Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
