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B3.7 · Solve ordinary simple annuity problems using technology

Learn to solve ordinary simple annuity problems using technology through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Exponential Functions

Finding the future value of regular savings deposits using a formula and technology

Many people save money by putting the same amount into a bank account on a regular schedule, such as every month. Over time, that account earns interest, and the interest itself earns more interest. This lesson teaches you how to calculate the total amount of money built up from this kind of regular saving plan. This type of savings plan is called an annuity, and you will learn how to solve these problems using a formula together with a calculator.

What you will learn

Explain what an ordinary simple annuity is and give real-life examples of one
Identify the payment amount, interest rate per period, and number of payments in a problem
Use the future value annuity formula, with technology, to solve for the total amount saved
Interpret the calculated future value in the context of a real problem
Check an annuity answer for reasonableness by comparing it to the total of raw deposits

Prerequisite Bridge: Percent, Interest Rate per Period, and Compound Growth

Before starting annuities, you need two ideas from earlier math courses. The first is percent as a decimal. For example, an annual interest rate of 6 percent is written as the decimal

0.06

in a formula. The second is compound interest, where interest is calculated on both the original amount and on interest already earned. In compound interest, money grows by repeated multiplication, not by repeated addition.

In this lesson, interest is almost always given per year, but deposits often happen more often than once a year, such as monthly. This means you must convert the annual interest rate into an interest rate per payment period. If interest compounds monthly, you divide the annual rate by 12 to get the monthly rate. This step is essential, because every annuity formula in this lesson uses the interest rate per period, not the annual rate.

Percent must be written as a decimal before it is used in a formula, for example 6 percent becomes $0.06$
Compound interest means interest is earned on interest already added to the account
The interest rate used in the annuity formula is the rate per payment period, found by dividing the annual rate by the number of periods in a year

What Is an Ordinary Simple Annuity?

An annuity is a series of equal payments made at regular, evenly spaced time intervals. A simple annuity is an annuity where the payment period and the interest compounding period are the same length. For example, if you deposit money every month, and the bank also compounds interest every month, that is a simple annuity, because the payments and the compounding line up.

An ordinary annuity is an annuity where each payment is made at the end of each payment period, not at the beginning. Putting these two ideas together, an ordinary simple annuity is a series of equal deposits, made at the end of each period, where the compounding period matches the payment period exactly. This is the most common type of savings plan you will see, such as a monthly savings deposit into an account that compounds monthly.

The future value of an annuity is the total amount of money in the account after all the payments have been made, including all the interest earned along the way. This is different from just adding up the deposits, because the future value also includes the extra money earned from compound interest.

An annuity is a series of equal payments made at regular time intervals
A simple annuity has payment periods equal in length to the compounding periods
An ordinary annuity places each payment at the end of its period
Future value means the total in the account after all deposits and all interest are combined

The Future Value Formula and How to Read It

To calculate the future value of an ordinary simple annuity, we use a formula that combines the payment amount, the interest rate per period, and the number of payments. In this formula,

FV

stands for future value,

R

stands for the regular payment amount,

i

stands for the interest rate per period written as a decimal, and

n

stands for the total number of payments.

The formula looks like this. Read it as: take one, add the interest rate per period, raise that whole quantity to the power of the number of payments, then subtract one, then divide by the interest rate per period, then multiply everything by the payment amount.

You will almost never calculate this formula by hand for large values of

n

. Instead, you use technology, such as a scientific or graphing calculator, or spreadsheet software, to compute the value inside the brackets and then multiply by

R

. The formula tells you what to calculate, and the technology does the arithmetic accurately and quickly.

It helps to organize the given information into a small table before you plug numbers into the formula. This way you clearly see the payment, the periodic rate, and the number of payments before you touch the calculator.

FV = R [

\frac{(1+i)^n - 1}{i}

]

$FV$ is the total future value, $R$ is the regular payment, $i$ is the interest rate per period, $n$ is the number of payments
The formula multiplies the payment by a growth factor built from compound interest
Technology is used to evaluate the power and the arithmetic accurately
Organizing given values in a table before substituting reduces errors

Guided Example: Monthly Savings Plan

Let us work through a full example together, from the plain language description, to organizing the data, to the formula, to the final answer in context.

Applying the Skill to a New Situation

Annuity problems can look different depending on the context, but the reasoning stays the same. You are always asked to find the total value built up from equal, regular deposits earning compound interest. The skill is in correctly identifying

R

i

, and

n

from the words in the problem, since these details determine everything else.

For example, if a problem describes quarterly deposits, then

n

must count the number of quarters, and

i

must be the interest rate per quarter, not per year. If a problem describes a savings period in years, you must convert years into the correct number of payment periods before using the formula. Always match the units of

i

and

n

to the payment period described in the problem.

Harder problems may ask you to work backward, such as finding the monthly payment

R

needed to reach a target future value

FV

. This still uses the same formula and the same three quantities, but now you rearrange the formula to isolate

R

before using technology to evaluate it. The relationship between the four quantities never changes, only which one is unknown.

Always match the payment period to the compounding period described in the problem
Convert time given in years into the correct number of payments before substituting
The same formula can be rearranged to solve for a different unknown quantity, such as the payment $R$

Organizing the Given Information

Quantity	Meaning	Value in Example
R	regular payment per period	150
i	interest rate per period as a decimal	0.005
n	total number of payments	36
FV	future value after all payments	approximately 5900.45

Worked example

Saving for a Car with Monthly Deposits

Amir deposits CAD 150 at the end of every month into a savings account. The account earns 6 percent interest per year, compounded monthly. How much money will Amir have after 3 years?

Identify the payment and the total time
The regular payment is $R = 150$ , since Amir deposits this amount every month. The savings plan lasts 3 years, and deposits happen monthly, so we will need to convert this time into a number of monthly payments.
$R = 150$
Find the number of payments $n$
Since deposits are monthly and the plan runs for 3 years, multiply the number of years by 12 months per year to get the total number of payments.
$n = 3 × 12 = 36$
Find the interest rate per period $i$
The annual interest rate is 6 percent, written as the decimal $0.06$ . Since interest compounds monthly, and payments are also monthly, this is a simple annuity, so divide the annual rate by 12 to get the interest rate per month.
$i = \frac{0.06}{12} = 0.005$
Substitute into the future value formula
Now place $R$ , $i$ , and $n$ into the ordinary simple annuity formula. This step only sets up the calculation; the arithmetic is done using a calculator in the next step. FV = 150 [ $\frac{(1+0.005)^{36} - 1}{0.005}$ ]
Evaluate using technology
Use a calculator to find $(1.005)^{36}$ first, since this is a power calculation that should not be estimated by hand. This gives approximately $1.19668$ . Subtract 1, divide by $0.005$ , then multiply by 150. FV \approx 150 [ $\frac{1.19668 - 1}{0.005}$ ] = 150(39.336) \approx 5900.45
State the answer in context
Round the answer to the nearest cent, since this is a money amount. After 3 years of monthly deposits, Amir will have approximately this much in his account, including all interest earned.
$FV \approx 5900.45$

Answer: Amir will have approximately CAD 5900.45 in his account after 3 years.

Check: Amir deposited 150 dollars a month for 36 months, which totals 150 times 36, or CAD 5400.00, with no interest. Since the future value of CAD 5900.45 is larger than CAD 5400.00, and the extra amount, about CAD 500.45, came from compound interest building up gradually, this answer is reasonable.

Common mistakes and how to avoid them

Using the annual interest rate directly in the formula instead of converting it to a rate per payment period.

Correction: Always divide the annual rate by the number of compounding periods per year before substituting it as

i

Counting the number of payments in years instead of converting to the actual number of payment periods.

Correction: Multiply the number of years by the number of payments per year to find the correct value of

n

Confusing the future value with the simple total of deposits, ignoring the effect of compound interest.

Correction: Remember that

FV

includes both the deposits and the interest earned, so it will always be larger than the plain sum of deposits.

Rounding the growth factor

(1+i)^n

too early, before completing the rest of the calculation.

Correction: Keep several decimal places for

(1+i)^n

during the calculation, and only round the final dollar answer.

Lesson summary

An ordinary simple annuity is a series of equal payments made at the end of each period, where the payment period matches the compounding period.
The future value formula is $FV = R \left[ \frac{(1+i)^n - 1}{i} \right]$ , where $R$ is the payment, $i$ is the interest rate per period, and $n$ is the number of payments.
Always convert the annual interest rate to the rate per payment period, and convert the total time to the correct number of payments.
Technology should be used to evaluate the power and complete the arithmetic accurately.
The future value is always greater than the simple total of deposits, because it includes compound interest.

Check your understanding

Question 1

Priya deposits CAD 200 at the end of every month for 2 years into an account earning 12 percent per year, compounded monthly. What are the correct values of

R

i

, and

n

to substitute into the future value formula?

$R = 200$ , $i = 0.12$ , $n = 2$
$R = 200$ , $i = 0.01$ , $n = 24$
$R = 2400$ , $i = 0.01$ , $n = 24$
$R = 200$ , $i = 0.12$ , $n = 24$

Show answer and explanation

R = 200

i = 0.01

n = 24

The payment is CAD 200 per month, so

R = 200

. The annual rate 0.12 must be divided by 12 to get the monthly rate, so

i = 0.01

. There are 2 years of monthly deposits, so

n = 2 \times 12 = 24

Question 2

Why is the future value of an ordinary simple annuity always greater than the sum of all the raw deposits?

Because the formula always rounds the answer upward
Because the number of payments $n$ is always larger than expected
Because compound interest adds extra money on top of the deposits over time
Because the payment $R$ increases automatically each period

Show answer and explanation

Because compound interest adds extra money on top of the deposits over time

The extra amount beyond the total deposits comes from compound interest, which grows the account balance beyond simply adding up each payment.

Question 3

A savings plan has quarterly deposits over 5 years. How many total payments,

n

, should be used in the formula?

$n = 5$
$n = 20$
$n = 60$
$n = 15$

Show answer and explanation

n = 20

Quarterly deposits happen 4 times per year, so multiply 5 years by 4 quarters per year to get

n = 20

total payments.

Question 4

A problem states that an account compounds monthly and deposits are made monthly, with each deposit made at the end of the month. Which term correctly describes this type of annuity?

Ordinary simple annuity
General annuity
Annuity due
Compound annuity

Show answer and explanation

Ordinary simple annuity

Since the payment period equals the compounding period, it is simple, and since payments occur at the end of each period, it is ordinary, making it an ordinary simple annuity.

Key terms

Annuity: A series of equal payments made at regular, evenly spaced time intervals.
Ordinary annuity: An annuity where each payment is made at the end of each payment period.
Simple annuity: An annuity where the payment period is the same length as the compounding period.
Future value (FV): The total amount of money in an account after all payments and all interest are added together.
Interest rate per period (i): The interest rate that applies to a single payment period, found by dividing the annual rate by the number of periods per year.
Payment (R): The fixed amount of money deposited or paid at each regular interval in an annuity.
Number of payments (n): The total count of equal payments made over the life of the annuity.
Compound interest: Interest calculated on both the original amount of money and on the interest already added to it.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.7. It is a study resource, not an official curriculum publication.

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