B2.5 Calculate the compound-interest cost of credit purchases

Ontario Grade 11 Mathematics

Personal Finance

MBF3C study topic B2.5: finding how much interest adds to a purchase

Buying on credit lets a person receive an item before paying its full price. If the unpaid balance earns compound interest, interest is added to the balance and later interest can be charged on that added amount too. This lesson uses a clear model: the balance is left unpaid for a stated period, the rate stays the same, and interest is compounded monthly. There are no extra fees or payments during that period. These assumptions matter because they tell us what the calculation represents.

What you will learn

Identify the amount of a purchase that is financed with credit.
Use a compound-interest model to calculate the balance after a stated time.
Find the interest cost and distinguish it from the total amount repaid.

1. Review the purchase and the interest rate

The cash price is the price paid if the buyer pays for the item without borrowing. A credit purchase may have a down payment, which is the amount paid at the time of purchase. The financed amount is the part of the price borrowed on credit.

For example, if an item costs CAD 900 and the buyer pays CAD 150 up front, the financed amount is CAD 750. Interest is calculated on the amount borrowed, not on the part already paid.

An annual interest rate describes interest for one year. If interest is compounded monthly, the annual rate must be changed to a monthly rate, and the time must be counted in months. For an annual rate of 12%, the monthly rate is 1%, because one year has 12 months. In calculations, write 1% as 0.01.

P=\text{cash price}-\text{down payment}

Financed amount = cash price − down payment.
Convert a percent rate to a decimal before using it in a calculation.
For monthly compounding, use a monthly rate and a number of months.

2. How monthly compounding grows a balance

Compound interest means that interest is added to the balance, and the next interest calculation uses the new balance. With monthly compounding, the lender calculates interest each month. The balance after one month is the starting balance multiplied by one plus the monthly rate.

The same change is repeated each month. Rather than calculate every month separately, use the compound-interest model. In the model,

P

is the financed amount,

i

is the monthly rate written as a decimal, and

n

is the number of months. The result,

A

, is the balance after those months.

This model assumes the financed balance is not reduced by payments during the stated period. It also assumes there are no fees and the rate does not change. It describes the compound-interest cost for that situation; it is not a payment schedule for a plan where regular payments reduce the balance.

A=P(1+i)^n

Monthly rate = annual rate ÷ 12, when the stated annual rate is used for monthly compounding.
Number of compounding periods = number of months.
The model gives the balance, including interest, not just the interest.

3. From the balance to the cost of credit

After finding the balance, subtract the financed amount to find the compound interest charged. This works because the balance contains the original borrowed amount plus the interest added to it.

If there was a down payment, add it to the final balance to find the total amount paid for the item under this model. Compare that total with the cash price: the difference is the interest cost. The down payment is not interest; it is part of the purchase price paid earlier.

Keep full calculator values during the calculation and round the final money amounts to the nearest cent. Check that the final balance is larger than the financed amount when the rate and time are both positive. Also check that the total paid equals the down payment plus the final balance.

\text{interest cost}=A-P

Interest cost = final balance − financed amount.
Total paid = down payment + final balance.
Under the stated assumptions, the difference between total paid and cash price is the interest cost.

4. A practical calculation routine

First identify the cash price and any down payment. Find the financed amount by subtracting the down payment from the cash price. Next convert the annual rate to a monthly decimal rate and count the months in the borrowing period.

Substitute these values into the compound-interest model. Use a calculator to evaluate the power and multiply by the financed amount. Then subtract the financed amount from the final balance to find the interest. If the question asks for the total amount paid, include the down payment as well.

A calculator can help with repeated monthly growth, but it cannot decide which amount is borrowed or what the question means by cost. Label the balance, interest, and total paid so they are not confused.

\text{total paid}=\text{down payment}+A

Translate the purchase details before entering values into a calculator.
Keep the monthly rate and time unit consistent.
State whether the answer is the interest cost, final balance, or total paid.

Worked example

A purchase with no down payment

A tablet costs CAD 1,200. The full price is put on credit at an annual interest rate of 18%, compounded monthly. The balance is left unpaid for 2 years, with no fees or payments during that time. Find the interest cost and the total amount due.

Identify the borrowed amount
There is no down payment, so the entire cash price is financed. This is the starting balance for the interest calculation.
$P=1200$
Convert the rate and time
There are 12 months in a year, so divide the annual rate by 12 to get the monthly rate. Two years contain 24 monthly compounding periods.
$i=0.18\div12=0.015,\quad n=2\times12=24$
Calculate the final balance
Use the monthly rate for each of the 24 periods. The balance includes both the original borrowed amount and all interest added.
$A=1200(1+0.015)^{24}\approx1715.40$
Find the interest cost
Subtract the original financed amount from the final balance. Since there was no down payment, the final balance is also the total amount due.
$1715.40-1200=515.40$

Answer: The compound-interest cost is CAD 515.40. The total amount due is CAD 1,715.40.

Check: The balance is greater than CAD 1,200, as expected for a positive rate over 24 months. Adding the interest to the starting amount gives CAD 1,715.40.

Worked example

A purchase with a down payment

A bicycle has a cash price of CAD 2,400. The buyer pays CAD 400 up front and puts the rest on credit at an annual rate of 12%, compounded monthly. The credit balance is left unpaid for 18 months, with no fees or payments during that time. Find the interest cost and the total amount paid, including the down payment.

Find the financed amount
The buyer has already paid CAD 400, so only the remainder of the cash price is borrowed. Interest applies to this financed amount.
$P=2400-400=2000$
Convert the rate and time
Divide the annual rate by 12 for monthly compounding. The borrowing period is already given in months.
$i=0.12\div12=0.01,\quad n=18$
Calculate the final balance
Apply the monthly growth for 18 periods to the financed amount. The result is the balance due after the stated period.
$A=2000(1+0.01)^{18}\approx2392.29$
Find the interest and total paid
Subtract the financed amount from the balance to find interest. Then add the down payment to the balance to find the total paid for the bicycle.
$2392.29-2000=392.29,\quad 2392.29+400=2792.29$

Answer: The compound-interest cost is CAD 392.29. Including the down payment, the total amount paid is CAD 2,792.29.

Check: The total paid is CAD 392.29 more than the CAD 2,400 cash price. That difference matches the interest cost.

Common mistakes and how to avoid them

Using the full cash price as the starting balance even when there is a down payment.

Correction: Subtract the down payment first. Compound interest applies to the amount financed.

Using the annual rate as the monthly rate.

Correction: For monthly compounding, divide the stated annual rate by 12 and write the monthly percent as a decimal.

Calling the final balance the interest cost.

Correction: The final balance includes the borrowed amount. Subtract the financed amount to isolate the interest.

Adding the down payment to the interest cost.

Correction: The down payment is part of the cash price, not interest. Add it only when finding the total paid.

Using this model for a balance that has regular payments without accounting for those payments.

Correction: This lesson's model assumes no payments during the stated period. Payments would change the balance on which later interest is calculated.

Lesson summary

Subtract any down payment from the cash price to find the amount financed.
For monthly compounding, use the monthly decimal rate and the number of months.
Calculate the final balance with the compound-interest model.
Subtract the financed amount from the balance to find interest cost.
Add the down payment to the final balance to find total paid.

Check your understanding

Question 1

An item costs CAD 800. A buyer pays CAD 200 up front. The remaining amount is left unpaid for 12 months at 12% annual interest compounded monthly, with no fees or payments. What is the interest cost, to the nearest cent?

CAD 72.00
CAD 74.63
CAD 96.00
CAD 274.63

Show answer and explanation

CAD 74.63

The financed amount is CAD 600, the monthly rate is 0.01, and there are 12 periods. The balance is about CAD 676.03, so the interest cost is about CAD 76.03. Correction: this calculation shows the correct option must be CAD 76.03; the listed options do not include it.

Question 2

An item costs CAD 800. A buyer pays CAD 200 up front. The remaining amount is left unpaid for 12 months at 12% annual interest compounded monthly, with no fees or payments. What is the interest cost, to the nearest cent?

CAD 72.00
CAD 76.03
CAD 96.00
CAD 276.03

Show answer and explanation

CAD 76.03

The financed amount is CAD 600. The monthly rate is 0.01, so the final balance is about CAD 676.03. Subtracting CAD 600 gives an interest cost of CAD 76.03.

Key terms

Cash price: The price of an item when it is paid for without borrowing.
Down payment: An amount paid at the time of purchase, before the rest is financed.
Financed amount: The part of the purchase price borrowed on credit.
Compound interest: Interest added to a balance, so later interest is calculated on the increased balance.
Compounded monthly: Interest is added to the balance once each month.
Final balance: The amount owed after the stated compounding periods, including interest.

Continue through MBF3C

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation B2.5. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

B2.5 · Calculate the compound-interest cost of credit purchases