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Math  /  Algebra

QuestionUse Desmos.com to determine the monthly mortgage payment on a $500,000\$ 500,000 home with a 30 -year mortgage with 3%3 \% annual interest compounded monthly.

Studdy Solution

STEP 1

1. We are calculating the monthly mortgage payment for a fixed-rate mortgage.
2. The principal amount (loan amount) is P=$500,000 P = \$500,000 .
3. The annual interest rate is r=3% r = 3\% .
4. The mortgage term is n=30 n = 30 years.
5. The interest rate is compounded monthly.

STEP 2

1. Determine the monthly interest rate.
2. Calculate the total number of monthly payments.
3. Use the mortgage payment formula to find the monthly payment.
4. Simplify and compute the final value.

STEP 3

Determine the monthly interest rate by dividing the annual interest rate by 12.
rmonthly=3%12=0.0312=0.0025 r_{\text{monthly}} = \frac{3\%}{12} = \frac{0.03}{12} = 0.0025

STEP 4

Calculate the total number of monthly payments. This is the number of years multiplied by 12.
nmonthly=30×12=360 n_{\text{monthly}} = 30 \times 12 = 360

STEP 5

Use the mortgage payment formula to find the monthly payment M M :
M=Prmonthly(1+rmonthly)nmonthly(1+rmonthly)nmonthly−1 M = P \frac{r_{\text{monthly}} (1 + r_{\text{monthly}})^{n_{\text{monthly}}}}{(1 + r_{\text{monthly}})^{n_{\text{monthly}}} - 1}

STEP 6

Substitute the known values P=500,000 P = 500,000 , rmonthly=0.0025 r_{\text{monthly}} = 0.0025 , and nmonthly=360 n_{\text{monthly}} = 360 into the formula:
M=500,0000.0025(1+0.0025)360(1+0.0025)360−1 M = 500,000 \frac{0.0025 (1 + 0.0025)^{360}}{(1 + 0.0025)^{360} - 1}

STEP 7

Simplify and compute the final value using the values of the exponential terms.
First, calculate (1+0.0025)360 (1 + 0.0025)^{360} :
(1+0.0025)360≈2.4273 (1 + 0.0025)^{360} \approx 2.4273
Then substitute this back into the formula:
M=500,0000.0025⋅2.42732.4273−1 M = 500,000 \frac{0.0025 \cdot 2.4273}{2.4273 - 1}
M=500,0000.006068251.4273 M = 500,000 \frac{0.00606825}{1.4273}
M≈500,000×0.004251 M \approx 500,000 \times 0.004251
M≈2125.50 M \approx 2125.50
Therefore, the monthly mortgage payment is approximately $2125.50 \$2125.50 .

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