The growth and decay formula

Finance and economics, grade 11 (3 units) Lesson 3 of 14Learning

The growth and decay formula

The formula A = A₀·qⁿ

When a value grows or shrinks by the same percent in every period, its value after $\displaystyle n$ periods is the value at the start times the factor $\displaystyle n$ times:

\displaystyle A

\displaystyle =

\displaystyle A_0\cdot q^{n}

$\displaystyle A_0$ – the value at the start;
$\displaystyle q$ – the factor of one period: greater than 1 in growth, between 0 and 1 in decay;
$\displaystyle n$ – the number of periods: years, months, quarters – the period the percent refers to;
$\displaystyle A$ – the value after $\displaystyle n$ periods.

Such a process is called exponential growth (or exponential decay): $\displaystyle n$ is in the exponent of the power.

A deposit for ten years

Maya deposited 8,000 ₪ for ten years, at a compound interest of 4% a year.
How much money will the deposit hold at the end?

$\displaystyle {A_0}={8000}$ , $\displaystyle {q}={1.04}$ , $\displaystyle {n}={10}$ :

A=8000\cdot1.04^{10}\approx11841.95

In ten years Maya earned almost 3,842 ₪.

An overdraft’s debt

A debt that is not paid back grows the same way: the bank adds interest to the debt, and in the next period – interest on the interest too.

Itai has an overdraft of 5,000 ₪ in his bank account.
The bank charges 1.2% interest a month.
What will the debt be after a year, if Itai puts no money in?

Here the period is a month, so $\displaystyle {q}={1.012}$ and $\displaystyle {n}={12}$ :

A=5000\cdot1.012^{12}\approx5769.47

In one year the debt grew by more than 769 ₪ – more than 15%, not $\displaystyle {12\cdot1.2\%}={14.4\%}$ .

The graph of exponential growth

The graph of exponential growth is a curve that rises more each year than the year before, and the graph of simple interest is a straight line.

Maya’s deposit, $\displaystyle {y}={8000\cdot1.04^{x}}$ , and beside its graph – the same deposit at a simple interest of 4%, $\displaystyle {y}={8000+320x}$ (the dashed line):

\displaystyle y=8000\cdot1.04^{x}

\displaystyle y=8000+320x

Both graphs start from the same point on the $\displaystyle y$ -axis: the principal, 8,000 ₪.
In the first years they are almost together; as the years go by, the curve moves away above the line.

Exercise 1

Tom deposited 15,000 ₪ for six years, at a compound interest of 3.5% a year.
How much money will the deposit hold at the end? Round to agorot.

/ – 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

Lia has an overdraft of 3,000 ₪ in her account.
The bank charges 1.5% interest a month.

a

What will the debt be after eight months, if Lia puts no money in? Round to agorot.

/ – 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 how many shekels did the debt grow?

/ – 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

Ron deposited 2,000 ₪ at a compound interest of 6% a year.
Write the sum in the deposit after $\displaystyle x$ years as a function of $\displaystyle x$ . Its graph will appear on the board.

Exercise 4

A flat in town costs 1,500,000 ₪ today.
The prices of flats in town rise by 6% a year.
What will the flat cost in four years, if prices go on rising like this?

/ – 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

Avi deposited 10,000 ₪ for ten years, at a compound interest of 3% a year.
Ben deposited 10,000 ₪ for five years, at a compound interest of 6% a year.

a

How much money will Avi have at the end? Round to agorot.

/ – 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 much money will Ben have at the end? Round to agorot.

/ – 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

c

In both cases $\displaystyle {3\cdot10}={6\cdot5}={30}$ . Why are the sums at the end different?

Formulas: finance and economics, grade 11 (3 units)

Growth and decay

Simple interest – $\displaystyle A_0$ the principal, $\displaystyle p$ the interest rate a year in percent, $\displaystyle n$ the number of years, $\displaystyle A$ the sum at the end
$\displaystyle A=A_0+n\cdot A_0\cdot\frac{p}{100}$
The growth factor – a rise of $\displaystyle p$ percent
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a fall of $\displaystyle p$ percent
$\displaystyle q=1-\frac{p}{100}$
Growth and decay – $\displaystyle A_0$ the value at the start, $\displaystyle q$ the factor for a period, $\displaystyle n$ the number of periods, $\displaystyle A$ the value at the end
$\displaystyle A$ $\displaystyle =$ $\displaystyle A_0\cdot q^{n}$
The factor of two changes, one after the other
$\displaystyle q_1\cdot q_2$
Today’s value of a sum $\displaystyle A$ in $\displaystyle n$ years, when prices grow $\displaystyle q$ times every year
$\displaystyle \frac{A}{q^{n}}$
The rule of 72 – about how many years until a sum doubles, growing by $\displaystyle p$ percent a year
$\displaystyle n\approx\frac{72}{p}$

The mean and the spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range
$\displaystyle x_{\max}-x_{\min}$
The variance – the mean of the squares of the deviations from the mean
$\displaystyle \sigma^{2}=\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}$
The standard deviation
$\displaystyle \sigma=\sqrt{\sigma^{2}}$

Quartiles and deciles

The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The interquartile range – $\displaystyle Q_1$ the lower quartile, $\displaystyle Q_3$ the upper quartile
$\displaystyle Q_3-Q_1$
The gap between the top and the bottom decile – how many times the mean income in decile 10 is that in decile 1
$\displaystyle \frac{\bar{x}_{10}}{\bar{x}_{1}}$