Question

Liam's goal is to save $20 000. What principal invested for 5 years at 6% per annum,compounded semi-annually,then for the next 3 Years at 6.5% per annum,compounded quarterly,achieves this goal in 8 years?

Answer 1

Liam's goal is to save $20 000. What principal invested for 5 years at 6% per annum, compounded semi-annually, then for the next 3 Years at 6.5% per annum, compounded quarterly, achieves this goal in 8 years?

$$x \left(1 + \frac {n \%}{100}\right) ^ {k}, x - \text{sum at start, } n - \text{interest rate, } k - \text{number of periods}.$$

Half year interest rate = (1/2)(6%) = 3%. There are 10 half year periods in 5 years.

Quarter year interest rate = (1/4)(6.5%) = 1.625%. There are 12 quarters in 3 years.

$$\begin{array}{l} x (1 + 0.03)^{10} + x (1 + 0.01625)^{12} = 20000, \\ (1 + 0.03)^{10} \approx 1.344, \quad (1 + 0.01625)^{12} \approx 1.213, \\ 1.344x + 1.213x = 20000 \\ 2.557x = 20000, \quad x = \frac{20000}{2.557} \approx 7822. \end{array}$$

Answer: 7882$.