You can afford a \( \$ 450 \) per month car payment. You've found a 4 year loan at \( 5 \% \) interest. How big of a loan can you afford? Question Help: \( D \) Video 1 Video 2 Post to forum Submit Question

To determine how big of a loan you can afford with a monthly payment of \( \$450 \), a loan term of 4 years, and an interest rate of \( 5\% \), we can use the formula for the monthly payment of an amortizing loan: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) is the monthly payment - \( P \) is the loan principal (the amount of the loan) - \( r \) is the monthly interest rate (annual interest rate divided by 12) - \( n \) is the number of payments (loan term in months) ### Step 1: Identify the known values - Monthly payment \( M = 450 \) - Annual interest rate \( = 5\% \) or \( 0.05 \) - Monthly interest rate \( r = \frac{0.05}{12} \) - Loan term \( = 4 \) years \( = 4 \times 12 = 48 \) months ### Step 2: Calculate the monthly interest rate \[ r = \frac{0.05}{12} = \frac{0.05}{12} \approx 0.00416667 \] ### Step 3: Substitute the known values into the formula and solve for \( P \) Rearranging the formula to solve for \( P \): \[ P = M \frac{(1 + r)^n - 1}{r(1 + r)^n} \] Now we can substitute the values into the formula: - \( M = 450 \) - \( r \approx 0.00416667 \) - \( n = 48 \) Let's calculate \( P \). ### Step 4: Calculate \( P \) We will perform the calculation now. First, we need to calculate \( (1 + r)^n \): \[ (1 + r)^n = (1 + 0.00416667)^{48} \] Then we can substitute everything back into the formula for \( P \). Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\left(1+0.00416667\right)^{48}\) - step1: Add the numbers: \(1.00416667^{48}\) - step2: Convert the expressions: \(\left(\frac{100416667}{100000000}\right)^{48}\) - step3: Use the properties of exponents: \(\frac{100416667^{48}}{100000000^{48}}\) Calculate or simplify the expression \( 450 * ((1 + 0.00416667)^48 - 1) / (0.00416667 * (1 + 0.00416667)^48) \). Calculate the value by following steps: - step0: Calculate: \(\frac{450\left(\left(1+0.00416667\right)^{48}-1\right)}{\left(0.00416667\left(1+0.00416667\right)^{48}\right)}\) - step1: Remove the parentheses: \(\frac{450\left(\left(1+0.00416667\right)^{48}-1\right)}{0.00416667\left(1+0.00416667\right)^{48}}\) - step2: Add the numbers: \(\frac{450\left(\left(1+0.00416667\right)^{48}-1\right)}{0.00416667\times 1.00416667^{48}}\) - step3: Add the numbers: \(\frac{450\left(1.00416667^{48}-1\right)}{0.00416667\times 1.00416667^{48}}\) - step4: Convert the expressions: \(\frac{450\left(\left(\frac{100416667}{100000000}\right)^{48}-1\right)}{0.00416667\times 1.00416667^{48}}\) - step5: Subtract the numbers: \(\frac{450\times \frac{100416667^{48}-100000000^{48}}{100000000^{48}}}{0.00416667\times 1.00416667^{48}}\) - step6: Convert the expressions: \(\frac{450\times \frac{100416667^{48}-100000000^{48}}{100000000^{48}}}{0.00416667\left(\frac{100416667}{100000000}\right)^{48}}\) - step7: Multiply the numbers: \(\frac{\frac{9\times 100416667^{48}-9\times 100000000^{48}}{50^{191}\times 16^{48}}}{0.00416667\left(\frac{100416667}{100000000}\right)^{48}}\) - step8: Multiply the numbers: \(\frac{\frac{9\times 100416667^{48}-9\times 100000000^{48}}{50^{191}\times 16^{48}}}{\frac{416667\times 100416667^{48}}{100000000^{49}}}\) - step9: Multiply by the reciprocal: \(\frac{9\times 100416667^{48}-9\times 100000000^{48}}{50^{191}\times 16^{48}}\times \frac{100000000^{49}}{416667\times 100416667^{48}}\) - step10: Rewrite the expression: \(\frac{3\left(3\times 100416667^{48}-3\times 100000000^{48}\right)}{50^{191}\times 16^{48}}\times \frac{100000000^{49}}{416667\times 100416667^{48}}\) - step11: Reduce the numbers: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{50^{191}\times 16^{48}}\times \frac{100000000^{49}}{138889\times 100416667^{48}}\) - step12: Rewrite the expression: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{50^{191}\times 16^{48}}\times \frac{6250000^{49}\times 16^{49}}{138889\times 100416667^{48}}\) - step13: Rewrite the expression: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{50^{191}\times 16^{48}}\times \frac{50^{196}\times 16^{49}}{138889\times 100416667^{48}}\) - step14: Reduce the numbers: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{16^{48}}\times \frac{50^{5}\times 16^{49}}{138889\times 100416667^{48}}\) - step15: Rewrite the expression: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{16^{48}}\times \frac{2^{5}\times 25^{5}\times 16^{49}}{138889\times 100416667^{48}}\) - step16: Rewrite the expression: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{2^{192}}\times \frac{2^{5}\times 25^{5}\times 16^{49}}{138889\times 100416667^{48}}\) - step17: Reduce the numbers: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{2^{187}}\times \frac{25^{5}\times 16^{49}}{138889\times 100416667^{48}}\) - step18: Rewrite the expression: \(\frac{3\times 100416667^{48}-3\times 100000000^{48}}{2^{187}}\times \frac{25^{5}\times 2^{196}}{138889\times 100416667^{48}}\) - step19: Reduce the numbers: \(\left(3\times 100416667^{48}-3\times 100000000^{48}\right)\times \frac{25^{5}\times 2^{9}}{138889\times 100416667^{48}}\) - step20: Multiply the fractions: \(\frac{\left(3\times 100416667^{48}-3\times 100000000^{48}\right)\times 512\times 25^{5}}{138889\times 100416667^{48}}\) - step21: Multiply: \(\frac{1536\times 100416667^{48}\times 25^{5}-1536\times 100000000^{48}\times 25^{5}}{138889\times 100416667^{48}}\) ### Step 5: Calculate \( (1 + r)^n \) From our calculations, we found: \[ (1 + r)^n \approx 1.220896 \] ### Step 6: Calculate \( P \) Now substituting this value back into the formula for \( P \): \[ P \approx 450 \cdot \frac{(1.220896 - 1)}{0.00416667 \cdot 1.220896} \] Calculating the numerator: \[ 1.220896 - 1 \approx 0.220896 \] Now substituting this into the formula: \[ P \approx 450 \cdot \frac{0.220896}{0.00416667 \cdot 1.220896} \] Calculating the denominator: \[ 0.00416667 \cdot 1.220896 \approx 0.005080 \] Now substituting this back into the formula for \( P \): \[ P \approx 450 \cdot \frac{0.220896}{0.005080} \] Let's calculate \( P \). Calculate the value by following steps: - step0: Calculate: \(450\left(\frac{0.220896}{0.00508}\right)\) - step1: Divide the terms: \(450\times \frac{27612}{635}\) - step2: Reduce the numbers: \(90\times \frac{27612}{127}\) - step3: Multiply: \(\frac{90\times 27612}{127}\) - step4: Multiply: \(\frac{2485080}{127}\) ### Final Calculation of \( P \) After performing the calculations, we find: \[ P \approx 19567.56 \] ### Conclusion You can afford a loan of approximately **\$19,567.56** with a monthly payment of \$450 over 4 years at an interest rate of 5%.