Maths IA · Algebra and functions · topic 4 of 97
The rule of 72: how good is it, and why 72? · IB Maths IA idea: guide, sample graph and step-by-step plan
test the rule, don't just use itDivide 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.
- ✓ you like testing a rule everyone quotes but nobody checks
- ✓ you enjoy error analysis more than long calculations
- ✓ you want a short, clean investigation with a surprising reason behind it
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.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Solve (1 + r)ⁿ = 2 with logarithms and tabulate n for rates from 1% to 40%.
→ An exponential equation, and the first set of numbers to test every rule against.Show that continuous compounding doubles in exactly ln 2 divided by the rate, and compare with annual compounding.
→ Two compounding models and a reason for the constant 0.693. Quantify the difference for several rates.Write ln(1 + r) as a Maclaurin series and keep one, two and three terms.
→ A correction term of about 0.35 per percentage point. Show how it moves 69.3 towards 72 (AA HL; AI students check it numerically).Plot the percentage error of 69.3, 70 and 72 against the rate and find where each crosses zero.
→ Three error curves and three crossing points. Say what range of rates each rule suits.Compare the divisors of 72 with those of 70 and 69.3, then the error over common rates.
→ A mix of accuracy and convenience. Decide which matters more and defend it.Build a rule for tripling, or for halving a currency's value by inflation. Test it on real published rates.
→ A new constant of your own, derived and tested on real data, with error bounds.Your Maths IA research question: run the check
which rates? which error?Write your own question first. Then make sure it ticks every box:
- 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.
The model
A balance grows by the same percentage each year. Mark where it reaches double its start, then ask how a shortcut compares with the true time.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this rule, where you met it, and what you expect: which rate suits 72 best?
The model, the logarithm step, and a table of exact times over a stated range of rates.
The series for ln(1 + r), the 69.3 and the correction term, each step explained.
Percentage error of each rule against the exact time, as a table and a graph with crossing points.
Published savings, loan or inflation rates, with the rules applied and the errors found.
Back to your aim: which rule is best where, and why 72 is the common choice.
Compounding frequency, taxes, changing rates, the limit of any one-number rule. Say which matters most.
Where every rate and every formula came from.
What your Maths IA graph could look like
mark where each rule is exact!Made-up numbers to show the form of a strong graph, not the answer: exact doubling time against the annual rate, with the rule of 72 on the same axes. Yours will look different. example only
- Rate on the x-axis and years on the y-axis, both with units.
- Exact curve and rule on one set of axes, so the gap is visible at once.
- Mark the crossing point. Calculate it, do not just read it off the graph.
- Points from your own table show your formula agrees with the exact times.
- Another graph to consider: the percentage error of each rule against the rate.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
How far off is each rule?
Rank the rules on one number: the error against the exact doubling time at one rate. Work it out for your own rate, and see whether the order holds at another.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind where the rule of 72 is used. Before any calculation, guess at which rate it is most accurate.
- Fix the modelState the compounding rule (yearly, monthly or continuous) and define every symbol. Say what you mean by doubling.
- Solve exactlySolve 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 constantsExpand 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 errorDefine percentage error, graph it for each rule and calculate where each rule is exact.
- Test on real ratesCollect published rates and compare the rule and the exact time. Say which rate range your data covers.
- Go further, with a reasonBuild a rule for tripling, or for monthly compounding. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a saver and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Exponential equation and logarithms – Solve exactly for , 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 and keep two or three terms to show , so a bigger constant fits larger rates.
- Continuous compounding – Show that gives 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
Syllabus links
- 1.4Compound interest and depreciation
- 1.5Exponents and an introduction to logarithms
- 1.7Rational exponents and laws of logarithms
- 2.9Exponential and logarithmic functions
- 5.13Limits with l'Hopital's rule or Maclaurin seriesHL
- 5.19Maclaurin seriesHL
- 1.4Compound interest and depreciation
- 1.5Exponents and an introduction to logarithms
- 1.6Approximation, bounds and percentage error
- 1.9Laws of logarithmsHL
- 2.3Graphs of functions
Read up first
Data and sources to start from
Compute exact times from the model, then test the rules on real savings, loan or inflation rates from published sources.
References to cite
- Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
- Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning. (Section 11.10, Taylor and Maclaurin series.)
- Federal Reserve Bank of St. Louis. (2026). Federal Reserve Economic Data (FRED). https://fred.stlouisfed.org/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
state the units of R- 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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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.