Question

Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?

 

 

Answer 1

Rachel deposits $ 3,600 into a retirement fund each year. Hence, a1 = 3,600

Total number of deposits will be 55 – 20 = 35. Hence, n = 35

 

Common ratio will be the interest rate plus 1. Annual interest rate is equal to 7.5% and it is compounded monthly. Hence,

r = 1 + 7.5/12%

   = 1 + 0.625%

   = 1 + 0.00625

   = 1.00625

 

Use the formula for the sum of first n terms of a geometric sequence,

Sn = a1(1 – rn)/(1 – r) ...... (1)

 

Substitute a1 = 3,600, n = 35 and r = 1.00625 in (1) and simplify,

S35 = 3,600(1 – 1.0062535)/(1 – 1.00625)

        = 3,600(1 – 1.24367317…)/-0.00625

       = 3,600(-0.24367317…)/-0.00625

       = 140355.75061495…

 

Rounding to two decimal places, the value of the annuity is $ 140,355.75

 

Therefore, Rachel will save $140,355.75 by the time she is 55.

 

Money deposited by Rachel in 35 years is equal to 3,600 × 35 = 126,000. So, interest earned by her will be,

140,355.75 – 126,000 = $14,355.75

Therefore, Rachel will save $140,355.75 by the time she is 55.