Edexcel International GCSE Mathematics A 4MA1: Fractions and decimals – Study Guide

This study guide teaches sections 1.2 Fractions and 1.3 Decimals of Topic 1, Numbers and the number system, in the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on that specification. Section 1.2 is the same on both tiers: the Higher tier content simply says “See Foundation Tier”. Section 1.3 has Foundation content that both tiers need, plus one extra statement, converting recurring decimals into fractions, which is Higher tier only. A calculator may be used on every 4MA1 paper, but fraction questions often ask you to show your working, so every method here is shown by hand.

For a wider view of the topic, read the Numbers and the number system study guide. When you have worked through this page, use the fractions and decimals revision notes and the fractions and decimals practice questions. The Edexcel IGCSE Mathematics course hub links the rest of the course, and the printable checklist lists every statement. If you are not sure where your gaps are, start with a free 10-minute diagnostic.

What this unit covers

Statement	What you must be able to do	Tier
1.2A	Use equivalent fractions; simplify by cancelling common factors (lowest terms)	Both
1.2B	Use mixed numbers and vulgar fractions	Both
1.2C	Identify common denominators	Both
1.2D	Order fractions; find a fraction of a quantity	Both
1.2E	Express one number as a fraction of another	Both
1.2F	Add and subtract fractions and mixed numbers	Both
1.2G	Convert a fraction to a decimal or a percentage	Both
1.2H	Use unit fractions as multiplicative inverses	Both
1.2I	Multiply and divide fractions and mixed numbers	Both
1.3A–C	Decimal notation, place value, ordering decimals	Both
1.3D–E	Convert a terminating decimal to a fraction or percentage	Both
1.3A (Higher)	Convert recurring decimals into fractions	Higher tier only

Equivalent fractions and simplifying (1.2A)

Equivalent fractions have the same value. You make one by multiplying or dividing the numerator and denominator by the same number: 2/5 = 4/10 = 12/30.

A fraction is in its simplest form (lowest terms) when the numerator and denominator have no common factor other than 1. Divide both by their highest common factor (HCF). If you cannot see the HCF, cancel in stages by any common factor until no more cancel.

Worked example 1. Write 36/84 in its simplest form.

HCF of 36 and 84 is 12
36 ÷ 12 = 3,  84 ÷ 12 = 7
36/84 = 3/7

Cancelling in stages works too: 36/84 = 18/42 = 9/21 = 3/7.

Mixed numbers and vulgar fractions (1.2B)

A vulgar fraction is a fraction written as one whole number over another, such as 3/8 or 35/8. When the numerator is larger than the denominator it is also called an improper fraction. A mixed number has a whole-number part and a fraction part, such as 4 3/8.

Worked example 2.

Mixed to improper:  4 3/8 = (4 × 8 + 3)/8 = 35/8
Improper to mixed:  47 ÷ 6 = 7 remainder 5,  so 47/6 = 7 5/6

Common denominators and ordering (1.2C, 1.2D)

A common denominator is a number that every denominator divides into. The lowest common multiple (LCM) of the denominators gives the smallest one, which keeps the numbers small.

To order fractions, rewrite them all over a common denominator and compare numerators. Converting each to a decimal also works and is often quicker with a calculator.

Worked example 3. Write 5/8, 7/12, 2/3 and 3/5 in order of size, smallest first.

LCM of 8, 12, 3 and 5 is 120
5/8 = 75/120    7/12 = 70/120    2/3 = 80/120    3/5 = 72/120
Order: 7/12, 3/5, 5/8, 2/3

Always give the answer in the original form the question used, not as the converted fractions 70/120, 72/120 and so on.

A fraction of a quantity and one number as a fraction of another (1.2D, 1.2E)

To find a fraction of a quantity, divide by the denominator, then multiply by the numerator.

Worked example 4. Find 7/12 of 156 kg.

156 ÷ 12 = 13
13 × 7 = 91 kg

To express one number as a fraction of another, write “part over whole”, make the units the same, then simplify.

Worked example 5. Write 40 minutes as a fraction of 2½ hours.

2½ hours = 150 minutes
40/150 = 4/15

Writing 40/2.5 mixes minutes with hours and gives a meaningless answer. Convert first.

Adding and subtracting fractions and mixed numbers (1.2F)

Fractions can only be added or subtracted when they have the same denominator. Find a common denominator, convert each fraction, then add or subtract the numerators. The denominator does not change.

Worked example 6. Work out 3/4 + 5/6.

Common denominator 12
3/4 = 9/12,  5/6 = 10/12
9/12 + 10/12 = 19/12 = 1 7/12

For mixed numbers, the safest method is to convert to improper fractions first. This avoids the problem of “borrowing” when the second fraction part is larger than the first.

Worked example 7. Work out 5 1/4 − 2 5/6.

5 1/4 = 21/4,   2 5/6 = 17/6
Common denominator 12:  21/4 = 63/12,  17/6 = 34/12
63/12 − 34/12 = 29/12 = 2 5/12

Check with an estimate: about 5.25 − 2.83 ≈ 2.4, and 2 5/12 ≈ 2.42.

Unit fractions as multiplicative inverses (1.2H)

A unit fraction has numerator 1, such as 1/5. Multiplying by 1/5 has the same effect as dividing by 5, so 1/5 is the multiplicative inverse (reciprocal) of 5: 5 × 1/5 = 1. That is why

3 ÷ 5 = 3 × 1/5 = 3/5
7 ÷ 4 = 7 × 1/4 = 7/4

The same idea extends to any fraction. The reciprocal of a/b is b/a, because (a/b) × (b/a) = 1. Dividing by 2/5 is the same as multiplying by 5/2. This is the reason the division method in the next section works.

Multiplying and dividing fractions and mixed numbers (1.2I)

Multiplying: convert mixed numbers to improper fractions, cancel any common factors between a numerator and a denominator, then multiply numerators and multiply denominators.

Worked example 8. Work out 2 1/4 × 1 3/5.

2 1/4 = 9/4,   1 3/5 = 8/5
9/4 × 8/5      cancel 4 and 8 by 4:  9/1 × 2/5
= 18/5 = 3 3/5

Dividing: convert to improper fractions, then multiply by the reciprocal of the second fraction.

Worked example 9. Work out 3 1/3 ÷ 1 7/9.

3 1/3 = 10/3,   1 7/9 = 16/9
10/3 ÷ 16/9 = 10/3 × 9/16
cancel 3 and 9 by 3, and 10 and 16 by 2:  5/1 × 3/8
= 15/8 = 1 7/8

Never multiply the whole numbers and fraction parts separately: 2 1/4 × 1 3/5 is not 2 × 1 plus 1/4 × 3/5.

Fractions to decimals and percentages (1.2G)

Divide the numerator by the denominator to get a decimal. Multiply the decimal by 100 to get a percentage.

7/8 = 7 ÷ 8 = 0.875 = 87.5%
5/11 = 5 ÷ 11 = 0.454545... = 45.4545...%

Some fractions give a terminating decimal (it stops, like 0.875). Others give a recurring decimal, where a digit or block of digits repeats forever. A dot over the repeating digit shows this: 0.454545… is written 0.4̇5̇, with dots over the first and last digits of the repeating block. If a question asks for a percentage to a given accuracy, round only at the end: 5/11 = 45.5% to 3 significant figures.

Decimal notation, place value and ordering (1.3A–C)

Each place to the right of the decimal point is worth one tenth of the place to its left: tenths, hundredths, thousandths and so on. In 47.306 the 3 is worth 3 tenths, the 0 holds the hundredths place, and the 6 is worth 6 thousandths (0.006).

To order decimals, line up the decimal points and compare digit by digit from the left. Filling empty places with zeros makes the comparison clearer.

Worked example 10. Order 0.45, 0.405, 0.5 and 0.054, smallest first.

0.450   0.405   0.500   0.054
Order: 0.054, 0.405, 0.45, 0.5

A longer decimal is not automatically larger. 0.405 has more digits than 0.45 but is smaller.

Terminating decimals to fractions and percentages (1.3D, 1.3E)

Every terminating decimal is a fraction with a denominator of 10, 100, 1000 and so on. The specification limits this skill to terminating decimals only. Write the digits over the place value of the last digit, then simplify. Multiply by 100 for a percentage.

Worked example 11.

0.84  = 84/100   = 21/25     = 84%
2.125 = 2125/1000 = 17/8 = 2 1/8 = 212.5%

Recurring decimals to fractions (Higher tier only)

This statement is Higher tier only. The method: call the decimal x, multiply by powers of 10 so that two numbers have identical recurring parts after the decimal point, then subtract so the recurring parts cancel.

Worked example 12. Write 0.1̇8̇ (0.181818…) as a fraction in its simplest form.

x    = 0.181818...
100x = 18.181818...
99x  = 18
x    = 18/99 = 2/11

Worked example 13. Write 0.58̇ (0.58888…) as a fraction in its simplest form.

Only the 8 recurs, so you need two multiples of x that both end in …888.

10x  = 5.8888...
100x = 58.8888...
90x  = 53
x    = 53/90

Worked example 14. Write 0.21̇5̇ (0.2151515…) as a fraction in its simplest form.

10x   = 2.151515...
1000x = 215.151515...
990x  = 213
x     = 213/990 = 71/330

The key decision is which powers of 10 to use. Their indices differ by the length of the repeating block: here 10x and 1000x are 10¹ and 10³, two apart, for the two-digit block 15. When a question says “prove” or “show that”, these lines of algebra are the evidence. A calculator conversion on its own does not prove anything.

Common errors

• Adding denominators: 3/4 + 5/6 written as 8/10.
• Leaving an answer unsimplified when the question asks for its simplest form.
• Multiplying or dividing mixed numbers without converting them to improper fractions first.
• Turning the first fraction upside down instead of the second when dividing.
• Comparing decimals by length, so 0.405 is wrongly placed above 0.45.
• Using different units in “one number as a fraction of another”.
• In recurring-decimal proofs, subtracting two multiples whose recurring parts do not line up, such as 100x − x for 0.58̇.

Next steps

Test your recall with the fractions and decimals revision notes, then try the fractions and decimals practice questions. For the rest of Topic 1, use the Numbers and the number system revision notes and the Numbers and the number system practice questions.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 1, Numbers and the number system, sections 1.2 Fractions and 1.3 Decimals (Foundation Tier content, and Higher Tier statement 1.3A).