Question

You plan to invest in securities pay more 11.2% compounded annually. if you invest $5000 today, how many years will it take for your investment to grow to $9,140.20

Answer 1

Answer:

$t\approx6\hspace{3}years$

Explanation:

When interest is compounded annually, we can use the following formula to calculate the amount in the account at the end of a given time period.:

$FV=PV(1+r)^t$

Where:

$FV=Future\hspace{3}value=9140.20\\PV=Present\hspace{3}value=5000\\r=Interest\hspace{3}rate=11.2\%=0.112\\t=Time$

Let's solve the previous equation for t:

Divide both sides by PV:

$\frac{FV}{PV} =\frac{PV}{PV} (1+r)^t\\\\\frac{FV}{PV} = (1+r)^t$

Take the natural logarithm of both sides:

$log(\frac{FV}{PV}) = log( (1+r)^t)\\\\Use\hspace{3}the\hspace{3}identity\hspace{5}log(a^b)=b*log(a)\\\\log(\frac{FV}{PV}) =t* log(1+r)\\\\Divide\hspace{3}both\hspace{3}sides\hspace{3}by\hspace{3}log(1+r)\\\\t=\frac{log(\frac{FV}{PV})}{log(1+r)}$

Replace the data provided by the problem:

$t=\frac{log(\frac{9140.20}{5000})}{log(1+0.112)}$

$t=5.682396777\approx 6\hspace{3}years$