Growth in percent

Science and society, grade 11 (3 units) Lesson 2 of 16Learning

Growth in percent

The growth factor of a growth in percent

When an amount grows by $\displaystyle p$ percent, it becomes $\displaystyle 100\%+p\%$ of what it was. So a growth of $\displaystyle p\%$ is multiplying by the growth factor:

\displaystyle q=1+\frac{p}{100}

A town has 20,000 residents.
The number of residents grows by 2% every year.

The growth factor: $\displaystyle {q}={1+\frac{2}{100}}={1.02}$ .
After a year: $\displaystyle {20000\cdot1.02}={20400}$ .
After 5 years: $\displaystyle {20000\cdot1.02^{5}}={22081.616\ldots}$

A number of people is a whole number, so we round: after 5 years the town will have about 22,082 residents. The rounded result is written after the sign $\displaystyle \approx$ :

20000\cdot1.02^{5}\approx22082

Followers on a social network

A young singer has 1,200 followers on a social network.
The number of followers grows by 15% every month.
How many followers will she have after 4 months?

The growth factor: $\displaystyle {q}={1.15}$ , the unit of time a month.
$\displaystyle {M_4}={1200\cdot1.15^{4}}={2098.8075\approx2099}$ .

After 4 months she will have about 2,099 followers – nearly twice as many as at the start.

Growth over several years in a row

A growth of 10% a year, for 3 years, is a growth of more than 30%: each year the amount added in the years before grows by 10% too.

1.1^{3}=1.331

After 3 years the amount has been multiplied by 1.331: it has grown by 33.1%.
100 plants in a greenhouse become 133.1 – not 130.

The percentage growth from a table

When a table is exponential, the growth factor is a value divided by the one before. The percentage growth is the part of $\displaystyle q$ above 1, in percent.

The number of members of a town library, at the end of each year:

Year	0	1	2
Members	5,000	5,300	5,618

$\displaystyle {\frac{5300}{5000}}={1.06}$ and $\displaystyle {\frac{5618}{5300}}={1.06}$ , so $\displaystyle {q}={1.06}$ .
$\displaystyle {1.06}={1+\frac{6}{100}}$ : the number of members grows by 6% every year.

Exercise 1

A town has 50,000 residents.
The number of residents grows by 3% every year.

a

What is the growth factor?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many residents will the town have after 4 years? Round to a whole number.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A lake holds 400 fish.
The number of fish grows by 25% every year.

How many fish will there be in the lake after 3 years? Round to a whole number.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The height of a seedling grows by 20% every week.

a

Work out how many times taller it is after two weeks.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

By what percentage did the height grow in two weeks? Why not by 40%?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The number of subscribers to a video channel, at the end of each month:

Month	0	1	2
Subscribers	2,500	2,700	2,916

a

What is the growth factor?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

By what percentage does the number of subscribers grow every month?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

The number of members of a sports club grew by 4% this year.
At the end of the year the club has 7,800 members.

Write an equation and solve it: how many members did the club have at the start of the year?

Formulas: science and society, grade 11 (3 units)

Growth and decay

An exponential process – $\displaystyle M_0$ the amount at the start, $\displaystyle q$ the factor the amount is multiplied by in each unit of time, $\displaystyle M_t$ the amount after $\displaystyle t$ units of time
$\displaystyle M_t=M_0\cdot q^{t}$
The growth factor – a growth of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a decay of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1-\frac{p}{100}$
Decay by half-life – $\displaystyle n$ the number of half-lives gone by
$\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}$
The number of half-lives – $\displaystyle t$ the time gone by, $\displaystyle T$ the half-life
$\displaystyle n=\frac{t}{T}$
Linear growth – the amount grows by $\displaystyle d$ in each unit of time
$\displaystyle M_t=M_0+d\cdot t$

Measures of spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range – the greatest datum minus the least
$\displaystyle x_{\max}-x_{\min}$
The deviation of the datum $\displaystyle x_i$ from the mean
$\displaystyle x_i-\bar{x}$
The standard deviation of $\displaystyle n$ data
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}$
The standard deviation from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}$

Probability

The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities on its branches
$\displaystyle P=p_1\cdot p_2\cdot p_3$
“At least once” in $\displaystyle n$ independent experiments – $\displaystyle p$ the probability that it happens in each
$\displaystyle 1-(1-p)^{n}$
Conditional probability – the probability of $\displaystyle A$ when $\displaystyle B$ is known to have happened; $\displaystyle A\cap B$ – both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
The probability of “both $\displaystyle B$ and $\displaystyle A$ ”
$\displaystyle P(A\cap B)=P(B)\cdot P(A|B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)=P(A)$
The expected value – $\displaystyle x_1, \dots, x_k$ the possible values, $\displaystyle p_1, \dots, p_k$ their probabilities
$\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k$