Years to reach the chosen multiple at each rate: the exact answer with annual compounding, the rule, and the rule’s error. Doubling also shows the rules of 70 and 69.3.
Rate
Exact years
Rule
Rule minus exact
Rule of 70
Rule of 69.3
1%
69.661
72.000
+2.339
70.000
69.300
2%
35.003
36.000
+0.997
35.000
34.650
4%
17.673
18.000
+0.327
17.500
17.325
6%
11.896
12.000
+0.104
11.667
11.550
8%
9.006
9.000
−0.006
8.750
8.662
10%
7.273
7.200
−0.073
7.000
6.930
12%
6.116
6.000
−0.116
5.833
5.775
15%
4.959
4.800
−0.159
4.667
4.620
20%
3.802
3.600
−0.202
3.500
3.465
25%
3.106
2.880
−0.226
2.800
2.772
30%
2.642
2.400
−0.242
2.333
2.310
Pick whether to solve for years or for the rate, and whether you want to double, triple or quadruple. The exact answer and the error sit beside the rule, and the table shows how the rule drifts as the rate moves away from about 8%.
Estimates only — not financial, tax, or professional advice.
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What it calculates: Rule estimate, Exact years, Rule minus exact, Error.
At 8% a year the rule of 72 says 9 years to double; the exact answer is 9.006
The rule of 72 turns a compounding question into one division. Money growing at a steady 8% a year doubles in about 72 ÷ 8 = 9 years. The exact answer comes from logarithms, ln 2 ÷ ln 1.08 = 9.006 years, so at 8% the shortcut is off by less than three days.
Years ≈ 72 ÷ rate%
Run it the other way to find a rate: to double in 10 years the rule asks for 72 ÷ 10 = 7.2% a year, while the exact rate is 2^(1/10) − 1 = 7.177%. Both forms are on the calculator, with the exact figure and the error shown beside the estimate.
Doubling at 8%
Rule: 72 ÷ 8 = 9.00 years
Exact: ln 2 ÷ ln 1.08
Exact = 9.006 years
The rule is exact at 7.85%: below that it runs long, above it short
The exact doubling time is ln 2 ÷ ln(1 + r). Divide 69.3 — that is, 100 × ln 2 — by the rate and you have the exact answer for continuous compounding. Annual compounding takes a little longer, and 72 is the round number that allows for it best across ordinary rates: it matches the exact figure at 7.85% a year.
Below 7.85% the rule gives slightly too many years (at 2%, 36 against 35.0); above it, too few (at 20%, 3.6 against 3.80; at 30%, 2.40 against 2.64). Seventy-two also divides evenly by 2, 3, 4, 6, 8, 9 and 12, which is the other half of why it survives. The table under the calculator prints the drift rate by rate.
At 2%: rule 36.0 years, exact 35.00
At 6%: rule 12.0 years, exact 11.90
At 10%: rule 7.2 years, exact 7.27
At 20%: rule 3.6 years, exact 3.80
Incomes growing 6% a year double in 72 ÷ 6 = 12 years — exactly 11.90
OpenStax’s Principles of Economics uses the rule to show why growth rates matter for a country’s standard of living: at 6% a year, incomes double in 72 ÷ 6 = 12 years. The exact figure is ln 2 ÷ ln 1.06 = 11.90 years — close enough that the textbook’s point, that a few points of growth transform living standards within a generation, stands either way.
Economists often use 70 instead of 72 because growth rates for economies are low, and at low rates 70 is closer: at 2% the rule of 70 gives 35.0 years against an exact 35.00, where 72 gives 36.
Incomes at 6%
Rule: 72 ÷ 6 = 12 years
Exact: ln 2 ÷ ln 1.06
Exact = 11.90 years
At 3% inflation, prices double — and cash buys half as much — in about 24 years
The rule works on anything that compounds, including prices. At 3% inflation prices double in about 72 ÷ 3 = 24 years (exactly ln 2 ÷ ln 1.03 = 23.45), which is the same as saying cash left under a mattress loses half its buying power in that time.
That is also why the rule should be run on a real rate when the question is about purchasing power: a 7% return with 3% inflation grows buying power at roughly 4%, doubling it in about 18 years rather than the 10 the nominal rate suggests.
A card charging 24% doubles an unpaid balance in about 3 years; monthly compounding makes it 2.92
Debt grows by the same arithmetic. At a 24% APR the rule of 72 says an untouched balance doubles in 72 ÷ 24 = 3 years. Compounded once a year the exact figure is 3.22 years, but cards charge interest monthly, at 2% a month here, and ln 2 ÷ (12 × ln 1.02) = 2.92 years — about 35 months.
At high rates the rule underestimates annual compounding and overestimates monthly compounding, so the shortcut lands between the two. It is still the fastest way to feel what an interest rate means.
24% APR
Rule: 72 ÷ 24 = 3.00 years
Annual: ln 2 ÷ ln 1.24 = 3.22 years
Monthly: ln 2 ÷ (12 ln 1.02) = 2.92 years
Tripling takes the rule of 114 and quadrupling the rule of 144: at 8%, 14.25 and 18 years
The same idea stretches to other multiples by swapping the rule number. To triple, divide 114 by the rate: at 8%, 114 ÷ 8 = 14.25 years, against an exact ln 3 ÷ ln 1.08 = 14.27. To quadruple, divide 144 — two doublings, 2 × 72 — by the rate: 18 years at 8%, exactly 18.01.
Choose the multiple on the calculator and the estimate, the exact answer and the error all switch to it.
This page is the shortcut and its error; the exact formula and CAGR are one click away
The Doubling Time Calculator works the exact formula for any growth rate and any period — days, months or years — and handles shrinking quantities too. This page is for the rule itself: the number you can work in your head, how far off it is at your rate, and its tripling and quadrupling cousins.
When the question starts from two values rather than a rate — revenue went from one figure to another in five years — the CAGR Calculator finds the rate first, and shows the rule-of-72 doubling time beside the exact one.
Frequently Asked Questions
What is the rule of 72?
A shortcut for how long a value takes to double at a steady yearly rate: divide 72 by the rate in percent. At 8% a year, 72 ÷ 8 = 9 years; the exact answer is 9.006.
How accurate is the rule of 72?
It is exact at about 7.85% and very close between roughly 5% and 12%. Below that it runs long (36 years at 2% against 35.0); above it, short (3.6 years at 20% against 3.80).
Rule of 72, 70 or 69.3 — which should I use?
Use 72 for everyday interest rates with annual compounding, 70 for low growth rates such as GDP, and 69.3 (100 × ln 2) for continuous compounding, where it is exact.
How do I find the rate needed to double in a given time?
Divide 72 by the years: to double in 10 years you need about 7.2% a year. The exact rate is 2^(1/10) − 1 = 7.177%.
Does the rule of 72 work for inflation and debt?
Yes. At 3% inflation prices double in about 24 years; at a 24% APR an unpaid balance doubles in about 3 years (2.92 with monthly compounding).
How do I do this in Excel?
The rule is =72/rate with the rate as a whole number. The exact doubling time is =LN(2)/LN(1+rate) with the rate as a decimal, or =NPER(rate, 0, -1, 2).
Sources & References
Figures on this page are checked against primary, authoritative sources. Links open in a new tab.
Nothing is added or withdrawn while the value grows.
Where it stops
The rule drifts as the rate moves away from about 8%: at 30% it is short by a quarter of a year.
It ignores taxes, fees and inflation; for real growth, subtract inflation from the rate first.
Monthly or daily compounding shortens the exact time; the continuous figure shows the limit.
Related Calculators
Doubling TimeHow long a quantity takes to double at a constant rate, or from two measurements — with the rule of 70 and rule of 72 measured against the exact answer.
CAGRThe compound annual growth rate between two values, or the end value a rate reaches and the years it needs, with the path year by year.
Compound InterestSee how savings grow as interest earns interest, with adjustable contributions and compounding frequency.
InflationWhat a US dollar from any year since 1913 is worth in another year, from the BLS CPI-U, with future inflation too.
Credit Card PayoffPlan up to 20 cards with issuer-style minimums, promo APRs, five payoff orders, and a balance-transfer scenario.
InvestmentProject lump-sum and regular-contribution growth, plan a goal, and solve future vs present value, with fees and inflation.
Regular InvestmentProject how regular monthly contributions grow over time — SIP-style investing, dollar-cost averaging, inflation-adjusted value, and long-term goals.
Retirement WithdrawalEstimate how long savings last under regular withdrawals (SWP) — drawdown, safe withdrawal rate, inflation, and a year-by-year schedule.
Returns are assumptions, not guarantees. Actual results may vary because of market performance, taxes, fees, inflation, and timing. This is an educational projection, not investment advice.
Published the rule of 72 calculator: years to double from a rate or the rate to double in a set time, with the exact answer and the error beside the rule, the rules of 70, 69.3, 114 and 144, and the OpenStax textbook case checked.
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