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Sagot :
To determine how much interest Brandon will save by consolidating the two credit card balances, let's calculate the total interest he would pay on each card over 8 years, both without consolidation and with consolidation.
### Step 1: Calculate the interest for each card separately over 8 years.
#### Card A:
- Amount on Card A: [tex]$\$[/tex]1463.82[tex]$ - APR for Card A: \(13\%\) or \(0.13\) - Time period: \(8\) years Interest for Card A is calculated using the formula: \[ \text{Interest}_A = \text{Amount}_A \times \text{APR}_A \times \text{Time} \] Substituting the values, we get: \[ \text{Interest}_A = 1463.82 \times 0.13 \times 8 = 1522.3728 \] #### Card B: - Amount on Card B: $[/tex]\[tex]$1157.98$[/tex]
- APR for Card B: [tex]\(17\%\)[/tex] or [tex]\(0.17\)[/tex]
- Time period: [tex]\(8\)[/tex] years
Interest for Card B is calculated using the formula:
[tex]\[ \text{Interest}_B = \text{Amount}_B \times \text{APR}_B \times \text{Time} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Interest}_B = 1157.98 \times 0.17 \times 8 = 1574.8528 \][/tex]
### Step 2: Calculate the total interest paid on both cards without consolidation.
[tex]\[ \text{Total Interest without Consolidation} = \text{Interest}_A + \text{Interest}_B \][/tex]
Substituting the values, we get:
[tex]\[ \text{Total Interest without Consolidation} = 1522.3728 + 1574.8528 = 3097.2256 \][/tex]
### Step 3: Calculate the interest if the balances are consolidated onto Card A (lower APR) over 8 years.
- Total consolidated amount: [tex]\(\$1463.82 + \$1157.98\)[/tex]
[tex]\[ \text{Consolidated Amount} = 1463.82 + 1157.98 = 2621.8 \][/tex]
- APR for consolidated amount: [tex]\(13\%\)[/tex] or [tex]\(0.13\)[/tex]
- Time period: [tex]\(8\)[/tex] years
The interest for the consolidated amount is calculated using the formula:
[tex]\[ \text{Consolidated Interest} = \text{Consolidated Amount} \times \text{APR}_A \times \text{Time} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Consolidated Interest} = 2621.8 \times 0.13 \times 8 = 2726.672 \][/tex]
### Step 4: Calculate the savings in interest by consolidating.
[tex]\[ \text{Savings} = \text{Total Interest without Consolidation} - \text{Consolidated Interest} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Savings} = 3097.2256 - 2726.672 = 370.5536 \][/tex]
### Step 5: Select the closest answer from the provided options.
Given the calculated savings of [tex]\(370.5536\)[/tex], the closest provided answer is:
c. [tex]\(\$256.32\)[/tex]
Therefore, the best answer from the provided choices is [tex]\( \boxed{256.32} \)[/tex].
### Step 1: Calculate the interest for each card separately over 8 years.
#### Card A:
- Amount on Card A: [tex]$\$[/tex]1463.82[tex]$ - APR for Card A: \(13\%\) or \(0.13\) - Time period: \(8\) years Interest for Card A is calculated using the formula: \[ \text{Interest}_A = \text{Amount}_A \times \text{APR}_A \times \text{Time} \] Substituting the values, we get: \[ \text{Interest}_A = 1463.82 \times 0.13 \times 8 = 1522.3728 \] #### Card B: - Amount on Card B: $[/tex]\[tex]$1157.98$[/tex]
- APR for Card B: [tex]\(17\%\)[/tex] or [tex]\(0.17\)[/tex]
- Time period: [tex]\(8\)[/tex] years
Interest for Card B is calculated using the formula:
[tex]\[ \text{Interest}_B = \text{Amount}_B \times \text{APR}_B \times \text{Time} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Interest}_B = 1157.98 \times 0.17 \times 8 = 1574.8528 \][/tex]
### Step 2: Calculate the total interest paid on both cards without consolidation.
[tex]\[ \text{Total Interest without Consolidation} = \text{Interest}_A + \text{Interest}_B \][/tex]
Substituting the values, we get:
[tex]\[ \text{Total Interest without Consolidation} = 1522.3728 + 1574.8528 = 3097.2256 \][/tex]
### Step 3: Calculate the interest if the balances are consolidated onto Card A (lower APR) over 8 years.
- Total consolidated amount: [tex]\(\$1463.82 + \$1157.98\)[/tex]
[tex]\[ \text{Consolidated Amount} = 1463.82 + 1157.98 = 2621.8 \][/tex]
- APR for consolidated amount: [tex]\(13\%\)[/tex] or [tex]\(0.13\)[/tex]
- Time period: [tex]\(8\)[/tex] years
The interest for the consolidated amount is calculated using the formula:
[tex]\[ \text{Consolidated Interest} = \text{Consolidated Amount} \times \text{APR}_A \times \text{Time} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Consolidated Interest} = 2621.8 \times 0.13 \times 8 = 2726.672 \][/tex]
### Step 4: Calculate the savings in interest by consolidating.
[tex]\[ \text{Savings} = \text{Total Interest without Consolidation} - \text{Consolidated Interest} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Savings} = 3097.2256 - 2726.672 = 370.5536 \][/tex]
### Step 5: Select the closest answer from the provided options.
Given the calculated savings of [tex]\(370.5536\)[/tex], the closest provided answer is:
c. [tex]\(\$256.32\)[/tex]
Therefore, the best answer from the provided choices is [tex]\( \boxed{256.32} \)[/tex].
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