How to Calculate Doubling Time with the Rule of 70
The rule of 70 estimates doubling time: divide 70 by the annual percentage growth rate. Growth of 3.5% a year doubles a quantity in about 20 years.
The Doubling Time formula
Calculator
Enter an annual growth rate to get the rule-of-70 doubling time and the exact answer it approximates.
Use the percent, not the decimal: 3.5 for 3.5% a year.
Rule of 70: 70 divided by the percent growth rate.
- Exact doubling time
- 20.1
- Rule's error
- −0.7%
ln(2) divided by ln(1 + r). The rule of 70 is closest when the rate is small.
How far the shortcut sits from the exact answer.
How to calculate Doubling Time, step by step
- 1Identify the annual percent growth rate. Use the rate as a whole percent, not a decimal. Real GDP growing at 3.5% a year uses the number 3.5.
- 2Divide 70 by that growth rate. Years to double = 70 ÷ growth rate. A 3.5% growth rate gives 70 ÷ 3.5 = 20 years.
- 3Round the result. The rule is an approximation, so round to the nearest year rather than reporting decimals.
- 4Check that the rate is annual and roughly constant. The rule assumes the same percent rate compounds every year. A rate that swings sharply from year to year makes the estimate unreliable.
Worked example: Doubling Time
If real GDP grows at 3.5% a year, it doubles in about 70 ÷ 3.5 = 20 years. If the price level rises at 4% a year, it doubles in about 70 ÷ 4 = 17.5 years, so a price of $100 today reaches roughly $200 in 17 to 18 years.
Where the 70 comes from
The rule is a shortcut for an exact formula: doubling time = ln(2) ÷ ln(1 + r), where r is the growth rate written as a decimal.
ln(2) is about 0.693. For small growth rates, ln(1 + r) sits very close to r itself, so doubling time is close to 0.693 ÷ r. Writing the rate as a percent instead of a decimal multiplies both sides by 100, turning 0.693 ÷ r into about 69.3 divided by the percent rate.
70 rounds that up slightly. It is a small overstatement of the constant in exchange for a number that divides evenly by 2, 5, 7, 10, and 14, so the arithmetic can be done without a calculator.
When it breaks down at high rates
The approximation ln(1 + r) close to r only holds for small r, and the rule of 70 drifts away from the true doubling time as the growth rate rises.
At 2% growth the rule gives 35 years and the exact value is also 35.0, a match. At 10% growth the rule gives 7 years against a true value of about 7.3, a small miss. At 50% growth the rule gives 1.4 years against a true value near 1.7, a gap that is a large share of the total. At 100% growth the rule gives 0.7 years while the true doubling time is exactly 1 year.
AP Economics rarely asks about growth above 10% a year, so the rule is safe on the exam. Treat any answer built on a growth rate above 15% as an approximation worth flagging, not a precise year count.
Doubling Time questions
Why is it 70 and not some other number?
The number comes from approximating the exact doubling time formula, ln(2) ÷ ln(1 + r), which is close to 0.693 ÷ r only when the growth rate is small. Converting to a percent turns that into about 69.3 ÷ the rate, and 70 rounds that up to a number that divides evenly by 2, 5, 7, 10, and 14.
Does the rule of 70 work at any growth rate?
Only for small rates, roughly under 10%. The gap between the rule's estimate and the true doubling time grows as the rate rises, and becomes substantial above 50%. AP questions use rates in the single digits, where the rule stays accurate to within a few percent.
Can the rule of 70 be used for shrinking quantities?
Yes, with the same division applied to a rate of decline: 70 divided by the rate gives a halving time instead of a doubling time. Real GDP shrinking by 2% a year takes about 70 ÷ 2 = 35 years to fall by half.
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