IB Maths IA idea · Money and personal finance

The rule of 72: how good is it, and why 72?

Divide 72 by the interest rate and you get the years to double your money. It's off by under 1% at 8%, and by about 13% at 40%. You're going to find where it works and why 72 was chosen.

Money and personal finance
SL ideal
HL ideal
Maths AA Maths AI Mixed data
Syllabus: AA 1.4, AA 1.5, AA 1.7, AA 2.9, AA 5.13, AA 5.19

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

Doubling time n solves (1 + r)ⁿ = 2, so n = ln 2 / ln(1 + r), an exact formula that a logarithm gives you straight away. The rule of 72 replaces it with 72 divided by the rate in percent. Why does that work at all? Write ln(1 + r) as a series: for a small rate it is close to r, which gives 69.3 divided by the rate, and the next term explains why a slightly bigger number such as 72 does better in practice. The calculation is short. The marks come from measuring the error, saying where each rule is best and why, and testing the rules on rates you found for yourself.

The maths you will use

Exponential equation and logarithms. Solve $(1+r)^n = 2$ exactly for $n$ , and give the doubling time as a function of the rate.
Percentage error. Define the error of each rule against the exact time and graph it against the rate; find where it is zero.
Maclaurin series. Expand $\ln(1+r)$ and keep two or three terms to show $nR \approx 69.3 + 0.35R$ , so a bigger constant fits larger rates.
Continuous compounding. Show that $e^{\rho t} = 2$ gives $t = \frac{\ln 2}{\rho}$ exactly, and compare with yearly compounding.
Solving numerically. Find the rates where each rule is exact with a GDC solver or bisection, and report to a stated accuracy.

Key equations

n = \frac{\ln 2}{\ln(1+r)}

nR \approx 100\ln 2\left(1 + \frac{r}{2} - \frac{r^{2}}{12}\right), \quad r = \frac{R}{100}

\text{error}(K) = \frac{K/R - n}{n} \times 100\%

The maths in action

Two panels. Left: rate times exact doubling time rising steadily from 69.3, crossed by horizontal lines at 69.3, 70 and 72. Right: percentage error of each rule against the rate, with the 72 curve crossing zero near 7.8% and the 69.3 curve always below zero. — (a) Rate times exact doubling time, $R \cdot n$ : each rule treats it as constant, but the exact value rises from 69.3. (b) Percentage error: 72 is exact at 7.8%, 70 at 2.0%, and 72 stays within 1% from 5.8% to 10.0%.

Example research question

To adapt, never to copy

How accurate are the rules 69.3, 70 and 72 divided by the annual rate for the doubling time of an investment growing at 1% to 40% a year, why does 72 do best near 8%, and does the same hold for monthly compounding?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the rules being tested (69.3, 70, 72 or others) and the range of rates, with a reason for the range.
States how error is measured: percentage error against the exact doubling time, and its sign.
Says what is derived and not just calculated: the exact formula or the series.
Includes a test against real rates, or says clearly why none is used.
States what you leave out: fees, tax, changing rates, compounding other than yearly.

Which course it fits

AA SL good
AA HL ideal
AI SL ideal
AI HL good

Syllabus links in Maths AA:

AA 1.4 Compound interest and depreciation
AA 1.5 Exponents and an introduction to logarithms
AA 1.7 Rational exponents and laws of logarithms
AA 2.9 Exponential and logarithmic functions
AA 5.13 Limits with l'Hopital's rule or Maclaurin series (HL only)
AA 5.19 Maclaurin series (HL only)

Steps

Read and predict. Find where the rule of 72 is used. Before any calculation, guess at which rate it is most accurate.
Fix the model. State the compounding rule (yearly, monthly or continuous) and define every symbol. Say what you mean by doubling.
Solve exactly. Solve the doubling equation with logarithms and tabulate exact times over your range of rates.
Trial run (1 lesson). Test 72 ÷ R at five rates by hand and by spreadsheet. Fix any mismatch now: percent against decimal, rounding.
Find the constants. Expand the logarithm as a series, or study the product of rate and doubling time numerically. Explain 69.3 and the shift towards 72.
Measure the error. Define percentage error, graph it for each rule and calculate where each rule is exact.
Test on real rates. Collect published rates and compare the rule and the exact time. Say which rate range your data covers.
Go further, with a reason. Build a rule for tripling, or for monthly compounding. HL: this is where the rigour for criterion E comes from.
Reflect as you go. After every result, write one line on what it means for a saver and what it assumes. These lines become your evaluation.
Cut, check, then send. Check every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

Make it yours

Find the exact doubling time. Solve (1 + r)ⁿ = 2 with logarithms and tabulate n for rates from 1% to 40%.
Explain the 69.3. Show that continuous compounding doubles in exactly ln 2 divided by the rate, and compare with annual compounding.
Expand the logarithm. Write ln(1 + r) as a Maclaurin series and keep one, two and three terms.
Where is each rule best?. Plot the percentage error of 69.3, 70 and 72 against the rate and find where each crosses zero.
Why 72 and not 70?. Compare the divisors of 72 with those of 70 and 69.3, then the error over common rates.
Other multiples and real rates. Build a rule for tripling, or for halving a currency's value by inflation. Test it on real published rates.

HL extension

Derive $nR \approx 69.3 + 0.35R$ from the Maclaurin series, bound the error of truncation, show why the constant that is best depends on the range of rates, and extend to continuous and monthly compounding.

The data you need

Compute exact times from the model, then test the rules on real savings, loan or inflation rates from published sources.

Traps

Putting the rate in as 0.08 in one place and 8 in another. State the units of R everywhere.
Judging a rule by one rate. A rule that is perfect at 8% can be poor at 2% or 20%.
Calling an error small without a percentage. Half a year matters more at 3 years than at 30.
Taking the series for ln(1 + r) outside its range. Say when it converges and how many terms you need.
Mixing continuous and yearly compounding. The constant 69.3 belongs to the continuous case.
Explaining 72 only by its many divisors. That is convenience, not accuracy; show both.

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