B2.5 Solve saving and investing problems involving compound interest

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

MEL3E — Specific expectation B2.5

Imagine putting CAD 500 into an account that pays interest. If the account earns interest and keeps it in the account, the next interest payment is based on a larger balance. This is compound interest. In this lesson, you will find balances by calculating one period at a time. You can do this with a calculator or a spreadsheet. The same process works for savings and investments when the interest rate and compounding schedule are known.

What you will learn

Explain how compound interest adds interest to a balance over time.
Calculate a new balance by repeating the interest calculation for each period.
Use a calculator or spreadsheet to solve saving and investing problems involving compound interest.
Compare results while keeping the starting amount, rate, and time in view.

1. What compound interest does

Interest is money an account earns for keeping money in it. The original amount deposited is called the principal. A balance is the total amount in the account at a particular time, including any interest already added.

With compound interest, earned interest stays in the account. In the next period, interest is calculated on the new balance, not just on the original deposit. That means the amount of interest can grow from one period to the next.

A compounding period is the time between interest calculations. If interest is added once each year, the period is one year. If it is added each month, the period is one month. Always check how often an account adds interest before calculating.

A rate is often written as a percent. To calculate interest, change the percent to a decimal by dividing by 100. For example, a yearly rate of 4% is 0.04. Interest for one year on CAD 500 at that rate is CAD 20, because 500 × 0.04 = 20.

I=B× r

Compound interest is interest calculated on a balance that includes earlier interest.
The period and rate must match: use a rate for the same length of time as the period.
Keep the interest in the account when the problem asks for the account balance.

2. Calculate one period at a time

For each period, multiply the starting balance for that period by the period's interest rate. This gives the interest earned. Add that interest to the starting balance to get the next balance. Then repeat with the new balance.

For annual compounding, use the annual rate once for each year. For monthly compounding, first find the monthly rate by dividing the annual rate by 12, if the account states that its annual rate is divided evenly across 12 monthly periods. Then use that monthly rate for each month.

A calculator helps with repeated multiplication and addition. Keep full calculator values between periods when possible, and round money to the nearest cent when reporting the final balance. Rounding after every period can cause small differences.

A spreadsheet can organize the same work. Put the period number in one column, the starting balance in the next, the interest earned in another, and the ending balance in the last. Each period's starting balance is the previous period's ending balance. This makes it easier to spot a copied or arithmetic error.

B_{\text{next}}=B_{\text{current}}+(B_{\text{current}}× r)

Find interest from the current balance, not always from the original deposit.
Add the interest before moving to the next period.
Match the interest rate to the length of the period.

3. Read the problem and choose the calculation

Before using a calculator, identify the starting balance, the interest rate, how often interest is added, and the number of periods. These details tell you how many times to repeat the calculation and which rate to use each time.

If a question asks how much interest was earned, subtract the original amount deposited from the final balance, provided no extra deposits or withdrawals were made. If there are extra deposits or withdrawals, account for their timing as well; a deposit made later does not earn interest for periods before it was added.

Compare accounts carefully. A higher rate does not by itself tell you the final balance if the starting amounts, time, or compounding schedules differ. To make a fair comparison, use the same starting amount and time, and follow each account's stated schedule.

The examples use no additional deposits or withdrawals. This keeps the focus on how compound interest changes the balance.

List the account details before calculating.
Count the compounding periods in the time given.
Do not compare balances without checking starting amount, time, rate, and schedule.

4. Use a calculator or spreadsheet as a check

A calculator is useful for each period's interest and updated balance. Write down the balance after each period so you can tell what amount was used next. When a result seems surprising, recalculate one period at a time rather than starting over with a different method.

In a spreadsheet, a cell can refer to the balance in the row above. For instance, if the starting balance is in cell B2 and the period rate is in cell C2, the interest cell can multiply those cells. The ending balance cell adds the interest to the starting balance. In the next row, use the previous ending balance as the new starting balance.

A spreadsheet does not decide whether the rate or period is correct. Check the account information first, then verify that the first and second rows use the balance you expect. Technology helps with repeated arithmetic, but you are responsible for setting up the calculation correctly.

Keep a record of each period's balance.
Check that the next period starts with the previous period's ending balance.
Use technology to calculate, then review the setup and result.

Mina's yearly balance

Year	Starting balance	Interest for year	Ending balance
1	CAD 800.00	CAD 24.00	CAD 824.00
2	CAD 824.00	CAD 24.72	CAD 848.72
3	CAD 848.72	CAD 25.46	CAD 874.18

Worked example

Savings account with yearly compounding

Mina deposits CAD 800 in a savings account earning 3% interest per year. Interest is added once each year. Find the balance after 3 years and the interest earned.

Identify the period rate
The account adds interest once each year, so use the 3% annual rate for each yearly calculation. As a decimal, the rate is 0.03.
$3%=0.03$
Calculate year 1
The first year's interest is based on the original CAD 800. Add that interest to find the balance that will earn interest in year 2.
$800\times0.03=24, 800+24=824$
Calculate year 2
Use CAD 824 because the first year's interest stayed in the account. Add the second year's interest to get the next balance.
$824\times0.03=24.72, 824+24.72=848.72$
Calculate year 3
Use the year 2 balance as the starting balance for year 3. Round the final money amount to the nearest cent.
$848.72\times0.03=25.4616, 848.72+25.4616=874.1816$
Find interest earned
With no other deposits or withdrawals, subtract the original deposit from the final balance. This isolates the amount added as interest.
$874.18-800=74.18$

Answer: After 3 years, Mina has CAD 874.18. The account earned CAD 74.18 in interest.

Check: Each year's interest is slightly larger because it is calculated on a larger balance: CAD 24, CAD 24.72, then about CAD 25.46. The final balance is greater than the original deposit.

Worked example

Investment with monthly compounding

A worker invests CAD 1,200 in an account with a stated annual interest rate of 6%, compounded monthly. Assume the annual rate is divided evenly across 12 months. Find the balance after 3 months, with no extra deposits or withdrawals.

Find the monthly rate
The account compounds monthly, so divide the annual decimal rate by 12 to get the rate for one month.
$0.06\div12=0.005$
Calculate month 1
Apply the monthly rate to CAD 1,200. Add the interest to get the amount used for month 2.
$1200\times0.005=6, 1200+6=1206$
Calculate month 2
The month 1 interest remains invested, so use CAD 1,206 as the next starting balance.
$1206\times0.005=6.03, 1206+6.03=1212.03$
Calculate month 3
Use the month 2 balance for the third calculation. Round the final result to the nearest cent.
$1212.03\times0.005=6.06015, 1212.03+6.06015=1218.09015$
Report the balance
The balance after three monthly periods is the ending balance from month 3. Rounding to cents gives the amount to report.
$1218.09015\approx1218.09$

Answer: After 3 months, the investment balance is CAD 1,218.09.

Check: The monthly rate is 0.5%, not 6%. Three monthly interest additions increase the balance by about CAD 18.09, which is reasonable for this short time.

Common mistakes and how to avoid them

Calculating every period's interest from the original deposit.

Correction: Use the latest balance each time. Earlier interest stays in the account and earns interest in later periods.

Using an annual rate as if it were the monthly rate.

Correction: For monthly compounding, follow the account's stated method to find a monthly rate. In the example, the annual rate is divided by 12.

Adding the interest to the balance but not using the new balance in the next period.

Correction: Carry each ending balance forward as the next period's starting balance.

Rounding every calculation too early.

Correction: Keep the calculator's full values during the repeated calculations when possible, then round the final money amount to cents.

Lesson summary

Compound interest means interest is added to an account balance and later interest is calculated on that larger balance.
For each period, calculate interest from the current balance, add it, and use the new balance for the next period.
Match the interest rate to the compounding period and use a calculator or spreadsheet to keep the repeated calculations organized.
To find interest earned with no other transactions, subtract the original deposit from the final balance.

Check your understanding

Question 1

An account starts with CAD 400 and earns 2% for one year. Interest is added yearly. What is the balance after the first year?

CAD 408
CAD 402
CAD 480
CAD 400.02

Show answer and explanation

CAD 408

Two percent of CAD 400 is CAD 8. Add the interest to the starting balance to get CAD 408.

Question 2

An account compounds monthly. What should you do with a stated annual rate of 4%, assuming it is divided evenly across the months?

Use 4% for each month.
Divide 4% by 12 to find the monthly rate.
Multiply 4% by 12 to find the monthly rate.
Use 4% only once, no matter how many months pass.

Show answer and explanation

Divide 4% by 12 to find the monthly rate.

The rate used for a period must match that period. Dividing the annual rate by 12 gives the monthly rate under the stated assumption.

Question 3

A balance is CAD 510 after one period. The next period earns interest on which amount?

Only the original deposit
The interest earned in the first period alone
CAD 510
Zero, because interest was already added

Show answer and explanation

CAD 510

The latest balance includes the original deposit and any interest already added. Compound interest uses that current balance.

Key terms

Principal: The original amount deposited or invested.
Balance: The total amount in an account at a particular time.
Interest: Money earned by keeping money in an account.
Compounding period: The length of time between interest calculations, such as one month or one year.
Compound interest: Interest calculated on a balance that includes interest added in earlier periods.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B2.5. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B2.5 · Solve saving and investing problems involving compound interest

Ontario Grade 11 Mathematics

MEL3E — Specific expectation B2.5

What you will learn

1. What compound interest does

2. Calculate one period at a time

3. Read the problem and choose the calculation

4. Use a calculator or spreadsheet as a check

Mina's yearly balance

Savings account with yearly compounding

Investment with monthly compounding

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MEL3E

About this lesson