Answer :
To determine the monthly payments Julius needs to make to pay off his credit card balance in 12 months, we start by calculating the monthly payment for the credit card balance and then check each modification option to see which one allows him to meet this payment goal with the least changes to his current expenses.
Based on the information:
1. Credit Card Balance: [tex]\( \$34.15 \)[/tex]
2. Annual Percentage Rate (APR): [tex]\( 20\% \)[/tex]
3. Months to Pay Off Balance: [tex]\( 12 \)[/tex]
4. Income: [tex]\( \$2,456.00 \)[/tex]
5. Total Monthly Expenses before any modification are calculated as follows:
[tex]\[ \$900.00 \, (\text{Rent}) + \$186.35 \, (\text{Utilities}) + \$298.00 \, (\text{Food/Clothes}) + \$330.00 \, (\text{Entertainment}) + \$385.00 \, (\text{Car}) + \$34.15 \, (\text{Credit Card}) + \$89.49 \, (\text{Cell Phone}) = \$2,223.00 \][/tex]
### Step-by-Step Solution:
1. Calculate Net Income:
[tex]\[ \text{Net Income} = \$2,456.00 \, (\text{Income}) - \$2,223.00 \, (\text{Total Expenses}) = \$233.00 \][/tex]
2. Credit Card Monthly Interest Rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{20\%}{12 \text{ months}} = 0.01666667 \][/tex]
3. Payment Calculation:
Using the formula for the monthly payment [tex]\( P \)[/tex] to pay off a balance [tex]\( B \)[/tex] with an interest rate [tex]\( r \)[/tex] over [tex]\( n \)[/tex] periods, we get:
[tex]\[ P = \frac{B \times r \times (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( B = 34.15 \)[/tex]
- [tex]\( r = 0.01666667 \)[/tex]
- [tex]\( n = 12 \)[/tex]
Using this formula, the result for the monthly payment is approximately:
[tex]\[ P \approx \$3.163468 \][/tex]
4. Modification Analysis:
We need to ensure the net income after modifications can cover the monthly payment. For each modification:
a. Entertainment: \[tex]$31, Food/Clothes: \$[/tex]55
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$31 + \$55) = \$2,137.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,137.00 = \$319.00 \][/tex]
Since [tex]\( \$319.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is sufficient.
b. Entertainment: \[tex]$80, Food/Clothes: \$[/tex]60
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$80 + \$60) = \$2,083.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,083.00 = \$373.00 \][/tex]
Since [tex]\( \$373.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is also sufficient.
c. Entertainment: \[tex]$18, Food/Clothes: \$[/tex]32
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$18 + \$32) = \$2,173.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,173.00 = \$283.00 \][/tex]
Since [tex]\( \$283.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is also sufficient.
Since we need to make the least cuts to his expenses and modification (a) requires him to eliminate \[tex]$86, which is the smallest adjustment, the best modification is: \[ \boxed{\text{a. Julius can eliminate } \$[/tex]31 \text{ from Entertainment and } \$55 \text{ from Food/Clothes.}}
\]
This will allow Julius to pay off his credit card balance within 12 months.
Based on the information:
1. Credit Card Balance: [tex]\( \$34.15 \)[/tex]
2. Annual Percentage Rate (APR): [tex]\( 20\% \)[/tex]
3. Months to Pay Off Balance: [tex]\( 12 \)[/tex]
4. Income: [tex]\( \$2,456.00 \)[/tex]
5. Total Monthly Expenses before any modification are calculated as follows:
[tex]\[ \$900.00 \, (\text{Rent}) + \$186.35 \, (\text{Utilities}) + \$298.00 \, (\text{Food/Clothes}) + \$330.00 \, (\text{Entertainment}) + \$385.00 \, (\text{Car}) + \$34.15 \, (\text{Credit Card}) + \$89.49 \, (\text{Cell Phone}) = \$2,223.00 \][/tex]
### Step-by-Step Solution:
1. Calculate Net Income:
[tex]\[ \text{Net Income} = \$2,456.00 \, (\text{Income}) - \$2,223.00 \, (\text{Total Expenses}) = \$233.00 \][/tex]
2. Credit Card Monthly Interest Rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{20\%}{12 \text{ months}} = 0.01666667 \][/tex]
3. Payment Calculation:
Using the formula for the monthly payment [tex]\( P \)[/tex] to pay off a balance [tex]\( B \)[/tex] with an interest rate [tex]\( r \)[/tex] over [tex]\( n \)[/tex] periods, we get:
[tex]\[ P = \frac{B \times r \times (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( B = 34.15 \)[/tex]
- [tex]\( r = 0.01666667 \)[/tex]
- [tex]\( n = 12 \)[/tex]
Using this formula, the result for the monthly payment is approximately:
[tex]\[ P \approx \$3.163468 \][/tex]
4. Modification Analysis:
We need to ensure the net income after modifications can cover the monthly payment. For each modification:
a. Entertainment: \[tex]$31, Food/Clothes: \$[/tex]55
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$31 + \$55) = \$2,137.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,137.00 = \$319.00 \][/tex]
Since [tex]\( \$319.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is sufficient.
b. Entertainment: \[tex]$80, Food/Clothes: \$[/tex]60
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$80 + \$60) = \$2,083.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,083.00 = \$373.00 \][/tex]
Since [tex]\( \$373.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is also sufficient.
c. Entertainment: \[tex]$18, Food/Clothes: \$[/tex]32
[tex]\[ \text{Modified Expenses} = \$2,223.00 - (\$18 + \$32) = \$2,173.00 \][/tex]
[tex]\[ \text{Remaining Income} = \$2,456.00 - \$2,173.00 = \$283.00 \][/tex]
Since [tex]\( \$283.00 \)[/tex] is greater than the monthly payment of [tex]\( \$3.163468 \)[/tex], this modification is also sufficient.
Since we need to make the least cuts to his expenses and modification (a) requires him to eliminate \[tex]$86, which is the smallest adjustment, the best modification is: \[ \boxed{\text{a. Julius can eliminate } \$[/tex]31 \text{ from Entertainment and } \$55 \text{ from Food/Clothes.}}
\]
This will allow Julius to pay off his credit card balance within 12 months.
