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Mathematics

Loan Payment Math

Amortization, APR, and the True Cost of Borrowing — A TLDR Primer

You just got approved for a car loan — or your teacher dropped amortization on you with zero context — and the numbers don't make sense. Why is the monthly payment exactly $312.47? Why does so much of it go to interest at first? And what does APR actually mean compared to the interest rate on the sticker?

This TLDR primer answers those questions directly, without the bloat. It covers the core vocabulary of borrowing — principal, interest, compounding, and the time value of money — then walks through exactly where the fixed monthly payment formula comes from, starting from a geometric series and ending at the equation your calculator uses. From there, you'll read a real amortization schedule row by row, seeing how each payment splits between interest and principal as the loan ages.

The second half tackles the rate alphabet soup that trips up every borrower: nominal rate, APR, and APY are not the same thing, and understanding the difference is how you spot a loan offer that looks cheaper than it is. The guide closes with the math of extra payments and refinancing — concrete numbers that show exactly how much total interest you save by paying a little more each month or shortening your term.

Written for high school and early college students studying personal finance, consumer math, or precalculus applications, and equally useful for parents helping kids understand how loan payment math works before they sign anything. Concise, worked examples throughout, no filler.

If you want to understand what you're actually agreeing to when you borrow money, start here.

What you'll learn
  • Understand how interest accrues and the difference between simple and compound interest
  • Derive and apply the fixed-payment amortization formula for mortgages, auto loans, and student loans
  • Read an amortization schedule and explain why early payments are mostly interest
  • Distinguish nominal rate, APR, and APY, and use them to compare loan offers
  • Calculate the impact of extra payments, refinancing, and loan term length on total interest paid
What's inside
  1. 1. Interest, Principal, and the Time Value of Money
    Sets up the core vocabulary of borrowing and shows the difference between simple and compound interest with worked numbers.
  2. 2. The Amortization Formula: Where It Comes From
    Derives the fixed monthly payment formula from a geometric series and explains each variable in plain terms.
  3. 3. Reading an Amortization Schedule
    Walks through a row-by-row amortization table to show how each payment splits between interest and principal over time.
  4. 4. APR vs APY vs Nominal Rate: Comparing Loans Honestly
    Clarifies the rate alphabet soup so students can compare loan offers and spot deceptive marketing.
  5. 5. Extra Payments, Refinancing, and Total Cost
    Quantifies how extra principal payments, shorter terms, and refinancing change the total interest paid over the life of a loan.
Published by Solid State Press
Loan Payment Math cover
TLDR STUDY GUIDES

Loan Payment Math

Amortization, APR, and the True Cost of Borrowing — A TLDR Primer
Solid State Press

Loan Payment Math

Amortization, APR, and the True Cost of Borrowing — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 Interest, Principal, and the Time Value of Money
  2. 2 The Amortization Formula: Where It Comes From
  3. 3 Reading an Amortization Schedule
  4. 4 APR vs APY vs Nominal Rate: Comparing Loans Honestly
  5. 5 Extra Payments, Refinancing, and Total Cost
Chapter 1

Interest, Principal, and the Time Value of Money

Money today is worth more than the same amount of money in the future. That one idea — the time value of money — is the foundation underneath every loan, mortgage, and credit card balance you will ever encounter.

Here is why it is true. If you have $1,000 right now, you can invest it, lend it to someone else, or simply spend it. A lender who hands you $1,000 today is giving up those possibilities. Interest is the price the borrower pays the lender for that sacrifice — compensation for time, risk, and opportunity cost.

The amount of money originally borrowed or lent is called the principal. The interest rate (usually written as an annual percentage) tells you how much interest accrues relative to the principal over a given period. Everything else in loan math is built from these two quantities.

Simple Interest

Simple interest accrues only on the original principal, not on any interest that has already built up. The formula is:

$I = P \cdot r \cdot t$

where $P$ is the principal, $r$ is the annual interest rate expressed as a decimal, $t$ is time in years, and $I$ is the interest owed.

Example. You borrow $5,000 at a simple interest rate of 6% per year for 3 years. How much interest do you owe?

Solution. $I = 5{,}000 \times 0.06 \times 3 = \$900$ You repay the principal plus interest: $5,000 + $900 = $5,900 total.

Simple interest shows up in some short-term personal loans and in the way daily interest is sometimes calculated on car loans. It is the easier case — and unfortunately, not the one that applies to most real-world borrowing.

Compound Interest

Compound interest accrues on the principal and on any interest that has already been added to the balance. Informally: you pay interest on your interest. This distinction matters enormously over time.

About This Book

If you're a high school student tackling personal finance math for teenagers in a consumer math or financial literacy course, a college freshman who just signed a student loan and wants to understand what you agreed to, or a parent helping your kid make sense of an auto loan offer, this book is for you.

It covers how loan payments are calculated — the math behind them, not just the button on a calculator. You'll work through the amortization formula explained step by step, learn to read an amortization schedule, and understand compound interest and borrowing explained simply enough to apply immediately. The APR vs. interest rate distinction gets its own section because it's the key to how to compare loan offers honestly. You'll also learn what total interest paid on a loan actually tells you. Short by design, no filler.

Read straight through in order — each section builds on the last. Work every example yourself before reading the solution, then test your understanding with the problem set at the end.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon