Calculate compound interest by repeated simple-interest steps

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

MEL3E study topic B2.2: Calculate compound interest by repeated simple-interest steps

Imagine putting savings in an account that earns interest each year. If the interest stays in the account, the next year's interest is calculated on a larger balance. That process is called compound interest. For this lesson, you will calculate it by repeating a simple-interest step for each period. You do not need a special compound-interest formula. You need the starting balance, the interest rate for one period, and the number of periods. A calculator or spreadsheet can help you keep the steps accurate.

What you will learn

Explain how compound interest is built by adding interest to a balance and calculating new interest on that balance.
Calculate compound interest by repeating a simple-interest calculation for each interest period.
Use a calculator or spreadsheet to keep track of changing balances and check a result.

Start with the balance for one period

A balance is the amount of money in an account at a particular time. The principal is the amount first put into the account. Interest is money earned for allowing the money to remain in the account. A rate tells you what part of the balance is earned as interest.

Simple interest for one period is found by multiplying the balance at the start of that period by the rate for that period. For example, at an annual rate of 4%, the rate as a decimal is 0.04. A balance of CAD 500 earns CAD 20 in one year because 500 times 0.04 is 20.

The key word is period. A rate might be given per year, while interest might be added monthly. The rate and the time used in a calculation must match. For a monthly calculation using an annual rate, first find the monthly rate by dividing the annual rate by 12. This lesson assumes the stated annual rate is divided evenly across the months.

I=P \times r \times t

Calculate interest using the balance at the start of the period.
Change a percentage to a decimal before multiplying.
Use a rate that matches the period being calculated.

Repeat the step to build compound interest

For compound interest, earned interest is left in the account. At the end of a period, add that period's interest to the balance. The new balance becomes the starting balance for the next period. Repeat these actions for every period.

This is different from calculating every period's interest on the original principal. The balance changes after each addition, so the amount of interest can change too. The method is still simple-interest arithmetic for each single period; the repeated steps make the overall result compound interest.

A useful record has one row for each period. Write the opening balance, interest earned, and closing balance. The closing balance in one row is the opening balance in the next row. Keep extra decimal places while calculating, then round money to the nearest cent at the end unless account instructions say to round each period.

\text{closing balance}=\text{opening balance}+\text{interest}

Find interest for one period from that period's opening balance.
Add the interest to get the closing balance.
Carry the closing balance forward as the next opening balance.

Use a calculator or spreadsheet carefully

A calculator can handle each multiplication and addition. Enter the rate as a decimal, such as 0.03 for 3%. After calculating one period's interest, add it to the balance before doing the next period.

A spreadsheet can make the repeated pattern easy to follow. Put the opening balance, interest, and closing balance in separate columns. In each new row, use the previous row's closing balance as the new opening balance. Check that the number of rows matches the number of periods.

Before trusting the result, make two quick checks. First, if the rate is positive and no money is withdrawn, each closing balance should be larger than the previous one. Second, the total interest should equal the final balance minus the amount first deposited. These checks can reveal a missed addition or a balance that was not carried forward.

Use one row for each period.
Carry forward the previous closing balance, not the original deposit.
Check that the final balance and total interest agree.

Choose the correct rate and number of steps

Read the account information before calculating. Identify the starting deposit, the annual rate, how often interest is added, and how long the money stays in the account. The number of steps is the number of times interest is added during that time.

If interest is added once each year for three years, complete three yearly steps. If it is added each month for six months, complete six monthly steps. For the monthly case, convert the annual rate to a monthly rate before calculating each step.

Do not add interest to the account more often than the instructions allow. For example, if the statement says interest is added annually, do not treat it as monthly. The calculation follows the account's stated timing.

Match the calculation periods to the account's interest timing.
Divide an annual rate by 12 for monthly steps when the rate is evenly divided.
Do not change the account's stated timing.

Yearly balance record for the first example

Year	Opening balance	Interest at 5%	Closing balance
1	CAD 800.00	CAD 40.00	CAD 840.00
2	CAD 840.00	CAD 42.00	CAD 882.00
3	CAD 882.00	CAD 44.10	CAD 926.10

Worked example

Savings account with yearly interest

A student deposits CAD 800 in an account paying 5% annual interest. Interest is added once each year. Find the balance after three years and the total interest earned.

Find the rate for one year
The rate is already annual, and each step is one year. Change 5% to a decimal by dividing by 100.
$5\%=0.05$
Calculate the first year's interest
Use the opening balance of CAD 800. The interest is 5% of that amount.
$800 \times 0.05=40$
Carry forward the first balance
Add CAD 40 interest to the opening balance. This closing balance is used in year two.
$800+40=840$
Repeat for years two and three
For each year, calculate 5% of the latest balance, then add that interest. The balances are shown to the nearest cent.
$840 \times 0.05=42;\quad 840+42=882;\quad 882 \times 0.05=44.10;\quad 882+44.10=926.10$
Find total interest
Subtract the original deposit from the final balance to find the total amount earned as interest.
$926.10-800=126.10$

Answer: The balance after three years is CAD 926.10. The total interest earned is CAD 126.10.

Check: The balance increased in each year, and the final balance minus the original CAD 800 deposit is CAD 126.10.

Worked example

Short-term savings with monthly interest

A worker deposits CAD 1,200 in an account paying 6% annual interest. Interest is added monthly, and the money stays in the account for four months. Assume the annual rate is divided evenly across 12 months. Find the balance after four months.

Find the monthly rate
The account adds interest monthly, so use a monthly rate. Divide the annual rate by 12, then write the result as a decimal.
$6\% \div 12=0.5\%=0.005$
Calculate month one's interest
Multiply the starting balance by the monthly decimal rate. Add the interest to get the balance that starts month two.
$1200 \times 0.005=6;\quad 1200+6=1206$
Calculate month two
Use CAD 1,206 rather than the original CAD 1,200. The balance has changed because the first month's interest stayed in the account.
$1206 \times 0.005=6.03;\quad 1206+6.03=1212.03$
Calculate months three and four
Repeat the same calculation from the latest balance each month. Keep the balance to the nearest cent.
$1212.03 \times 0.005=6.06015;\quad 1212.03+6.06015=1218.09015;\quad 1218.09015 \times 0.005=6.09045075;\quad 1218.09015+6.09045075=1224.18060075$
Round the final amount
Round the final balance to the nearest cent. The total interest is the final balance minus the original deposit.
$1224.18060075 \approx 1224.18;\quad 1224.18-1200=24.18$

Answer: The balance after four months is CAD 1,224.18. The total interest earned is CAD 24.18.

Check: Four monthly steps were completed. Each month's interest used the balance at the start of that month, and the balance rose each time.

Common mistakes and how to avoid them

Using the original deposit to calculate interest in every period.

Correction: Use the opening balance for the current period. After adding interest, carry the closing balance into the next period.

Using an annual rate in a monthly calculation without changing it.

Correction: When the annual rate is evenly divided and interest is monthly, divide the annual rate by 12 and use that monthly rate for each step.

Doing the right calculation the wrong number of times.

Correction: Count how often interest is added. Complete one step for each of those periods.

Rounding every intermediate amount too early.

Correction: Keep extra decimal places during the repeated calculations and round the final money amount to the nearest cent, unless the account's instructions require a different approach.

Lesson summary

Compound interest can be calculated as repeated simple-interest steps.
For each period, multiply the opening balance by the rate for that period.
Add the interest to the opening balance, then use the new balance in the next step.
Match the rate and number of steps to the account's interest timing.
Check the final balance by comparing it with the original deposit.

Check your understanding

Question 1

A balance of CAD 600 earns 2% for one year, with interest added yearly. What is the balance after that year?

CAD 602
CAD 612
CAD 620
correctIndex

Show answer and explanation

CAD 612

Two percent of CAD 600 is CAD 12. Add it to the opening balance: CAD 600 plus CAD 12 is CAD 612.

Question 2

An account pays 12% annual interest, added monthly. Assuming the annual rate is divided evenly, what monthly rate should be used?

1%
6%
12%
correctIndex

Show answer and explanation

1%

Divide 12% by 12 months. The monthly rate is 1%.

Question 3

At the start of a period, an account balance is CAD 500. It earns CAD 10 interest, which stays in the account. What balance starts the next period?

CAD 500
CAD 510
CAD 490
correctIndex

Show answer and explanation

CAD 510

Add the interest to the opening balance. The next period starts with CAD 510.

Key terms

Balance: The amount of money in an account at a particular time.
Principal: The amount first deposited or invested.
Interest: Money earned on an account balance.
Rate: The percent used to calculate interest for a stated period.
Period: One time interval used for an interest calculation, such as one month or one year.
Compound interest: Interest calculated by repeating interest steps while adding earned interest to the balance.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B2.2. 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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