# Greg bought a gold coin for $9,000. If the value of the coin increases at a constant rate of 12% every 5 years, how many years will it take for the coin to be worth$20,000?

Aug 24, 2017

Approximately 35 years (35.23).

#### Explanation:

If the price increases by 12% every five years, to get the value after five years we multiply by $1.12$. To get the value after another five years we multiply by $1.12$ again, ie we multiply the original value by ${1.12}^{2}$.

So the value after $5 n$ years will be given by

$9000 \cdot {\left(1.12\right)}^{n}$

So we now want to find out when the value will be $20000. So $20000 = 9000 \cdot {1.12}^{n}$$\implies \frac{20000}{9000} = \frac{20}{9} = {1.12}^{n}$Take logs of both sides and use rules of logs on LHS to bring down the n: $\log \left(\frac{20}{9}\right) = n \cdot \log \left(1.12\right)$$\therefore n = \log \frac{\frac{20}{9}}{\log} \left(1.12\right) \cong 7.046$The coin reaches$20000 value after roughly 35 years.