Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our 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 stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
