When does the balance first exceed $500? and Find the balance after this amount of time?

Question

A deposit $400 in an account that pays 2.4% annual interest compounded monthly.

Answer 1

The formula for this type of TVM (time value of money) problem is:

 

Future Value = Present Value * (1+annual interest/12)^n*12 , the interest rate is divided by 12 and the number of periods is multiplied by 12 since we are compounding monthly

 

Since we know the Present Value, $400, and the Future Value, $500, and the monthly interest rate, .024/12, We can solve for the number of periods (notice that I used the word period, not years, because our answer will be in years*12, or months.

 

First, we define a new variable, M, where M = 12n

 

The equation becomes

 

$500 = $400 * (1+.024/12)^M

 

Future Value = Present Value * (1+annual interest/12)^M

 

First step is to divide both sides by $400

 

500/400 = (1.002)^M

 

Next we take natural log of both sided

 

ln(1.25) = ln(1.002)^M

 

the law of logarithms says that

 

ln(1.25) = M * ln(1.002)

 

Dividing both sides by ln(1.002) gives

 

ln(1.25)/ln(1.002) = M =111.6833 months

 

If we round up to 112 months, the balance will be

 

Future Value = Present Value * (1+annual interest/12)^M

 

Future Value = $400* (1.002)¹¹² = $500.32 which occurs in month 112