Arithmetic

Simple and Compound Interest formulas

Learn Simple and Compound Interest step by step

At a glance

SI = PRT/100            A = P(1 + R/100)^T
k times a year: R/k per period, kT periods
CI − SI, 2 yr = PR²/100²    3 yr = PR²(300 + R)/100³
SI doubles in 100/R yr     CI doubles in ≈ 72/R yr
Loan = x/(1 + r) + x/(1 + r)²
Depreciation: P(1 − R/100)^n
CAGR = (F/I)^(1/n) − 1  <  simple annual rate = total % ÷ n

Simple interest: the same slice every year

P = principal, R = rate % per year, T = time in years.

SI = P × R × T / 100
Amount A = P + SI

Shortcut: SI for T years at R% is simply (R × T)% of the principal. 8% for 5 years is 40% of P; 12.5% for 4 years is 50% of P.

Compound interest: interest on interest

Compound interest: each year's interest is the last one × (1 + R/100)

A = P × (1 + R/100)^T
CI = A − P

SI vs CI: a line against a curve

SI is a straight line; CI curves away above it after year 1

Shortcut: SI for 2 years tells you P × R. Then CI for 2 years = SI × (1 + R/200). At 10%, CI for 2 years is 1.05 × the SI.

Half-yearly and quarterly compounding

Compounded k times a year:

A = P × (1 + R/(100k))^(kT)
Half-yearly: R/2 per period, 2T periods
Quarterly: R/4 per period, 4T periods

Shortcut: 20% a year compounded half-yearly is 10% twice: 10 + 10 + 10×10/100 = 21% a year. It is the successive-change formula from Percentages.

CI − SI for 2 and 3 years

2 years:  CI − SI = P × (R/100)² = PR² / 100²
3 years:  CI − SI = PR²(300 + R) / 100³

Shortcut: The 2-year gap is just one year's SI earning interest for one year: CI − SI = (one year's SI) × R/100.

Doubling, tripling and the rule of 72

CI: doubling time ≈ 72 / R years (rule of 72)
SI: doubling time = 100 / R years
CI: x× in n years  →  x^k × in kn years

Equal instalments (in brief)

Each instalment, discounted back to today, adds up to the loan:

Loan = x/(1 + r) + x/(1 + r)²   (r = R/100)

Population growth and depreciation

Growth:        P × (1 + R/100)^n
Depreciation:  P × (1 − R/100)^n
n years ago:   P / (1 + R/100)^n
Different rates: P × (1 + a/100)(1 + b/100)(1 + c/100)

Growth rates: absolute, average and CAGR

Initial value I, final value F, over n years.

absolute growth = F − I        total growth % = (F − I) / I × 100
simple (average) annual growth = total growth % ÷ n
CAGR = (F / I)^(1/n) − 1
yearly rates a, b, c, …:  CAGR = [(1 + a)(1 + b)(1 + c)…]^(1/n) − 1

Growth factor (1 + R/100)ⁿ: read CAGR backwards from it.

Rate	2 years	3 years	4 years	5 years
5%	1.1025	1.158	1.216	1.276
10%	1.21	1.331	1.464	1.611
15%	1.3225	1.521	1.749	2.011
20%	1.44	1.728	2.074	2.488
25%	1.5625	1.953	2.441	3.052

Revenue up 1.75 times in 4 years? The 4-year column puts that at about 15% CAGR.

Shortcut: Find F/I, then find the row in the growth-factor table whose n-year factor is closest. For a doubling, use the rule of 72: doubling in n years means CAGR ≈ 72/n %. For small growth over a few years, CAGR is a little under the simple rate, so if the options are far apart, pick the one just below total growth ÷ n.

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