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B3.5 · Explain annuities using numeric and graphical representations

Learn to explain annuities using numeric and graphical representations through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Exponential Functions

Explaining regular savings and payments using numbers and graphs

You already know that money in a savings account can earn interest, and that interest can be simple or compound. In earlier work, you saw that compound interest makes an amount of money grow following a curved, exponential pattern, because each new amount of interest is calculated on a bigger total than before. An annuity uses this same idea of compound interest, but it adds one new feature: instead of depositing one lump sum of money and leaving it alone, you deposit the same fixed amount of money over and over again, at regular time intervals, such as every month or every year. This lesson explains what an annuity is and shows how to describe it using a table of values and a graph, so you can see and explain the pattern of growth without needing any new formulas beyond what compound interest already gave you.

What you will learn

Explain what an annuity is, in plain language, using a real-life example
Represent an annuity using a table of values that shows deposits and growing totals
Represent an annuity using a graph and describe its shape
Connect the numeric and graphical representations to explain how an annuity grows over time

Prerequisite Bridge: Compound Interest Review

Before we talk about annuities, let's remind ourselves what compound interest means. Compound interest is interest that is calculated on the original amount of money plus any interest that has already been added. This is different from simple interest, where interest is only ever calculated on the original amount.

Because compound interest keeps adding interest on top of interest, a single deposit left alone grows using an exponential pattern. An exponential pattern means the amount increases by a fixed percentage each period, not by a fixed dollar amount. On a graph, this produces a curve that gets steeper as time goes on, not a straight line.

An annuity uses this same compounding idea, but it changes one thing: money is added regularly, not just once. This means the total in an annuity grows both because old money is earning interest and because new deposits keep arriving. Understanding this combination is the whole point of this lesson.

Compound interest calculates interest on the growing total, producing an exponential curve for a single deposit
An annuity involves regular, equal deposits or payments made at fixed time intervals
An annuity combines regular new deposits with compound interest on money already saved

What Is an Annuity, in Plain Language?

An annuity is a series of equal payments made at regular time intervals, where each payment can earn interest until the end of the annuity. The payments could be deposits into a savings account, or they could be regular payments toward a big purchase. In this lesson, we focus on the savings type, where someone deposits the same amount every period and lets the account grow.

Three features define an annuity: the payments are always the same fixed amount, they always happen after equal time gaps such as every month or every year, and the money already in the account keeps earning interest each period. If any of these three features is missing, it is not an annuity.

A simple everyday example is a student who deposits CAD 50 into a savings account at the end of every year, and the bank pays 4 percent interest per year on the total in the account. Every year, two things happen: the old total earns 4 percent interest, and a new CAD 50 deposit is added. The account balance after several years is the annuity total.

An annuity is a sequence of equal payments made at equal time intervals
Money already saved keeps earning interest while new equal payments are added
All three features together, equal payment, equal time gap, and ongoing interest, must be present

Representing an Annuity with a Table of Values

The clearest way to begin explaining an annuity is with a table of values, because a table shows exactly what happens period by period, without hiding any steps. Using our example of CAD 50 deposited at the end of every year at 4 percent annual interest, we can build the balance year by year.

At the end of year 1, only the first deposit has been made, so there has been no time yet for interest to apply to it. At the end of year 2, the amount from year 1 earns interest for one year, and a new deposit is added. This pattern repeats: each year, take the previous total, add 4 percent interest to it, then add another CAD 50 deposit.

Reading down the table, you should notice that the yearly increase in the total is not constant. It grows slightly larger each year, because the interest is being calculated on a bigger and bigger total. This is the numeric signal that the annuity is growing in an exponential-style pattern, not a straight-line pattern, even though the deposits themselves are always the same fixed amount.

A table lists the balance at the end of each period, showing interest added plus a new deposit
The yearly increase in total grows larger over time, since interest applies to a larger balance each year
Reading a table carefully lets you explain the annuity's growth without needing a formula first

Representing an Annuity with a Graph

Once we have a table of values, we can plot the year number on the horizontal axis and the account balance on the vertical axis. Each point represents the total saved at the end of that year.

When you plot these points and look at the overall shape, you see a curve that rises and gets steeper as the years pass. This curve looks similar to the compound interest curve you studied earlier, because the same compounding idea is at work. The difference is that an annuity graph rises a bit more steadily near the start, since new deposits are being added on top of the growing interest, but the long-term shape still curves upward more and more steeply, showing exponential-style growth rather than straight-line growth.

Comparing the table and the graph together is powerful. The table gives you exact numbers you can check by hand, while the graph gives you the overall shape and lets you see the pattern of accelerating growth at a glance. Both representations describe the very same annuity; they just show it in different ways.

Plotting time on the horizontal axis and balance on the vertical axis produces a rising, curving graph
The curve becomes steeper over time because interest is calculated on an ever-larger balance
Table and graph are two views of the same annuity information, one exact and one visual

Maria's Annuity: CAD 200 per year at 5 percent interest

Year	Interest earned that year	Deposit added	Balance at year end
1	0.00	200.00	200.00
2	10.00	200.00	410.00
3	20.50	200.00	630.50
4	31.53	200.00	862.03

Worked example

Building and Reading an Annuity Table and Graph

Maria deposits CAD 200 at the end of every year into a savings account that pays 5 percent interest per year, compounded yearly. Build a table of values for the first 4 years, describe the graph, and explain how the balance is growing.

Set up the yearly rule
Each year, the previous balance earns 5 percent interest, and then a new CAD 200 deposit is added at the end of the year. We apply this same rule every year in order.
$\text{new balance} = \text{old balance} × 1.05 + 200$
Year 1 balance
At the end of year 1, only the first deposit exists. There has been no earlier balance to earn interest on yet, so the balance is simply the deposit itself.
$B_1 = 200$
Year 2 balance
The year 1 balance of CAD 200 earns 5 percent interest, giving CAD 210, and then a new CAD 200 deposit is added at the end of year 2.
$B_2 = 200(1.05) + 200 = 410$
Year 3 balance
The year 2 balance of CAD 410 earns 5 percent interest, giving CAD 430.50, and then another CAD 200 deposit is added.
$B_3 = 410(1.05) + 200 = 630.50$
Year 4 balance
The year 3 balance of CAD 630.50 earns 5 percent interest, giving about CAD 662.03, and then the fourth CAD 200 deposit is added.
$B_4 = 630.50(1.05) + 200 = 862.03$
Describe the table pattern
Listing the balances gives 200, 410, 630.50, and 862.03. The yearly increases are 210, 220.50, and 231.53. Each increase is larger than the one before, because interest is now being earned on a bigger total each year, on top of the same fixed CAD 200 deposit.
Describe the graph
If we plot year number on the horizontal axis and balance on the vertical axis, the four points rise and curve slightly upward. Over many more years, this curve would bend upward more sharply, matching the exponential-style growth pattern seen in compound interest.

Answer: The balances at the end of years 1 through 4 are CAD 200, CAD 410, CAD 630.50, and CAD 862.03. The table shows increasing yearly growth amounts, and the graph shows a rising curve that gets steeper, because both new deposits and compounding interest are adding to the total.

Check: Recomputing year 4 directly: start from 0, add 200 at year 1; multiply by 1.05 and add 200 for year 2 to get 410; multiply by 1.05 and add 200 for year 3 to get 630.50; multiply by 1.05 and add 200 for year 4 to get 862.025, which rounds to 862.03, matching the step above.

Common mistakes and how to avoid them

Thinking the annuity graph is a straight line because the deposits are always the same fixed amount.

Correction: Remember that the deposits are equal, but the total balance still curves upward, because interest is earned on a growing balance each period, not just on the fixed deposit.

Forgetting that the very first deposit has not yet earned any interest at the end of its own first period.

Correction: In a table, always show the first period's balance as just the deposit itself, since no time has passed yet for interest to apply.

Applying interest to the new deposit before adding it, instead of applying interest only to the old balance.

Correction: Follow the order in the yearly rule: multiply only the previous balance by the interest factor, then add the fresh deposit afterward.

Confusing an annuity with a single lump sum invested once.

Correction: Check that payments happen repeatedly at equal time gaps; a one-time deposit left alone is compound interest, not an annuity.

Lesson summary

An annuity is a series of equal payments made at equal time intervals, where existing money continues to earn interest.
A table of values shows the annuity period by period, revealing that the yearly growth amount increases over time.
A graph of balance versus time shows a curve that rises and becomes steeper, similar in shape to compound interest growth.
The rule for updating an annuity balance is to multiply the previous balance by the interest factor, then add the fixed new deposit.
Table and graph representations describe the same annuity information in two complementary ways, one exact and one visual.

Check your understanding

Question 1

Which set of features correctly describes an annuity?

A single deposit left alone to earn compound interest
Equal payments made at equal time intervals, with interest earned on the growing balance
Payments of different amounts made at random times
Equal payments made at equal time intervals but with no interest at all

Show answer and explanation

Equal payments made at equal time intervals, with interest earned on the growing balance

An annuity needs three features together: equal payment amounts, equal time gaps between payments, and ongoing interest on the balance. Only the second option includes all three.

Question 2

In an annuity table, why does the yearly increase in balance get larger over time, even though the deposit amount never changes?

Because the deposit amount secretly increases each year
Because interest is calculated on a balance that is growing larger each year
Because the interest rate increases automatically every year
Because earlier years are recorded incorrectly on purpose

Show answer and explanation

Because interest is calculated on a balance that is growing larger each year

The deposit stays fixed, but the balance it is added to keeps growing, so the same interest rate produces a larger dollar amount of interest each year, making the total increase grow.

Question 3

A graph of an annuity's balance over time is best described as

A straight line with constant slope
A curve that rises and becomes steeper over time
A curve that rises and then falls back down
A horizontal line that never changes

Show answer and explanation

A curve that rises and becomes steeper over time

Because interest compounds on a growing balance while new deposits are also added, the balance rises faster and faster, producing a curve that becomes steeper, not a straight line.

Question 4

In the rule new balance equals old balance times 1.05 plus 200, what does the 200 represent?

The interest rate written as a percentage
The total balance after one year
The fixed regular deposit added at the end of each period
The number of years the account has been open

Show answer and explanation

The fixed regular deposit added at the end of each period

The 200 is the amount added fresh at the end of every period, after interest has already been applied to the previous balance. It stays the same size every period, which is what makes it an annuity payment.

Key terms

Annuity: A series of equal payments made at equal time intervals, where money already saved continues to earn interest.
Compound interest: Interest calculated on the original amount plus any interest already added, causing exponential-style growth.
Simple interest: Interest calculated only on the original amount of money, never on interest already earned.
Interest rate: The percentage used to calculate how much interest is added to a balance in each time period.
Table of values: A list showing, for each period, the exact balance and the amounts added, used to track a pattern step by step.
Exponential-style growth: A pattern where an amount increases by a fixed percentage each period, producing a curve that gets steeper over time.
Period: A fixed length of time, such as one month or one year, between two payments in an annuity.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.5. It is a study resource, not an official curriculum publication.

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