Edexcel International GCSE Mathematics A 4MA1: Fractions and decimals – Revision Notes

These notes condense sections 1.2 Fractions and 1.3 Decimals of 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 and the Foundation part of 1.3 apply to both tiers. Converting recurring decimals into fractions (statement 1.3A in the Higher Tier content) is Higher tier only. A calculator is allowed on every paper, but questions often say “show your working”, so learn the written methods.

For full explanations and longer worked examples, use the fractions and decimals study guide. Then test yourself with the fractions and decimals practice questions. For the rest of Topic 1, see the Numbers and the number system revision notes. The course hub and the printable checklist cover the whole specification, and a free 10-minute diagnostic shows which topics need work.

Key words

Term	Meaning	Example
Equivalent fractions	Same value, different numerator and denominator	3/4 = 15/20
Simplest form (lowest terms)	Numerator and denominator have no common factor except 1	15/20 → 3/4
Vulgar fraction	One whole number over another; improper if top > bottom	11/4
Mixed number	Whole number plus a fraction	2 3/4
Common denominator	A number all the denominators divide into	24 for 3/8 and 5/6
Unit fraction	Numerator 1	1/6
Reciprocal (multiplicative inverse)	The number you multiply by to get 1	1/6 is the reciprocal of 6; 5/2 of 2/5
Terminating decimal	Stops after a finite number of digits	0.375
Recurring decimal	A digit or block repeats forever	0.3̇ = 0.333…

Methods in steps

Task	Steps
Simplify (1.2A)	Divide top and bottom by the HCF, or cancel in stages until nothing more cancels
Mixed → improper (1.2B)	whole × denominator + numerator, over the same denominator
Improper → mixed (1.2B)	Divide; quotient is the whole part, remainder over the denominator
Order fractions (1.2C, D)	Common denominator (LCM), compare numerators, answer in original form
Fraction of an amount (1.2D)	÷ denominator, × numerator
A as a fraction of B (1.2E)	Same units, then A/B, then simplify
Add or subtract (1.2F)	Improper fractions → common denominator → combine numerators → simplify
Multiply (1.2I)	Improper fractions → cancel across → multiply tops, multiply bottoms
Divide (1.2H, 1.2I)	Improper fractions → keep the first, flip the second, multiply
Fraction → decimal → % (1.2G)	Top ÷ bottom; then × 100
Terminating decimal → fraction (1.3D, E)	Digits over 10, 100, 1000…; simplify; × 100 for %

Small reminders

Divide by a fraction:   5/6 ÷ 10/9 = 5/6 × 9/10 = 45/60 = 3/4
Unit fraction:          8 ÷ 3 = 8 × 1/3 = 8/3
Terminating decimal:    0.035 = 35/1000 = 7/200 = 3.5%

Equivalents worth knowing by heart (1.2G, 1.3D)

Knowing these saves time when ordering, estimating or checking a calculator answer.

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4, 3/4	0.25, 0.75	25%, 75%
1/5, 2/5	0.2, 0.4	20%, 40%
1/8, 3/8	0.125, 0.375	12.5%, 37.5%
1/10, 1/20	0.1, 0.05	10%, 5%
1/3, 2/3	0.3̇, 0.6̇	33.3̇%, 66.6̇%
1/9	0.1̇	11.1̇%

For any fraction not in the table, divide the numerator by the denominator. For example, 7/40 = 0.175 = 17.5%, which terminates, but 7/30 = 0.23̇ = 23.3̇%, which recurs.

Using your calculator sensibly

• The fraction key gives a quick check of any fraction sum, product or quotient. Use it to check, not to replace, working that the question asks for.
• The fraction–decimal toggle (S⇔D on many models) switches between fraction and decimal form, which is useful for ordering.
• Keep the full calculator value when you convert to a percentage, and round once at the end to the accuracy asked for.
• Estimate first: 3 1/3 ÷ 1 7/9 is roughly 3.3 ÷ 1.8, a bit less than 2. An answer of 5 or 0.5 is clearly wrong.

Place value and ordering decimals (1.3A–C)

Places after the point are tenths, hundredths, thousandths. In 6.0952 the 9 is 9 hundredths and the 2 is 2 ten-thousandths.

To order decimals, pad with zeros to the same number of places, then compare from the left.

1.7, 1.07, 1.707, 1.077  →  1.700, 1.070, 1.707, 1.077
smallest first:  1.07, 1.077, 1.7, 1.707

To order a mix of fractions and decimals, convert everything to decimals, order, then write the answer using the original numbers.

Recurring decimals (Higher tier only)

Notation. One dot over a single repeating digit: 0.6̇ = 0.666… Dots over the first and last digits of a repeating block: 0.1̇4̇ = 0.141414… and 0.3̇12̇ = 0.312312…

Fraction → recurring decimal (both tiers, from 1.2G): divide. 1/3 = 0.3̇, 7/9 = 0.7̇, 4/11 = 0.3̇6̇.

Recurring decimal → fraction (Higher tier only):

1. Let x = the decimal, written out with several repeats.
2. Multiply by powers of 10 to get two numbers with the
   SAME digits after the point.
3. Subtract to remove the recurring part.
4. Solve for x and simplify.

Pattern	Use	Example
All digits recur, block length 1	10x − x = 9x	0.5̇ = 5/9
All digits recur, block length 2	100x − x = 99x	0.0̇7̇ = 7/99
One digit before the recurring digit	100x − 10x = 90x	0.24̇: 90x = 24.44… − 2.44… = 22, so x = 22/90 = 11/45

The pattern table is a quick check. When a question says “prove” or “show that”, write the full algebra: the x, the two multiples, the subtraction and the simplifying.

Must-know distinctions

• Terminating vs recurring. Statement 1.3D (decimal → fraction) is for terminating decimals only. Recurring → fraction is the separate Higher tier statement.
• Fraction OF vs fraction AS. “3/5 of 40” means multiply (24). “24 as a fraction of 40” means divide (24/40 = 3/5).
• Reciprocal vs negative. The reciprocal of 4 is 1/4, not −4.
• Simplest form vs mixed number. 29/12 is in simplest form. The mixed number 2 5/12 is the same value written another way. If the question asks for a particular form, give that form.

Quick self-test

1. Write 42/56 in its simplest form.
2. Write 5 2/7 as an improper fraction.
3. Find 3/8 of 96.
4. Write 25 cm as a fraction of 2 m, in its simplest form.
5. Work out 5/6 − 3/8.
6. Work out 1 1/2 × 2 2/9. Give your answer as a mixed number.
7. Work out 4/5 ÷ 2 2/5.
8. Write 2/9 as a decimal and as a percentage.
9. Write 0.075 as a fraction in its simplest form and as a percentage.
10. Write 2.09, 2.9, 2.019 and 2.1 in order, smallest first.
11. (Higher) Write 0.6̇3̇ as a fraction in its simplest form.
12. (Higher) Write 0.83̇ as a fraction in its simplest form.

Answers

1. HCF 14, so 3/4.
2. 5 × 7 + 2 = 37, so 37/7.
3. 96 ÷ 8 = 12, 12 × 3 = 36.
4. 2 m = 200 cm; 25/200 = 1/8.
5. 20/24 − 9/24 = 11/24.
6. 3/2 × 20/9 = 60/18 = 10/3 = 3 1/3.
7. 4/5 × 5/12 = 20/60 = 1/3.
8. 0.2̇ (0.222…) and 22.2̇% (22.2% to 3 s.f.).
9. 75/1000 = 3/40; 7.5%.
10. 2.019, 2.09, 2.1, 2.9.
11. 100x − x = 63, so x = 63/99 = 7/11.
12. 100x = 83.33…, 10x = 8.33…, 90x = 75, x = 75/90 = 5/6.

Where marks are usually lost

• Adding the denominators as well as the numerators.
• Not converting mixed numbers to improper fractions before multiplying or dividing.
• Flipping the wrong fraction in a division, or flipping both.
• Giving an answer like 45/60 when the question asks for the simplest form.
• Leaving ordered fractions in their converted form (for example over 120) instead of the original form.
• Mixing units when writing one quantity as a fraction of another (minutes over hours, cm over m).
• Ranking decimals by number of digits, so 2.019 ends up above 2.1.
• Rounding a recurring decimal too early, then giving a percentage that is slightly wrong.
• (Higher) Subtracting 100x − x when only one digit recurs after a non-recurring digit, so the tails do not cancel.
• (Higher) Giving a calculator conversion with no algebra on a “prove” question.

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).