# Doubling Time Calculator

Calculate the time it takes for an amount to double in value given the growth rate per period using the doubling time calculator below.

## Doubling Time:

### Growth Over Time

## On this page:

## How to Calculate Doubling Time

Doubling time is the amount of time that it takes for an amount to double in size at a constant growth rate. It is the inverse of half-life, which is the amount of time for a value to be reduced in half.

The study of doubling time can be dated back to the Babylonians in 2000 BC. Scribes have been found where students were tasked with calculating how long it takes for a mina of silver to double at the typical Mesopotamian rate.^{[1]}

### Doubling Time Formula

You can use the following formula to calculate the time it takes an amount to double.

t = ln(2) / ln(1 + r)

**where:**

r = growth rate

t = number of periods

Thus, the time it takes to double a value *t* is equal to the natural logarithm of 2 divided by the natural logarithm of 1 plus the growth rate *r*.

**For example,** let’s calculate the time it takes to double your savings that grow at a 5% interest rate.

Substitute the rate in the formula above and solve. Remember to divide the 5% interest rate by 100 to convert it to a decimal.

t = log(2) / log(1 + 0.05)

t = 0.301 / 0.0212

t = 14.2067 years

So, it would take 14.2067 years for your savings to double at a 5% interest rate.

## The Rule of 72

The doubling time formula above is very precise, but it’s not very convenient to use. A formula was derived from the doubling time formula called the Rule of 72.

The Rule of 72 formula is not as precise as the doubling time formula, but it is very easy to use. The Rule of 72 formula is:

periods × rate = 72

The formula states that the number of periods times the growth rate is equal to 72. By rearranging the formula, you can solve for the number periods or the growth rate.

The Rule of 72 is used because it has many factors (2, 3, 4, 6, 12, 24, etc.), but using 69 or 70 is actually much more accurate. For this reason, you might see the Rule of 69 or the Rule of 70 used in its place, but the concept is the same.

## Doubling Time Chart

Growth Rate | Periods Using Doubling Time Formula | Periods Using the Rule of 72 |
---|---|---|

1% | 69.66 | 72 |

2% | 35 | 36 |

3% | 23.45 | 24 |

4% | 17.67 | 18 |

5% | 14.21 | 14.4 |

10% | 7.27 | 7.2 |

20% | 3.8 | 3.6 |

30% | 2.64 | 2.4 |

40% | 2.06 | 1.8 |

50% | 1.71 | 1.44 |

75% | 1.24 | 0.96 |

100% | 1 | 0.72 |

## Doubling Time and Exponential Growth

So you might be wondering why doubling time is so important? Doubling time is useful for calculating exponential growth. Much of the growth that we need to measure is exponential.^{[2]}

Albert Einstein once said, “the power of compound interest is the most powerful force in the universe.”^{[3]} Compound interest is so powerful because of the exponential growth effect over time.

So, doubling time and exponential growth are used heavily in finance. Many formulas for interest and loans rely on the principles of doubling time and exponential growth.

It’s also used widely in physics and statistics, such as calculating the half-life of radioactive elements or the population growth of bacteria.

Doubling time is also used in economics to estimate the value of currency, accounting for inflation, and measuring national GDP.

As you can see, doubling time and the formula above are used in a variety of ways.

## References

- Bakir, S. T., Compound Interest Doubling Time Rule:

Extensions and Examples from Antiquities,*Communications in Mathematical Finance*, 2016, 5(2), 1-11. http://www.scienpress.com/Upload/CMF/Vol%205_2_1.pdf - Hewitt, P., Exponential Growth and Doubling Time,
*The Science Teacher*, July/August 2020, https://www.nsta.org/science-teacher/science-teacher-julyaugust-2020/exponential-growth-and-doubling-time - Seagull, B., The Life-Changing Magic of Numbers,
*Penguin Random House UK*, 2018, https://www.penguin.co.uk/books/111/1116561/the-life-changing-magic-of-numbers/9780753552803.html