Answer:
(a) Annual Compounding: $35462.98
(b) Semiannual Compounding: $35613.78
(c) Monthly Compounding: $35744.53
(d) Daily Compounding: $35770.38
Step-by-step explanation:
Let's calculate the value of the investment at the end of 5 years using the different compounding methods, where:
(a) Annual Compounding:
We can use formula:
[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{100}\right)^t}} [/tex]
Where
Substitute the value and simplify:
[tex]\begin{aligned} \textsf{Compound Amount} &= 26500 \left(1 + \dfrac{6}{100}\right)^5\\\\ & = 26500 \left(1 + 0.06\right)^5\\\\ & = 26500 \times (1.06)^5 \\\\ & = 26500 \times 1.3382255776 \\\\ & = 35462.9778064 \\\\ & = 35462.98 \textsf{(in nearest cent)} \end{aligned} [/tex]
[tex]\dotfill[/tex]
(b) Semiannual Compounding:
We can use formula:
[tex]\large\boxed{\boxed{\textsf{Compound Amount} = P \left(1 + \dfrac{r}{200}\right)^{2t}}} [/tex]
Where
Substitute the value and simplify:
[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{200}\right)^{2 \times 5} \\\\ & = 26500 \left(1 + 0.03\right)^{10} \\\\ & = 26500 \times (1.03)^{10} \\\\ & = 26500 \times 1.34391637934412192049 \\\\ & = 35613.784052619230892985 \\\\ & = 35613.78 \textsf{(in nearest cent)}\end{aligned} [/tex]
[tex]\dotfill[/tex]
(c) Monthly Compounding:
We can use formula:
[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{1200}\right)^{12t} }}[/tex]
Where
Substitute the value and simplify:
[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{1200}\right)^{12 \times 5} \\\\ & = 26500 \left(1 + 0.005\right)^{60} \\\\ & = 26500 \times (1.005)^{60} \\\\ & = 26500 \times 1.3488501525493 \\\\ & = 35744.529042556 \\\\ & = 35744.53 \textsf{(in nearest cent)}\end{aligned} [/tex]
[tex]\dotfill[/tex]
(d) Daily Compounding:
We can use formula:
[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{36500}\right)^{365t}}} [/tex]
Where
Substitute the value and simplify:
[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{36500}\right)^{365 \times 5} \\\\ & = 26500 \left(1 + 0.0001643835616\right)^{1825} \\\\ & = 26500 \times (1.00016438356)^{1825} \\\\ & = 26500 \times 1.3498255274436 \\\\ & = 35770.376477255 \\\\ & = 35770.38 \textsf{(in nearest cent)} \end{aligned} [/tex]
Therefore, the value of the investment at the end of 5 years for each compounding method (rounded to the nearest cent) are:
(a) Annual Compounding: $35462.98
(b) Semiannual Compounding: $35613.78
(c) Monthly Compounding: $35744.53
(d) Daily Compounding: $35770.38
Answer:
(a) $35,462.98
(b) $35,613.78
(c) $35,744.53
(d) $35,770.38
Step-by-step explanation:
To find the value of the investment at the end of 5 years for each compounding method, we can use the formula for compound interest:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]
In this case:
Substitute these values into the formula:
[tex]A=26500\left(1+\dfrac{0.06}{n}\right)^{5n}[/tex]
Now, calculate for each compounding method by substituting the corresponding value of n into the equation.
(a) Annual compounding: n = 1
[tex]A=26500\left(1+\dfrac{0.06}{1}\right)^{5\cdot 1}\\\\\\A=26500\left(1.06\right)^{5}\\\\A=35462.9778064\\\\A=\$35462.98[/tex]
(b) Semi-annual compounding: n = 2
[tex]A=26500\left(1+\dfrac{0.06}{2}\right)^{5\cdot 2}\\\\\\A=26500\left(1.03\right)^{10}\\\\A=35,613.7840526...\\\\A=\$35613.78[/tex]
(c) Monthly compounding: n = 12
[tex]A=26500\left(1+\dfrac{0.06}{12}\right)^{5\cdot 12}\\\\\\A=26500\left(1.005\right)^{60}\\\\A=35,744.52904255...\\\\A=\$35744.53[/tex]
(d) Daily compounding: n = 365
[tex]A=26500\left(1+\dfrac{0.06}{365}\right)^{5\cdot 365}\\\\\\A=26500\left(1.00016438...\right)^{1825}\\\\A=35770.3764772...\\\\A=\$35770.38[/tex]