# Beyond the Rule of 72

May 3, 2017

The Rule of 72 is a useful shortcut to determine the approximate time required to double one's investment. By this rule, It takes roughly `72/r` years to double your principal/investment, where `r` is the annual rate of compounding interest. So, with `12% p.a.` interest rate (compounded annually), you can double your investment in 6 (`72/12`) years. The rule can also be used to calculate the rate of interest `r` required to double a sum of money in `y` years (`r = 72/y`).

This is a useful rule. But, a general rule of thumb that estimates the time required to convert an amount `P` into `X*P` would be better. In this article we will learn to do that.

### PROBLEM

Say, we need to convert `50,000 \$` to `1,000,000 \$` at `8 % p.a.` How many years will that take? Our goal here is to be able to solve this problem in our head (easily).

### SOLUTION

First we will use our knowledge of Rule of 72, to get a rough estimate. Then we will use a clever trick to solve the problem in an elegant manner. Finally, we will use the concept of approximation to make the elegant solution feasible to be computed in our heads.

#### 1. Use Rule of 72 to get a rough estimate

`1,000,000` is `20` times `50,000`.

We know that the amount will double in `72/8=9` years. At the end of `9` years, the amount is `100,000`. Crossing `18 (2*9)` years it becomes `200,000 = 22 * (50,000)`. After `27 (3*9)` years, it becomes `400,000 = 23 * (50,000)`.

Notice the pattern that after `n*9` years the total amount is `2n * (50,000)`. We know that `20` lies between `24 (16)` and `25 (32)`. In `9*4` years our amount grows to `800,000 (24*50,000=16*50,000)` and in `9*5` years our amount grows to `1,600,000 (25*50,000=32*50,000)`. `1,00,000` falls in between `800,000` and `1,600,000`. So the time it takes to get a million dollars will be anywhere between `36 (9*4)` and `45 (9*5)` years.

To get a more accurate estimate we can use simple interest calculation. The Simple Interest for `800,000` is `64,000 (0.08 * 800,000) `. Thrice the Simple Interest is approximately `200,000` (`196,000` exactly). Compound interest for this three years will be more than Simple Interest as the interest earns interest while compounding. So we get a million in `36+3` years approximately.

The exact value after `39` years is `1,005,764.88` (which can be calculated here ). We need to be careful of this small difference of `5764.88 \$`(`1,005,764.88-1,00,000`). It will grow to a large amount when compounding for more years due to exponential nature of compounding. That aside, the problem with this approach is that the estimated `36-45` years has a range/gap of `11` years. This gap widens as the rate of interest gets smaller. And calculating Simple Interest leads to larger errors when this range grows. We need a better solution that can reduce this range/gap and one which does not resort to calculation of Simple Interest.

#### 2. An elegant solution

Just as we have a rule to double our money, we also have a rule to triple our money. It takes `110/r` years to triple. To make our money `4` times initial, it takes `139/r` years. To make it `5` times it takes `161/r` years and so on ... to make it `20` times it takes `300/r` years. We need to remember 20 rules to be able to estimate the number of years for multiplying our principal for up to 20 times. To calculate for a multiple of up to 100 times, we should remember 100 such rules. Remembering so many numbers is a challenge for many. Luckily, there is a neat mathematical trick that eases the load on our memory.

We all have learnt in school that any positive number can be written as a product of just prime numbers. Yes. It is called the Fundamental theorem of Arithmetic. Take out a paper and work it out. For example `20=22*5`, `53=53` (it is a prime number), `45=33*5`.

Here are rules for the first 15 prime numbers, which we will be referencing shortly. Just these 15 prime numbers can be used to represent any integer from 1 to 50.

Prime NumbersRule (`r` is the rate of interest)
272/r
3110/r
5161/r
7195/r
11240/r
13257/r
17284/r
19295/r
23314/r
29337/r
31344/r
37362/r
41372/r
43377/r
47386/r

We use the above knowledge to solve our problem as below:

1. Represent the multiplier `X` (`20` in our problem) as a product of its primes.
`20=22*5=22*51`.
2. The prime factors of `X` will be of the form `primen`. Using the table above apply the corresponding rule and find the number of years, `y`. Multiply `y` with `n`.
3. Repeat Step 2. for all the prime factors.
4. Sum up all the years from above. This is the result we need.

Now, lets apply the steps to our problem.

1. The desired/final amount (`1,000,000`) is `20` times the initial/principal amount (`50,000`).`20=22*51`.
2. For factor `22`, we apply Rule of 72, and multiply the result by `2`. `72/8=9` and `9*2=18` years.
3. We repeat it for `51`.For factor `51`, we apply Rule of 161, and multiply the result by `1`.`161/8=20` (approx) and `20*1=20` years.
4. So our answer is `18+20=38` years.The accurate answer is a touch below `39` years.

I prefer to look at the above procedure this way:
`Multiplier, X=(prime1)a * (prime2)b * (prime3)c * ...`
becomes
`Required years, Y=(Rule for prime1) * a + (Rule for prime2) * b +(Rule for prime3) * c + ...`

#### 3. The faster solution

The above solution is quite accurate. But then there is the problem of remembering 15 rules for multipliers till 50 (as we need 15 prime numbers to represent any integer from 1 to 50). We can forego a bit of accuracy by using an approximation trick. The trick is to approximate the multiplier (`X`) using a small set of prime factors, rather than all. We will try to approximate any `X` as a product of prime factors `2,3,5,` and `7`; 53 as approximately 56 (`23*7=56`), 29 as 30 (`21*31*51=30`) and so on. Now we need to remember only 4 rules ( Rule of 72, Rule of 110, Rule of 161 and Rule of 195) to get a good approximation.

Prime NumbersRule (`r` is the rate of interest)
272/r
3110/r
5161/r
7195/r

Lets quickly work through another problem using this method.
Problem: Years to make `2,650,000 \$` from `50,000 \$` at `6 % p.a.` compounded annually.
Solution: `2,650,000 \$` is `53 (2,650,000/50,000=53)` times `50,000 \$`. Taking 56 as an approximation for 53 results in `56=23*7`. Required years, `Y=(Rule of 72) * 3 + (Rule of 195) * 1`.`Y=(72/6) * 3 + (195/6) * 1=12*3 + 32.5*1 = 68.5`.

At the end of `68.5` years the actual value is `2,628,868.38 \$`, which is close to our desired value.

With little practise you should be able to work out the answer in your head. You will need just the four rules above.