Suppose you want to borrow ​$110 comma 000 and you find a bank offering a​ 20-year term for a loan of that​ amount, with an APR of 4​%.
a. Find your regular payments if you make them​ yearly, monthly, biweekly​ (every 2​ weeks), or weeklylong dashthat ​is, for nequals​1, ​12, 26, 52.
b. Compute the total payment for each case in part​ (a).
c. Compare the total payments computed in part​ (b). b. For nequals​1, the total amount paid is ​$
  
enter your response here​, for nequals​12, the total amount paid is ​$
  
enter your response here​, for nequals​26, the total amount paid is ​$
  
enter your response here​, and for nequals​52, the total amount paid is ​$
  
enter your response here.
​(Round to the nearest​ dollar.)

We will use the loan payment formula to solve this problem, which is based on the annuity formula:

\[
P = \dfrac{A \cdot r}{1 - (1 + r)^{-nt}}
\]

Where:
- $ P $ is the regular payment.
- $ A $ is the loan amount ($110,000).
- $ r $ is the interest rate per period.
- $ n $ is the number of payments per year (1 for yearly, 12 for monthly, 26 for biweekly, 52 for weekly).
- $ t $ is the term of the loan in years (20 years).
- $ APR $ is the annual interest rate (4% or 0.04).

### Part a: Finding Regular Payments

We need to compute the payments for $ n = 1 $, $ n = 12 $, $ n = 26 $, and $ n = 52 $. First, let's calculate the interest rate per period ($ r $) for each scenario:

\[
r_{\text{yearly}} = \dfrac{0.04}{1} = 0.04
\]
\[
r_{\text{monthly}} = \dfrac{0.04}{12} = 0.0033333
\]
\[
r_{\text{biweekly}} = \dfrac{0.04}{26} = 0.00153846
\]
\[
r_{\text{weekly}} = \dfrac{0.04}{52} = 0.00076923
\]

Now we can use the loan payment formula for each case.

I'll compute the payments for each case step-by-step.

### Part b: Total Payments
Once the periodic payments are calculated, we can find the total payment by multiplying the regular payment by the total number of payments over the loan's lifetime, which is $ n \times t $.

Let's calculate each case next.### Part a: Regular Payments
- Yearly payments ($ n = 1 $): $8,094
- Monthly payments ($ n = 12 $): $667
- Biweekly payments ($ n = 26 $): $307
- Weekly payments ($ n = 52 $): $154

### Part b: Total Payments
- For $ n = 1 $ (yearly), total amount paid: **$161,880**
- For $ n = 12 $ (monthly), total amount paid: **$159,979**
- For $ n = 26 $ (biweekly), total amount paid: **$159,885**
- For $ n = 52 $ (weekly), total amount paid: **$159,845**

### Part c: Comparison of Total Payments
The total amount paid decreases slightly as the payment frequency increases, with weekly payments resulting in the lowest total amount paid.

Do you need further details or have any questions?

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Here are five related questions for practice:
1. How would the payments change if the interest rate was higher, say 5%?
2. What would the payments be if the loan term was reduced to 15 years?
3. How does the frequency of payments affect the total interest paid over time?
4. If you wanted to borrow a higher amount, say $150,000, how would the payments change?
5. How would the payments differ if you opted for biweekly payments but a 30-year loan term?

**Tip**: Making more frequent payments typically reduces the total interest paid over the life of the loan.

Suppose you want to borrow $110,000 and you find a bank offering a​ 20-year term for a loan of that​ amount, with an APR of 4​%. a. Find your regular payments if you make them​ yearly, monthly, biweekly​ (every 2​ weeks), or weeklylong dashthat ​is, for nequals​1, ​12, 26, 52. b. Compute the total payment for each case in part​ (a). c. Compare the total payments computed in part​ (b). b. For nequals​1, the total amount paid is ​$ enter your response here​, for nequals​12, the total amount paid is ​$ enter your response here​, for nequals​26, the total amount paid is ​$ enter your response here​, and for nequals​52, the total amount paid is ​$ enter your response here. ​(Round to the nearest​ dollar.)

Learn how to calculate loan payments for a $110,000 loan with a 20-year term and a 4% APR. This problem involves calculating yearly, monthly, biweekly, and weekly payments, as well as comparing total payments. Suitable for students in Grades 10-12.

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