Connect compound interest with exponential growth

Ontario Grade 11 Mathematics

Personal Finance

MBF3C study topic B1.2: seeing how repeated interest creates a growth pattern

Suppose you deposit money in an account that earns interest. If the interest is added to the account, the next interest calculation uses the new, larger balance. This repeated process is called compound interest. It creates exponential growth: the amount is multiplied by the same factor over and over. In this lesson, you will connect the account’s changing balance to a table, repeated multiplication, and a formula. The goal is to understand the connection, not just to substitute numbers.

What you will learn

Describe how compound interest is added to an amount over time.
Connect repeated multiplication in a compound-interest model to exponential growth.
Use a table and a formula to find an account balance after a number of compounding periods.
Explain what the starting amount, growth factor, and number of periods mean in a financial situation.

1. Prerequisite bridge: repeated multiplication

A percentage is a part out of

100

. For example, an interest rate of 5% means that the interest for one period is

0.05

times the balance at the start of that period. The decimal

0.05

is the rate written as a decimal.

If an amount increases by 5%, the new amount is the original amount plus 5% of it. That is the same as multiplying the original amount by

1.05

. The factor

1.05

includes the original 100% and the extra 5%. This multiplier is called the growth factor.

Exponential growth occurs when an amount is multiplied by the same growth factor for equal time periods. Compound interest is one practical example. The balance does not rise by the same fixed number each period, because the interest is calculated from a balance that can keep increasing.

\text{growth factor}=1+\text{rate as a decimal}

Convert a percentage rate to a decimal before using it in a calculation.
For a rate of r% per period, the growth factor is $1+\frac{r}{100}$ .
In exponential growth, equal periods use the same multiplier.

2. From interest to an exponential pattern

Consider an account that starts with CAD 100 and earns 5% interest each year. Assume the interest is added to the account at the end of each year, and the rate stays the same. After the first year, the balance is CAD 105. The next year’s interest is calculated on CAD 105, not just on the original CAD 100.

The balance after each year is found by multiplying the previous balance by

1.05

. This is why each new balance is

1.05

times the one before it. The interest earned in a year is the increase from one balance to the next. In this example, that increase gets larger because it is calculated on a growing balance.

The number of times interest is added is the number of compounding periods. In this example, there is one compounding period per year. If interest is added more than once a year, the rate used for each period and the number of periods must match that schedule. The annual rate is not automatically the rate for each shorter period.

A=P(1+i)^n

Compound interest means interest is added to the balance, so later interest can be earned on earlier interest.
A compounding period is one interval at which interest is added.
Use a rate per period together with the number of periods; keep the time units consistent.

3. Read the formula and representations

In the formula,

P

is the starting principal, or starting amount. The letter

i

is the interest rate written as a decimal for one compounding period. The letter

n

is the number of compounding periods. The letter

A

is the balance after those periods. The expression

(1+i)^n

means multiply the growth factor by itself

n

times.

A table shows how the balance changes period by period. It helps make the repeated multiplication visible. The formula is a compact way to find a balance after many periods without writing every multiplication separately.

The formula gives the ending balance, including the starting principal and interest. To find only the interest earned, compare the ending balance with the starting principal by subtracting the starting amount. Check that the answer makes sense: for a positive rate, each period’s balance should be greater than the previous one.

A=P(1+i)^n

$P$ is the initial amount; $i$ is the rate per period as a decimal; $n$ counts periods; $A$ is the resulting balance.
The exponent records how many times the growth factor is used.
An answer for total interest is different from an answer for the full account balance.

4. Applying the connection

Before using the formula, identify the starting amount, the rate for one compounding period, and the number of periods. Convert a percentage to a decimal. Then check that the period used for the rate matches the period counted by

n

.

For example, if interest is compounded quarterly, each quarter is one period. If a problem gives an annual rate and asks for a quarterly model, establish the rate per quarter as directed by the situation before using the formula. Do not count years as periods when the interest is being added quarterly.

A calculator can help evaluate a power, but the entries still need to match the situation. Keep enough digits during the calculation and round a money amount to the nearest cent at the end. State whether the result is a balance or interest earned.

Match the rate unit and the period count before calculating.
Keep the meaning of the answer clear: balance or interest earned.
Round money only after evaluating the model.

Annual balances for Example 1

Year	Balance (CAD)	How the balance is found
0	200.00	Starting amount
1	208.00	200.00 multiplied by 1.04
2	216.32	208.00 multiplied by 1.04
3	224.97	216.32 multiplied by 1.04

Worked example

Example 1: Follow the balance year by year

A savings account begins with CAD 200 and earns 4% interest compounded annually. Find the balance after three years using a table, and connect the pattern to exponential growth.

Identify the repeated multiplier
The annual rate is 4%, or $0.04$ as a decimal. Since the interest is added yearly, one year is one period. Adding the interest to the balance is equivalent to multiplying by $1.04$ .
$1+0.04=1.04$
Build the balance table
Start with CAD 200. Multiply each year’s balance by $1.04$ to get the next year’s balance. This works because each year’s interest is based on the current account balance.
$200(1.04)=208,\quad 208(1.04)=216.32,\quad 216.32(1.04)=224.9728$
Interpret and round
After three years, the unrounded model gives CAD 224.9728. Rounding to the nearest cent gives CAD 224.97. Each year’s balance is the previous balance multiplied by the same factor, so the pattern is exponential growth.

Answer: The balance after three years is CAD 224.97.

Check: The balance has increased from CAD 200, as expected for a positive interest rate. The year-three calculation is also

200(1.04)^3=224.9728

.

Worked example

Example 2: Use the formula for monthly compounding

An account starts with CAD 500 and earns a stated annual interest rate of 6%, compounded monthly. For this model, use a monthly rate of 0.5%. Find the balance after one year.

Set the period rate and count
There are twelve monthly periods in one year. The given monthly rate is 0.5%, which is $0.005$ as a decimal. Using a monthly rate with twelve periods keeps the units consistent.
$i=0.005,\quad n=12$
Substitute in the growth model
The starting amount is CAD 500. Each month the balance is multiplied by $1.005$ . Raising this factor to the twelfth power represents twelve repeated monthly increases.
$A=500(1.005)^{12}$
Evaluate and state the result
Evaluating the expression gives approximately CAD 530.84 when rounded to the nearest cent. This is the full balance after one year, not just the interest. The increase above the starting amount is about CAD 30.84.
$500(1.005)^{12}\approx 530.84$

Answer: The balance after one year is approximately CAD 530.84.

Check: The result is greater than CAD 500, and its increase is about CAD 30.84. The monthly multiplier was used twelve times, matching the twelve compounding periods.

Common mistakes and how to avoid them

Adding the same dollar amount of interest each period.

Correction: With compound interest, calculate each period’s interest on the current balance. The same growth factor multiplies each period’s balance.

Using a percentage such as 4% as though it were the decimal

4

.

Correction: Convert the percentage to a decimal:

4\%=0.04

. The corresponding growth factor is

1.04

.

Counting years when the account compounds monthly.

Correction: Count compounding periods. One year of monthly compounding has twelve periods, so use the monthly rate with

n=12

.

Calling the full ending balance the interest earned.

Correction: The ending balance includes the starting amount. Subtract the starting amount to find the interest earned.

Lesson summary

Compound interest adds interest to the account balance, allowing later interest to be calculated on a larger amount.
A constant rate per equal period gives a constant growth factor, which is repeated multiplication and therefore exponential growth.
In $A=P(1+i)^n$ , match the decimal rate $i$ to each period counted by $n$ .
The formula gives the ending balance; subtract the principal if you need only the interest earned.

Check your understanding

Question 1

An account starts at CAD 300 and earns 2% per year, compounded annually. Which expression represents its balance after four years?

$300(1.02)^4$
$300(0.02)^4$
$300+1.02(4)$
$300(1.2)^4$

Show answer and explanation

300(1.02)^4

The rate as a decimal is

0.02

, so the growth factor is

1.02

. Four annual periods mean the factor is used four times.

Question 2

An account earns 1% interest each month. What does the factor

1.01

represent?

The starting balance
The multiplier for one monthly period
The number of months in a year
The total interest after one year

Show answer and explanation

The multiplier for one monthly period

A monthly rate of 1% is

0.01

as a decimal. Adding that rate to the original 100% gives a one-month growth factor of

1.01

.

Question 3

A balance is CAD 420 before interest and CAD 432.60 after one compounding period. What is the interest earned for that period?

CAD 12.60
CAD 420.00
CAD 432.60
CAD 10.00

Show answer and explanation

CAD 12.60

The interest earned is the increase in the balance: CAD 432.60 minus CAD 420.00 equals CAD 12.60.

Key terms

Compound interest: Interest added to an amount, so future interest can be calculated on the increased balance.
Principal: The starting amount deposited or invested.
Compounding period: One interval at the end of which interest is added to the balance.
Growth factor: The multiplier used to find the amount after one growth period.
Exponential growth: A pattern in which an amount is repeatedly multiplied by the same factor over equal periods.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation B1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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