Academics · High School · Financial math

The math behind
every balance
and payment.

Grow savings with monthly deposits, work a loan payment and its schedule to the cent, and turn an APR into the APY you actually earn. Every answer shows the formula with your numbers in it.

Open the interest labWhat you get
3parts: savings, loans and rates
5ways to compound, from yearly to continuous
11plans and loans checked a second way

The interest lab

Change one number and watch the balance, the payment or the APY move. Your numbers stay in this browser.

Loading the interest lab. If nothing appears, turn on JavaScript.

Three ideas that run all of it

Every number on this page comes from one of these.

Compounding

Interest earns interest

At 7% a year, $1,000 grows to $1,070, then $1,144.90, because the second year’s 7% is taken on $1,070. Over 30 years that is $7,612.26, against $3,100 if only the first $1,000 earned interest.

Rule of 72

A quick doubling estimate

Divide 72 by the yearly rate to estimate the years to double. At 6% compounded monthly: 72 ÷ 6 = 12 years, against an exact 11.6. The rule is close for rates from about 4% to 12%.

Amortization

Early payments are mostly interest

On the example car loan the first payment of $303.43 carries $98.75 of interest; the last carries $1.98. Money paid early on the balance saves interest every month after.

What you get

Academics · High SchoolAlso for Enrichment · STEM ExplorationsFree

What you finish with
Worked answers to real money questions with the formula shown: what savings grow to and how much of that is interest, what a loan costs each month and in total, how much an extra payment saves, and which of two rates is better once compounding is counted. A printed summary and .csv tables.
Who it is for
High school students in algebra, precalculus, financial math or personal finance; middle school students in a STEM enrichment course; teachers who want live examples with real formulas; families comparing a savings account or a loan.
Format
Browser lab in three parts: Grow savings, Borrow and Compare rates. Works on a phone, tablet or computer, by touch or keyboard. Prints a one-page summary; downloads the year table or the loan schedule as a .csv.
Access
Free. No account, no sign-in and nothing to install. Everything you type stays in this browser; nothing is sent to Caveat.
Where it lives
The High School space in Caveat Academics; also listed for STEM Explorations (the same title in the STEM Explorations catalog for middle school).
Support
Questions, a correction or a request: team@caveatacademic.com.
Simple against compound interest: the same 7% on the same $1,000. The gap is interest earned on interest.

How to use it

  1. Load the example, or type a starting amount, a monthly deposit, a rate and the years.
  2. Read the formula with your numbers, then change the compounding and watch the monthly rate move.
  3. Open Borrow, type a loan and an extra monthly amount, and scroll the schedule to see interest shrink.
  4. Open Compare rates and type two offers as printed to see which grows money faster.

Open the interest lab

How the numbers are worked out

Standard formulas, whole cents for loans, and a second calculation behind every result.

$0$100$200$300Interest in month 1: $98.75month 60: $1.98Month 1Month 60interestpaying down the balance$0$100$200$300Interest in month 1: $98.75month 60: $1.98Month 1Month 60interestpaying down the balance
Where each payment goes on the example loan, $15,000.00 at 7.9% for 60 months. The interest share shrinks as the balance falls.

How the page computes

Savings use the future value of a starting amount plus the future value of an annuity, with deposits at the end of each month. When the rate compounds other than monthly, the monthly rate is the one that grows the same in a year: (1 + r/k)k/12 − 1. Loans use the standard payment formula with i = APR ÷ 12, then run the schedule month by month in whole cents, so the last payment absorbs the rounding, the way a lender’s schedule does. APY is (1 + r/k)k − 1, and er − 1 for continuous compounding.

Checked a second way

Every result was computed again in Python before the page was built (fin_ref.py): 5 savings plans, 6 loans with their schedules and 5 rates, including textbook values such as $1,000 at 5% for 10 years ($1,628.89) and a $10,000 loan at 6% for 30 years ($59.96 a month). The page is compared with those on every test run.

Sources

  1. Consumer Financial Protection Bureau, Regulation DD (Truth in Savings), 12 CFR part 1030, Appendix A: annual percentage yield, and Regulation Z (Truth in Lending), 12 CFR part 1026: annual percentage rate.
  2. Common Core State Standards for Mathematics (2010): A-SSE.B.4 (the sum of a finite geometric series, used to calculate mortgage payments), F-LE.A.1c and F-LE.A.2 (exponential growth), and F-IF.C.8b (interpreting growth rates in exponential functions).
  3. Jump$tart Coalition, National Standards for Personal Financial Education (2021): saving (compound interest, the time value of money) and managing credit (the cost of borrowing, amortization).
  4. The arithmetic was checked against an independent Python version written for this page (guide/suite/fin_ref.py).

A math lab, not financial advice. Real accounts and loans have fees, variable rates and terms this page does not model; a lender’s schedule can differ by a few cents. Example numbers are made up. Support: team@caveatacademic.com.