Answer :
To determine how much Marcia would save by consolidating her two credit card balances into one balance with the lower interest rate, let's calculate the interest she would pay on each card over a period of 4 years and compare that with the interest she would pay if she consolidates the balances.
### Step 1: Calculate Interest for Each Card Individually
First, we'll find the interest accrued on each card over 4 years.
Interest for Card A:
- Balance: \[tex]$1,389.47 - Annual Percentage Rate (APR): 16% (or 0.16 as a decimal) - Time period: 4 years The interest for Card A is calculated using the formula: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] \[ \text{Interest}_A = 1389.47 \times 0.16 \times 4 \] \[ \text{Interest}_A = 889.2608 \] Interest for Card B: - Balance: \$[/tex]1,065.32
- Annual Percentage Rate (APR): 12% (or 0.12 as a decimal)
- Time period: 4 years
The interest for Card B is calculated using the same formula:
[tex]\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \][/tex]
[tex]\[ \text{Interest}_B = 1065.32 \times 0.12 \times 4 \][/tex]
[tex]\[ \text{Interest}_B = 511.3536 \][/tex]
Total Interest Without Consolidation:
[tex]\[ \text{Total Interest without Consolidation} = \text{Interest}_A + \text{Interest}_B \][/tex]
[tex]\[ \text{Total Interest without Consolidation} = 889.2608 + 511.3536 \][/tex]
[tex]\[ \text{Total Interest without Consolidation} = 1400.6144 \][/tex]
### Step 2: Calculate Interest for Consolidated Balance
If Marcia consolidates the balances, the combined balance would be:
- Combined Balance: \[tex]$1,389.47 (Card A) + \$[/tex]1,065.32 (Card B)
[tex]\[ \text{Combined Balance} = 1389.47 + 1065.32 \][/tex]
[tex]\[ \text{Combined Balance} = 2454.79 \][/tex]
The new APR would be the lower APR, which is 12% (or 0.12 as a decimal).
Interest for Consolidated Balance:
[tex]\[ \text{Interest (consolidated)} = \text{Combined Balance} \times \text{Rate} \times \text{Time} \][/tex]
[tex]\[ \text{Interest (consolidated)} = 2454.79 \times 0.12 \times 4 \][/tex]
[tex]\[ \text{Interest (consolidated)} = 1178.2992 \][/tex]
### Step 3: Calculate Savings by Consolidating
To find the savings from consolidating, we subtract the total interest with consolidation from the total interest without consolidation:
[tex]\[ \text{Savings} = \text{Total Interest without Consolidation} - \text{Interest (consolidated)} \][/tex]
[tex]\[ \text{Savings} = 1400.6144 - 1178.2992 \][/tex]
[tex]\[ \text{Savings} = 222.3152 \][/tex]
Therefore, Marcia will save approximately \[tex]$222.32 in interest by consolidating her balances into the card with the lower interest rate. The correct answer is not explicitly listed among the given choices but the best match for Marcia's savings is close to option (b) $[/tex]133.92[tex]$, albeit it is actually incorrect as the exact calculated savings are \$[/tex]222.32.
### Step 1: Calculate Interest for Each Card Individually
First, we'll find the interest accrued on each card over 4 years.
Interest for Card A:
- Balance: \[tex]$1,389.47 - Annual Percentage Rate (APR): 16% (or 0.16 as a decimal) - Time period: 4 years The interest for Card A is calculated using the formula: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] \[ \text{Interest}_A = 1389.47 \times 0.16 \times 4 \] \[ \text{Interest}_A = 889.2608 \] Interest for Card B: - Balance: \$[/tex]1,065.32
- Annual Percentage Rate (APR): 12% (or 0.12 as a decimal)
- Time period: 4 years
The interest for Card B is calculated using the same formula:
[tex]\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \][/tex]
[tex]\[ \text{Interest}_B = 1065.32 \times 0.12 \times 4 \][/tex]
[tex]\[ \text{Interest}_B = 511.3536 \][/tex]
Total Interest Without Consolidation:
[tex]\[ \text{Total Interest without Consolidation} = \text{Interest}_A + \text{Interest}_B \][/tex]
[tex]\[ \text{Total Interest without Consolidation} = 889.2608 + 511.3536 \][/tex]
[tex]\[ \text{Total Interest without Consolidation} = 1400.6144 \][/tex]
### Step 2: Calculate Interest for Consolidated Balance
If Marcia consolidates the balances, the combined balance would be:
- Combined Balance: \[tex]$1,389.47 (Card A) + \$[/tex]1,065.32 (Card B)
[tex]\[ \text{Combined Balance} = 1389.47 + 1065.32 \][/tex]
[tex]\[ \text{Combined Balance} = 2454.79 \][/tex]
The new APR would be the lower APR, which is 12% (or 0.12 as a decimal).
Interest for Consolidated Balance:
[tex]\[ \text{Interest (consolidated)} = \text{Combined Balance} \times \text{Rate} \times \text{Time} \][/tex]
[tex]\[ \text{Interest (consolidated)} = 2454.79 \times 0.12 \times 4 \][/tex]
[tex]\[ \text{Interest (consolidated)} = 1178.2992 \][/tex]
### Step 3: Calculate Savings by Consolidating
To find the savings from consolidating, we subtract the total interest with consolidation from the total interest without consolidation:
[tex]\[ \text{Savings} = \text{Total Interest without Consolidation} - \text{Interest (consolidated)} \][/tex]
[tex]\[ \text{Savings} = 1400.6144 - 1178.2992 \][/tex]
[tex]\[ \text{Savings} = 222.3152 \][/tex]
Therefore, Marcia will save approximately \[tex]$222.32 in interest by consolidating her balances into the card with the lower interest rate. The correct answer is not explicitly listed among the given choices but the best match for Marcia's savings is close to option (b) $[/tex]133.92[tex]$, albeit it is actually incorrect as the exact calculated savings are \$[/tex]222.32.
