It costs money to borrow money from financial institutions (like banks). The extra money that these lenders charge is called interest. Interest can also refer to money earned from investing money, such as in a savings accounts.

The starting amount, either borrowed or invested, is called the principal. The interest is usually described as a rate (percentage) per annum. For example, an investment of \$100 at a rate of 3\% per annum. 3\% of 100 is 3, so this investment produces \$3 every year.

Calculate simple interest

Simple, or straight line interest is a method where the interest amount is fixed (i.e. it doesn't change). The interest is based on the original principal.

It is calculated using the formula: I = \dfrac{PrT}{100}, where I is the total interest earned, P is the principal (the initial amount borrowed/invested), r\% is the interest rate, expressed as a whole number, T is the number of time periods (the duration of the loan/investment).

Alternatively, when the interest rate is expressed as a decimal or fraction, the following formula can be used I = PRT, where R = \dfrac{r}{100}\%.

The percentage symbol \% means per cent, which is a number out of 100. For example, 10\% = \dfrac{10}{100}. Great care must be taken when substituting interest rates into either formula above. The first formula, the interest rate can be entered as a whole number. For example, for 10\% interest, using formula I = \dfrac{PrT}{100}, the interest rate substitution would be r = 10.

Whereas the second formula, the interest rate must be entered as a fraction or decimal. For example, for 10\% interest, using the formula I = PRT, the interest rate substitution would be R = 10\% = \dfrac{10}{100} = \dfrac{1}{10} = 0.10.

The interest rate is often given as a percentage per time period. For example percentage(4.5) p.a. where "p.a." is an abbreviation of per annum, which means every year. However, the unit of time used for T must match the unit of time used in the interest rate r. Sometimes this means a conversion is required. For example, the time of a loan / investment may be given as a number of days, which would then need to be converted into a number (or fraction) of years.

Using any three known pieces of information from the simple interest formula it is possible to find the remaining unknown variable.

The formula can be rearranged to calculate the principle, the rate or the time. As long as you know the simple interest rule you don't need to remember all the variations but they are useful to see.

Rearranging the simple interest formula:

To find the interest rate (r\%): r=\dfrac{100I}{PT}

To find the time (T): T=\dfrac{100I}{Pr}

To find the principal amount (P): P=\dfrac{100I}{rT}

For simple interest, once the interest is calculated, it can then be added to the principal amount to calculate the total amount of the loan / investment, represented by A. This is found using the following formula: A = P + I.

In some cases you may know the final amount of the investment A, and not I, the interest that has been earned. In this case use the formula: P=\dfrac{A}{(1+RT)}, where R is the interest rate expressed as a decimal or fraction.

Examples

Example 1

Calculate the simple interest on a loan of \$8000 at 8\% p.a. for 6 years. Round your answer to the nearest dollar.

Worked Solution

Create a strategy

Use the simple interest formula: I=PRT

Apply the idea

We are given: P=\$8000 ,\, R = 8\% and T=6

\displaystyle I	\displaystyle =	\displaystyle 8000 \times 0.08 \times 6	Substitute the values
	\displaystyle =	\displaystyle \$3840	Evaluate

Example 2

For a simple interest rate of 6\% p.a. , calculate the number of years T needed for an interest of \$1174.20 to be earned on the investment \$1957. Round your answer as a whole number of years.

Worked Solution

Create a strategy

Use the simple interest formula: I=PRT

Apply the idea

We are given: I=\$1174.20 ,\, R = 6\% and P=1957

\displaystyle 1174.2	\displaystyle =	\displaystyle 1957 \times 0.06 \times T	Substitute the values
\displaystyle T	\displaystyle =	\displaystyle \dfrac{1174.2}{1957 \times 0.06}	Make T the subject of the equation
\displaystyle T	\displaystyle =	\displaystyle 10	Evaluate

Idea summary

Simple interest is calculated as:

\displaystyle I = PRT

\bm{P}

is the principal amount invested (or borrowed)

\bm{R}

is the interest rate per year expressed as a decimal or fraction

\bm{T}

is the time in years

The total value of an investment or loan A is then calculated as the principal plus interest: A = P + I

Graph simple interest

Simple interest can be modelled as a linear graph.

We can use simple interest graphs to compare different scenarios, such as:

the interest, I, earned over time, t
the interest earned for different principals, P
the total value of a loan or investment, A, over time, t

The graph of simple interest is a straight line, since the interest rate is constant. The gradient of the line indicates how much interest is earned in each time period. If the graph shows interest I against t then the y-intercept will be zero, and if the graph shows total amount A against t then the y-intercept will represent the initial amount invested or borrowed.

When presented with a graph, make sure to read the description of the given graph and the axis labels to understand the context of the question.

Examples

Example 3

Below is the graph showing the amount of interest earned over time at a particular interest rate.

\text{Time (years)}

200

400

600

800

1000

1200

1400

\text{Interest earned (dollars)}

How much interest is earned after 5 years?

Worked Solution

Create a strategy

We need to point on the line that corresponds to 5 years.

Apply the idea

The point on the line that corresponds to 5 years on the horizontal axis:

\text{Time (years)}

200

400

600

800

1000

1200

1400

\text{Interest earned (dollars)}

Reading across, we can see that this point corresponds to 600 on the vertical axis.

So the total interest earned after 5 years is \$600.

Find the gradient of the line.

Worked Solution

Create a strategy

We can use the formula: \text{gradient}=\dfrac{\text{rise}}{\text{run}}, where \text{rise}=600 and \text{run}=5.

Apply the idea

\displaystyle \text{gradient}	\displaystyle =	\displaystyle \dfrac{600}{5}	Substitute \text{rise}=600 and \text{run}=5
	\displaystyle =	\displaystyle 120	Evaluate

Reflect and check

This tells us that \$120 of interest is earned each year.

How long will it take to earn \$1800 of interest?

Worked Solution

Create a strategy

Divide the \$1800 by the gradient we found in part (b).

Apply the idea

\displaystyle \text{Number of year}	\displaystyle =	\displaystyle \dfrac{1800}{120}	Divide 1800 by 120
	\displaystyle =	\displaystyle 15	Evaluate

Example 4

The graph shows the amount of simple interest charged each year by a particular bank, on a 3-year loan:

1000

2000

3000

4000

5000

\text{Principal}

100

150

200

250

300

\text{Interest}

Find the total amount of simple interest charged on a loan of \$5000 over a 3 year period.

Worked Solution

Create a strategy

We need to point on the line that corresponds to \$5000, and then multiply it by 3.

Apply the idea

The point on the line that corresponds to \$5000 on the horizontal axis:

1000

2000

3000

4000

5000

\text{Principal}

100

150

200

250

300

\text{Interest}

Reading across, we can see that this point corresponds to 300 on the vertical axis.

\displaystyle \text{Total interest}	\displaystyle =	\displaystyle 300 \times 3	Multiply 300 by 3
	\displaystyle =	\displaystyle \$900	Evaluate

Determine the simple interest rate per year,r, charged by the bank on these loans. Give your answer as a percentage.

Worked Solution

Create a strategy

Use the simple interest formula: I=PRT

Apply the idea

We are given: P = \$5000 ,\, I = \$900 and t = 3

\displaystyle 900	\displaystyle =	\displaystyle 5000 \times r \times 3	Substitute the values
\displaystyle r	\displaystyle =	\displaystyle \dfrac{900}{5000 \times 3}	Make r the subject of the equation
\displaystyle r	\displaystyle =	\displaystyle \dfrac{900}{15\,000}	Evaluate the multiplication
\displaystyle r	\displaystyle =	\displaystyle 0.06	Evaluate
\displaystyle r	\displaystyle =	\displaystyle 6\%	Convert to percent

Idea summary

We can use simple interest graphs to compare different scenarios, such as:

the interest, I, earned over time, t
the interest earned for different principals, P
the total value of a loan or investment, A, over time, t

Outcomes

U1.AoS2.6

concepts of ratio, proportion, percentage, percentage change and rate, and unitary method

4.06 Simple interest

Introduction

Ideas

Calculate simple interest

Examples

Example 1

Example 2

Graph simple interest

Examples

Example 3

Example 4

Outcomes

U1.AoS2.6

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