# Is Rule of 72 always correct?

By Nick

### Quick Peek:

Want to know how long it will take for your money to double? The Rule of 72 can help you estimate it quickly, but it’s not always accurate. Derived from a more complex calculation, it’s an approximation that works best for an 8% interest rate. The farther you go from 8%, the less precise the results will be. While it’s useful for mental calculations, for more precise results, use the full formula for exponential growth.

## The Rule of 72: Is it Always Correct?

Have you ever heard of the Rule of 72? It’s a quick and easy way to estimate how long it will take for your money to double, based on a given interest rate. The rule states that if you divide 72 by the interest rate, the result will be the number of years it takes for your money to double. For example, if you have an interest rate of 8%, it will take approximately 9 years for your money to double (72 divided by 8 equals 9).

However, the Rule of 72 is not always perfectly accurate. It is actually an approximation derived from a more complex calculation. The most accurate results from the Rule of 72 are based on an 8% interest rate, and the farther you go from 8% in either direction, the less precise the results will be.

### Why is the Rule of 72 an Approximation?

The Rule of 72 is based on the mathematical concept of exponential growth. In finance, exponential growth refers to the compounding of interest on an investment over time. The formula for calculating exponential growth is:

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A = P(1 + r/n)^(nt)

Where:
A = the final amount
P = the principal amount
r = the annual interest rate
n = the number of times the interest is compounded per year
t = the number of years

This formula can be simplified for the purposes of the Rule of 72. By assuming that the interest rate and the number of times the interest is compounded per year are equal, the formula becomes:

A = P(1 + r)^t

To find out how long it takes for the investment to double, we can set A equal to 2P and solve for t:

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

Using the natural logarithm, we can simplify this equation further:

t = 0.6931/r

This is where the Rule of 72 comes in. By dividing 72 by the interest rate, we get an approximation of how long it will take for the investment to double:

t ≈ 72/r

This approximation works best when the interest rate is close to 8%, because that is the interest rate that makes the simplified formula and the more complex formula give the same result. As the interest rate deviates from 8%, the approximation becomes less accurate.

### When is the Rule of 72 Useful?

Despite its limitations, the Rule of 72 can be a useful tool for estimating how long it will take for your money to double. It is particularly useful for mental calculations and quick estimations. For example, if you are considering two different investments with different interest rates, you can use the Rule of 72 to compare how long it will take for your money to double in each investment.

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However, if you need more precise calculations, you should use the full formula for exponential growth. This formula takes into account the compounding of interest, which can significantly affect the final amount of your investment.

### Conclusion

In conclusion, the Rule of 72 is a quick and easy way to estimate how long it will take for your money to double, but it is not always perfectly accurate. The most accurate results from the Rule of 72 are based on an 8% interest rate, and the farther you go from 8% in either direction, the less precise the results will be. The Rule of 72 is useful for mental calculations and quick estimations, but if you need more precise calculations, you should use the full formula for exponential growth.

## References for « Is Rule of 72 always correct? »

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