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Maths years 8–10: free practice, theory and problems

When a calculation has several operations, everyone must work them out in the same order – otherwise we get different answers.

8 parts76 problemsPractice examFree

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Numbers and order of operations
Fractions, percent and growth factor
Powers and square roots
Algebra and equations
Linear functions
Geometry
Probability and statistics
Personal finance

1. Numbers and order of operations

What is it about?

When a calculation has several operations, everyone must work them out in the same order – otherwise we get different answers.

Key points

Order of operations: 1) brackets, 2) powers, 3) multiply and divide, 4) add and subtract.
Negative numbers: minus times minus is plus. $(-3) \cdot (-4) = 12$ , while $(-3) \cdot 4 = -12$ .
Subtracting a negative number is the same as adding: $5 - (-2) = 7$ .
A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 …).
Prime factorisation: $60 = 2 \cdot 2 \cdot 3 \cdot 5$ .

Example

$3 + 4 \cdot 2 = 3 + 8 = 11$ – not 14, because multiplying comes before adding.

Not sure? Put brackets around what should be done first.

Practise numbers and order of operations in the app →

2. Fractions, percent and growth factor

What is it about?

Percentages are everywhere: discounts, interest, pay and statistics. The growth factor makes it easy to calculate increases and decreases.

Key points

Fractions with different denominators: find a common denominator. $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .
Percent of: $p\,\%$ of $x$ is $\frac{p}{100} \cdot x$ .
Growth factor: increase by $p\,\%$ → multiply by $1 + \frac{p}{100}$ . Decrease by $p\,\%$ → multiply by $1 - \frac{p}{100}$ .
25 % increase: growth factor 1.25. 20 % discount: growth factor 0.80.
Percentage change = $\frac{\text{change}}{\text{original value}} \cdot 100\,\%$ .

Example

A jacket for 800 kr is reduced by 30 %. New price: $800 \cdot 0.70 = 560$ kr.

Percentage points are not the same as percent: from 10 % to 12 % is 2 percentage points, but a 20 % increase.

Practise fractions, percent and growth factor in the app →

3. Powers and square roots

What is it about?

Powers are a short way to write repeated multiplication. The square root is the opposite of squaring.

Key points

$2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ . The number below is the base, the number above is the exponent.
$a^m \cdot a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$ .
$a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ .
Standard form: $a \cdot 10^n$ where $1 \le a < 10$ . $45\,000 = 4.5 \cdot 10^4$ .
$\sqrt{49} = 7$ because $7^2 = 49$ .

Example

$3^2 \cdot 3^3 = 3^5 = 243$ .

(-2)^2 = 4

, but

-2^2 = -4

: the brackets decide what is squared.

Practise powers and square roots in the app →

4. Algebra and equations

What is it about?

Algebra is calculating with letters. With equations you can find unknown numbers in everything from recipes to phone plans.

Key points

Like terms can be combined: $3x + 2x = 5x$ , but $3x + 2$ cannot be simplified.
Expanding brackets: $3(x + 4) = 3x + 12$ .
Factorising: $6x + 9 = 3(2x + 3)$ .
Solving equations: collect the $x$ terms on one side and the numbers on the other. Do the same on both sides.
Always check by putting the solution into the equation.

Example

$5x - 3 = 2x + 9$ . Subtract $2x$ : $3x - 3 = 9$ . Add 3: $3x = 12$ . Divide by 3: $x = 4$ .

When you move a term across the equals sign, it changes sign.

Practise algebra and equations in the app →

5. Linear functions

What is it about?

A function links each $x$ to exactly one $y$ . A linear function gives a straight line.

Key points

$f(x) = ax + b$ . The slope $a$ says how much $y$ increases when $x$ increases by 1. The constant term $b$ is where the line crosses the $y$ -axis.
Slope from two points: $a = \frac{y_2 - y_1}{x_2 - x_1}$ .
$a > 0$ : the line rises. $a < 0$ : the line falls.
Proportional quantities: $y = ax$ (the line goes through the origin). Twice as much $x$ gives twice as much $y$ .
Make a table of values and plot the points to see the graph.

Example

Taxi: 50 kr starting fee and 15 kr per km. $f(x) = 15x + 50$ . 8 km costs $15 \cdot 8 + 50 = 170$ kr.

The slope is "how much per unit" – kr per km, litres per minute.

Practise linear functions in the app →

6. Geometry

What is it about?

Geometry is about shapes, angles, area and volume – from drawing a house to working out how much paint you need.

Key points

The angles in a triangle add up to 180°. In a quadrilateral they add up to 360°.
Pythagoras: in a right-angled triangle $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse (the longest side).
Circle: circumference $C = 2\pi r$ , area $A = \pi r^2$ . $\pi \approx 3.14$ .
Volume of a prism: base area · height. Cylinder: $V = \pi r^2 h$ .
Similar shapes have equal angles, and their sides are in the same ratio.

Example

The legs are 6 cm and 8 cm. $c^2 = 36 + 64 = 100$ , so $c = 10$ cm.

Pythagoras only applies in triangles with a right angle (90°).

Practise geometry in the app →

7. Probability and statistics

What is it about?

Probability says how likely something is, from 0 (impossible) to 1 (certain). Statistics describes data with a few numbers.

Key points

$P(A) = \frac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}$ when all outcomes are equally likely.
A die: $P(\text{six}) = \frac{1}{6}$ .
$P(\text{not } A) = 1 - P(A)$ .
Two independent events: $P(A \text{ and } B) = P(A) \cdot P(B)$ .
Mean, median, mode and range describe a data set.

Example

The probability of two sixes in a row: $\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$ .

The die has no memory: after five ones the chance of a one is still

\frac{1}{6}

Practise probability and statistics in the app →

8. Personal finance

What is it about?

Being in control of your own money: pay, tax, budgets, saving and loans.

Key points

A budget lists income and expenses. Surplus = income $-$ expenses.
Gross pay is pay before tax, net pay is what you are paid after tax.
Interest on savings: the money grows by the growth factor every year. $10\,000 \cdot 1.04^3$ after three years at 4 % interest.
Compound interest: you also earn interest on the interest from the year before.
Loans cost interest. A longer repayment time gives lower instalments, but more interest in total.

Example

You earn 4000 kr and spend 3200 kr a month. Surplus: 800 kr. In a year you save $12 \cdot 800 = 9600$ kr.

Consumer loans often have very high interest. Save first instead.

Practise personal finance in the app →

Example problems with solutions

Here are some of the problems in maths years 8–10. In the app, calculation problems get new numbers every time, so you can practise until it sticks – and take a graded practice exam before the real one.

Numbers and order of operations: What is

2 + 3 \cdot 5

Answer: 17

Multiply first: $3 \cdot 5 = 15$ , then $2 + 15 = 17$ .

Fractions, percent and growth factor: What is the growth factor when something increases by 15 %?

Answer: 1.15

$1 + 0.15 = 1.15$ .

Powers and square roots: What is

5^0

Answer: 1

Any number (except 0) to the power 0 is 1.

Algebra and equations: Simplify

4x + 3 - x + 5

Answer: $3x + 8$

$4x - x = 3x$ and $3 + 5 = 8$ .

Practise all the problems →

Maths years 8–10: free practice, theory and problems

Contents

1. Numbers and order of operations

What is it about?

Key points

Example

2. Fractions, percent and growth factor

What is it about?

Key points

Example

3. Powers and square roots

What is it about?

Key points

Example

4. Algebra and equations

What is it about?

Key points

Example

5. Linear functions

What is it about?

Key points

Example

6. Geometry

What is it about?

Key points

Example

7. Probability and statistics

What is it about?

Key points

Example

8. Personal finance

What is it about?

Key points

Example

Example problems with solutions