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Edexcel IGCSE Maths A (4MA1) · Foundation tier · Revision notes

IGCSE Maths 4MA1 Foundation Number Notes

Revision notes for numbers and the number system at Foundation tier (Papers 1F and 2F, grades 5 to 1). Each sub-topic has what you need to know, the key methods, a common mistake and a worked example. Cover the worked solution and try the example first.

Number questions with answers →

1.1 Integers

What you need to know: place value and negative numbers in context; the order of operations; factors, multiples and primes; prime factors, common factors and common multiples.

  • Place value tells you what each digit is worth; negative numbers sit to the left of zero on a number line, so −11 is smaller than −3.
  • Work out brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right.
  • A prime number has exactly two factors, 1 and itself, so 1 is not prime and 2 is the only even prime.
  • Find prime factors with a factor tree or by dividing by primes in turn, and write the answer as a product such as 2 × 2 × 3 × 7.

Watch out: Do not add before multiplying: in 5 + 4 × 6 the multiplication comes first.

Worked example (2 marks, grade 3):

Write \(252\) as a product of its prime factors.

  1. 252 ÷ 2 = 126
  2. 126 ÷ 2 = 63
  3. 63 ÷ 3 = 21
  4. 21 ÷ 3 = 7, and 7 is prime.
  5. So 252 = 2 × 2 × 3 × 3 × 7 = 2² × 3² × 7.

Answer: \(2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7\)

Practise: Integers questions with answers · Unit 1 practice (Foundation) · Higher level: HCF, LCM and Prime Factors questions

1.2 Fractions

What you need to know: simplifying and comparing fractions; mixed numbers; fractions of amounts; adding, subtracting, multiplying and dividing fractions; fractions as decimals and percentages.

  • Simplify a fraction by dividing the numerator and the denominator by their highest common factor.
  • To add or subtract, write the fractions with a common denominator first; deal with whole numbers and fractions separately in mixed numbers.
  • To multiply, multiply the numerators and multiply the denominators; to divide, multiply by the reciprocal of the second fraction.
  • To find a fraction of an amount, divide by the denominator and multiply by the numerator.

Watch out: Never add the denominators: 1/2 + 1/3 is 5/6, not 2/5.

Worked example (3 marks, grade 3):

Work out \(2\frac{1}{4} + 1\frac{5}{6}\)

Give your answer as a mixed number in its simplest form.

  1. Add the whole numbers: 2 + 1 = 3.
  2. Add the fractions using 12ths: 1/4 = 3/12 and 5/6 = 10/12, so 3/12 + 10/12 = 13/12 = 1 1/12.
  3. Total: 3 + 1 1/12 = 4 1/12.

Answer: \(4\frac{1}{12}\)

Practise: Fractions questions with answers · Unit 1 practice (Foundation) · Higher level: Fractions and Decimals questions

1.3 Decimals

What you need to know: place value and ordering of decimals; terminating decimals as fractions and percentages.

  • Compare decimals digit by digit from the left; writing them with the same number of decimal places helps.
  • A terminating decimal is a whole number of tenths, hundredths or thousandths, so it can be written as a fraction and simplified.
  • To multiply decimals, multiply as whole numbers and then count the decimal places; to divide by a decimal, multiply both numbers by a power of 10 first.

Watch out: 0.5 is larger than 0.35: more digits does not mean a bigger number.

Worked example (2 marks, grade 3):

Work out \(3.6 \times 0.25\)

  1. 36 × 25 = 900.
  2. 3.6 has 1 decimal place and 0.25 has 2, so the answer has 3: 0.900 = 0.9.

Answer: \(0.9\)

Higher tier only: convert recurring decimals to fractions.

Practise: Decimals questions with answers · Unit 1 practice (Foundation) · Higher level: Fractions and Decimals questions

1.4 Powers and roots

What you need to know: squares, cubes and their roots; index laws with whole-number, zero and negative powers; products of prime factors; HCF and LCM.

  • Square numbers are 1, 4, 9, 16, …; cube numbers are 1, 8, 27, 64, ….
  • When you multiply powers of the same number, add the indices; when you divide, subtract them.
  • Any non-zero number to the power 0 is 1, and a negative power means one over: 2⁻³ = 1/8.
  • The HCF uses the prime factors that two numbers share; the LCM uses every prime factor to its highest power.

Watch out: 3² × 3⁴ is 3⁶, not 9⁶: the base stays the same.

Worked example (3 marks, grade 3):

Work out the value of (a) \(5^0\) (b) \(2^{-3}\) (c) \(\frac{3^6 \times 3^2}{3^5}\)

  1. (a) Any non-zero number to the power 0 is 1.
  2. (b) 2⁻³ = 1/2³ = 1/8.
  3. (c) 3⁶ × 3² = 3⁸, and 3⁸ ÷ 3⁵ = 3³ = 27.

Answer: (a) \(1\) (b) \(\frac{1}{8}\) (c) \(27\)

Higher tier only: surds, rationalising a denominator, fractional powers.

Practise: Powers and roots questions with answers · Unit 1 practice (Foundation) · Unit 2 practice (Foundation) · Higher level: HCF, LCM and Prime Factors questions · Indices and Standard Form questions

1.5 Set language and notation

What you need to know: sets, members and the empty set; union, intersection and complement; Venn diagrams.

  • A set is a collection of members written in curly brackets; ∈ means 'is a member of' and ∉ means 'is not a member of'.
  • A ∩ B (intersection) holds the members in both sets; A ∪ B (union) holds the members in either set, each listed once.
  • The universal set ℰ holds everything being considered, A′ (the complement) is everything in ℰ not in A, and ∅ is the empty set.
  • In a Venn diagram, fill in the overlap first, then the rest of each circle, then the region outside the circles.

Watch out: Members of A ∩ B also belong to A and to B, so do not count them twice.

Worked example (3 marks, grade 3):

\(\mathscr{E} = \{a, b, c, d, e, f, g, h\}\), \(X = \{a, b, c, d\}\) and \(Y = \{c, d, e, f\}\)

List the members of (a) \(X \cap Y\) (b) \((X \cup Y)'\)

  1. (a) c and d are in both sets.
  2. (b) X ∪ Y = {a, b, c, d, e, f}. The complement is what is left in ℰ: g and h.

Answer: (a) \(\{c, d\}\) (b) \(\{g, h\}\)

Higher tier only: subsets, n(A), sets defined in algebraic terms and sets in practical problems.

Practise: Set language and notation questions with answers · Unit 6 practice (Foundation) · Higher level: Sets and Venn Diagrams questions

1.6 Percentages

What you need to know: percentages of amounts and percentage change; percentages as fractions, decimals and multipliers; reverse percentages; compound interest and depreciation.

  • A percentage is a number of parts per hundred, so 35% = 35/100 = 0.35.
  • Use a multiplier: an increase of 3.5% multiplies by 1.035 and a decrease of 20% multiplies by 0.8.
  • Percentage change = change ÷ original amount × 100.
  • For a reverse percentage, divide the new amount by the multiplier; for compound interest or depreciation, multiply by the multiplier once for each year.

Watch out: In a reverse percentage, do not take the percentage off the sale price: divide by the multiplier instead.

Worked example (2 marks, grade 3):

Ben's salary was £24 000 a year. It increased by \(3.5\%\).

Work out his new salary.

  1. 3.5% of £24 000 = 0.035 × 24 000 = £840.
  2. New salary: £24 000 + £840 = £24 840.

Answer: £24 840

Higher tier only: repeated percentage change and harder compound interest problems.

Practise: Percentages questions with answers · Unit 1 practice (Foundation) · Higher level: Percentages questions

1.7 Ratio and proportion

What you need to know: simplifying ratios and the form 1 : n; sharing in a ratio; direct proportion and the unitary method; maps and scale diagrams.

  • Simplify a ratio by dividing every part by the same number; for the form 1 : n, divide both parts by the first part.
  • To share in a ratio, add the parts, find the value of one part, then multiply.
  • In direct proportion, find the value for one (the unitary method) or use a multiplier.
  • A map scale 1 : 25 000 means 1 cm on the map is 25 000 cm (250 m) in real life.

Watch out: Check the order of the ratio: boys : girls = 3 : 5 is not the same as 5 : 3.

Worked example (2 marks, grade 3):

Three friends share 72 sweets in the ratio \(2 : 3 : 4\).

How many sweets does each friend get?

  1. Total parts: 2 + 3 + 4 = 9.
  2. One part: 72 ÷ 9 = 8.
  3. 2 × 8 = 16, 3 × 8 = 24, 4 × 8 = 32.

Answer: \(16, 24, 32\)

Practise: Ratio and proportion questions with answers · Unit 1 practice (Foundation) · Unit 5 practice (Foundation) · Higher level: Ratio questions · Bearings and Constructions questions

1.8 Degree of accuracy

What you need to know: rounding to decimal places and significant figures; estimating by rounding to 1 significant figure; upper and lower bounds.

  • Look at the digit after the last one you keep: 5 or more rounds up, 4 or less rounds down.
  • Significant figures start at the first non-zero digit, so 0.03058 to 2 significant figures is 0.031.
  • To estimate, round each number to 1 significant figure before you calculate.
  • A measurement to the nearest unit can be up to half a unit either side: the lower bound is included and the upper bound is not.

Watch out: Do not count leading zeros as significant figures.

Worked example (2 marks, grade 3):

The length of a pencil is 14 cm, correct to the nearest centimetre.

Write down (a) the lower bound (b) the upper bound of the length.

  1. To the nearest centimetre means within half a centimetre either side.
  2. 14 − 0.5 = 13.5 and 14 + 0.5 = 14.5.

Answer: (a) \(13.5\) cm (b) \(14.5\) cm

Higher tier only: problems that use upper and lower bounds in calculations.

Practise: Degree of accuracy questions with answers · Unit 1 practice (Foundation) · Unit 4 practice (Foundation) · Higher level: Rounding and Bounds questions

1.9 Standard form

What you need to know: writing and calculating with numbers in standard form.

  • A number in standard form is a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
  • Large numbers have a positive power of 10 and small numbers (less than 1) a negative power.
  • To multiply or divide, work with the numbers and the powers of 10 separately, then adjust so that a is between 1 and 10.
  • To add or subtract, it is usually easiest to write the numbers out in full first.

Watch out: 18 × 10⁹ is not in standard form: write it as 1.8 × 10¹⁰.

Worked example (2 marks, grade 3):

Work out \((3 \times 10^4) \times (6 \times 10^5)\). Give your answer in standard form.

  1. 3 × 6 = 18 and 10⁴ × 10⁵ = 10⁹, giving 18 × 10⁹.
  2. 18 = 1.8 × 10, so the answer is 1.8 × 10¹⁰.

Answer: \(1.8 \times 10^{10}\)

Higher tier only: harder problems in standard form.

Practise: Standard form questions with answers · Unit 1 practice (Foundation) · Higher level: Indices and Standard Form questions

1.10 Applying number

What you need to know: metric units of length, mass, area, volume and capacity; time calculations; money, best buys and exchange rates.

  • Metric conversions: 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g and 1 litre = 1000 ml = 1000 cm³.
  • Area units use the square of the length factor (1 m² = 10 000 cm²) and volume units the cube (1 m³ = 1 000 000 cm³).
  • For time, count on in steps to the next hour, and remember there are 60 minutes in an hour.
  • To compare best buys, find the cost of one unit (one gram, one ml) for each option; multiply by the exchange rate to change pounds into the other currency and divide to change back.

Watch out: 1.5 hours is 1 hour 30 minutes, not 1 hour 50 minutes.

Worked example (3 marks, grade 3):

Shampoo is sold in two sizes.

Small: 250 ml for £2.40

Large: 400 ml for £3.68

Which size is the better value for money? Show your working.

  1. Small: 240p ÷ 250 ml = 0.96p per ml.
  2. Large: 368p ÷ 400 ml = 0.92p per ml.
  3. The large bottle costs less per ml, so it is the better value.

Answer: Large (0.92p per ml compared with 0.96p per ml)

Practise: Applying number questions with answers · Unit 1 practice (Foundation) · Higher level: Exchange Rates and Best Buys questions · Units and Compound Measures questions

1.11 Electronic calculators

What you need to know: using a scientific calculator efficiently and checking answers.

  • Use brackets (or the fraction key) so the calculator works out the whole numerator and the whole denominator before dividing.
  • Write down the full calculator display before you round, then round as the question asks.
  • Check an answer with a quick estimate to spot a key-pressing error.
  • Use the calculator's fraction key when the question asks for an exact fraction.

Watch out: Typing 12 + 8 ÷ 4 gives 14, not 5: the calculator divides first unless you use brackets.

Worked example (2 marks, grade 3):

Work out \(\sqrt{\frac{48.7 - 12.3}{2.6}}\)

Give your answer correct to 3 significant figures.

  1. 48.7 − 12.3 = 36.4.
  2. 36.4 ÷ 2.6 = 14.
  3. √14 = 3.741657… = 3.74 to 3 significant figures.

Answer: \(3.74\)

Practise: Electronic calculators questions with answers · Unit 1 practice (Foundation) · Higher level: Rounding and Bounds questions

Topic list for Edexcel IGCSE Maths 4MA1 Foundation number: integers, fractions, decimals, powers and roots, set language and notation, percentages, ratio and proportion, degree of accuracy, standard form, applying number, electronic calculators
The 11 Foundation sub-topics of number.

Practise number