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B3.4 · Investigate how repayment time affects principal and interest

Learn to investigate how repayment time affects principal and interest through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

A Grade 11 MEL3E lesson on comparing loan repayment plans

Imagine borrowing money to replace a work tool or cover an unexpected expense. A longer repayment plan may make each payment smaller, but the loan stays unpaid for more months. Interest is the cost of borrowing. While a balance remains, interest can be added, so a longer repayment time can mean more interest overall. In this lesson, you will compare repayment plans by tracking how each payment reduces the amount still owed. Principal means the part of the loan that was borrowed, not including interest. The starting principal matters: to see the effect of repayment time fairly, compare plans with the same starting principal and interest rate.

What you will learn

Explain how repayment time affects the amount of interest paid.
Use repeated calculations or a spreadsheet to track a loan balance.
Compare repayment plans while keeping the starting loan and interest rate the same.
Use both total interest and payment size when making a repayment decision.

1. Follow where each payment goes

A loan balance is the amount still owed. In a typical repayment schedule, interest for a period is worked out from the balance at the start of that period. The payment first covers that interest. The rest reduces the principal. This explains why making a payment does not always reduce the balance by the full payment amount.

For example, suppose a balance is CAD 1,000 and the interest for the month is CAD 10. If the payment is CAD 260, CAD 10 covers interest and CAD 250 reduces the amount borrowed. The new balance is CAD 750. The same steps are repeated for the next month, using the new balance.

When the balance gets small, the final payment is usually less than the regular payment. It only needs to cover the remaining balance and that period's interest. This prevents paying more than is owed.

\text{new balance} = \text{old balance} + \text{interest} - \text{payment}

Interest is charged while money is still owed.
Subtract the interest portion from the payment to find how much principal is repaid.
A lower balance usually means less interest in the next period when the rate is unchanged.

2. Use repeated calculations or a spreadsheet

A repayment schedule is a list of payment periods and the amounts that make up each payment. A calculator is enough for a short schedule. For many periods, a spreadsheet can repeat the same calculation in rows.

A useful spreadsheet can have columns for the starting balance, interest, payment, principal repaid, and ending balance. In each row, calculate interest from the starting balance. Subtract that interest from the payment to find the principal repaid. Then subtract the principal repaid from the starting balance to get the ending balance. Carry that ending balance into the next row as its starting balance.

For a fair comparison, keep the starting principal and interest rate the same. Change the payment plan so one loan takes more periods to repay. If the rate is given annually but interest is calculated monthly, use the monthly rate provided by the lender or assignment. In the examples here, the monthly rate is given directly. Real loan agreements may use different fees, rates, and calculation rules, so check the actual terms.

Repeat the same calculation for each period.
Use the ending balance of one period as the starting balance of the next.
Compare plans with the same loan amount and interest rate to focus on repayment time.

3. Decide what the comparison shows

A longer repayment time often comes with smaller regular payments. But because the balance remains for more periods, more interest may build up before the loan is cleared. The total paid is the sum of all payments. To find the total interest, subtract the original principal from the total paid, assuming there are no extra fees.

Do not judge a plan by the regular payment alone. A smaller payment can be easier to manage each month, but it may cost more in total interest. A larger payment can clear the loan sooner and lower total interest, but it must still fit the borrower's budget.

The word often matters: the comparison depends on the loan's terms. Use the schedule or lender's information rather than assuming every loan works the same way. When examining a repayment plan, check the regular payment, the number of periods, the total paid, and the total interest.

\text{total interest} = \text{total paid} - \text{starting principal}

More repayment periods can mean more total interest.
A smaller regular payment is not automatically the lower-cost choice.
Balance affordability against the total cost shown by the repayment schedule.

Example 2 repayment comparison

Monthly payment	Payoff time	Total paid	Total interest
CAD 510	2 months	CAD 1,015.00	CAD 15.00
CAD 260	4 months	About CAD 1,024.90	About CAD 24.90

Worked example

Example 1: Compare two plans for a work laptop

A worker borrows CAD 1,200. The monthly interest rate is 1%, and there are no fees. Compare a CAD 220 monthly payment plan with a CAD 130 monthly payment plan. Find how long each takes and compare the total interest.

Calculate the first month's interest
Interest for the first month is 1% of CAD 1,200. One percent is the same as 0.01, so the first month's interest is CAD 12.
$1{,}200 \times 0.01 = 12$
Track the CAD 220 plan
In month 1, CAD 12 of the payment covers interest, leaving CAD 208 to reduce the balance to CAD 992. Repeat the calculation using each new balance. The first five payments are CAD 220; after the fifth, the balance is about CAD 138.99. In month 6, interest is about CAD 1.39, so the final payment is about CAD 140.38.
$1{,}200 \rightarrow 992 \rightarrow 781.92 \rightarrow 569.74 \rightarrow 355.44 \rightarrow 138.99 \rightarrow 0$
Find the total for the faster plan
There are five regular payments of CAD 220 and one final payment of about CAD 140.38. Add them to find the total paid, then subtract the original CAD 1,200 to find total interest.
$5(220) + 140.38 = 1{,}240.38$
Track and total the CAD 130 plan
Repeat the same monthly steps with a CAD 130 payment. After nine regular payments, the balance is about CAD 94.51. The tenth month's interest is about CAD 0.95, making the final payment about CAD 95.46. The plan takes longer and has more total interest.
$9(130) + 95.46 = 1{,}265.46$

Answer: The CAD 220 plan takes 6 months and costs about CAD 40.38 in interest. The CAD 130 plan takes 10 months and costs about CAD 65.46 in interest. The longer plan has smaller regular payments, but its total interest is about CAD 25.08 more.

Check: For each plan, subtract the original CAD 1,200 from the total paid. The results are CAD 40.38 and CAD 65.46. The final payments include the last period's interest.

Worked example

Example 2: Choose between two plans for a repair

A student borrows CAD 1,000 for an essential repair. The monthly interest rate is 1%, with no fees. A spreadsheet shows two choices: CAD 510 per month or CAD 260 per month. Use repeated calculations to compare their payoff time and interest.

Calculate the first period for each plan
Both plans start with the same CAD 1,000 balance, so the first month's interest is CAD 10. With a CAD 510 payment, CAD 500 reduces principal, leaving CAD 500. With a CAD 260 payment, CAD 250 reduces principal, leaving CAD 750.
$1{,}000 + 10 - 510 = 500$
Finish the CAD 510 plan
In month 2, interest on CAD 500 is CAD 5. The remaining amount owed is CAD 505, so the final payment is CAD 505. The total paid is CAD 1,015, which is CAD 15 more than the original principal.
$1{,}000 + 15 = 1{,}015$
Finish the CAD 260 plan
Continue using the new balance each month. After three CAD 260 payments, the balance is about CAD 242.48. Month 4 interest is about CAD 2.42, so the final payment is about CAD 244.90. This plan takes 4 months, and its total is about CAD 1,024.90.
$3(260) + 244.90 = 1{,}024.90$
Compare time and cost
The CAD 510 plan takes 2 months and has CAD 15 in interest. The CAD 260 plan takes 4 months and has about CAD 24.90 in interest. The faster plan costs less interest, but its regular payment is much larger. A suitable choice must also fit the borrower's budget.
$1{,}024.90 - 1{,}000 = 24.90$

Answer: The CAD 510 plan takes 2 months and costs CAD 15 in interest. The CAD 260 plan takes 4 months and costs about CAD 24.90 in interest. The longer plan costs about CAD 9.90 more in interest, while requiring smaller regular payments.

Check: The two plans use the same starting principal and rate. Their different payoff times and total interest can therefore be compared directly.

Common mistakes and how to avoid them

Subtracting the entire payment from the balance.

Correction: First account for that period's interest. Only the rest of the payment reduces principal.

Assuming a smaller monthly payment means a cheaper loan.

Correction: Add all payments and compare total interest. A smaller payment may mean more repayment periods and more interest.

Comparing plans with different starting amounts or interest rates and blaming the difference only on repayment time.

Correction: To study the effect of repayment time fairly, hold the starting principal and interest rate constant.

Making the final payment equal to every earlier payment.

Correction: Calculate the remaining balance and final period's interest. The final payment is only the amount still owed.

Lesson summary

Each period, interest is added to the balance and the payment is subtracted.
The part of the payment left after interest reduces principal.
Compare repayment time by tracking the balance repeatedly or using a spreadsheet.
A longer plan often lowers regular payments but can increase total interest.
Use total interest and payment affordability together when comparing plans.

Check your understanding

Question 1

Two plans have the same starting principal and monthly interest rate. One takes more months to repay. What is the best way to compare their cost?

Compare only the regular payments.
Compare total paid and total interest from the repayment schedules.
Choose the plan with the most payment periods.
Ignore interest if the monthly payments are affordable.

Show answer and explanation

Compare total paid and total interest from the repayment schedules.

The regular payment alone does not show the full cost. Summing payments and finding total interest makes the comparison clearer.

Question 2

A loan balance is CAD 600. The period's interest is CAD 6 and the payment is CAD 106. How much principal does the payment reduce?

CAD 6
CAD 100
CAD 106
CAD 112

Show answer and explanation

CAD 100

CAD 6 covers interest, so CAD 100 reduces principal. The new balance is CAD 500.

Question 3

Why might a longer repayment plan have more total interest?

Interest may be charged in more periods while a balance remains.
The starting principal automatically increases each month.
A longer plan always has a higher interest rate.
The final payment is always larger than the first payment.

Show answer and explanation

Interest may be charged in more periods while a balance remains.

When a balance remains for more periods, interest can be charged for more periods. The rate does not have to change for total interest to rise.

Key terms

Balance: The amount still owed on a loan.
Interest: The cost of borrowing money, calculated under the loan's terms.
Principal: The amount borrowed, not including interest.
Repayment schedule: A list showing payments, interest, and the balance over time.
Total paid: The sum of all payments made to repay the loan.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B3.4. 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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