MAT 144 Exam 2 Review — Problem 13: Down Payment First, Then the Loan

TITLE
MAT 144 Exam 2 Review — Problem 13: Down Payment First, Then the Loan

DESCRIPTION
Donna buys a $31,000 work truck. The dealer requires 20% down, and she finances the rest at 9.5% annual interest with monthly payments for 6 years. (a) the down payment, (b) the loan amount, (c) the monthly payment — three parts, and each one feeds the next. Part (a) is one line: $31,000 × 0.20 = $6,200, and the gold slice splits off the front of the price bar. Part (b) looks like busywork and is not: $31,000 − $6,200 = $24,800 is the number part (c) needs, and writing it down is the whole point. Then part (c) opens with the formula itself, written the way the course formula sheet writes it: PMT = [P × (r/n)] / [1 − (1 + r/n)^(−n·t)] — and every letter gets labelled against Donna's numbers before anything is typed (P = the loan = $24,800, r = 0.095, n = 12 because the payments are monthly, t = 6 years, PMT = the unknown). That act closes on one concept line and no arithmetic at all: the rate is yearly, the payments are monthly, and that is exactly what r/n and n·t are there for. Then it goes straight to the calculator — nothing is evaluated in between, because identifying the values and plugging them in IS the method. The solve is one act: write the substituted line in that same sheet shape — [24,800 × (0.095/12)] / [1 − (1 + 0.095/12)^(−12·6)] — and that is the last line you write by hand. Then one sentence: this whole thing goes into the calculator. It hands back 453.212332…, rounded exactly once to $453.21. Setup, then readout, and nothing in between: r/n stays 0.095/12 inside the parentheses and the exponent stays −12·6, so no long decimal is ever written down by hand. The money shot: run the sticker price through the identical formula and it hands back $566.52 — a clean-looking number with cents, in a perfectly ordinary range, with no minus sign and no alarm of any kind. That is why using the price instead of the loan is the most common error on this exam. Finally the fact that makes it land, and the payment count is built on screen before it is used: 12 payments a year × 6 years = 72 payments, so 72 × $453.21 = $32,631.12 in payments, plus $6,200 down, is $38,831.12 for a $31,000 truck. Plus four traps: the $31,000 substitution, chopping the repeating rate to 0.0079 (which is 25 cents off and marked wrong), using N = 6 instead of 72, and reporting the down payment as the loan.

You'll learn:
• down payment = price × the down payment rate; loan = price − down payment
• PMT = [P × (r/n)] / [1 − (1 + r/n)^(−n·t)] in course-sheet form — and why P is the LOAN, not the purchase price
• How to label P, r, n and t against the problem BEFORE you touch a calculator
• Why part (b) exists: to force you to write $24,800 down before part (c) starts
• Identify the values, then plug in: r/n stays 0.095/12 and the exponent stays −12·6, with no intermediate decimals written down
• Setup, then readout: write the substituted sheet line, put that whole thing into the calculator, and round exactly once at the end
• Why $566.52 is the dangerous wrong answer: nothing about it looks wrong
• Where the payment count comes from: 12 payments a year × 6 years = 72 payments, written down before it is ever multiplied by anything
• The all-in cost: $38,831.12 handed over for a $31,000 truck

Chapters
0:00 The problem — three parts in a chain
1:08 Part (a) — the down payment
1:55 Part (b) — the loan amount
2:49 Part (c) — the formula, letter by letter
4:27 The solve — the setup line, then the readout
5:21 Which price? $453.21 vs $566.52
6:41 What the truck actually costs — 72 payments, then the total
7:42 The traps
9:03 Recap & the Problem 10 note

ORBITAL — Watch it click.

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