Question

(1 point) (Exercise 2.11) AA deposits $300 today and another $400 in five years into a fund paying simple interest of 7% per year. BB will make the same two deposits, but the $300 will be deposited nn years from today and the $400 will be deposited 2n2n years from today. BB's deposits earn an annual effective rate of 8.5%. At the end of 9 years, the accumulated value of BB's deposits equals the accumulated value of AA's deposits. Calculate nn.

Answer 1

Accumulated amount of AA’ account in 9 years can be computed using simple interest as:

A = P (1+ rt)

P = Principal amount = $ 300, $ 400

r = Interest rate = 0.07

t = Time period = 9 years and 4 (i.e. 9 -5) years

A = [$ 300 (1+ 9 x 0.07)] + [$ 400 (1+ 4 x 0.07)]

    = $ 300 (1 + 0.63) + $ 400 (1+0.28)

= $ 300 x 1.63 + $ 400 x 1.28

= $ 489 + $ 512

= $ 1,001

Accumulated amount of BB’s account after 9 years is $ 1,001

[$ 300 (1+ (9 -nn) x 0.085))] + [$ 400 (1 + (9-2n2n) x 0.085))] = $ 1,001

[$ 300 (1 + (0.765 – 0.085 nn)] + [$ 400 (1+ (0.765 - 0.085 x 2n2n)] = $ 1,001

[$ 300 + $ 300 x (0.765 – 0.085 nn)] + [$ 400 + $ 400 (0.765 - 0.085 x 2n2n)] = $ 1,001

$ 300 + $ 229.5 – $ 25.5 nn + $ 400 + $ 306 – $ 34 x 2n2n = $ 1,001

$ 529.5 – $ 25.5 nn + $ 706 – $ 34 x 2n2n = $ 1,001

$ 529.5 + $ 706 – $ 1,001 = $ 25.5 nn + $ 34 x 2n2n

$ 1,235.5 - $ 1,001 = nn ($ 25.5 + $ 34 x 2)

$ 234.5 = nn x ($ 25.4 + $ 68)

$ 234.5 = nn x $ 93.5

nn = $ 234.5 / $ 93.5

     = 2.508021 or 2.51

nn is 2.51 years