Number — IGCSE Mathematics 0580

1 · Types of number 2 · Sets 3 · FDP 4 · Ordering 5 · Four operations 6 · Bounds 7 · Percentages 8 · Time 9 · Money 10 · Growth and decay 11 · Surds 12 · Mixed set

What your check says about this topic

Topic 1 is the biggest topic on the syllabus and it is the one Paper 2 leans on hardest. Your Foundations Check split it cleanly into things you already own and things that will fight back, so this guide is honest about which is which.

Likely to be hard for you:

E1.18 Surds. Your indices strand broke at step 2, the lowest break of all eleven strands. Surds sit directly on top of powers and roots, so expect this section to be the slowest one here.
E1.4 and E1.6, fractions and the four operations. Fractions were secure only to step 5, so the Extended rungs are unbuilt. Both sections start below step 5 and climb, rather than assuming fluency.
E1.13 Percentages, and reverse percentage in particular. Decimals and percentages ran out at step 7. Reverse percentage is the single most reliably lost mark in the whole topic.
Sign errors. These recurred across several unrelated strands in your check. They are not a percentage problem or an algebra problem, they are a habit, so they get flagged wherever they bite.

Probably easier than you expect: E1.2 Sets and E1.17 exponential growth and decay. Sets is almost entirely notation and picture-drawing, and growth and decay is one formula used repeatedly.

Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. Everything here is done by hand, every line shown.

The part of your check that was perfect

Number sense and the four operations: solid on all seven rungs. Not one gap, from the primary rung all the way to Extended hard. Your arithmetic mechanics work.

That is the foundation this entire topic is built on, and you have it. Adding, subtracting, multiplying, dividing, handling negatives, holding a calculation together without losing your place — all working. So when a percentage or a surd question goes wrong, it will not be because you cannot do the arithmetic. It will be because of the layer above: which method, in which order. That is a far quicker thing to fix.

How to use this guide

Every idea appears four times, with less help each time.

Worked in full — every step with a reason. Read it, do not skim it.
Last step is yours — work out the final line before pressing the button.
Last two are yours — the same, harder.
All yours — type an answer and check it.

Sections 1 to 11 are the sub-topics of syllabus topic 1 that your repair guides do not already cover. Section 12 is a mixed set, deliberately unlabelled and out of order, because in an exam nobody tells you which idea applies.

E1.1 · high risk1 · Types of number, HCF and LCM▼

▶ Watch: E1.1 Types of number

Short hand-worked explanations for exactly this sub-topic. Opens on YouTube in a new tab. Watch one, then come straight back and try the questions below — watching without testing yourself feels like learning but is not.

This section is mostly vocabulary, and vocabulary is worth marks. If a question says “write down a prime factor of 84” and you are unsure what a prime factor is, no amount of arithmetic saves you. Learn the words first, then the two methods that use them: HCF and LCM by prime factorisation.

Word	What it means	Example
Natural number	The counting numbers, from 1 upwards	1, 2, 3, 4, …
Integer	A whole number, positive, negative or zero	−3, 0, 7
Prime	Exactly two factors: 1 and itself	2, 3, 5, 7, 11, 13, 17, 19, 23
Square number	A whole number multiplied by itself	1, 4, 9, 16, 25, 36, 49
Cube number	A whole number cubed	1, 8, 27, 64, 125
Factor	A number that divides into it exactly	Factors of 12: 1, 2, 3, 4, 6, 12
Multiple	What you get in its times table	Multiples of 12: 12, 24, 36, …
Rational	Can be written as one integer over another	0.75 = ³⁄₄, −5, 0.333… = ¹⁄₃
Irrational	Cannot be. Decimal never stops and never repeats	√2, √3, π
Reciprocal	1 divided by it. Flip the fraction	Reciprocal of ²⁄₅ is ⁵⁄₂

Reciprocals of every kind of number

The reciprocal of a number is 1 ÷ that number, so a number times its reciprocal is always 1.

the number	what to do	example
a whole number n	it becomes 1/n	reciprocal of 7 is ¹⁄₇
a fraction	turn it upside down	³⁄₈ → ⁸⁄₃
a decimal	write it as a fraction first	0.25 = ¹⁄₄, so its reciprocal is 4
a mixed number	make it improper first	2½ = ⁵⁄₂, so its reciprocal is ²⁄₅
a negative number	the sign stays	−³⁄₇ → −⁷⁄₃
zero	has no reciprocal	1 ÷ 0 cannot be done

Last step is yours

Write down the reciprocal of 1¾.

1¾ = ⁷⁄₄

Mixed number to improper fraction: 1 × 4 + 3 = 7 quarters.

reciprocal = ⁴⁄₇

Check: ⁷⁄₄ × ⁴⁄₇ = 1.

All yours

Write down the reciprocal of 2½. Type it like 3/7.

Write down the reciprocal of 0.125.

Write down the reciprocal of −3/7. Type it like -5/2.

Numbers in words and in figures

Read a number in groups of three digits from the right: ones, thousands, millions, billions (1 billion = 1 000 000 000 = 10⁹). Say each group, then its name. Every empty place needs a zero: a missing zero is the usual mistake.

Worked in full

Write “two million and forty” in figures.

millions group: 2

Two million.

thousands group: 000

There are no thousands, but the three places must still be filled.

ones group: 040

Forty is 40, padded to three digits.

2 000 040

Read back: two million, (no thousands), forty ✓

The other way: 10 007 is “ten thousand and seven”, and 6 000 000 000 is “six billion”.

All yours

Write “twelve thousand and nine” in figures.

Write “seven billion, two hundred million” in figures.

How many zeros are there in one billion written in figures?

The mistake: calling 1 prime. It is not — it has only one factor, and prime means exactly two. And 2 is prime, the only even one. Both of these get tested directly.

The second mistake: assuming a square root is always irrational. √9 = 3, which is rational. Only the roots of non-square numbers are irrational. If a question asks you to pick the irrational one from a list, check each root against the squares table first.

Non-calculator: the primes below 30 — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 — should be recall, not working out. To test whether a number under 300 is prime you only need to try dividing by 2, 3, 5, 7, 11 and 13, because once your divisor squared passes the number you have finished.

Prime factorisation — the division ladder

Every method below runs on this. Divide by the smallest prime that goes in, repeatedly, until you reach 1. The divisors down the side are your prime factors.

Worked in full

Write 84 as a product of its prime factors

84 ÷ 2 = 42

Start with the smallest prime, 2. It goes in, so use it.

42 ÷ 2 = 21

Try 2 again before moving on. It still goes in.

21 ÷ 2 does not work, so try 3: 21 ÷ 3 = 7

Move up to the next prime only when the current one fails.

7 ÷ 7 = 1

7 is itself prime, so it divides out and you have reached 1. Stop.

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Collect the divisors and write repeats as a power. This is the standard form of the answer.

Last step is yours

Write 180 as a product of its prime factors

180 ÷ 2 = 90

Smallest prime first.

90 ÷ 2 = 45

Still even, so 2 again.

45 ÷ 3 = 15, then 15 ÷ 3 = 5

45 is odd so 2 fails. 3 works twice.

5 ÷ 5 = 1

And 5 finishes it.

180 = 2² × 3² × 5

Two 2s, two 3s, one 5. Write it with powers.

HCF and LCM from the prime factors

The method that works: factorise both numbers, then
HCF = multiply the primes they share, each to the lower power.
LCM = multiply every prime that appears in either, each to the higher power.
H for Highest but Lower power; L for Lowest but Higher power. It feels backwards, and it is correct.

Worked in full

Find the HCF and the LCM of 60 and 84

60 = 2² × 3 × 5

Ladder: 60, 30, 15, 5, 1 dividing by 2, 2, 3, 5.

84 = 2² × 3 × 7

From the worked example above.

Shared primes: 2 and 3

5 appears only in 60 and 7 only in 84, so neither is shared.

HCF = 2² × 3 = 4 × 3 = 12

Lower power of each shared prime. Both have 2² and both have 3¹.

LCM = 2² × 3 × 5 × 7 = 12 × 35 = 420

Every prime that appears anywhere, at its higher power.

Check: 12 divides both 60 and 84; 420 is in both times tables

A ten-second check that catches a swapped answer.

Last two are yours

Find the HCF and the LCM of 24 and 36

24 = 2³ × 3

Ladder: 24, 12, 6, 3, 1.

36 = 2² × 3²

Ladder: 36, 18, 9, 3, 1.

Shared primes: 2 and 3. Lower powers are 2² and 3¹

24 has 2³, 36 has 2², so take 2².

HCF = 4 × 3 = 12

Multiply the lower powers together.

LCM = 2³ × 3² = 8 × 9 = 72

Higher power of each: 2³ from 24, 3² from 36.

All yours

Write 90 as a product of prime factors. Type it like 2^2 x 3 x 5.

Find the HCF of 45 and 75

Find the LCM of 8 and 12

Which of these is irrational: √16, √20, ³√27, 0.25? Type the one that is.

Write down the reciprocal of ³⁄₈. Type it like 5/2.

E1.2 SETS AND VENN DIAGRAMS

▶ IGCSE Maths 0580 : Set Notation and Venn Diagrams | Cambridge IGCSE Maths | IGCSE Maths | 0580 MathsPW International Board · IGCSE 0580 ↗ ▶ Intro to Set Notation and Venn Diagrams IGCSEKevin Olding - Mathsaurus · IGCSE 0580 ↗ ▶ Set Notation - Corbettmathscorbettmaths · GCSE, content overlaps ↗

E1.2 · low risk2 · Sets, notation and Venn diagrams▼

▶ Watch: E1.2 Sets

🎙 Tutor Live — hear this section insteadA voice lesson that talks you through sets and Venn diagrams step by step, asks you questions as it goes, and draws a fresh picture when you get stuck. Same content as below — just out loud. (Pilot — tell Appa what you think.)

Sets look unfamiliar and are almost entirely notation. Once you can read the symbols, the questions are picture questions: draw the Venn diagram, fill it from the middle outwards, read off the answer. This should be one of the quicker sections in the guide.

If the symbols look like alien writing, start here. Every set symbol is just a picture instruction — it tells you which part of a Venn diagram to look at. Learn the six pictures below and the whole table above becomes readable. (And yes — these are exactly the symbols on the official 0580 syllabus list, nothing extra.)

A ∪ B — union

everything in either circle (or both)

the CUP holds everything

A ∩ B — intersection

only the overlap

the CAP: only what fits under it

A′ — complement

everything OUTSIDE circle A

“not A” — shade the rest of the box

ξ — universal set

the whole rectangle

everyone in the question lives in this box

3 ∈ A — element of

3 is a dot inside circle A

∈ just means “is inside”

B ⊆ A — subset

circle B sits entirely inside A

every member of B is already in A

Concrete example to hold onto. In a class, A = people who play hockey, B = people who swim. Then A ∪ B = plays hockey or swims (or both); A ∩ B = does both; A′ = does not play hockey; n(A) just counts the hockey players; and ξ is the whole class, including people who do neither.

Quick check — type the word or number:

Which region is A ∩ B — type union or intersection

A = {2, 4, 6, 8}. What is n(A)?

Is 5 ∈ {1, 3, 5, 7}? Type yes or no

A = hockey players. In one word, who is in A′? Type players or non-players

n(ξ) = 30 and n(A) = 12. What is n(A′)?

{1, 2} and {1, 2, 3}: type yes or no — is {1, 2} ⊆ {1, 2, 3}?

Symbol	Read it as	Meaning
{2, 4, 6}	the set containing 2, 4 and 6	Curly brackets list the members
∈	is an element of	3 ∈ A means 3 is in set A
∉	is not an element of	3 ∉ A means 3 is not in A
n(A)	the number of elements in A	If A = {2, 4, 6} then n(A) = 3
∪	union	Everything in A or B or both
∩	intersection	Only what is in A and B
A′	the complement of A	Everything in the universal set that is not in A
ξ	the universal set	Everything under discussion — the outer rectangle
∅	the empty set	No members at all. n(∅) = 0
⊆	is a subset of	Every member of the first set is also in the second

The empty set, “is not a subset”, and sets described by a rule

written	means	example
∅ or { }	the empty set: no members, n(∅) = 0	the even numbers that are also odd
A ⊆ B	every member of A is also in B (the A circle sits inside B)	{4, 8} ⊆ {2, 4, 6, 8}
A ⊈ B	at least one member of A is NOT in B: name it as the reason	{4, 5} ⊈ {2, 4, 6, 8}, because 5 ∉ {2, 4, 6, 8}

Two traps: {0} is not empty, because it has one member, the number 0. And every set is a subset of itself, and ∅ is a subset of every set.

If A ∩ B = ∅, the two sets have nothing in common, and on a Venn diagram the circles do not overlap:

Set-builder notation. {x: rule} reads “the set of all x such that rule”. {x: x is a natural number} = {1, 2, 3, …}. {x: a ≤ x ≤ b} is every number from a to b; add “x is an integer” and you can list it. {(x, y): y = mx + c} is a set of points: every point on the line. Read each inequality sign: < leaves the end out, ≤ keeps it.

Worked in full

List the elements of {x: x is an integer, −2 < x ≤ 3}.

−2 is left out (<) and 3 is kept (≤)

Decide the two ends first.

{−1, 0, 1, 2, 3}

Every integer in between. n = 5.

All yours

ξ = {1, 2, …, 10}, A = {even numbers}, B = {odd numbers}. Write down n(A ∩ B).

List the elements of {x: x is an integer, −2 < x ≤ 3}. Type them like 1,2,3.

A = {x: x is a factor of 12}, B = {x: x is a multiple of 3, x ≤ 12}. List A ∩ B, like 1,2,3.

Does the point (2, 5) belong to {(x, y): y = 3x − 1}? Answer yes or no.

The two you will mix up: ∪ is a cup, and a cup holds everything — union. ∩ is a cap, it sits on top of both, and only the small overlap fits — intersection. Union is bigger; intersection is smaller.

The mistake: forgetting the fourth region. Two circles inside a rectangle make four regions, not three — only A, both, only B, and neither. The “neither” region sits outside both circles and inside the rectangle, and questions about n(ξ) or a complement live there.

Filling a two-set Venn diagram — always start in the middle

Worked in full

In a class of 30, 18 study French, 14 study Spanish and 7 study both. How many study neither?

Middle region = 7

Always fill the intersection first, because both other circle counts include it.

Only French = 18 − 7 = 11

The 18 counts everyone doing French, including the 7 doing both. Take them out.

Only Spanish = 14 − 7 = 7

Same reasoning on the other side.

Inside the circles altogether = 11 + 7 + 7 = 25

Add the three inner regions.

Neither = 30 − 25 = 5

Everything in ξ that is not in a circle. So n((F ∪ S)′) = 5.

Last step is yours

n(ξ) = 40, n(A) = 22, n(B) = 17, n(A ∩ B) = 9. Find n(A′ ∩ B′).

The middle is 9

Given, so write it in first.

Only A = 22 − 9 = 13

Strip the overlap out of A.

Only B = 17 − 9 = 8

Strip the overlap out of B.

13 + 9 + 8 = 30 inside the circles

The union of A and B.

n(A′ ∩ B′) = 40 − 30 = 10

Not in A and not in B — that is the outside region.

Three sets

Three circles make eight regions. The rule does not change: fill the very middle first, then work outwards one ring at a time, subtracting what you have already placed.

Last two are yours

60 students. 30 play tennis (T), 27 play hockey (H), 25 play squash (S). 12 play T and H, 10 play H and S, 11 play T and S, and 6 play all three. How many play none?

Centre = 6

All three. Always first.

T and H only = 12 − 6 = 6

The 12 includes the 6 in the centre.

H and S only = 10 − 6 = 4, T and S only = 11 − 6 = 5

Same subtraction on the other two petals.

T only = 30 − (6 + 6 + 5) = 13

Take the three regions already inside T away from the total for T.

H only = 27 − (6 + 6 + 4) = 11

The three regions inside H are the centre 6, the T-and-H 6, and the H-and-S 4.

S only = 25 − (6 + 4 + 5) = 10, so total inside = 13 + 11 + 10 + 6 + 4 + 5 + 6 = 55, and none = 5

60 − 55 = 5 play none of the three.

All yours

n(ξ) = 25, n(A) = 14, n(B) = 13, n(A ∩ B) = 6. Find n(A ∪ B).

Using the numbers above, find n(A′).

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, find n(A ∩ B).

A = {prime numbers less than 12}. Write down n(A).

In a group of 20, 12 like tea, 9 like coffee and 3 like neither. How many like both?

Challenge worked examples — exam-style, every step shown

These six are at the hard end of what the exam can ask. Do not rush them. Each part is its own card, each step is one small move, and the diagram is redrawn every time it changes so you can watch it fill up. The habit that makes all of them easy: fill the middle first and work outwards.

Challenge example 1

Two-set Venn from four given numbers

n(ξ) = 50, n(A) = 27, n(B) = 19, n(A ∩ B) = 11.
(a) Fill in the Venn diagram. (b) Find n((A ∪ B)′). (c) Find n(A ∩ B′) — the number in A only.

(a)Fill the diagram — middle first, then outwards

Middle region = 11

n(A ∩ B) is given. ALWAYS write the overlap in first — n(A) and n(B) both include it, so nothing else is safe to place until it is down.

A only = 27 − 11 = 16

The 27 counts EVERYONE in A, including the 11 in the overlap. Take the overlap out to get the left crescent.

B only = 19 − 11 = 8

Same move on the right-hand side.

Neither = 50 − (16 + 11 + 8) = 50 − 35 = 15

ξ (the rectangle) holds everyone, so whoever is not inside a circle sits outside both. Never forget this fourth region.

Check the total: 16 + 11 + 8 + 15 = 50 ✓

The four regions must add to n(ξ). Ten seconds, and it catches almost every slip.

(b)Find n((A ∪ B)′)

Translate the symbols: (A ∪ B)′ = “NOT in (A or B)”

∪ is the cup (either circle), ′ means “everything outside that”. So this is the region outside both circles.

Read it off the finished diagram: n((A ∪ B)′) = 15

It is the “neither” number you already found. Translating first, then reading off, beats staring at symbols.

Two roads, same answer — use whichever your school taught

Road 1 — fill the diagram

Middle 11 → A only 16 → B only 8.
Inside the circles: 16 + 11 + 8 = 35.
Outside: 50 − 35 = 15.

Road 2 — the formula

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 27 + 19 − 11 = 46 − 11 = 35.
Then 50 − 35 = 15. Same number ✓

(c)Find n(A ∩ B′) — A only

Translate: A ∩ B′ = “in A AND not in B”

∩ is the cap (and), B′ is “outside B”. In A but outside B is the left crescent only.

Read it off: n(A ∩ B′) = 16

NOT 27. The trap is answering with the whole of A. “A only” always means the overlap has been taken out.

Challenge example 2

Two-set Venn with algebra in the regions

In a year group of 44 students, the Venn diagram shows the numbers studying Art (A) and Biology (B): the region for Art only contains 2x + 3, the overlap contains x, the region for Biology only contains 3x − 2, and 7 students study neither.
(a) Form an equation in x. (b) Solve it. (c) Find n(A). (d) Your turn: find n(B).

(a)Form an equation in x

The four regions must add to n(ξ) = 44

This is the ONLY fact you need. Every region of the rectangle, added up, is everybody.

(2x + 3) + x + (3x − 2) + 7 = 44

Write all four regions down before touching anything. The brackets are just packaging — drop them next step.

Collect the x terms: 2x + x + 3x = 6x

One kind of thing at a time. Letters first.

Collect the numbers: 3 − 2 + 7 = 8

3 − 2 = 1, then 1 + 7 = 8. Watch the minus sign on the 2 — this is where sign errors live.

So the equation is 6x + 8 = 44

That is the answer to (a). In the exam, this line alone earns a mark.

(b)Solve the equation

6x + 8 = 44

Copy it down again so you are solving from a clean line.

Take 8 from both sides: 6x = 44 − 8 = 36

Undo the + 8 first, then deal with the × 6.

Divide both sides by 6: x = 36 ÷ 6 = 6

So x = 6.

Sense check: regions are 15, 6, 16, 7 → 15 + 6 + 16 + 7 = 44 ✓

2(6) + 3 = 15, and 3(6) − 2 = 16. A negative or fraction region means the x is wrong — go back.

(c)Find n(A)

n(A) = everything inside circle A = (A only) + (the overlap)

n(A) is the WHOLE circle. The overlap is part of A, so it goes back in for this question.

n(A) = 15 + 6 = 21

Read both numbers straight off the finished diagram.

(d)Your turn — find n(B)

Using the finished diagram above, n(B) = ?

Challenge example 3

Three-set Venn — the classic sports question

80 students were asked which sports they play. 35 play football (F), 30 play cricket (C) and 28 play basketball (B). 12 play football and cricket, 9 play cricket and basketball, 10 play football and basketball, and 5 play all three.
(a) Fill in the centre and the three petals. (b) Fill in the three outer regions. (c) How many play exactly one sport? (d) Your turn: how many play none?

(a)Centre first, then the three petals

Centre = 5

“All three” is the very middle. It goes in FIRST, every time — all three pair-counts include it.

F and C only = 12 − 5 = 7

“12 play football and cricket” INCLUDES the 5 who play all three. Subtract them to get the petal between F and C.

C and B only = 9 − 5 = 4

Same subtraction, next petal.

F and B only = 10 − 5 = 5

And the third petal.

(b)Now the three outer regions

F only = 35 − (7 + 5 + 5) = 35 − 17 = 18

Circle F already holds three filled regions: the two petals touching it (7 and 5) and the centre (5). Whatever is left of the 35 goes in the outer part.

C only = 30 − (7 + 4 + 5) = 30 − 16 = 14

The petals touching C are 7 (with F) and 4 (with B), plus the centre.

B only = 28 − (5 + 4 + 5) = 28 − 14 = 14

The petals touching B are 5 and 4, plus the centre.

(c)How many play exactly one sport?

“Exactly one” = the three outer regions only

Not the petals (those play two) and not the centre (those play three). Point at the three single regions on the diagram.

18 + 14 + 14 = 46

18 + 14 = 32, then 32 + 14 = 46.

(d)Your turn — how many play none of the three?

n(ξ) = 80. Use the finished diagram. How many play no sport at all?

Challenge example 4

Reading set notation — the translation table

Exam questions quietly test whether you can turn symbols into a region. Work through each card: cover the picture, translate the symbols yourself, then check. (a) The four two-set expressions you must know. (b) One three-set expression. (c) Put numbers on all of them.

(a)The four two-set translations

(A ∪ B)′

“not in A or B” — outside both circles

this is the “neither” region

A′ ∩ B

“not in A, AND in B” — the right crescent

this is “B only”

A ∩ B′

“in A, AND not in B” — the left crescent

this is “A only”

A ∩ B

“in A AND in B” — the overlap only

“both” in word problems

How to translate ANY expression: read it inside-out and word by word. ∪ says “or”, ∩ says “and”, ′ says “not”. Then shade what the words describe. (A ∪ B)′ and A′ ∩ B′ come out as the SAME region — “not (A or B)” is the same people as “not A and not B”.

(b)A three-set expression: A ∩ B ∩ C′

Inside-out: (in A) AND (in B) AND (NOT in C)

Three conditions joined by ∩, so ALL must hold at once.

In A and in B → start from the A–B overlap

That is the lens between A and B — which the C circle cuts into two pieces.

…and NOT in C → keep only the piece OUTSIDE C

That is the petal at the top: the “A and B but not C” region. In Example 3 language: plays football and cricket but not basketball.

(c)Put numbers on them — using Example 1’s diagram

Recall Example 1: A only = 16, overlap = 11, B only = 8, neither = 15, n(ξ) = 50

Same finished diagram as before — now we read four different expressions off it.

n((A ∪ B)′) = 15

The “neither” region, straight off the picture.

n(A′ ∩ B) = 8

B only — the right crescent.

n(A ∩ B′) = 16

A only — the left crescent.

n(A′) = 50 − n(A) = 50 − 27 = 23

Everything outside circle A: that is B only + neither = 8 + 15 = 23 — same answer, good check.

Challenge example 5

Word problem — “at least”, “only” and “neither”

50 people were surveyed. 27 own a bike, 22 own a skateboard, and 8 own neither.
(a) How many own at least one of the two? (b) How many own both? (c) How many own only a bike? (d) Your turn: only a skateboard.

(a)“At least one” — translate it before you touch a number

“At least one” means “one or both” = anywhere inside the circles = n(B ∪ S)

TRAP: “at least one” does NOT mean “exactly one”. It includes the people who own both.

Everyone is either inside the circles or in the “neither” region

Two pieces that together make n(ξ) = 50. We know the neither piece: 8.

n(B ∪ S) = 50 − 8 = 42

So 42 own at least one.

(b)How many own both?

Add the two circle totals: 27 + 22 = 49

But only 42 people are actually inside the circles — so 49 is TOO BIG. Why? The “both” people got counted twice, once in each total.

The double-count is the overlap: both = 49 − 42 = 7

This is the formula n(B ∩ S) = n(B) + n(S) − n(B ∪ S), said in words.

(c)How many own only a bike?

“Only a bike” = in B but NOT in S = B ∩ S′

TRAP: “only” always throws the overlap out. The question is not asking for n(B).

Only bike = 27 − 7 = 20

Circle total minus overlap, exactly as in Example 1.

(d)Your turn — only a skateboard

How many own only a skateboard? (Then check all four regions add to 50.)

Challenge example 6

Past-paper style — notation, listing and a “show that”

ξ = {1, 2, 3, …, 12}. A = {multiples of 3} B = {factors of 12}.
(a) List the elements of A and B, and place every number on a Venn diagram. (b) List A ∩ B and find n((A ∪ B)′). (c) Show that n(A) + n(B) − n(A ∩ B) = n(A ∪ B). (d) True or false, with a reason: 9 ∈ A ∩ B′, and {3, 6} ⊆ A ∩ B.

(a)List the sets, then place every number once

A = {multiples of 3 up to 12} = {3, 6, 9, 12}

Count up in threes and stop at 12, because ξ stops at 12.

B = {factors of 12} = {1, 2, 3, 4, 6, 12}

Numbers that divide into 12 exactly. Work in pairs so you miss none: 1×12, 2×6, 3×4.

In both lists: 3, 6, 12 → these go in the overlap

Any number in both lists must be written ONCE, in the middle — never once in each circle.

Left over: 9 goes in A only; 1, 2, 4 go in B only; 5, 7, 8, 10, 11 go outside both

Every one of the 12 numbers appears exactly once somewhere in the rectangle. Count them at the end: 1 + 3 + 3 + 5 = 12 ✓

(b)List A ∩ B, then find n((A ∪ B)′)

A ∩ B = {3, 6, 12}, so n(A ∩ B) = 3

Read the overlap straight off the diagram.

A ∪ B = {1, 2, 3, 4, 6, 9, 12}, so n(A ∪ B) = 7

Everything inside either circle — list it once, no repeats.

n((A ∪ B)′) = 12 − 7 = 5

They are the numbers outside both circles: {5, 7, 8, 10, 11}. Listing them is the safest possible check.

(c)“Show that” n(A) + n(B) − n(A ∩ B) = n(A ∪ B)

Left side with OUR numbers: 4 + 6 − 3 = 10 − 3 = 7

A “show that” means: compute BOTH sides separately and show they match. Never start by assuming they are equal.

Right side: n(A ∪ B) = 7 (counted from the list in part b)

7 = 7 ✓ — both sides agree.

Say WHY in one sentence: adding n(A) and n(B) counts the overlap twice, so subtracting n(A ∩ B) once puts it right

The numbers 3, 6, 12 sit in both lists. That sentence is what the reasoning mark is for.

(d)True or false — with the reason written down

9 ∈ A ∩ B′ — TRUE

9 is a multiple of 3 (so 9 ∈ A) and 9 is not a factor of 12 (so 9 ∈ B′). Both conditions hold. On the diagram, 9 is the number sitting alone in the left crescent.

{3, 6} ⊆ A ∩ B — TRUE

A ∩ B = {3, 6, 12}. Is EVERY member of {3, 6} in that set? 3 ✓ and 6 ✓ — yes. ⊆ asks nothing about 12; the small set does not have to be the whole overlap.

How you lose the mark: a bare “true”

The reason IS the answer. Always name the set membership: “9 ∈ A because 3 × 3 = 9, and 9 ∉ B because 9 does not divide 12”.

Exam checklist for Sets — read it before every paper:
1. Fill the Venn from the MIDDLE outwards — centre first, petals next, singles last.
2. “Only” means the overlap is thrown out; a circle total n(A) always keeps it in.
3. ξ includes the “neither” people — two circles make FOUR regions, three circles make EIGHT.
4. Before answering anything, check every region adds up to n(ξ). Ten seconds, many marks.
5. ∪ cup = “or” (holds everything), ∩ cap = “and” (only the overlap), ′ = “not”.

E1.4 · high risk3 · Fractions, decimals and percentages▼

▶ Watch: E1.4 Fractions, decimals and percentages

Your check had fractions secure to step 5, so this section starts at step 4 and climbs. Converting between fractions, decimals and percentages is not three separate skills. It is one triangle with three one-way trips in it, and each trip is a single move.

From → to	What you do	Example
Fraction → decimal	Divide top by bottom (short division)	³⁄₈ = 3 ÷ 8 = 0.375
Decimal → fraction	Put it over 10, 100, 1000 … then simplify	0.35 = ³⁵⁄₁₀₀ = ⁷⁄₂₀
Decimal → percentage	× 100 (move the point two places right)	0.375 → 37.5%
Percentage → decimal	÷ 100 (move the point two places left)	62% → 0.62
Fraction → percentage	Go via the decimal, or scale the bottom to 100	⁷⁄₂₀ = ³⁵⁄₁₀₀ = 35%
Percentage → fraction	Over 100, then simplify	64% = ⁶⁴⁄₁₀₀ = ¹⁶⁄₂₅

Non-calculator: these eight are worth knowing on sight, because they turn a 30-second conversion into a 2-second one.
¹⁄₂ = 0.5 = 50% · ¹⁄₄ = 0.25 = 25% · ³⁄₄ = 0.75 = 75% · ¹⁄₅ = 0.2 = 20%
¹⁄₈ = 0.125 = 12.5% · ¹⁄₃ = 0.333… = 33¹⁄₃% · ¹⁄₁₀ = 0.1 = 10% · ¹⁄₂₀ = 0.05 = 5%

The mistake: converting a fraction to a percentage by writing the numerator with a percent sign — reading ³⁄₅ as 3%. A fraction becomes a percentage only by scaling the bottom to 100 or by dividing then multiplying by 100. ³⁄₅ = ⁶⁰⁄₁₀₀ = 60%.

Worked in full

Write ⁷⁄₈ as a decimal and as a percentage, without a calculator

7 ÷ 8 by short division: 8 into 7 does not go, so write 0.

Set it out as 7.000 ÷ 8. Carrying zeros is what makes this work by hand.

8 into 70 goes 8 times, 8 × 8 = 64, remainder 6

First decimal place is 8, carry the 6 to make 60.

8 into 60 goes 7 times, 8 × 7 = 56, remainder 4

Second decimal place is 7, carry the 4 to make 40.

8 into 40 goes 5 times exactly, remainder 0

Third place is 5 and the division has terminated.

⁷⁄₈ = 0.875

Read the digits off in order.

0.875 × 100 = 87.5%

Multiplying by 100 moves the decimal point two places right.

Last step is yours

Write 0.24 as a fraction in its simplest form

0.24 has two decimal places, so it is 24 hundredths

Two places means denominator 100. Three places would mean 1000.

0.24 = ²⁴⁄₁₀₀

Write it down before simplifying — do not try to do both at once.

Both are even: divide by 4, the HCF of 24 and 100

24 ÷ 4 = 6 and 100 ÷ 4 = 25.

0.24 = ⁶⁄₂₅

Check 6 and 25 share no factor, so this is simplest form.

Last two are yours

Write ⁵⁄₁₆ as a percentage

5 ÷ 16, so set out 5.0000 ÷ 16

Fraction to percentage goes through the decimal.

16 into 50 goes 3, 48, remainder 2 → 0.3

First place 3, carry 2 to make 20.

16 into 20 goes 1, 16, remainder 4 → 0.31

Second place 1, carry 4 to make 40.

16 into 40 goes 2, 32, remainder 8 → 0.312, then 16 into 80 goes 5 exactly

Third place 2, fourth place 5, remainder 0.

⁵⁄₁₆ = 0.3125

The division terminates after four places.

As a percentage: 31.25%

Multiply by 100, so the point moves two places right.

All yours

Write ³⁄₈ as a decimal

Write 0.45 as a fraction in its simplest form. Type it like 3/4.

Write 0.6% as a decimal

Write ¹³⁄₂₀ as a percentage. Type just the number.

Write 175% as a mixed number in its simplest form. Type it like 1 3/4.

Proper, improper and mixed

kind	means	example
proper fraction	top smaller than bottom	³⁄₅
improper fraction	top bigger than bottom	¹⁷⁄₅
mixed number	a whole number and a proper fraction	3²⁄₅

Improper → mixed: divide, and the remainder stays over the same bottom: 17 ÷ 5 = 3 remainder 2, so ¹⁷⁄₅ = 3²⁄₅. Mixed → improper: whole × bottom + top: 4³⁄₈ = ^{4 × 8 + 3}⁄₈ = ³⁵⁄₈.

Recurring decimals

A recurring decimal repeats for ever. One repeating digit gets a dot over it: 0.1777… is written 0.17̇. A repeating block gets a dot over its first and last digits: 0.1232323… is 0.12̇3̇, and 0.123123… is 0.1̇23̇ (or a bar over 123). Dividing a fraction gives a recurring decimal when the remainders start to repeat: 1/6 = 0.1666… = 0.16̇. Every recurring decimal is a fraction, so it is rational.

Recurring decimal → fraction. Call the decimal x. Multiply by powers of 10 until two numbers have exactly the same tail after the point. Subtract, so the tails cancel, and solve. The method marks are for the subtraction.

Worked in full

Write 0.17̇ (= 0.1777…) as a fraction in its simplest form.

x = 0.1777…

Name the decimal.

10x = 1.777… and 100x = 17.777…

Both now have the tail .777… after the point.

100x − 10x = 17.777… − 1.777…, so 90x = 16

The tails cancel exactly.

x = 16/90 = 8/45

Divide, then simplify by 2.

Last step is yours

Write 0.3̇6̇ (= 0.363636…) as a fraction in its simplest form.

x = 0.3636…, 100x = 36.3636…

A two-digit block repeats, so multiply by 100.

100x − x = 36, so 99x = 36

x has the same tail .3636… already.

x = 36/99 = 4/11

Divide top and bottom by 9.

All yours

Write ¹⁷⁄₅ as a mixed number. Type it like 2 1/4.

Write 4 3/8 as an improper fraction. Type it like 9/4.

Write 0.363636… as a fraction in its simplest form.

Write 0.2111… (= 0.21̇) as a fraction in its simplest form.

5/11 = 0.454545… Which block of digits repeats?

E1.5 · medium risk4 · Ordering and the inequality symbols▼

▶ Watch: E1.5 Ordering

Ordering questions are cheap marks that get thrown away for one reason: people compare a fraction against a decimal against a percentage by eye. Do not. Convert everything to decimals first, order the decimals, then write the answer back in the original forms.

Symbol	Means	Example
=	is equal to	0.5 = ¹⁄₂
≠	is not equal to	0.33 ≠ ¹⁄₃
>	is greater than	7 > 3
<	is less than	−7 < −3
≥	is greater than or equal to	x ≥ 5 allows x = 5
≤	is less than or equal to	x ≤ 5 allows x = 5

Reading the symbol: the wide open end always faces the bigger number, and the point faces the smaller. 7 > 3 opens towards the 7. Write the numbers first, then decide which way the symbol opens.

The mistake, and it is a sign-error mistake: saying −7 > −3 because 7 is bigger than 3. On a number line −7 sits further left, so it is smaller. With negatives, the one that looks bigger is smaller. Your check flagged recurring sign errors, and this is one of the places they show up.

Non-calculator: when comparing decimals, pad them to the same number of places first. 0.4, 0.35 and 0.409 become 0.400, 0.350 and 0.409, and now they order by eye. Without padding, 0.35 looks longer than 0.4 and people call it bigger.

Worked in full

Put in order, smallest first: 0.62, ³⁄₅, 63%, ⁵⁄₈

Choose one form to compare in — decimals

Never compare a fraction directly against a percentage.

³⁄₅ = 3 ÷ 5 = 0.6

Or scale to ⁶⁄₁₀.

63% = 0.63

Divide by 100, so the point moves two places left.

⁵⁄₈ = 5 ÷ 8 = 0.625

Short division: 8 into 50 is 6 r 2, 8 into 20 is 2 r 4, 8 into 40 is 5.

Pad to three places: 0.620, 0.600, 0.630, 0.625

Same number of digits, so now they compare by eye.

Order: ³⁄₅, 0.62, ⁵⁄₈, 63%

Write them back in their original forms — the question asked for those, not the decimals.

Last step is yours

Put in order, smallest first: −3, −0.5, −⁷⁄₂, −2.9

Convert to decimals: −3, −0.5, −3.5, −2.9

−⁷⁄₂ is −7 ÷ 2 = −3.5.

Picture a number line. More negative means further left, which means smaller

This is the step that stops the sign error.

−3.5 is furthest left, then −3, then −2.9, then −0.5

Ignore the minus signs to judge distance, then reverse the order.

Order: −⁷⁄₂, −3, −2.9, −0.5

Back into the original forms, smallest first.

Last two are yours

Write the correct symbol between each pair: (a) −8 □ −11 (b) ²⁄₃ □ 0.67

(a) −8 and −11: which is further left on the line?

−11 is further left, so −11 is the smaller one.

So −8 is the bigger number

The open end of the symbol must face −8.

(a) −8 > −11

Greater than. The bigger-looking digit belongs to the smaller number here.

(b) ²⁄₃ = 0.6666… and the other is 0.6700

Pad both to four places to compare.

(b) ²⁄₃ < 0.67

0.6666 is less than 0.6700, so less than.

All yours

Put the correct symbol between −5 and −2. Type > or <.

Which is largest: 0.7, ³⁄₄, 72%? Type it as a decimal, a fraction or a percentage exactly as listed.

Which is smallest: 0.3, 0.29, 0.301? Type it.

x is an integer and 2 < x ≤ 5. How many values can x take?

True or false: −0.4 > −0.04. Type true or false.

E1.6 · high risk5 · The four operations, negatives and fractions▼

▶ Watch: E1.6 The four operations

Your number sense scored full marks, so the arithmetic underneath this is not the problem. Two things are: the order operations happen in, and signs. Your check flagged sign errors recurring across several different topics, and this is where they start.

The order, and why brackets are first: Brackets, Indices, Division and Multiplication together left to right, then Addition and Subtraction together left to right. Division and multiplication are the same rank — you do not do all the multiplying first. 12 ÷ 3 × 2 = 8, not 2.

The sign mistake, number one: −7 − (−3). Two minus signs next to each other become a plus, so this is −7 + 3 = −4. Not −10. Read “subtract negative three” as “add three” every single time.

The sign mistake, number two: −3² against (−3)². Without a bracket the power belongs to the 3 only, so −3² = −9. With the bracket the minus is inside, so (−3)² = +9. Cambridge tests this pair directly.

Non-calculator sign rule: for multiplying and dividing only, same signs give a positive and different signs give a negative. That rule does not apply to adding and subtracting, which is where most people misuse it. For adding, picture the number line instead.

Negatives and order of operations

Worked in full

Work out 5 − 3 × (−4) + (−2)³

Brackets first: (−4) and (−2) are already single values

Nothing to simplify inside them, so move on.

Indices next: (−2)³ = −2 × −2 × −2 = −8

Two negatives give +4, then × −2 gives −8. An odd power keeps the sign negative.

Multiplication: 3 × (−4) = −12

Different signs, so the answer is negative.

Now the expression is 5 − (−12) + (−8)

Rewrite the whole line before adding anything. This is the step that prevents the error.

5 − (−12) = 5 + 12 = 17

Minus a negative becomes plus.

17 + (−8) = 17 − 8 = 9

Plus a negative becomes minus. Final answer 9.

Negative numbers in real life: temperature

A temperature question is a number line stood on its end. A rise means add; a fall means subtract. The difference between two temperatures is the higher one minus the lower one, and a difference is never negative.

Count through zero. From −5 °C up to 8 °C: 5 degrees to reach zero, then 8 more. 5 + 8 = 13. The calculation 8 − (−5) gives the same 13, because subtracting a negative is adding.

Worked in full

At 6 a.m. the temperature is −7 °C. By midday it has risen by 12 °C. By midnight it has fallen by 9 °C from the midday temperature. Find the temperature at midday and at midnight.

midday: −7 + 12 = 5 °C

A rise is an addition. On the scale: up 7 to reach zero, then 5 more.

midnight: 5 − 9 = −4 °C

A fall is a subtraction. Down 5 to reach zero, then 4 below it.

check: −7 + 12 − 9 = −4 ✓

The whole day in one line gives the same answer.

Last step is yours

Find the difference between −5 °C and 8 °C.

higher − lower = 8 − (−5)

Always the higher temperature first, so the answer is positive.

8 − (−5) = 8 + 5 = 13 degrees

Minus a negative is plus. The picture says the same: 5 up to zero, then 8 more.

The mistake: 8 − 5 = 3. That treats −5 as if it were 5 and finds the gap from 5 to 8, not from −5. If one temperature is below zero and the other above, the gap must be bigger than both numbers.

All yours

At midnight it is −3 °C. By 6 a.m. the temperature has fallen by 9 °C. What is the temperature at 6 a.m., in °C? Number only.

It is −14 °C in Moscow and 23 °C in Delhi. How many degrees warmer is Delhi? Number only.

The temperature was 4 °C at 6 p.m. and −6 °C at midnight. By how many degrees did it fall? Number only.

Decimals by hand

to	do this	example
multiply	ignore the points and multiply the whole numbers; the answer has as many decimal places as the two numbers had between them	0.3 × 0.04: 3 × 4 = 12, and 1 + 2 = 3 places, so 0.012
divide	multiply BOTH numbers by 10, 100 or 1000 until you are dividing by a whole number; the answer does not change	4.2 ÷ 0.06 = 420 ÷ 6 = 70

Worked in full

Work out 0.3 × 0.04.

3 × 4 = 12

Multiply the digits as whole numbers first.

decimal places: 0.3 has 1, 0.04 has 2, so 1 + 2 = 3

Count the places in the question, not in your 12.

12 → 0.012

Give 12 three decimal places by putting in a zero in front.

check the size: 0.3 × 0.04 is less than 0.04 ✓

Multiplying by 0.3 makes a number smaller.

Worked in full

Work out 4.2 ÷ 0.06.

× 100 both: 420 ÷ 6

Two decimal places in 0.06, so multiply both numbers by 100. The answer is unchanged, like scaling both sides of a fraction.

420 ÷ 6 = 70

Now an ordinary whole-number division.

check: 70 × 0.06 = 4.2 ✓

Dividing by a number less than 1 makes the answer bigger, so 70 being bigger than 4.2 is right.

Last step is yours

Work out 2.5 × 1.6.

25 × 16 = 400

25 × 16 = 25 × 4 × 4 = 100 × 4.

1 + 1 = 2 decimal places: 400 → 4.00 = 4

Two places in 400 gives 4.00, which is just 4.

The mistake: 4.2 ÷ 0.06 = 0.7. That moves the point in only one number. Whatever you multiply the second number by, you must multiply the first number by too.

All yours

Work out 7.2 ÷ 0.3.

Work out 0.6 × 0.07.

Work out 1.44 ÷ 0.012.

Work out 3.4 × 0.25.

Fractions — rebuilding from step 5

Your fractions strand was secure to step 5 and broke at step 6, so start with the rule for each of the four operations stated plainly, then climb into mixed numbers.

Operation	Rule	Example
Add / subtract	Common denominator first, then add the tops only	¹⁄₄ + ¹⁄₆ = ³⁄₁₂ + ²⁄₁₂ = ⁵⁄₁₂
Multiply	Tops together, bottoms together. Cancel first if you can	²⁄₃ × ⁹⁄₁₀ = ³⁄₅
Divide	Flip the second fraction and multiply	³⁄₄ ÷ ²⁄₅ = ³⁄₄ × ⁵⁄₂ = ¹⁵⁄₈
Mixed numbers	Turn into improper fractions before doing anything	2³⁄₄ = ¹¹⁄₄

The mistake: adding fractions by adding tops and bottoms — ¹⁄₂ + ¹⁄₃ = ²⁄₅. That is wrong, and you can see it is wrong: ²⁄₅ is smaller than ¹⁄₂, yet you added something to it. The real answer is ⁵⁄₆.

The mixed-number trap: subtracting the whole parts and the fraction parts separately. 3¹⁄₄ − 1²⁄₃ is not 2 − something small, because ¹⁄₄ − ²⁄₃ is negative. Convert to improper fractions first and the trap disappears.

Worked in full

Work out 2³⁄₄ − 1⁵⁄₆

2³⁄₄ = ¹¹⁄₄

2 × 4 = 8, plus the 3 on top gives 11 quarters.

1⁵⁄₆ = ¹¹⁄₆

1 × 6 = 6, plus 5 gives 11 sixths.

Common denominator of 4 and 6 is 12

The LCM of 4 and 6. Use the LCM, not 24, to keep the numbers small.

¹¹⁄₄ = ³³⁄₁₂ and ¹¹⁄₆ = ²²⁄₁₂

Multiply top and bottom by 3 and by 2 respectively.

³³⁄₁₂ − ²²⁄₁₂ = ¹¹⁄₁₂

Subtract the tops only. The bottom does not change.

Answer ¹¹⁄₁₂

Less than 1, which is sensible: 2.75 − 1.83 is about 0.92.

Last step is yours

Work out ²⁄₃ ÷ 1¹⁄₅

1¹⁄₅ = ⁶⁄₅

Convert the mixed number before touching the division.

²⁄₃ ÷ ⁶⁄₅ = ²⁄₃ × ⁵⁄₆

Flip the second fraction and change the sign to multiply.

Cancel: 2 and 6 share a factor of 2, giving ¹⁄₃ × ⁵⁄₃

Cancelling before multiplying keeps the numbers small.

= ⁵⁄₉

1 × 5 on top, 3 × 3 on the bottom.

Last two are yours

Work out (−³⁄₄) + ⁵⁄₆ × ³⁄₁₀

Multiplication comes before addition, so do ⁵⁄₆ × ³⁄₁₀ first

The order of operations applies to fractions exactly as it does to whole numbers.

Cancel 5 with 10 and 3 with 6: ¹⁄₂ × ¹⁄₂ = ¹⁄₄

Cancel diagonally across the multiplication before multiplying.

Now −³⁄₄ + ¹⁄₄

Same denominator already, so no scaling needed.

−3 + 1 = −2 on the top

Adding a positive to a negative moves right along the line, towards zero.

Answer −¹⁄₂

−²⁄₄ simplifies to −¹⁄₂. Keep the minus sign in the answer.

All yours

Work out −6 − (−9)

Work out −4²

Work out 20 − 4 × 3 + 2³

Work out ³⁄₅ + ¹⁄₄. Type it like 7/8.

Work out 1¹⁄₂ × 2²⁄₃. Type it like 7/2 or 3 1/2.

Work out (−2)³ × (−5)

Work out ⁷⁄₈ ÷ ⁷⁄₂. Type it like 1/4.

E1.10 · medium risk6 · Limits of accuracy and bounds▼

▶ Watch: E1.10 Limits of accuracy

A measurement written as 24 cm to the nearest centimetre is not exactly 24. It is anything that rounds to 24. Bounds questions ask for the two ends of that range, and there is one rule.

The rule: take half of the rounding unit and go half down and half up. Rounded to the nearest 1 → ±0.5. Nearest 10 → ±5. Nearest 0.1 → ±0.05. To 2 decimal places → ±0.005. To 1 significant figure on a number like 300 → ±50.

The mistake: giving the upper bound as 24.4 or 24.49 because “24.5 rounds up, so it cannot be included”. Cambridge wants 24.5. The convention is 23.5 ≤ x < 24.5, and you write the upper bound as the exact number 24.5 even though it is not attained. Writing 24.49 loses the mark.

Bounds of a calculation — which bound to use:
Biggest sum: upper + upper. Smallest sum: lower + lower.
Biggest difference: upper − lower. Smallest difference: lower − upper.
Biggest product: upper × upper. Smallest product: lower × lower.
Biggest quotient: upper ÷ lower. Smallest quotient: lower ÷ upper.
Subtraction and division are the two that cross over. To make a gap big, start big and take away as little as possible.

Worked in full

A rod is 24 cm to the nearest cm. Write down its lower and upper bounds.

The rounding unit is 1 cm

Nearest centimetre, so the unit is 1.

Half the unit is 0.5

Every bounds question starts with this number.

Lower bound = 24 − 0.5 = 23.5 cm

Anything from 23.5 upwards rounds to 24.

Upper bound = 24 + 0.5 = 24.5 cm

Write 24.5, not 24.49.

So 23.5 ≤ L < 24.5

Note the strict inequality on the right: 24.5 itself would round to 25.

Last step is yours

A mass is given as 3.6 kg to 1 decimal place. Find its bounds.

1 decimal place means the rounding unit is 0.1

The last digit shown is a tenth.

Half the unit is 0.05

Half of 0.1. Not 0.5, and not 0.01.

Lower bound = 3.6 − 0.05 = 3.55 kg

Subtract half the unit.

Upper bound = 3.6 + 0.05 = 3.65 kg

Add half the unit. Both bounds carry an extra decimal place.

Last two are yours

A rectangle measures 8 cm by 5 cm, each to the nearest cm. Find the largest possible area and the smallest possible perimeter.

Bounds for 8: 7.5 to 8.5. Bounds for 5: 4.5 to 5.5

Half of 1 is 0.5 for both.

Largest area needs both sides as large as possible

Multiplying, so upper × upper.

Largest area = 8.5 × 5.5

8.5 × 5.5 = 8.5 × 5 + 8.5 × 0.5 = 42.5 + 4.25.

Largest area = 46.75 cm²

Keep the units squared.

Smallest perimeter = 2 × (7.5 + 4.5) = 24 cm

Adding, so smallest needs lower + lower. 2 × 12 = 24.

All yours

A length is 150 cm to the nearest 10 cm. Write down the lower bound.

A mass is 4.72 kg to 2 decimal places. Write down the upper bound.

x = 12 and y = 7, each to the nearest whole number. Find the largest possible value of x − y.

A square has side 9 cm to the nearest cm. Find the smallest possible area in cm squared.

A car travels 80 km (to the nearest km) in 2 hours (to the nearest hour). Find the largest possible average speed in km/h, to 1 decimal place.

E1.13 · high risk7 · Percentages, reverse percentage and interest▼

▶ Watch: E1.13 Percentages

🎙 Tutor Live — hear this section insteadA voice lesson on the five percentage types, including reverse percentages. Same content as below, just out loud, and it asks you questions as it goes.

Percentages are the most examined idea in topic 1 and the one with the most ways to go wrong. Almost all of it collapses into a single technique: the multiplier. Learn it once and five different question types become the same question.

The multiplier: a percentage change is a single multiplication.
Increase by 15% → × 1.15 Decrease by 15% → × 0.85
Increase by 7% → × 1.07 Decrease by 40% → × 0.6
Build it as 100% ± the change, then divide by 100. Never work out the change and add it on separately — that is two steps and two chances to slip.

The mistake that costs the most marks in this whole topic: answering a reverse percentage question by taking the percentage off the new figure. A price is £60 after a 20% increase — the original is not 60 − 20% = 48. The 20% was of the old price, not the new one. 60 is 120% of the old price, so the old price is 60 ÷ 1.2 = 50. Check: 50 + 20% = 60. Correct.

Non-calculator percentages of an amount: build them from 10%, 1% and halves. 10% is the amount ÷ 10. 1% is ÷ 100. 5% is half of 10%. So 35% of 240 = 24 + 24 + 24 + 12 = 84, using three lots of 10% plus a 5%. Faster and safer than any other route by hand.

Worked in full

Increase 480 by 15%, without a calculator

Multiplier = 100% + 15% = 115% = 1.15

Build it first, before touching the 480.

480 × 1.15 — split it as 480 × 1 + 480 × 0.15

By hand, split the multiplier into parts you can do mentally.

480 × 0.1 = 48

That is 10%.

480 × 0.05 = 24

Half of the 10%, so that is the 5%.

480 + 48 + 24 = 552

The original plus the 15%.

Check: 552 is a bit more than 480, and 15% of 480 is roughly 70

Sanity check the size before writing it down.

One amount as a percentage of another

Worked in full

A test is marked out of 80 and Tara scores 68. What percentage is that?

Write it as a fraction: ⁶⁸⁄₈₀

The part goes on top, the whole goes on the bottom. Getting these the wrong way round is the usual error.

Multiply by 100: ⁶⁸⁄₈₀ × 100

This is the definition of turning a fraction into a percentage.

Cancel: 100 ÷ 80 = ⁵⁄₄, so 68 × ⁵⁄₄

Cancel before multiplying to keep the arithmetic small.

68 ÷ 4 = 17, then 17 × 5 = 85

Divide first, then multiply — much easier by hand than 68 × 5 ÷ 4.

85%

Sensible: 68 out of 80 is clearly more than three quarters.

Reverse percentage

Last step is yours

A coat costs £90 in a sale after a 25% reduction. Find the original price.

The 25% was taken off the original, not off 90

This sentence is the whole question.

So £90 represents 100% − 25% = 75% of the original

Write down what percentage the given figure actually is.

Multiplier is 0.75, and it was applied to the original to get 90

Original × 0.75 = 90.

So original = 90 ÷ 0.75

Reverse the multiplication by dividing.

90 ÷ 0.75 = 9000 ÷ 75 = £120

Multiply both by 100 to clear the decimal, then 75 × 120 = 9000. Check: 120 − 25% = 90.

Simple and compound interest

Simple interest is the same amount added every year, worked out on the starting amount only: interest = P × r × t ÷ 100. Compound interest earns interest on the interest, so you multiply repeatedly: final = P × (multiplier)ⁿ. Compound always gives more after year 1.

Last two are yours

£2000 is invested at 5% per year compound interest for 3 years. Find the total value, and how much more it is than simple interest at the same rate.

Compound multiplier = 1.05, applied 3 times

100% + 5% = 105% = 1.05.

Year 1: 2000 × 1.05 = 2100

5% of 2000 is 100.

Year 2: 2100 × 1.05 = 2205

5% of 2100 is 105 — more than last year, which is the whole point of compounding.

Year 3: 2205 × 1.05 = £2315.25

5% of 2205 is 110.25, and 2205 + 110.25 = 2315.25.

Simple interest = 2000 × 5 × 3 ÷ 100 = £300, so the simple total is £2300

The same £100 every year, three times.

Difference = 2315.25 − 2300 = £15.25

Small over 3 years, large over 30. That is the exam point.

All yours

Work out 35% of 240

Increase 250 by 12%

Decrease 640 by 15%

Write 27 out of 60 as a percentage. Type just the number.

After a 20% increase a price is £72. Find the original price in pounds. Type just the number.

After a 30% reduction a jacket costs £56. Find the original price in pounds.

£500 at 4% per year simple interest for 5 years. How much interest is earned, in pounds?

£1000 at 10% per year compound interest for 2 years. What is the total value in pounds?

Challenge worked examples — exam-style, every step shown

Six harder questions of the kind the unit exam will actually ask, each one worked line by line. Nothing here is skipped or “left as an exercise” except where the box says it is yours.

About the two methods: your school teaches the unitary method (find 1%, then scale up to 100%). This site mostly uses the multiplier method (× 1.15 for a 15% increase, ÷ 1.15 to reverse it). They are the same maths written two ways, so wherever they differ this page shows both, side by side. Pick one, but be able to read the other — mark schemes accept either.

Example 1 · part (a)

A jacket costs ₹500. The shop raises the price by 20%. Find the new price.

School way (unitary)

10% of 500 = 500 ÷ 10 = 50

Ten percent is just ÷ 10.

20% = 50 × 2 = 100

Double the 10%.

500 + 100 = 600

Add the increase on.

₹600

Multiplier way

100% + 20% = 120% = 1.2

Build the multiplier first.

500 × 1.2 = 500 + 500 × 0.2

Split it to do it by hand.

500 + 100 = 600

Same arithmetic, one multiplication.

₹600

Both are correct. Use the one your teacher uses in class; they will always agree.

Example 1 · part (b)

A month later the shop cuts the new price by 20%. Find the final price.

The 20% cut is 20% of ₹600, the current price

A percentage is always OF something — here it is of the new price, not the old one.

10% of 600 = 60, so 20% = 120

Build from 10% as usual.

600 − 120 = 480

Or in one go: 600 × 0.8 = 480.

Final price is ₹480 — not back to ₹500

Up 20% then down 20% does NOT cancel out. Keep reading.

Example 1 · part (c)

Explain why the price did not return to ₹500, and find the overall percentage change.

The increase was 20% of 500 = ₹100 added

20% of the smaller number.

The decrease was 20% of 600 = ₹120 taken off

20% of a BIGGER number takes off more than was added. That is the whole story.

Multipliers: 1.2 × 0.8 = 0.96

Successive changes multiply. 12 × 8 = 96, so 1.2 × 0.8 = 0.96.

0.96 means 96% of the start: a 4% decrease overall

Check: 4% of 500 = 20, and 500 − 20 = 480. Matches part (b).

Example 2 · part (a)

Tara revises for 45 minutes out of a 2-hour evening. First, put both times in the same unit.

2 hours = 2 × 60 = 120 minutes

You cannot make a percentage out of minutes and hours mixed. Convert FIRST, before any fraction.

The two numbers to compare are 45 and 120

Writing 45 over 2 would be nonsense — the units must match.

Example 2 · part (b)

Write 45 minutes as a percentage of 2 hours.

Fraction: ⁴⁵⁄₁₂₀

The part on top, the whole underneath.

Cancel: divide top and bottom by 15 → ³⁄₈

45 ÷ 15 = 3 and 120 ÷ 15 = 8. Small numbers are safe numbers.

³⁄₈ × 100 = 300 ÷ 8

Turn the fraction into a percentage by multiplying by 100.

300 ÷ 8: 8 × 37 = 296, remainder 4, and 4 ÷ 8 = 0.5

So 300 ÷ 8 = 37.5, done entirely by hand.

45 minutes is 37.5% of 2 hours

Sensible: 45 min is a bit more than a third of 120 min, and 37.5% is a bit more than 33%.

Example 2 · part (c)

Quick second run: write 1 hour 30 minutes as a percentage of 2 hours.

1 h 30 min = 90 minutes; the whole is 120 minutes

Same first move: everything into minutes.

⁹⁰⁄₁₂₀ = ³⁄₄

Divide top and bottom by 30.

³⁄₄ = 75%

A fraction you should know on sight.

Example 3 · part (a) — reverse percentage, the big one

After a 15% price increase, a phone costs ₹4,600. Find the original price.

Read this line before any working: the 15% was of the original price, which you do not know yet. So you may NOT take 15% of 4,600. The figure ₹4,600 is 115% of the original, and the question is: what is 100%?

School way (unitary)

115% = 4600

Say what the given number IS as a percentage.

1% = 4600 ÷ 115 = 40

115 × 40 = 4600, because 115 × 4 = 460.

100% = 40 × 100 = 4000

Scale the 1% back up to the whole.

₹4,000

Multiplier way

original × 1.15 = 4600

The increase was one multiplication.

original = 4600 ÷ 1.15

Reverse a multiplication by dividing.

= 460000 ÷ 115 = 4000

Multiply both by 100 to clear the decimal.

₹4,000

Both are correct. Use the one your teacher uses in class; they will always agree. (They must: dividing by 1.15 is exactly “÷ 115, × 100” done in one step.)

Example 3 · part (b)

Check the answer, and see exactly what the wrong method gives.

Check forwards: 15% of 4000 = 400 + 200 = 600

10% is 400, 5% is 200.

4000 + 600 = 4600 ✓

A reverse-percentage answer can ALWAYS be checked forwards. Always do it.

The wrong way: 15% of 4600 = 690, and 4600 − 690 = 3910 ✗

Taking 15% off the NEW price gives 3910, not 4000 — and the check exposes it: 3910 × 1.15 = 4496.50, not 4600.

Same idea after a decrease: a bag costs ₹3,300 after a 12% reduction

Now the given figure is 100% − 12% = 88% of the original.

School way: 88% = 3300, 1% = 3300 ÷ 88 = 37.5, 100% = 3750

88 × 37.5 = 3300 because 88 × 37 = 3256 and 88 × 0.5 = 44.

Multiplier way: 3300 ÷ 0.88 = 330000 ÷ 88 = 3750

Same answer, one line. Check: 12% of 3750 = 450, and 3750 − 450 = 3300 ✓

Example 3 · part (c) — your turn

After a 25% increase, a ticket costs ₹615. Find the original price. Type just the number.

Example 4 · part (a)

₹8,000 is invested at 5% per year. Build the value year by year for 3 years, simple and compound side by side.

Simple interest: 5% of 8000 = 400, the SAME ₹400 every year

Simple interest is always worked on the starting amount only.

Compound: multiply by 1.05 each year, so the interest itself earns interest

Year 2 pays 5% of 8400, not 5% of 8000.

End of year	Simple (+400 each year)	Compound (×1.05 each year)
start	8000	8000
1	8000 + 400 = 8400	8000 × 1.05 = 8400
2	8400 + 400 = 8800	8400 × 1.05 = 8820
3	8800 + 400 = 9200	8820 × 1.05 = 9261

After 3 years: simple ₹9,200, compound ₹9,261 — compound is ₹61 ahead

Identical after year 1, then compound pulls away a little more every year.

Example 4 · part (b)

Get the same compound answer with the formula, then find how many whole years until the investment first exceeds ₹10,000.

Formula: final = P × (multiplier)ⁿ = 8000 × 1.05³

The table above IS this formula, unrolled one year at a time.

1.05 × 1.05 = 1.1025, and 1.1025 × 1.05 = 1.157625

So 1.05³ = 1.157625.

8000 × 1.157625 = 9261 ✓ — matches the table exactly

8000 × 1.157625 = 8 × 1157.625 = 9261.

“How many years until it exceeds 10,000?” — keep multiplying and WATCH

At IGCSE you answer this by trial, year by year. Not with logarithms.

Year 4: 9261 × 1.05 = 9724.05 — not yet

5% of 9261 is 463.05.

Year 5: 9724.05 × 1.05 = 10210.2525 — over 10,000. Answer: 5 years

5% of 9724.05 is 486.2025. Write the trial lines down; they are the method marks.

Example 4 · part (c) — your turn

₹2,000 is invested at 10% per year compound interest. After how many whole years does it first exceed ₹2,600? Type just the number of years.

Example 5 · part (a) — exam composite

A laptop’s price rises by 10%, then falls by 15% in a sale. Build the single multiplier for the whole journey.

Rise of 10% → × 1.1. Fall of 15% → × 0.85

One multiplier per change, built as 100% ± the change.

Combined: 1.1 × 0.85 = 0.935

11 × 85 = 935, then place the decimal: 0.935.

0.935 is NOT the same as “down 5%” (that would be 0.95)

+10% then −15% is not −5%, because the 15% is taken off a bigger number. Same trap as Example 1.

Example 5 · part (b)

The sale price is ₹18,700. Find the price before either change.

School way (unitary)

The sale price is 93.5% of the original

Because the combined multiplier is 0.935.

93.5% = 18700, so 1% = 18700 ÷ 93.5 = 200

93.5 × 200 = 18700 — spot it, or do 187000 ÷ 935.

100% = 200 × 100 = 20000

Scale back up.

₹20,000

Multiplier way

original × 0.935 = 18700

The whole journey was one multiplication.

original = 18700 ÷ 0.935 = 18700000 ÷ 935

Multiply both by 1000 to clear the decimal.

935 × 20000 = 18700000, so the original = 20000

One division reverses BOTH changes at once.

₹20,000

Both are correct. Use the one your teacher uses in class; they will always agree.

Never reverse the steps one at a time in the wrong order — and never “add the percentages back on”. Reversing means dividing by the combined multiplier (or 1% → 100% on the combined percentage), in one clean move.

Example 5 · part (c)

Check the answer forwards, and state the overall percentage change.

20000 × 1.1 = 22000

The 10% rise: 10% of 20000 is 2000.

22000 × 0.85 = 18700 ✓

The 15% fall: 15% of 22000 = 2200 + 1100 = 3300, and 22000 − 3300 = 18700.

Overall: 0.935 means a 6.5% decrease from the starting price

100% − 93.5% = 6.5%. Check: 6.5% of 20000 = 1300, and 20000 − 1300 = 18700 ✓

Example 6 · part (a) — profit and loss

A shop buys a bag for ₹800 (the cost price) and sells it for ₹1,000. Find the percentage profit.

Profit = 1000 − 800 = ₹200

Selling price minus cost price.

Percentage profit = ^profit⁄_{COST price} × 100 = ²⁰⁰⁄₈₀₀ × 100

The denominator is what the SHOP PAID. This is the single decision that earns or loses the marks.

²⁰⁰⁄₈₀₀ = ¹⁄₄, so the profit is 25%

Divide top and bottom by 200.

Example 6 · part (b) — the denominator trap, then a reverse

Why is 20% the wrong answer to part (a)? Then: a watch is sold for ₹1,092 at a 30% profit — find the cost price.

The trap: ²⁰⁰⁄₁₀₀₀ × 100 = 20% uses the SELLING price underneath

Same ₹200 profit, wrong denominator. Profit percent is measured against what was paid out, so it is 200 out of 800, giving 25%.

Now the reverse: sold at 30% profit means selling price = 130% of cost

The 30% was of the COST, which is the unknown — this is a reverse percentage in disguise.

School way (unitary)

130% = 1092

The selling price as a percentage of cost.

1% = 1092 ÷ 130 = 8.4

130 × 8 = 1040, and 130 × 0.4 = 52; 1040 + 52 = 1092.

100% = 8.4 × 100 = 840

Scale up to the cost.

₹840

Multiplier way

cost × 1.3 = 1092

A 30% profit is × 1.3 on the cost.

cost = 1092 ÷ 1.3 = 10920 ÷ 13

Multiply both by 10 to clear the decimal.

13 × 840 = 10920, so cost = 840

Check: 30% of 840 = 252, and 840 + 252 = 1092 ✓

₹840

Both are correct. Use the one your teacher uses in class; they will always agree.

Example 6 · part (c) — your turn

A phone case is sold for ₹690 at a LOSS of 8%. Find the cost price. Type just the number.

Exam checklist for Percentages

Reverse percentage means DIVIDE by the multiplier (or 1% → 100%). Never take the percentage of the new amount.
Before any fraction, get both quantities into the same unit (minutes with minutes, rupees with rupees).
Compound interest = multiply repeatedly (× 1.05 each year). Never add the same interest every year — that is simple interest.
Successive changes multiply: +10% then −15% is ×1.1 × 0.85 = ×0.935, a 6.5% fall — not a 5% fall. Up 20% then down 20% is NOT back to the start.
Profit or loss percent is on the COST price, the amount the buyer paid out — never on the selling price.
Check every answer forwards: after an increase the original must be SMALLER than the new figure; after a decrease, bigger. Ten seconds, and it catches nearly everything.

E1.15 · medium risk8 · Time, the 24-hour clock and timetables▼

▶ Watch: E1.15 Time

🎙 Tutor Live — hear this section insteadA voice lesson on durations, the next-day trap and reading timetables. Same content as below, just out loud, and it asks you questions as it goes.

Time is the one place where your solid arithmetic can still let you down, because time is not base ten. There are 60 minutes in an hour, not 100, so 2.5 hours is 2 hours 30 minutes and 2 hours 50 minutes is not 2.5 hours.

Unit	Equals
1 minute	60 seconds
1 hour	60 minutes = 3600 seconds
1 day	24 hours
1 week	7 days
1 year	365 days (366 in a leap year) = 52 weeks and 1 day

The mistake: writing 3 hours 45 minutes as 3.45 hours. Minutes convert to a decimal by dividing by 60, so 45 minutes is ⁴⁵⁄₆₀ = 0.75 hours, and the time is 3.75 hours. Any speed or rate calculation done with 3.45 is simply wrong.

Minutes to a decimal, the four you meet constantly: 15 min = 0.25 h, 20 min = ¹⁄₃ h, 30 min = 0.5 h, 45 min = 0.75 h. For anything else, divide by 60. Going back the other way, multiply the decimal part by 60: 2.4 h = 2 h and 0.4 × 60 = 24 min.

Non-calculator: never subtract times as if they were decimals. 13:15 − 09:40 is not 3.75. Bridge through the whole hour as in the diagram above — up to 10:00, across the whole hours, then on to the finish. It is three easy steps instead of one borrow you will get wrong.

24-hour clock: after midday, add 12. 3:20 pm becomes 15:20. Going back, subtract 12: 19:45 is 7:45 pm. Midnight is 00:00 and noon is 12:00. Always four digits, always a colon.

Worked in full

A train leaves at 09:40 and arrives at 13:15. How long is the journey?

Do not subtract the digits — minutes are out of 60, not 100

This is where the marks go.

09:40 to 10:00 is 20 minutes

Bridge up to the next whole hour first.

10:00 to 13:00 is 3 hours

Now count the whole hours, which is easy.

13:00 to 13:15 is 15 minutes

Then the leftover minutes at the end.

20 min + 3 h + 15 min = 3 h 35 min

20 + 15 = 35 minutes, which is under 60, so no extra hour.

Last step is yours

A film starts at 19:50 and lasts 2 hours 25 minutes. When does it end?

Add the hours first: 19:50 + 2 h = 21:50

Hours are safe to add straight on.

Now add the 25 minutes: 50 + 25 = 75 minutes

This is over 60, so it will roll into the next hour.

75 minutes = 1 hour 15 minutes

Take one 60 out and keep the remainder.

21:50 + 25 min = 22:15

21 + 1 = 22 hours, and 15 minutes left over.

Last two are yours

A bus leaves Bangalore at 22:45 and the journey takes 8 hours 40 minutes. Give the arrival time and say whether it is the next day.

Add the hours: 22:45 + 8 h = 30:45

Go past 24 for now — it is easier than splitting at midnight.

Add the minutes: 45 + 40 = 85 min = 1 h 25 min

85 is over 60, so one hour rolls over.

30:45 + 40 min = 31:25

30 + 1 = 31 hours, 25 minutes.

31:25 is past 24:00, so subtract 24: 07:25

Anything of 24 or more means you have crossed midnight.

It arrives on the next day

One rollover of 24 hours means one day later.

All yours

Write 4:35 pm using the 24-hour clock. Type it like 16:35.

Write 2 hours 12 minutes as a decimal number of hours

How many minutes from 08:55 to 11:20? Type just the number.

A journey starts at 23:30 and lasts 3 hours 50 minutes. What time does it end? Type it like 03:20.

Convert 1.75 hours into minutes

Challenge worked examples — exam-style, every step shown

Five exam-style time questions, harder than the warm-ups above, with every single step written out. Work through them slowly — each one is built around a trap that costs real marks on Paper 2.

Example 1 · A duration that crosses midnight

An overnight coach leaves at 23:35 and arrives the next morning at 06:20.

Part (a) — why ordinary subtraction fails

Explain why you cannot just do 06:20 − 23:35.

Try it: 06:20 − 23:35 would be a negative answer

The finish number is SMALLER than the start number, because the clock started again at midnight.

And even 24-hour times are not decimals: minutes run 00–59, not 00–99

Column subtraction of 620 − 2335 as if they were ordinary numbers is meaningless.

So we go the safe way: hop along a number line, past midnight

One easy hop at a time. This is the method that never breaks.

Part (b) — the hop method

Find the length of the journey from 23:35 to 06:20 the next day.

Hop 1: 23:35 → 00:00 (midnight)

35 + ? = 60, so this hop is 60 − 35 = 25 minutes.

Hop 2: 00:00 → 06:00

Whole hours are easy: 6 hours.

Hop 3: 06:00 → 06:20

The leftover minutes: 20 minutes.

Add the hops: 25 min + 6 h + 20 min

Minutes first: 25 + 20 = 45. That is under 60, so nothing rolls over.

Journey time = 6 h 45 min

As minutes if a question asks: 6 × 60 + 45 = 360 + 45 = 405 minutes.

School method — column subtraction (borrowing 60): some schools set this out like a column sum instead. Add 24 h to the finish time first because it is the next day: 06:20 + 24:00 = 30:20. Now subtract 30:20 − 23:35. You cannot take 35 from 20, so borrow one hour = 60 minutes (never 100!): 30:20 becomes 29:80. Then 29:80 − 23:35 = 6:45 — the same 6 h 45 min as the hop method. Both methods are fine in the exam; use whichever feels safer, but if you borrow, you must borrow 60.

Part (c) — 12-hour and 24-hour, both ways

Write the departure and arrival using the 12-hour clock.

23:35 is after midday, so subtract 12 from the hours: 23 − 12 = 11

24-hour times of 13:00 or more are pm times.

23:35 = 11:35 pm

The minutes never change — only the hours.

06:20 is before midday, so it stays as it is: 6:20 am

Morning times 01:00–11:59 keep their hours; just drop the leading zero and write am.

Check the special cases: midnight is 00:00 (12:00 am), noon is 12:00 (12:00 pm)

These two are where am/pm mistakes happen — learn them cold.

Example 2 · Reading a train timetable

Three trains run from Ashford to Eastgate each morning. The timetable shows the time each train leaves each station (Eastgate times are arrivals).

Station	Train A	Train B	Train C
Ashford	07:12	08:47	09:35
Brenley	07:31	09:06	09:54
Corton	07:58	09:33	10:23
Dunwich	08:24	09:59	10:51
Eastgate	08:46	10:21	11:15

Part (a) — how long does the 08:47 take?

How long does the 08:47 train take from Ashford to Eastgate?

Find the right column: the train leaving Ashford at 08:47 is Train B

Always read DOWN a column for one train, ACROSS a row for one station.

Read its Eastgate arrival: 10:21

Start 08:47, finish 10:21.

Hop 1: 08:47 → 09:00 is 13 minutes

47 + 13 = 60.

Hop 2: 09:00 → 10:00 is 1 hour

Hop 3: 10:00 → 10:21 is 21 minutes

13 min + 1 h + 21 min = 1 h 34 min

13 + 21 = 34 minutes, under 60, so no rollover.

Part (b) — the latest train to arrive by a deadline

Tara must reach Dunwich by 10:30. What is the latest train she can catch from Ashford?

Go to the Dunwich ROW and read across: 08:24, 09:59, 10:51

The question is about arriving at Dunwich, so that row decides everything.

Test the latest train first: Train C reaches Dunwich at 10:51

10:51 is AFTER 10:30 — too late. Cross it off.

Test Train B: it reaches Dunwich at 09:59

09:59 is before 10:30 — she arrives in time.

Answer: Train B, leaving Ashford at 08:47

Give the departure time from HER station, not the arrival time — read back up the column.

The mistake: answering “09:59” (an arrival time) or picking Train C because it is “closest to 10:30”. Closest is not good enough — 10:51 misses the deadline. Latest that still arrives in time = check from the right-hand end and cross off the ones that fail.

Part (c) — waiting time, then arrival

Meera gets to Corton station at 09:40. How long does she wait for the next train, and when does she reach Eastgate?

Corton row: trains leave Corton at 07:58, 09:33, 10:23

The 09:33 has already gone — it is before 09:40.

The next train is Train C at 10:23

First departure AFTER 09:40 in that row.

Wait: 09:40 → 10:00 is 20 min, then 10:00 → 10:23 is 23 min

Hop through the whole hour as usual.

Wait = 20 + 23 = 43 minutes

Train C reaches Eastgate at 11:15

Read down Train C’s column to the Eastgate row — no arithmetic needed, the timetable already says it.

Example 3 · The decimal-hours trap

This is the single most common time error on Paper 2: treating the decimal part of an hour as if it were minutes. 2.4 hours is NOT 2 h 40 min.

Part (a) — decimal hours → hours and minutes

Write 2.4 hours in hours and minutes.

Split off the whole hours: 2.4 h = 2 h + 0.4 h

Only the decimal part needs converting.

0.4 h means 0.4 OF an hour: 0.4 × 60 = 24 minutes

An hour has 60 minutes, so multiply the decimal by 60 — never just read the digits.

2.4 h = 2 h 24 min

Not 2 h 40 min. The digit 4 was tenths of an hour, and a tenth of an hour is 6 minutes: 4 × 6 = 24.

The mistake: reading 2.4 h as 2 h 40 min. Decimals are out of 10 or 100; minutes are out of 60. The bridge between them is always × 60 (hours → minutes) or ÷ 60 (minutes → hours).

Part (b) — hours and minutes → decimal hours

Write 3 h 36 min as a decimal number of hours.

Keep the whole hours: 3 h

Convert the minutes: 36 ÷ 60 = 0.6

Divide by 60 going this way. Check it feels right: 36 min is a bit over half an hour, and 0.6 is a bit over 0.5.

3 h 36 min = 3.6 hours

Not 3.36 — that would be 3 h 21.6 min.

Part (c) — where the trap bites: speed

A cyclist rides for 2 h 45 min at a steady 24 km/h. How far does she ride?

Distance = speed × time, and the time MUST be in hours (because the speed is per hour)

Units must match before any formula is used.

Convert: 45 ÷ 60 = 0.75, so 2 h 45 min = 2.75 h

If you had typed 2.45 the whole question would be wrong.

Distance = 24 × 2.75

Split it to do it without a calculator: 24 × 2 = 48 and 24 × 0.75 = 18.

48 + 18 = 66 km

24 × 0.75 is three quarters of 24: half of 24 is 12, a quarter is 6, so 12 + 6 = 18.

Part (d) — your turn

A car travels 126 km in 2 h 15 min. Work out its average speed in km/h.

Example 4 · Time zones: Bengaluru to London

A flight leaves Bengaluru at 02:15 IST and lands in London at 08:05 local time. In summer London is on UTC+1; India is always UTC+5:30. Find the flight time.

The mistake: doing 08:05 − 02:15 = 5 h 50 min. Those two times are on DIFFERENT clocks — subtracting them compares apples with oranges. Convert everything to ONE zone first.

Part (a) — how far apart are the two clocks?

Work out the time difference between Bengaluru (UTC+5:30) and London in summer (UTC+1).

Bengaluru is 5 h 30 min ahead of UTC; London is 1 h ahead of UTC

Both offsets are measured from the same reference, UTC, so they can be compared directly.

Difference = 5 h 30 min − 1 h = 4 h 30 min

So Bengaluru is 4 h 30 min AHEAD of London

When it is 08:05 in London, it is already 12:35 in Bengaluru. Ahead means the Indian clock shows a LATER time.

Part (b) — convert to one zone, then hop

Find the flight time.

Pick one zone and move BOTH times into it. Use IST

Departure 02:15 is already IST — only the landing needs converting.

Landing in IST = 08:05 + 4 h 30 min

London time + the 4:30 the Indian clock is ahead.

08:05 + 4 h = 12:05, then + 30 min = 12:35 IST

Add the hours first, then the minutes.

Now both times are on one clock: 02:15 → 12:35

An ordinary duration question at last.

Hops: 02:15 → 03:00 is 45 min; 03:00 → 12:00 is 9 h; 12:00 → 12:35 is 35 min

45 min + 9 h + 35 min = 9 h 80 min = 10 h 20 min

45 + 35 = 80 minutes = 1 h 20 min, so one hour rolls over: 9 + 1 = 10.

Check by a second route (UTC): departure 02:15 IST − 5:30 = 20:45 UTC the evening BEFORE; landing 08:05 − 1:00 = 07:05 UTC. From 20:45 to 07:05: 3 h 15 min to midnight, then 7 h 5 min = 10 h 20 min. Same answer, so the conversion was right. Converting both times to one zone always works — it does not matter which zone you pick.

Part (c) — your turn

The return flight leaves London at 13:40 local time and takes 9 h 35 min. At what time (IST, 24-hour clock) does it land in Bengaluru? Type it like 05:10.

Example 5 · A multi-leg journey, worked backwards

Tara must be at the exam hall by 09:00. Her journey: walk 12 min from home to the station, she wants 5 min at the station before the train leaves, the train ride is 26 min, then a 9 min walk to the hall. Trains leave at 08:05, 08:25 and 08:45.

Part (a) — which train? Work backwards from the deadline

What is the latest train she can catch?

Start at the deadline and undo the last leg: she must reach the far station by 09:00 − 9 min = 08:51

Working FORWARDS from three possible trains means three calculations; working backwards from the one fixed deadline means one.

Undo the train ride: the train must leave by 08:51 − 26 min

51 − 26 = 25, so it must leave by 08:25.

Check the timetable: 08:45 leaves after 08:25 — too late. The 08:25 is exactly on the limit

08:25 + 26 = 08:51, + 9 min walk = 09:00 on the dot.

Latest train = the 08:25

Part (b) — when must she leave home?

What is the latest time she can leave home?

She wants to be at the station 5 min before the train: 08:25 − 5 = 08:20

Undo the walk to the station: 08:20 − 12 min

20 − 12 = 8, so 08:08.

Latest time to leave home = 08:08

Sensible check going forwards: 08:08 + 12 = 08:20, + 5 = 08:25 train, + 26 = 08:51, + 9 = 09:00. It all chains up.

Why backwards? Whenever a question fixes the FINISH (“arrive by…”, “be there for…”), start at the finish and subtract the legs one at a time. Each subtraction is small and checkable. Then run the whole chain forwards once as a check, like step 3 above.

Part (c) — your turn

Disaster: she misses the 08:25 and catches the 08:45 instead. At what time does she reach the exam hall? Type it like 09:05.

Exam checklist for Time
1. Never subtract clock times like decimals — minutes are out of 60, so hop through the next whole hour (or borrow 60, never 100).
2. Decimal hours: multiply the decimal by 60. 0.4 h = 24 min, not 40 min — and 2 h 15 min = 2.25 h before it goes anywhere near a speed formula.
3. Crossing midnight? Hop to 00:00 first, then onwards — or add 24 h to the finish time before subtracting.
4. Time zones: convert EVERYTHING to one zone before finding any duration, then check by converting to the other zone.
5. Converting 12↔24-hour: pm means add 12 to the hours; check midnight (00:00) and noon (12:00) specially; minutes never change.

E1.16 · medium risk9 · Money and currency conversion▼

▶ Watch: E1.16 Money

🎙 Tutor Live — hear this section insteadA voice lesson on best value, currency and percentage profit. Same content as below, just out loud, and it asks you questions as it goes.

Money questions are ordinary arithmetic wearing a currency symbol. Two things make them their own sub-topic: always two decimal places in an answer, and currency conversion, which is one multiplication done in the right direction.

Getting the direction right, every time: write the exchange rate as a sentence with the currency you have on the left.
£1 = 105 rupees. Have pounds, want rupees → multiply by 105.
Have rupees, want pounds → divide by 105.
Then check the size: there are many rupees to a pound, so a rupee answer should be a big number. If it comes out small, you divided when you should have multiplied.

The mistake: writing a money answer as £4.5 or £12.7. Money always carries two decimal places: £4.50 and £12.70. And 3.6 hours of pay is not £3.60 — keep the units straight.

Non-calculator division by a rate: to divide by 1.25, multiply by 100 and divide by 125. To divide by 0.8, multiply by 10 and divide by 8. Clearing the decimal from the divisor first turns an awkward division into an easy one, and it is always allowed because you scale both numbers equally.

Worked in full

The exchange rate is £1 = 105 rupees. Convert 4500 rupees into pounds.

I have rupees and want pounds

State this first. It fixes the direction.

£1 = 105 rupees, so rupees → pounds means divide by 105

Going towards the currency worth more per unit means dividing.

4500 ÷ 105 — cancel a factor of 15 from both

4500 ÷ 15 = 300 and 105 ÷ 15 = 7.

= 300 ÷ 7 = 42.857…

7 into 30 is 4 r 2, 7 into 28 is 4 exactly, and so on.

£42.86 to the nearest penny

Round to 2 decimal places, and write both of them.

Last step is yours

1 euro = 1.20 US dollars. Convert 96 dollars into euros.

I have dollars and want euros

The rate is written with euros on the left, so the given direction is euros to dollars.

Euros → dollars is × 1.20, so dollars → euros is ÷ 1.20

Reverse a multiplication with a division.

96 ÷ 1.2 = 960 ÷ 12

Multiply both numbers by 10 to clear the decimal from the divisor.

= 80 euros

12 × 80 = 960. Sensible, because a euro is worth more than a dollar so the euro figure must be smaller.

Last two are yours

A shop sells 750 g of rice for £2.10 and 1.2 kg for £3.48. Which is better value, and by how much per kilogram?

Put both into the same units: 750 g = 0.75 kg

You cannot compare grams against kilograms.

Small pack: 2.10 ÷ 0.75 per kg

Cost divided by mass gives cost per kilogram.

2.10 ÷ 0.75 = 210 ÷ 75 = £2.80 per kg

Multiply both by 100 to clear the decimals, then 75 × 2.8 = 210.

Large pack: 3.48 ÷ 1.2 = 34.8 ÷ 12 = £2.90 per kg

12 × 2.9 = 34.8.

So the 750 g pack is better value, by 10p per kg

2.90 − 2.80 = 0.10. The bigger pack is not automatically cheaper, and that is the point of the question.

All yours

£1 = 1.15 euros. Convert £60 into euros.

1 US dollar = 83 rupees. Convert 4150 rupees into dollars.

A jacket costs £48. In a sale it is reduced by £7.20. Write the sale price with two decimal places, without the pound sign.

6 pens cost £7.50. How much do 10 pens cost, in pounds?

£1 = 105 rupees. A bag costs 3675 rupees. Find the cost in pounds.

Challenge worked examples — exam-style, every step shown

Five money questions of the kind the unit exam actually asks, in rising order of difficulty. Every part is worked in numbered micro-steps. One rule runs through all of them: a money answer always has two decimal places and its currency symbol — £150.00, not £150; ₹99.82, not ₹99.8.

Example 1 · One rate, both directions

The exchange rate is £1 = ₹105.40. (a) Convert £260 into rupees. (b) Convert ₹15,810 into pounds. (c) Convert ₹8,200 into pounds.

The arrow picture — decide multiply-or-divide before touching a number. Draw one arrow from the currency in the rate that equals 1 (here £) to the other (₹), and write the rate on it. Travelling along the arrow: multiply. Travelling against it: divide. That is the whole rule.

1(a) · pounds → rupees

Convert £260 into rupees.

I have pounds, I want rupees — that is travelling ALONG the arrow, so multiply

Say the direction out loud before any arithmetic. It earns the method mark.

260 × 105.40 = 260 × 105 + 260 × 0.40

Split the rate into an easy whole part and an easy decimal part.

260 × 105 = 260 × 100 + 260 × 5 = 26,000 + 1,300 = 27,300

Hundreds first, then fives — no carrying needed.

260 × 0.40 = 26 × 4 = 104

×0.4 is the same as ×4 then ÷10.

27,300 + 104 = ₹27,404.00

Two decimal places and the symbol, even when the paise are zero.

1(b) · rupees → pounds, two layouts

Convert ₹15,810 into pounds — shown by BOTH school methods.

I have rupees, I want pounds — that is AGAINST the arrow, so divide

Sense check waiting at the end: pounds are worth more each, so the answer must be a smaller number than 15,810.

Direct method: 15,810 ÷ 105.40 = 158,100 ÷ 1,054

Multiply top and bottom by 10 to clear the decimal from the divisor — always allowed.

1,054 × 150 = 105,400 + 52,700 = 158,100, so 158,100 ÷ 1,054 = 150

1,054 × 100 = 105,400 and 1,054 × 50 = 52,700. It divides exactly.

Unitary method: ₹105.40 = £1, so ₹1 = £(1 ÷ 105.40)

Find what ONE rupee is worth first — that is the whole idea of the unitary layout.

₹15,810 = 15,810 × (1 ÷ 105.40) = 15,810 ÷ 105.40 = 150

Multiplying by 1⁄105.40 IS dividing by 105.40 — the two layouts are the same sum in different clothes.

Both methods: £150.00

Same boxed answer either way. Use your teacher’s layout in the unit exam — the marks are for a clear method, and both are clear.

1(c) · when it does not divide exactly

Convert ₹8,200 into pounds.

Rupees → pounds is against the arrow: 8,200 ÷ 105.40 = 82,000 ÷ 1,054

Clear the decimal first, exactly as in (b).

1,054 × 77 = 73,780 + 7,378 = 81,158, remainder 82,000 − 81,158 = 842

1,054 × 70 = 73,780 and 1,054 × 7 = 7,378. So the answer starts 77.something.

Carry on: 8,420 ÷ 1,054 = 7 r 1,042; then 10,420 ÷ 1,054 = 9 r 934; then 9,340 ÷ 1,054 = 8…

Bring down a zero each time. That gives 77.798… — three decimals is enough to round safely.

77.798… rounds to £77.80

Round to 2 decimal places at the END, and write the zero: £77.8 loses the accuracy mark.

Example 2 · A flat fee — the order matters

A bureau converts pounds to rupees at £1 = ₹105.40. It charges a flat fee of £4, taken off before converting. Meera hands over £320. (a) How many rupees does she receive? (b) She changes it all straight back at the same rate, no second fee. How much does she get, and what has the trip cost her? (c) Your turn.

2(a) · fee first, then convert

How many rupees does Meera receive?

The fee is in POUNDS, so it comes off while the money is still pounds: 320 − 4 = 316

Match the fee’s currency to the moment you subtract it. That one sentence is the whole trick.

316 × 105.40 = 316 × 105 + 316 × 0.40

Same split as Example 1(a).

316 × 105 = 31,600 + 1,580 = 33,180; and 316 × 0.40 = 126.40

316 × 100 = 31,600; 316 × 5 = 1,580; 316 × 4 = 1,264 so ×0.4 gives 126.4.

33,180 + 126.40 = ₹33,306.40

Two decimal places, symbol on.

THE TRAP — wrong order, shown deliberately: convert first, subtract after: 320 × 105.40 = 33,728, then 33,728 − 4 = ₹33,724.00 ✗. That subtracts £4 from a rupee amount — a £4 fee is worth ₹421.60, not ₹4 — so this answer is ₹417.60 too big. The examiner writes the fee’s currency into the question precisely to catch this.

2(b) · convert it back

She converts ₹33,306.40 back to pounds at ₹105.40 per £1. What does she get, and what did the round trip cost?

Rupees → pounds is against the arrow: 33,306.40 ÷ 105.40

Direction first, every time.

33,306.40 ÷ 105.40 = 333,064 ÷ 1,054

Multiply both by 10. Now spot it: 333,064 = 316 × 1,054, because step 2(a) built it that way.

= £316.00 exactly

Converting there and back at the SAME rate undoes itself perfectly.

Cost of the trip: 320 − 316 = £4.00 — exactly the fee

Nothing else could have been lost, because the rate cancelled. Compare Example 5, where the rate itself takes a bite.

2(c) · Your turn

Same bureau, same rules (£1 = ₹105.40, flat fee £4 taken first). Rohan hands over £130. How many rupees does he receive? (Number only, 2 decimal places.)

Example 3 · Best rate — compare, then SAY it

Tara’s family wants to change ₹50,000 into pounds. Two bureaus, each takes its fee off the rupees first: Bureau A: £1 = ₹105.40, fee ₹500. Bureau B: £1 = ₹104.20, fee ₹1,500. (a) How many pounds from each? (b) Which is better, and by how much? (c) Your turn.

Careful: when you are BUYING pounds with rupees, a smaller number of rupees per pound is the better rate — each pound costs you less. B has the better rate but the bigger fee. That tension is the whole question, so work both out in full.

3(a) · both bureaus, in full

Pounds received from A and from B.

A: fee off first: 50,000 − 500 = 49,500 rupees to convert

The fee is in rupees, so it comes off the rupees — Example 2’s rule.

49,500 ÷ 105.40 = 495,000 ÷ 1,054

Against the arrow, decimal cleared.

1,054 × 469 = 494,326, remainder 674; carrying on gives 469.639… so A pays £469.64

1,054 × 400 = 421,600; 1,054 × 69 = 72,726; total 494,326. Round at the end.

B: 50,000 − 1,500 = 48,500; then 48,500 ÷ 104.20 = 485,000 ÷ 1,042

Same two moves: fee off in the right currency, then divide against the arrow.

1,042 × 465 = 484,530, remainder 470; carrying on gives 465.451… so B pays £465.45

1,042 × 400 = 416,800; 1,042 × 65 = 67,730; total 484,530.

3(b) · the decision SENTENCE

Which bureau should they use?

Difference: 469.64 − 465.45 = £4.19

Line up the decimal points and subtract.

“Bureau A is better: it gives £469.64, which is £4.19 more than Bureau B’s £465.45.”

Cambridge gives the final mark for a SENTENCE naming the winner with the figures — two correct calculations and no sentence loses it. Notice B’s better rate lost to its bigger fee.

3(c) · Your turn

If Bureau B scrapped its fee (rate still ₹104.20 per £1), how many pounds would ₹50,000 buy? (Number only, 2 decimal places.)

Example 4 · Bill + discount + tax + change — round at the END

A shop sells: a backpack for ₹1,240.00, notebooks at ₹85.00 each, and a water bottle for ₹399.50. Tara buys the backpack, 3 notebooks and the bottle. The shop gives 15% off the subtotal, then 18% GST is added to the discounted amount. (a) Find the subtotal. (b) Find the final bill. (c) How much change from ₹2,000?

4(a) · subtotal

Add up the bill before discount.

3 notebooks: 3 × 85 = 255.00

Quantity times unit price, before anything else.

1,240.00 + 255.00 + 399.50

Line up the decimal points in a column.

1,240 + 255 = 1,495; then 1,495 + 399.50 = ₹1,894.50

1,495 + 400 would be 1,895, so 1,495 + 399.50 is 50 paise less: 1,894.50.

4(b) · discount, then tax

15% off, then 18% GST on what remains. Find the final bill.

15% off means paying 85%: multiplier 0.85. Then 18% added: multiplier 1.18

Build each multiplier from 100% ± the rate, exactly as in the percentages section.

1,894.50 × 0.85 = 1,894.50 × 0.8 + 1,894.50 × 0.05 = 1,515.60 + 94.725 = 1,610.325

KEEP all three decimals. Do not round yet — that half-paisa is about to matter.

1,610.325 × 1.18 = 1,610.325 + 1,610.325 × 0.18 = 1,610.325 + 289.8585 = 1,900.1835

18% of 1,610.325: ×0.1 gives 161.0325, ×0.08 gives 128.826, sum 289.8585.

NOW round: ₹1,900.18

Round once, at the end, to 2 decimal places, with the symbol.

Check with one combined multiplier: 0.85 × 1.18 = 1.003, and 1,894.50 × 1.003 = 1,894.50 + 5.6835 = 1,900.1835 ✓

A 15% discount then 18% tax is ×1.003 overall — almost back where you started. A one-line check like this catches slips.

THE TRAP — rounding early, shown deliberately: round step 2 to ₹1,610.33 and carry on: 1,610.33 × 1.18 = 1,900.1894 → ₹1,900.19 ✗ — one paisa too much, and the accuracy mark is gone. Keep full digits in the working; round only the final answer.

4(c) · the change

Tara pays with ₹2,000. How much change?

2,000.00 − 1,900.18

Change is worked from the actual amount charged — the rounded bill.

2,000.00 − 1,900.18 = 99.82, so the change is ₹99.82

1,900.18 + 0.82 = 1,901, and 1,901 + 99 = 2,000 — adding up to the target is the safest way to subtract money.

Example 5 · There and back — why the bank always wins

A bank sells US dollars at ₹84.90 per $1 and buys them back at ₹83.10 per $1. (Two different rates — the gap is called the spread, and it is how the bank earns without charging a fee.) Tara’s family changes ₹42,450 into dollars for a trip. The trip is cancelled and they change every dollar straight back. (a) How many dollars did they buy? (b) How many rupees do they get back, and how much is lost? (c) The loss as a percentage. (d) Your turn.

5(a) · buying the dollars

How many dollars does ₹42,450 buy at ₹84.90 per $1?

The bank SELLS dollars, so the selling rate applies: $1 costs ₹84.90

First decision in any spread question: which of the two rates is for THIS direction? You are buying, so the bank is selling.

Rupees → dollars is against the arrow: 42,450 ÷ 84.90 = 424,500 ÷ 849

Decimal cleared, as always.

849 × 500 = 424,500, so they get $500.00 exactly

Sense check: about 85 rupees a dollar, and 500 × 85 = 42,500 — right size.

5(b) · selling them back

They sell all $500 back. How many rupees, and what is lost?

Now the bank BUYS, so the buying rate applies: each $1 fetches only ₹83.10

Same bank, other rate. Never reuse 84.90 here — that is the classic error.

Dollars → rupees is along that arrow: 500 × 83.10 = 500 × 83 + 500 × 0.10 = 41,500 + 50 = ₹41,550.00

Whole part then decimal part, digit by digit.

Lost: 42,450 − 41,550 = ₹900.00

They did nothing wrong — no fee was charged. The spread alone took ₹1.80 on each of the 500 dollars: 500 × 1.80 = 900. ✓

5(c) · the loss as a percentage

What percentage of the original money was lost?

Percentage loss = loss ÷ ORIGINAL × 100 = 900 ÷ 42,450 × 100

Always divide by the starting amount, never the final one.

900 ÷ 42,450 = 90,000 ÷ 4,245,000 = 0.021201…, so the loss is 2.1201…% = 2.12%

Round the percentage to 2 decimal places at the end. Not catastrophic — but it happened for holding the money one afternoon.

5(d) · Your turn

Same bank (sells at ₹84.90, buys back at ₹83.10). A friend changes ₹16,980 into dollars and immediately changes them all back. How many rupees does she end up with? (Number only, 2 decimal places.)

Exam checklist for Money — five lines, five traps:

Decide multiply-or-divide with the arrow picture before touching a number — along the arrow multiply, against it divide, then size-check the answer.
A fee comes off in its own currency, at the stated moment — a £4 fee is not ₹4, and the wrong order gives a confidently wrong answer.
Keep every digit in the working; round money to 2 decimal places at the END only — early rounding cost a paisa (and the accuracy mark) in Example 4.
A best-value or best-rate answer needs a sentence naming the winner and the saving (“Bureau A is better, by £4.19”) — the numbers alone drop the final mark.
Write the currency symbol and both decimal places every time: £77.80, ₹99.82 — never 77.8.

E1.17 · low risk10 · Exponential growth and decay▼

▶ Watch: E1.17 Exponential growth and decay

This is compound interest with a different cover story. Populations growing, cars losing value, bacteria doubling, radioactive material decaying — all the same single formula, so this section should be quick.

final = starting amount × (multiplier)ⁿ

The multiplier is built exactly as in section 7: 100% ± the rate, over 100. And n is the number of times the change happens — years, hours, days, whatever the question is counting.

Wording	Multiplier	Why
grows by 8% a year	1.08	108% of what it was
falls by 8% a year	0.92	100 − 8 = 92%
depreciates by 20% a year	0.8	Depreciate means lose value
doubles every hour	2	200% of what it was
halves every 5 years	0.5	And n counts 5-year periods, not years

The mistake: using 0.2 as the multiplier for a 20% decrease. Multiplying by 0.2 leaves you with 20% of the value, not 80% of it. A decrease of 20% is × 0.8. The quick check: after a decrease the answer should be a bit smaller, not five times smaller.

The second mistake, and it is a sign-and-counting one: getting n wrong. “Halves every 5 years, find the amount after 20 years” means n = 20 ÷ 5 = 4, not 20. Count the number of changes, not the number of years.

Non-calculator: the powers you can realistically do by hand are small, so exam questions keep n at 2, 3 or 4, or make the multiplier friendly (2, 0.5, 1.1, 0.9). 1.1³ is 1.1 × 1.1 = 1.21, then 1.21 × 1.1 = 1.331 — each step is a multiply-by-10-and-add-a-tenth.

Worked in full

A car worth £12000 depreciates by 10% each year. Find its value after 3 years.

Multiplier = 100% − 10% = 90% = 0.9

Depreciation means a decrease, so the multiplier is below 1.

n = 3, so value = 12000 × 0.9³

Three years means three applications.

Year 1: 12000 × 0.9 = 10800

10% of 12000 is 1200, and 12000 − 1200 = 10800.

Year 2: 10800 × 0.9 = 9720

10% of 10800 is 1080.

Year 3: 9720 × 0.9 = 8748

10% of 9720 is 972, and 9720 − 972 = 8748.

£8748

Note it is not 12000 − 30% = 8400. Each year takes 10% of a smaller amount.

Last step is yours

A colony of 400 bacteria doubles every hour. How many are there after 5 hours?

Multiplier = 2

Doubling is × 2.

n = 5, so total = 400 × 2⁵

Five hours, five doublings.

2⁵ = 32

The powers of 2: 2, 4, 8, 16, 32.

400 × 32 = 12800

4 × 32 = 128, then put the two zeros back.

Last two are yours

A radioactive sample of mass 640 g halves every 20 years. Find its mass after 80 years.

Multiplier = 0.5

Halving.

n counts 20-year periods, not years: 80 ÷ 20 = 4

This is the step the question is really testing.

Mass = 640 × 0.5⁴

Four halvings.

0.5⁴ = ¹⁄₁₆

Halving four times means dividing by 2 four times, which is dividing by 16.

640 ÷ 16 = 40 g

Or halve four times directly: 320, 160, 80, 40.

All yours

A population of 5000 grows by 20% per year. Find it after 2 years.

A machine worth £8000 loses 25% of its value each year. Find its value after 2 years, in pounds.

An investment of £2000 grows by 10% a year. Find its value after 3 years, in pounds.

A sample halves every 3 days. What fraction of the original is left after 12 days? Type it like 1/8.

Write down the multiplier for a decrease of 7% per year

E1.18 · high risk11 · Surds, and rationalising a denominator▼

▶ Watch: E1.18 Surds

Expect this to be the slowest section here. Your indices strand broke at step 2, the lowest break of the eleven, and this material sits directly on top of powers and roots. Nothing below is hard in itself, but it will only feel automatic once roots feel automatic, so go through it slowly and come back to it.

A surd is a root that cannot be written exactly as a fraction — √2, √3, √5. Leaving an answer as 3√2 rather than 4.2426… is not laziness, it is the exact answer, and in Paper 2 it is the only answer available to you.

The three rules, and that is all there are:
√a × √b = √(ab) — roots multiply straight through
√a ÷ √b = √(a⁄b) — and divide straight through
√a × √a = a — a root times itself undoes the root. This one does all the work.

The mistake: writing √(a + b) = √a + √b. It is false and it is easy to prove false: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Roots split over multiplying and dividing, never over adding and subtracting.

Simplifying — hunt for a square factor

To simplify √50, split 50 into a square number times something else. The square number comes out of the root; the rest stays in. The squares to hunt for are 4, 9, 16, 25, 36, 49, 64, 100.

Non-calculator: take the largest square factor or you will have to simplify twice. √72 as √4 × √18 gives 2√18, which is not finished, because 18 still holds a 9. Going straight for √36 × √2 gives 6√2 in one move. Scan 100, 64, 49, 36, 25, 16, 9, 4 downwards.

Worked in full

Simplify √72

Look for the largest square number that divides 72

Check downwards: 64 no, 49 no, 36 yes.

72 = 36 × 2

Write the split before touching the root sign.

√72 = √36 × √2

Roots split over multiplication — the first rule.

√36 = 6

The square number comes out as a whole number.

√72 = 6√2

The 2 has no square factor, so this is fully simplified.

Adding and subtracting

Treat the root like a letter. 3√5 + 4√5 = 7√5, in exactly the way 3x + 4x = 7x. Only like roots combine: 3√2 + 4√5 cannot be simplified at all. If they do not look alike, simplify each one first — they often become alike.

Last step is yours

Simplify √18 + √8

They are not like terms yet, so simplify each one

Never try to add them as they stand.

√18 = √9 × √2 = 3√2

Largest square factor of 18 is 9.

√8 = √4 × √2 = 2√2

Largest square factor of 8 is 4.

Now they match: 3√2 + 2√2

Both are lots of √2, so they combine.

= 5√2

3 + 2 = 5, and the √2 stays exactly as it is. It does not become √4.

Multiplying, and rationalising the denominator

A fraction is not considered tidy while a root sits on the bottom. Rationalising means moving it to the top, and you do it by multiplying top and bottom by that root — which is multiplying by 1, so the value is unchanged.

6⁄√3 = 6⁄√3 × √3⁄√3 = 6√3⁄3 = 2√3

Worked in full

Rationalise the denominator of 10⁄√5

The bottom is √5, so multiply top and bottom by √5

Multiplying by √5⁄√5 is multiplying by 1, so nothing changes value.

Top: 10 × √5 = 10√5

Just carry the root along.

Bottom: √5 × √5 = 5

A root times itself undoes the root. This is why the method works.

So the fraction is 10√5⁄5

The bottom is now a whole number.

10 ÷ 5 = 2, so the answer is 2√5

Always cancel at the end. Leaving it as 10√5⁄5 loses the final mark.

Last two are yours

Simplify √3(√12 − 2) and rationalise 8⁄√2

Expand the bracket: √3 × √12 − √3 × 2

Multiply the outside term into each term inside.

√3 × √12 = √36 = 6

Roots multiply straight through, and 36 is a perfect square.

So the first part is 6 − 2√3

The second term is just 2√3, written with the number in front.

For 8⁄√2, multiply top and bottom by √2: top becomes 8√2, bottom becomes 2

√2 × √2 = 2.

8√2 ÷ 2 = 4√2

Cancel the numbers, leave the root alone.

All yours

Simplify √50. Type it like 5root2.

Simplify √48. Type it like 5root2.

Work out √7 × √7

Simplify 5√3 − 2√3. Type it like 3root2.

Simplify √20 + √45. Type it like 5root2.

Rationalise 12⁄√3. Type it like 5root2.

Work out √2 × √8

Rationalising a denominator such as 3 − √5

When the bottom is a ± √b, multiply top and bottom by the conjugate: the same two numbers with the sign between them changed. The bottom then has no root, because (a + √b)(a − √b) = a² − b.

Worked in full

Rationalise the denominator of 1/(−1 + √3).

multiply top and bottom by (√3 + 1)

The bottom is √3 − 1, so its conjugate is √3 + 1.

bottom: (√3 − 1)(√3 + 1) = 3 − 1 = 2

The two middle terms cancel. No root left.

top: 1 × (√3 + 1) = 1 + √3

answer: (1 + √3)/2

This is the syllabus’s own example.

The mistake: multiplying by the same bracket, (√3 − 1)(√3 − 1) = 4 − 2√3, which still has a root. Change the sign.

All yours

Rationalise 4/(3 − √5) and simplify. Type it like 2+root3.

Rationalise 6/(√7 + 2) and simplify. Type it like 2root3-1.

mixed · no labels12 · Mixed set — no labels, no order▼

Fifteen questions drawn from all eleven sections above, shuffled and unlabelled. Ordinary revision does one sub-topic at a time, which quietly does the hardest part for you — working out which method applies. Here nobody tells you. Before you calculate anything, name the method out loud.

nothing answered yet

1. Work out −5 − (−8)

2. A price rises by 25% to £75. Find the original price in pounds.

3. Simplify √27. Type it like 5root2.

4. Find the LCM of 6 and 10

5. Write ⁹⁄₄₀ as a decimal

6. n(ξ) = 30, n(A) = 16, n(B) = 20, n(A ∩ B) = 9. Find n(A′ ∩ B′).

7. A length is 6.4 cm to 1 decimal place. Write down the upper bound.

8. Work out 1¹⁄₄ + ²⁄₃. Type it like 7/4 or 1 3/4.

9. A train leaves at 14:35 and the journey takes 2 h 50 min. Give the arrival time like 16:20.

10. £1 = 1.25 US dollars. Convert 150 dollars into pounds.

11. Which is smallest: ⁵⁄₈, 0.63, 62%? Type it exactly as listed.

12. A car worth £20000 loses 15% of its value each year. Find its value after 2 years, in pounds.

13. Rationalise 14⁄√7. Type it like 5root2.

14. Work out (−3)² − 3²

15. Write 84 and 126 as products of primes and hence find their HCF.

When you have finished: look at which numbers you got wrong, not at the score. If two or more came from the same section, go back to that section and redo its four fading stages before anything else. One wrong out of fifteen is arithmetic slippage. Two from the same section is a method you do not own yet.