# Rule of 72 Calculator

## Calculator Use

Calculate the number of years to double your investment at a given interest rate. Or, calculate the interest rate required to double your money within a certain time frame.

The Rule of 72 is a simple mental or pencil and paper method to estimate time or interest required to double your money. It is a simplified compound interest calculation.

- Interest Rate
- the annual nominal interest rate of your investment in percent
- Time Period in Years
- number of years for this investment. (you can interchange months or any period as long as your interest rate entered is in the same time period)
- Compounding
- the frequency of compounding coincides with periods. Once per period.

## Rule of 72 Formula

The rule of 72 is a simple way to estimate a compound interest calculation that doubles an investment without applying the actual formula. This allows you to do an estimate, pretty well, in your head or just using a pencil and paper.

The rule says that in order to double an amount of money at a given rate and time, the interest percentage per period times the number of periods is equal to 72. We can write it as

**R * t = 72**

where

R = interest rate per period as a percentage, not a decimal

r = interest rate in decimal form, r = R/100

t = number of periods

Commonly, periods are years so R would be the rate per year and t is the number of years. Now we can calculate the number of years to double our investment at some known interest rate by solving for t;** t=72/R**. And, we can calculate the interest rate required to double our money within a known time frame by solving for R; **R=72/t**.

Generally, periods is in years but you can use any time period as long as the interest rate you use is the corresponding periodic rate.

## Derivation of the Rule of 72 Formula

The basic compound interest formula is

**A = P(1 + r) ^{t}** (equation 1)

where A is the accrued amount, P is the principal investment, r is the interest rate per period t in decimal, and t is the number of periods. If we want to reformat this formula to show that the accrued amount is double of our principal investment, P, then we have A = 2P. Rewriting;

2P = P(1 + r)^{t} , and cancelling P from both sides we have;

**(1 + r) ^{t} = 2**

We can solve this equation for t by taking the natural log, ln(), of both sides,

\[ t \times ln(1+r)=ln(2) \]and isolating t on the left;

\[ t = \frac{ln(2)}{ln(1+r)} \]it should be easy to show that we can rewrite this to an equivalent form

\[ t = \frac{ln(2)}{r}\times\frac{r}{ln(1+r)} \]Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8% (r=0.08)*

\[ t = \frac{0.69}{r}\times\frac{0.08}{ln(1.08)}=\frac{0.69}{r}(1.0395) \]Solving this equation for r times t

\[ rt=0.69\times1.0395\approx0.72 \]finally putting r the decimal rate into R the percent rate by multiplying both sides by 100,

**R*t = 72**

*8% is used as a common average and makes this formula most accurate for interest rates from 6% to 10%.

### Example Calculations in Years

If you invest money at 6% interest per year, how long will it take you to double your investment?

**t=72/R = 72/6 = 12 years**

What compounding interest rate do you need to double your money in 10 years?

**R = 72/t = 72/10 = 7.2%**

### Example Calculation in Months

If you invest money at 0.5% interest per month, how long will it take you to double your investment?

**t=72/R = 72/0.5 = 144 months** (since R is a monthly rate our answer is in months not years)

144 months = 144 months / 12 months per years = 12 years

## References

Vaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition) Washington DC: The Mathematical Association of Amercia, 2009, page 75.

Weisstein, Eric W. "Rule of 72." From *MathWorld*--A Wolfram Web Resource. Rule of 72.

**Cite this content, page or calculator as:**

Furey, Edward "Rule of 72 Calculator"; from *http://www.calculatorsoup.com* - Online Calculator Resource.