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?

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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**

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.**

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