How accurate is the Rule of 72: exact doubling time vs 72 divided by the rate

The Rule of 72 says money doubles in about 72 divided by the annual rate of return, in years. At 6% that gives 12 years; the exact answer is 11.90 years. The rule is within about 1% of the exact value for rates from 6% to 10%, within 2% from 4% to 12%, and it is exact near 7.85%.

Informational only, not financial advice. Real returns vary and are not guaranteed.

What is the exact doubling time formula?

For a constant annual rate r (as a percentage), the exact time to double is:

years = ln(2) / ln(1 + r/100)

This is the same formula Mathify's Rule of 72 Calculator uses for its exact comparison. Each row below was also checked by compounding 1 unit for exactly that many years and confirming the result is 2. The generating script is scripts/content-data/rule-of-72-accuracy.ts.

How does 72 divided by the rate compare with the exact answer?

Assumptions: one constant annual effective rate, interest compounded once a year, no fees, taxes, deposits or withdrawals. "Error" is the estimate minus the exact value, as a percentage of the exact value. A plus sign means the shortcut overestimates the time.

1%: exact 69.66 yr | 72/r = 72.00 yr (+3.36%) | 70/r = 70.00 yr (+0.49%)
2%: exact 35.00 yr | 72/r = 36.00 yr (+2.85%) | 70/r = 35.00 yr (-0.01%)
3%: exact 23.45 yr | 72/r = 24.00 yr (+2.35%) | 70/r = 23.33 yr (-0.50%)
4%: exact 17.67 yr | 72/r = 18.00 yr (+1.85%) | 70/r = 17.50 yr (-0.98%)
5%: exact 14.21 yr | 72/r = 14.40 yr (+1.36%) | 70/r = 14.00 yr (-1.45%)
6%: exact 11.90 yr | 72/r = 12.00 yr (+0.88%) | 70/r = 11.67 yr (-1.93%)
7%: exact 10.24 yr | 72/r = 10.29 yr (+0.40%) | 70/r = 10.00 yr (-2.39%)
8%: exact 9.01 yr | 72/r = 9.00 yr (-0.07%) | 70/r = 8.75 yr (-2.85%)
9%: exact 8.04 yr | 72/r = 8.00 yr (-0.54%) | 70/r = 7.78 yr (-3.30%)
10%: exact 7.27 yr | 72/r = 7.20 yr (-1.00%) | 70/r = 7.00 yr (-3.75%)
12%: exact 6.12 yr | 72/r = 6.00 yr (-1.90%) | 70/r = 5.83 yr (-4.63%)
15%: exact 4.96 yr | 72/r = 4.80 yr (-3.22%) | 70/r = 4.67 yr (-5.90%)
18%: exact 4.19 yr | 72/r = 4.00 yr (-4.49%) | 70/r = 3.89 yr (-7.14%)
20%: exact 3.80 yr | 72/r = 3.60 yr (-5.31%) | 70/r = 3.50 yr (-7.94%)
24%: exact 3.22 yr | 72/r = 3.00 yr (-6.90%) | 70/r = 2.92 yr (-9.48%)
30%: exact 2.64 yr | 72/r = 2.40 yr (-9.16%) | 70/r = 2.33 yr (-11.68%)
36%: exact 2.25 yr | 72/r = 2.00 yr (-11.28%) | 70/r = 1.94 yr (-13.74%)
50%: exact 1.71 yr | 72/r = 1.44 yr (-15.77%) | 70/r = 1.40 yr (-18.11%)

The shortcut is slightly long at low rates, nearly perfect around 8%, and increasingly short at high rates.

Why does 72 work at all?

For small rates, ln(1 + r/100) is close to r/100, so the exact time is close to ln(2) x 100 / r, or 69.31 / r. The number 72 is a convenient choice for two reasons. It sits a little above 69.31, which compensates for the curve at everyday rates, and it divides evenly by many common rates: 1%: 72 yr, 2%: 36 yr, 3%: 24 yr, 4%: 18 yr, 6%: 12 yr, 8%: 9 yr, 9%: 8 yr, 12%: 6 yr.

You can see the drift by multiplying the exact doubling time by the rate. For a perfect "Rule of N" that product would be constant:

2%: exact years x rate = 70.01
5%: exact years x rate = 71.03
8%: exact years x rate = 72.05
12%: exact years x rate = 73.40
20%: exact years x rate = 76.04
36%: exact years x rate = 81.15

The product rises from about 70 at 2% to 72 near 8% and keeps climbing. So 70 fits low rates best and 72 fits rates near 8%.

Worked example: how long to double at 6% and 8%?

At 6%: 72 / 6 = 12.00 years. Exact: 11.90 years, roughly 143 months.
At 8%: 72 / 8 = 9.00 years. Exact: 9.01 years.

Which rate do you need to double in a set time?

Flip the rule: rate is about 72 divided by years. For 10 years, 72 / 10 = 7.20%, and the exact rate is 7.18%. For 5 years, 72 / 5 = 14.40%, and the exact rate is 14.87%. The shortcut is closer for the 10-year case because 7% is near the rule's sweet spot.

Common mistakes

Using the rule at very high rates, where the error exceeds 5%.
Entering the rate as a decimal (0.06) instead of a whole number (6).
Applying it to a changing or negative rate; it assumes one constant rate.
Ignoring inflation, so "doubling" refers to nominal money, not purchasing power.
Forgetting that fees and taxes reduce the effective rate.

Key takeaways

72 / rate is a mental shortcut, not an exact answer.
It is within about 1% from 6% to 10% and within 2% from 4% to 12%.
For rates up to 4%, 70 is closer; above 12%, both shortcuts run short.
For exact figures, use the formula or the Rule of 72 Calculator.

To go further, the Compound Interest Calculator shows the balance year by year, and the CAGR Calculator works out the constant rate behind a known start and end value. See also what $1,000 grows to at different rates.

FAQ

How long does it take to double money at 7%?

The shortcut gives 72 / 7 = 10.29 years. The exact value is 10.24 years.

Is the Rule of 72 or the Rule of 70 more accurate?

It depends on the rate. In the table, 70 is closer for rates up to 4% and 72 is closer from 5% upward in the rows shown.

Does the Rule of 72 work for inflation?

Yes, for a constant rate. At 3% inflation, prices double in about 24 years (exact: 23.45). It is the same arithmetic.

Can it be used for tripling?

Not directly. Tripling uses ln(3) instead of ln(2), so the constant would be about 110 rather than 72.

Why is the exact value different from 72 divided by the rate?

Because compound growth is a curve. The shortcut treats it as a straight line scaled by 72.

How accurate is the Rule of 72: exact doubling time vs 72 divided by the rate