Introduction
While borrowing money can be useful for buying things that we can't currently afford, it's not a free service. A loan is a service that allows you to borrow money and then repay that over time, plus a bit extra. This extra amount is called interest which is calculated as a percentage of the amount that needs to be repaid.
Ideas
- Simple interest
- Loan repayments
Simple interest
Simple interest is calculated as a flat percentage of the amount that was borrowed.
We can calculate the simple interest of a loan using the formula: I=Prn where I is the interest accumulated, P is the principal amount borrowed, r is the rate of interest per period and n is the number of periods.
Examples
Example 1
What is the total interest to be paid on a 2-year \$3000 loan at 17\% p.a. flat interest?
Worked Solution
Create a strategy
We can use the simple interest formula: I=Prn.
Apply the idea
| \displaystyle I | \displaystyle = | \displaystyle Prn | Write the formula |
| \displaystyle = | \displaystyle 3000 \times 17\% \times 2 | Substitute P,\,r,\, and n | |
| \displaystyle = | \displaystyle \$1020 | Evaluate |
Example 2
The simple interest on a loan of \$6600 over 33 months is \$1252.35.
If the annual interest rate is r, find r as a percentage to the nearest one decimal place.
Worked Solution
Create a strategy
We can use the simple interest formula: I=Prn.
Apply the idea
Since we are looking for the annual interest rate be sure to convert the time to years.
| \displaystyle 1252.35 | \displaystyle = | \displaystyle 6600 \times r \times \dfrac{33}{12} | Substitute I,\,P,\, and n |
| \displaystyle 1252.35 | \displaystyle = | \displaystyle 18\,150 r | Evaluate the multiplication |
| \displaystyle r | \displaystyle = | \displaystyle \dfrac{1252.35}{18\,150} | Divide both sides by 18\,150 |
| \displaystyle = | \displaystyle 0.069 | Evaluate the division | |
| \displaystyle = | \displaystyle 6.9\% | Convert to a percentage |
Example 3
\$906 is invested at 5\% p.a simple interest. Dave wants to know the number of years it will take the investment to grow to \$1132.50.
a
Calculate the interest that will be earned on the investment.
Worked Solution
Create a strategy
Subtract the initial value of the investment from the final value of the investment.
Apply the idea
| \displaystyle I | \displaystyle = | \displaystyle 1132.50 - 906 | Subtract the initial value from the final value |
| \displaystyle = | \displaystyle \$226.50 | Evaluate the difference |
b
Calculate the number of years it will take the investment to grow to \$1132.50.
Worked Solution
Create a strategy
We can use the simple interest formula: I=Prn.
Apply the idea
| \displaystyle 226.50 | \displaystyle = | \displaystyle 906 \times 5\% \times n | Substitute I,\,P,\, and r |
| \displaystyle 226.50 | \displaystyle = | \displaystyle 45.3 n | Evaluate the multiplication |
| \displaystyle n | \displaystyle = | \displaystyle \dfrac{226.50}{45.3} | Divide both sides by 45.3 |
| \displaystyle = | \displaystyle 5 | Evaluate the division |
It will take 5 years for the investment to grow to \$1132.50.
Idea summary
We can calculate the simple interest of a loan using the formula:
\displaystyle I=Prn
\bm{I}
is the interest accumulated
\bm{P}
is the principal amount borrowed
\bm{r}
is the rate of interest per period
\bm{n}
is the number of periods
Loan repayments
When repaying loans, calculations are required so that the number of repayments and the size of a repayment multiply to match the total amount to be repaid. The total amount to be repaid on a loan is equal to the sum of the principal amount borrowed and the total interest accumulated.
\text{Total amount}=\text{Principal amount} + \text{ Interest accrued}
\text{Total amount}=\text{No. repayments} \times \text{ Size of repayments}
Since the 'total amount' appears in both equations, we can relate the number and size of repayments to the principal amount and interest. As such, knowing any three of these values will allow us to find the fourth.
Examples
Example 4
Katrina takes out a loan to purchase a surround sound system. She makes 19 equal loan repayments. The total loan amount paid is \$95\,000.
What is the value of each repayment?
Worked Solution
Create a strategy
We can rearrange the formula \text{Total amount}=\text{No. repayments} \times \text{ Size of repayments} to find the number of loan repayments.
Apply the idea
| \displaystyle \text{Size of repayments} | \displaystyle = | \displaystyle \dfrac{\text{Total amount}}{\text{No. repayments}} | Rearrange the formula |
| \displaystyle = | \displaystyle \dfrac{95\,000}{19} | Substitute the values | |
| \displaystyle = | \displaystyle \$ 5000 | Evaluate |
Idea summary
\text{Total amount}=\text{Principal amount} + \text{ Interest accrued}
\text{Total amount}=\text{No. repayments} \times \text{ Size of repayments}
