Proportionality to find spent years for price drop

Question

Well, the title's kinda messy, but this is a concrete example of what I'm trying to find out:

Lets say there is a price of 40.000 USD, if the price drops at half, how many years does it take for the price to become 5 USD?

I believe that this has something to do with proportionality. Am I mistaken? What' the solution to this?

Answer 1

Assuming you mean the price drops in half each year, then starting at 40: $40\to20\to10\to5$, so it takes 3 years. Notice that this is repeated multiplication by $0.5$, i.e. $0.5^3\times40 = 5$

With larger numbers, and you may have meant starting at 40000 (I know some countries have different conventions for the decimal point), it's easier to write an equation: $$\begin{align} 0.5^n\times40000 &= 5,\;\;\;\mbox{but $0.5 = 1/2$ and $(1/x)^n = x^{-n}$}\\ 2^{-n}\times40000 &= 5,\;\;\;\mbox{take logs to base 2}\\ -n + \log_2 40000 &= \log_2 5\\ n &= \log_2 40000 - \log_2 5\\ &= 12.96578 \end{align}$$

So 13 years in round numbers.

Answer 2

3 years. After one year the price is $\$20$. After two years the price is 10. Then after 3 years the price is $\$5$.