Question

James deposits $12,500 in a simple interest account with an annual interestrate of 1.5%. After a few years, he notices that he has earned $1,687.50 ininterest. How long has James had this account?

Answer 1

Recall that the simple interest means that you will calculate interest only based on the initial deposit amount. This means that the generated interests do not generate any interests.

To find the number of years James has had this account, we will first find a formula for the amount available in the account each year.

At year 0, James deposits 12500. So, at year 0 the account has 12500

At year 1, James earns 1.5% over the 12500. So we add to what we had at year 0, the interest. That is

$12500+\text{ 12500}\cdot i$

where i is the interest annual rate of 1.5%.

At year 2, James earns 1.5% over 12500 again. So we add this value to what we had at year 1, so at year 2 he has

$12500+12500\cdot i+12500\cdot i\text{ = 12500+12500}\cdot2\cdot i$

Finally, at year 3, James earns another 1.5% over the 12500. So we add this value to what he had at year 2, so he has

$12500+12500\cdot2\cdot i+12500\cdot i\text{ = 12500+12500}\cdot3\cdot i$

In general, from this we can see a pattern. At year n, the amount available would be

$12500+12500\cdot i\cdot n\text{ = 12500}\cdot(1+i\cdot n)$

In this formula the amount of interest is given by the expression

$12500\cdot i\cdot n$

We are told that this amount is 1687.5. So we have the following equation

$12500\cdot i\cdot n=1687.5$

so, if we divide both sides by 12500*i, we get

$n=\frac{1687.5}{12500\cdot i}$

we know that i=1.5%. = 0.015. So we have

$n=\frac{1687.5}{12500\cdot0.015}=9$

so James has had the account for 9 years.