B3.1 Compare simple and compound interest | Ontario Grade 11 Mathematics

Ontario Grade 11 Mathematics

Exponential Functions

How money grows two different ways when you save or borrow

Imagine two friends each put CAD 1000 into a savings account. One account pays simple interest. The other pays compound interest. After one year, both friends have almost the same amount. After ten years, one friend has noticeably more money than the other. Why does this happen? The answer lies in how interest is calculated each year. In this lesson, you will learn what simple interest and compound interest mean, how to calculate each one, and how to compare them using numbers, tables, and graphs. This connects to your earlier work with linear and exponential relationships, since simple interest grows in a straight line and compound interest grows along a curve."

What you will learn

Explain the difference between simple interest and compound interest in plain language.
Calculate simple interest using the formula $I = Prt$ .
Calculate compound interest using repeated growth and the formula $A = P(1+i)^n$ .
Compare the total amounts and growth patterns produced by simple versus compound interest over time.
Choose the correct interest model for a given real-life savings or loan situation.

Prerequisite Bridge: Percent, Rate, and Growth Patterns

Before comparing the two types of interest, let's review two ideas from earlier grades. First, a percent is a way to express a part out of 100. For example, 5 percent means 5 out of every 100, or the decimal

0.05

. Second, you already know two kinds of patterns from Grade 9 and 10: a linear pattern increases by the same fixed amount each step, and an exponential pattern increases by the same fixed factor (multiplier) each step. Interest calculations use exactly these two patterns. Simple interest follows a linear pattern. Compound interest follows an exponential pattern.

In money problems, we use some key words. Principal means the original amount of money invested or borrowed. Interest rate is the percent used to calculate how much interest is earned or owed, usually written as an annual (yearly) rate. Time is usually measured in years for these calculations.

A percent rate can be written as a decimal for calculations, such as 5 percent equals $0.05$ .
Linear patterns add the same amount every step; exponential patterns multiply by the same factor every step.
Principal is the starting amount of money before any interest is added.

Simple Interest: The Same Amount Added Every Year

Simple interest is calculated only on the original principal, every single year. The amount of interest earned never changes from year to year, because it is always based on the same starting principal, not on any interest already earned.

The formula for simple interest is

I = Prt

, where

I

is the interest earned,

P

is the principal,

r

is the annual interest rate written as a decimal, and

t

is the time in years. Once you find

I

, the total amount of money after

t

years, called the future value or total amount

A

, is found by adding the interest to the principal:

A = P + I

.

Because the same interest amount is added every year, simple interest produces a linear growth pattern. If you plotted the total amount against time on a graph, the points would form a straight line.

I = Prt

Simple interest is calculated only on the original principal, not on previously earned interest.
The formula $I = Prt$ gives the interest earned over $t$ years.
Total amount is found using $A = P + I$ .
Simple interest grows in a straight-line (linear) pattern over time.

Compound Interest: Interest Earned on Interest

Compound interest is calculated on the principal plus any interest that has already been added. This means each year's interest amount grows, because it is calculated on a larger and larger balance.

To understand this without complex notation, think of it as repeated multiplication. Suppose the annual interest rate is

r

, written as a decimal. Each year, the balance is multiplied by the growth factor

(1 + r)

. After one year, the new balance is

P(1+r)

. After two years, that new balance is multiplied by

(1+r)

again, giving

P(1+r)(1+r)

, which is

P(1+r)^2

. After

n

years, the pattern continues, giving the compound interest formula for total amount:

A = P(1+i)^n

, where

i

is the interest rate per compounding period written as a decimal, and

n

is the number of compounding periods.

When interest compounds once per year,

i

equals the annual rate and

n

equals the number of years. Because each year's interest is calculated on a growing balance, compound interest produces an exponential growth pattern. On a graph, the points curve upward more steeply as time passes, unlike the straight line for simple interest.

To find just the interest earned with compounding, subtract the principal from the total amount:

I = A - P

.

A = P(1+i)^n

Compound interest is calculated on the principal plus previously earned interest.
The growth factor per period is $(1+i)$ , applied repeatedly through multiplication.
The formula $A = P(1+i)^n$ gives the total amount after $n$ compounding periods.
Compound interest grows in a curved (exponential) pattern, increasing faster over time than simple interest.

Comparing the Two Models Side by Side

The clearest way to see the difference is to track the same principal under both models over several years. Suppose CAD 1000 is invested at 5 percent per year, once under simple interest and once under compound interest (compounded annually). In year 1, both amounts are almost identical, since compounding has not had time to build up. But by year 5, the compound interest total is noticeably higher, because the interest has been earning interest on itself.

A table makes this comparison easy to read, and it shows numerically what the linear versus exponential graph shapes mean in terms of real dollars. Notice that the simple interest column increases by exactly the same amount, CAD 50, every year. The compound interest column increases by a growing amount each year.

Both models start at the same principal, so early-year totals look similar.
Simple interest increases by a constant dollar amount every year.
Compound interest increases by a larger dollar amount each successive year.
Over long time periods, compound interest always produces a greater total amount than simple interest at the same rate.

CAD 1000 at 5 Percent: Simple vs Compound Interest by Year

Year	Simple Interest Total (CAD)	Compound Interest Total (CAD)	Difference (CAD)
1	1050.00	1050.00	0.00
2	1100.00	1102.50	2.50
3	1150.00	1157.63	7.63
4	1200.00	1215.51	15.51
5	1250.00	1276.28	26.28

Worked example

Comparing CAD 1000 at 5 Percent Over 5 Years

Wendy invests CAD 1000 for 5 years at an annual rate of 5 percent. Find the total amount she has after 5 years under simple interest, and under compound interest (compounded annually). Then compare the two results.

Identify the known values
The principal is CAD 1000, the annual rate is 5 percent, which as a decimal is $0.05$ , and the time is 5 years. These values will be used in both formulas.
$P = 1000,\ r = i = 0.05,\ t = n = 5$
Calculate simple interest
Use the simple interest formula, multiplying the principal by the rate and by the number of years. This gives the total interest earned over the whole 5 years, calculated only on the original CAD 1000.
$I = 1000 × 0.05 × 5 = 250$
Find the simple interest total amount
Add the interest to the original principal to find the total amount after 5 years under simple interest.
$A_{simple} = 1000 + 250 = 1250$
Calculate the compound interest total amount
Use the compound interest formula. The balance is multiplied by the growth factor $(1+0.05)$ once for each of the 5 years, since interest compounds annually.
$A_{compound} = 1000(1.05)^5$
Evaluate the compound total
Calculate $(1.05)^5$ first, which equals approximately $1.27628$ . Multiply this by the principal to get the total amount, then round to the nearest cent.
$A_{compound} \approx 1000 × 1.27628 = 1276.28$
Compare the two results
Subtract the simple interest total from the compound interest total to see how much more Wendy earns with compounding over these 5 years.
$1276.28 - 1250.00 = 26.28$

Answer: After 5 years, simple interest gives a total of CAD 1250.00, while compound interest gives a total of about CAD 1276.28. Compounding earns Wendy about CAD 26.28 more over the 5 years, even though both start with the same principal and rate.

Check: Check the simple interest by confirming CAD 50 is earned each year for 5 years, which is

50 \times 5 = 250

, matching the interest found. Check the compound total by recalculating

1.05^5

using repeated multiplication:

1.05 \times 1.05 \times 1.05 \times 1.05 \times 1.05 \approx 1.27628

, which matches the value used.

Common mistakes and how to avoid them

Using the growing balance in the simple interest formula, for example calculating year-two simple interest based on year one's total instead of the original principal.

Correction: Always calculate simple interest using only the original principal

P

, never a previous year's total. The formula

I = Prt

always uses the same

P

for every year.

Forgetting to convert the percent rate into a decimal before using it in either formula.

Correction: Always divide a percent by 100 before calculating, so 5 percent becomes

0.05

. Using 5 instead of

0.05

will make the answer far too large.

Adding the growth factor incorrectly, such as writing

A = P(1+i^n)

instead of

A = P(1+i)^n

.

Correction: The exponent applies to the entire growth factor

(1+i)

, not just to

i

alone. Compute

(1+i)

first, then raise that whole result to the power

n

.

Assuming compound interest always gives a much larger total than simple interest even after only one compounding period.

Correction: In the first period, simple and compound interest give the same result, since no interest has yet earned additional interest. The difference only grows in later years, as shown in the comparison table.

Lesson summary

Simple interest is earned only on the original principal, following the formula $I = Prt$ .
Compound interest is earned on the principal plus previously added interest, following the formula $A = P(1+i)^n$ .
Simple interest produces a linear (straight-line) growth pattern, while compound interest produces an exponential (curved) growth pattern.
Over short time periods the two models give similar totals, but compound interest grows increasingly larger than simple interest as time passes.
Comparing totals in a table or graph helps show clearly how the constant additions of simple interest differ from the increasing additions of compound interest.

Check your understanding

Question 1

CAD 2000 is invested at 4 percent per year for 3 years under simple interest. What is the total amount after 3 years?

CAD 2080
CAD 2240
CAD 2000
CAD 2400

Show answer and explanation

CAD 2240

Using

I = Prt = 2000 \times 0.04 \times 3 = 240

, then

A = 2000 + 240 = 2240

. So the total amount is CAD 2240.

Question 2

Which statement correctly describes the difference between simple and compound interest?

Simple interest is calculated on the growing balance, while compound interest is calculated only on the principal.
Simple interest is calculated only on the original principal, while compound interest is calculated on the principal plus previously earned interest.
Simple interest and compound interest always give exactly the same total amount for any number of years.
Compound interest only applies to loans, while simple interest only applies to savings accounts.

Show answer and explanation

Simple interest is calculated only on the original principal, while compound interest is calculated on the principal plus previously earned interest.

Simple interest always uses the same original principal for every calculation. Compound interest recalculates interest using the previous balance, which includes earlier interest.

Question 3

A graph shows the total amount of an investment over time. The plotted points form a curve that rises more and more steeply as time increases. Which type of interest does this graph most likely represent?

Simple interest, because the balance increases by a constant amount
Compound interest, because the growth factor is applied repeatedly, creating exponential growth
Neither, because interest graphs are always straight lines
Simple interest, because percentages always create curves

Show answer and explanation

Compound interest, because the growth factor is applied repeatedly, creating exponential growth

A curve that rises increasingly steeply over time is the visual signature of exponential growth, which matches compound interest, where each period's interest is calculated on a larger balance than the period before.

Question 4

CAD 500 is invested at 6 percent per year, compounded annually, for 2 years. What is the total amount after 2 years, rounded to the nearest cent?

CAD 560.00
CAD 561.80
CAD 530.00
CAD 500.60

Show answer and explanation

CAD 561.80

Using

A = P(1+i)^n = 500(1.06)^2 = 500 \times 1.1236 = 561.80

. So the total amount after 2 years is CAD 561.80.

Key terms

Principal: The original amount of money invested or borrowed, before any interest is added.
Interest rate: The percent, usually written per year, used to calculate how much interest is earned or owed.
Simple interest: Interest calculated only on the original principal, giving the same dollar amount of interest every period.
Compound interest: Interest calculated on the principal plus any interest already earned, so each period's interest amount grows.
Future value (total amount): The amount of money you have after interest has been added to the principal, represented by $A$ .
Growth factor: The multiplier $(1+i)$ used to increase a balance by one compounding period under compound interest.
Linear pattern: A pattern that increases by the same fixed amount at every step, producing a straight-line graph.
Exponential pattern: A pattern that increases by the same fixed multiplying factor at every step, producing a curved graph.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B3.1 · Compare simple and compound interest

Ontario Grade 11 Mathematics

How money grows two different ways when you save or borrow

What you will learn

Prerequisite Bridge: Percent, Rate, and Growth Patterns

Simple Interest: The Same Amount Added Every Year

Compound Interest: Interest Earned on Interest

Comparing the Two Models Side by Side

CAD 1000 at 5 Percent: Simple vs Compound Interest by Year

Comparing CAD 1000 at 5 Percent Over 5 Years

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCF3M

About this lesson