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Mathematics 1P: free practice, theory and problems

Percent means "per hundred". Sales, interest, pay rises and inflation are all percentage calculations. With the growth factor you do it all in one step.

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Percentages and growth factors
Personal finance
Geometry and scale
Linear models and graphs
Probability

1. Percentages and growth factors

What is it about?

Percent means "per hundred". Sales, interest, pay rises and inflation are all percentage calculations. With the growth factor you do it all in one step.

Concepts and formulas

$p\,\% = \dfrac{p}{100}$ . For example $25\,\% = 0.25$ .
Part: $\text{part} = \text{whole} \cdot \dfrac{p}{100}$ .
Growth factor for an increase of $p\,\%$ : $1 + \dfrac{p}{100}$ . For a decrease: $1 - \dfrac{p}{100}$ .
New value = old value · growth factor. Old value = new value / growth factor.
Several changes in a row: multiply the growth factors. $+10\,\%$ then $-10\,\%$ gives $1.1 \cdot 0.9 = 0.99$ , i.e. $1\,\%$ below the start.
Percentage points are the difference between two percentages: from 20 % to 25 % is 5 percentage points, but a 25 % increase.

Example

A jacket costs 800 NOK and is reduced by 30 %. The growth factor is 0.70, so the new price is $800 \cdot 0.70 = 560$ NOK.

New value = old value · growth factor.

Practise percentages and growth factors in the app →

2. Personal finance

What is it about?

Pay, tax, budgets and loans are maths you use your whole life. A budget gives you an overview, and interest calculations show what a loan really costs.

Concepts and formulas

Gross pay: pay before tax. Net pay: what you receive after tax.
Tax deduction in percent: $\text{net} = \text{gross} \cdot (1 - \text{tax rate})$ (simplified).
Budget: planned income and expenses. Surplus = income − expenses.
Interest: what you pay to borrow (or earn to save). After $n$ years at a fixed rate: $K_n = K_0 \cdot \left(1 + \dfrac{p}{100}\right)^n$ .
Effective interest rate includes fees, and is the one to compare loans with.
Credit cards and consumer loans often have very high interest.

Example

You save 10,000 NOK at 3 % interest. After 5 years: $10\,000 \cdot 1.03^5 = 11\,592.74$ NOK.

Compare loans by effective rate, not nominal rate.

Practise personal finance in the app →

3. Geometry and scale

What is it about?

The floor area of a flat, paint for a wall, the volume of a tank or distances on a map: geometry and scale are everywhere.

Concepts and formulas

Rectangle: $A = l \cdot w$ . Triangle: $A = \dfrac{b \cdot h}{2}$ . Circle: $A = \pi r^2$ , circumference $C = 2\pi r$ .
Prism and cylinder: $V = B \cdot h$ (base times height). Cylinder: $V = \pi r^2 h$ .
Pythagoras: in a right triangle $a^2 + b^2 = c^2$ .
Scale 1 : 50,000 means 1 cm on the map is 50,000 cm = 500 m in reality.
Units: 1 m² = 10,000 cm². 1 m³ = 1000 litres. 1 dm³ = 1 litre.

Example

Two towns are 7 cm apart on a map at scale 1 : 200,000. In reality the distance is $7 \cdot 200\,000 = 1\,400\,000$ cm = 14 km.

Convert to the same unit before calculating.

Practise geometry and scale in the app →

4. Linear models and graphs

What is it about?

Many everyday things increase by the same amount all the time: taxi price per km, electricity per kWh, pay per hour. Then a linear model $y = ax + b$ fits.

Concepts and formulas

$b$ is the starting value (where the graph crosses the $y$ -axis). $a$ is the slope: how much $y$ changes when $x$ increases by 1.
Slope between two points: $a = \dfrac{y_2 - y_1}{x_2 - x_1}$ .
Proportional quantities: $y = kx$ (no starting value). Twice the $x$ gives twice the $y$ .
Inversely proportional: $x \cdot y = k$ . Twice as many workers take half the time.
The intersection of two lines: where the two models give the same value.

Example

An electricity plan costs 49 NOK a month plus 1.20 NOK per kWh: $C(x) = 1.2x + 49$ . Using 500 kWh costs $1.2 \cdot 500 + 49 = 649$ NOK.

The slope is the change per unit.

Practise linear models and graphs in the app →

5. Probability

What is it about?

What is the chance of winning the lottery, rolling a six or rain tomorrow? A probability is a number between 0 (impossible) and 1 (certain).

Concepts and formulas

Uniform model (all outcomes equally likely): $P(A) = \dfrac{\text{favourable outcomes}}{\text{possible outcomes}}$ .
Complement rule: $P(\text{not } A) = 1 - P(A)$ .
Independent events: $P(A \text{ and } B) = P(A) \cdot P(B)$ .
Relative frequency: how often something happens in an experiment. With many trials it approaches the probability.
Tables and tree diagrams give an overview when there are several stages.

Example

The probability of at least one six in two dice rolls is $1 - \left(\dfrac{5}{6}\right)^2 = \dfrac{11}{36} \approx 0.31$ .

"At least one" is easiest with the complement.

Practise probability in the app →

Example problems with solutions

Here are some of the problems in mathematics 1P. 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.

Percentages and growth factors: What is 15 % of 600?

Answer: 90

$600 \cdot 0.15 = 90$ .

Personal finance: Gross pay is 30,000 NOK and the tax deduction 30 %. What is the net pay?

Answer: 21000 NOK

$30\,000 \cdot 0.70 = 21\,000$ NOK.

Geometry and scale: A room is 4 m long and 3.5 m wide. What is the area?

Answer: 14 m²

$4 \cdot 3.5 = 14$ m².

Linear models and graphs:

y = 3x + 5

. What is the starting value?

Answer: 5

The constant term $b = 5$ .

Practise all the problems →

Mathematics 1P: free practice, theory and problems

Contents

1. Percentages and growth factors

What is it about?

Concepts and formulas

Example

2. Personal finance

What is it about?

Concepts and formulas

Example

3. Geometry and scale

What is it about?

Concepts and formulas

Example

4. Linear models and graphs

What is it about?

Concepts and formulas

Example

5. Probability

What is it about?

Concepts and formulas

Example

Example problems with solutions