Hao made a single investment which,after 5 years invested at 12% compounded semiannually,has accumulated to $214,900.How much did Hao invest initially? (PV of $1,FV of $1,PVA of $1,and FVA of $1) $\bold{\text{(Use appropriate factor(s) from the tables provided.) }}$

A) $214,896
B) $160,584
C) $211,476
D) $120,000
E) $ 21,486

Question

Hao made a single investment which,after 5 years invested at 12% compounded semiannually,has accumulated to $214,900.How much did Hao invest initially? (PV of $1,FV of $1,PVA of $1,and FVA of $1)

\bold{\text{(Use appropriate factor(s) from the tables provided.) }}

A) $214,896
B) $160,584
C) $211,476
D) $120,000
E) $ 21,486

Ravindra Nagar · Accepted Answer

Final Answer : D, Explanation :   The future value (FV) of an investment is calculated using the formula   F V = P V ( 1 + r n ) n t FV = PV(1 + \frac{r}{n})^{nt} F V = P V ( 1 + n r ​ ) n t  , where PV is the present value (initial investment), r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. Given a 12% annual interest rate compounded semiannually for 5 years, we have   r = 0.12 r = 0.12 r = 0.12  ,   n = 2 n = 2 n = 2  , and   t = 5 t = 5 t = 5  . The future value is given as $214,900. We need to solve for PV.Rearranging the formula to solve for PV gives us   P V = F V ( 1 + r n ) n t PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} P V = ( 1 + n r ​ ) n t F V ​  .Substituting the given values:   P V = 214 , 900 ( 1 + 0.12 2 ) 2 ∗ 5 PV = \frac{214,900}{(1 + \frac{0.12}{2})^{2*5}} P V = ( 1 + 2 0.12 ​ ) 2 ∗ 5 214 , 900 ​  .This simplifies to   P V = 214 , 900 ( 1 + 0.06 ) 10 PV = \frac{214,900}{(1 + 0.06)^{10}} P V = ( 1 + 0.06 ) 10 214 , 900 ​  .Calculating this gives us   P V = 214 , 900 1.790847 PV = \frac{214,900}{1.790847} P V = 1.790847 214 , 900 ​  .Therefore,   P V ≈ 120 , 000 PV ≈ 120,000 P V ≈ 120 , 000  .This calculation shows that Hao initially invested approximately $120,000.

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