 So, notice the amount ACT is directly proportional to the investment C, or at least that's what we've been assuming. So, it makes sense to ask the question, how long will it take before the money grows by a factor of K? While K could be any value, we often ask for the doubling time. For example, suppose an investment earns 5% per year, how long will it take before the amount doubles? We have the interest rate, 0.05, but we don't know C, T, or ACT. But since we want the amount to double, we want ACT to be 2C, that's twice the initial investment, and so we can write, and solve, and so T is approximately 14.21 years. We can apply this generally, so if we want to find the doubling time, we have our amount given by our compound interest formula, the amount at time T should be 2C, and we can solve, and this gives us our doubling time. Now, while we don't encourage the use of approximations, we notice the following. Log of 2 is approximately, meanwhile log of 1 plus the interest rate, 0.05, is approximately 0.05. So if we use the approximation, log of 1 plus I is approximately I, we get a doubling time that can be computed, or if we want to express things in terms of the actual percentage rate, we can multiply by 100. Consequently, you can find the approximate doubling time by dividing 69 by the interest rate. For example, at 5% interest, the doubling time is approximately 69.5, 13.8 years. Now while we could use 69 as a numerator, we often use other numbers, and not for the obvious reasons. Classically, this is known as the rule of 70, and so you can find the approximate doubling time by dividing 70 by the interest rate. But the problem is neither 69 nor 70 are easy to divide into, and since we're only looking for an approximation, another version is you can find the approximate doubling time by dividing 72 by the interest rate. So for example, let's find the approximate doubling time for an investment earning 12% annually, and then let's find the exact value. So we could use the rules of 69, 70, or 72, and let's use all of them, dividing 12 into 69, dividing 12 into 70, and dividing 12 into 72. If we want the amount to double, we find. And so our doubling time is about 6.12 years. And notice that the rule of 72 actually gives the best approximation. This always happens. Or does it? So it never hurts to try out a concrete example, suppose we have an interest rate of 0.5%. So again, using the rule of 69, 70, and 72, we find. And we can find the exact value by, and so our doubling time is about 138.98 years, and this time the rule of 69 gives us the best approximation.