How Simple and Compound Interest Work

Simple interest grows in a straight line. You apply the rate only to the original principal for each period. Past interest never earns more interest.

Formula

I = P × r × t, where I is interest earned (or owed), P is principal, r is the annual rate as a decimal, and t is time in years. Final amount A = P + I = P(1 + r·t).

Worked example

$5,000 at 6% simple for 3 years: I = 5,000 × 0.06 × 3 = $900. A = $5,900. Year 1, 2, and 3 each add the same $300 because the base never changes.

Where you still see it

Some short-term consumer products, certain notes, and textbook problems use simple interest. Many retail savings accounts and CDs do not — they compound. Always read the disclosure before you assume “6% for 3 years” means $900 flat.

If you need “what is 6% of 5,000?” as a one-shot percent, use the free percentage calculator.

Compound interest adds earned interest back into the balance, then charges (or pays) the rate on the new total. Growth curves upward instead of in a straight line. That is why long horizons matter so much for savings and investments.

Standard formula

A = P(1 + r/n)^n·t. A is the ending amount, P principal, r annual nominal rate (decimal), n compounding periods per year, and t years. Interest earned is A − P.

Worked example (monthly)

$5,000 at 6% compounded monthly for 3 years: r = 0.06, n = 12, t = 3. A = 5,000 × (1 + 0.06/12)³⁶ ≈ 5,000 × 1.19668 ≈ $5,983.40. Interest ≈ $983.40 — about $83 more than the simple case above.

Why the gap grows with time

At 3 years the difference looks modest. Stretch the same 6% monthly compound to 20 years: A ≈ 5,000 × (1.005)²⁴⁰ ≈ $16,551 versus simple A = 5,000 × (1 + 0.06×20) = $11,000. Compounding’s edge is mostly a time story, not a “magic rate” story.

Frequency is n in the formula: how many times per year interest is credited and added to the base.

Frequency	n	Typical use
Annually	1	Some bonds, textbook defaults
Quarterly	4	Some bank products
Monthly	12	Savings, many loans’ internal math
Daily	365	High-yield savings, many CDs

Same rate, different n

$10,000 at 5% for 5 years:

• Annual (n=1): A ≈ $12,762.82
• Monthly (n=12): A ≈ $12,833.59
• Daily (n=365): A ≈ $12,840.03

Moving from annual to monthly is the bigger jump. Daily versus monthly is usually a few dollars to tens of dollars at moderate rates and terms. Always match the product’s stated frequency; guessing “daily” when the bank compounds monthly will slightly overstate growth.

Locked fixed-term products often quote APY already. For maturity value on a certificate, also see the free CD calculator and our CD interest guide.

A growth chart plots balance (or interest earned) against time. Simple interest is a straight line. Compound interest bends up. Side-by-side comparison makes the long-run gap obvious without staring at formulas.

What to look for

• Principal line vs total balance — the gap is interest earned
• Year markers — early years look flat; later years accelerate under compounding
• Rate sensitivity — a 1-point rate change compounds into a large dollar gap over decades
• Frequency — higher n lifts the curve slightly for the same nominal rate

Year-by-year tables

Tables show ending balance and interest for each year. Use them when you need exact planning numbers (tax estimates, goal checks). Charts are better for intuition; tables are better for “what is year-7 interest?” questions.

Run both views in the free interest calculator: set principal, rate, years, and frequency, then switch between simple and compound to see the split.

Pick the model that matches the product or problem — not the one that looks friendlier.

• Use simple when the agreement says interest is calculated only on original principal, or a short-term note defines I = P×r×t explicitly.
• Use compound for savings accounts, CDs, reinvested investments, and most multi-year growth projections.
• Use amortization formulas for loans when you make fixed monthly payments that blend principal and interest — that is related math, not pure “leave principal alone and watch it grow.”

Loans vs savings

Borrowing costs are often quoted as APR with monthly payment schedules. The free loan calculator and loan payment guide cover installment math. Home loans use the same family of ideas at larger balances — see the mortgage calculator and mortgage payment guide.

Retirement horizons

Multi-decade savings with ongoing contributions need compound growth plus deposit schedules. That is a step beyond a single lump-sum interest run — pair this guide with our retirement savings guide when contributions and withdrawal rates enter the picture.

Enter principal, annual rate, time, choose simple or compound, set frequency if compound, then read ending balance and total interest.

1. Open the free interest calculator.
2. Enter starting principal (the amount invested or borrowed as a lump sum).
3. Enter the annual rate as a percent (for example 5.25, not 0.0525, if the tool expects percent form).
4. Set the time horizon in years (or convert months: 18 months = 1.5 years).
5. Pick simple or compound interest.
6. If compound, choose daily, monthly, quarterly, or annually to match the product.
7. Compare total interest and ending balance; use the growth chart for a multi-year view.

Sanity checks

If compound results look lower than simple at the same inputs, you likely mistyped rate or years. If daily and monthly differ by hundreds on a small balance and short term, recheck the rate (you may have entered 50 instead of 5.0).

Most errors are unit and definition problems, not broken algebra.

• Rate as decimal vs percent — 5% is 0.05 in the formula; plugging in 5 explodes the result
• Months as years — 6 months is t = 0.5, not t = 6
• Wrong n — using annual compounding on a daily product understates APY-style growth
• APR vs APY mix-up — APR is nominal; APY already includes compounding for comparison shopping
• Ignoring contributions — pure interest formulas assume one deposit; monthly savings need a future-value-of-annuity model
• Comparing loan APR to savings APY without context — fees, taxes, and payment structure change the real cost or return

This guide is educational, not personalized financial advice. Product disclosures and tax rules control what you actually earn or pay.