B2.1 Investigate and solve simple-interest problems

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

MEL3E, Grade 11 | Study-guide label: B2.1

A worker might borrow money to replace a tool, or save money toward a purchase. In either case, interest is an amount connected to the money and the time involved. Simple interest is calculated on the original amount each period. The interest does not become part of the amount used to calculate later interest. You can solve these problems with repeated calculations, a calculator, or a spreadsheet. The main tasks are to identify the information given, keep the time units clear, and answer the question that was asked.

What you will learn

Explain how simple interest is calculated from the original amount saved or borrowed.
Use repeated calculations, a calculator, or a spreadsheet to find interest and a total amount.
Solve simple-interest problems by identifying the original amount, rate, and time.
Check whether a simple-interest answer makes sense in its context.

1. Understand the amounts and rate

The principal is the original amount saved or borrowed. The interest rate tells what percentage of the principal is earned or charged in a stated period. The time tells how long the money is saved or borrowed. Interest is the extra money earned or paid. The total amount is the principal together with the interest.

A rate of 5% per year means 5 for every 100 of the principal for one year. To find the interest, first find 1% of the principal by dividing the principal by 100. Then multiply that amount by 5. For example, if the principal is CAD 600, then 1% of the principal is CAD 6, so 5% is CAD 30.

The percentage itself is not a money amount. The amount of money that is 1% depends on the principal. If the principal changes, the value of 1% changes too.

With simple interest, each year's interest is based on the original principal. If CAD 30 is earned in the first year on CAD 600, the second year's interest is still calculated from CAD 600. It is not calculated from CAD 630. When the principal and rate stay the same, the interest for each equal period stays the same.

1% of principal=

\frac{principal}{100}

Principal means the original amount.
A yearly rate applies to each year of the stated time.
With simple interest, each period's interest is calculated from the original principal.

2. Build a repeated-calculation record

A year-by-year record helps show how simple interest works. Find the interest for one year from the principal. Add that same interest to the running total at the end of each year. A running total is the amount after each new addition. The total changes, but the principal used to calculate the interest does not.

For example, if one year adds CAD 24 in interest, the next year also adds CAD 24 when the principal and rate are unchanged. After two years, the total interest is CAD 24 plus CAD 24. Writing each year in a row makes it easier to spot an incorrect calculation.

A spreadsheet can keep the principal, yearly interest, and total in separate columns. Enter the principal and the interest for one year. In the next row, add that yearly interest to the previous total. Repeat for as many years as needed. A calculator can be used for the same arithmetic.

For a time given in months, use a matching part of the yearly interest when the problem says to count equal months. Divide the yearly interest by 12 to find one month's interest, then multiply by the number of months. For instance, 6 months is half of 12 months. This method assumes the stated rate is annual and the problem uses equal monthly parts.

\text{new total}=\text{previous total}+\text{period interest}

Calculate one period's interest from the original principal.
Add the same interest for each equal period.
For equal monthly parts, divide yearly interest by 12 before finding interest for a number of months.

3. Choose the calculation and check it

Read the whole problem before calculating. Identify the principal, the rate and its time unit, and the length of time. Then decide what the question asks for: interest alone, the total amount, or another value that can be found from the information given. Keeping the units beside the values helps you notice whether the rate is yearly and the time is in years or months.

For a whole number of years, find the interest for one year and repeat that amount for each year. For a part of a year, find the matching part of the yearly interest if the problem specifies how to count that time. If the question asks for the total saved or repaid, add the interest to the principal. If it asks only for interest, report only the interest.

Check the answer against the situation. With a positive principal, positive rate, and positive time, the interest should be positive. For one year, it should equal the stated percentage of the principal. For two years of simple interest, it should be twice the one-year interest. When asked for a total, the total should be greater than the principal in these positive-rate examples.

A document may give a rate but not say whether it is yearly or how to count part of a year. Do not guess about missing terms. Ask for the needed information before using the numbers to compare a cost or a saving.

\text{total amount}=\text{principal}+\text{interest}

Keep the rate's time unit consistent with the problem's time.
Report interest or total according to what the question asks.
Use the size and repeated pattern of the interest to check your result.

Savings record for CAD 750 at 4% simple interest per year

End of year	Interest added that year	Total in account
Start	—	CAD 750
1	CAD 30	CAD 780
2	CAD 30	CAD 810
3	CAD 30	CAD 840

Worked example

Saving for work boots

A student deposits CAD 750 in an account that pays 4% simple interest per year. Find the interest earned after 3 years and the total amount in the account.

Find one year's interest
Find 1% of the original CAD 750 by dividing by 100. Then multiply that amount by 4 because the rate is 4% per year.
$750\div100=7.50;\quad7.50\times4=30$
Repeat for three years
Each year's interest is calculated from the original CAD 750. Add CAD 30 once for each of the three years.
$30+30+30=90$
Find the account total
The question asks for both interest and the total. Add the interest to the original deposit to find the total amount.
$750+90=840$

Answer: The interest earned is CAD 90. The total after 3 years is CAD 840.

Check: One year earns CAD 30, so three years should earn three times that amount. CAD 90 is three times CAD 30. The total is CAD 90 more than the original deposit.

Worked example

A short-term repair loan

A repair shop borrows CAD 1,200 at 6% simple interest per year. The loan is repaid after 9 months. Find the interest and the total repayment. Assume the yearly interest is counted in equal monthly parts.

Find one year's interest
Find 1% of CAD 1,200 by dividing by 100. Multiply that money amount by 6 to find the interest for one year.
$1{,}200\div100=12;\quad12\times6=72$
Find one month's interest
The problem says to count equal monthly parts. Divide the yearly interest by 12 to find the interest for one month.
$72\div12=6$
Find the interest for nine months
The same monthly interest applies for each of the nine months because the calculation uses the original loan amount. Multiply the monthly interest by 9.
$6\times9=54$
Find the repayment total
Add the interest to the amount borrowed. This gives the total repayment under the terms in the problem.
$1{,}200+54=1{,}254$

Answer: The interest is CAD 54. The total repayment is CAD 1,254.

Check: Nine months is less than one year, so the interest should be less than the full-year interest of CAD 72. CAD 54 is less than CAD 72.

Common mistakes and how to avoid them

Treating 1% as the same money amount for every principal.

Correction: Find 1% from the principal in the problem by dividing that principal by 100. The resulting money amount depends on the principal.

Calculating the next year's interest from the previous year's total.

Correction: For simple interest, calculate each period's interest from the original principal. The interest stays the same when the principal and rate stay the same.

Giving a total when the question asks only for interest.

Correction: Read the final question carefully. Report interest by itself, and add it to the principal only when asked for the total.

Using a full year's interest for a number of months without adjusting it.

Correction: When the problem specifies equal monthly parts, divide the yearly interest by 12 and multiply by the number of months.

Lesson summary

Simple interest is calculated from the original principal each period.
To find a percentage amount, first find 1% of the principal, then use the stated rate.
Repeat the same period interest for each equal period; adjust yearly interest for months when equal monthly parts are specified.
Add interest to the principal only when the total amount is requested.
Check the time units, the repeated pattern, and whether the answer fits the situation.

Check your understanding

Question 1

A deposit of CAD 500 earns 3% simple interest per year. How much interest does it earn in 2 years?

CAD 15
CAD 30
CAD 530
CAD 1,030

Show answer and explanation

CAD 30

One year earns CAD 15 because 1% of CAD 500 is CAD 5 and 3% is CAD 15. Two years earn CAD 15 plus CAD 15, for CAD 30.

Question 2

A loan of CAD 900 has 4% simple interest per year. The one-year interest is CAD 36. What is the interest for 6 months if months are counted equally?

CAD 6
CAD 18
CAD 36
CAD 936

Show answer and explanation

CAD 18

Six months is half of 12 months. Half of the yearly interest of CAD 36 is CAD 18.

Key terms

Principal: The original amount of money saved or borrowed.
Interest: The extra money earned on savings or paid for borrowing.
Interest rate: The percentage used to find interest for a stated period, such as one year.
Simple interest: Interest calculated on the original principal for each period.
Total amount: The principal together with the interest earned or charged.

Continue through MEL3E

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B2.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

B2.1 · Investigate and solve simple-interest problems