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Question

# To what sum will \$2000 amount in eight years if invested at a 6% effective rate for the first four years and at 6% compounded semiannually thereafter?

Solution

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\begin{align*} \text{Principal amount}&=\2000\\ \text{Time period}&=8\text{years} \end{align*}

$\textbf{Part 1}$: For the first 4 years $r_e=6\%$

\begin{align*} S&=P(1+r)^n\\ &=2000(1+0.06)^4&\textcolor{#4257b2}{\text{Substitute in the values}}\\ &=2000(1.06)^4 &\textcolor{#4257b2}{\text{Evaluate the power}}\\ &=2000\times 1.2625 &\textcolor{#4257b2}{\text{Multiplying}}\\ &=2524.95 &\textcolor{#4257b2}{\text{Simplify}} \end{align*}

Now the Principal amount is $2524.95$ after 4 years. Total compounded amount after 8 years will be

$\textbf{Part 2}$: for the last 4 years rate $6\%$ semiannually

\begin{align*} S&=P\left(1+\dfrac{r}{2}\right){4\times 2}\\ &=2524.95 \left(1+\dfrac{0.06}{2}\right){8} &\textcolor{#4257b2}{\text{Substitute in the values}}\\ &=2524.95 (1.03)^8 &\textcolor{#4257b2}{\text{Evaluate the power}}\\ &=2524.95 \times 1.2668 &\textcolor{#4257b2}{\text{Multiplying}}\\ &=3198.53 &\textcolor{#4257b2}{\text{Simplify}} \end{align*}

$\boxed{\text{The compounded amount after 8 years will be }\3198.53}$

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