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MAT-144 · Topic 04 · Cheat Sheet

Loans

The formulas, the moves, and the traps for Topic 4.

Compute, don't memorize

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Key formulas

The amortization formula is the only ALEKS dictionary formula we didn't already use in T3. Everything else is per-payment arithmetic built on top of it.

Loan amortization (the central T4 formula)
M = P(r/12) / (1 − (1 + r/12)−12t)
P = loan amount, r = annual rate (decimal), t = years. Output: fixed monthly payment M.
Total cost & total interest
total cost = M × 12 × t
total interest = total cost − P
A 30-year mortgage at 7% on $300K pays roughly $419K in interest alone — more than the loan itself.
Per-payment split (mortgage / amortization schedule)
interest portion = balance × (r/12)
principal portion = M − interest portion
new balance = balance − principal portion
Apply once for ALEKS Q5 (one mortgage payment). Apply 360 times to build a 30-year amortization schedule.
Credit card statement (one statement period)
interest = balance × (r/12)
new balance = balance + interest − payment
Same per-payment split as a mortgage, but applied to a revolving balance with no fixed term. Minimum payments are typically 1-3% of balance, often barely covering the interest charge.
Down payment & loan amount
loan amount = sticker price − down payment
loan amount also = down% × sticker (when given as %)
P in the amortization formula is the amount financed, not the sticker price. Auto loan: 10-20% down typical. Mortgage: 20% conventionally, less with PMI.
Carry-forward from Topic 3
A = P(1 + r/n)nt (compound interest)
Used in Lesson 5 for unsubsidized student loan interest capitalization. Run T3's compound formula across the in-school years to find the new P for the amortization formula. Full version on the Topic 3 cheat sheet.

Variable legend

P = principal (loan amount)  ·  r = annual rate (decimal)
t = term in years  ·  M = monthly payment
r/12 = monthly periodic rate  ·  12t = total monthly payments
balance = remaining principal at any point during the loan

Common slips

Forgot the negative exponent
The formula has (1 + r/12)−12t in the denominator. In Excel that's (1+r/12)^(-12*t). Drop the minus sign and the answer is wildly off.
Used the sticker price as P
P is the amount financed, not the sticker price. Subtract the down payment first.
Used annual rate where monthly was needed
In credit card and mortgage per-payment split: interest = balance × (r/12), not balance × r. The per-period rate is monthly.
Computed total cost as just M × t (missing × 12)
There are 12 payments per year. Total cost = M × 12 × t, not M × t.
MAT-144 · Topic 04 · Cheat Sheet · mat144.com