Interest is the extra money that banks and lenders charge us to borrow money. It may also refer to additional money you earn from depositing money, such as in a savings account. There are two different types of interest simple interest and compound interest.
We can solve problems involving simple and compound interest in the following ways:
- Using the simple interest or compound interest formula
- Using sequences - recursive or explicit forms
- Using the financial solver online
- Using a spreadsheet
Simple interest formula
Simple interest is a method where the interest amount is fixed (i.e. it doesn't change). This fixed interest charge is based on the original amount, which is called the principal. Simple interest can be calculated using the formula below. Note that this formula calculates the interest, not the final balance.
$I=Pin$I=Pin
where $P$P is the principal (the initial amount borrowed or invested)
$i$i is the interest rate per time period, expressed as a decimal or fraction
$n$n is the number of time periods (the duration of the loan or deposit)
Practice question
Question 1
Calculate the simple interest earned on an investment of $\$6350$$6350 at a rate of $6%$6% p.a. for $13$13 months.
Round your answer to the nearest cent.
Simple interest using sequences
With simple interest the balance is increased or decreased by adding or subtracting the same amount every time, therefore simple interest problems can also be modelled using an arithmetic sequence. You can think of it as "next value equals the value before plus the simple interest".
Consider the following problem: James invests $\$15000$$15000 into an investment account that pays simple interest of $3.2%$3.2% per annum. The table below shows the value of the investment over the first four years.
| Month ($n$n) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Balance ($V_n$Vn) | $15000$15000 | $15480$15480 | $15960$15960 | $16440$16440 | $16920$16920 |
We can see this is an arithmetic sequence with $a=15000$a=15000 and $d=480$d=480.
The recursive form is $A_{n+1}=A_n+480$An+1=An+480, where $A_0=15000$A0=15000.
Note: When creating a sequence to generate the value of the investment at the end of each period we let $A_0$A0 equal to the initial amount so that $A_1$A1 is then the amount at the end of the first instalment period. This makes it easier to answer questions involving the value of the investment over time.
For a principal investment/loan, $P$P, at the simple interest rate of $i$i per period, the sequence of the value of the investment over time forms an arithmetic sequence with a starting value of $P=a$P=a and a common difference $d=a\times r$d=a×r.
The sequence which generates the value, $A_n$An, of the investment/loan at the end of each instalment period is:
- Recursive form:
$A_{n+1}=A_n+d$An+1=An+d, where $A_0=a$A0=a
- Explicit form:
$A_n=a+nd$An=a+nd
The sequence which generates the value, $A_n$An, of the investment/loan at the beginning of each instalment period is:
- Recursive form:
$A_{n+1}=A_n+d$An+1=An+d, where $A_1=a$A1=a
- Explicit form:
$A_n=a+\left(n-1\right)d$An=a+(n−1)d
Sequences with a scientific calculator
To create create a list of the value of an investment/loan at each period make use of the answer key 
.
- Enter the principal amount and press enter (
or equivalent). - Type in the next step using the answer key, in this case + interest earned per period.
- Continue to press to obtain the next term in the sequence.
Practice question
Question 2
Manpreet lives in India and invests $56000$56000 INR into an investment account that pays $6.9%$6.9% simple interest per annum.
By what amount will the account increase each year?
Complete the recurrence relation for Manpreet's situation, where $t_n$tn is the balance at the end of the $n$nth year and $t_0$t0 is the initial investment.
$t_{n+1}=t_n$tn+1=tn$+$+$\editable{}$, where $t_0$t0$=$=$\editable{}$.
Complete and then simplify the explicit rule that can be used to find the balance at the end of $n$n years.
$t_n=$tn=$\editable{}$$+\left(n-1\right)\times$+(n−1)×$\editable{}$ which simplifies to $t_n=$tn=$\editable{}$.
Determine the balance after $7$7 years.
Determine how many whole years it takes for the balance to exceed $108164$108164 INR.
Compound interest formula
Most of the time, when banks and financial institutions calculate interest, they are using compound interest.
Compound interest is calculated at the end of each compounding period, which is typically a day, month, quarter, or year. At the end of each compounding period, the total amount (principal plus interest) from previous compounding periods is used to calculate the new quantity of interest. We multiply the total amount by the interest and then add it to the total. Note that the compound interest formula calculates the final balance, or amount. To find the amount of interest we need to subtract the principal from the final balance.
$A=P\times(1+i)^n$A=P×(1+i)n
where:
$A$A is the final amount of money (principal and interest together)
$P$P is the principal (the initial amount of money invested)
$i$i is the interest rate per compounding period, expressed as a decimal
$n$n is the total number of compounding periods
Practice question
Question 3
A $\$1110$$1110 investment earns interest at $2.2%$2.2% p.a. compounded annually over $18$18 years. Use the compound interest formula to calculate the value of this investment to the nearest cent.
Compound interest using sequences
With compound interest the balance is increased by multiplying the same amount every time, therefore compound interest problems can be modelled using a geometric sequence. The "next term" is made by increasing the term before by the interest rate percentage. Therefore you can think of it as "next value equals the value before multiplied by ($1$1 + the interest rate as a decimal)".
Consider the following problem: Emma puts $\$5000$$5000 into an investment account paying a compound interest rate of $4.2%$4.2% pa. The table below shows the value of the investment over the first four years.| Year ($n$n) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $5000\times1.042=$5000×1.042= | $5210\times1.042=$5210×1.042= | $5428.82\times1.042=$5428.82×1.042= | $5656.83\times1.042=$5656.83×1.042= | ||
| Amount ($V_n$Vn) | $5000$5000 | $5210$5210 | $5428.82$5428.82 | $5656.83$5656.83 | $5894.42$5894.42 |
For a principal investment/loan, $P$P, at the compound interest rate of $i$i per period, the sequence of the value of the investment over time forms a geometric sequence with a starting value of $P=a$P=a and a common ratio of $r=(1+i)$r=(1+i).
The sequence which generates the value, $A_n$An, of the investment/loan at the end of each instalment period is:
- Recursive form:
$A_{n+1}=rA_n$An+1=rAn, where $A_0=a$A0=a
- Explicit form:
$A_n=ar^n$An=arn
The sequence which generates the value, $A_n$An, of the investment/loan at the beginning of each instalment period is:
- Recursive form:
$A_{n+1}=rA_n$An+1=rAn, where $A_1=a$A1=a
- Explicit form:
$A_n=ar^{n-1}$An=arn−1
Worked example
Example 1
Holly invests $\$2000$$2000 at $5%$5% p.a. compounded annually.
(a) Write a recurrence relation for this situation, where $A_n$An is the balance at the end of the $n$nth month and $A_0$A0 is the initial investment.
Think: Using the form $A_{n+1}=rA_n$An+1=rAn , where $A_0=$A0= the initial investment and $r=\left(1+i\right)$r=(1+i). Substitute in the values for $r$r and $A_0$A0.
Do: $A_{n+1}=\left(1.05\right)\times A_n$An+1=(1.05)×An, where $A_0=\$2000$A0=$2000
(b) Find the value of the investment after $5$5 years.
Think: Using our recursive rule we can see each step we multiply the previous value by $1.05$1.05. We can use the answer key on the calculator to find the balance at the end of each year for five years.
Do:
- Enter the principal amount $2000$2000 and press enter ( or equivalent). This is the starting balance
- Type in the next step using the answer key, in this case $1.05\times$1.05×. This is the balance after $1$1 year.
- Continue to press to obtain the next term in the sequence. One press: balance after $2$2 years, Two presses: balance after $3$3 years, $\ldots$…. Count until you get to the balance after $5$5 years.
The balance after $5$5 years is $\$2431.01$$2431.01
(c) Determine how many whole years it takes for the balance to exceed $\$2800$$2800.
Think: Continue using the sequence as above in the calculator and count until the balance first exceeds the given amount.
Do: At $5$5 years the balance was $\$2431.01$$2431.01, pressing three more times we see the balance first exceed $\$2800$$2800, so it will take $8$8 years.
Practice questions
Question 4
$\$3000$$3000 is invested at the beginning of the year in an account that earns $12%$12% per annum interest, compounded quarterly.
How much money is in the account at the end of the first year?
Give your answer to the nearest cent.
Write a recursive rule for $V_n$Vn that gives the balance in the account at the end of the $n$nth quarter.
Write both parts of the rule (including for $V_0$V0) on the same line, separated by a comma. Express all necessary values as decimals.
Question 5
The balance of an investment, in dollars, at the end of the $n$nth year where interest is compounded annually is given by $A_n=1.061A_{n-1}$An=1.061An−1, $A_0=15000$A0=15000.
State the annual interest rate.
State the amount invested in dollars.
Determine the balance at the end of the first year.
Use your calculator to determine the balance at the end of $20$20 years.
Round your answer to the nearest cent.
Using a spreadsheet to model compound interest
We can use a spreadsheet to track the balance of investments and loans, together with the amount of interest incurred. This allows us to efficiently perform numerous calculations and "what if analysis" to explore different options. Let's explore this in the interactive compound interest spreadsheet below:
You can change the amount invested (the blue cell) to any value you'd like to invest.
You can change the annual interest rate (the green cell) to any value.
You can change the number of compounding periods (the pink cell) to quarterly ($4$4), monthly ($12$12), weekly ($52$52) or perhaps daily ($365$365).
Investigate
- What happens as you increase the number of compounding periods?
- What happens as you increase the annual interest rate?
- How has the value in cell C10 been calculated?
- How has the value in D12 been calculated?
Worked example
Example 2
David invests $\$8000$$8000 in the bank with an interest rate of $3%$3% p.a. compounded monthly. He creates the following spreadsheet to help him do "what if analysis" to examine the problem:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | Principal | $\$8000$$8000 | ||
| 2 | Annual interest rate | $3%$3% | ||
| 3 | Compounds per year | $12$12 | ||
| 4 | ||||
| 5 | Month | Balance start of month | Interest | Balance end of month |
| 6 | 1 | $\$8000$$8000 | $\$20$$20 | $\$8020$$8020 |
| 7 | 2 | $\$8020$$8020 | $\$20.05$$20.05 | $\$8040.05$$8040.05 |
Some of the formulae David used to create this spreadsheet are shown in the table below:
| A | B | C | D | |
|---|---|---|---|---|
| 5 | Month | Balance start of month | Interest | Balance end of month |
| 6 | 1 | =B1 | =$B$2/$B$3*B6 | =B6+C6 |
| 7 | =A6+1 | =D6 | ||
| 8 |
Note:
- The spreadsheet is designed so that if the values in cells B1, B2 and B3 are changed, the whole page instantly updates. This makes it quick and easy to investigate different investment options.
- The $ signs in the cell references make the reference absolute. That means the cell name will not change as the formula is copied down the column.
- The month numbers here have been calculated using a formula in cell A7.
(a) What is the purpose of the formula in cell C6?
Think: Ignore the $ signs. The formula is =B2/B3 * B6, which is calculating $3%\div12\times\$8000$3%÷12×$8000.
Do: This calculates the amount of monthly interest by taking the yearly interest rate (cell B2) and dividing by the number of compounds per year (cell B3) then multiplying by the amount at the start of the month (cell B6).
(b) What is the purpose of the formula in cell D6?
Think: The formula is =B6+C6 so it is added two values together.
Do: Write that it calculates the balance at the end of the month by calculating start balance + interest.
(c) Use a spreadsheet program to recreate this spreadsheet. How long does it take David to save the $\$9000$$9000?
Reflect: The spreadsheet should show that it takes him $48$48 months for his balance to first exceed $\$9000$$9000 and hence, it will take David approximately $4$4 years if he doesn't make any additional payments.
Practice question
Question 6
The following spreadsheet shows the balance (in dollars) in a savings account in 2013, where interest is compounded monthly.
| A | B | C | D | |
| 1 | Month | Balance at beginning of month | Interest | Balance at end of month |
| 2 | July | $9000$9000 | $180$180 | $X$X |
| 3 | August | $9180$9180 | $183.60$183.60 | $9363.60$9363.60 |
| 4 | September | $9363.60$9363.60 | $Y$Y | $9550.87$9550.87 |
| 5 | October | $Z$Z | $191.02$191.02 | $9741.89$9741.89 |
| 6 | November | $9741.89$9741.89 | $194.84$194.84 | $9936.73$9936.73 |
Calculate the value of $X$X.
Use the numbers for July to calculate the monthly interest rate.
Calculate the value of $Y$Y.
Calculate the value of $Z$Z.
Write a recursive rule, $B_n$Bn, that gives the balance at the end of the $n$nth month, with July being the first month.
Write both parts of the rule (including for $B_0$B0) on the same line, separated by a comma.
Write an explicit rule for $B_n$Bn, the balance at the end of the $n$nth month, with July being the first month.