Simple and compound interest

Finance and economics, grade 11 (3 units) Lesson 1 of 14Learning

Simple and compound interest

A deposit and interest

When you put money in a bank for a fixed time, the bank pays interest: a percent of the sum, for every year it holds the money.

A deposit – money put in a bank for a fixed time.
The principal – the sum deposited at the start.
The interest rate – the percent the bank pays a year: “4% interest a year”.

Simple interest

With simple interest, the interest is worked out every year on the principal alone, so the same sum is added each year.

Noa deposited 10,000 ₪ for three years, at a simple interest of 4% a year.
Each year’s interest: $\displaystyle {10000\cdot0.04}={400}$ ₪.
After three years the deposit holds $\displaystyle {10000+3\cdot400}={11200}$ ₪.

\displaystyle A

\displaystyle =

\displaystyle A_0+n\cdot A_0\cdot\frac{p}{100}

$\displaystyle A_0$ – the principal;
$\displaystyle p$ – the interest rate a year, in percent;
$\displaystyle n$ – the number of years;
$\displaystyle A$ – the sum at the end.

Compound interest

With compound interest, each year’s interest joins the deposit, and the next year it earns interest too. So each year more is added than the year before.

Omer deposited 10,000 ₪ for three years, at a compound interest of 4% a year:

Year	At the start of the year	The interest	At the end of the year
1	10,000	400	10,400
2	10,400	416	10,816
3	10,816	432.64	11,248.64

Adding 4% is multiplying by $\displaystyle 1.04$ : the sum at the end of the year is 104% of the sum at its start. So it can be worked out without a table:

10000\cdot1.04\cdot1.04\cdot1.04=11248.64

After three years Omer has 48.64 ₪ more than Noa.

The gap between the two over time

In simple interest the deposit grows each year by the same sum, and in compound interest – by the same percent. So as time goes by, the gap between them grows.

10,000 ₪ at 4% interest a year, for 25 years:

Simple interest: a straight line – 400 ₪ every year.
Compound interest: a curve bending upwards. After 25 years the deposit holds almost 26,700 ₪, more than 2.5 times the principal.

Rounding to agorot

A sum of money is rounded to agorot – two digits after the point – and $\displaystyle \approx$ (“is about equal to”) is written before the rounded number:

3000\cdot1.06^{5}=4014.6767\ldots\approx4014.68

Here $\displaystyle 1.06^{5}$ is $\displaystyle 1.06\cdot1.06\cdot1.06\cdot1.06\cdot1.06$ : on a calculator, the power key.

Exercise 1

Dana deposited 5,000 ₪ for two years, at 3% interest a year.

a

How much money will the deposit hold at the end, with simple interest?

/ – 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 the deposit hold at the end, with compound interest?

/ – 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

By how many shekels does compound interest give more?

/ – 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

Yoav’s grandmother opened a savings plan for him with 20,000 ₪.
The plan pays a compound interest of 5% a year.

a

How much money will the plan hold after three years?

/ – 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 the plan hold after six years? 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

Yoav says: “In the second three years I earned exactly as much as in the first three.” Is he right? Explain.

Exercise 3

Two banks offer a deposit of 10,000 ₪:
Bank A – a simple interest of 5% a year.
Bank B – a compound interest of 4.5% a year.

a

How much money will the deposit hold in each bank after two years? Round to agorot.

Bank A:

/ – 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

Bank B:

/ – 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 the deposit hold in each bank after ten years? Round to agorot.

Bank A:

/ – 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

Bank B:

/ – 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

Which bank is the better choice? Explain why the answer depends on the time.

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