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Problem Summary:

1. Given Data:

   • Credit card balance: $6200
   • Annual interest rate: 13% (or 0.13)
   • Current monthly payment: $166.37 (to pay off in 4 years, with $1785.76 in total interest)
   • New scenario: Pay off the balance over 1 year instead of 4 years.
   • Goal: Calculate:
     • How much more you must pay per month.
     • How much less you will pay in total interest.

2. Formula Provided: $PMT = \frac{P \cdot \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}$ where:

   • $P$: Loan amount ($6200)
   • $r$: Annual interest rate (13%)
   • $n$: Number of payments per year (12)
   • $t$: Loan term in years (1 or 4)

Solution:

We calculate the monthly payment ($PMT$) for both 1-year and 4-year plans and find the differences.

Step 1: Monthly Payment for 1 Year ($t = 1$):

$PMT_{1} = \frac{6200 \cdot \frac{0.13}{12}}{1 - \left(1 + \frac{0.13}{12}\right)^{-12}}$

Step 2: Monthly Payment for 4 Years ($t = 4$):

$PMT_{4} = \frac{6200 \cdot \frac{0.13}{12}}{1 - \left(1 + \frac{0.13}{12}\right)^{-48}}$

Step 3: Total Interest for Each Plan:

• Total Interest = ($PMT \cdot n \cdot t$) - $P$.

Let me calculate these values step by step for clarity. Given data P = 6200 # Loan amount r = 0.13 # Annual interest rate n = 12 # Number of payments per year

Loan term for the two scenarios

t_1 = 1 # 1-year term t_4 = 4 # 4-year term

Formula for PMT calculation

def calculate_pmt(P, r, n, t): monthly_rate = r / n num_payments = n * t PMT = (P * monthly_rate) / (1 - (1 + monthly_rate) ** -num_payments) return PMT

Monthly payments for 1 year and 4 years

PMT_1 = calculate_pmt(P, r, n, t_1) PMT_4 = calculate_pmt(P, r, n, t_4)

Total interest calculation

def calculate_total_interest(PMT, P, n, t): total_paid = PMT * n * t total_interest = total_paid - P return total_interest

Total interest for 1 year and 4 years

interest_1 = calculate_total_interest(PMT_1, P, n, t_1) interest_4 = calculate_total_interest(PMT_4, P, n, t_4)

Differences in monthly payment and total interest

monthly_payment_difference = PMT_1 - PMT_4 interest_difference = interest_4 - interest_1

PMT_1, PMT_4, monthly_payment_difference, interest_1, interest_4, interest_difference### Results:

1. Monthly Payments:

   • 1-year plan: $553.77
   • 4-year plan: $166.33
   • Difference: You must pay $387.44 more per month to finish in 1 year.

2. Total Interest:

   • 1-year plan: $445.21
   • 4-year plan: $1783.86
   • Difference: You will save $1338.66 in total interest by paying off the balance in 1 year instead of 4 years.

Would you like further details or explanations on the calculations?

Related Questions:

1. How is the formula for calculating monthly payments derived?
2. What happens if the annual interest rate increases to 15%?
3. How would the calculations change for semi-annual payments instead of monthly payments?
4. What would the savings be if the loan term was reduced to 2 years instead of 1 year?
5. How does compound interest impact the total interest paid over time?

Tip:

When reducing the loan term, it's important to verify that the new monthly payment is affordable based on your current budget.