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Mathematics

Simple vs. Compound Interest

Principal, APR vs. APY, and the Rule of 72 — A TLDR Primer

Your exam has a problem about compound interest and you're not sure whether to use the simple interest formula or the compound one — or what APY even means. Your first credit card statement arrives and the math doesn't add up the way you expected. This guide is built for exactly those moments.

**Simple vs. Compound Interest** covers everything a high school or early college student needs to handle interest problems with confidence. You'll learn how principal, rate, and time interact in the straight-line simple interest formula, then see step by step why compounding produces a fundamentally different — and much steeper — growth curve. The guide explains the difference between APR and APY in plain language, shows you how to convert between them, and introduces the Rule of 72 as a fast mental shortcut for estimating how long any investment takes to double. It then pushes the idea of compounding frequency to its logical limit to arrive at continuous compounding and the number *e*, the same constant that shows up in calculus and science courses. The final section applies all of it to real products: credit cards, student loans, and savings accounts.

Every section leads with the one idea you actually need, backs it with worked numbers, and calls out the specific mistakes students make on exams and in real life — like mixing up time units in the simple interest formula or confusing APR with APY on a loan disclosure.

Concise, no filler, and built around the formulas that matter. If you need to get oriented fast, this is the place to start.

What you'll learn
  • Compute simple interest and compound interest from the formulas without memorization tricks
  • Translate between APR, APY, and effective interest rates over different compounding periods
  • Use the Rule of 72 to estimate doubling time and sanity-check answers
  • Recognize when a real-world problem (loan, savings, credit card) is simple vs. compound
  • Set up and solve word problems involving continuous compounding using the e formula
What's inside
  1. 1. What Interest Actually Is
    Defines principal, rate, and time, and frames interest as the price of borrowing or the reward for lending.
  2. 2. Simple Interest: The Straight-Line Case
    Develops I = Prt and A = P(1 + rt) with worked examples on loans and short-term savings, and clears up the unit-of-time trap.
  3. 3. Compound Interest: Interest on Interest
    Builds the compound formula step by step from repeated multiplication, shows why compounding frequency matters, and contrasts the growth curve with simple interest.
  4. 4. APR vs. APY and the Rule of 72
    Distinguishes nominal from effective rates, derives APY from APR, and introduces the Rule of 72 as a fast doubling-time estimate.
  5. 5. Continuous Compounding and the Number e
    Pushes compounding frequency to its limit, arrives at A = Pe^(rt), and explains when this formula actually shows up in finance and science.
  6. 6. Where This Shows Up: Credit Cards, Student Loans, and Savings
    Applies the formulas to real products students will encounter, names the most common misconceptions, and shows how small rate differences become large dollar differences over time.
Published by Solid State Press
Simple vs. Compound Interest cover
TLDR STUDY GUIDES

Simple vs. Compound Interest

Principal, APR vs. APY, and the Rule of 72 — A TLDR Primer
Solid State Press

Simple vs. Compound Interest

Principal, APR vs. APY, and the Rule of 72 — 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 What Interest Actually Is
  2. 2 Simple Interest: The Straight-Line Case
  3. 3 Compound Interest: Interest on Interest
  4. 4 APR vs. APY and the Rule of 72
  5. 5 Continuous Compounding and the Number e
  6. 6 Where This Shows Up: Credit Cards, Student Loans, and Savings
Chapter 1

What Interest Actually Is

Every time you borrow money or deposit it in a bank, a price gets attached to that transaction. That price is interest — the extra amount paid by whoever is using someone else's money.

The setup always involves two parties. The lender is the one supplying the money; the borrower is the one receiving and using it. When you deposit cash in a savings account, you are the lender and the bank is the borrower — the bank uses your money and pays you interest in return. When you take out a car loan, the roles flip: the bank lends, you borrow, and you pay interest for the privilege. Same mechanism, opposite seats.

Three quantities determine how much interest changes hands.

Principal ($P$) is the original amount of money — the starting balance on a loan or deposit before any interest is added. If you deposit $500 in a savings account, the principal is \$500. If you borrow $8,000 for a used car, the principal is \$8,000. Principal is always a dollar amount, never a percentage.

Interest rate ($r$) measures how much interest accumulates relative to the principal, expressed as a percentage per unit of time. A rate of 6% per year means that for every $100 of principal, \$6 of interest builds up each year. In formulas, rates are always converted to decimal form: 6% becomes $r = 0.06$. A common mistake is to plug the percentage directly into a formula — writing $r = 6$ instead of $r = 0.06$ — which overstates interest by a factor of 100.

About This Book

If you are taking a personal finance math high school course, working through an algebra class that covers interest formulas, or reviewing for a standardized exam that tests financial math, this book was written for you. It also fits parents helping a student work through a problem set and tutors who need a clean, fast reference.

This guide covers how compound interest works — from the high school basics through the real-world mechanics that trip students up. You will find simple and compound interest math explained side by side, APR vs. APY explained for students in plain terms, a rule of 72 doubling time study guide section, and a treatment of continuous compounding and the number e that actually makes sense. Short by design, with no filler.

Read straight through once to build the conceptual thread, then work each example as you encounter it. At the end, the problem set is there to test whether the ideas have actually landed — do not skip it.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon