please help to solve the problem

Question

Problem 1. Compound interest. This morning Paul and Mark invest their money. Paul invests 20 000$ at a 8.8% annual interest rate. Interest accrues each night, but compounds only semiannually. Mark invests 24 000$ at a 4.8% annual interest rate. Interest accrues each night, but compounds only quarterly. Who among them will get 30 000$ earlier? Compute the difference in days/

Answer 1

The formula for compound interest is:

 

P(t) = P₀(1 + r/n)^nt

where:

• P(t) = the principle after time t
• P0 = the initial investment (principle)
• r = annual interest rate as a decimal
• n = number of compounding per year
• t = time (years)

 

For Paul:

      30,000 = 20,000(1 + 0.088/2)^2t

      1.5 = (1.044)^2t

Solve by using the log.  I used log base 10 but any base will do.

      log(1.5) = log(1.044)^2t

      log(1.5) = 2t*log(1.044)           Used the identity log(a^b) = b*log(a)

      log(1.5)/log(1.044) = 2t

      log(1.5)/(2log(1.044)) = t         Use your calculator here

      4.7 years = t

 

Do the same for Mark with P(t) = 30,000, P₀ = 24,000, r = 0.048, n = 4.  Solve for t using logs like we did for Paul.