Question

Anne plans to save $40 a week, starting next week,for ten years and earn a rate of return of 4.6 percent, compounded weekly. after the ten years, she will discontinue saving and invest her account at 6.5 percent, compounded annually. how long from now will it be before she has accumulated a total of $50,000? 10.32 years 21.14 years 15.08 years 11.14 years 20.32 years

Answer 1

Week 1: She has $40

Week 2: She adds $40 and the other $40 have earned an interest: $40*(100+4.6)/100 = $40*1.046 = $41.84. So now she has $80.84.

Week 3: She adds $40 and the other $80.84 have earned an interest: $80.84*1.046 = $84.55864

Let’s put all the math together:

$40+40*1.046+40*1.046*1.046

Sounds like adding the terms of a geometric progression. There is a formula for that: Sn = a1(1-r^n)/(1-r)

Ten years have 10*365 days = 3650 days

That is 3650/7 weeks = 521 weeks (I rounded to the bottom, considering half a week doesn’t count)

So last week will be week 521

Sn = $40*(1-1.046^521)/(1-1.046) = $13,040,967,134,112.40

(You can also verify this number using Exel).

So she has already got much more tan 50,000 in ten years… Looks like none of the options are correct.

Let’s find the correct number of weeks:

40(1-1.046^n)/(1-1.046)= 50,000
(1-1.046^n)/(1-1.046)= 50,000/40
1-1.046^n= 50,000/40*(1-1.046)
1.046^n-1= 50,000/40*(1.046-1)
1.046^n= 50,000/40*(1.046-1)+1
log(1.046^n)= log(50,000/40*(1.046-1)+1)
n*log(1.046)= log(50,000/40*(1.046-1)+1)
n= log(50,000/40*(1.046-1)+1)/ log(1.046)

n = 90.4763674

So the answer is 90 weeks.

1 year = 365 days

1 week = 7 days

So 1 year = 365/7 = 52,2 weeks

So 90 weeks is 1.72 years.