Question

Holly wants to save money for an emergency. Holly gets $600 in an account that pays an interest rate of 7.5%.

How many years will it take for their account to reach $7,600?round your answer to the nearest hundredth.

Has to be in decimal form like 00.00

Answer 1

now, we're assuming the interest is compound interest with a 7.5 APR or namely 7.5% compounding annually, so

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$7600\\ P=\textit{original amount deposited}\dotfill &\$600\\ r=rate\to 7.5\%\to (7.5)/(100)\dotfill &0.075\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years \end{cases}`

`7600=600\left(1+(0.075)/(1)\right)^(1\cdot t)\implies \cfrac{7600}{600}=1.075^t\implies \cfrac{38}{3}= 1.075^t \\\\\\ \log\left( \cfrac{38}{3} \right)=\log(1.075^t)\implies \log\left( \cfrac{38}{3} \right)=t\log(1.075) \\\\\\ \cfrac{\log\left( (38)/(3) \right)}{\log(1.075)}=t\implies {\LARGE \begin{array}{llll} 35.11\approx t \end{array}}\qquad \textit{about 35 years and 40 days}`