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  1. 019A plant's output grows 10% every month. Roughly how many months until output doubles, and how long until it is eight times today's level? Use the rule of 72, then check it against the exact answer.Compounding and time valueWarm upConsulting-style caseCorporate FP&A

    Try it first

    How long until output is eight times today's level?

    Show the worked solution

    Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.

    Why does eight times take three doublings and not 70 months?

    Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.

    The relationship
    t2x=ln⁡2ln⁡1.10=0.6930.0953=7.27rule of 72: 7210=7.2t_{2x} = \frac{\ln 2}{\ln 1.10} = \frac{0.693}{0.0953} = 7.27 \qquad \text{rule of 72: } \frac{72}{10} = 7.2
    ln 2the natural log of 2, about 0.693
    ln 1.10the natural log of one month's growth factor, about 0.0953
    72a convenient number close to 100 times ln 2, with many divisors
    What it says in wordsDoubling time is the log of two divided by the log of one period's growth factor; the rule of 72 approximates that with one division.
    Output at 10% a month: three doublings in about 22 months2x4x8x1x06121824MonthsOutput, multiple of todayexact 7.27rule of 72: 7.2exact 14.55rule of 72: 14.4exact 21.82rule of 72: 21.6
    At 10% a month, output reaches twice today's level after 7.27 months, four times after 14.5 and eight times after 21.8. The rule of 72 marks at 7.2, 14.4 and 21.6 months sit just short of each, so the shortcut is close enough to say out loud.

    Where does the rule of 72 work, and where does it slip?

    The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.

    Growth per periodRule of 72ExactGap
    2%36.0035.00+2.8%
    5%14.4014.21+1.4%
    10%7.207.27-1.0%
    25%2.883.11-7.3%
    50%1.441.71-15.8%
    Periods to double, with the rule's error as a share of the exact answer. It is within about 1% at 5% and 10% a period, and drifts further at very low and very high rates.

    Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.

    Where candidates lose it

    The common loss is answering about 70 months for eight times, treating the growth as simple rather than compound. It is the same instinct that underestimates how quickly compounding builds.

    The second loss is reaching for a calculator-style log in your head and stalling. Give the rule of 72 answer first, then say the exact number is a little longer, about 7.3 months.

    What the interviewer asks next

    • Using the same idea, how long does it take for prices to halve in value at 6% inflation?
    • Why do some people use 69 or 70 instead of 72?
    • Output grows 10% a month for a year. What is the annual growth rate?