Doubling Time

Question

You deposit $2500 in an account that pays 7.5% annual interest, compounds quarterly. How long will it take for your balance to become $3000? Would a larger deposit increase by $500 in a shorter time? Explain.

Answer 1

The formula for compound interest is:

 

A = P(1 + r/c)^ct

 

where:

 

A = the amount after applying the interest

P = the initial amount -- $2500

r = the annual interest rate, as a decimal -- 7.5% = 0.075

c = the rate of compounding -- quarterly means 4 times a year, so this is 4

t = the number of years -- unknown

 

So in this case you have:

 

3000 = 2500 (1 + 0.075 / 4)^4t

 

And we have to solve for t:

3000 = 2500 (1 + 0.075 / 4)^4t

3000 / 2500 = (1 + 0.01875)^4t

1.2 = 1.01875 ^4t

ln 1.2 = ln 1.01875 ^4t

ln 1.2 = 4t ln 1.01875

ln 1.2 / ln 1.01875 = 4t

(ln 1.2 / ln 1.01875) / 4 = t

2.454 = t

 

So the answer is 2.454 years.

 

Yes a larger deposit would increase by $500 in a shorter time.  The interest each is a multiple of the amount in the account.  The larger the amount in the account, the larger the interest, so the interest stream would be larger each period.  If you are adding larger numbers, it takes fewer of them to reach a specific amount than if you are adding smaller numbers.  So the total of the interest stream would reach $500 more quickly.