Compare simple and compound interest using tables and graphs

Ontario Grade 11 Mathematics

Personal Finance

MBF3C study topic B1.1: Use tables and graphs to compare how savings grow over time.

Suppose two accounts start with the same amount and earn interest at the same annual rate. One account adds interest only to the starting amount. The other adds interest to the growing balance. Will they always have the same balance? A table and a graph help us compare them. Before comparing, we will review the starting amount, interest rate, and time.

What you will learn

Explain the difference between simple interest and compound interest.
Use a table to compare amounts when the starting amount, rate, and time are the same.
Read and describe graphs of simple and compound interest growth.

1. Review the parts of an interest model

The principal is the amount placed in an account at the start. The interest rate tells what fraction of an amount is added as interest in one year. Time is the number of years the money earns interest. For a fair comparison, use the same principal, rate, and time for both accounts.

A percent can be written as a decimal to calculate an amount. For example, an annual rate of 6% is

0.06

. The interest for one year on a principal of CAD 1,000 at that rate is CAD 60, because

1000(0.06)=60

.

The balance is the principal together with all interest earned so far. A year-by-year table records the balance at each time. On a graph, time is usually shown along the horizontal axis and balance along the vertical axis.

6\%=0.06

Keep the starting amount, rate, and time the same when comparing models.
A table or graph can show how the balance changes over time.

2. Understand the two ways interest is added

With simple interest, the interest is calculated on the original principal each year. If the principal is CAD 1,000 and the rate is 6%, the account earns CAD 60 each year. The added amount stays the same, so the balance increases by equal steps.

With compound interest, interest is added to the balance. In the next year, interest is calculated on the new balance, not just the original principal. This is called earning interest on interest. The amount added each year can grow as the balance grows.

For annual interest, a simple-interest balance can be found by adding the same yearly interest for each year. A compound-interest balance can be found by multiplying the previous balance by one plus the rate each year. In the formulas,

P

means principal,

r

means annual rate written as a decimal, and

t

means time in years. The formulas describe the balance after

t

years.

A_{\text{simple}}=P+Prt,\qquad A_{\text{compound}}=P(1+r)^t

Simple interest adds the same dollar amount each year.
Compound interest calculates each year's interest from the current balance.

3. Compare balances in a table

A useful comparison starts with matching conditions. Make a row for each year, including year zero. At year zero, neither account has earned interest, so both balances equal the principal.

For simple interest, the change from one row to the next is the same each year. For compound interest, apply the rate to the balance in the previous row. Keep cents in the calculations and round displayed balances to the nearest cent. Small differences may appear because of rounding.

Compare corresponding rows rather than comparing different times. If the compound balance is greater in a later row, that shows it has grown more by that time under the conditions in the table.

\text{yearly simple-interest increase}=Pr

Year zero provides a shared starting point.
Equal yearly increases signal simple interest; growing yearly increases signal compound interest.

4. Show the comparison on a graph

Plot each table balance at its matching year. Label the horizontal axis with time in years and the vertical axis with balance in dollars. Use the same scales and axes for both sets of points so the comparison is clear.

Simple-interest points lie on a straight line because the same amount is added each year. Compound-interest points rise by larger amounts over time, so they bend upward rather than following a straight line. In a short time period, the lines or curves may look close together. Read the plotted values or table to make a careful comparison.

A graph shows the overall pattern, while a table gives exact balances for the listed years. Use both: the table supports precise comparisons, and the graph makes the pattern of growth easier to see.

A straight-line pattern represents equal increases over equal time intervals.
An upward-bending compound-interest pattern represents increases that get larger.

Balances for Example 1

Time (years)	Simple interest (CAD)	Compound interest (CAD)
0	1,000.00	1,000.00
1	1,060.00	1,060.00
2	1,120.00	1,123.60
3	1,180.00	1,191.02
4	1,240.00	1,262.48

Worked example

Example 1: Compare two accounts in a table

Two accounts each start with CAD 1,000 and earn 6% interest annually for four years. One uses simple interest and the other uses compound interest. Find and compare the balances at the end of each year.

Find the simple-interest increase
Simple interest uses the original CAD 1,000 every year. The annual interest is CAD 60, so add CAD 60 to the balance for each year.
$1000(0.06)=60$
Build the simple-interest values
Start at year zero with CAD 1,000. Add CAD 60 each time you move to the next year because the interest is based on the same original principal.
$1000,\ 1060,\ 1120,\ 1180,\ 1240$
Build the compound-interest values
For each new year, multiply the previous balance by $1.06$ . This factor keeps the full previous balance and adds interest equal to 6% of it.
$1000,\ 1060,\ 1123.60,\ 1191.02,\ 1262.48$
Compare at the same time
At four years, compare the two year-four balances. The compound account is higher. The difference is larger than it was at year two because compound interest is calculated on a growing balance.
$1262.48-1240=22.48$

Answer: At the end of four years, the simple-interest balance is CAD 1,240.00 and the compound-interest balance is CAD 1,262.48. Compound interest is higher by CAD 22.48.

Check: The simple balance rises by CAD 60 in every year. The compound increases are CAD 60.00, CAD 63.60, CAD 67.42, and CAD 71.46 after rounding, so the increases grow as expected.

Worked example

Example 2: Interpret a graph comparison

A graph compares two accounts that each start at CAD 800 and earn 4% annually. One account earns simple interest and the other compound interest. The graph shows values from year zero to year three. Use the corresponding values to describe the graph and decide which account is higher at year three.

Calculate the simple-interest points
The simple account earns CAD 32 each year, since 4% of CAD 800 is CAD 32. Its balance therefore increases by the same amount between every pair of years.
$800,\ 832,\ 864,\ 896$
Calculate the compound-interest points
For the compound account, multiply each year's balance by $1.04$ to get the next year's balance. Round the displayed amounts to the nearest cent.
$800,\ 832,\ 865.28,\ 899.89$
Connect the values to the graph
The simple points make a straight-line pattern because each yearly increase is CAD 32. The compound points bend upward because their increases get larger. At year three, compare the two points at the same horizontal position.
$899.89-896=3.89$

Answer: At year three, the compound-interest account is higher by CAD 3.89. The simple-interest graph is straight, while the compound-interest graph bends upward.

Check: The compound increases are CAD 32.00, CAD 33.28, and about CAD 34.61. These are increasing amounts, consistent with the upward bend.

Common mistakes and how to avoid them

Calculating compound interest on the original principal every year.

Correction: For compound interest, calculate the next year's interest from the current balance.

Comparing one account at year three with the other at year four.

Correction: Compare balances at the same time. Match the year before deciding which is higher.

Assuming both graph patterns are straight lines because both accounts grow.

Correction: Simple interest has equal yearly increases and a straight-line pattern. Compound interest has increases that grow over time.

Lesson summary

Simple interest is calculated on the original principal, so it adds equal amounts each year.
Compound interest is calculated on the current balance, so the yearly additions can grow.
Use matching starting amounts, rates, and times to make a fair comparison.
Tables show balances at specific times; graphs show how the patterns of growth compare.

Check your understanding

Question 1

An account starts with CAD 500 and earns simple interest at 4% per year. How much interest is added each year?

CAD 4
CAD 20
CAD 40
CAD 520

Show answer and explanation

CAD 20

Find 4% of the original CAD 500:

500(0.04)=20

. Simple interest adds CAD 20 each year.

Question 2

Two accounts have the same principal and annual rate. Which pattern suggests compound interest on a balance-versus-time graph?

A straight line with equal increases each year
A horizontal line with no change
An upward-bending pattern with increases that get larger
A line that decreases by the same amount each year

Show answer and explanation

An upward-bending pattern with increases that get larger

Compound interest is calculated on the growing balance, so the amount added can increase over time.

Question 3

An account starts at CAD 1,000 and earns simple interest at 5% annually. What is its balance after two years?

CAD 1,050
CAD 1,100
CAD 1,102.50
CAD 1,250

Show answer and explanation

CAD 1,100

The account adds CAD 50 each year because

1000(0.05)=50

. After two years, the balance is CAD 1,000 plus CAD 100, or CAD 1,100.

Key terms

Principal: The amount in an account at the start, before interest is added.
Interest rate: The percentage used to calculate interest for a stated period, such as one year.
Balance: The principal plus the interest earned so far.
Simple interest: Interest calculated on the original principal each period.
Compound interest: Interest calculated on the current balance, including interest already added.

Continue through MBF3C

View the complete MBF3C Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation B1.1. 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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