The growth factor and the decay factor

Finance and economics, grade 11 (3 units) Lesson 2 of 14Learning

The growth factor and the decay factor

The growth factor

A rise of $\displaystyle p$ percent is multiplying by $\displaystyle 1+\frac{p}{100}$ : the new value is 100% of the old one plus $\displaystyle p$ percent of it. That number is the growth factor, $\displaystyle q$ .

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

A rise of 4%: $\displaystyle {q}={1.04}$ .
A rise of 12%: $\displaystyle {q}={1.12}$ .
A rise of 0.5%: $\displaystyle {q}={1.005}$ .
A rise of 100% – the price doubled: $\displaystyle {q}={2}$ .

The decay factor

A fall of $\displaystyle p$ percent is multiplying by $\displaystyle 1-\frac{p}{100}$ : $\displaystyle 100-p$ percent of the value is left. That number is the decay factor.

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

A fall of 15%: $\displaystyle {q}={0.85}$ .
A fall of 3%: $\displaystyle {q}={0.97}$ .
A fall of 40%: $\displaystyle {q}={0.6}$ .

A factor greater than 1 – the value grows; a factor between 0 and 1 – the value shrinks.

From the factor back to the percent

What the factor adds to 1 is the percent of the rise, and what it lacks from 1 is the percent of the fall:

$\displaystyle {q}={1.07}$ – a rise of 7%.
$\displaystyle {q}={1.3}$ – a rise of 30%.
$\displaystyle {q}={0.92}$ – a fall of 8%.
$\displaystyle {q}={0.6}$ – a fall of 40%.

Two changes, one after the other

When two changes come one after the other, multiply by the factor of each – and the factor of both together is the product.

\displaystyle q_1\cdot q_2

The Levy family’s rent is 4,000 ₪ a month.
It rose by 5%, and the year after by 5% again:

4000\cdot1.05\cdot1.05=4000\cdot1.1025=4410

The factor of the two rises together is $\displaystyle 1.1025$ : a rise of 10.25%, not of 10%. The second rise is 5% of the new, larger rent.

A rise and a fall by the same percent

A rise and a fall by the same percent do not cancel out: the fall is worked out from the value after the rise.

The price of a share rose by 20%, and then fell by 20%:

1.2\cdot0.8=0.96

The price at the end is 96% of the price at the start – 4% lower.

A price that rises every year

When a price rises by the same percent every year, multiply by the same factor once for each year.

A falafel costs 25 ₪, and the price rises by 4% every year:

After a year: $\displaystyle {25\cdot1.04}={26}$ ₪.
After two years: $\displaystyle {25\cdot1.04\cdot1.04}={27.04}$ ₪.
After three years: $\displaystyle {25\cdot1.04\cdot1.04\cdot1.04}={28.1216\approx28.12}$ ₪.

Exercise 1

Write the factor of each change.

A rise of 6%.

/ – 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

A fall of 25%.

/ – 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

A rise of 2.5%.

/ – 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

A fall of 0.5%.

/ – 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

For each factor, write whether it stands for a rise or a fall, and by how many percent:

$\displaystyle {q}={1.15}$
$\displaystyle {q}={0.75}$
$\displaystyle {q}={1.008}$
$\displaystyle {q}={0.99}$
$\displaystyle {q}={3}$

Exercise 3

Shira’s pay is 8,000 ₪ a month.
In January she got a rise of 3%, and in July – another rise of 4%.

What is Shira’s pay after the two rises?

/ – 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

By how many percent did the pay rise altogether?

/ – 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

At the end of the season the price of a coat fell by 30%.
At the start of the next season the price rose by 30%.

Work out the factor of the two changes together.

/ – 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

Is the coat back at the price it had at the start? If not – by how many percent does its price differ from it?

/ – 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

A bus ticket costs 8 ₪.
The price rises by 5% every year.

What will the ticket cost after two years?

/ – 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

What will the ticket cost after three years? Round to agorot.

/ – 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

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

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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
$\displaystyle \frac{\bar{x}_{10}}{\bar{x}_{1}}$