for ever. The syllabus notation puts a dot over the first and last digit of the repeating block:

• 0.4̇ means 0.4444… (one digit repeats)
• 0.26̇ means 0.2666… (only the 6 repeats)
• 0.3̇6̇ means 0.363636… (the block 36 repeats)
• 0.2̇16̇ means 0.216216216… (dots on the 2 and the 6: the block 216 repeats)

When there are two dots, every digit from the first dot to the second is in the repeating block, including any digits between them.

Fraction to recurring decimal. Divide as usual and stop when a remainder repeats, because from then on the digits repeat too.

5/11:  50 ÷ 11 = 4 r 6,  60 ÷ 11 = 5 r 5,  50 ÷ 11 = 4 r 6 ...   ->  0.454545... = 0.4̇5̇
7/12:  70 ÷ 12 = 5 r 10, 100 ÷ 12 = 8 r 4, 40 ÷ 12 = 3 r 4 ...   ->  0.58333...  = 0.583̇

Recurring decimal to fraction. Let x be the decimal. Multiply by powers of 10 so that two versions of x have identical recurring tails, then subtract to remove the tail.

Worked example 3 (Extended). Write 0.3̇6̇ as a fraction in its simplest form.

x      = 0.363636...
100x   = 36.363636...       (block of 2 digits, so multiply by 100)
100x - x = 36
99x    = 36
x      = 36/99 = 4/11

Worked example 4 (Extended). Write 0.26̇ as a fraction in its simplest form.

Only the 6 repeats, so first move the non-repeating 2 past the decimal point.

x      = 0.2666...
10x    = 2.666...
100x   = 26.666...
100x - 10x = 26.666... - 2.666...
90x    = 24
x      = 24/90 = 4/15

Check: 4 ÷ 15 = 0.2666… The two lines you subtract must have the same digits after the point, so that the difference is a whole number.

Worked example 5 (Extended). Write 0.2̇16̇ as a fraction in its simplest form.

x       = 0.216216...
1000x   = 216.216216...      (block of 3 digits, so multiply by 1000)
999x    = 216
x       = 216/999 = 8/37     (divide top and bottom by 27)

1.5 Ordering

The symbols

Symbol	Meaning	Example
=	is equal to	0.4 = 2/5
≠	is not equal to	0.4 ≠ 0.04
>	is greater than	0.71 > 0.705
<	is less than	-2.4 < -2.04
⩾	is greater than or equal to	“at least 18” is x ⩾ 18
⩽	is less than or equal to	“no more than 30” is n ⩽ 30

“At least” includes the value (⩾); “more than” does not (>).

Ordering numbers in different forms

Convert everything to decimals, write them to the same number of decimal places, then compare digit by digit from the left.

Worked example 6. Write these in order of size, smallest first: 0.71, 7/10, 72%, 5/7, 0.705

0.71    = 0.7100
7/10    = 0.7000
72%     = 0.7200
5/7     = 0.7142...
0.705   = 0.7050

Smallest first: 7/10, 0.705, 0.71, 5/7, 72%. Answer in the original forms.

Negative numbers

On a number line, numbers get smaller as you move left. A negative number further from zero is smaller.

Worked example 7. Order -2.4, -2¼, -2.04, -2⅓, smallest first.

-2.4 = -2.400     -2 1/4 = -2.250     -2.04 = -2.040     -2 1/3 = -2.333...

Smallest first: -2.4, -2⅓, -2¼, -2.04.

Inequalities and lists of integers

Worked example 8. List the integers n such that -3 ⩽ n < 2.

-3 is included (⩽), 2 is not (<): -3, -2, -1, 0, 1.

Ordering recurring decimals (Extended only)

Write out enough digits to see where the numbers differ.

Worked example 9 (Extended). Place these in order, smallest first, and state which are equal: 0.4̇5̇, 0.45̇, 0.454, 5/11

0.4̇5̇  = 0.454545...
0.45̇   = 0.455555...
0.454   = 0.454000
5/11    = 0.454545...

Order: 0.454 < 0.4̇5̇ = 5/11 < 0.45̇. The = sign is correct here: 0.4̇5̇ and 5/11 are the same number.

1.6 The four operations

Order of operations

Work in this order:

1. Brackets
2. Indices (powers and roots)
3. Multiplication and division, from left to right
4. Addition and subtraction, from left to right

× and ÷ have equal priority, as do + and -.

Worked example 10. Work out 18 - 12 ÷ 4 × 3.

12 ÷ 4 = 3       (÷ and × left to right)
3 × 3  = 9
18 - 9 = 9

Worked example 11. Work out (5 - 2³) × 4 + 6 ÷ (-2).

brackets:  2³ = 8, so 5 - 8 = -3
then:      -3 × 4 = -12   and   6 ÷ (-2) = -3
finally:   -12 + (-3) = -15

Worked example 12. Insert one pair of brackets to make 6 + 9 ÷ 3 - 1 = 4 true.

Without brackets the value is 6 + 3 - 1 = 8. Try grouping the addition: (6 + 9) ÷ 3 - 1 = 15 ÷ 3 - 1 = 5 - 1 = 4. So (6 + 9) ÷ 3 - 1 = 4.

Negative numbers

• Subtracting a negative is the same as adding: -7 - (-12) = -7 + 12 = 5.
• Multiplying or dividing: same signs give a positive, different signs give a negative. (-4) × (-6) ÷ 3 = 24 ÷ 3 = 8.

Worked example 13 (temperature). (a) At 06 00 the temperature is -8 °C. By noon it has risen by 13 degrees. Find the temperature at noon. (b) A freezer is at -18 °C and the kitchen is at 21 °C. Find the difference.

(a) -8 + 13 = 5            ->  5 °C
(b) 21 - (-18) = 21 + 18   ->  39 degrees

Fractions and mixed numbers

• Add or subtract: change mixed numbers to improper fractions (or deal with whole parts separately), then use a common denominator.
• Multiply: change mixed numbers to improper fractions, cancel common factors, then multiply tops and bottoms.
• Divide: multiply by the reciprocal of the second fraction.

Worked example 14. Work out 3⅝ - 1⅚.

3 5/8 = 29/8      1 5/6 = 11/6      common denominator 24
29/8 = 87/24      11/6 = 44/24
87/24 - 44/24 = 43/24 = 1 19/24

Worked example 15. Work out 2⅖ × 1⅞.

12/5 × 15/8
cancel 12 and 8 by 4:   3/5 × 15/2
cancel 15 and 5 by 5:   3/1 × 3/2
= 9/2 = 4 1/2

Worked example 16. Work out 4½ ÷ 1⅕.

9/2 ÷ 6/5 = 9/2 × 5/6 = 45/12 = 15/4 = 3 3/4

Decimals

• Add and subtract: line up the decimal points. 12.5 - 3.87 = 12.50 - 3.87 = 8.63.
• Multiply: multiply the digits as whole numbers, then count the decimal places. 36 × 7 = 252, and 3.6 × 0.07 has 1 + 2 = 3 decimal places, so the answer is 0.252.
• Divide: multiply both numbers by the same power of 10 so that you divide by a whole number. 4.83 ÷ 0.3 = 48.3 ÷ 3 = 16.1.

Practical problems

Worked example 17. A ribbon 7½ m long is cut into pieces each ⅝ m long. How many pieces are there?

7 1/2 ÷ 5/8 = 15/2 × 8/5 = 120/10 = 12 pieces

Check: ⅝ m = 0.625 m and 12 × 0.625 = 7.5 m.

Common errors

• Leaving a fraction answer unsimplified, such as 36/99 instead of 4/11.
• (Extended) Putting the dots on the wrong digits. 0.26̇ (0.2666…) is not the same as 0.2̇6̇ (0.2626…).
• (Extended) Multiplying by the wrong power of 10. For 0.3̇6̇, 10x - x = 3.6363… - 0.3636… does not cancel the tail; a 2-digit block needs 100x.
• Giving an ordering answer in decimals when the question listed fractions and percentages.
• Saying -2.04 < -2.4 because 2.04 < 2.4.
• Doing × before ÷ (or + before -) instead of working left to right.
• Multiplying mixed numbers part by part: 2⅖ × 1⅞ is not 2 × 1 plus ⅖ × ⅞.
• Dividing by a fraction without turning the second fraction upside down.

Next steps

Test yourself with the Number practice questions and, for Extended, the Number (Extended) practice questions. Tick these sections off on the 0580 checklist.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Cambridge University Press & Assessment – subject content sections C1.4, C1.5 and C1.6 (Core) and E1.4, E1.5 and E1.6 (Extended).