Question

Find the time required for an investment of 5000 dollars to grow to 8500 dollars at an interest rate of 7.5 percent per year, compounded quarterly

Answer 1

Given:

`\begin{gathered} Pr\text{ incipal, P=\$5000} \\ \text{Amount, A=\$8500} \\ \text{Rate, r=7.5\%=}(7.5)/(100)=0.075 \\ \text{Number of compounding (quarterly), n =4} \\ t=\text{?} \end{gathered}`

The formula for finding the amount in compound interest is given by;

`A=P(1+(r)/(n))^(nt)`

Substituting the values to get the time it will take,

`\begin{gathered} A=P(1+(r)/(n))^(nt) \\ 8500=5000(1+(0.075)/(4))^(4t) \\ (8500)/(5000)=(1+0.01875)^(4t) \\ 1.7=1.01875^(4t) \\ \text{Taking the logarithm of both side to get t,} \end{gathered}`
`\begin{gathered} 1.7=1.01875^(4t) \\ \log 1.7=\log 1.01875^(4t) \\ \text{Applying the law of logarithm below;} \\ \log a^x=x\log a \\ \text{Then,} \\ \log 1.7=4t\log 1.01875 \\ \text{Dividing both sides by log1.01875} \\ (\log 1.7)/(\log 1.01875)=4t \\ (0.2304)/(0.008068)=4t \\ 28.557=4t \\ \text{Dividing both sides by 4 to get the value of t,} \\ t=(28.557)/(4) \\ t=7.13925 \\ t\approx7.14years \end{gathered}`

Therefore, the time is approximately 7.14years.