MAT-144 · Mathematical Reasoning Topic 04 · Loans
Lesson 03 · Auto loans

Auto loans: the standard amortizing loan.

Fixed monthly payment, fixed term (typically 3-7 years). Two new pieces enter: down payment, and the difference between sticker price and amount financed.

01Down payment 02Loan amount = price − down 03Compare 5-yr vs 7-yr
▸ THE HOOK

Walk into the dealership. The sticker says $35,000. You can't write that check, so you finance most of it. The salesperson asks two questions:

"How much down?" · "How many years?"

Those two answers, combined with the rate the bank quotes you, fully determine your monthly payment via the L2 formula. They also fully determine the total you'll pay, which is the number you should actually be optimizing — and which the dealership has no incentive to talk about. This lesson teaches you to read both numbers before you sign.

The simplest amortizing loan most people see first.

Auto loans are the cleanest application of the amortization formula. Fixed term (3-7 years), fixed monthly payment, fixed rate. Two new pieces enter the picture: a down payment (money you pay up front) and the resulting amount financed (sticker price minus down payment). The amount financed is what goes into the formula as P; the sticker price is what you pay in total once you add the down payment and all 12·t monthly payments together.
$30,000 AT 6% — TERM TRADEOFF stretch the term, lower the monthly, pay more in total 5-YEAR TERM monthly $579.98 /month total cost $34,799 +$4,799 interest 7-YEAR TERM monthly $438.27 /month (lower!) total cost $36,815 +$6,815 interest 7-yr: $141.71 less per month · but $2,016 more total interest paid

Two sides of the same $30,000 loan at 6%. The 5-year term (left) has a higher monthly but lower total cost. The 7-year term (right) drops your monthly $142 but adds $2,016 in interest. The lower monthly is the dealership's argument; the higher total is yours.

▸ DEFINITION

An auto loan is a fixed-term, fixed-rate amortizing loan, typically 3-7 years. Loan amount = sticker price − down payment. Monthly payment comes from the L2 formula with P = loan amount.

Vocabulary you'll see in word problems

  • Auto loan A fixed-term loan to buy a vehicle. Term typically 3-7 years; rate depends on credit score and current rates.
  • Down payment Money paid up front. Reduces the principal you finance and the monthly payment that results. Usually 10-20% of sticker price.
  • Sticker price (total cost) The full price of the vehicle. Equals down payment plus the loan amount.
  • Amount financed (P) The principal of the loan. Sticker price minus down payment. This is the P that goes into the amortization formula.
  • Term length The loan duration in years. Longer term means lower monthly payment but more total interest paid.

$35K SUV, 15% down, 5 years at 5.5%.

End-to-end: from sticker price to total out-of-pocket cost. Five steps.

"Maria buys a $35,000 SUV. She makes a 15% down payment and finances the rest with a 5-year auto loan at 5.5% APR. What's her monthly payment, total interest paid, and total cost of the SUV?"

1

Compute the down payment.

15% of the sticker price.

down payment = 0.15 × $35,000 = $5,250
2

Compute the loan amount (P).

Sticker minus down payment. This is the P that goes into the amortization formula.

P = $35,000 − $5,250 = $29,750
3

Apply the amortization formula.

r = 0.055, t = 5, so r/12 ≈ 0.004583 and n = 12t = 60. Plug in.

M = (29,750 × 0.004583) / (1 − 1.004583^(−60)) ≈ $568.26
→ monthly payment
4

Total paid over the loan + total interest.

Sum of all 60 monthly payments, then subtract principal for interest.

total paid = M × 60 = $568.26 × 60 ≈ $34,096 total interest = $34,096 − $29,750 = $4,346
5

Total cost (the number that matters).

Add the down payment back. This is what Maria actually pays out of pocket for the SUV.

total cost = down + total paid = $5,250 + $34,096 = $39,346
→ ~$4,346 over sticker, all of it interest

Three problems. Down payment, formula, comparison.

Each builds on the last. By the third problem you'll have a feel for how down payment size shifts the monthly payment.
PROBLEM 01 ☆ ☆   warm-up · loan amount

A car has a sticker price of $42,000. The buyer puts 18% down. What's the loan amount?

P = $
PROBLEM 02 ★ ★ ☆   monthly payment

A $20,000 auto loan at 4.5% APR for 5 years. What's the monthly payment?

M = $
PROBLEM 03 ★ ★ ★   down-payment comparison

$30,000 car at 6% APR over 5 years. With a 10% down payment (P = $27,000), M ≈ $521.99. Now run it with a 20% down payment. What's the new M?

M = $

Three fast questions before you move on.

Tap an answer. You'll see right away whether it stuck.

Q1. If the sticker price is $40,000 and you put 15% down, the loan amount (P) is...

Why B? Loan amount = sticker − down payment = $40,000 − $6,000 = $34,000. (Or equivalently, 85% of sticker.)

Q2. Stretching an auto loan from 5 years to 7 years at the same rate...

Why B? Spreading payments over more months shrinks each one, but interest keeps accruing on the balance for two extra years. Lower monthly, higher total. The dealership's favorite tradeoff to highlight (the lower monthly) and the one you should care about most (the higher total).

Q3. A 0% APR auto loan deal ($24,000 over 4 years) means...

Why B? With r = 0, the amortization formula collapses to M = P / (12t) — just the loan amount divided by the number of months, no interest. $24,000 / 48 = $500/mo. (Real-world note: 0% deals usually require a higher sticker price or come with restrictions, so "0% APR" doesn't always mean "free money.")
▸ UP NEXT — LESSON 04

The term-length tradeoff.

One specific intuition this lesson aims to build: stretching a loan from 5 years to 7 years lowers your monthly payment but increases total interest paid by hundreds or thousands of dollars. The monthly looks better; the total cost gets worse. This is the most common "wait, what?" moment students have on auto loans, and the next DQ asks you to compute it explicitly.

Next: Credit cards. Same amortization mechanics underneath, but with a twist — no fixed term, and minimum payments that look small but trap you for decades.

Continue to Lesson 04

Different angle? Need another rep? These are optional — tap any that look helpful.

▸ Browse all Topic 4 resources