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Ch 3: Savings·§3.0 Chapter overview
3
Chapter 3

Savings

Simple vs. compound interest, the CPI and inflation, and annuity formulas for planning how money grows over time.

This week is about how money grows over time. By Sunday you'll know the difference between simple and compound interest, how an annuity adds up over years of monthly deposits, and why starting earlier matters more than saving more.

You'll meet three formulas you'll use the rest of the course: simple interest (FV = P(1+rt)), compound interest (FV = P(1+r/n)nt), and annuity future value. The shapes look heavy on the page; the moves are small.

By the end of this chapter, you'll be able to…

  • 3.1Use simple and compound interest formulas to analyze financial issues.
  • 3.2Explore the relationship between the Consumer Price Index (CPI) and inflation.
  • 3.3Use the lump sum and annuity formulas to explore methods of financial savings.
  • 3.4Analyze systems of investments to help develop financial literacy skills.

Sections

  1. 3.1What interest actually is.Before any formula, four words: principal, rate, time, interest. Once you name them, every calculation in this topic becomes a recipe.Read →
  2. 3.2Simple interest, the warmup formula.Multiply principal by rate by time. That's it. Simple interest grows in a straight line — no compounding, no surprises.Read →
  3. 3.3Compound interest, and where it leads.Interest earning interest. Run the same formula at four different compounding frequencies and watch the values converge — toward continuous compounding, the limit case.Read →
  4. 3.4The rate you really earn.The advertised rate isn't what you get. APY adjusts for compounding frequency. Real return adjusts for inflation. Two flavors of the same insight.Read →
  5. 3.5Lump sum vs. annuity.One deposit or many. Both grow with compound interest, but the formulas look different because of how the deposits stack up over time.Read →
  6. 3.6Build a savings plan.Three goals, three formulas, three time horizons. Emergency fund, mid-term goals, retirement — each one a savings problem with the math you've already learned.Read →
Start with §3.1 →

Chapter summary

Formulas, key concepts, and the kind of one-page reference you'd want during a problem set.

This chapter is the math of money growing for you: simple interest, compound interest, continuous compounding, annual percentage yield (APY), and the future value of regular contributions (annuities). By the end, you can read any savings instrument — checking account, CD, 401(k), Roth IRA — and tell the story of how money grows over time. The companion Compound Interest Lab and Retirement Planner tools on the course site let you change the inputs and watch the curves move.

Chapter glossary

All key terms introduced across this chapter, in the order they appear in the reading.

Principal (P)
The starting amount: what you borrowed or deposited. Every formula starts here.
Rate (r)
The annual interest rate, written as a decimal. 5% becomes 0.05 before it goes into a formula.
Time (t)
The term of the loan or investment, in years. Six months is 0.5; a decade is 10.
Interest (I)
The dollar amount of interest itself — the growth in the pile, separate from the principal.
Future value (A)
The whole pile after interest: principal plus interest. Most ALEKS questions ask for A, not I.
Simple interest
Interest computed only on the original principal. Doesn't compound.
Linear growth
Equal additions in equal time periods. The graph is a straight line.
Treasury bill (T-bill)
Short-term U.S. government debt — typically 4, 13, 26, or 52 weeks. Standard real-world use of simple interest.
Term
How long the principal is lent or invested, in years. Convert months and days to years before plugging into the formula.
Accumulated amount (A)
Principal plus interest. A = P + I, which factors to P(1 + rt).
Compound interest
Interest paid on principal plus previously earned interest. The pile grows on itself.
Compounding period
The interval at which interest is added to the principal. Annual, monthly, daily, or continuous.
Compounding frequency (n)
Number of compounding periods per year. Annual = 1, semiannual = 2, quarterly = 4, monthly = 12, daily = 365.
Continuous compounding
The limit case where interest is added instantaneously and continuously. Uses A = Pert.
Euler's number (e)
An irrational constant, approximately 2.71828. Appears whenever something grows or decays continuously.
APY (Effective Annual Yield)
What an account actually pays in a year, with compounding factored in. Always ≥ APR for any n > 1.
APR (Nominal rate)
The advertised annual rate, before compounding effect. Also called the stated or nominal rate.
CPI (Consumer Price Index)
The U.S. Bureau of Labor Statistics index tracking the cost of a basket of household goods. The data source for inflation.
Inflation
The annual percent increase in prices. Computed from CPI: (CPInew / CPIold) − 1.
Nominal vs. real return
Nominal is the headline rate. Real is what's left after subtracting inflation. Real is what your purchasing power actually did.
Lump sum
One deposit, left to grow. Future value comes from the compound interest formula.
Annuity
A series of equal periodic payments — weekly, monthly, or annually — into a savings or investment account.
Periodic payment (M)
The amount of each regular deposit. The ALEKS dictionary uses M; some textbooks use PMT.
Time horizon
How long the money has to grow. Lengthening t beats raising r for long-term saving.
Emergency fund
3 to 6 months of expenses, kept liquid. The first savings goal — before any longer-term investing makes sense.
HYSA (high-yield savings account)
An online savings account paying APYs near the federal funds rate. Where short-term savings live.
CD (certificate of deposit)
A time deposit with a fixed term and rate. Can't withdraw early without penalty. Used for mid-term goals.
IRA / 401(k)
Retirement accounts with tax advantages. Where long-term savings compound for decades. Topic 7 covers what to put inside them.
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