Simple vs Compound Interest

Side-by-side interest model comparison.

Worked example — Simple vs Compound Interest

Simple interest value	24000.00
Compound interest value	40387.39
Compound minus simple	16387.39

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Simple vs Compound Interest

Simple interest is paid on the original sum only, so it adds the same amount every year — a straight line. Compound interest is paid on the balance, which includes the interest already earned, so each year’s interest is larger than the last. One is linear growth; the other is exponential.

What each symbol means

r: the annual rate, 7% here, entered as a percentage and used as 0.07.
n: the compounds per year, 12. It appears only in the compound formula — simple interest has no such knob, which is itself the difference in miniature.

Where the formula comes from

Simple interest never revisits its base: after t years you have the principal plus t lots of P·r, so A = P(1 + r·t). At the defaults, 10000 × (1 + 0.07 × 20) = 24,000.
Compound interest multiplies instead of adding. Each period the balance is multiplied by (1 + r/n), and over t years there are n·t such periods — so A = P(1 + r/n)^n·t.
With n = 12 and t = 20 that is 240 multiplications, giving 40,387.39. The gap between the two, 16,387.39, is larger than the entire simple-interest gain of 14,000 — the interest earned on interest has out-earned the interest on the principal.

Assumes: A constant rate, no deposits or withdrawals after the start, no tax and no fees. Everything is nominal, so none of these figures is adjusted for inflation.
Breaks when: Set the time to a single year and the two are nearly equal — the whole advantage of compounding is a story about duration, not about rate. The comparison is also slightly unfair by construction: simple interest here compounds never, while its rival compounds monthly, so part of the gap is the frequency and not the principle. Set the compounds per year to 1 and the honest like-for-like gap is smaller.

The compounding slider runs out of road almost immediately 🖖

Compounding more often always helps, but it helps by less and less, and the tool's own frequency slider shows where that stops mattering. Take its long preset — 10,000 at 7% for 20 years. Compounded once a year you finish with 38,696.84. Switch to monthly and you gain 1,690.54. Switch from monthly to daily and you gain 159.17. Go from daily to compounding infinitely often and you gain 5.44. The slider goes to 365, but everything worth winning is already won by about the twelfth step, because the whole sequence is converging on a hard ceiling: Pe^rt = 40,552.00. That is where e comes from here — not a choice of base, but the limit the doubling-up runs into.

Interest on interest 🖖

Simple interest pays a fixed amount every year based only on your starting principal, so the total climbs in a straight line. Compound interest also pays interest on the interest already earned, so each year's gain is slightly bigger than the last and the curve bends upward. With annual compounding both match after the first year; every bit of the widening gap is money that past interest earned on your behalf.

A penny from year zero 🖖

In 1772 the actuary Richard Price calculated that one penny lent at the start of the Christian era at 5% simple interest would, after roughly 1,800 years, be worth only about seven shillings — less than a pound. The very same penny at 5% compound interest would be worth more gold than the entire solar system could hold. Same coin, same rate: the only difference is whether interest is allowed to earn interest.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Leave the defaults — 10,000 at 7%, compounded monthly. Guess how many years it takes the compound value to reach 20,000, then hunt for it with the years box.

Show answer

Just over 9.93 years: at exactly 9.93 the compound row reads 19998.67, a whisker short. The rule of 72 predicts 72 / 7 = 10.3 years, and by 10.3 you are already at 20521.85. The rule assumes interest added once a year; this page adds it twelve times, so the doubling lands early. Compounding more often never changes the rate — it shortens the wait.
Twenty years at 7% compounded monthly turns 10,000 into 40387.39. What simple rate would land on the same figure over the same twenty years? Move the rate box and read the simple row.

Show answer

About 15.19%, where simple interest reads 40380.00 — and 40400.00 once you nudge it to 15.2%. Seven per cent, allowed to earn on itself for twenty years, is worth more than double that rate paid on the original sum alone. The two rows are only comparable at the rate you stopped on, mind: compounding has run off to 205099.01 by then.

Problem solved in full

10 000 for 20 years at 7% compounding monthly 5 steps
10 000 for 20 years at 7%. Work out both interest schemes, then derive where the “rule of 72” comes from. Compounding monthly.
1. Simple interest pays only on the original sum, so the balance grows along a straight line. Twenty years of 7% adds 140% and nothing more.
  
  $simple = P (1 + r t) = 10000 (1 + 0.07 \times 20) = 24000$
2. Compound interest pays on the balance, including previously paid interest. That makes the growth exponential, and the exponent counts periods, not years — monthly compounding over 20 years is 240 of them.
  
  $compound = P (1 + \frac{r}{m})^{m t} = 10000 (1 + \frac{0.07}{12})^{240}$
3. Evaluate. The comparison above prints both of these, and the second is not 40% larger than the first by accident.
  
  $= 10000 \times (1.0058 \overline{3})^{240} = 40 387.39$
4. The gap is the interest that itself earned interest. Over 20 years it exceeds the entire original investment.
  
  $gap = 40 387.39 - 24000 = 16 387.39$
5. Now derive the shortcut everybody quotes. Doubling means the growth factor reaches 2, so take logs.
  
  $t_{2} = \frac{ln 2}{ln ( 1 + r / m ) m} = \frac{0.6931}{0.06976} = 9.93 yr (rule of 72 : \frac{72}{7} = 10.3)$
Answer

24 000 against 40 387.39 — a gap of 16 387.39, more than the money you started with. The doubling time works out at 9.93 years, and the rule of 72 gives 10.3: the rule is just ln 2 ≈ 0.693 rewritten as 69.3, rounded up to 72 because 72 divides neatly by 2, 3, 4, 6, 8, 9 and 12. It is a mental-arithmetic convenience hiding a logarithm, and it is accurate to about 1% for rates between 5 and 10%.

$compound - simple = 16 387.39 t_{double} \approx \frac{72}{r %}$

Learning path

Money over time

Leads to Annuity future value the growth factor (1 + r) and the habit of applying it once per period.

References (1)

On compounding conventions and why the frequency has to be stated with the rate: C. Attwood, "Compound Interest Functions," in Compound Interest Functions, 1–165. Pergamon, 1967.

By the 42Math team · Updated 2026-07-29

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Statistics & Finance lessons →

Example problems

short horizon - Short horizon where simple and compound outcomes remain close.
long horizon - Long horizon where compounding gap becomes pronounced.
high rate - Higher rates and time horizon amplify compound outperformance.