GMAT Math : The rule of 72

The rule of 72 is a great approximation tool for GMAT Math questions involving compounding (e.g., compound interest, bacterial growth rate etc)

Let’s start with an example:

John has an option of depositing $100 he saved with his friend or with the Bank. His friends, he will give John back 4 times the initial deposit money along with the principal in 10 years. The bank will provide him 12% pa on his deposit compounded annually. What is the closest number of years John would have to leave his money with the Bank to get back a return equivalent to what his friend will provide him?

A. 10

B. 12

C. 13

D. 18

E. 21

Readers are encouraged to take some time to tackle the above question 🙂

Let’s look at the traditional way of solving this question:

Sum provided by John’s friend at the end of 10 years = 4 x $100 + $100 = $500

Compound interest returns = (Principal) x (1+ Rate/100)# of years

= ($100) x (1 + 12/100)# of years

= $100 x (1.12)# of years

This should equate to return from John’s friend

$100 x (1.12)# of years = $500

1.12# of years = 5

Now, using trial and error and acute mathematical prowess we may be able to figure out # of years. However, this technique is a bit time consuming.

Here’s is the rule of 72 technique:

Trick: The # of years it takes for something to double when compounded at a certain interest rate R is roughly equal to 72/R

Using the above, at 12% interest rate, it would take approximately 72/12 = 6 years for John’s money to double. Another 6 years and John’s money would become 4 times (i.e, 2 x 2)…a bit more time and we would have 5 times the money (i.e., $500)

In total we have accounted for 6 years + 6 years + a bit more = 12 years + bit more

Looking at the options, this can only be Option (C).

It cannot be Option (D) as John’s money would become 8 times till then (18 years = 3 spans of 6 years each)

So, we go with Option (C)

Told you, this was going to be fast 🙂