B3.3 Connect compound interest with exponential growth

Ontario Grade 11 Mathematics

Exponential Functions

Connecting repeated percentage growth to an exponential model

A savings balance can grow because interest is added to it. With compound interest, later interest is calculated on the earlier interest as well as on the original deposit. This creates repeated percentage growth, the same pattern described by an exponential function. In this lesson, we will connect the money context to the table, equation, and graph of exponential growth.

What you will learn

Explain how compound interest produces repeated percentage growth.
Represent compound interest with a table, an equation, and a graph description.
Identify the initial amount, growth factor, and number of compounding periods in a model.
Use a compound-interest model to find an amount after a stated number of periods.

1. Prerequisite bridge: repeated percentage change

A percentage is a rate out of one hundred. For example, a rate of 5% is the decimal

0.05

. If a quantity increases by 5%, the new quantity is the old quantity plus 5% of the old quantity. That is the same as multiplying the old quantity by

1.05

.

The number

1.05

is the growth factor. A growth factor is the multiplier used for each equal time step. For a 5% increase, the factor is

1+0.05=1.05

. For a 5% decrease, the factor would be

1-0.05=0.95

. Compound interest uses a growth factor greater than one when the interest rate is positive.

\text{growth factor}=1+\frac{r}{100}

Change a percent rate to a decimal before using it as a multiplier.
For an increase of r%, the growth factor is $1+\frac{r}{100}$ .

2. What compounding means

Interest is money added to savings or charged on a loan. The principal is the starting amount of money. In compound interest, interest is added at regular intervals called compounding periods. The next interest amount is calculated using the new balance, not just the original principal.

Suppose a balance starts at CAD 100 and earns 5% interest each year. After one year, the interest is CAD 5, so the balance is CAD 105. In the next year, 5% is calculated on CAD 105. The balance becomes CAD 110.25, not CAD 110. This extra CAD 0.25 comes from interest earned on the earlier interest.

Each year, the balance is multiplied by the same factor,

1.05

. Repeating the same multiplication makes the balance grow exponentially. Exponential growth occurs when a quantity is multiplied by a constant factor over equal time steps.

Compound interest is interest calculated on the current balance.
The balance is multiplied by a constant factor for each compounding period.
A fixed positive rate produces exponential growth.

3. Tables, equations, and graphs

A table shows how the balance changes period by period. An equation gives a rule that works for any whole number of periods. A graph shows the pattern visually: for positive interest, the balance rises and the increases become larger over time because each period starts from a larger balance.

In the model,

P

means the principal, or starting balance. The letter

i

means the interest rate per compounding period written as a decimal. The letter

n

means the number of compounding periods. The amount

A

is the balance after those periods. For an annual rate compounded once each year, the annual rate is also the rate per period. If the compounding period differs, use the rate for one such period and count those periods.

For instance, with a starting balance of CAD 100 and a yearly rate of 5%, the first few balances are shown below. The entries are rounded to the nearest cent. The repeated multiplier in the table is the key connection to exponential growth.

A=P(1+i)^n

The initial value of the exponential model is the principal.
The growth factor is one plus the interest rate per period.
The exponent counts how many times the growth factor is applied.

4. Reading and using the model

The equation says to start with the principal and multiply by the growth factor once for every compounding period. The exponent is a compact way to write repeated multiplication. For example, a factor applied for three periods means multiplying by it three times.

To use the model, first identify the starting amount and the rate for one compounding period. Convert the rate from a percent to a decimal. Next, count the number of periods and substitute the values. Finally, evaluate the power and multiply by the principal. If the situation involves money, round the final amount to the nearest cent unless another rounding rule is stated.

This model assumes the stated rate stays the same for all periods and that interest is added at the end of each period. It describes the balance at the end of a whole number of periods. The model also works backward in the sense that the table and graph help show how the balance changes as the period count increases; this lesson uses it to calculate future balances.

Match the rate and period count: a yearly rate compounded yearly uses years as periods.
Do not add a percent directly to the balance as a fixed dollar amount.
The model represents repeated multiplication, not repeated addition.

A balance growing by five percent per year

Years elapsed	Balance (CAD)	How the new balance is found
0	100.00	Starting balance
1	105.00	100.00 multiplied by 1.05
2	110.25	105.00 multiplied by 1.05
3	115.76	110.25 multiplied by 1.05

Worked example

Balance after three years

A student deposits CAD 240 into an account that earns 4% interest compounded annually. Find the balance after three years. Assume the rate stays constant.

Identify the starting amount and rate
The principal is CAD 240. The annual rate is 4%, which is $0.04$ as a decimal. Since interest is compounded annually, one period is one year.
Find the number of periods
Three years of annual compounding means the growth factor is applied three times.
$n=3$
Substitute in the compound-interest model
The growth factor is one plus the rate per period. Use the same factor for each of the three periods.
$A=240(1+0.04)^3$
Evaluate and round
The three-period growth factor is $1.124864$ . Multiplying by the starting amount gives CAD 269.96736. Rounding to the nearest cent gives CAD 269.97.
$A=240(1.04)^3=269.96736$

Answer: The balance after three years is CAD 269.97.

Check: The result is greater than the starting amount, as expected for a positive interest rate. The yearly balances are CAD 249.60, CAD 259.58, and CAD 269.97 to the nearest cent. Each year applies the same factor,

1.04

.

Common mistakes and how to avoid them

Using the percent number as the multiplier, such as multiplying by

5

for a 5% rate.

Correction: Convert the percent to a decimal and add it to one. A 5% increase uses the factor

1.05

.

Calculating each year's interest only on the original principal.

Correction: With compound interest, use the balance at the start of each period. The previous interest is already part of that balance.

Adding the same dollar amount each period.

Correction: A fixed percentage gives a changing dollar increase. The increase grows as the balance grows.

Using years for the exponent when the compounding period is not one year.

Correction: Count the actual compounding periods and use the interest rate for one of those periods.

Lesson summary

Compound interest adds interest to the balance, so later interest is calculated on earlier interest too.
A positive fixed rate gives repeated multiplication by a constant growth factor.
The compound-interest model is $A=P(1+i)^n$ , where $P$ is the principal, $i$ is the rate per period as a decimal, and $n$ is the number of periods.
This repeated multiplication connects compound interest directly to exponential growth.

Check your understanding

Question 1

An account earns 3% per year. What growth factor should be used for each year?

$0.03$
$1.03$
$3.00$
$1.30$

Show answer and explanation

1.03

Convert 3% to

0.03

, then add one. The factor is

1.03

.

Question 2

A balance of CAD 500 earns 2% interest compounded annually. Which expression gives the balance after four years?

$500(1.02)^4$
$500+0.02(4)$
$500(0.02)^4$
$500(1.04)^2$

Show answer and explanation

500(1.02)^4

The principal is 500, the growth factor is

1.02

, and four annual periods mean the factor is raised to the fourth power.

Question 3

Why does the dollar amount of interest usually increase from one year to the next in a compound-interest account with a fixed positive rate?

The rate automatically rises every year.
The balance used to calculate interest becomes larger.
The account adds the same fixed amount each year.
The number of years is part of the interest rate.

Show answer and explanation

The balance used to calculate interest becomes larger.

The rate stays fixed, but it is applied to a larger balance each period. Therefore, the dollar amount of interest increases.

Key terms

Principal: The starting amount deposited, borrowed, or invested.
Interest: Money added to savings or charged on a loan for a period of time.
Compounding period: A regular time interval after which interest is added to the balance.
Growth factor: The multiplier that changes a quantity from one equal time step to the next.
Exponential growth: Growth in which a quantity is multiplied by the same factor over equal time steps.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B3.3 · Connect compound interest with exponential growth

Ontario Grade 11 Mathematics

Connecting repeated percentage growth to an exponential model

What you will learn

1. Prerequisite bridge: repeated percentage change

2. What compounding means

3. Tables, equations, and graphs

4. Reading and using the model

A balance growing by five percent per year

Balance after three years

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson