MAT-144 · Mathematical Reasoning Topic 07 · Taxes & Stocks
Topic 07 · Review · Q23

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 Worked Example Video

Every step of Comparing monthly payments and total costs of two loans explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE
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

Olivia is taking a $200,000 mortgage.

Credit union: 30-year at 4.5% APR.
Online lender: 15-year at 6.0% APR.

By how much is the winning lender cheaper over the lifetime? Type just the number, rounded to the nearest cent.

savings = $
Yusuf's mortgage choice

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

Bank A: 30-year at 5.0% APR.
Bank B: 20-year at 5.75% APR.

By how much is the winning offer cheaper over the lifetime? Type just the number.

savings = $
Camila's mortgage choice

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

Lender A: 30-year at 3.9% APR.
Lender B: 15-year at 5.4% APR.

By how much is the winning offer cheaper over the lifetime? Type just the number.

savings = $
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 The Rule
Same P, two different (r, t) compute M for each offer
total = M × 12t compare totals, not monthlies
Two parallel calculations. The lower monthly loan is often the higher total because it usually has a longer term. Compare lifetime cost (M × 12t) to pick the winner — and remember, a bigger monthly can win overall. See it live: Offer A ($150K, 30-yr, 5%) → then swap in years=15, rate=6 to see Offer B and compare the two totals.
Common slips
(1) Picked the lower monthly and stopped. Monthly ≠ lifetime cost. A 30-year loan almost always has a lower monthly than a 15-year at the same P — but the 30-year usually costs way more over the full term. (2) Used the wrong n. Each loan has its own n = 12t. The 30-year uses 360; the 15-year uses 180. (3) Compared totals in the wrong direction. The winner is the SMALLER total; subtract smaller from bigger to get the savings.
04 Practice — Step by Step
Practice this problem step by step
A $150,000 mortgage. Offer A: 30-year at 5%. Offer B: 15-year at 6%. Which is cheaper over the lifetime, and by how much?
Step 1

Offer A monthly (30-year at 5%)

P = 150,000, r/12 ≈ 0.004167, n = 360. Compute the monthly payment. Round to the nearest cent.
M<sub>A</sub> = $
Step 2

Offer A lifetime total

Multiply the monthly by the total months: $805.23 × 360.
total<sub>A</sub> = $
Step 3

Offer B monthly (15-year at 6%)

P = 150,000, r/12 = 0.005, n = 180. Compute the monthly payment. Round to the nearest cent.
M<sub>B</sub> = $
Step 4

Offer B lifetime total

Multiply: $1,265.79 × 180.
total<sub>B</sub> = $
Q22 Q24
MAT-144 · TOPIC 07 · REVIEW · Q23 OF 38