# Amla's goal is to save \$ 20,000, What principal invested for 5 years @ 6% per annum, compounded semi-annually, then......  ....for the next 3 years @ 6.5% per annum compounded quaterly, achieves...

Amla's goal is to save \$ 20,000, What principal invested for 5 years @ 6% per annum, compounded semi-annually, then......

....for the next 3 years @ 6.5% per annum compounded quaterly, achieves this goal in 8 years?

lfryerda | High School Teacher | (Level 2) Educator

Posted on

To determine the amount saved requires two steps, since the savings method is using two different interest rates.

To find the first step, we have an amount A that has been saved semi-annually at 6% for 5 years.  This is finding the present-value of the amount, where the interest per saving period is `1+0.06/2=1.03` , to get:

`A(1.03)^10`

Note that there are `2 times 5 = 10` interest periods since it is semi-annual interest.

From the second step, we have the amount from the first step, but now it is compounded quarterly at 6.5% for three years.  This is also finding present value, but using the future value of the first step, where the interest rate is `1+0.065/4=1.01625` .

The two steps can now be combined, knowing that they have to equal \$20000 to get:

`A(1.03)^10(1.01625)^12=20000`   Now solve for A

`A=20000/{(1.03)^10(1.01625)^12}`    evaluate

`A=12664.53`

The amount required to invest now is \$12664.53.

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Amla's goal is to have \$20000 in savings after 8 years. An amount invested by her today yields 6% per annum compounded semi-annually for 5 years and for the subsequent 3 years yields 6.5% per annum compounded quarterly.

If the principal invested right now is P, its value after 8 years is equal to `P*(1 + 0.06/2)^10*(1+0.065/4)^12` . As this has to equal \$20000:

`P*(1 + 0.06/2)^10*(1+0.065/4)^12 = 20000`

=> `P = 20000/((1 + 0.06/2)^10*(1+0.065/4)^12)`

=> `P ~~ 12264.53`

The investment that Amla should make right now to achieve her goal is \$12264.53