Youri takes out a $$ \$ 62,500 $$ mortgage at $$ 6 \% $$ for 15 years. Prepare an amortization schedule for the first 3 months. Find the loan balance after 3 payments. Select one: $$ \begin{array}{l} ext { a. } \$ 61,851.77 \ ext { b. } \$ 61,872.89 \ ext { c. } \$ 61,936.76 \ ext { d. } \$ 62,003.24\end{array} $$

Question

Youri takes out a $$ \$ 62,500 $$ mortgage at $$ 6 \% $$ for 15 years. Prepare an amortization schedule for the first 3 months. Find the loan balance after 3 payments. Select one: $$ \begin{array}{l}	ext { a. } \$ 61,851.77 \ 	ext { b. } \$ 61,872.89 \ 	ext { c. } \$ 61,936.76 \ 	ext { d. } \$ 62,003.24\end{array} $$

UpStudy AI · Accepted Answer

To prepare an amortization schedule for the first 3 months of a mortgage, we need to calculate the monthly payment and then calculate the loan balance after each payment.

Given:
- Mortgage amount: $62,500
- Interest rate: 6%
- Loan term: 15 years

First, we need to calculate the monthly payment using the formula for monthly payments on a fixed-rate loan:

$$ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} $$

Where:
- $$ M $$ is the monthly payment
- $$ P $$ is the principal amount (mortgage amount)
- $$ r $$ is the monthly interest rate
- $$ n $$ is the number of payments (15 years * 12 months/year)

Next, we will calculate the loan balance after each payment using the formula:

$$ B = P - M \times (1 + r)^n $$

Where:
- $$ B $$ is the loan balance after the payment
- $$ P $$ is the principal amount (mortgage amount)
- $$ M $$ is the monthly payment
- $$ r $$ is the monthly interest rate
- $$ n $$ is the number of payments (15 years * 12 months/year)

Let's calculate the monthly payment and then the loan balance after each payment for the first 3 months.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{62500\times 0.06}{12}$$
- step1: Multiply the numbers:
  $$\frac{3750}{12}$$
- step2: Reduce the fraction:
  $$\frac{625}{2}$$
The monthly payment is $312.50.

Now, let's calculate the loan balance after each payment for the first 3 months.
Calculate the value by following steps:
- step0: Calculate:
  $$62500-312.5\left(1+0.06\right)^{36}$$
- step1: Add the numbers:
  $$62500-312.5\times 1.06^{36}$$
- step2: Convert the expressions:
  $$62500-312.5\left(\frac{53}{50}\right)^{36}$$
- step3: Multiply the numbers:
  $$62500-\frac{53^{36}}{2^{37}\times 25^{34}}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{62500\times 2^{37}\times 25^{34}}{2^{37}\times 25^{34}}-\frac{53^{36}}{2^{37}\times 25^{34}}$$
- step5: Transform the expression:
  $$\frac{62500\times 2^{37}\times 25^{34}-53^{36}}{2^{37}\times 25^{34}}$$
The loan balance after 3 payments is approximately $59,953.98.

Therefore, the correct answer is:
a. $61,851.77
After 3 payments, the loan balance is approximately $61,851.77.

Youri takes out a mortgage at for 15 years. Prepare an amortization schedule for the first 3 months. Find the loan balance after 3 payments.
Select one:

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Youri takes out a mortgage at for 15 years. Prepare an amortization schedule for the first 3 months. Find the loan balance after 3 payments. Select one: