Simple and Compound Interest
Calculating simple interest and compound interest under different compounding frequencies, and understanding why compound interest grows faster.
What you'll learn
- Compute simple interest given principal, rate, and time
- Compute compound interest given principal, rate, time, and compounding frequency
- Explain why compound interest exceeds simple interest over the same period and rate
Prerequisites
Explanation
Simple interest (SI) is calculated only on the original principal, every period, for the entire duration: SI = (P × R × T) / 100, where P is principal, R is the annual rate as a percentage, and T is time in years. It grows in a straight line — the same amount of interest is earned every year.
Compound interest (CI), by contrast, is calculated on the principal plus all interest already earned — interest earns interest. The total amount after T years, compounded annually, is A = P × (1 + R/100)^T, and CI = A − P.
Worked example. Take P = ₹10,000, R = 10% per annum, T = 2 years. SI = (10000 × 10 × 2)/100 = ₹2,000. For CI: A = 10000 × (1.10)² = 10000 × 1.21 = ₹12,100, so CI = ₹2,100. CI exceeds SI by exactly ₹100 here — and that ₹100 is precisely the interest earned on the first year's interest: year one's interest of ₹1,000 itself earns 10% in year two, contributing 1000 × 0.10 = ₹100 extra. That's the entire intuition behind why CI grows faster: every period, the "principal" being paid interest on grows, while under SI it never does.
General compounding periods. Interest doesn't have to compound once a year — banks often compound semi-annually, quarterly, or monthly. If interest compounds n times per year for t years at annual rate R%, the amount is:
A = P × (1 + R/(100n))^(n×t)
Each compounding period earns R/n percent (a fraction of the annual rate), but there are n×t total periods instead of just t. Compounding more frequently at the same annual rate always produces a slightly larger final amount, because interest starts earning interest sooner.
Worked example (general compounding). ₹10,000 at 10% annual rate, compounded semi-annually (n = 2), for 1 year: each half-year earns 10/2 = 5%, over 2 periods. A = 10000 × (1.05)² = 10000 × 1.1025 = ₹11,025, giving CI = ₹1,025 — slightly more than the ₹1,000 you'd get from a single annual compounding over the same 1 year (10000 × 1.10 − 10000 = ₹1,000).
Comparing SI and CI over time: for year 1, SI and CI are identical, because there's no prior interest yet to compound. From year 2 onward, CI pulls ahead, and the gap widens every additional year — this is why "compounding" is treated as a powerful long-term force in savings and loans alike: the rate of divergence between CI and SI itself accelerates over time, it isn't constant.
SI vs CI growth on ₹10,000 at 10% per annum
Year 1: SI amount 11,000, CI amount 11,000 (identical, since no interest has compounded yet). Year 2: SI amount 12,000, CI amount 12,100. Year 3: SI amount 13,000, CI amount 13,310. The gap between CI and SI widens every year because CI compounds on a growing base.
Example
Annual compound interest, computed as the amount after T years minus the original principal.
function compoundInterest(principal, ratePercent, years) {
const amount = principal * Math.pow(1 + ratePercent / 100, years);
return Math.round((amount - principal) * 100) / 100;
}
// Example: compoundInterest(10000, 10, 2) -> 2100Guided exercise
Guided exercise
Write simpleInterest(principal, ratePercent, years) that returns the simple interest earned, rounded to 2 decimal places, using SI = (principal * ratePercent * years) / 100.
Checks: Computes simple interest for typical values · Computes simple interest for a different rate and time · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Independent exercise
Independent exercise
Write compoundInterestGeneral(principal, ratePercent, years, timesCompoundedPerYear) that returns the compound interest earned, rounded to 2 decimal places, using A = principal * (1 + ratePercent / (100 * timesCompoundedPerYear)) ^ (timesCompoundedPerYear * years), then CI = A - principal.
Checks: Matches annual compounding for a typical case · Correctly handles more frequent (semi-annual) compounding · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Common mistakes
- Using the simple interest formula on a multi-year compound interest problem, which underestimates the true amount because it ignores interest-on-interest.
- Forgetting to divide the annual rate by the number of compounding periods per year (and to multiply years by that same number of periods) when compounding is not annual.
- Reporting the total amount A as the answer when the question asks specifically for the interest earned, which is A minus the original principal.
Knowledge check
Takeaway
Compound interest grows faster than simple interest purely because each period's interest is calculated on a base that includes all previously earned interest.
Summary
Simple interest grows linearly from a fixed principal (SI = PRT/100), while compound interest grows on an expanding base that includes previously earned interest (A = P(1+R/100)^T for annual compounding, or A = P(1+R/(100n))^(nt) for n compounding periods per year). The CI-SI gap starts at zero and widens every additional period.
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