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Horatio transferred a balance of [tex]$\$[/tex]2600[tex]$ to a new credit card at the beginning of the year. The card offered an introductory APR of $[/tex]4.3\%[tex]$ for the first 5 months and a standard APR of $[/tex]13.7\%[tex]$ thereafter. If the card compounds interest monthly, which of these expressions represents Horatio's balance at the end of the year? (Assume that Horatio will make no payments or new purchases during the year, and ignore any possible late payment fees.)

A. $[/tex](\[tex]$2600)\left(1+\frac{0.043}{12}\right)^{12}\left(1+\frac{0.137}{12}\right)^{12}$[/tex]

B. [tex]$(\$[/tex]2600)\left(1+\frac{0.043}{5}\right)^{12}\left(1+\frac{0.137}{7}\right)^{12}[tex]$

C. $[/tex](\[tex]$2600)\left(1+\frac{0.043}{12}\right)^5\left(1+\frac{0.137}{12}\right)^7$[/tex]

D. [tex]$(\$[/tex]2600)\left(1+\frac{0.043}{5}\right)^5\left(1+\frac{0.137}{7}\right)^7$

Sagot :

GinnyAnswer
To determine Horatio’s balance at the end of the year, let’s go through the problem step by step:

1. Initial Principal Balance:
Horatio starts with a balance of [tex]\(\$ 2600\)[/tex].

2. Introductory Period:
- For the first 5 months, the introductory APR is [tex]\(4.3\%\)[/tex].
- Given that the APR is compounded monthly, we need to divide the APR by 12 to get the monthly interest rate:
[tex]\[ \text{Monthly interest rate for introductory APR} = \frac{4.3\%}{12} = \frac{0.043}{12} \][/tex]
- We then calculate the balance at the end of the 5-month introductory period using the formula for compound interest:
[tex]\[ \text{Balance after 5 months} = \$2600 \left(1 + \frac{0.043}{12}\right)^5 \][/tex]

3. Standard Period:
- After the introductory period, the standard APR of [tex]\(13.7\%\)[/tex] applies for the remaining 7 months.
- Similarly, we calculate the monthly interest rate for the standard APR:
[tex]\[ \text{Monthly interest rate for standard APR} = \frac{13.7\%}{12} = \frac{0.137}{12} \][/tex]
- We then use the balance obtained after the 5 months as the principal for the next 7 months and again apply the compound interest formula:
[tex]\[ \text{Balance at the end of the year} = \left(\$2600 \left(1 + \frac{0.043}{12}\right)^5\right) \left(1 + \frac{0.137}{12}\right)^7 \][/tex]

4. Combining the calculations:
Combining both periods, we get the final balance at the end of the year:
[tex]\[ \text{Balance at the end of the year} = \$2600 \left(1 + \frac{0.043}{12}\right)^5 \left(1 + \frac{0.137}{12}\right)^7 \][/tex]

5. Conclusion:
Comparing this expression to the choices given in the multiple-choice question, we see that it matches option C.

Therefore, the correct expression that represents Horatio’s balance at the end of the year is:
[tex]\[ C. (\$ 2600)\left(1+\frac{0.043}{12}\right)^5\left(1+\frac{0.137}{12}\right)^7 \][/tex]

Additionally, based on given information, after performing the calculations, we find:
- Balance after introductory period: [tex]\( \$2646.918378986316 \)[/tex]
- Balance at the end of the year: [tex]\( \$2865.8357153477655 \)[/tex]
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