NBT Mathematics (MAT) — Complete Study Guide

Read this first — it is written for you.

If maths has never quite "clicked" for you, this guide is built exactly for that. Nothing here assumes you already understand it. Every rule is explained in plain English before you see the formula, every symbol is spelled out in words, and every method comes with a fully worked example where every single step is shown — no steps are skipped and nothing is left for you to "just know."

You do not have to be naturally good at maths to do well on this test. You have to understand a small number of ideas properly, and practise them until they feel familiar. That is a skill anyone can build. Take it one small idea at a time, and be patient with yourself.

The National Benchmark Test in Mathematics (NBT MAT) checks whether you are ready for university-level maths. It is not the same as your matric exam. Matric often rewards long calculations. The NBT rewards understanding — knowing why a rule works, so you can use it on a question you have never seen before.

No calculator is allowed. That sounds scary, but it is actually good news: it means the numbers are almost always kept small and friendly on purpose. You will rarely need ugly arithmetic.

How This Book Is Organised

This opening guide is the map; the ten chapters after it are the territory. Here is how the whole book fits together:

• This Study Guide — every definition, formula, symbol and strategy in one place. Read it first, then keep returning to it as a reference while you work through the chapters.
• Chapters 1–8 and 10 — the practice chapters. Each one covers one topic area and follows the same four-part shape: 1. Questions — attempt these first, on paper, without peeking. 2. Answers and full explanations — every question in this section re-taught from the ground up: the idea, the steps, and for every one of them why each wrong option is wrong (each one is a real mistake real students make — learning to recognise them is half the training), and a Takeaway worth remembering. 3. Mixed practice — 30 questions from all topics, unlabelled, like the real test. 4. Answer key for the mixed practice.
• Chapter 9 — three full mock exams, 60 questions each, in the real test's format and topic mix. Save these for the final weeks (the study plan below tells you when).

How to work a question — the routine that makes this book work: attempt it honestly first, even if you only get partway. Then read the full explanation even when you got it right — the explanations teach the traps, shortcuts and cross-connections that turn one question into ten marks' worth of understanding. If you got it wrong, find your error among the "why the others are wrong" notes; recognising your own mistake pattern is the fastest improvement there is.

The five areas the NBT tests — and where to practise each

The NBT MAT draws every paper from the same five competence areas, named on the NBT's own website. This table is your route map:

How much of the paper is each one worth? Nobody outside the NBT knows. The NBT publishes the areas and the four cognitive levels, but it does not publish how many of the 60 questions fall in each area, and its National Report gives performance by subdomain rather than counts. Any book or website that prints an exact split is guessing — so this one does not. Practise all six rows; the test is only 60 questions and it mixes them.

(Chapter 10 covers introductory differential calculus, which the NBT's own reporting places under functions and graphs. Chapter 9's mock exams mix every area, in no claimed proportion.)

How to Use This Guide

The rest of this guide has five parts. You do not have to read them in order, and you definitely do not have to read them all in one sitting.

• The Language of Maths — a dictionary of the words and symbols that appear everywhere. If a question ever confuses you because of a word, this is where you look. Start here.
• Part 1 — The Building Blocks — the actual maths, topic by topic. Each topic has: a plain-English explanation, the formulas you need (in tables), a worked example, and a "watch out" warning.
• Part 2 — Doing Maths in Your Head — tricks for calculating without a calculator.
• Part 3 — Exam Strategy — how to actually get marks on the day, including how to make good guesses.
• Part 4 — Your Study Plan — what to do in the weeks before the test, built around this book's chapters and mock exams.

A promise about this test: there is no negative marking. You are never punished for a wrong answer. So you must answer every single question, even if you are guessing. A blank is a guaranteed zero; a guess has a 1-in-4 chance of being right. Never leave a bubble empty.

If you only remember five things from this whole guide: 1. Answer every question — a blank scores 0, a guess might score 1. 2. Learn the times tables, the special-angle table, and the basic formulas until you know them without thinking. 3. Turn "increase by 15%" into "multiply by 1.15" — percentages are just multiplying. 4. When stuck, try the four options one by one and see which fits. 5. Read the question slowly, twice. Half of all mistakes are just misreading.

The Language of Maths — Words Decoded

Most people who struggle with maths are not actually bad at maths — they get lost because of the words and symbols, which nobody ever explained in plain language. Here they are. Keep coming back to this list.

Everyday maths words

Number words

Fraction and power words

Symbols you will see

A habit that will save you again and again: whenever a question confuses you, read it out loud in words, replacing every symbol with its meaning. Suddenly it is just a sentence you can understand.

The Test at a Glance: 60 Questions, 3 Hours, No Calculator

Is a calculator allowed in the NBT? No. The NBT's own rules list a calculator among the things you may not bring, along with a dictionary, notebooks and any phone, tablet or smart watch. That sounds harsh and is actually in your favour: it means the numbers are chosen to be workable by hand, and Part 2 of this guide is nothing but ways to do that quickly.

And there is no scrap paper either. You work in the white space of the question booklet, so get used to laying out your working small and neat while you practise. Bathroom breaks are not allowed during the session, so plan for three hours in your seat.

What this means for you: you have plenty of time (3 minutes each is generous), so do not rush. Do the easy ones first to bank those marks, then come back for the hard ones.

Part 1 — The Building Blocks: Your NBT Formula Sheet

This is the actual maths. Taken together, the tables below are a formula sheet for the NBT — every formula, rule and fact this test draws on, gathered in one place so that you can learn them and check them. You walk into the test with nothing but pencils, so the sheet has to be in your head; that is exactly what these tables are for.

Each topic below follows the same friendly pattern:

• What this is really about — the idea, in plain words.
• The formulas / facts — in tables, so they are easy to find.
• Worked example — a full question solved with every step shown.
• Watch out — the mistake most people make.

You do not need to master everything at once. Pick one topic, understand it, practise it, then move on. Little by little.

1. Number Sense

What this is really about: being comfortable with fractions, decimals, percentages, and different kinds of numbers. This is the most common topic on the test, and the good news is it is the most learnable. Get this solid and you have already earned a big chunk of the marks.

1.1 Fraction–Decimal–Percentage Table

A fraction, a decimal, and a percentage are three different ways of writing the same amount. $\tfrac{1}{2}$, $0.5$, and $50\%$ are all "half." Learn this table by heart — it saves you from doing conversions under pressure.

(The little bar in $0.\overline{3}$ means the 3 repeats forever: $0.3333\dots$)

To turn a fraction into a percentage: divide top by bottom, then multiply by 100. To turn a percentage into a fraction: put it over 100 and simplify.

1.2 Number Facts to Know Instantly

You will use these constantly. Flash-card them until they are automatic.

Perfect squares (a number times itself), from $1^2$ to $15^2$:

$$1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ 121,\ 144,\ 169,\ 196,\ 225$$

Perfect cubes (a number times itself, times itself again), $1^3$ to $5^3$:

$$1,\ 8,\ 27,\ 64,\ 125$$

Powers of 2 (keep doubling), up to $2^{10}$:

$$2,\ 4,\ 8,\ 16,\ 32,\ 64,\ 128,\ 256,\ 512,\ 1024$$

Prime numbers up to 50 (only divisible by 1 and themselves):

$$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29,\ 31,\ 37,\ 41,\ 43,\ 47$$

The 10% ladder — powers of $1.1$, which the NBT uses constantly in interest and growth questions (notice the digits are the rows of Pascal's triangle):

$$1.1^2 = 1.21 \qquad 1.1^3 = 1.331 \qquad 1.1^4 = 1.4641$$

Triangular numbers — the running totals $1+2+3+\cdots+n$ (formula $\tfrac{n(n+1)}{2}$). Know at least: $n=5 \to 15$, $\ n=10 \to 55$, $\ n=20 \to 210$, $\ n=40 \to 820$.

Sum of the first $n$ odd numbers is always $n^2$: $1+3+5+7 = 16 = 4^2$. This one fact is a confirmed exam favourite.

Parity in one breath (odd/even rules): odd ± odd = even; even ± even = even; odd ± even = odd; odd × odd = odd; even × anything = even.

1.3 Percentages — the one idea that unlocks everything

In plain English: "per cent" literally means "out of 100." So $30\%$ just means $\tfrac{30}{100} = 0.3$. The single most useful trick in the whole test is this:

Any percentage change is just a multiplication. To increase by $15\%$, multiply by $1.15$. To decrease by $15\%$, multiply by $0.85$.

Why? If you have R100 and add 15%, you now have the original 100% plus 15% $= 115\%$ of what you had $= 1.15 \times 100 = 115$. To take away 15%, you keep $100\% - 15\% = 85\% = 0.85$ of it.

Worked example — a shirt in a sale: A shirt costs R240. It is increased by 15%. What is the new price?

• Step 1 — turn the percentage into a multiplier: increase of 15% means multiply by $1 + 0.15 = 1.15$.
• Step 2 — multiply: $240 \times 1.15$.
• Step 3 — do it in easy pieces: $240 \times 1 = 240$, and $240 \times 0.15 = 36$. Add them: $240 + 36 = 276$.
• Answer: R276.

Watch out — the "back to the start" trap: A price goes down 10%, then up 10%. You are not back where you started. Down 10% means $\times 0.9$; up 10% means $\times 1.1$. Together: $0.9 \times 1.1 = 0.99$, which is a 1% loss. Whenever percentages happen one after another, multiply the multipliers — never add or subtract the percentages.

1.4 Types of Numbers

Numbers live in nested groups, like boxes inside boxes:

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$

Read left to right, that says: natural numbers are inside the integers, which are inside the rationals, which are inside the reals.

• $\mathbb{N}$ Natural numbers: $1, 2, 3, \dots$ (counting numbers)
• $\mathbb{Z}$ Integers: $\dots, -2, -1, 0, 1, 2, \dots$ (whole numbers, including negatives and zero)
• $\mathbb{Q}$ Rational numbers: anything you can write as a fraction — this includes all decimals that stop ($0.25$) or repeat ($0.333\dots$)
• Irrational numbers: cannot be written as a fraction; decimals go on forever with no pattern, like $\sqrt{2}$, $\pi$, $e$
• $\mathbb{R}$ Real numbers: everything above, all together

Quick test for a square root: $\sqrt{n}$ is a nice rational number only if $n$ is a perfect square. $\sqrt{25} = 5$ (rational), but $\sqrt{5}$ is irrational.

1.5 Scientific Notation

Scientific notation writes a number as $a \times 10^n$, where $a$ is between $1$ and $10$ and $n$ is a whole number. The exponent counts the places the decimal point moves — positive for big numbers, negative for small ones.

$$4\,500\,000 = 4.5 \times 10^6 \qquad\qquad 0.00045 = 4.5 \times 10^{-4}$$

To multiply or divide, deal with the coefficients and the powers of ten separately:

• Multiplying: multiply the coefficients, add the exponents.
• Dividing: divide the coefficients, subtract the exponents.

If the coefficient lands outside $1$ to $10$, fix it: $37 \times 10^4 = 3.7 \times 10^5$.

1.6 Speed, Distance and Time

One relationship, read three ways:

$$\text{distance} = \text{speed} \times \text{time} \qquad \text{speed} = \frac{\text{distance}}{\text{time}} \qquad \text{time} = \frac{\text{distance}}{\text{speed}}$$

Two rules stop nearly every mistake here:

1. Make the units agree first. With a speed in km/h, $30$ minutes is $0.5$ hours — divide minutes by $60$, never treat them as decimals.
2. Average speed is total distance ÷ total time. It is not the average of the speeds. If you travel equal distances at two different speeds, more time is spent at the slower one, so the true average always comes out below the halfway point between them.

2. Exponents and Logarithms

What this is really about: an exponent (or "power") is just a shortcut for repeated multiplying. $2^4$ means "multiply four 2's together": $2 \times 2 \times 2 \times 2 = 16$. A logarithm is the reverse question: "how many 2's did I multiply to get 16?" — the answer is 4. That is all a log is: an exponent, asked backwards.

2.1 Laws of Exponents

These rules let you combine powers quickly. The trick to all of them: they only work when the base (the big number) is the same.

Worked example: Simplify $\dfrac{x^5 \cdot x^3}{x^4}$.

• Step 1 — top: same base multiplying, so add the powers: $x^5 \cdot x^3 = x^{5+3} = x^8$.
• Step 2 — now dividing, same base, so subtract: $\dfrac{x^8}{x^4} = x^{8-4} = x^4$.
• Answer: $x^4$.

2.2 Laws of Logarithms

A logarithm answers "what power?" Written $\log_a N = x$, it means "$a$ to the power $x$ gives $N$," i.e. $a^x = N$. The two are the same fact written two ways:

$$y = a^x \iff x = \log_a y$$

Worked example: Evaluate $\log_2 32$.

• In words this asks: "2 to what power gives 32?"
• Count the doublings: $2, 4, 8, 16, 32$ — that is $2^5 = 32$.
• Answer: 5.

Watch out — the biggest log/exponent trap: you can only split a log when things are multiplied or divided inside. $\log(a+b)$ does NOT equal $\log a + \log b$. There is no rule for adding inside a log. (The same idea catches people with roots: $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$.)

3. Algebra and Surds

What this is really about: algebra is arithmetic with letters standing in for numbers. "Factorising" means writing an expression as a multiplication (breaking it into pieces multiplied together); "expanding" is the reverse (multiplying the brackets out).

3.1 Special Identities — patterns worth memorising

These come up so often that recognising them instantly saves huge time.

The one everyone forgets: $(a+b)^2$ is not $a^2 + b^2$. You must include the middle term $2ab$. Check with numbers: $(2+3)^2 = 25$, but $2^2 + 3^2 = 13$. Different! The missing piece is the $2ab = 2\times2\times3 = 12$, and $13 + 12 = 25$. ✓

3.2 Factorising — always try these in order

When asked to factorise, run down this checklist from the top:

1. Common factor first — is there a number or letter in every term you can pull out? Always check this first. $6x + 9 = 3(2x + 3)$.
2. Difference of two squares — does it look like $a^2 - b^2$? Then $= (a-b)(a+b)$.
3. Sum or difference of cubes — $a^3 \pm b^3$.
4. Trinomial (three terms, $x^2 + bx + c$) — find two numbers that multiply to $c$ and add to $b$.
5. Grouping — for four terms, group them in pairs.
6. Factor theorem — for cubics: if putting $x = p$ into the expression gives $0$, then $(x - p)$ is a factor.

Worked example — a trinomial: Factorise $x^2 + 5x + 6$.

• We need two numbers that multiply to 6 and add to 5.
• List pairs that multiply to 6: $(1,6)$ and $(2,3)$. Which pair adds to 5? $2 + 3 = 5$. ✓
• So $x^2 + 5x + 6 = (x + 2)(x + 3)$.
• Check by expanding: $(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$. ✓

3.3 Surds (leftover square roots)

A surd is a root that does not simplify to a whole number, like $\sqrt{2}$. Rules:

$$\sqrt{ab} = \sqrt{a}\cdot\sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

Simplifying a surd — pull out the biggest perfect-square factor hiding inside:

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\times\sqrt{2} = 6\sqrt{2}$$

Adding surds — you can only add "like" surds (same number under the root), exactly like adding $2x + 3x = 5x$:

$$\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$$

Rationalising the denominator — getting rid of a root on the bottom of a fraction. Multiply top and bottom by the conjugate (the same expression with the middle sign flipped):

$$\frac{3}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1} = \frac{3(\sqrt{5} + 1)}{5 - 1} = \frac{3(\sqrt{5} + 1)}{4}$$

3.4 Completing the Square

This rewrites $x^2 + bx + c$ into the form $(x + p)^2 + q$, which instantly shows the turning point of a parabola. The recipe:

• Halve the number in front of $x$ (call it $b$), to get $p = \dfrac{b}{2}$.
• Then $q = c - p^2$.

Worked example: Write $x^2 + 8x + 3$ as $(x+p)^2 + q$.

• Halve the 8: $p = 4$.
• $(x+4)^2$ expands to $x^2 + 8x + 16$ — that is $16$ too big, so subtract 16 and add back the original 3: $q = 3 - 16 = -13$.
• Answer: $(x+4)^2 - 13$.

4. Quadratic Equations

What this is really about: a "quadratic" is any equation with an $x^2$ in it (and no higher power). Its graph is a U-shape (or upside-down U) called a parabola. Solving it means finding where it crosses the x-axis.

4.1 Three Ways to Solve

1. Factorising — write it as $(x - p)(x - q) = 0$. Because anything times zero is zero, either $x - p = 0$ or $x - q = 0$, so $x = p$ or $x = q$. Always try this first.
2. The quadratic formula — works for every quadratic $ax^2 + bx + c = 0$, even ugly ones:

$$\boxed{\,x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\,}$$

3. Completing the square — sometimes handy, but the formula usually wins.

Worked example — by factorising: Solve $x^2 - 7x + 10 = 0$.

• Find two numbers that multiply to $+10$ and add to $-7$: those are $-2$ and $-5$.
• So $(x - 2)(x - 5) = 0$.
• Therefore $x = 2$ or $x = 5$.
• Answer: $x = 2$ or $x = 5$.

4.2 The Discriminant — a shortcut to the "nature of the roots"

Sometimes a question does not want you to solve, only to say how many real answers there are. The part under the root in the formula, $\Delta = b^2 - 4ac$ (called the discriminant), tells you without doing the whole calculation:

(A negative under a square root has no real answer — that is your signal for "no real roots.")

4.3 Sum and Product of the Roots

For $ax^2 + bx + c = 0$ with answers $x_1$ and $x_2$, without solving you already know:

$$x_1 + x_2 = -\frac{b}{a} \qquad\qquad x_1 \cdot x_2 = \frac{c}{a}$$

These let you check answers or solve clever questions without ever finding $x_1$ and $x_2$ separately.

4.4 Quadratic Inequalities — solve with a picture

To solve something like $x^2 - x - 6 > 0$: factorise, find the two roots, then think of the parabola. An upward parabola is below zero between its roots and above zero outside them.

• $x^2 - x - 6 = (x-3)(x+2)$, roots at $x = 3$ and $x = -2$.
• "$> 0$" wants the parabola above the axis → outside the roots: $x < -2$ or $x > 3$.
• "$< 0$" would want between: $-2 < x < 3$.

Watch out: before any of this, ask whether the statement can even be false — $x^2 + 4 < 0$ has no solutions (a square is never negative), and the NBT loves testing exactly that.

5. Functions and Graphs

What this is really about: a function is a rule that takes an input number and gives back one output number — like a machine. $f(x) = 2x + 1$ means "double it and add one." The graph is just a picture of every input-output pair plotted as points.

Two bits of language:

• $f(x)$ ("f of x") is just the name of the output. $f(3)$ means "put $3$ in as the input."
• Intercept = where a graph crosses an axis. Gradient (or slope) = how steep a line is.
• Asymptote = an invisible line the graph gets closer and closer to but never quite touches.

5.1 The Main Graph Types

For a parabola $y = ax^2 + bx + c$, the turning point sits on the line

$$x = -\frac{b}{2a}.$$

Find that $x$, then substitute it back in to get the $y$ of the turning point.

Worked example — turning point: Find the turning point of $y = x^2 - 6x + 5$.

• Here $a = 1$, $b = -6$. The turning point's $x$: $x = -\dfrac{-6}{2(1)} = 3$.
• Substitute back: $y = 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4$.
• Answer: $(3, -4)$ — and since $a > 0$ the parabola opens upward, so this is a minimum.

Why asymptotes happen (plain English): for a hyperbola, you can never divide by zero — so at the $x$ that makes the bottom zero, the graph "breaks" (vertical asymptote). And as $x$ gets huge, the fraction shrinks to almost nothing, so $y$ flattens out towards $q$ (horizontal asymptote).

5.2 Moving Graphs Around (Transformations)

These work for any graph $y = f(x)$. Learn the pattern once and it applies everywhere.

5.3 Inverse Functions

The inverse undoes what the function did. To find it: swap $x$ and $y$, then solve for $y$. On a graph, the inverse is the original reflected in the line $y = x$. Handy fact: the inverse of $y = b^x$ is $y = \log_b x$.

6. Trigonometry

What this is really about: trigonometry ("trig") connects the angles of a right-angled triangle to the lengths of its sides. It sounds intimidating but rests on one memory trick: SOHCAHTOA.

6.1 SOHCAHTOA — the whole foundation

In a right-angled triangle, name the three sides from the angle you care about:

• Hypotenuse — the longest side, always opposite the right angle.
• Opposite — the side across from your angle.
• Adjacent — the side next to your angle (that is not the hypotenuse).

$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$$

The nonsense word SOH-CAH-TOA stores all three: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.

Worked example: A right triangle has an opposite side of 3 and a hypotenuse of 5. Find $\sin\theta$.

• $\sin$ uses Opposite over Hypotenuse (the "SOH" part).
• $\sin\theta = \dfrac{3}{5}$.
• Answer: $\tfrac{3}{5}$.

6.2 Special Angles — memorise this table

For a few "nice" angles, the answers are exact. These appear constantly (remember, no calculator):

6.3 The CAST Diagram — which sign in which quadrant

Angles bigger than 90° live in different "quadrants" (quarters) of a circle. In each quarter, some of sin/cos/tan are positive and some negative. CAST tells you which are positive:

Read the letters going round: A, S, T, C — remember it as "All Students Take Calculus."

6.4 The Identities You Must Know

An identity is a trig equation that is always true. The most important one:

$$\sin^2\theta + \cos^2\theta = 1 \qquad\qquad \tan\theta = \frac{\sin\theta}{\cos\theta}$$

(The notation $\sin^2\theta$ means $(\sin\theta)^2$ — square the whole thing.) From these you can also get $1 + \tan^2\theta = \sec^2\theta$.

The double-angle formulas — confirmed favourites on recent papers (practise them in Chapter 4):

$$\sin 2\theta = 2\sin\theta\cos\theta \qquad\qquad \cos 2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1 = \cos^2\theta - \sin^2\theta$$

How they appear in disguise: an expression like $2 - 4\sin^2 3x$ is really $2(1 - 2\sin^2 3x) = 2\cos 6x$ — factor until the bracket matches the identity, then remember the formula doubles whatever angle is inside ($3x$ becomes $6x$).

6.5 Reduction Formulae (angles outside 0–90°)

These convert a big or awkward angle back to a friendly one:

How to use them: find the "reference angle" (the small acute angle to the horizontal axis), then use CAST to decide whether the answer is $+$ or $-$.

6.6 Rules for Any Triangle (not just right-angled)

When a triangle has no right angle, use these:

$$\text{Sine rule:}\quad \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

$$\text{Cosine rule:}\quad c^2 = a^2 + b^2 - 2ab\cos C$$

$$\text{Area of a triangle:}\quad \text{Area} = \tfrac{1}{2}\,ab\sin C$$

(In these, a small letter is the side opposite the matching capital-letter angle.) Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule for most other cases.

7. Analytical Geometry

What this is really about: doing geometry using coordinates — points written as $(x, y)$ on a grid. Every question here is really just Pythagoras or straight lines in disguise.

Worked example — distance: Find the distance between $(1, 2)$ and $(4, 6)$.

• Across-gap: $4 - 1 = 3$. Up-gap: $6 - 2 = 4$.
• Square and add: $3^2 + 4^2 = 9 + 16 = 25$.
• Square root: $\sqrt{25} = 5$.
• Answer: 5. (This is the famous 3-4-5 triangle.)

Watch out: for a perpendicular gradient you need both changes — flip it over and swap the sign. The perpendicular of $\tfrac{2}{3}$ is $-\tfrac{3}{2}$, not just $-\tfrac{2}{3}$ or $\tfrac{3}{2}$.

8. Geometry and Measurement

What this is really about: shapes, and how to find their area (the space inside), perimeter (the distance around), and for 3-D solids, volume (how much fits inside) and surface area (the total skin).

8.1 Flat Shapes — Area and Perimeter

(Here $h$ always means the perpendicular height — straight up, at a right angle to the base — never a slanted side.)

8.2 Solid Shapes — Volume and Surface Area

(For the cone, $\ell$ is the slant height — the sloped distance up the side.) Notice cones and pyramids both have a $\tfrac{1}{3}$ — they hold exactly one-third of the box or cylinder they fit inside.

Worked example — cylinder volume: A tin can has radius 3 cm and height 10 cm. Find its volume (leave $\pi$ in the answer).

• Volume of a cylinder = (area of the circular base) × (height) $= \pi r^2 h$.
• $V = \pi \times 3^2 \times 10 = 90\pi$ cm³.
• Answer: $90\pi$ cm³ — and notice the options in a no-calculator test will usually keep the $\pi$, exactly like this. Do not convert to a decimal unless the options do.

8.3 Theorems Worth Knowing

Pythagoras' theorem (right-angled triangles only): the two shorter sides squared, added, equal the longest side squared:

$$a^2 + b^2 = c^2 \quad (c \text{ is the hypotenuse, the longest side})$$

Handy triples to recognise instantly: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and any multiple, like 6-8-10).

Similar shapes and scale factor: if two shapes are the same shape but different sizes, and lengths are scaled by $k$, then areas scale by $k^2$ and volumes by $k^3$. (Double the length → 4× the area → 8× the volume.)

Circle theorems to recognise:

• The angle at the centre is double the angle at the edge (standing on the same arc).
• The angle in a semicircle is always 90°.
• Opposite angles of a cyclic quadrilateral (4-sided shape inside a circle) add up to 180°.
• A tangent (a line just touching the circle) meets the radius at 90°.

9. Sequences and Series

What this is really about: a sequence is a list of numbers following a pattern (like $2, 5, 8, 11, \dots$). A series is what you get when you add the numbers of a sequence up. There are two patterns to know.

9.1 Arithmetic — adding the same amount each time

An arithmetic sequence goes up (or down) by a fixed step, called the common difference $d$. Example: $5, 8, 11, 14, \dots$ has $d = 3$.

$$T_n = a + (n-1)d \qquad\qquad S_n = \frac{n}{2}\bigl[2a + (n-1)d\bigr]$$

• $T_n$ = the $n$-th term (the number in position $n$). $a$ = the first term. $d$ = the step.
• $S_n$ = the sum of the first $n$ terms.

Worked example: Find the 10th term of $5, 8, 11, 14, \dots$

• First term $a = 5$; step $d = 3$; we want $n = 10$.
• $T_{10} = 5 + (10 - 1)\times 3 = 5 + 9\times 3 = 5 + 27 = 32$.
• Answer: 32. (The most common mistake here is using $10\times3$ instead of $9\times3$ — it is always $(n-1)$ steps, not $n$.)

9.2 Geometric — multiplying by the same amount each time

A geometric sequence multiplies by a fixed number, the common ratio $r$. Example: $3, 6, 12, 24, \dots$ has $r = 2$.

$$T_n = a\,r^{\,n-1} \qquad\qquad S_n = \frac{a(r^n - 1)}{r - 1}$$

Worked example: Find the 5th term of $2, 6, 18, \dots$

• First term $a = 2$; ratio $r = 6 \div 2 = 3$; we want $n = 5$.
• $T_5 = 2 \times 3^{5-1} = 2 \times 3^4 = 2 \times 81 = 162$.
• Answer: 162. (Same trap as arithmetic: the exponent is $n-1 = 4$, not $n = 5$ — you take four multiplication "steps" to reach the 5th term.)

Sum to infinity — if the numbers are shrinking (that is, $|r| < 1$), an endless sum can still reach a finite total:

$$S_\infty = \frac{a}{1 - r} \qquad (\text{only works when } |r| < 1)$$

9.3 Sigma Notation (the $\sum$ symbol)

$\sum$ (the Greek letter "sigma") just says "add up." $\displaystyle\sum_{i=1}^{5} i$ means "add the whole numbers from 1 to 5" $= 1+2+3+4+5 = 15$. Useful shortcuts:

$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2} \qquad \sum_{i=1}^{n} (2i-1) = n^2 \qquad \sum_{i=1}^{n} c = c\,n$$

(That middle one says: the sum of the first $n$ odd numbers is exactly $n^2$ — the same fact from the number-facts list, now in sigma costume. The exam bank uses it in exactly this costume.)

10. Financial Mathematics

What this is really about: money growing (interest on savings) or shrinking (depreciation — things losing value). It is really just percentages applied over several years.

($P$ = starting amount, $i$ = interest rate as a decimal — so 8% becomes 0.08 — $n$ = number of years.)

Three more money situations the NBT tests (all practised in Chapter 7):

• Hire purchase: deposit first, then simple interest on the remaining balance for the full term, then divide by the number of payments. The interest never shrinks as you pay — that is why HP is expensive.
• Exchange rates: the rate is the price of one unit of foreign currency. Rand → dollars: divide by the rate. Dollars → rand: multiply. Decide before calculating whether your answer should be bigger or smaller.
• Real return: what your money actually gains after inflation ≈ interest rate − inflation rate. If inflation is higher than your interest rate, your savings are losing buying power.

Simple vs compound, in one line: simple interest grows in a straight line; compound interest snowballs (each year's interest is bigger than the last). Over many years, compound is much larger.

Worked example — compound growth: R10 000 is invested at 10% per year, compounded, for 2 years.

• Rate as a decimal: $i = 0.10$, so the multiplier is $1.10$.
• $A = 10\,000 \times (1.10)^2 = 10\,000 \times 1.21 = 12\,100$.
• Answer: R12 100. (Note $1.1^2 = 1.21$, not $1.20$ — that extra R100 is the "interest on the interest.")

11. Data Handling and Probability

What this is really about: two things — statistics (describing a set of numbers, like an average) and probability (measuring how likely something is).

11.1 Averages and Spread