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Mathematics 2P: free practice, theory and problems

Statistics is collecting, ordering and interpreting numbers: grades, pay, temperatures. Measures of centre say what is "typical", measures of spread how much the numbers vary.

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Statistics
Exponential growth
Powers and standard form

1. Statistics

What is it about?

Statistics is collecting, ordering and interpreting numbers: grades, pay, temperatures. Measures of centre say what is "typical", measures of spread how much the numbers vary.

Concepts and formulas

Mean: the sum divided by the count. Strongly affected by extreme values.
Median: the middle value once the numbers are sorted (the average of the two middle ones for an even count).
Mode: the most frequent value.
Range: largest minus smallest value.
Standard deviation $\sigma$ : the typical distance from the mean. A large standard deviation = large spread.
Quartiles: $Q_1$ and $Q_3$ split the sorted data into four. Interquartile range = $Q_3 - Q_1$ .
Grouped data: use class midpoints when calculating the mean.

Example

Salaries 25, 27, 28, 30 and 90 (thousand NOK): the mean is 40, but the median 28. The median describes the typical case better because 90 pulls the mean up.

The median is robust to extreme values. The mean is not.

Practise statistics in the app →

2. Exponential growth

What is it about?

When something grows by the same percentage each period, it grows faster and faster: savings, population, bacteria. When it shrinks by a fixed percentage, like the value of a car, we get exponential decay.

Concepts and formulas

Exponential model: $f(x) = a \cdot b^x$ . $a$ is the starting value, $b$ the growth factor.
$b > 1$ gives growth, $0 < b < 1$ gives decay.
Doubling time: the time it takes for the amount to double. Rule of thumb: $\approx \dfrac{70}{p}$ periods at $p\,\%$ growth.
Half-life: the time it takes for the amount to halve.
Linear growth: a fixed amount per period. Exponential growth: a fixed percentage per period.

Example

A town has 20,000 inhabitants and grows by 2 % a year. After 10 years: $20\,000 \cdot 1.02^{10} \approx 24\,380$ .

A fixed percentage gives an exponential model, a fixed amount a linear one.

Practise exponential growth in the app →

3. Powers and standard form

What is it about?

Very large and very small numbers, like the distance to the Sun or the size of a virus, are easiest to write in standard form.

Concepts and formulas

$a^n$ means $a$ multiplied by itself $n$ times. $a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ .
Rules: $a^m \cdot a^n = a^{m+n}$ , $\dfrac{a^m}{a^n} = a^{m-n}$ , $(a^m)^n = a^{mn}$ .
Standard form: $k \cdot 10^n$ with $1 \le k < 10$ . For example $150\,000\,000 = 1.5 \cdot 10^8$ .
Small numbers: $0.000\,03 = 3 \cdot 10^{-5}$ .
Prefixes: kilo $= 10^3$ , mega $= 10^6$ , giga $= 10^9$ , milli $= 10^{-3}$ , micro $= 10^{-6}$ .

Example

The distance to the Sun is about $1.5 \cdot 10^{8}$ km. Light travels $3 \cdot 10^{5}$ km/s, so it takes $\dfrac{1.5 \cdot 10^8}{3 \cdot 10^5} = 500$ seconds, just over 8 minutes.

Standard form: one digit before the decimal point, then a power of ten.

Practise powers and standard form in the app →

Example problems with solutions

Here are some of the problems in mathematics 2P. 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.

Statistics: Find the median of 3, 8, 5, 10, 7.

Answer: 7

Sorted: 3, 5, 7, 8, 10. The middle one is 7.

Exponential growth:

f(x) = 500 \cdot 1.04^x

. What is the starting value?

Answer: 500

$a = 500$ .

Powers and standard form: Write 45,000 in standard form.

Answer: $4.5 \cdot 10^4$

One digit before the point: 4.5, and the point moves 4 places.

Statistics: Find the mean of 4, 6, 8, 10.

Answer: 7

$\dfrac{28}{4} = 7$ .

Practise all the problems →

Mathematics 2P: free practice, theory and problems

Contents

1. Statistics

What is it about?

Concepts and formulas

Example

2. Exponential growth

What is it about?

Concepts and formulas

Example

3. Powers and standard form

What is it about?

Concepts and formulas

Example

Example problems with solutions