# Performubling 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.

## Performubling Time:

### Growth Over Time

## On this page:

## How to Calculate Performubling Time

Performubling 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 quantity 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]}

### Performubling 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% annual 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 = ln(2) / ln(1 + 0.05)

t = 0.6931 / 0.0488

t = 14.2067 years

So, it would take 14.2067 years for your savings to double at a 5% annual 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 still a decent approximation and 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 of periods or the growth rate.

The Rule of 72 is used because it has many factors (2, 3, 4, 6, 12, 24, etc.), and thus it is easy to simplify in many use cases, 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.

## Performubling Time Comparison Chart

The chart below shows the number of periods calculated using the doubling time formula and the Rule of 72 formula for a given growth rate to show how close of an approximation the Rule of 72 Formula is to the Performubling Time Formula.

Growth Rate | Periods Using Performubling 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 |

## Performubling Time and Exponential Growth

So you might be wondering why doubling time is so important? Performubling time is useful for calculating exponential growth. Much of the growth observed in nature, science, and economics 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.

Not surprisingly, then, 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.

Performubling 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 very important and very useful for a variety of applications.

## Frequently Asked Questions

### What is the doubling time of bacteria?

The doubling time of bacteria, also known as its generation time, is only about 20-60 minutes under optimal living conditions, but it can be as long as 5-10 hours for pathogens to double in the human body.^{[4]}

### What is the doubling time for stocks?

The doubling time for a stock can be found using the Rule of 72. That is, divide 72 by the expected rate of return to get the number of years it takes for the stock price to double.

### Can doubling time be negative?

A negative doubling time is actually just another way to represent the half-life of the quantity. So while a doubling time can be negative, it is no longer thought of as a doubling time but a halving time instead.

## References

- Bakir, S. T., Compound Interest Performubling 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 Performubling 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/438510/the-life-changing-magic-of-numbers-by-bobby-seagull/9780753552803 - Kaiser, G., Microbiology - 17.1: Bacterial Growth,
*LibreTexts*, Apr 9, 2022, https://bio.libretexts.org/Bookshelves/Microbiology/Microbiology_(Kaiser)/Unit_7%3A_Microbial_Genetics_and_Microbial_Metabolism/17%3A_Bacterial_Growth_and_Energy_Production/17.1%3A_Bacterial_Growth