Simple and Compound Interest

Expert reviewed • 21 July 2024 • 11 minute read

HSC Maths Advanced Syllabus

solve compound interest problems involving financial decisions, including a home loan, a savings account, a car loan or superannuation
- identify an annuity (present or future value) as an investment account with regular, equal contributions and interest compounding at the end of each period, or a single-sum investment from which regular, equal withdrawals are made
- use technology to model an annuity as a recurrence relation and investigate (numerically or graphically) the effect of varying the interest rate or the amount and frequency of each contribution or a withdrawal on the duration and/or future or present value of the annuity
- use a table of interest factors to perform annuity calculations, eg calculating the present or future value of an annuity, the contribution amount required to achieve a given future value or the single sum that would produce the same future value as a given annuity
verify entries in tables of future values or annuities by using geometric series

Note:

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Simple Interest

Simple interest is a method of calculating the interest charged on a loan. When working with simple interest, the interest is only accumulated on the original principal, not on the interest which has been added to the original amount. The original principal refers to the initial sum of money. The formula to calculate simple interest is as follows:

I=PRt

where,

$I$ is the interest accumulated after $t$ units of time
$P$ is the principal (initial amount of money)
$R$ is the yearly interest rate
$t$ is the units of time

If we look closely at the formula for simple interest, we can see that consecutive values of $n$ will result in the formation of an AP with first term $PR$ , and common difference $PR$ . This is important to know when dealing with difficult simple interest problems, as formula’s relating to AP’s can be used to solve problems.

Practice Question 1

Suppose you lend $5,000 to a friend who agrees to pay you an annual simple interest rate of 6%. However, instead of paying you back annually, they pay you back in full after 18 months. How much will they pay you back in total?

First we must identify all known variables:

P=5000\\R=6\% = 0.06 \\t=\frac{18}{12}=1.5

We have converted months to years in this situation, as we are given the annual interest rate and not the monthly interest rate.

Now we calculate the interest using the formula:

I=PRt\\=5000\times0.06\times1.5\\=450

$\therefore$ We now know over the course of 1.5 years, $\$ 450$ of interest was accumulated. We now need to add this value to the principal, to find the total amount being payed back.

Total=5000+450=\$5450

Compound Interest

Compound interest is another way of calculating the interest charged on a loan. Unlike simple interest, compound interest introduces the concept of adding accumulated interest back to the principal sum. In simple terms this means, that interest is continuously gathered on previous interest + the principal amount. The formula to calculate compound interest is as follows:

A_n=P(1+R)^{nt}

where,

$A$ is the amount of money accumulated after $t$ years
$P$ is the principal (initial amount of money)
$R$ is the annual interest rate
$n$ is the number of times interest is compounded per year
$t$ is time in years

If we look closely at the formula for compound interest, we can see the formation of an GP with first term $P(1+R)$ , and ratio $1+R$ . This is important to know when dealing with difficult compound interest questions, as formula’s relating to GP’s can be used to solve problems.

Practice Question 2

You invest $2,000 in a bank account that offers an annual interest rate of 4%, compounded monthly. How much money will you have in the account after 5 years?

First we must identify all known variables:

P=2000\\R=4\%=0.04\\n=12\\t=5

However, we are given the annual interest rate, but the money is being compounded monthly. Thus we must find the monthly interest rate.

R=\frac{0.04}{12}

Now, we can substitute these values into the compound interest formula:

A=P(1+R)^{nt}\\=2000(1+\frac{0.04}{12})^{12\times5}\\=2000(1.22)\\=2441.99

$\therefore$ You will have $\$ 2441.99$ in your account after 5 years.

Depreciation

Depreciation is the loss in value of an asset, item or tangible object over a period of time. In finance, this can be seen as the reverse effect of compound interest. Similar to compound interest, the operation of depreciation involves the use of GP’s. This is made evident in the formula for depreciation, which is fairly similar to compound interest.

A_n=P(1-R)^{nt}

where,

$A$ is the remaining value of the asset after $t$ years
$P$ is the principal (initial amount of money)
$R$ is the annual interest rate
$n$ is the number of times interest is compounded per year
$t$ is time in years

This formula in other terms is known as the formula for exponential decay. We can also see from the formula, the formation of an GP with first term $P(1-R)$ , and ratio $1-R$ . As such, we apply the same steps to solving a formula for compound interest, except we use a slightly different formula.

Geometric Sequences with Logarithms

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Investing with Regular Instalments

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