Comparing monthly payments and total costs of two loans

Two parallel calculations: monthly payment for each offer, then multiply by total months for the lifetime cost. The lower-rate loan isn't always the cheaper one.

01 Mini-Lesson Video

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A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Olivia's mortgage choice v1

Olivia is taking out a mortgage for $200,000 to buy a new house and is deciding between the offers from two lenders. She wants to know which one would be the better deal over the life of the mortgage loan, and by how much.

(a) A credit union has offered her a 30-year mortgage loan at an annual interest rate of 4.5%. Find the monthly payment.
(b) An online lending company has offered her a 15-year mortgage loan at an annual interest rate of 6.0%. Find the monthly payment.
(c) Suppose Olivia pays the monthly payment each month for the full term. Which lender's mortgage loan would have the lowest total amount to pay off, and by how much?

Yusuf's mortgage choice v2

Yusuf compares two offers on a $140,000 mortgage:

(a) Bank A: 30-year at 5.0%. Find the monthly.
(b) Bank B: 20-year at 5.75%. Find the monthly.
(c) Which has the lower lifetime cost, and by how much?

Camila's mortgage choice v3

Camila compares two offers on a $280,000 mortgage:

(a) Lender A: 30-year at 3.9%. Find the monthly.
(b) Lender B: 15-year at 5.4%. Find the monthly.
(c) Which has the lower lifetime cost, and by how much?

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 How to Solve It

Olivia's two offers, side by side. The credit union's lower monthly ($1,013) hides a higher lifetime cost. The online lender wins on total by $61,025 — even though its rate is higher and its monthly is bigger.

M = P(r/12) / (1 − (1 + r/12)^−12t)
total = M × 12t (compute for each offer)
winner: the smaller total

The same formula twice, with different (r, t) pairs. Compare totals, not monthlies — the bigger monthly often wins on total cost because of the shorter term.

Credit union: 30-year @ 4.5% (part a).

r/12 ≈ 0.00375, n = 360.

M = (200,000 \times 0.00375) / (1 - 1.00375 -360) \approx $1,013.37

Online lender: 15-year @ 6.0% (part b).

r/12 = 0.005, n = 180.

M = (200,000 \times 0.005) / (1 - 1.005 -180) \approx $1,687.71

Compute and compare totals (part c).

Multiply each monthly by its total payments. Subtract the smaller from the bigger.

credit union total = 1,013.37 \times 360 \approx $364,813 online total = 1,687.71 \times 180 \approx $303,788 difference = 364,813 - 303,788 \approx $61,025

Online lender wins by $61,025 — even though its rate is higher and its monthly is bigger. Time matters more than rate when terms differ this much.

▸ COMMON SLIPS(1) Compared monthlies. The lower monthly isn't the better deal — the lower TOTAL is. (2) Multiplied both M's by the same n. Different terms have different totals of payments (360 vs 180). (3) Assumed lower rate wins. Rate matters, but term length matters more when the gap is big.

04 Practice — Step by Step

Try Yusuf's mortgage choice: $140,000, 30-year @ 5.0% vs 20-year @ 5.75%.

Compute Bank A's monthly (30-year @ 5.0%).

$140,000 at 5.0% APR for 30 years. What's the monthly payment?

M_A = $

Compute Bank B's monthly (20-year @ 5.75%).

Same $140,000 at 5.75% APR for 20 years. What's the monthly?

M_B = $

Compare totals and find the difference.

Bank A total = $751.55 × 360 ≈ $270,558. Bank B total = $982.97 × 240 ≈ $235,913. Which is the better deal, and by how much?

savings = $

▸ NICE WORK

You've walked through the whole problem.

That's the move. ALEKS will give you a different version with different numbers — but the steps are the same.

Q5 Topic overview