I need help on this math question

Question

Nancy wants to invest $2000 in saving certificates that bear an interest rate of 8.75% per year, compounded semiannually. How long a time period should she choose to save an amount to save an amount of $3000? ( round your answer to two decimal places.)

Answer 1

Hi Alexia:

Here's how to tackle this one. We use the compound interest formula:

A = P(1 + i/2)²ⁿ [A=amount, P=principal invested; i=interest rate per period; n = # of years

Here P = $2,000 (the principal invested; A = $3,000 (desired final amount); i = 0.0875/2; n = # of years

3,000 = 2,000 (1 + 0.0875/2)²ⁿ [Note: 8.75% interest per year = 0.0875/2 per half-year

3/2 = 1.04375²ⁿ

1.5 = 1.04375²ⁿ

Since the unknown, n, is in an exponent (power), we take logs on both sides

ln(1.5) = 2n ln(1.04375) [by the laws of logs, exponents become multiplication

=> n = ln(1.5) / 2 ln(1.04375)

= 4.73 years

= 5 years, rounded up to nearest half-year