Question

Jean wants to save up to put a $2,000 down pay payment on a car. If he deposits $1,800 in an account that earns 4.8% interest compounded monthly, how long will it take for the account to. reach $2,000 Round to the hundredths place.

I am lost

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 2000\\ P=\textit{original amount deposited}\dotfill &\$1800\\ r=rate\to 4.8\%\to (4.8)/(100)\dotfill &0.048\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years \end{cases}`

`2000 = 1800\left(1+(0.048)/(12)\right)^(12\cdot t) \implies \cfrac{2000}{1800}=(1.004)^(12t)\implies \cfrac{10}{9}=1.004^(12t) \\\\\\ \log\left( \cfrac{10}{9} \right)=\log(1.004^(12t))\implies \log\left( \cfrac{10}{9} \right)=t\log(1.004^(12)) \\\\\\ \cfrac{\log\left( (10)/(9) \right)}{\log(1.004^(12))}=t\implies 2.20\approx t\qquad \textit{about 2 years and 73 days}`