Math is more than plugging in a formula, dammit

On Quora, I answered this question: How long does it take for your money to grow 10 times its original value if the rate of interest is 5% per annum?

In my answer, I pointed out that there's a missing piece of information: how often interest is compounded (or if it is compounded at all!)

Let's take a look at the other two answers there currently:

47 years and 2 months to the nearest month

Where did this answer come from? We can use the compound interest formula

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

Where P is the principal amount (current value), r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

If we assume yearly compounding, then:

10P = P(1+0.05) ^ t

Solving for t (using logarithms) gives t = log(10)/log(1.05) which is approximately 47.194. Hence the answer 47.2.

I think this is wrong. If you assume interest is paid yearly, you can't assume that it can instead be paid midyear because it's mathematically convenient. Maybe there's a penalty for early withdrawal. Maybe it's dependent upon a dividend payment that won't be paid in the first quarter of the year. Your model can't first assume that you only get interest at the end of the year, and then switch to assuming you get interest after only 2/10th of the year.

(If you could take your interest at midyear, then presumably you could put it back in and get better compounding!)

Now, charitably we might say that the answer is assuming monthly compounding, i.e., "to the nearest month" and the 5% is the compound interest rate. That is, the yearly interest rate is 4.89%, compounded monthly. Does this work?

That gives t = 566.202 months, so 567 monthly periods. But that's 47 years and 3 months.

The second answer is worse:

According to the “rule of 72”, 72 divided by the interest rate = the number of years required to double if returns are compounded. 72 divided by 5 equals 14.4 years.

Doesn't answer the question, and makes the same mistake. It's not possible for the answer to be a fractional year. This is not even a particularly good rule of thumb because the answer is not 5 times 14.4 (that's way too long.) The answer is closer to log base 2 of 10 times 14.4 years, and who has that particular logarithm in their head?

(Well, OK, it's about 3.32 and that's a useful number to know, but the end result is 47.8, or 48 years.)

When you build a mathematical model, you're reducing from one problem (compound interest) to another (an exponential equation.) Everybody who's gone through algebra and gotten negative answers to their quadratic equation should understand that the two are not always equivalent. A solution to an equation is only good if it respects the constraints of the original problem.