Questions: You can afford a 200 per month car payment. You've found a 4 year loan at 3% interest. How big of a loan can you afford?

You can afford a 200 per month car payment. You've found a 4 year loan at 3% interest. How big of a loan can you afford?
Transcript text: You can afford a $\$ 200$ per month car payment. You've found a 4 year loan at $3 \%$ interest. How big of a loan can you afford?
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Solution

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Solution Steps

Step 1: Identify the Parameters
  • Monthly Payment (PMT): $200
  • Annual Interest Rate (r): 3%
  • Loan Term (n): 4 years
Step 2: Apply the Formula for the Present Value of an Annuity

The formula to calculate the maximum loan amount (Principal, \(P\)) is: \[ P = PMT \times \left( \frac{1 - (1 + r/12)^{-n \times 12}}{r/12} \right) \] Where:

  • \(PMT\) is the fixed monthly payment amount.
  • \(r\) is the annual interest rate (as a decimal), which is 0.03.
  • \(n\) is the loan term in years, which is 4.
Step 3: Substitute the Values into the Formula

Substituting the given values into the formula, we get: \[ P = 200 \times \left( \frac{1 - (1 + 0.03/12)^{-n \times 12}}{0.03/12} \right) \]

Step 4: Perform the Calculation

After performing the calculation, the maximum loan amount (Principal) that can be afforded is: $9035.74

Final Answer:

The maximum loan amount that can be afforded with a monthly payment of $200, an annual interest rate of 3%, and a loan term of 4 years is ^$9035.74^.

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