Question

A small business owner contributes $2000 at the end of each quarter to a retirement account that earns 4% compounded quarterly.

(a) How long will it be until the account is worth $150,000? (Round your answer UP to the nearest quarter.)

(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $8000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.)

Answer 1

a) Here, P = $2000 and R = 4% = 0.04 and A = 150000

Now, b y condition we have, P[1+(1+R/4)+(1+R/4)²+.....(1+R/4)ⁿ] = A

i.e., 1+(1+R/4)+(1+R/4)²+.....(1+R/4)ⁿ = A/P

i.e., [(1+R/4)ⁿ⁺¹-1]/[(1+R/4)-1] = A/P

i.e., (1+R/4)ⁿ⁺¹-1 = (A/P)(R/4)

i.e., (1+R/4)ⁿ⁺¹ = (A/P)(R/4)+1

i.e., (1.01)ⁿ⁺¹ = (150000*0.01/2000)+1

i.e., (1.01)ⁿ⁺¹ = 1.75

i.e., (n+1)*log(1.01) = log(1.75)

i.e., n+1 = log(1.75)/log(1.01)

i.e., n 56

Therefore, the required number of quarters is = 56.

b) Now, after 15 more years, the account balance will be =

$[150000*(0.04)*15+8000*{(1+R/4)⁶⁰⁺¹-1}/{(1+R/4)-1}] [n = 60 quarters in 15 years]

= $[240000+8000*{(1.01)⁶¹-1}/0.01]

= $[240000+667890.9324]

$907891

Therefore, after 15 more years, the account balance will be $907891.