Let's break down the problem step by step: --- ### **Step 1: Find the Amount Lent (Principal, \( P \))** - Cost of machine = **$350,000** - Bank lends **80%** of purchase price: \[ P = 350,000 \times 0.8 = \$280,000 \] --- ### **Step 2: Calculate the Monthly Interest Rate (\( r \))** - Annual rate = **7% per year compounded monthly** - Monthly rate (\( r \)) = \( \frac{7\%}{12} = 0.07 / 12 = 0.0058333 \) --- ### **Step 3: Number of Monthly Payments (\( n \))** - Life = **10 years** - \( n = 10 \times 12 = 120 \) months --- ### **Step 4: Amortization with Balloon Payment** - Monthly payment (\( PMT \)) = **$2,000** - Final balloon payment at the end. The loan formula with a balloon payment is: \[ P = PMT \times \frac{1 - (1+r)^{-n}}{r} + \text{Balloon} \times (1+r)^{-n} \] We need to solve for the **Balloon payment**. \[ P - PMT \times \frac{1 - (1+r)^{-n}}{r} = \text{Balloon} \times (1+r)^{-n} \] \[ \text{Balloon} = \frac{P - PMT \times \frac{1 - (1+r)^{-n}}{r}}{(1+r)^{-n}} \] --- ### **Step 5: Plug in the Values** - \( P = 280,000 \) - \( PMT = 2,000 \) - \( r = 0.0058333 \) - \( n = 120 \) Calculate: #### **First, Calculate \( (1 + r)^{-n} \):** \[ (1+r)^{-n} = (1.0058333)^{-120} \] Let's compute \( (1.0058333)^{120} \) first: \[ (1.0058333)^{120} \approx e^{120 \times \ln(1.0058333)} \approx e^{120 \times 0.005816} \approx e^{0.698} \approx 2.009 \] So, \[ (1.0058333)^{-120} \approx \frac{1}{2.009} \approx 0.4976 \] #### **Next, Calculate \( \frac{1 - (1+r)^{-n}}{r} \):** \[ 1 - 0.4976 = 0.5024 \] \[ \frac{0.5024}{0.0058333} \approx 86.14 \] #### **Multiply by Monthly Payment:** \[ 2,000 \times 86.14 = 172,280 \] #### **Calculate the Present Value of Balloon Payment:** \[ \text{Balloon} = \frac{280,000 - 172,280}{0.4976} \] \[ = \frac{107,720}{0.4976} \] \[ = 216,536 \] --- ### **Step 6: Final Answer** **The balloon payment at the end of the period will be:** \[ \boxed{216,536} \] --- ### **Summary of Steps** 1. Find loan amount: $280,000 2. Find monthly rate: 0.0058333 3. Find number of payments: 120 4. Use amortization formula with balloon payment 5. Solve for balloon payment **Final Answer: $216,536**

Let's work through this problem carefully, step by step. --- ### **Given Data:** - **Cost of machine:** \$350,000 - **Bank loan percentage:** 80% of purchase price - **Annual interest rate:** 7% per annum (p.y.c.m. = per year compounded monthly) - **Loan term:** 10 years - **Monthly payment:** \$2,000 - **Balloon payment:** to be calculated --- ### **Step 1: Determine the principal amount (Loan amount, \( P \))** \[ P = 80\% \text{ of } \$350,000 = 0.8 \times 350,000 = \$280,000 \] --- ### **Step 2: Find the monthly interest rate (\( r \))** Annual interest rate = 7%, compounded monthly: \[ r = \frac{7\%}{12} = \frac{0.07}{12} \approx 0.0058333 \] --- ### **Step 3: Calculate total number of payments (\( n \))** \[ n = 10 \text{ years} \times 12 \text{ months/year} = 120 \text{ months} \] --- ### **Step 4: Use the amortization formula with balloon payment** The general formula for the present value of a loan with fixed monthly payments and a balloon payment at the end is: \[ P = \text{PV of annuity} + \text{Present value of balloon payment} \] Expressed mathematically: \[ P = PMT \times \frac{1 - (1 + r)^{-n}}{r} + \text{Balloon} \times (1 + r)^{-n} \] We are solving for **Balloon**, so: \[ \text{Balloon} = \frac{P - PMT \times \frac{1 - (1 + r)^{-n}}{r}}{(1 + r)^{-n}} \] --- ### **Step 5: Compute the necessary components** #### **(a) Calculate \( (1 + r)^{n} \) and \( (1 + r)^{-n} \):** \[ (1 + r)^{n} = (1.0058333)^{120} \] Calculate using logarithms: \[ \ln(1.0058333) \approx 0.005816 \] \[ \Rightarrow \ln((1.0058333)^{120}) = 120 \times 0.005816 = 0.69792 \] \[ (1.0058333)^{120} = e^{0.69792} \approx 2.009 \] Thus, \[ (1 + r)^{-n} = \frac{1}{2.009} \approx 0.4976 \] #### **(b) Calculate \( \frac{1 - (1 + r)^{-n}}{r} \):** \[ 1 - 0.4976 = 0.5024 \] \[ \frac{0.5024}{0.0058333} \approx 86.14 \] #### **(c) Calculate the present value of the annuity:** \[ PMT \times \frac{1 - (1 + r)^{-n}}{r} = 2000 \times 86.14 \approx \$172,280 \] --- ### **Step 6: Calculate the balloon payment** \[ \text{Balloon} = \frac{P - \text{PV of annuity}}{(1 + r)^{-n}} = \frac{280,000 - 172,280}{0.4976} \] \[ = \frac{107,720}{0.4976} \approx \$216,536 \] --- ### **Final Answer:** \[ \boxed{\$216,536} \] **This matches one of the options provided.** --- ### **Summary:** - Loan amount = \$280,000 - Monthly interest rate = 0.0058333 - Number of payments = 120 - Monthly payment = \$2,000 - Calculated balloon payment = **\$216,536** --- **Hence, the amount of the balloon payment is approximately \$216,536.**

The balloon payment is calculated by subtracting the present value of the total monthly payments from the initial loan amount, then adjusting for the discounting over the loan period. This ensures the remaining balance after all scheduled payments is correctly determined at the end of the loan term.