Compare the cost of carrying credit-card balances

Ontario Grade 11 Mathematics

Saving, Investing, and Borrowing

Use clear assumptions, repeated calculations, and a calculator or spreadsheet to compare interest costs.

Imagine you need to put a CAD 800 work uniform and equipment purchase on a credit card. You cannot pay the full amount right away, so some of the balance remains after the payment due date. That remaining amount is a carried balance. The card issuer may charge interest on it. A lower rate can reduce the cost, but the amount borrowed and how long it stays unpaid also matter. In this lesson, you will compare costs using simple, clearly stated examples. Real card agreements may calculate interest differently, so use the rate and instructions shown on the statement or agreement when making a real comparison.

What you will learn

Explain what it means to carry a credit-card balance.
Estimate monthly interest using a stated annual interest rate and monthly rate.
Compare the interest cost of two credit-card choices over the same period.
Recognize why the balance, interest rate, payment, and number of months matter.

1. Understand what you are comparing

A credit-card balance is the amount you owe on the card. Carrying a balance means you do not pay the full amount by the due date and some debt remains. Interest is the charge for borrowing that money.

The annual interest rate, often called the annual rate, is the rate shown for a full year. For an estimate in this lesson, divide that rate by 12 to get a monthly rate. For example, an annual rate of 18% gives a monthly rate of 1.5%. This is a classroom estimate, not a promise that every card calculates interest this way.

To make a fair comparison, use the same starting balance and the same number of months when comparing rates. If you compare different payment plans, also keep the starting balance and annual rate the same. Otherwise, you may not know which difference caused the cost to change.

This estimate leaves out purchases, fees, and changes in the issuer’s calculation method. It assumes the balance stays the same during each month until interest is added. For an actual card, check the agreement for how interest is calculated and whether other charges apply.

\text{monthly interest} = \text{balance} \times \text{monthly rate}

Carrying a balance can lead to interest charges.
State the assumptions before estimating a cost.
A fair comparison changes one factor at a time.

2. Estimate interest with repeated calculations

A calculator is enough for a short estimate. First change a percentage into a decimal: 1.5% is 0.015. Multiply the balance by the monthly rate to estimate one month’s interest.

If the interest is added to the balance and no payment is made, the next month’s interest is slightly higher because it is calculated on a larger balance. This is why it is useful to repeat the calculation month by month instead of multiplying one month’s interest by the number of months.

A spreadsheet can do the same steps. Put the starting balance in one cell. In the next row, calculate interest by multiplying that month’s balance by the monthly rate, then add the interest to the balance. Repeat for each month. Keep a separate total of the interest amounts if you want to compare the cost of borrowing.

For a simplified estimate that includes a payment, subtract the payment from the balance after adding that month’s interest. If the payment is larger than the amount owed, the balance becomes zero; do not treat it as a negative debt. This lesson uses a payment made at the end of each month. A real card may use different timing or rules.

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

Convert a percent to a decimal before multiplying.
When interest is added and no payment is made, use the new balance for the next month.
For a payment comparison, use the same payment timing and amount in each option.

3. Decide which option costs less

Compare the total interest paid or added over the same number of months. Do not compare only the annual rates if the balances or payment plans are different. A lower annual rate is helpful, but it does not by itself tell the full cost when the amount owed or time period changes.

Round money to the nearest cent at each step, or follow the rounding method requested by the card agreement. Use the same method for both choices. Small rounding differences can appear if a calculator or spreadsheet keeps extra decimal places.

Before choosing, ask: What is the starting balance? What rate applies? How many months will the balance be carried? What payment will be made, and when? Are there fees or other card rules not included in the estimate? These questions help you avoid treating an estimate as an exact statement amount.

Compare like with like over the same time period.
The total interest is the borrowing cost in these examples.
Check the actual agreement before relying on an estimate.

Example 2 balance and interest record

Plan	Month 1 interest	Balance after month 1 payment	Month 2 interest	Two-month interest total
CAD 100 monthly payment	CAD 6.00	CAD 506.00	CAD 5.06	CAD 11.06
CAD 50 monthly payment	CAD 6.00	CAD 556.00	CAD 5.56	CAD 11.56

Worked example

Example 1: Compare two rates for the same balance

A person carries CAD 900 for two months and makes no payment. Card A has an annual rate of 18%; Card B has an annual rate of 24%. Estimate and compare the interest added. Assume interest is added monthly, and round each month to the nearest cent.

Find the monthly rates
Divide each annual percentage rate by 12 because this estimate uses one-twelfth of the annual rate for each month. Change the monthly percentages to decimals for multiplying.
$18\% \div 12 = 1.5\% = 0.015;\quad 24\% \div 12 = 2\% = 0.02$
Calculate Card A
For month one, multiply the CAD 900 balance by 0.015. Add that interest to the balance. For month two, use the updated balance because the first month’s interest remains unpaid.
$900.00(0.015)=13.50;\quad 913.50(0.015)=13.70$
Calculate Card B
Use the same process for Card B, but use its monthly rate of 0.02. The second month’s interest is based on the balance after the first month’s interest was added.
$900.00(0.02)=18.00;\quad 918.00(0.02)=18.36$
Compare total interest
Add the two monthly interest amounts for each card. The starting balance and time period are the same, so the totals show which rate costs less under these assumptions.
$13.50+13.70=27.20;\quad 18.00+18.36=36.36$

Answer: Card A adds an estimated CAD 27.20 in interest. Card B adds an estimated CAD 36.36. Card A costs CAD 9.16 less over the two months under these assumptions.

Check: Card B’s rate is higher, so its estimated monthly interest is higher on the same balance. The difference in total interest is CAD 36.36 − CAD 27.20 = CAD 9.16.

Worked example

Example 2: Compare payment plans on one card

A worker owes CAD 600 on a card with an annual rate of 12%. Compare paying CAD 100 at the end of each month with paying CAD 50 at the end of each month for two months. Estimate total interest added during the two months. Use a monthly rate of 1%, round each month to the nearest cent, and do not let the balance go below zero.

Set the monthly rate
Divide the annual rate by 12, then write the result as a decimal. Both plans use the same rate and starting balance, so the payment amount is the factor being compared.
$12\% \div 12 = 1\% = 0.01$
Follow the CAD 100 plan
Month one interest is CAD 6.00. Add it to the balance, then subtract the CAD 100 payment. In month two, calculate interest on the resulting balance, add it, and subtract the next payment.
$600.00(0.01)=6.00;\quad 600.00+6.00-100.00=506.00;\quad 506.00(0.01)=5.06;\quad 506.00+5.06-100.00=411.06$
Follow the CAD 50 plan
Repeat the same steps, changing only the payment. The smaller payment leaves a larger balance for the second month’s interest calculation.
$600.00(0.01)=6.00;\quad 600.00+6.00-50.00=556.00;\quad 556.00(0.01)=5.56;\quad 556.00+5.56-50.00=511.56$
Compare interest added
Add the interest amounts for each plan. This compares the interest cost for the same two-month period, although the plans also leave different balances unpaid.
$6.00+5.06=11.06;\quad 6.00+5.56=11.56$

Answer: The CAD 100 payment plan adds CAD 11.06 in estimated interest over two months. The CAD 50 payment plan adds CAD 11.56. The larger payment plan adds CAD 0.50 less interest during this period and also leaves a lower balance.

Check: Both plans have the same first month’s interest because they start with the same balance. The second month’s interest is lower with the CAD 100 payment because more of the balance was paid down.

Common mistakes and how to avoid them

Using the annual rate as the monthly rate.

Correction: For these estimates, divide the annual rate by 12 before calculating monthly interest.

Using the starting balance for every month even after unpaid interest is added.

Correction: When no payment is made, calculate the next month’s interest on the updated balance.

Comparing two cards with different starting balances and deciding the rate alone caused the difference.

Correction: Keep the balance and time period the same when comparing rates, or clearly identify every difference.

Treating a classroom estimate as the exact amount a card issuer will charge.

Correction: Check the card agreement for its rate, calculation method, payment timing, and any other charges.

Lesson summary

Carrying a balance means some card debt remains unpaid and may be charged interest.
Estimate the monthly rate from the stated annual rate, then calculate interest using the balance for that month.
Repeat the calculation when interest is added or payments change the balance.
Compare total interest over the same period and state any assumptions.

Check your understanding

Question 1

A balance of CAD 400 is carried for one month at a stated monthly rate of 1.5%, with no payment. What is the estimated interest for that month?

CAD 0.60
CAD 6.00
CAD 15.00
CAD 60.00

Show answer and explanation

CAD 6.00

Change 1.5% to 0.015 and multiply: CAD 400 × 0.015 = CAD 6.00.

Question 2

Two cards have the same starting balance and are compared for the same number of months. One has a lower rate. Which statement is best?

The lower rate must cost more because it is smaller.
The lower rate generally gives less interest under the same assumptions.
The rates do not matter if the balance is the same.
The card with the lower rate always has no fees.

Show answer and explanation

The lower rate generally gives less interest under the same assumptions.

With the same balance, time period, and calculation assumptions, the lower rate produces less estimated interest. The rate alone does not tell you whether there are fees.

Question 3

A person pays more toward the same balance at the end of month one. Why can that lower month two’s interest?

The payment makes the annual rate disappear.
The payment reduces the balance used for the next interest estimate.
The payment changes the number of days in the month.
The payment means interest is always refunded.

Show answer and explanation

The payment reduces the balance used for the next interest estimate.

A larger payment leaves less debt unpaid, so the next month’s estimated interest is calculated on a smaller balance.

Key terms

Balance: The amount owed on a credit-card account.
Carrying a balance: Leaving some of the amount owed unpaid instead of paying the full balance.
Interest: A charge for borrowing money.
Annual interest rate: The interest rate stated for a year; in these examples it is divided by 12 to estimate a monthly rate.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MEL3E), expectation B3.1. 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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