Questions: wants to buy a cabin that costs 75,000. The bank requires a 10% down payment. The rest is financed with a 15-year, fixed-rate mortgage at 3.5% annual interest with monthly payments. (a) Find the required down payment. (b) Find the amount of the mortgage (c) Find the monthly payment

wants to buy a cabin that costs 75,000. The bank requires a 10% down payment. The rest is financed with a 15-year, fixed-rate mortgage at 3.5% annual interest with monthly payments. (a) Find the required down payment. (b) Find the amount of the mortgage (c) Find the monthly payment
Transcript text: wants to buy a cabin that costs $\$ 75,000$. The bank requires a $10 \%$ down payment. The rest is financed with a 15 -year, fixed-rate mortgage at $3.5 \%$ al interest with monthly payments. (a) Find the required down payment. $\$$ ～ (b) Find the amount of the mortgage \$ $]$ (c) Find the monthly payment \$ $]$
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Solution

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Solution Steps

Solution Approach
  1. Find the required down payment: Calculate 10% of the total cost of the cabin.
  2. Find the amount of the mortgage: Subtract the down payment from the total cost of the cabin.
  3. Find the monthly payment: Use the formula for monthly mortgage payments, which involves the principal amount (mortgage), the monthly interest rate, and the number of payments (months).
Step 1: Calculate the Down Payment

The down payment is calculated as \(10\%\) of the total cost of the cabin: \[ \text{Down Payment} = 0.10 \times 75000 = 7500 \]

Step 2: Calculate the Mortgage Amount

The mortgage amount is the total cost of the cabin minus the down payment: \[ \text{Mortgage Amount} = 75000 - 7500 = 67500 \]

Step 3: Calculate the Monthly Payment

The monthly payment is calculated using the formula for monthly mortgage payments: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where:

  • \(P\) is the principal amount (mortgage amount),
  • \(r\) is the monthly interest rate,
  • \(n\) is the number of payments (months).

Given: \[ P = 67500, \quad r = \frac{0.035}{12} \approx 0.00291667, \quad n = 15 \times 12 = 180 \]

Substituting these values into the formula: \[ M = 67500 \frac{0.00291667(1+0.00291667)^{180}}{(1+0.00291667)^{180} - 1} \approx 482.5457 \]

Final Answer

\[ \boxed{\text{Down Payment} = \$7500.00} \] \[ \boxed{\text{Mortgage Amount} = \$67500.00} \] \[ \boxed{\text{Monthly Payment} = \$482.55} \]

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