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Sagot :
To determine which equation correctly equates the annual costs of the two credit cards, we need to consider their respective interest rates and fees.
1. Interest Rates Conversion to Monthly:
- APR (Annual Percentage Rate) for card A is [tex]\(0.143\)[/tex].
- APR for card B is [tex]\(0.171\)[/tex].
- Since interest is compounded monthly, we convert these APRs to monthly interest rates by dividing by 12:
[tex]\[ \text{Monthly Rate for A} = \frac{0.143}{12} = 0.011916666666666666 \][/tex]
[tex]\[ \text{Monthly Rate for B} = \frac{0.171}{12} = 0.01425 \][/tex]
2. Compute Annual Growth Factor:
- We calculate the amount that the principal [tex]\(P\)[/tex] would grow to over one year with each monthly rate:
[tex]\[ \text{Growth Factor for A} = (1 + \text{Monthly Rate for A})^{12} = (1 + 0.011916666666666666)^{12} = 1.1527549283620337 \][/tex]
[tex]\[ \text{Growth Factor for B} = (1 + \text{Monthly Rate for B})^{12} = (1 + 0.01425)^{12} = 1.185059610158403 \][/tex]
3. Annual Cost Consideration:
- Credit card A has an annual fee of \$36, which affects its overall cost.
- Credit card B has no annual fee.
4. Formulate the Cost Equivalence Equation:
- To find the principal [tex]\(P\)[/tex] where both cards offer the same deal, equate the total costs for one year:
- For card A, the total amount after one year, considering the annual fee:
[tex]\[ P \times \text{Growth Factor for A} - 36 \][/tex]
- For card B, the total amount after one year:
[tex]\[ P \times \text{Growth Factor for B} \][/tex]
- Setting these equal gives:
[tex]\[ P \times (1 + 0.011916666666666666)^{12} - 36 = P \times (1 + 0.01425)^{12} \][/tex]
5. Compare with Given Options:
- Compare with the given options:
- Option A:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option B:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}+36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option C:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-\frac{36}{12} = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option D:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}+\frac{36}{12} = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
The correct equation that balances the total cost between the two credit cards over the course of a year is:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{A} \][/tex]
1. Interest Rates Conversion to Monthly:
- APR (Annual Percentage Rate) for card A is [tex]\(0.143\)[/tex].
- APR for card B is [tex]\(0.171\)[/tex].
- Since interest is compounded monthly, we convert these APRs to monthly interest rates by dividing by 12:
[tex]\[ \text{Monthly Rate for A} = \frac{0.143}{12} = 0.011916666666666666 \][/tex]
[tex]\[ \text{Monthly Rate for B} = \frac{0.171}{12} = 0.01425 \][/tex]
2. Compute Annual Growth Factor:
- We calculate the amount that the principal [tex]\(P\)[/tex] would grow to over one year with each monthly rate:
[tex]\[ \text{Growth Factor for A} = (1 + \text{Monthly Rate for A})^{12} = (1 + 0.011916666666666666)^{12} = 1.1527549283620337 \][/tex]
[tex]\[ \text{Growth Factor for B} = (1 + \text{Monthly Rate for B})^{12} = (1 + 0.01425)^{12} = 1.185059610158403 \][/tex]
3. Annual Cost Consideration:
- Credit card A has an annual fee of \$36, which affects its overall cost.
- Credit card B has no annual fee.
4. Formulate the Cost Equivalence Equation:
- To find the principal [tex]\(P\)[/tex] where both cards offer the same deal, equate the total costs for one year:
- For card A, the total amount after one year, considering the annual fee:
[tex]\[ P \times \text{Growth Factor for A} - 36 \][/tex]
- For card B, the total amount after one year:
[tex]\[ P \times \text{Growth Factor for B} \][/tex]
- Setting these equal gives:
[tex]\[ P \times (1 + 0.011916666666666666)^{12} - 36 = P \times (1 + 0.01425)^{12} \][/tex]
5. Compare with Given Options:
- Compare with the given options:
- Option A:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option B:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}+36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option C:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-\frac{36}{12} = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
- Option D:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}+\frac{36}{12} = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
The correct equation that balances the total cost between the two credit cards over the course of a year is:
[tex]\[ P\left(1+\frac{0.143}{12}\right)^{12}-36 = P\left(1+\frac{0.171}{12}\right)^{12} \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{A} \][/tex]
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