Question

Find the time required for an investment of 5000 dollars to grow to 6000 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Your answer is t=

Answer 1

2.5 years required for an investment of 5000 dollars to grow to 6000 dollars at an interest rate of 7.5 percent per year, compounded quarterly.

Step-by-step explanation:

The given is,

               Initial investment - $5000

               Future amount - $6000

               Interest rate - 7.5% (compounded quarterly)

Step:1

           Formula to calculate the Future amount with compound interest,

                                   $F = P(1+\frac{r}{n} )^{nt}$...................................(1)

          Where, F - Future amount

                           P - Initial amount

                            r - Rate of interest

                           n - No. of compounding in a year

                            t - Time period

          From given,

                      F = $6000

                      P = $5000

                       r = 7.5%

                       n = 4 (compounded quarterly)

        Equation (1) becomes,

                              $6000=5000(1+\frac{0.075}{4} )^{(t)(4)}$

                               $\frac{6000}{5000} =(1+0.01875)^{4t}$

                                 $1.2 = (1.01875)^{4t}$

         Take log on both sides,

                            $log 1.2 = 4(t) log 1.01875$

         Substitute log values,

                       0.07918 = 4(t) (0.0080676)

                                      = (t) (0.0322705)

                                   $t = \frac{0.07918}{0.0322705}$

                                      = 2.45

                                   t ≅ 2.5 years

Result:

         2.5 years required for an investment of 5000 dollars to grow to 6000 dollars at an interest rate of 7.5 percent per year, compounded quarterly.