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Sagot :
Let's solve the problem step-by-step:
1. Identify the amounts and APRs for both cards:
- Card A: [tex]$1,879.58 with an APR of 14% - Card B: $[/tex]861.00 with an APR of 10%
2. Determine the interest accrued on each card over 5 years:
- For Card A:
[tex]\[ \text{Interest}_{A} = \text{Amount}_{A} \times \text{APR}_{A} \times \text{Years} = 1,879.58 \times 0.14 \times 5 = 1,315.706 \][/tex]
- For Card B:
[tex]\[ \text{Interest}_{B} = \text{Amount}_{B} \times \text{APR}_{B} \times \text{Years} = 861.00 \times 0.10 \times 5 = 430.50 \][/tex]
3. Calculate total interest without consolidation:
- Total Interest without Consolidation:
[tex]\[ \text{Total Interest} = \text{Interest}_{A} + \text{Interest}_{B} = 1,315.706 + 430.50 = 1,746.206 \][/tex]
4. Determine the total amount if the balances are consolidated to the card with the lower APR (Card B):
- Total Amount After Consolidation:
[tex]\[ \text{Total Amount} = \text{Amount}_{A} + \text{Amount}_{B} = 1,879.58 + 861.00 = 2,740.58 \][/tex]
- Consolidated Interest (using Card B's APR):
[tex]\[ \text{Consolidated Interest} = \text{Total Amount} \times \text{APR}_{B} \times \text{Years} = 2,740.58 \times 0.10 \times 5 = 1,370.29 \][/tex]
5. Calculate the savings in interest by consolidating:
- Interest Savings:
[tex]\[ \text{Interest Savings} = \text{Total Interest without Consolidation} - \text{Consolidated Interest} = 1,746.206 - 1,370.29 = 375.916 \][/tex]
6. Match the interest savings to the closest answer choice:
Based on the savings calculated:
- The possible options given were [tex]$\$[/tex] 1,526.40[tex]$, $[/tex]\[tex]$ 2,422.80$[/tex], [tex]$\$[/tex] 105.00[tex]$, and $[/tex]\[tex]$ 227.40$[/tex].
Given the savings calculated are \[tex]$375.92, none of the provided options match this amount exactly. Therefore, it seems there might be an issue with the provided choices or the setup. However, assuming the choices given are supposed to cover common scenarios, we should conclude that based on the computed interest savings, none of the provided choices are a perfect match. Thus, the final answer for the interest savings is approximately: \[ \$[/tex]375.92
\]
Since none of the options ([tex]\(\$ 1,526.40\)[/tex], [tex]\(\$ 2,422.80\)[/tex], [tex]\(\$ 105.00\)[/tex], [tex]\(\$ 227.40\)[/tex]) match exactly, the right choice is, unfortunately, not listed.
1. Identify the amounts and APRs for both cards:
- Card A: [tex]$1,879.58 with an APR of 14% - Card B: $[/tex]861.00 with an APR of 10%
2. Determine the interest accrued on each card over 5 years:
- For Card A:
[tex]\[ \text{Interest}_{A} = \text{Amount}_{A} \times \text{APR}_{A} \times \text{Years} = 1,879.58 \times 0.14 \times 5 = 1,315.706 \][/tex]
- For Card B:
[tex]\[ \text{Interest}_{B} = \text{Amount}_{B} \times \text{APR}_{B} \times \text{Years} = 861.00 \times 0.10 \times 5 = 430.50 \][/tex]
3. Calculate total interest without consolidation:
- Total Interest without Consolidation:
[tex]\[ \text{Total Interest} = \text{Interest}_{A} + \text{Interest}_{B} = 1,315.706 + 430.50 = 1,746.206 \][/tex]
4. Determine the total amount if the balances are consolidated to the card with the lower APR (Card B):
- Total Amount After Consolidation:
[tex]\[ \text{Total Amount} = \text{Amount}_{A} + \text{Amount}_{B} = 1,879.58 + 861.00 = 2,740.58 \][/tex]
- Consolidated Interest (using Card B's APR):
[tex]\[ \text{Consolidated Interest} = \text{Total Amount} \times \text{APR}_{B} \times \text{Years} = 2,740.58 \times 0.10 \times 5 = 1,370.29 \][/tex]
5. Calculate the savings in interest by consolidating:
- Interest Savings:
[tex]\[ \text{Interest Savings} = \text{Total Interest without Consolidation} - \text{Consolidated Interest} = 1,746.206 - 1,370.29 = 375.916 \][/tex]
6. Match the interest savings to the closest answer choice:
Based on the savings calculated:
- The possible options given were [tex]$\$[/tex] 1,526.40[tex]$, $[/tex]\[tex]$ 2,422.80$[/tex], [tex]$\$[/tex] 105.00[tex]$, and $[/tex]\[tex]$ 227.40$[/tex].
Given the savings calculated are \[tex]$375.92, none of the provided options match this amount exactly. Therefore, it seems there might be an issue with the provided choices or the setup. However, assuming the choices given are supposed to cover common scenarios, we should conclude that based on the computed interest savings, none of the provided choices are a perfect match. Thus, the final answer for the interest savings is approximately: \[ \$[/tex]375.92
\]
Since none of the options ([tex]\(\$ 1,526.40\)[/tex], [tex]\(\$ 2,422.80\)[/tex], [tex]\(\$ 105.00\)[/tex], [tex]\(\$ 227.40\)[/tex]) match exactly, the right choice is, unfortunately, not listed.
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