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Questionwebassign.net/web/Student/Assignment-Rosponses/submit?dep=348685768tags=autosave\#question5304511_0 Relsunch to update ： PRACTICE ANOTHER
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TANAPMATH7 4.3.025. The Taylors have purchased a $340,000\$ 340,000 house. They made an initial down payment of $30,000\$ 30,000 and secured a mortgage with interest charged at the rate of 6%/6 \% / year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over 30 years, what monthly payment will the Taylors be required to make? (Round your answer to the nearest cent.) \ \squareWhatistheirequity(disregardingappreciation)after5years?After10years?After20years?(Roundyouranswerstothenearestcent.)5years$ What is their equity (disregarding appreciation) after 5 years? After 10 years? After 20 years? (Round your answers to the nearest cent.) 5 years \$ \square10years$ 10 years \$ \square20years$ 20 years \$ \square$ Need Help? Readit Watch II

Studdy Solution

STEP 1

1. The Taylors purchased a house for \$340,000.
2. They made an initial down payment of \$30,000.
3. The mortgage interest rate is 6% per year, compounded monthly.
4. The loan is to be amortized over 30 years.
5. We need to calculate the monthly payment and the equity after 5, 10, and 20 years.

STEP 2

1. Calculate the principal amount of the mortgage.
2. Determine the monthly interest rate.
3. Calculate the monthly payment using the amortization formula.
4. Calculate the remaining balance after 5, 10, and 20 years.
5. Determine the equity after 5, 10, and 20 years.

STEP 3

Calculate the principal amount of the mortgage.
The principal amount is the total cost of the house minus the down payment:
Principal=$340,000−$30,000=$310,000 \text{Principal} = \$340,000 - \$30,000 = \$310,000

STEP 4

Determine the monthly interest rate.
The annual interest rate is 6%, so the monthly interest rate is:
Monthly interest rate=6%12=0.5%=0.005 \text{Monthly interest rate} = \frac{6\%}{12} = 0.5\% = 0.005

STEP 5

Calculate the monthly payment using the amortization formula.
The formula for the monthly payment M M is:
M=Pr(1+r)n(1+r)n−1 M = P \frac{r(1+r)^n}{(1+r)^n - 1}
where P P is the principal, r r is the monthly interest rate, and n n is the total number of payments (months).
For this problem:
P=310,000,r=0.005,n=30×12=360 P = 310,000, \quad r = 0.005, \quad n = 30 \times 12 = 360
Substitute these values into the formula:
M=310,0000.005(1+0.005)360(1+0.005)360−1 M = 310,000 \frac{0.005(1+0.005)^{360}}{(1+0.005)^{360} - 1}
Calculate M M to find the monthly payment.

STEP 6

Calculate the monthly payment M M .
M≈310,0000.005×6.0225755.022575 M \approx 310,000 \frac{0.005 \times 6.022575}{5.022575}
M≈310,000×0.005×1.197 M \approx 310,000 \times 0.005 \times 1.197
M≈1860.70 M \approx 1860.70
The monthly payment is approximately:
\$ \boxed{1860.70}

STEP 7

Calculate the remaining balance after 5, 10, and 20 years.
Use the formula for the remaining balance B B after t t years:
B=P(1+r)n−M(1+r)n−1r B = P(1+r)^n - M \frac{(1+r)^n - 1}{r}
Calculate for 5, 10, and 20 years:
For 5 years (n=60 n = 60 ):
B5=310,000(1+0.005)60−1860.70(1+0.005)60−10.005 B_5 = 310,000(1+0.005)^{60} - 1860.70 \frac{(1+0.005)^{60} - 1}{0.005}
For 10 years (n=120 n = 120 ):
B10=310,000(1+0.005)120−1860.70(1+0.005)120−10.005 B_{10} = 310,000(1+0.005)^{120} - 1860.70 \frac{(1+0.005)^{120} - 1}{0.005}
For 20 years (n=240 n = 240 ):
B20=310,000(1+0.005)240−1860.70(1+0.005)240−10.005 B_{20} = 310,000(1+0.005)^{240} - 1860.70 \frac{(1+0.005)^{240} - 1}{0.005}

STEP 8

Calculate the remaining balances.
Calculate B5 B_5 , B10 B_{10} , and B20 B_{20} using the formulas above.

STEP 9

Determine the equity after 5, 10, and 20 years.
Equity is the original loan amount minus the remaining balance.
For 5 years:
Equity5=310,000−B5 \text{Equity}_5 = 310,000 - B_5
For 10 years:
Equity10=310,000−B10 \text{Equity}_{10} = 310,000 - B_{10}
For 20 years:
Equity20=310,000−B20 \text{Equity}_{20} = 310,000 - B_{20}
Calculate these values to find the equity.

STEP 10

Calculate the equity values.
Calculate the equity after 5, 10, and 20 years using the formulas above.

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