How long to quadruple

Question

How long does it take to quadruple your money at 4.5% interest rate? I've already used the Rule of 144, divided 144 by 4.5 and got 32 and it was marked incorrect.

Answer 1

Let's say you have $x$ dollars. You want to know how long it will take to get to $4x$ dollars. For simplicity let's also assume the interest is recieved annually (although any other unit of time can be taken and the result will change units accordingly).

Given the $4.5%$ interest rate, you know that every year, your money will multiply by $1.045$. Therefore, after $T$ years, you will have:

$$Money(T\space\text{years})=x\underbrace{\cdot1.045\cdot1.045\cdot\dots\cdot1.045}_{T\space\text{times}}=1.045^{T}\cdot x$$

(To get how much money you have you multiply it by $1.045$ after each year, so a total of $T$ times). In general, the formula for this type of growth is: $$M(t)=M_0\cdot(1+\frac{\text{growth percentage}}{100})^{t}$$ Where $M_0$ is the initial amount (in this case the growth percentage is $4.5$ and the initial amount is $x$).

Anyway, now that we know the amount of money we have after $T$ years, we would like to check when exactly we would have $4x$ dollars. So, we set:

$$Money(T)=4x \implies 1.045^{T}\cdot x=4x \implies 1.045^{T}=4$$

This equation can be solved by taking $log_{1.045}$ on both sides. If your calculator can calculate this - great. Otherwise (hopefully it can calculate natural logs) by laws of logrithms:

$$T=log_{1.045}{4}=\frac{\ln4}{\ln1.045}\approx31.49\space\text{years}$$ Since the interest is only recieved at the end of each year, we must round up to $32\space \text{years}$. This result can obviously change depending on how often interest is recieved. If it is recieved monthly, the units would change to months and so on. If we (unrealistically) assume the growth rate is continuous, which may have been the intention of the problem, no rounding up shall occur, and the result will be about $31.49 \space\text{months\years\weeks}$...

In general, we can repeat this process for any interest rate, and conclude that with a growth percentage of $p$, the time it would take to multiply by $k$ a certain amount of money will be:

$${\text{Time}=\frac{\ln{k}}{\ln(1+\frac{p}{100})}}$$ This formula will solve all problems of this type (in this case, $k=4$, $p=4.5$).

About the rule of 144, I am not very familiar with it, but I believe it would give you good estimates (but won't yield precise results and may slightly error in some cases). In this case it appears to be correct, unless the problem intended on the more exact $31.49 \space\text{time units}$ for continuous interest returns.