MAT-144 · Mathematical Reasoning
Topic 04 · Loans
Study card
The formulas, the moves, and the traps for Topic 4, in one printable page.
Compute, don't memorize
Plug in your numbers and get the answer in either direction. The web tool solves for any variable; the Excel workbook is yours to keep.
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
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
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
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 %)
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
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.