F. Financial Literacy · Financial Decisions: Interest, Borrowing, Appreciation, Depreciation and Budgets

How the Interest Rate and the Length of Borrowing Change the Cost

F1.3 A1About 70 minutes

Learning goals

I can explain how an installment loan works: each payment covers that month's interest plus some principal.
I can use a payment formula or a digital tool to find a monthly loan payment.
I can find the total amount paid and the total interest for a loan.
I can compare how different interest rates and lengths of borrowing time change the overall cost of a purchase.

Words to know

installment loan
A loan paid back in equal regular payments (often monthly) over a set time. Car loans and student loans are examples.

term (of a loan)
The length of time you have to pay the loan back, such as 36 or 60 months.

monthly payment
The fixed amount paid each month. Part pays the month's interest and the rest reduces the principal.

total cost of borrowing
The total amount of interest (and any fees) you pay over the whole loan.

financing
Paying for a purchase with a loan, often arranged by the store or dealer.

credit score
A number lenders use to judge how likely you are to repay. A better score usually gets a lower interest rate.

Hook: "Drive it home for only $302 a month!"

A car dealer's ad shows a $20 000 car for "only $301.85 a month". It sounds affordable. But the small print says 84 months at 7%. That is 7 years of payments. How much will the car really cost?

In this lesson you will see how two choices, the interest rate and the length of time you borrow for, change the total cost. (All prices and rates here are examples.)

How an installment loan works

With a car loan you make the same payment every month. Each payment does two jobs:

It pays the interest for that month: balance × i (where i = annual rate ÷ 12).
The rest pays down the principal, so the balance gets smaller.

Because the balance shrinks, the interest part gets smaller each month and the principal part gets bigger. The loan is finished after the last payment.

Worked example

Example 1: The first payment

For a $20 000 loan at 7% per year (compounded monthly) for 36 months, the payment is $617.54. How much of the first payment is interest and how much is principal?

i = 0.07 ÷ 12 ≈ 0.0058333.
Interest for month 1: 20 000 × 0.07 ÷ 12 ≈ $116.67.
Principal paid: 617.54 − 116.67 = $500.87.
New balance: 20 000 − 500.87 = $19 499.13. Next month's interest is a little smaller.

$116.67 is interest and $500.87 goes to the principal.

Finding the payment

Lenders use this formula for the monthly payment (PMT):

PMT = P × i ÷ (1 − (1 + i)⁻ⁿ)

where P is the amount borrowed, i is the rate per month, and n is the number of payments. In Grade 9 you do not need to prove this formula. Use it as a tool, along with other tools that do the same job: a spreadsheet's PMT function (Lesson 7) or a bank's online loan calculator. A negative exponent means "one over": (1 + i)⁻ⁿ = 1 ÷ (1 + i)ⁿ.

Worked example

Example 2: Use the formula

Find the monthly payment for $20 000 at 7% compounded monthly for 36 months. Then find the total paid and the total interest.

i = 0.07 ÷ 12 ≈ 0.0058333 and n = 36.
(1 + i)⁻³⁶ ≈ 0.811079, so 1 − 0.811079 = 0.188921.
P × i = 20 000 × 0.0058333 ≈ 116.667.
PMT ≈ 116.667 ÷ 0.188921 ≈ $617.54.
Total paid = 617.54 × 36 = $22 231.44. Total interest = 22 231.44 − 20 000 = $2 231.44.

Payment $617.54, total paid $22 231.44, total interest $2 231.44.

Term	Monthly payment	Total paid	Total interest
36 months (3 years)	$617.54	$22 231.44	$2 231.44
60 months (5 years)	$396.02	$23 761.20	$3 761.20
84 months (7 years)	$301.85	$25 355.40	$5 355.40

$20 000 borrowed at 7% per year, compounded monthly. Total paid = payment × number of payments.

A longer term makes the monthly payment smaller. This is why ads often show a long term: the payment looks friendly.

But a longer term makes the total interest much bigger. The 7-year loan costs about $3 124 more in interest than the 3-year loan for the same car.

Changing the interest rate

Now keep the term at 5 years and change the rate. Your rate depends on the lender, the economy (remember Lesson 1) and your credit score: people with a strong history of paying on time are usually offered lower rates.

Rate	Monthly payment	Total interest (60 months)
5%	$377.42	$2 645.20
7%	$396.02	$3 761.20
9%	$415.17	$4 910.20

$20 000 borrowed for 60 months. Each extra 2 percentage points adds over $1 100 of interest.

Total interest for the same $20 000 over 5 years at three rates. The 9% loan costs almost twice as much interest as the 5% loan.

Worked example

Example 3: Rate or term: which matters more here?

Option A: 5% for 84 months. Option B: 7% for 60 months. Both are for $20 000. The payment for A is $282.68. Which costs less in total interest?

A: total paid = 282.68 × 84 = $23 745.12, so interest = $3 745.12.
B: from the table, interest = $3 761.20.
Difference: 3761.20 − 3745.12 = $16.08.

They are almost the same: A costs about $16 less in interest, but A keeps you paying for 2 more years. A lower rate saves money, but stretching the time can cancel out the savings.

Questions to ask before you borrow

What is the interest rate, and is it fixed or variable?
How many payments, and how often?
What is the total amount I will pay? (payment × number of payments + fees)
Can I pay extra or pay it off early without a penalty?
Will the thing I am buying still be worth something when the loan ends? (You will study depreciation in Lessons 8 to 10.)

Hands-on activity

Ad detective

You need: a newspaper, flyer or website ad that shows a monthly or bi-weekly payment (cars, furniture, electronics), a calculator, and a grown-up to help you find one.

Write down the price, the payment, how often it is paid, the number of payments and the rate (look in the fine print).
Calculate the total paid and the total interest.
Use the PMT formula or an online loan calculator to check the payment.
Recalculate with a term that is 2 years shorter. How do the payment and the total interest change?
Write one sentence the ad should have said more clearly.

Model answer: sofa $2 400 at 9.9% for 36 months, payment $77.33. Total paid: 77.33 × 36 = $2 783.88, so interest is $383.88. For 12 months instead, the payment is about $210.89 and the interest only about $130.68.

Just for fun

If you finance a car for 96 months (8 years), a baby born the day you sign the papers could be in Grade 2 or 3 by the time you make the last payment.

Your turn

Find the first month's interest on a $15 000 loan at 6% per year compounded monthly.

Use the formula to find the monthly payment for $15 000 at 6% for 48 months. Then find the total interest.

An $8 000 loan at 8% can be paid over 24 months ($361.82 a month) or 36 months ($250.69 a month). Find the total interest for each.

A $12 000 loan for 60 months costs $225.91 a month at 4.9% or $242.74 a month at 7.9%. How much more interest does the 7.9% loan cost?

An ad says "$420.25 a month for 72 months" on a $25 000 car. Find the total paid and the total interest.

Explain why a longer term gives a smaller payment but more total interest.

More practice: warm-up

What is i for 6% per year compounded monthly?

How many monthly payments are in a 5-year loan?

A loan payment is $300 a month for 48 months. What is the total paid?

You borrowed $12 000 and paid $14 400 in total. How much interest did you pay?

Same amount, same rate: which costs less total interest, a 3-year loan or a 5-year loan?

More practice: on your own

$6 000 at 10%: 12 months costs $527.50 a month and 36 months costs $193.60 a month. Find the total interest for each.

$25 000 at 6.5%: 48 months ($592.87 a month) or 72 months ($420.25 a month). How much more interest does the 72-month loan cost?

Rowan's budget allows at most $400 a month for a car. Does either option in the previous question fit? What could Rowan do?

For $20 000 at 7% for 36 months (payment $617.54), find the interest and principal in the first payment and the balance after it.

$30 000 at 5.99%: 60 months ($579.84) or 96 months ($394.10). Find the total interest for each and the difference.

Challenge

For $20 000 at 7%, the 36-month payment is $617.54. Find the 72-month payment with the formula. Is it half of $617.54? Explain.

Start with $20 000 at 5% for 60 months (interest $2 645.20). Which adds more interest: raising the rate to 7% (same 60 months) or stretching to 84 months (same 5%, payment $282.68)?

Rowan can pay $400 a month for 60 months at 7%. What is the most Rowan can borrow? (Rearrange the formula: P = PMT × (1 − (1 + i)⁻ⁿ) ÷ i.)

Check your understanding

Quiz

Same amount and same rate, but a longer term. What happens?

A $20 000 loan has 60 payments of $396.02. What is the total interest?

Same amount and same term, but a higher rate. What happens to the total interest?

What is the first month's interest on $12 000 at 6% per year, compounded monthly?

An ad says "Only $199 a month!" What is the most important thing to find before deciding?

Which $10 000 loan costs the least total interest?

For parents and teachers

Big idea: F1.3 asks students to compare the effects of different interest rates and lengths of borrowing time on the overall cost of purchasing goods or services, using appropriate tools. The payment formula is given as a tool, not something to memorize or derive. Spreadsheets and online calculators are just as appropriate (Lesson 7 builds a spreadsheet).

Common difficulties:

Judging a loan by its monthly payment. Always calculate total paid = payment × number of payments.
Calculator entry of (1 + i)⁻ⁿ. Use brackets and the negative exponent, or compute 1 ÷ (1 + i)ⁿ.
Thinking twice the time means half the payment. It does not, because interest keeps adding up.

How to help: Next time you see a car, furniture or phone financing ad, ask your teen to find the total cost from the fine print. If your family has a loan you are comfortable discussing (even with rounded numbers), compare its payment with a shorter or longer term using an online calculator. Rates in this lesson are examples only.