We first need to determine the monthly payment, PMT, on the original mortgage.

$PMT = \frac{\text{Principal}}{\left( \frac{1 - (1 + r)^{-n}}{r} \right)}$

Where:

• PMT = amortized amount for each month
• r = monthly interest rate (annual rate / 12) = 0.06 / 12 = 0.005
• n = total number of months = 15 × 12 = 180

PMT = \frac{500{,}000}{\left( \frac{1 - (1 + 0.005)^{-180}}{0.005} \right)} \approx \`$4`{,}219.28

Step 2: Determine the outstanding balance after 10 years (120 payments)

After 10 years, there are 180 − 120 = 60 payments remaining. The outstanding balance is the present value of these remaining payments at the same monthly interest rate:

$\text{Outstanding Balance} = PMT \times \left( \frac{1 - (1 + r)^{-60}}{r} \right)$

\text{Outstanding Balance} = 4{,}219.28 \times \left( \frac{1 - (1 + 0.005)^{-60}}{0.005} \right) \approx \`$218`{,}245

Step 3: Amount received by the lender

When the borrower pays off the outstanding principal, the lender receives the remaining loan balance, which is approximately $218,245.