Math / Algebra

QuestionThe price of a condominium is $\$ 142,000$ . The bank requires a $5 \%$ down payment and one point at the time of closing. The cost of the condominium is financed with a 30 -year fixed-rate mortgage at $8 \%$ . Use the following formula to determine the regular payment amount. Complete parts (a) through (e) below. $P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$ \ $1,349 (Round to the nearest dollar as needed.) d. Find the monthly payment (excluding escrowed taxes and insurance).$ \ $990$ (Round to the nearest dollar as needed.) e. Find the total cost of interest over 30 years. \\square$ (Round to the nearest dollar as needed.)

Studdy Solution

STEP 1

What is this asking? We need to figure out the total interest paid over 30 years for a condo mortgage, given the price, down payment, interest rate, and loan term. Watch out! Don't forget about the down payment – we're not borrowing the entire price!
Also, remember that "one point" is an upfront fee, and it's a percentage of the loan amount, not the condo price.

STEP 2

1. Calculate Loan Amount
2. Calculate Monthly Payment
3. Calculate Total Payments
4. Calculate Total Interest

STEP 3

The down payment is $5\%$ of the condo price, which is $\$142,000$ .
So, the down payment is $0.05 \cdot \$142,000 = \$7,100$ .

STEP 4

The loan amount before points is the condo price minus the down payment: $\$142,000 - \$7,100 = \$134,900$ .

STEP 5

One point is $1\%$ of the loan amount (before points): $0.01 \cdot \$134,900 = \$1,349$ .

STEP 6

The final loan amount is the loan amount before points plus the points cost: $\$134,900 + \$1,349 = \$136,249$ .

STEP 7

$P = \$136,249$ (Loan Amount), $r = 0.08$ (Annual Interest Rate), $n = 12$ (Number of Payments per Year), $t = 30$ (Number of Years).

STEP 8

$PMT = \frac{P \cdot \left(\frac{r}{n}\right)}{\left[1 - \left(1 + \frac{r}{n}\right)^{-n \cdot t}\right]}$ $PMT = \frac{\$136,249 \cdot \left(\frac{0.08}{12}\right)}{\left[1 - \left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30}\right]}$

STEP 9

$\$136,249 \cdot \left(\frac{0.08}{12}\right) \approx \$908.326666...$ .

STEP 10

$\left(1 + \frac{0.08}{12}\right) \approx 1.006666...$ $\left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30} \approx 0.09167...$ $1 - \left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30} \approx 0.9083...$

STEP 11

$PMT \approx \frac{\$908.326666...}{0.9083...} \approx \$999.996 \approx \$1,000$ (Rounded to the nearest dollar).

STEP 12

Total payments = Monthly Payment $\cdot$ Number of Years $\cdot$ Payments per Year Total payments = $\$1,000 \cdot 30 \cdot 12 = \$360,000$ .

STEP 13

Total interest = Total payments - Loan amount Total interest = $\$360,000 - \$136,249 = \$223,751$ .

STEP 14

The total cost of interest over 30 years is $\$223,751$ .

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount
2. Calculate Monthly Payment
3. Calculate Total Payments
4. Calculate Total Interest

STEP 3

The down payment is $5\%$ of the condo price, which is $\$142,000$ .
So, the down payment is $0.05 \cdot \$142,000 = \$7,100$ .

STEP 4

The loan amount before points is the condo price minus the down payment: $\$142,000 - \$7,100 = \$134,900$ .

STEP 5

One point is $1\%$ of the loan amount (before points): $0.01 \cdot \$134,900 = \$1,349$ .

STEP 6

The final loan amount is the loan amount before points plus the points cost: $\$134,900 + \$1,349 = \$136,249$ .

STEP 7

$P = \$136,249$ (Loan Amount), $r = 0.08$ (Annual Interest Rate), $n = 12$ (Number of Payments per Year), $t = 30$ (Number of Years).

STEP 8

$PMT = \frac{P \cdot \left(\frac{r}{n}\right)}{\left[1 - \left(1 + \frac{r}{n}\right)^{-n \cdot t}\right]}$ $PMT = \frac{\$136,249 \cdot \left(\frac{0.08}{12}\right)}{\left[1 - \left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30}\right]}$

STEP 9

$\$136,249 \cdot \left(\frac{0.08}{12}\right) \approx \$908.326666...$ .

STEP 10

$\left(1 + \frac{0.08}{12}\right) \approx 1.006666...$ $\left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30} \approx 0.09167...$ $1 - \left(1 + \frac{0.08}{12}\right)^{-12 \cdot 30} \approx 0.9083...$

STEP 11

$PMT \approx \frac{\$908.326666...}{0.9083...} \approx \$999.996 \approx \$1,000$ (Rounded to the nearest dollar).

STEP 12

Total payments = Monthly Payment $\cdot$ Number of Years $\cdot$ Payments per Year Total payments = $\$1,000 \cdot 30 \cdot 12 = \$360,000$ .

STEP 13

Total interest = Total payments - Loan amount Total interest = $\$360,000 - \$136,249 = \$223,751$ .

STEP 14

The total cost of interest over 30 years is $\$223,751$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount2. Calculate Monthly Payment3. Calculate Total Payments4. Calculate Total Interest

STEP 3

The down payment is 5%5\%5% of the condo price, which is $142,000\$142,000$142,000.So, the down payment is 0.05⋅$142,000=$7,1000.05 \cdot \$142,000 = \$7,1000.05⋅$142,000=$7,100.

STEP 4

The loan amount before points is the condo price minus the down payment: $142,000−$7,100=$134,900\$142,000 - \$7,100 = \$134,900$142,000−$7,100=$134,900.

STEP 5

One point is 1%1\%1% of the loan amount (before points): 0.01⋅$134,900=$1,3490.01 \cdot \$134,900 = \$1,3490.01⋅$134,900=$1,349.

STEP 6

The final loan amount is the loan amount before points plus the points cost: $134,900+$1,349=$136,249\$134,900 + \$1,349 = \$136,249$134,900+$1,349=$136,249.

STEP 7

P=$136,249P = \$136,249P=$136,249 (**Loan Amount**), r=0.08r = 0.08r=0.08 (**Annual Interest Rate**), n=12n = 12n=12 (**Number of Payments per Year**), t=30t = 30t=30 (**Number of Years**).

STEP 8

STEP 9

$136,249⋅(0.0812)≈$908.326666...\$136,249 \cdot \left(\frac{0.08}{12}\right) \approx \$908.326666...$136,249⋅(120.08​)≈$908.326666....

STEP 10

STEP 11

PMT≈$908.326666...0.9083...≈$999.996≈$1,000PMT \approx \frac{\$908.326666...}{0.9083...} \approx \$999.996 \approx \$1,000PMT≈0.9083...$908.326666...​≈$999.996≈$1,000 (**Rounded to the nearest dollar**).

STEP 12

Total payments = Monthly Payment ⋅\cdot⋅ Number of Years ⋅\cdot⋅ Payments per Year Total payments = $1,000⋅30⋅12=$360,000\$1,000 \cdot 30 \cdot 12 = \$360,000$1,000⋅30⋅12=$360,000.

STEP 13

Total interest = Total payments - Loan amount Total interest = $360,000−$136,249=$223,751\$360,000 - \$136,249 = \$223,751$360,000−$136,249=$223,751.

STEP 14

The total cost of interest over 30 years is $223,751\$223,751$223,751.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate Loan Amount
2. Calculate Monthly Payment
3. Calculate Total Payments
4. Calculate Total Interest

The down payment is $5\%$ of the condo price, which is $\$142,000$ .
So, the down payment is $0.05 \cdot \$142,000 = \$7,100$ .

The loan amount before points is the condo price minus the down payment: $\$142,000 - \$7,100 = \$134,900$ .

One point is $1\%$ of the loan amount (before points): $0.01 \cdot \$134,900 = \$1,349$ .

The final loan amount is the loan amount before points plus the points cost: $\$134,900 + \$1,349 = \$136,249$ .

$P = \$136,249$ (Loan Amount), $r = 0.08$ (Annual Interest Rate), $n = 12$ (Number of Payments per Year), $t = 30$ (Number of Years).

$\$136,249 \cdot \left(\frac{0.08}{12}\right) \approx \$908.326666...$ .

$PMT \approx \frac{\$908.326666...}{0.9083...} \approx \$999.996 \approx \$1,000$ (Rounded to the nearest dollar).

Total payments = Monthly Payment $\cdot$ Number of Years $\cdot$ Payments per Year Total payments = $\$1,000 \cdot 30 \cdot 12 = \$360,000$ .

Total interest = Total payments - Loan amount Total interest = $\$360,000 - \$136,249 = \$223,751$ .

The total cost of interest over 30 years is $\$223,751$ .