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

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

S=P(1+r)n=2000(1+0.06)4Substitute in the values=2000(1.06)4Evaluate the power=2000×1.2625Multiplying=2524.95Simplify\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 2524.952524.95 after 4 years. Total compounded amount after 8 years will be

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

S=P(1+r2)4×2=2524.95(1+0.062)8Substitute in the values=2524.95(1.03)8Evaluate the power=2524.95×1.2668Multiplying=3198.53Simplify\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*}

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

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