Time Value of Money

Concept:Time for a sum to grow at compound interest converted quarterly.Explanation:Let $n$ be the number of quarters.Quarterly rate $= \frac{5\%}{4} = 1.25\% = 0.0125$.Amount $A = P(1+r)^n$ gives $2000 = 1200(1.0125)^n$.Thus $(1.0125)^n = \frac{2000}{1200} = \frac{5}{3} \approx 1.6667$.Take logarithms: $n \log(1.0125) = \log(5/3)$.$\log(1.0125) \approx 0.005395$, $\log(5/3) \approx 0.22185$.So $n = \frac{0.22185}{0.005395} \approx 41.12$ quarters.$41.12$ quarters $= \frac{41.12}{4} = 10.28$ years $= 10$ years $+ 0.28 \times 12$ months $\approx 10$ years $3.37$ months.The closest full quarter is $10$ years $3$ months (since $41$ quarters = $10$ years $3$ months).Thus, the approximate time is $10$ years $3$ months.Answer:10 years 3 months