Question

Suppose you have $45 to invest in a CD that pays simple interest calculated semi-annually, the annual interest rate is 6% how much will you earn after 2 years

Answer 1

Since the interest is calculated semi-annually, it is calculated every six months.

This means half of the interest rate is earned every six months.

We can consider the period to be four periods of six months.

The formula to calculate the amount earned with simple interest is:

`A=P(1+r\cdot t),`

where A is the final amount, P is the initial amount, r is the interest rate and t is the time period. This formula is obtained from the fact that, in simple interest, every time the interest period comes up we add an ammount equal to the initial investment times the interest rate to our initial investment.

For example, if we start with $100 and we have a 2% semi-anual interest rate, after the first 6 months we have,

`100+100\cdot0.02=100+2=102.`

After another 6 months we get,

`102+100\cdot0.02=104.`

If we keep doing this, we'll realize that we are adding the interest rate as many times as time periods have passed to the previous amount we had.

This can be seen as the following:

`A=100+100\cdot0.02t,`

in which we can factor a 100 to obtain

`A=100(1+0.02t),`

which if we analyze properly we can see it's the same as the formula given before:

`A=P(1+r\cdot t)\text{.}`

In this case, we have:

`A=45(1+(0.06)/(2)\cdot4)=50.40,`

so we would have a total of $50.40 after two years. In order to determine how much we earned, we simply calculate:

`A-P=50.40-45=5.40,`

so we earned $5.40.