The Cost of Borrowing

When you can’t pay for something all at once, you can borrow: a phone on a monthly plan, a car loan, or a credit card. Borrowing lets you have something now, but it almost always costs more than paying cash. This page shows how to find the real total cost and which choices make borrowing cheaper.

Key ideas

Total cost and the cost of borrowing

When you buy something with a loan or payment plan, add up everything you pay:

$$\text{total cost} = \text{down payment} + \text{all the payments}$$

• The down payment is the money you pay up front.
• The payments are usually monthly: monthly payment $\times$ number of months.

The cost of borrowing is the extra you pay compared with paying cash. It’s mostly interest (sometimes fees too):

$$\text{cost of borrowing} = \text{total cost} - \text{cash price}$$

Four things that change the cost

If you…	the cost of borrowing…	because…
get a lower interest rate	goes down	each dollar borrowed costs less per year
borrow for a shorter time	goes down	you pay interest for fewer months
borrow with simple instead of compound interest	goes down	there’s no interest on interest
make a bigger down payment	goes down	you borrow less, so there’s less to pay interest on

A longer loan has smaller monthly payments, which can look attractive. But you pay for more months, so the total cost is usually higher. Always compare total costs, not just monthly payments.

Simple vs. compound interest

With simple interest, interest is charged only on the amount borrowed: $I = Prt$. With compound interest, interest is added to what you owe, and then that interest gets charged interest too. For the same rate and a time longer than one compounding period (for example, more than one year with yearly compounding), compound interest costs more, and the gap grows the longer you borrow.

Real car loans and mortgages are worked out with a more advanced method, so on this page the lender tells you the monthly payment. Your job is to find the totals and compare.

Buy now, pay later

Some stores offer “buy now, pay later” plans that split a purchase into a few equal payments. Some have no interest if every payment is on time, but they may charge a fee for each late or missed payment, or high interest after a promotional period. Read the terms before you agree, and only use a plan if you’re sure you can make every payment.

A cautionary tale: credit card minimum payments

Credit cards in Canada often charge around $20\%$ interest per year on any balance you don’t pay off, compounded every month. If you pay only a small amount each month, most of your payment goes to interest and the balance barely shrinks.

Here is a $1000 balance on a card charging $19.99\%$ per year, if you make no new purchases (results from a month-by-month calculation, rounded):

Payment each month	Time to pay it off	Total interest paid
$25	$67$ months (over $5$ years)	about $661
$50	$25$ months	about $226
$100	$12$ months	about $103

Paying $25 a month, you’d pay about $1661 for $1000 of stuff. The best plan is to pay the full balance every month, so you pay no interest at all.

Worked examples

Example 1: A phone on a payment plan

A phone costs $1200 if you pay cash. A phone company offers it for $150 down plus $48 per month for $24$ months. Find the total cost and the cost of borrowing.

Solution.

$$\begin{aligned} \text{total cost} &= 150 + 48 \times 24 \\ &= 150 + 1152 \\ &= 1302 \end{aligned}$$ $$\text{cost of borrowing} = 1302 - 1200 = 102$$

The total cost is $1302, so the plan costs $102 more than paying cash.

Example 2: Changing the rate and the time

Keisha borrows $5000 for a used car. Suppose the lender charges simple interest. Compare these three loans:

• (a) $6\%$ per year for $3$ years
• (b) $6\%$ per year for $5$ years
• (c) $9\%$ per year for $3$ years

Solution. Use $I = Prt$ for each.

Loan	Interest $I = Prt$	Total repaid	Monthly payment
(a) $6\%$, $3$ years	$5000(0.06)(3) = 900$	$5900$	$5900 \div 36 \approx 163.89$
(b) $6\%$, $5$ years	$5000(0.06)(5) = 1500$	$6500$	$6500 \div 60 \approx 108.33$
(c) $9\%$, $3$ years	$5000(0.09)(3) = 1350$	$6350$	$6350 \div 36 \approx 176.39$

Loan (a) is the cheapest overall at $900 of interest. Loan (b) has the smallest monthly payment (about $108.33), but borrowing for $2$ extra years costs $600 more in interest. Loan (c)‘s higher rate costs $450 more than (a) over the same $3$ years.

Example 3: Simple vs. compound interest

Omar borrows $2000 from a family member at $8\%$ per year for $3$ years and repays it all at the end. How much more would he owe with interest compounded annually than with simple interest?

Solution. Simple interest:

$$I = 2000(0.08)(3) = 480 \quad\Rightarrow\quad \text{owes } 2000 + 480 = 2480$$

Compound interest: multiply by $1.08$ each year.

$$\begin{aligned} \text{Year 1: } & 2000 \times 1.08 = 2160 \\ \text{Year 2: } & 2160 \times 1.08 = 2332.80 \\ \text{Year 3: } & 2332.80 \times 1.08 = 2519.424 \approx 2519.42 \end{aligned}$$

With compound interest he owes $2519.42, which is $2519.42 - 2480 = 39.42$ dollars more. The extra comes from interest charged on earlier interest.

Example 4: The effect of a down payment

A used car costs $12 000. The dealer offers three loans at the same interest rate (the dealer has worked out the monthly payments):

Option	Down payment	Monthly payment	Months
A	$0	$243.26	$60$
B	$3000	$182.44	$60$
C	$3000	$281.99	$36$

Find the total cost and the cost of borrowing for each option.

Solution.

Option	Total cost	Cost of borrowing
A	$0 + 243.26 \times 60 = 14\,595.60$	$14\,595.60 - 12\,000 = 2595.60$
B	$3000 + 182.44 \times 60 = 13\,946.40$	$13\,946.40 - 12\,000 = 1946.40$
C	$3000 + 281.99 \times 36 = 13\,151.64$	$13\,151.64 - 12\,000 = 1151.64$

The car price is the same in every option. The down payment and the length of the loan change how much interest is added on top.

The $3000 down payment saves $2595.60 - 1946.40 = 649.20$ dollars (A vs. B), because less money is borrowed. Paying it off in $36$ months instead of $60$ saves another $1946.40 - 1151.64 = 794.76$ dollars (B vs. C). Option C has the highest monthly payment, but it’s the cheapest overall.

Common mistakes

Forgetting the down payment. The total cost includes the money you paid up front. In Example 1, $48 \times 24 = 1152$ is not the total: you must add the $150 down payment.

Comparing only monthly payments. A smaller monthly payment often means a longer loan and more interest. In Example 2, loan (b) has the smallest payment but costs the most interest. Compare total costs.

Calling the total cost the interest. The cost of borrowing is the total cost minus the cash price. In Example 1 it’s $102, not $1302.

Using the yearly rate for months. Rates are usually per year. A loan for $18$ months uses $t = 1.5$ years in $I = Prt$, not $t = 18$.

Assuming “no interest” means free. A buy now, pay later plan or a “0% financing” offer can still have fees, or high interest if you miss a payment. Read the terms.

Paying only the credit card minimum. Small payments on a card at about $20\%$ interest can stretch a debt over years and add hundreds of dollars of interest. Pay the full balance whenever you can.

Practice

1. (Warm-up) A bike costs $520 cash. A store plan is $100 down plus $40 per month for $12$ months. Find the total cost and the cost of borrowing.

Solution

Total cost: $100 + 40 \times 12 = 100 + 480 = 580$, so $580.

Cost of borrowing: $580 - 520 = 60$, so $60.

2. (Warm-up) Find the simple interest and the total repaid on a loan of $3000 at $5\%$ per year for $2$ years.

Solution

$I = 3000(0.05)(2) = 300$, so the interest is $300 and the total repaid is $3300.

3. (Warm-up) Which plan costs less in total: $45 per month for $24$ months, or $60 per month for $15$ months? (Neither has a down payment.)

Solution

$45 \times 24 = 1080$ and $60 \times 15 = 900$. The $60 plan costs less in total ($900 vs. $1080), even though its monthly payment is higher.

4. (Core) A laptop costs $1500 cash. Compare the cost of borrowing for these plans:

• Plan A: $0 down, $68.50 per month for $24$ months.
• Plan B: $300 down, $54 per month for $24$ months.

Solution

Plan A: total $= 68.50 \times 24 = 1644$. Cost of borrowing $= 1644 - 1500 = 144$, so $144.

Plan B: total $= 300 + 54 \times 24 = 300 + 1296 = 1596$. Cost of borrowing $= 1596 - 1500 = 96$, so $96.

Plan B costs $48 less, mainly because the down payment means less is borrowed.

5. (Core) You borrow $4000 at $6\%$ per year for $4$ years, repaid at the end. Find the interest with simple interest and with interest compounded annually. How much more is the compound interest?

Solution

Simple: $I = 4000(0.06)(4) = 960$, so $960.

Compound: multiply by $1.06$ four times.

$$4000 \times 1.06^4 = 4000 \times 1.26247696 \approx 5049.91$$

The compound interest is $5049.91 - 4000 = 1049.91$, so $1049.91. That’s $1049.91 - 960 = 89.91$ dollars more.

6. (Core) Jasmine needs an $8000 car loan. The bank gives her the monthly payments for three choices:

Loan	Monthly payment
$5.9\%$ for $48$ months	$187.51
$8.9\%$ for $48$ months	$198.70
$8.9\%$ for $72$ months	$143.81

Find the total repaid and the interest for each. What do the results show?

Solution

Loan	Total repaid	Interest
$5.9\%$, $48$ months	$187.51 \times 48 = 9000.48$	$9000.48 - 8000 = 1000.48$
$8.9\%$, $48$ months	$198.70 \times 48 = 9537.60$	$9537.60 - 8000 = 1537.60$
$8.9\%$, $72$ months	$143.81 \times 72 = 10\,354.32$	$10\,354.32 - 8000 = 2354.32$

The higher rate adds $537.12 of interest over the same $48$ months. Stretching the loan to $72$ months lowers the payment but adds another $816.72 of interest. The lowest rate for the shortest time is cheapest.

7. (Core) A store offers headphones for $240, split into $4$ payments of $60 every two weeks with no interest. The plan charges a $10 fee for each late payment.

• (a) What is the total cost if every payment is on time?
• (b) What is the total cost if two payments are late? By what percent does that increase the cost?

Solution

(a) $4 \times 60 = 240$, so $240, the same as the cash price.

(b) $240 + 2 \times 10 = 260$, so $260.

$$\frac{20}{240} \times 100\% \approx 8.3\%$$

The late fees raise the cost by about $8.3\%$.

8. (Challenge) Liam owes $600 on a credit card that charges $19.99\%$ per year, compounded monthly.

• (a) About how much interest is charged in the first month? (Hint: the monthly rate is the yearly rate divided by $12$.)
• (b) If his monthly payment is $15, how much of the first payment actually reduces what he owes?
• (c) Paying $15 per month, it takes him $67$ months to pay off the card, and he pays $996.61 in total. How much interest did he pay?

Solution

(a) Monthly rate: $0.1999 \div 12 \approx 0.016658$. Interest: $600 \times 0.1999 \div 12 \approx 9.995$, so about $10.00.

(b) About $15 - 10 = 5$ dollars. Two-thirds of his payment goes to interest.

(c) $996.61 - 600 = 396.61$, so $396.61 of interest. That’s about two-thirds of the original $600, and it takes over $5$ years.

9. (Challenge) Sofia wants a $900 game console. She can buy it now on a store plan at $45 per month for $24$ months, or save $45 per month and buy it with cash when she has enough.

• (a) What is the total cost of the store plan?
• (b) How many months would she need to save? What does buying now cost her, in dollars, compared with waiting?

Solution

(a) $45 \times 24 = 1080$, so $1080.

(b) $900 \div 45 = 20$ months of saving. Buying now costs $1080 - 900 = 180$ dollars more. So she’d be paying $180 to have the console $20$ months sooner (and the price might also change while she waits).