Let's analyze each offer over a 12-month period to determine the average Annual Percentage Rate (APR) and identify the option with the lowest average APR.
### Option A: Constant 15% APR
The APR for option A is straightforward:
[tex]\[ \text{APR}_{A} = 15\% \][/tex]
### Option B: 6% for the first 6 months, then 23% for the next 6 months
We need to calculate the average APR over the 12-month period:
[tex]\[ \text{APR}_{B} = \frac{(6\% \times 6 \text{ months}) + (23\% \times 6 \text{ months})}{12 \text{ months}} \][/tex]
Calculate the sum of the APRs for each period:
[tex]\[ \text{APR}_{B} = \frac{(6 \times 6) + (23 \times 6)}{12} \][/tex]
[tex]\[ \text{APR}_{B} = \frac{36 + 138}{12} \][/tex]
[tex]\[ \text{APR}_{B} = \frac{174}{12} \][/tex]
[tex]\[ \text{APR}_{B} = 14.5\% \][/tex]
### Option C: 10% for the first 6 months, then it increases by 1% each month
For the first 6 months, the APR is constant at 10%. Then, it increases by 1% each month for the next 6 months:
- 1st month after 6 months: 11%
- 2nd month after 6 months: 12%
- 3rd month after 6 months: 13%
- 4th month after 6 months: 14%
- 5th month after 6 months: 15%
- 6th month after 6 months: 16%
Calculate the average APR:
[tex]\[ \text{APR}_{C} = \frac{(10\% \times 6 \text{ months}) + (11\% + 12\% + 13\% + 14\% + 15\% + 16\%)}{12 \text{ months}} \][/tex]
Calculate the sum of the increments:
[tex]\[ \text{Sum of increments} = 11 + 12 + 13 + 14 + 15 + 16 = 81 \][/tex]
[tex]\[ \text{Sum of total APR for the first 6 months} = 10 \times 6 = 60 \][/tex]
Now add both sums and divide by 12:
[tex]\[ \text{APR}_{C} = \frac{60 + 81}{12} \][/tex]
[tex]\[ \text{APR}_{C} = \frac{141}{12} \][/tex]
[tex]\[ \text{APR}_{C} = 11.75\% \][/tex]
### Option D: 19% APR with points earned towards a gift card
For simplicity, we'll only consider the APR since the points earned are subjective and not directly quantifiable in terms of APR:
[tex]\[ \text{APR}_{D} = 19\% \][/tex]
### Comparing the Options
- Option A: 15%
- Option B: 14.5%
- Option C: 11.75%
- Option D: 19%
From these calculations, the average APRs for each option over a 12-month period are:
1. Option A: 15%
2. Option B: 14.5%
3. Option C: 11.75%
4. Option D: 19%
### Conclusion
Option C, with an average APR of 11.75%, offers the best long-term deal among the given options.
