B2.3 Compare simple and compound interest | Ontario Grade 11 Mathematics

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

MEL3E study topic: B2.3

Suppose you can leave money in a savings account or borrow money for a purchase. Interest is money paid for the use of money. When comparing options, it helps to know whether interest is calculated only on the starting amount or on the growing balance. This lesson compares simple and compound interest using step-by-step calculations. The examples use an annual rate and one calculation at the end of each year, so the comparison stays clear.

What you will learn

Explain how simple interest and compound interest are calculated over time.
Use repeated calculations or a calculator to find a balance.
Compare two interest options when the starting amount, rate, and time are the same.

Start with the same basic terms

The starting amount is called the principal. The interest rate tells you what part of an amount is added as interest for a set time, such as one year. A balance is the amount in the account or still owed at a particular time. The number of years matters because interest can be added more than once.

For a fair comparison, keep the principal, rate, and length of time the same. Also check how often interest is calculated. In this lesson, interest is calculated once at the end of each year. If an account uses a different schedule, follow the schedule shown in its information.

Principal means the starting amount.
A rate of 4% means CAD 4 of interest for each CAD 100 for one year.
Compare options using the same principal, rate, time, and calculation schedule.

Simple interest: use the starting amount each year

With simple interest, each year's interest is based on the original principal. The interest added each year is the same when the rate and principal stay the same. To find a later balance, add that year's interest to the previous balance.

For example, 4% of CAD 500 is CAD 20. Under simple interest, another year also adds CAD 20, not a percentage of the larger balance. A calculator can find the yearly interest by multiplying the principal by the rate written as a decimal. For 4%, enter 0.04.

A spreadsheet can make the repeated steps easier to check. Put the year in one column, the balance at the start of the year in another, and the interest added in a third. For simple interest, calculate the interest from the original principal in every row.

\text{yearly interest} = \text{principal} \times \text{rate as a decimal}

Simple interest is calculated from the original principal each year.
When the principal and rate do not change, the interest added each year stays the same.
Keep the previous balance visible so each new year's balance is clear.

Compound interest: use the current balance

With compound interest, the interest for a year is calculated from the balance at the start of that year. After the interest is added, the new balance becomes the starting balance for the next year. This means later interest can include interest added in earlier years.

This added interest on earlier interest is sometimes called interest on interest. The important comparison is practical: when the rate and starting amount are positive, compound interest usually adds more over time than simple interest at the same rate, because its yearly calculation uses a growing balance.

You do not need a shortcut formula to make this comparison. Repeat the same steps for each year: find the interest from the appropriate amount, add it, and record the new balance. A calculator or spreadsheet can help, but check that the amount used for interest is the correct one.

\text{new balance}

=

\text{starting balance}

+ interest for the period

Compound interest is calculated from the current balance at the start of each period.
Each new balance becomes the amount used for the next period.
The difference between the methods can grow over time.

Make a fair comparison

Compare the final balances after the same number of years. For an investment, a higher final balance means more money accumulated. For a loan, a higher amount owed means more money to repay, if all other terms are the same.

The calculation method is only one part of a real account or loan. Fees, payment dates, and the schedule for adding interest can also matter. This lesson isolates the interest method so you can compare simple and compound interest without mixing in other terms.

Before deciding what a result means, label the situation. A larger balance is helpful for savings but may be costly for borrowing. Check your arithmetic by estimating: the result should be close to the starting amount plus the interest added over the years.

Keep the starting amount, rate, and time the same in the comparison.
A larger savings balance is generally better; a larger loan balance is generally worse.
State whether the balance is money saved or money owed.

How the yearly calculation differs

Method	Amount used to find interest	What happens next
Simple interest	Original principal	Add the same interest amount each year
Compound interest	Balance at the start of the year	Add interest, then use the new balance next year

Worked example

Compare savings over three years

A student deposits CAD 500. One account pays 4% simple interest per year. Another pays 4% compound interest once per year. Find and compare the balances after three years, assuming no deposits or withdrawals.

Find the yearly simple interest
For simple interest, use the original CAD 500 each year. Convert 4% to 0.04, then multiply. This gives the same interest amount in each of the three years.
$500 \times 0.04 = 20$
Build the simple-interest balance
Add CAD 20 to the balance at the end of each year. Repeating the addition gives the balance after year three.
$500 \rightarrow 520 \rightarrow 540 \rightarrow 560$
Calculate the first compound year
For compound interest, calculate the first year's interest from CAD 500. Add it to get the balance used at the start of year two.
$500 \times 0.04 = 20,\quad 500 + 20 = 520$
Continue compounding
Use CAD 520 for year two, then use the resulting balance for year three. Round each year's balance to the nearest cent.
$520 \times 0.04 = 20.80,\quad 520 + 20.80 = 540.80$
Finish the comparison
Calculate 4% of CAD 540.80 for year three, then compare the final compound balance with the simple-interest balance.
$540.80 \times 0.04 = 21.632,\quad 540.80 + 21.63 = 562.43$

Answer: After three years, the simple-interest balance is CAD 560.00 and the compound-interest balance is CAD 562.43. The compound account has CAD 2.43 more.

Check: The compound balance is slightly higher because year two and year three interest are calculated from balances that already include earlier interest.

Worked example

Compare two borrowing costs

A person borrows CAD 1,200 for two years at 5% per year. Compare the amount owed after two years if interest is simple or compounded once per year. Assume no payments or fees.

Calculate simple interest each year
Simple interest uses the original CAD 1,200 for both years. Find 5% of that principal. The same amount is added in each year.
$1{,}200 \times 0.05 = 60$
Find the simple-interest amount owed
Add CAD 60 for each of two years to the amount borrowed. This gives the total owed under simple interest.
$1{,}200 + 60 + 60 = 1{,}320$
Calculate the first compound year
For the first year, 5% is calculated from CAD 1,200. Add that interest to find the amount used for the next year's calculation.
$1{,}200 \times 0.05 = 60,\quad 1{,}200 + 60 = 1{,}260$
Calculate the second compound year
The second year's interest is based on CAD 1,260, not the original CAD 1,200. Add the interest to find the final amount owed.
$1{,}260 \times 0.05 = 63,\quad 1{,}260 + 63 = 1{,}323$

Answer: The simple-interest amount owed is CAD 1,320. The compound-interest amount owed is CAD 1,323. With the same borrowing terms, compound interest costs CAD 3 more over these two years.

Check: The compound amount is higher because the second year's interest includes the CAD 60 added during the first year.

Common mistakes and how to avoid them

Using the growing balance to calculate simple interest.

Correction: Use the original principal for simple interest in every year.

Using the original principal for every year of compound interest.

Correction: For compound interest, use the balance at the start of each year.

Comparing accounts with different time periods or rates.

Correction: Match the principal, rate, time, and interest schedule before comparing.

Thinking a larger balance is always better.

Correction: A larger balance helps when saving, but it means more money owed when borrowing.

Lesson summary

Simple interest is calculated from the original principal each year.
Compound interest is calculated from the current balance at the start of each year.
Use repeated calculations, a calculator, or a spreadsheet to find each new balance.
Compare final balances under matching terms, and interpret the result based on whether the money is saved or borrowed.

Check your understanding

Question 1

CAD 800 earns 3% interest for one year. What is the interest for that year?

CAD 24
CAD 3
CAD 80
correctIndex

Show answer and explanation

CAD 24

Convert 3% to 0.03 and multiply: CAD 800 times 0.03 is CAD 24.

Question 2

A CAD 1,000 balance earns 5% compound interest once per year. After one year, what starting balance is used to calculate the second year's interest?

CAD 1,000
CAD 1,050
CAD 50
correctIndex

Show answer and explanation

CAD 1,050

The first year's interest is CAD 50, so the balance becomes CAD 1,050. Compound interest uses this new balance for the second year.

Key terms

Interest: Money paid for the use of money.
Principal: The starting amount saved or borrowed.
Balance: The amount in an account or the amount still owed at a particular time.
Simple interest: Interest calculated from the original principal each period.
Compound interest: Interest calculated from the current balance, which includes interest added earlier.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B2.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B2.3 · Compare simple and compound interest