How long until your money doubles?
Tonight's idea
Money that earns 6% interest a year is multiplied by 1.06 every year. The interest earns interest too, so the growth speeds up. How long until the money doubles?
There is a shortcut called the rule of 72.
Doubling time in years \(\approx 72 \div r\), where \(r\) is the interest rate in percent a year.
At 6% the rule says \(72 \div 6 = 12\) years. To test it, find the true doubling time the slow way. Start with 1, multiply by 1.06 again and again, and count the years until you pass 2. Here is that test for seven rates.
| Rate \(r\) | 2% | 3% | 4% | 6% | 8% | 9% | 12% |
|---|---|---|---|---|---|---|---|
| \(72 \div r\) | 36 | 24 | 18 | 12 | 9 | 8 | 6 |
| True years | 35.0 | 23.4 | 17.7 | 11.9 | 9.0 | 8.0 | 6.1 |
Across these rates the rule is never more than a year out. It runs slightly long at low rates and slightly short at high ones.
Where it comes from. For small rates the exact figure is close to \(69.3 \div r\). People use 72 because it divides easily: by 2, 3, 4, 6, 8, 9 and 12.
Watch one: $500 at 6%
The rule says \(72 \div 6 = 12\) years. Check by multiplying by 1.06, year after year.
| Year | 0 | 2 | 4 | 6 | 8 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|
| Amount ($) | 500 | 561.80 | 631.24 | 709.26 | 796.92 | 895.42 | 949.15 | 1,006.10 |
The money passes $1,000 during year 12. From $949.15 up to $1,006.10 is a step of $56.95, and $1,000 is about nine tenths of the way along it. So the true doubling time is about 11.9 years.
The starting amount does not matter. Doubling means the multiplier \(1.06^n\) reaches 2, whatever it multiplies.
Your turn
- Use the rule of 72 to estimate the doubling time at (a) 4%; (b) 9%; (c) 1.5%; (d) 18%.
- Test the rule at 8%. Start with 1 and multiply by 1.08 nine times, keeping the full number in your calculator. Write each result to 3 decimal places. How close to 2 are you after 9 years? What does the rule say?
- (a) A savings account doubled in 18 years. Roughly what was the interest rate? (b) At 6%, about how long until the money is 4 times as large? 8 times? (c) Prices rise 3% a year. About how long until prices have doubled, so that each dollar buys half as much?
- The rule has limits. (a) A debt grows at 24% a year. What does the rule say? Work out \(1.24^3\) and \(1.24^4\). Is the rule too short or too long? (b) At 1%, the amount is 1.987 after 69 years and 2.007 after 70. Which fits better, \(72 \div r\) or \(69.3 \div r\)? (c) Why is a rule of 72 still handier than a rule of 69.3?
Stretch: a rule for tripling
At 4% the amount is 2.999 after 28 years. At 6% it is 2.854 after 18 years and 3.026 after 19. Propose a "rule of \(N\)" for the time to triple. Use it to predict the tripling time at 8%, then check by multiplying.
Answers
- (a) \(72 \div 4 = 18\) years. (b) 8 years. (c) \(72 \div 1.5 = 48\) years. (d) 4 years.
- 1.080, 1.166, 1.260, 1.360, 1.469, 1.587, 1.714, 1.851, 1.999. After 9 years you are at 1.999, a whisker under 2. The rule says \(72 \div 8 = 9\) years. At 8% it is almost exact.
- (a) \(72 \div 18 = 4\), so about 4%. (b) Doubling takes about 12 years. Four times is two doublings, about 24 years. Eight times is three doublings, about 36 years. (c) \(72 \div 3 = 24\) years. The rule works for anything that grows by a fixed percent a year, not only money.
- (a) The rule says \(72 \div 24 = 3\) years. But \(1.24^3 = 1.907\) and \(1.24^4 = 2.364\), so the debt doubles a little after 3 years. The rule is too short. (b) The true time is between 69 and 70 years. \(69.3 \div 1 = 69.3\) fits; 72 is more than two years too long. (c) 72 divides exactly by the rates people meet most, so you can do it in your head, and for rates from 2% to 12% it is within a year of the truth.
Stretch. At 4%, tripling takes 28 years, and \(4 \times 28 = 112\). At 6% it takes about 18.9 years, and \(6 \times 18.9 \approx 113\). Pick \(N = 112\), which is close to both and divides easily: a rule of 112. At 8% it predicts \(112 \div 8 = 14\) years. Check: \(1.08^{14} = 2.937\) and \(1.08^{15} = 3.172\), so the amount triples early in year 15. The rule holds. Like 72, the best \(N\) creeps up with the rate: about 110 for tiny rates, 112 at 4% and 114 at 8%.