Learning activity 3.5: Financial literacy: simple and compound interest

In this learning activity, you will learn to identify the difference between simple and compound interest. You will be able to solve problems that involve calculating simple and compound interest.

Pre-assessment review:

Check your prior knowledge of the following terms and conversion techniques related to finance that you will encounter in this learning activity.

Try it!

How would you define the term ‘invest’ in relation to finance?

When you invest you put money into something (for example a property, financial product, gold, etc) with the hope of increasing its value.

What does the term ‘interest’ mean in relation to finance?

Whenever you invest or borrow money you will either earn money on the investment or have to pay money on the loan. The money that is earned or paid is called interest.

What does the term ‘loan’ mean in relation to finance?

A loan is a sum of money that is borrowed from a lender with the expectation that the money will be repaid by the borrower to the lender over time. Typically, the lender will charge the borrower interest on the initial amount loaned.

How do you convert a percentage into a decimal?

To convert a percentage into a decimal you divide the percentage by $100$ . For example, $8 %$ can also be expressed as $0.08 (8 / 100 = 0.08)$ .

Explore this!

Let's explore the following videos which explain how to distinguish debts that have simple interest and debts with compound interest.

Summer savings with simple interest

A person sits at a desk displaying a piggy bank wih carefully stacked coins next to it.

Mara and Aram are a young couple who want to invest for their future. Concept of simple and compound interest has been explained by using their situations as examples. Aram worked during the summer vacation doing minor landscaping and cutting lawns. Instead of spending the paycheques, they decided to deposit the $$ 1,000$ in a bank account that earns $4 %$ interest each year for six years.

At the end of each year, Aram received a cheque for the interest earned. Find out how much interest Aram earned in the following activity.

What is simple interest?

The interest that is either earned (or paid) on the original sum of money invested (or borrowed) is simple interest.

The sum of money that is borrowed or invested is called principal.

Explore this!

Explore the following video to find out more about simple interest.

What is the formula for simple interest?

The interest earned depends on three factors:

Principal, $P$ (the money invested).
Rate, $r$ (the annual interest rate).
Time, $t$ (the time for which the money is invested).

This is the formula for calculating simple interest:

Formula for simple interest: $I = P r t$

$I$ is the interest earned in dollars.
$P$ is the principal invested in dollars.
$r$ is the annual interest rate, in decimal form.
$t$ is the time in years.

Take note of the following equation, as you’ll need it to calculate the total amount of an investment over time.

When money is invested, the sum of the principal and the interest is called the amount of an investment.

Amount is Principal $+$ Interest
Formula: $A = P + I$

$A = P + I$

To find out Aram’s interest each year for six years: first, we must convert $4 %$ to a decimal, $0.04 (4 % = 4 / 100 = 0.04)$ . This is called the rate of interest. We then have to multiply the rate with $$ 1,000$ to find the interest earned at the end of each year.

The following table shows the amount of interest Aram will earn each year.

Year	Amount in account ($)	Suggested Solutions
1	$1,000$	$1,000 \times 0.04 = 40$
2	$1,000$	$1,000 \times 0.04 = 40$
3	$1,000$	$1,000 \times 0.04 = 40$
4	$1,000$	$1,000 \times 0.04 = 40$
5	$1,000$	$1,000 \times 0.04 = 40$
6	$1,000$	$1,000 \times 0.04 = 40$

Notice that the amount of interest stays the same each year and so, it represents simple interest because the interest is not reinvested.

Only the original amount of $$ 1,000$ , called the principal, earns interest each year. The interest earned is paid separately at the end of each year.

To determine the accumulated interest, add the interest earned at the end of each year. The last column in the following table shows the accumulated interest.

Year	Amount in account ($)	Interest received ($)	Suggested Solutions
1	$1000$	$1,000 \times 0.04 = 40$	$40$
2	$1000$	$1,000 \times 0.04 = 40$	$80$
3	$1000$	$1,000 \times 0.04 = 40$	$120$
4	$1000$	$1,000 \times 0.04 = 40$	$160$
5	$1000$	$1,000 \times 0.04 = 40$	$200$
6	$1000$	$1,000 \times 0.04 = 40$	$240$

Notebook

In your notebook, determine the pattern for the accumulated interest in each year and compare with the suggested solution.

Each value can be determined by multiplying 40 with the year number. Notice the following:

For year 1, the accumulated interest is $$ 40 \times 1 = $ 40$ , or $($ 1,000 \times 0.04) \times 1 = $ 40$
For year 2, the accumulated interest is $$ 40 \times 2 = $ 80$ , or $($ 1,000 \times 0.04) \times 2 = $ 80$
For year 3, the accumulated interest is $$ 40 \times 3 = $ 120$ , or $($ 1,000 \times 0.04) \times 3 = $ 120$
If the money was invested for 12 years, the accumulated interest would be $$ 40 \times 12 = $ 480$ or $($ 1,000 \times 0.04) \times 12 = $ 480$

Using the year number and accumulated interest values from the following table produce a graph of simple interest earned over time.

Year	Simple interest ($)
1	$40$
2	$80$
3	$120$
4	$160$
5	$200$
6	$240$

Because the points form a straight line, simple interest represents a linear function. The slope of the line, which is 40, indicates that the simple interest increases at a constant rate of $40 per year.

Investing $2,000 prize using simple interest

An older person in a very classy suit gives an excited younger person a novelty blank cheque.

Aram is a talented individual. To earn more money, Aram submitted a short story to a creative writing contest and won the top prize: a cheque for $$ 2,000$ .

A term deposit is a fixed-term investment that locks your principal away until the term is over. Consider a situation in which Aram invested the $$ 2,000$ in an 18-month term deposit account that paid $3.5 %$ per year. As noted earlier:

$I = P r t$

To find how much interest $(I)$ Aram earned, you need to identify the following elements:

principal(p)
rate(r)
time(t)

Try it!

The principal is the initial amount invested. How much is Aram’s principal?

The principal is $P = 2,000$ .

What is the interest rate? Convert this to a decimal.

The interest rate is $3.5 %$ .

As a decimal, $r = 0.035$ .

What is the time? (This is time in years, so you may have to determine the fraction of a year.)

The time is 18 months.

Since there are 12 months in a year, then in years $t = \frac{18}{12} = \frac{3}{2}$ , or one and a half years.

Now you can use $I = P r t$ to find the interest. What is the interest?

$I = P r t$

$I = 2,000 \times 0.035 \times \frac{3}{2}$

$I = 105$

Therefore, Aram earned $105 in interest.

In some problems you may have to solve for $P$ , $r$ , or $t$ .

Borrowing from a bank

Stylish depiction of a plant growing out of coins spilling from a jar.

Consider a situation in which Aram paid $165 in interest for borrowing a sum of money at an interest rate of $2.75 %$ annually for four years. Based on these values, calculate how much money Aram borrowed initially.

To calculate how much Aram borrowed you will have to first find the interest rate, the time, and the interest.

What was Aram's rate?

The interest rate is $2.75 %$ .

As a decimal, $r = 0.0275$ .

What was Aram’s time?

The time is $t = 4$ years.

What was Aram’s interest?

The interest is $I = 165$ .

Use the formula $I = P r t$ to find the principal.

The principal is:

$165 = P \times 0.0275 \times 4$

$165 = 0.11 P$

$P = \frac{165}{0.11}$

$P = 1,500$

Therefore, Aram borrowed $$ 1,500$ .

The death of Mara’s great-aunt

Depiction of a last will and testament featuring a gavel, two sacks of money and legal documents with old timey lettering. — Press here for long description(Open in new window)

Mara tells Aram that an inheritance is due from a great-aunt. Consider a situation in which Mara invests the inheritance of $$ 5,000$ .

Mara decides to buy a seven-year, $$ 5,000$ Guaranteed Investment Certificate (GIC) that earns $5.8 %$ per year.

Try it!

Determine the interest that Mara earned on the GIC.

Use the formula $I = P r t$ .

Substitute: $P = 5,000$ , $r = 0.058$ , and $t = 7$ .

$I = 5,000 \times 0.058 \times 7$

$I = 2,030$

Therefore, Mara earned $$ 2,030$ in interest.

What is the amount of the GIC at the end of seven years?

$A = P + I$

$A = 5,000 + 2,030$

$A = 7,030$

The amount of the GIC is $$ 7,030$ .

Summer savings with compound interest

Someone crawls through the grass desperately trying to grab a series of increasingly large piggy banks ahead of him. — Press here for long description(Open in new window)

Earlier we learned that Aram deposited $$ 1,000$ in an account that pays $4 %$ simple interest for six years. Let’s consider the interest received if interest had been reinvested back in Aram's account.

Each year, Aram receives $$ 40$ , hence, at the end of six years total received interest is $$ 240$ .

Suppose Aram reinvested the interest back into the account. Explore how this situation is different from simple interest. The table below represents the calculations for the amounts earned each year if Aram had invested in a $4 %$ compound interest account.

Year	Principal for year (P)	Interest earned (I)	Accumulated interest ($)
$1$	$1,000$	$1,000 \times 0.04 = 40$	$40$
$2$	$1,000 + 40 = 1,040$	$1,040 \times 0.04 = 41.60$	$40 + 41.60 = 81.60$
$3$	$1,040 + 41.60 = 1,081.60$	$1,081.60 \times 0.04 = 43.26$	$81.60 + 43.26 = 124.86$
$4$	$1,081.60 + 43.26 = 1,124.86$	$1,124.86 \times 0.04 = 45.00$	$124.86 + 45.00 = 169.86$
$5$	$1,124.86 + 45.00 = 1,169.86$	$1,169.86 \times 0.04 = 46.79$	$169.86 + 46.79 = 216.65$
$6$	$1,169.86 + 46.79 = 1,216.65$	$1,216.65 \times 0.04 = 48.67$	$216.65 + 48.67 = 265.32$

When Aram invested $$ 1,000$ in a simple interest account, and earned a total of $$ 240$ in interest over 6 years. In this new situation, by reinvesting the interest earned at the end of each year into the account, Aram now earns $$ 265.32$ in interest over 6 years.

The interest difference between the two investments is $$ 265.32 - $ 240 = $ 25.32$ . Therefore, Aram earns $$ 25.32$ more in interest by adding it to the principal each year.

When interest is earned on interest, that interest compounds. This describes another type of interest—, compound interest.

In other words, interest calculated at regular periods and added to the principal for the next period is called compound interest.

Explore this!

Explore the following video to deepen your understanding of compound interest.

Compare simple interest and compound interest by investigating the amount of interest Aram earned on his $$ 1,000$ in each situation. Examine the following two tables. Table A indicates the change in simple interest each year. Table B indicates the change in compound interest each year.

Table A: $$ 1,000$ invested at $4 %$ simple interest.

Year	Total interest	Change in interest
$1$	$$ 40$	----------
$2$	$$ 80$	$$ 40$
$3$	$$ 120$	$$ 40$
$4$	$$ 160$	$$ 40$
$5$	$$ 200$	$$ 40$
$6$	$$ 240$	$$ 40$

Table B: $$ 1,000$ invested at $4 %$ compound interest.

Year	Total interest	Change in interest
$1$	$$ 40$	----------
$2$	$$ 81.60$	$$ 41.60$
$3$	$$ 124.86$	$$ 43.26$
$4$	$$ 169.86$	$$ 45.00$
$5$	$$ 216.65$	$$ 46.79$
$6$	$$ 265.32$	$$ 48.67$

For simple interest, the interest increases by the same amount each year.

For compound interest, the interest increases by a greater amount each year. It is compounded annually. This means the yearly interest is added to the principal amount and reinvested, therefore gaining even more interest.

In each table, the third column, change in interest, represents the finite differences (specifically, the first differences). You may want to review finite differences.

You already know that simple interest represents linear growth, so it makes sense that the finite differences are constant.

In Table B, the values in the third column are not constant, which indicates that compound interest is not linear. You can confirm this by graphing Simple interest vs. Compound interest.

Notebook

Sketch the following graph in your notebook, use the values in columns 1 and 2 of Tables A and B.

Observe that in the first year, the blue diamond covers the green dot. The interest earned under simple interest and compound interest is the same.

After the first year, the compound interest earned is greater than the simple interest. Notice that the difference increases each year.

The graph confirms that compound interest does not represent linear growth.

Determine the ratio of finite difference to determine the type of function it represents (divide the change in interest for a given year by the change in interest from the year before) For example, the ratio of finite differences between year 2 and year 1 is $$ 43.26 / $ 41.60 = 1.04$ .

Year	Total interest	Change in interest	Suggested solutions
1	$$ 40$	$$ 41.60$	$1.04$
2	$$ 81.60$	$$ 43.26$	$1.04$
3	$$ 124.86$	$$ 45.00$	$1.04$
4	$$ 169.86$	$$ 46.79$	$1.04$
5	$$ 216.65$	$$ 48.67$	$1.04$
6	$$ 265.32$

Since the ratio of the change in compound interest is constant, then compound interest represents exponential growth.

Mara purchases a Guaranteed Investment Certificate (GIC)

Series of stacks of money increasing in size with a squiggly arrow indicating financial growth over time. — Press here for long description(Open in new window)

Money grows more rapidly when interest is compounded because it grows exponentially. Here’s an exercise involving compound interest.

Mara purchases a $$ 1,500$ GIC that earns $6.25 %$ interest each year for eight years.

Determine the total interest earned at the end of each year using simple interest. Organize your calculations using the provided table.

Year	Principal (P)	Simple interest (I)	Accumulated simple interest ($)
$1$	$1,500$	$1,500 \times 0.0625 = 93.75$	$93.75$
$2$	$1,500$	$1,500 \times 0.0625 = 93.75$	$187.50$
$3$	$1,500$	$1,500 \times 0.0625 = 93.75$	$281.25$
$4$	$1,500$	$1,500 \times 0.0625 = 93.75$	$375.00$
$5$	$1,500$	$1,500 \times 0.0625 = 93.75$	$468.75$
$6$	$1,500$	$1,500 \times 0.0625 = 93.75$	$562.50$
$7$	$1,500$	$1,500 \times 0.0625 = 93.75$	$656.25$
$8$	$1,500$	$1,500 \times 0.0625 = 93.75$	$750.00$

Next, determine the total interest earned at the end of each year using compound interest. Organize your calculations using the following table.

Year	Principal for year ($)	Compound interest ($)	Accumulated compound interest ($)
$1$	$1,500$	$1,500 \times 0.0625 = 93.75$	$93.75$
$2$	$1,500 + 93.75 = 1,593.75$	$1,593.75 \times 0.0625 = 99.61$	$93.75 + 99.61 = 193.36$
$3$	$1,593.75 + 99.61 = 1,693.36$	$1,693.36 \times 0.0625 = 105.84$	$193.36 + 105.84 = 299.20$
$4$	$1,693.36 + 105.84 = 1,799.20$	$1,799.20 \times 0.0625 = 112.45$	$299.20 + 112.45 = 411.65$
$5$	$1,799.20 + 112.45 = 1,911.65$	$1,911.65 \times 0.0625 = 119.48$	$411.65 + 119.48 = 531.13$
$6$	$1,911.65 + 119.48 = 2,031.13$	$2,031.13 \times 0.0625 = 126.95$	$531.13 + 126.95 = 658.08$
$7$	$2,031.13 + 126.95 = 2,158.08$	$2,158.08 \times 0.0625 = 134.88$	$658.08 + 134.88 = 792.96$
$8$	$2,158.08 + 134.88 = 2,292.96$	$2,292.96 \times 0.0625 = 143.31$	$792.96 + 143.31 = 936.27$

Try it!

Determine the amount of the investment under simple interest.

Use the formula $A = P + I$ .

The amount for simple interest is:

$A = $ 1,500 + $ 750$

$= $ 2,250$

Determine the amount of the investment under compound interest.

The amount for compound interest is:

$A = $ 1,500 + $ 936.27$

$= $ 2,436.27$

How much extra interest is earned under compound interest?

From earlier, you know that the total amount of accumulated simple interest is $$ 750$ and the total amount of accumulated compound interest is $$ 936.27$ .

Subtract these two amounts: $$ 936.27 - $ 750.00 = $ 186.27$

Therefore, $$ 186.27$ extra interest is earned under compound interest.

Solving compound interest problems

Here is a handy formula for solving compound interest problems when you want to know the total amount. With this formula, you do not need to add the interest earned to the principal because the formula does that step for you, automatically.

The formula of an investment when the interest is compounded is:

$A = P (1 + i)^{n}$ where:

$A$ is the amount.
$P$ is the principal.
$i$ is the compounded interest rate as a decimal.
$n$ is the number of compounding periods. For interest compounded annually, this is the number of years.

In the following examples, you will use the annual compounded interest formula and the Time Value Money (TVM) Solver.

Borrowing money to buy a laptop

A young professional sits at a laptop during the day.

To buy a new laptop for a home business, Mara borrows $$ 1,600$ at an interest rate of $3.5 %$ per annum compounded annually. Mara plans to pay back the loan in three years.

Method 1: Using the formula

$A = P (1 + i)^{n}$

How much will Mara owe in 3 years?
How much interest will Mara owe after three years?

Use the formula $A = P (1 + i)^{n}$ .

$P = 1,600$ , $i = 3.5 % = 0.035$ , $n = 3$

$A = 1,600 (1 + 0.035)^{3}$

$A = 1,600 (1.035)^{3}$

$A = 1,773.95$

So, at the end of three years, Mara will owe $$ 1,773.95$ .

How much interest will Mara pay for the loan?

The interest is the amount paid after three years less the money borrowed.

$$ 1,773.95 - $ 1,600 = $ 173.95$

Therefore, the interest for the loan is $$ 173.95$ .

Method 2: Using TVM solver

This type of problem can also be solved with a financial calculator. This method is particularly helpful when solving more difficult problems such as a payment deal in which the lender is misleading you about the actual interest rate. In your case, you’ll get to use the TVM solver.

Start

The TVM solver can help you calculate the various parameters related to compound interest. It’s based on the future value of TVM equation:

$F V = P V \times (1 + I)^{N \times p e r i o d s p e r y e a r}$

This is similar to the equation for compound interest.

$A = P (1 + i)^{n}$

This is not by accident!

Let’s examine each component of the TVM equation to find out how they relate to what you already know about the compound interest equation.

FV – Future value

The variable A (the total amount of an investment or debt load) is called the “future value” or FV, because the total amount is the value at the end of the term, after it has accumulated interest.

PV – Present value

P (the principal) is called “present value” or PV, because the principal is the amount of money at the beginning of an investment or loan, before any interest has accumulated.

I – Interest rate (annual)

The variable I stands for the annual interest rate. This is independent of the number of compounding periods. The interest rate percentage is entered directly into the TVM solver.

N – Number of years

The variable $N$ stands for the number of years interest is accumulating. This is the term of the investment or loan.

Periods per year

The “periods per year” refers to the number of compounding periods per year. The value you enter will depend on how the investment or loan is being compounded (yearly = 1, semi-annually = 2, quarterly = 4, monthly = 12, and so on).

Now, let’s take the TVM solver out for a test drive! The following is the same problem you saw previously. To buy a new laptop for a home business, Mara borrows $$ 1,600$ at an interest rate of $3.5 %$ per annum compounded annually. The plan is to pay back the loan in three years.

How much will Mara owe after three years?
How much interest will Mara pay for the loan?

Try it!

How much will Mara owe after three years?

To find this number, you enter the following values into the TVM Solver:

Present value: $1,600$
Annual interest: $3.5$
Number of years: $3$
Periods per year: $1$ (recall that the interest is compounded annually)

Then press the FV button to find the Future Value. The future value (FV) is $$ 1,773.95$ .

How much interest will Mara pay for the loan?

The interest is the amount paid after three years less the money borrowed:

$$ 1,773.95 - $ 1,600 = $ 173.95$

Therefore, the interest for the loan is $$ 173.95$ .

Mara dreams of university

Someone in a graduation outfit is putting coins in a piggy bank.

In some problems, you want to know how much you should invest today to have a certain amount of money in the future.

In the following situation, Mara wants to know how much to invest today to cover tuition fees for her first year of university.

To find the principal or present value of an investment, you can rewrite the formula $A = P (1 + i)^{n}$ so $P$ is isolated. Try to rearrange the formula for $P$ , if you can.

$A = P (1 + i)^{n}$

$\frac{A}{(1 + i)^{n}} = \frac{P (1 + i)^{n}}{(1 + i)^{n}}$

$\frac{A}{(1 + i)^{n}} = P$

$P = \frac{A}{(1 + i)^{n}}$

In five years, Mara will need $$ 8,000$ to cover tuition fees for first year of university. How much money should Mara invest today, at a rate of $7.25 %$ compounded annually, so that there is enough money to pay tuition fees?

Use the formula $P = \frac{A}{(1 + i)^{n}}$

Substitute $A = 8,000$
Substitute $I = 7.25 % = 0.0725$
Substitute $n = 5$

$P = \frac{8,000}{(1 + 0.0725)^{5}}$

$= \frac{8,000}{(1.0725)^{5}}$

$= 5,637.72$

Therefore, Mara should invest $$ 5,637.72$ at $7.25 %$ today to have $$ 8,000$ in five years.

Different compounding periods

For some investments or loans, the interest is compounded more than once a year. For instance, if you have money invested in a daily interest account, then the interest earned is compounded daily, or 365 times a year. The following table is a summary of possible compounding periods.

Frequency of compounding	Number of times interest is added during a year
Annually	1 (every year)
Semi-annually	2 (every six months)
Quarterly	4 (every three months)
Monthly	12 (every month)
Daily	365 (every day)

For these questions, the formula $A = P (1 + i)^{n}$ can still be used to determine amounts when the interest is compounded more than once a year; however, the interest rate must be divided by the number of compounding periods in order to find the amount of interest per compounding period.

Three different compounding periods for Aram’s investment

Aram now considers the option of three different compounding periods for his $$ 1,000$ investment at 4% interest over 6 years: semi-annually, quarterly and monthly.

Semi-annually

The principal is $$ 1,000$ . The annual rate is $4 %$ . The time is six years.

When the interest rate is compounded semi-annually, it is added twice a year.

The semi-annual rate is $\frac{1}{2}$ of $4 % = 2 % = 0.02$ .

In six years, there are $6 \times 2$ , or $12$ compounding periods.

To find the amount, use $A = P (1 + i)^{n}$ .

Substitute $P = 1,000$ , $i = 0.02$ , and $n = 12$ .

$A = 1,000 (1 + 0.02)^{12}$

$= 1,000 (1.02)^{12}$

$= 1,268.24$

The amount after six years compounded semi-annually is $$ 1,268.24$ .

Quarterly

The principal is $$ 1,000$ . The annual rate is $4 %$ . The time is six years.

When the interest rate is compounded quarterly, it is added four times a year.

The quarterly rate is $\frac{1}{4}$ of $4 % = 1 % = 0.01$ .

In six years, there are $6 \times 4$ , or $24$ compounding periods.

To find the amount, use $A = P (1 + i)^{n}$ .

Substitute $P = 1,000$ , $i = 0.01$ , and $n = 24$ .

$A = 1,000 (1 + 0.01)^{24}$

$= 1,000 (1.01)^{24}$

$= 1,269.73$

The amount after six years compounded quarterly is $$ 1,269.73$ .

Try it!

Try determining the final amount when compounded monthly and compare your answer with the suggested solution.

What is the amount after six years, compounded monthly?

The principal is $$ 1,000$ . The annual rate is $4 %$ . The time is six years.

When the interest rate is compounded monthly, it is added 12 times a year.

The monthly rate is $\frac{1}{12}$ of $4 %$ $= 0.3344 % = 0.003333$ .

In six years, there are $6 \times 12$ , or $72$ compounding periods.

To find the amount, use $A = P (1 + i)^{n}$ .

Substitute $P = 1,000$ , $i = 0.003333$ , and $n = 72$ .

$A = 1,000 (1 + 0.003333)^{72}$

$= 1,000 (1.003333)^{72}$

$= 1,270.74$

The amount after six years compounded monthly is $$ 1,270.74$ .

Which compounding period should Aram select? Justify your choice.

Since the largest amount occurs when the interest is compounded monthly, Aram should choose the monthly option to invest his $$ 1,000$ .

Try it!

Aram is investing $$ 5,000$ in an account that earns $6.95 %$ compounded monthly for seven years. What is the amount in the account at the end of this time?

To find this number, you enter the following values into the TVM Solver:

Present value: $5,000$
Annual interest: $6.95$
Number of years: $7$
Periods per year: $12$ (recall that the interest is compounded monthly)

The future value (FV) is $$ 8,121.66$ .

Aram plans for a big anniversary

Aram attends a major sporting arena and wins the 50/50 charity drawing happening that night. As a result, $$ 10,000$ goes to helping children with special needs sporting groups and $$ 10,000$ is the prize that was awarded to Aram, who really wants a motorcycle.

Two young people taking a selfie by a cliff near their motorcycle.

Aram now has $$ 10,000$ to invest for five years. At the end of five years, Aram would like to have $$ 16,000$ to buy a new motorcycle. What rate of interest, to the nearest hundredth of a percent, compounded quarterly, does Aram need to achieve the goal?

To find this rate of interest, you enter the following values into the TVM Solver:

Present value: $10,000$
Future value: $16,000$
Number of years: $5$
Periods per year: $4$ (recall that the interest is compounded quarterly)

Aram needs a rate of interest of $9.51 %$ to achieve his goal.

Simple and compound interest practice problems

International Express (a non-existent credit card company) charges an annual interest rate of $18 %$ on unpaid account balances. Calculate the amount of interest that the company would charge when a balance of $$ 2,200$ is paid $43$ days late.

The principal is $P = 2,200$ .

The interest rate is $18 %$ . As a decimal, $r = 0.18$ .

The time is $43$ days. Since there are $365$ days in a year, then in years,

$t = \frac{43}{365}$

Substitute these values into the formula $I = P r t$ and solve for $I$ .

$I = 2,200 \times 0.18 \times \frac{43}{365}$

$I = 46.65$

The company would charge $$ 46.65$ in interest.

Graphic depiction of a past due bill notification.

Aram borrowed $$ 1,350$ for eight months to buy a new car and paid $$ 38.25$ in interest on the loan. What was the annual interest rate of Aram’s loan?

Determine the interest rate, $r$ .

The principal is $P = 1,350$ .

The time is $t = 8$ months. Since there are $12$ months in a year, then in years $t = \frac{8}{12} = \frac{2}{3}$

The interest is $I = 38.25$ .

Substitute these values into the formula $I = P r t$ and solve for $r$ .

$38.25 = 1,350 \times r \times \frac{2}{3}$

$38.25 = 900 r$

$r = \frac{38.25}{900}$

$r = 0.0425$

$r = 4.25 %$

The annual rate of interest of Aram’s loan was $4.25 %$ .

In your notebook, describe the difference between simple and compound interest. What type of growth does each represent? Explain.

Simple interest is paid annually on the principal and is not reinvested. Compound interest is earned annually but is reinvested with the principal, so interest is earned on interest. Simple interest accumulates at the same rate and represents linear growth. Compound interest accumulates at a rate that has a constant ratio and represents exponential growth.

Aram purchases a $$ 2,500$ Canadian Savings Bond that earns $7.5 %$ interest each year for five years.

If you would like to learn about Canadian Saving Bonds, use your favourite internet search engine and enter the terms “Canada Savings Bonds” and “interest earned”. 2021 is a unique year to be learning about these bonds!

Determine the total interest earned at the end of each year under simple interest. Organize your calculations using the following table.

Year	Principal ($)	Simple interest ($)	Accumulated simple interest ($)
$1$	$2,500$	$2,500 \times 0.075 = 187.50$	$187.50$
$2$	$2,500$	$2,500 \times 0.075 = 187.50$	$375.00$
$3$	$2,500$	$2,500 \times 0.075 = 187.50$	$562.50$
$4$	$2,500$	$2,500 \times 0.075 = 187.50$	$750.00$
$5$	$2,500$	$2,500 \times 0.075 = 187.50$	$937.50$

Determine the total interest earned at the end of each year under compound interest. Organize your calculations using the following table.

Year	Principal for Year ($)	Compound Interest ($)	Accumulated Compound Interest ($)
$1$	$2,500$	$2,500 \times 0.075 = 187.50$	$187.50$
$2$	$2,500 + 187.50 = 2,687.50$	$2,687.50 \times 0.075 = 201.56$	$187.50 + 201.56 = 389.06$
$3$	$2,687.50 + 201.56 = 2,889.06$	$2,889.06 \times 0.075 = 216.68$	$389.06 + 216.68 = 605.74$
$4$	$2,889.06 + 216.68 = 3,105.74$	$3,105.74 \times 0.075 = 232.93$	$605.74 + 232.93 = 838.67$
$5$	$3,105.74 + 232.93 = 3,338.67$	$3,338.67 \times 0.075 = 250.40$	$838.67 + 250.40 = 1,089.07$

How much extra interest is earned under compound interest?

From earlier, you calculated the total amount of accumulated simple interest as $$ 937.50$ . You also calculated the total amount of accumulated compound interest as $$ 1,089.07$ .

Subtract these two amounts: $$ 1,089.07 - $ 937.50 = $ 151.57$ .

Therefore, the $$ 151.57$ extra interest is earned under compound interest.

Determine the amount of the investment under simple interest.

Use the formula $A = P + i$ .

The amount for simple interest is:

$A = $ 2,500 + $ 937.50$

$A = $ 3,437.50$

Determine the amount of the investment under compound interest.

Use the formula $A = P + i$ .

The amount for compound interest is:

$A = $ 2,500 + 1,089.07$

$A = $ 3,589.07$

Determine the change in interest for simple interest and compound interest. Organize your calculations using the following tables.

Table A – Simple interest: $$ 2,500$ invested at $7.5 %$

Year	Total interest	Change in interest
$1$	$$ 187.50$	$$ 187.50$
$2$	$$ 375.00$	$$ 187.50$
$3$	$$ 562.50$	$$ 187.50$
$4$	$$ 750.00$	$$ 187.50$
$5$	$$ 937.50$	-

Table B – Compound interest: $$ 2,500$ invested at $7.5 %$

Year	Total interest	Change in interest
$1$	$$ 187.50$	$$ 201.56$
$2$	$$ 389.06$	$$ 216.68$
$3$	$$ 605.74$	$$ 232.93$
$4$	$$ 838.67$	$$ 250.40$
$5$	$$ 1,089.07$

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Determine the ratio of the change in compound interest.

To find the ratio of the change in compound interest, use the tables above and divide consecutive values in column 3 as shown here:

$216.68 / 201.56 = 1.075$ ,

$232.93 / 216.68 = 1.075$ ,

$250.40 / 232.93 = 1.075$

Describe the type of growth represented by each table. Justify your answer.

Using a pencil and paper, plot Year against Total interest for simple and compound interest on the same grid.

Aram invested $$ 6,800$ at $5.2 %$ compounded semi-annually.

Determine the amount of the investment after four years.

When the interest is compounded semi-annually, it is added twice a year.

The semi-annual rate is $12$ of $5.2 % = 2.6 % = 0.026$ .

In 4 years, there are $4 \times 2$ , or $8$ compounding periods.

To find the amount, use $A = P (1 + i)^{n}$ .

Substitute $P = 6,800$ , $i = 0.026$ , and $n = 8$ .

$A = 6,800 (1 + 0.026)^{8}$

$A = 6,800 (1.026)^{8}$

$A = 8,350.03$

The amount after four years is $$ 8,350.03$ .

What was the amount of the investment after seven years?

When the interest is compounded semi-annually, it is added twice a year.

The semi-annual rate is $12$ of $5.2 % = 2.6 % = 0.026$ .

In 7 years, there are $7 \times 2$ , or $14$ compounding periods.

To find the amount, use $A = P (1 + i)^{n}$ .

Substitute $P = 6,800$ , $i = 0.026$ , and $n = 14$ .

$A = 6,800 (1 + 0.026)^{14}$

$A = 6,800 (1.026)^{14}$

$A = 9,740.29$

The amount after seven years is $$ 9,740.29$ .

How much interest was earned on the investment between the fourth and the seventh year? Explain.

To determine the interest that was earned on the investment between the fourth year and the seventh year, subtract:

$$ 9,740.29 - $ 8,350.03 = $ 1,390.26$

Therefore, $$ 1,390.26$ was earned in interest between the fourth and seventh year.

How much money should Mara invest now at a rate of $6.3 %$ compounded monthly, to have $$ 10,000$ in five years?

You know that the future value of the investment is $$ 10,000$ .

Determine the amount that Mara must invest today (the present value).

Use the formula $P = \frac{A}{{(1 + i)}^{n}}$ .

Substitute $A = 10,000$ .

The monthly rate is $1 / 12$ of $6.3 % = 0.53 % = 0.0053$ .

In 5 years, there are $5 \times 12$ , or $60$ compounding periods.

$P = \frac{10,000}{{(1 + 0.0053)}^{60}}$

$P = \frac{10,000}{{(1.0053)}^{60}}$

$P = 7,282.14$

Therefore, Mara should invest $$ 7,282.14$ at $6.3 %$ today to have $$ 10,000$ in five years.

Aram has $$ 5,000$ to invest in a GIC that earns $6 %$ per year, compounded daily. How long will it take for Aram’s investment to double? Verify your answer using the TVM Solver.

The principal is $P = 5,000$ . Since Aram wants to double the investment, then $A = 10,000$ .

The daily interest rate is $\frac{1}{365}$ of $6 % = 0.0164 % = 0.000164$ .

Let $x$ represent the number of years it takes for the investment to double in value.

In $x$ years, there are $x \times 365$ , or $365 x$ compounding periods.

To find the number of years, use $A = P (1 + i)^{n}$ .

Substitute $P = 5,000$ , $A = 10,000$ , $i = 0.000164$ , and $n = 365 x$ .

$10,000 = 5,000 (1 + 0.000164)^{365 x}$

$2 = (1.000164)^{365 x}$

Use trial and error to find the value of $365 x$ .

Since $(1.000164)^{4,230} \approx 2.001$ which is very close to $2$ , then $365 x \approx 4,230$ and so $x \approx 11.6$ .

Convert $11.6$ to years and months.

$11.6$ is $11$ years and $0.6 \times 12 = 7$ months.

Therefore, it takes approximately $11$ years and $7$ months for Aram’s investment to double.

Connections

Aram compares saving now vs. saving later

Explore the following video to better understand savings over time.

If you start at age 20, deposit $$ 1,000$ per year at $6 %$ compounded annually, you will have $$ 226, 508$ at age 65. If you start at age 50, deposit $$ 1,000$ per year at $6 %$ compounded annually, you will have $$ 25, 672.55$ at age 65.

Obviously, the first option is more profitable, so the sooner you can start saving, the better.

Rate your understanding on a scale of five to one.

Five means “I have a thorough understanding.” One means “I am confused.”

Are you able to:

Reflection questions:

Now that you’ve completed this activity relating to simple and compound interest, contemplate the following questions to help you extend your understanding and consider how this knowledge and set of skills may relate to and impact your own life and the lives of others.

Why is it important to you to learn about some of the financial applications of mathematical functions?
How might you apply what you’ve learned in this activity to impact your own: education, home-life, work-life and community?
Consider some ways that you could use information and skills from this activity to help improve the lives of others both locally and globally.
Reflect on how an understanding of simple and complex interest could help you achieve potential future career goals.

Join the discussion

Share your thoughts and answers to the reflection questions with your classmates to show your learning.

Press the “Join The Discussion” button when you’re ready to engage.

Statement	1	2	3	4	5
I understand what simple and compound interest is.
I can calculate interest.

Pre-assessment review:

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Summer savings with simple interest

What is simple interest?

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What is the formula for simple interest?

Notebook

Investing $2,000 prize using simple interest

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Borrowing from a bank

The death of Mara’s great-aunt

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Summer savings with compound interest

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Table A: $1,000 invested at 4% simple interest.

Table B:$1,000 invested at 4% compound interest.

Notebook

Mara purchases a Guaranteed Investment Certificate (GIC)

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Solving compound interest problems

Borrowing money to buy a laptop

Method 1: Using the formula

Method 2: Using TVM solver

FV – Future value

PV – Present value

I – Interest rate (annual)

N – Number of years

Periods per year

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Mara dreams of university

Different compounding periods

Three different compounding periods for Aram’s investment

Semi-annually

Quarterly

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Aram uses a TVM solver to dream big!

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Aram plans for a big anniversary

Simple and compound interest practice problems

Table A – Simple interest: $2,500 invested at 7.5%

Table B – Compound interest: $2,500 invested at 7.5%

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Connections

Aram compares saving now vs. saving later

Reflection questions:

Join the discussion

Table A: $$ 1,000$ invested at $4 %$ simple interest.

Table B: $$ 1,000$ invested at $4 %$ compound interest.

Table A – Simple interest: $$ 2,500$ invested at $7.5 %$

Table B – Compound interest: $$ 2,500$ invested at $7.5 %$