Middle Years

# 4.09 Simple interest

Lesson

It costs money to borrow money. The extra money that banks and other lenders charge us to borrow money is called interest. Interest may also refer to the additional money that is earned from investing money, such as into a savings account. There are different types of interest and in this lesson we are going to look at simple interest.

### What is simple interest?

Simple interest, or flat rate interest, describes a method of calculating interest where the interest amount is fixed (i.e. it doesn't change). The interest charge is always based on the original amount borrowed (or invested), and does not take into account any interest earned along the way (that is, interest on interest is not included).

Many financial institutions express their interest rates as a percentage "per annum", often abbreviated to "p.a.", which means "per year". For example, an interest rate might be given as $3%$3% p.a.

Simple interest

Simple interest is calculated as:

$I=PRT$I=PRT

• $P$P is the principal amount invested (or borrowed)
• $R$R is the interest rate per year expressed as a decimal or fraction
• $T$T is the time in years

The total total value of an investment or loan $A$A is then calculated as the principal plus interest:

$A=P+I$A=P+I

#### Worked example

##### example 1

An investment of $\$8580$$8580 is deposited at 2%2% p.a. flat rate for 44 years. (a) Find the interest earned in 11 year. Think: 2%2% interest is earned each year. So we need to find 2%2% of \8580$$8580. Can you see this in the same as using the formula with $P=\$8580$P=$8580, $R=2%$R=2% and $T=1$T=1 year? For the formula express $R$R as a decimal or fraction, thus $R=\frac{2}{100}$R=2100 or $0.02$0.02.

Do:

 $I$I $=$= $PRT$PRT $=$= $\$8580\times0.02\times1$$8580×0.02×1 == \171.60$$171.60 For financial questions give answers to $2$2 decimal places.

Thus, $\$171.60$$171.60 in interest would be earned in 11 year for this investment. (b) Find the interest earned in 44 years. Think: For simple interest the same amount is earned each year, so we could multiply the figure found in part (a) by 44. Can you see this is the same as using the formula with the values P=\8580P=8580, R=0.02R=0.02, T=4T=4 years? Do:  II == PRTPRT == 8580\times0.02\times48580×0.02×4 == \686.60$$686.60

(c) What is the total value of the investment after $4$4 years?

Think: The total value is the original investment plus the interest that has accrued over $4$4 years.

Do:

 $A$A $=$= $P+I$P+I $=$= $\$8580+\$686.60$$8580+686.60 == \9266.40$$9266.40 #### Practice questions ##### Question 1 Calculate the simple interest earned on an investment of$\$2540$$2540 at 9%9% p.a. for 22 years. Give your answer to the nearest cent. ##### Question 2 Buzz takes out 66-year \50000$$50000 loan at$8%$8% p.a. flat interest. 1. What is the total interest he will have to pay? 2. What is the total amount he will have to pay? ## Different rates and time periods In the formula given above the rate was an annual rate and the time was given in years. If we are given a rate which is not annual we can convert it to annual. For example, a rate of$1.5%$1.5% per quarter is the same as an annual rate of$1.5%\times4=6%$1.5%×4=6% p.a., since there are$4$4 quarters in a year. When calculating simple interest for time periods that are not years, such as months, weeks or days, we can express the time given as a fraction of a year. For example, for an investment of$10$10 months$T=\frac{10}{12}$T=1012 or for a loan for$30$30 days$T=\frac{30}{365}$T=30365 Remember! $1$1 year$=$=$12$12 months$=$=$52$52 weeks$=$=$4$4 quarters (of$3$3 months each)$=$=$365$365 days #### Worked example ##### example 2 A loan of$\$2000$$2000 is taken out at 5%5% p.a. flat rate for 1818 months. Find the interest due on the loan after 1818 months. Think: We can substitute the values into the formula using P=\2000P=2000, R=5%=0.05R=5%=0.05 and T=\frac{18}{12}T=1812 years. Do:  II == PRTPRT == \2000\times0.05\times\frac{18}{12}$$2000×0.05×1812​$=$=$\$150$$150 #### Practice questions ##### Question 3 Calculate the simple interest generated on a loan of \3860$$3860 at a rate of$9%$9% p.a. for$13$13 months. 1. Round your answer to the nearest cent. ##### Question 4 Calculate the simple interest earned on an investment of$\$7000$$7000 at 1.8%1.8% per quarter for 99 years. Give your answer to the nearest cent. ##### Question 5 Calculate the simple interest earned on an investment of \5320$$5320 at$6%$6% p.a. for$95$95 weeks. Assume that a year has$52$52 weeks. Give your answer to the nearest cent. ## Finding rate, principal or time The formula for finding interest can also be rearranged to find an unknown rate, principal amount or time. Simple interest Rearranging$I=PRT$I=PRT for each variable we obtain: $P=\frac{I}{R\times T}$P=IR×T​$T=\frac{I}{P\times R}$T=IP×R​$R=\frac{I}{P\times T}$R=IP×T​ Unknown principal Unknown time Unknown rate Note: • The formula is the similar for each with the interest earned on top of the fraction and the two known variables on the bottom •$T$T is the time in years •$R$R is the interest rate per year expressed as a decimal or fraction, multiply by$100%$100% to convert to a percentage #### Worked example ##### EXAMPLE 3 An investment of$\$2800$$2800 increases to a total value of \3052$$3052 in$2$2 years. What simple interest rate was the investment earning? Think: For the formula to find a rate we need the interest earned by the investment, the principal and the time in years. In this case, we have not been given the interest earned directly but we can find it. The interest earned on the investment is the difference between the principal and final value. Hence, $I$I$=$=$\$3052-\$2800$$3052−2800 == \252$$252

Then we have the values necessary for our formula, with $I=\$252$I=$252$P=\$2800$P=$2800 and $T=2$T=2 years.

Do: