Math / Algebra

Question $\frac{P\left(\frac{r}{n}\right)}{\left.1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \text { to }$
Use PMT $=\frac{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}{[t o}$ determine the regular payment amount, rounded to the nearest dollar. The price of a small cabin is $\$ 40,000$ . The bank requires $5.5 \%$ down payment. The buyer is offered two mortgage options: 20 -year fixed at $8 \%$ or 30 -year fixed at $8 \%$ . Calculate the amount of interest paid for each opion. How much does the buyer save in interest with the 20-year option?
Find the monthly payment for the 20 -year option. \\square$ (Round to the nearest dollar as needed.)

Studdy Solution

STEP 1

What is this asking? We're comparing two mortgage options for a cabin, a 20-year loan and a 30-year loan, both at 8% interest, to see how much interest is saved with the shorter loan and what the monthly payment would be for that shorter loan. Watch out! Don't forget to calculate the loan amount after the down payment, not on the full cabin price!
Also, remember to round to the nearest dollar at the end.

STEP 2

1. Calculate Loan Amount
2. Calculate 20-Year Monthly Payment
3. Calculate Total 20-Year Interest
4. Calculate 30-Year Monthly Payment
5. Calculate Total 30-Year Interest
6. Calculate Interest Savings

STEP 3

Alright, let's start by figuring out the loan amount.
The cabin costs $\$40,000$ and the down payment is $5.5\%$ .

STEP 4

To find the down payment amount, we multiply the cabin price by the down payment percentage: $\$40,000 \cdot 0.055 = \$2,200$ .

STEP 5

Now, subtract the down payment from the cabin price to get the loan amount: $\$40,000 - \$2,200 = \$37,800$ .
This is the principal amount we'll use for our calculations.

STEP 6

Let's use the formula given: $PMT = \frac{P(\frac{r}{n})}{1 - (1 + \frac{r}{n})^{-nt}}$ .
Here, $P$ is the principal (\$37,800), $r$ is the annual interest rate (0.08), $n$ is the number of payments per year (12 for monthly payments), and $t$ is the loan term in years (20).

STEP 7

Plugging in the values, we get: $PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 20}}$ .

STEP 8

Let's simplify the expression inside the parentheses: $1 + \frac{0.08}{12} \approx 1.0066667$ .

STEP 9

Now, calculate the exponent: $-12 \cdot 20 = -240$ .
So, $(1.0066667)^{-240} \approx 0.202765$ .

STEP 10

Next, we have $1 - 0.202765 \approx 0.797235$ .

STEP 11

In the numerator, $\frac{0.08}{12} \approx 0.0066667$ , and $\$37,800 \cdot 0.0066667 \approx \$252$ .

STEP 12

Finally, divide the numerator by the denominator: $\frac{\$252}{0.797235} \approx \$316.10$ .
Rounded to the nearest dollar, the monthly payment for the 20-year loan is $\$316$ .

STEP 13

The total amount paid over 20 years is the monthly payment multiplied by the total number of payments: $\$316 \cdot 12 \cdot 20 = \$75,840$ .

STEP 14

The total interest paid is the total amount paid minus the principal: $\$75,840 - \$37,800 = \$38,040$ .

STEP 15

We use the same formula, but with $t = 30$ : $PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 30}}$ .

STEP 16

Following similar steps as before, we get $PMT \approx \$275.54$ , which rounds to $\$276$ .

STEP 17

Total amount paid: $\$276 \cdot 12 \cdot 30 = \$99,360$ .

STEP 18

Total interest paid: $\$99,360 - \$37,800 = \$61,560$ .

STEP 19

The interest savings with the 20-year option is the difference between the 30-year interest and the 20-year interest: $\$61,560 - \$38,040 = \$23,520$ .

STEP 20

20-year monthly payment: $\$316$ Interest saved with the 20-year option: $\$23,520$

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Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount2. Calculate 20-Year Monthly Payment3. Calculate Total 20-Year Interest4. Calculate 30-Year Monthly Payment5. Calculate Total 30-Year Interest6. Calculate Interest Savings

STEP 3

Alright, let's **start** by figuring out the loan amount.The cabin costs $40,000\$40,000$40,000 and the down payment is 5.5%5.5\%5.5%.

STEP 4

To find the down payment amount, we multiply the cabin price by the down payment percentage: $40,000⋅0.055=$2,200 \$40,000 \cdot 0.055 = \$2,200 $40,000⋅0.055=$2,200.

STEP 5

Now, subtract the down payment from the cabin price to get the loan amount: $40,000−$2,200=$37,800\$40,000 - \$2,200 = \$37,800$40,000−$2,200=$37,800.This is the **principal** amount we'll use for our calculations.

STEP 6

STEP 7

Plugging in the values, we get: PMT=$37,800(0.0812)1−(1+0.0812)−12⋅20PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 20}}PMT=1−(1+120.08​)−12⋅20$37,800(120.08​)​.

STEP 8

Let's simplify the expression inside the parentheses: 1+0.0812≈1.00666671 + \frac{0.08}{12} \approx 1.00666671+120.08​≈1.0066667.

STEP 9

Now, calculate the exponent: −12⋅20=−240-12 \cdot 20 = -240−12⋅20=−240.So, (1.0066667)−240≈0.202765(1.0066667)^{-240} \approx 0.202765(1.0066667)−240≈0.202765.

STEP 10

Next, we have 1−0.202765≈0.7972351 - 0.202765 \approx 0.7972351−0.202765≈0.797235.

STEP 11

In the numerator, 0.0812≈0.0066667\frac{0.08}{12} \approx 0.0066667120.08​≈0.0066667, and $37,800⋅0.0066667≈$252\$37,800 \cdot 0.0066667 \approx \$252$37,800⋅0.0066667≈$252.

STEP 12

Finally, divide the numerator by the denominator: $2520.797235≈$316.10\frac{\$252}{0.797235} \approx \$316.100.797235$252​≈$316.10.Rounded to the nearest dollar, the **monthly payment** for the 20-year loan is $316\$316$316.

STEP 13

The total amount paid over 20 years is the monthly payment multiplied by the total number of payments: $316⋅12⋅20=$75,840\$316 \cdot 12 \cdot 20 = \$75,840$316⋅12⋅20=$75,840.

STEP 14

The total interest paid is the total amount paid minus the principal: $75,840−$37,800=$38,040\$75,840 - \$37,800 = \$38,040$75,840−$37,800=$38,040.

STEP 15

We use the same formula, but with t=30t = 30t=30: PMT=$37,800(0.0812)1−(1+0.0812)−12⋅30PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 30}}PMT=1−(1+120.08​)−12⋅30$37,800(120.08​)​.

STEP 16

Following similar steps as before, we get PMT≈$275.54PMT \approx \$275.54PMT≈$275.54, which rounds to $276\$276$276.

STEP 17

Total amount paid: $276⋅12⋅30=$99,360\$276 \cdot 12 \cdot 30 = \$99,360$276⋅12⋅30=$99,360.

STEP 18

Total interest paid: $99,360−$37,800=$61,560\$99,360 - \$37,800 = \$61,560$99,360−$37,800=$61,560.

STEP 19

The interest savings with the 20-year option is the difference between the 30-year interest and the 20-year interest: $61,560−$38,040=$23,520\$61,560 - \$38,040 = \$23,520$61,560−$38,040=$23,520.

STEP 20

20-year monthly payment: $316\$316$316 Interest saved with the 20-year option: $23,520\$23,520$23,520

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1. Calculate Loan Amount
2. Calculate 20-Year Monthly Payment
3. Calculate Total 20-Year Interest
4. Calculate 30-Year Monthly Payment
5. Calculate Total 30-Year Interest
6. Calculate Interest Savings

Alright, let's start by figuring out the loan amount.
The cabin costs $\$40,000$ and the down payment is $5.5\%$ .

To find the down payment amount, we multiply the cabin price by the down payment percentage: $\$40,000 \cdot 0.055 = \$2,200$ .

Now, subtract the down payment from the cabin price to get the loan amount: $\$40,000 - \$2,200 = \$37,800$ .
This is the principal amount we'll use for our calculations.

Plugging in the values, we get: $PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 20}}$ .

Let's simplify the expression inside the parentheses: $1 + \frac{0.08}{12} \approx 1.0066667$ .

Now, calculate the exponent: $-12 \cdot 20 = -240$ .
So, $(1.0066667)^{-240} \approx 0.202765$ .

Next, we have $1 - 0.202765 \approx 0.797235$ .

In the numerator, $\frac{0.08}{12} \approx 0.0066667$ , and $\$37,800 \cdot 0.0066667 \approx \$252$ .

Finally, divide the numerator by the denominator: $\frac{\$252}{0.797235} \approx \$316.10$ .
Rounded to the nearest dollar, the monthly payment for the 20-year loan is $\$316$ .

The total amount paid over 20 years is the monthly payment multiplied by the total number of payments: $\$316 \cdot 12 \cdot 20 = \$75,840$ .

The total interest paid is the total amount paid minus the principal: $\$75,840 - \$37,800 = \$38,040$ .

We use the same formula, but with $t = 30$ : $PMT = \frac{\$37,800(\frac{0.08}{12})}{1 - (1 + \frac{0.08}{12})^{-12 \cdot 30}}$ .

Following similar steps as before, we get $PMT \approx \$275.54$ , which rounds to $\$276$ .

Total amount paid: $\$276 \cdot 12 \cdot 30 = \$99,360$ .

Total interest paid: $\$99,360 - \$37,800 = \$61,560$ .

The interest savings with the 20-year option is the difference between the 30-year interest and the 20-year interest: $\$61,560 - \$38,040 = \$23,520$ .

20-year monthly payment: $\$316$ Interest saved with the 20-year option: $\$23,520$