This question aims to find the total finance charges after **15** years at a given annual interest rate of **11%**.

A finance charge is the total cost charged due to borrowing an amount, which is the amount to be paid as interest over the whole period of the mortgage taken. It includes the interest rates and also other charges. It is calculated by removing the debt amount from the payment to be made over the given period.

**Expert Answer**

The principal amount **P** that is the mortgage is $\$48000$. The annual payment is calculated by the formula given below:

\[ M = P \dfrac{(r (1+r)^n )}{(1+r)^n -1} \]

Here,

\[ M = \text{Annual payment} \]

\[ P = Mortgage \]

\[ r = \text{annual interest rate} \]

\[ n = \text{number of payments to be made} \]

$r$ is converted into decimal by dividing it by $100$.

\[ M = 48000 \times \dfrac{(0.11(1+0.11)^{15})}{(1+0.11)^{15} – 1} \]

\[ M = 48000 \times \dfrac{(0.11*(1.11)^{15})}{(1.11)^{15} – 1} \]

\[ M = 48000 \dfrac{(0.11*4.784)}{4.784 – 1} \]

\[ M = \$ 6675.13 \]

**The annual payment is $\$6675.13$.**

To find the total finance charge, the **total mortgage** for **15 years** needs to be calculated. This can be done by multiplying the annual payment by several payments to be made.

\[ \text{Total payment required} = \text{Annual Payment} \times n \]

\[ \text{Total payment required} = 6675.13 \times 15 = \$ 100126.97 \]

Hence,

The **finance charge** will be calculated by subtracting the mortgage value from the **total payment for 15 years**.

\[ \text{Finance charge} = \$ 100126.97 – \$ 48000 = \$ 52126.97 \]

**Numerical Result**

The formula to find the annual payment is given below:

\[ M = P \dfrac{(r (1+r)^n )}{(1+r)^n} – 1 \]

Putting the value of mortgage $P= 48000$, interest rate $r=11\%$, and time period $n=15$ years in the above equation, we get:

\[ M = 48000 \dfrac{(0.11(1+0.11)^{15})}{(1+0.11)^{15} – 1} \]

\[ M = \$ 6675.13 \]

\[ \text{Total Payment required} = \text{Annual Payment} \times n \]

\[ = \$ 6675.13 \times 15 = \$ 100126.97 \]

\[ \text{Finance Charge} = \$ 100126.97 – \$48000 = \$ 52126.97 \]

**The total finance charge for $15$ years with a loan of $ \$ 48000$ at an $11\%$ interest rate is $ \$ 52126.972$.**

**Example:**

Given a mortgage of $\$6,000$ for $3$ years with a rate of $5\%$, what are the total finance charges?

\[ M = P \dfrac{ (r (1+r)^n )}{ (1+r)^n – 1} \]

Putting the value of mortgage $\$6000$, interest rate $5\%$, and time period $3$ years in the above equation:

\[ M = 6000\dfrac{(0.05(1+0.05)^3)}{ (1+0.05)^3 – 1} \]

\[ M = \$ 2,778,300 \]

\[ \text{Total Payment required} =\text{Annual Payment} \times n \]

\[ = \$ 2,778,300 \times 3 = \$ 8,334,900 \]

\[ \text{Finance Charge} = \$ 8,334,900 – \$ 6000 = \$ 8,328,900 \]

**The total finance charge for $3$ years with a loan of $\$6000$ at a $5\%$ interest rate is $\$8,328,900$.**

*Image/Mathematical drawings are created in Geogebra*