Li Ting deposited $$ \$ 15000 $$ in an account that pays $$ 4.12 \% $$ per annum compound interest. Find the total amount in the account after 3 years if the interest is compounded $$ \begin{array}{ll} ext { (a) monthly, } & ext { (b) half-yearly. }\end{array} $$

Question

Li Ting deposited $$ \$ 15000 $$ in an account that pays $$ 4.12 \% $$ per annum compound interest. Find the total amount in the account after 3 years if the interest is compounded $$ \begin{array}{ll}	ext { (a) monthly, } & 	ext { (b) half-yearly. }\end{array} $$

UpStudy AI · Accepted Answer

$$
\textbf{Step 1. Define the variables:}
$$
Let the principal be $$ P = 15000 $$, the annual interest rate be $$ r = 0.0412 $$ (in decimal form), and the time period be $$ t = 3 $$ years.

$$
\textbf{Step 2. Compound Interest Formula:}
$$
The amount $$ A $$ after $$ t $$ years with $$ n $$ compounding periods per year is given by
$$
A = P \left(1 + \frac{r}{n}\right)^{n t}.
$$

$$
\textbf{(a) Compounded Monthly:}
$$
For monthly compounding, $$ n = 12 $$. Substitute the values into the formula:
$$
A = 15000 \left(1 + \frac{0.0412}{12}\right)^{12 \times 3} = 15000 \left(1 + \frac{0.0412}{12}\right)^{36}.
$$
First, compute the monthly interest rate:
$$
\frac{0.0412}{12} \approx 0.0034333.
$$
Thus, the formula becomes
$$
A = 15000 \left(1.0034333\right)^{36}.
$$
Taking natural logarithms or using a calculator, we compute:
$$
\left(1.0034333\right)^{36} \approx 1.1313.
$$
Then, the total amount is approximately:
$$
A \approx 15000 \times 1.1313 \approx 16969.50.
$$

$$
\textbf{(b) Compounded Half-Yearly:}
$$
For half-yearly compounding, $$ n = 2 $$. Substitute the values into the formula:
$$
A = 15000 \left(1 + \frac{0.0412}{2}\right)^{2 \times 3} = 15000 \left(1 + \frac{0.0412}{2}\right)^{6}.
$$
First, compute the half-yearly interest rate:
$$
\frac{0.0412}{2} = 0.0206.
$$
Thus, the formula becomes
$$
A = 15000 \left(1.0206\right)^{6}.
$$
Computing the factor:
$$
\left(1.0206\right)^{6} \approx 1.1303.
$$
Then, the total amount is approximately:
$$
A \approx 15000 \times 1.1303 \approx 16954.50.
$$

$$
\textbf{Final Answers:}
$$
$$
\text{(a) Monthly Compounding: } A \approx \$16,\!969.50.
$$
$$
\text{(b) Half-Yearly Compounding: } A \approx \$16,\!954.50.
$$
After 3 years: - **Monthly Compounding:** \$16,969.50 - **Half-Yearly Compounding:** \$16,954.50

Li Ting deposited in an account that pays
per annum compound interest. Find the
total amount in the account after 3 years if the
interest is compounded

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Solución

The Deep Dive

(a) Monthly Compounding

(b) Half-Yearly Compounding

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Li Ting deposited in an account that pays per annum compound interest. Find the total amount in the account after 3 years if the interest is compounded