Trouble finding interest rate!

Question

A woman who is now 28 years old invests $30,000 in a retirement account What interest rate, compounded monthly, would she have to earn in order for the account to be worth $500,000 when she retires at age 65? The formula I’m using doesn’t want to seem to work F= p(1+r/n)^nt  and I’m having a hard time figuring out the time. Whether that should be in years or months. The gap between her age now and her age at 65… Do I put 38 years? 

Answer 1

The formula is correct

 

F = p(1+r/n)^nt

F = future value = 500000

p = principal = 30000

r = interest rate (as a decimal)

n = number of times compounded per year = 12

t = time in years = 65-28 = 37

 

500000 = 30000(1+r/12)¹²⁽³⁷⁾

500000/30000 = (1+r/12)⁴⁴⁴

50/3 = (1+r/12)⁴⁴⁴

⁴⁴⁴√(50/3) = 1+ r/12

12( ⁴⁴⁴√(50/3) - 1) = r

 

r = 0.076279545

rounding we have:

r = 7.63%

 

check:

30000(1+0.0763/12)⁴⁴⁴ = $500,376

 

The extra is due to rounding on the value of r.

Without the rounding we get exactly $500,000

Answer 2

The time is in years, and 65 - 28 = 37.

 

500,000 = 30,000(1 + r/12)¹²⁽³⁷⁾

500,000/30,000 = (1 + r/12)⁴⁴⁴

(500,000/30,000)^1/444 = 1 + r/12

(500,000/30,000)^1/444 - 1 = r/12

r = 12((500,000/30,000)^1/444 - 1)

 

Put the right side in your calculator, then multiply by 100 for the percentage.