To solve this problem, we need to calculate both the rewards and the interest for the first month on a balance of $1,685.90 with an interest rate of 14.75%.
Step 1: Calculate the reward amount.
The credit card gives 2% cash back on purchases. To find the reward amount:
\[ \text{Reward} = \text{Total Purchase} \times \text{Cash Back Percentage} \]
\[ \text{Reward} = 1685.90 \times 0.02 = 33.718 \text{ or } 33.72 \text{ (rounded to two decimal places)} \]
Step 2: Calculate the interest for the first month.
The monthly interest rate is calculated by dividing the annual interest rate by 12:
\[ \text{Monthly Interest Rate} = \frac{14.75%}{12} = 1.2291667% = 0.012291667 \]
Now, we calculate the interest for the balance of $1,685.90 for one month:
\[ \text{Interest} = \text{Balance} \times \text{Monthly Interest Rate} \]
\[ \text{Interest} = 1685.90 \times 0.012291667 \approx 20.71 \text{ (rounded to two decimal places)} \]
Step 3: Calculate the difference between the interest and the reward.
Now, we find the difference between the interest accrued and the reward earned:
\[ \text{Difference} = \text{Interest} - \text{Reward} \]
\[ \text{Difference} = 20.71 - 33.72 = -13.01 \]
Thus, the difference between the interest and the reward is approximately -$13.01. This means that the rewards outweigh the interest.
To summarize:
- Reward amount = $33.72
- Interest amount = $20.71
- Difference = $20.71 - $33.72 = -$13.01
The difference is negative, indicating the rewards exceeded the interest by about $13.01.
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