Growth & projection
Doubling, compounding, and why losses need bigger gains.
In your head, jot if needed · no calculator why?
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the idea
Growth that builds on itself compounds: each step multiplies what you already have instead of adding a fixed amount. That is why small steady rates end up huge.
Three facts cover most of it. At r% a year, a thing doubles in about 72 ÷ r years. Ten doublings make 1,024 times, about a thousandfold. And a fall needs a bigger rise to recover, because the rise works on a smaller amount.
techniques
Rule of 72
- Years to double: about 72 ÷ the yearly rate in percent.
- Rate that doubles it in n years: about 72 ÷ n, in percent.
worked example
Revenue grows 6% a year. Using the rule of 72, about how many years does it take to double?
- 72 ÷ 6 = 12 years.
Answer: 12
Count the doublings
- Count the doublings: total time ÷ doubling time.
- Multiply by 2 that many times. Ten doublings make 1,024.
- Multiply by the starting amount, if there is one.
- For a small rate, get the doubling time from the rule of 72 first.
worked example
A colony of 300 bacteria doubles every 20 minutes. How many bacteria are there after 2 hours?
- 2 hours is 120 minutes: 120 ÷ 20 = 6 doublings.
- 2 × 2 × 2 × 2 × 2 × 2 = 64.
- 300 × 64 = 19,200.
Answer: 19,200
Recover a loss from 100
- Start at 100 and apply the fall: a 20% fall leaves 80.
- The rise needed is the amount lost divided by what is left: 20 ÷ 80.
- Write it as a percent: 25%.
worked example
A shop's sales fall 25%. What percent increase brings them back to where they started? Round to one decimal place.
- 100 falls to 75.
- Getting back needs 25 more on 75.
- 25 ÷ 75 ≈ 33.3%.
Answer: 33.3%
watch out for
- Dividing 100 by the rate instead of 72, as if growth did not compound.
- Adding 2 for each doubling instead of multiplying by 2. Ten doublings make 1,024 times, not 20.
- Expecting a 40% rise to undo a 40% fall. The rise works on the smaller amount, so it must be 66.7%.
- Adding a small rate instead of compounding it. 1% a day for 365 days is about 38 times, not 4.65.
practice
Rule of 72
worked example
Using the rule of 72, what yearly growth rate, in percent, doubles a town's population in 4 years?
Answer: 18 %
- Divide 72 by the number of years: 72 ÷ 4 = 18, so about 18% a year.
Repeated doubling
worked example
A colony of 200 bacteria doubles every 15 minutes. How many bacteria are there after 105 minutes?
Answer: 25,600 bacteria
- 105 minutes is 105 ÷ 15 = 7 doublings.
- 2⁷ = 128, and 200 × 128 = 25,600.
Recovering a loss
worked example
An investment falls 20%. What percent gain brings it back to where it started?
Answer: 25%
- After a 20% fall, 100 becomes 80.
- Getting back to 100 needs +20 on 80: 20 ÷ 80 = 25%.
Small gains, compounded
worked example
You get 2% better every day for 300 days, and each day's gain builds on the last. About how many times your starting level are you at the end?
- about 38×
- about 380×
- about 3,800×
- about 7×
Answer: about 380×
- Compounding multiplies: 1.02³⁰⁰ ≈ 380.
- Adding 2% of the starting amount each time would give only 1 + 300 × 0.02 = 7×. Gains that build on each other end up far bigger.