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Compound interest is the greatest mathematical discovery of all time. — Albert Einstein

Whenever we borrow money from the bank or any other source, the *interest* is charged. It may be simple interest or compound interest depending on the conditions.

For example, If you invest $1000 for 10 years at an annual rate of 5%.
With simple interest, $5 is added each year to the principal amount of $1000. Thus, after 10 years, you’ll have $1500.
While with compound interest, if you have P amount at the start of the year, then during the year 5% of P is added. Thus, at the end of the year, you’ll have P(1 + 5/100) amount. This happens every year. Hence, after 10 years, you’ll have $1000(1 + 5/100)^{100} i.e. $1628.89.

Now, let’s calculate the time required for the principal amount to double. Suppose, you borrow C amount from a bank, on which an annual interst rate of r% is charged, then your debt will double at the time t when

Taking logarithm on both sides and solving, we get

This is known as the **rule of 72**. If no repayments are made, then at an annual interest rate of r%, it’ll take approx. 72/r years for a debt to double. Also, it you invest money at r% interest rate, then it’ll double in 72/r years.

Although we got the value 69.3, we’re using 72 as it has many small divisors: 1, 2, 3, 4, 6, 12. Thus, it is a good choice for numerator. But, the rule of 72 becomes less accurate at higher interest rates.

As is clear from the graph, there is a little difference b/w graphs of 69.3/r and 72/r. Thus, the rule of 72 provides a good approximation.

More accurately, the rule underestimates the doubling time when the interest rate is larger than 7.8% and overestimates when the rate is lower than 7.8%. It is the point where the graphs of 72/r and 69.3/r intersects.

Now, we can calculate the time for an amount to double in simple way. For example, when the interest is 6% p.a., we divide 72 by 6, thus, in 12 years, the amount will be doubled. The rule of 72 comes handy in everyday life.

**Further Reading:**

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