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### Problem Summary:
1. Find the average daily balance for a credit card account using transactions within a billing cycle.
2. Calculate the finance charge using the average daily balance.
3. Determine the balance at the end of the billing cycle.

### Steps to Solve:

1. **Find the Average Daily Balance:**
   - The billing cycle is from May 5 to June 2.
   - Identify each transaction and its impact on the balance.
   - Calculate the daily balance for each segment (between transactions) and multiply by the number of days the balance stayed the same.
   - Sum these values and divide by the total number of days in the billing cycle (29 days).

2. **Calculate the Finance Charge:**
   - Use the formula for finance charge:
     \[
     \text{Finance Charge} = \text{Average Daily Balance} \times \text{Monthly Interest Rate}
     \]
   - The interest rate is 2.7% per month.

3. **Find the Final Balance:**
   - Adjust the balance based on the calculated finance charge.
   - Add or subtract each transaction as it occurs to arrive at the final balance at the end of the billing cycle.

### Let's go through the details for each step:

#### Step 1: Calculate the Average Daily Balance
- May 5: Billing date, balance is $756.12.
- May 8: Mail Order of $27.67, new balance is $ 756.12 + 27.67 = 783.79 $.
- May 14: Returns of $26.67, new balance is $ 783.79 - 26.67 = 757.12 $.
- May 16: Wine purchase of $33.79, new balance is $ 757.12 + 33.79 = 790.91 $.
- May 20: Payment of $123, new balance is $ 790.91 - 123 = 667.91 $.
- May 25: Returns of $70.25, new balance is $ 667.91 - 70.25 = 597.66 $.
- May 30: Birthday Gift of $130, new balance is $ 597.66 + 130 = 727.66 $.
- June 2: Cash Advance of $100, new balance is $ 727.66 + 100 = 827.66 $.

Now, calculate the daily balances:
- From May 5 to May 8 (3 days): $756.12
- From May 8 to May 14 (6 days): $783.79
- From May 14 to May 16 (2 days): $757.12
- From May 16 to May 20 (4 days): $790.91
- From May 20 to May 25 (5 days): $667.91
- From May 25 to May 30 (5 days): $597.66
- From May 30 to June 2 (3 days): $727.66

Now, calculate the average daily balance using these values:
\[
\text{Average Daily Balance} = \frac{(756.12 \times 3) + (783.79 \times 6) + (757.12 \times 2) + (790.91 \times 4) + (667.91 \times 5) + (597.66 \times 5) + (727.66 \times 3)}{29}
\]

Let’s calculate these values to get the average daily balance.

#### Step 2: Calculate the Finance Charge
Use the formula:
\[
\text{Finance Charge} = \text{Average Daily Balance} \times 0.027
\]

#### Step 3: Calculate the Final Balance
\[
\text{Final Balance} = \text{Balance at the end of billing cycle} + \text{Finance Charge}
\]

Let me do the calculations for these steps.The calculations are as follows:

1. **Average Daily Balance**: \$695.17 (rounded to the nearest cent)
2. **Finance Charge**: \$18.77 (rounded to the nearest cent)
3. **Balance at the End of the Billing Cycle**: \$846.43 (rounded to the nearest cent)

Would you like a more detailed explanation of these calculations or have any questions? 

Here are some related questions for further exploration:
1. How does changing the interest rate affect the finance charge?
2. What happens to the average daily balance if a large payment is made early in the billing cycle?
3. How would the calculations change if the billing cycle covered a different number of days?
4. What are some strategies to minimize finance charges?
5. How can you use balance transfer offers to lower finance charges?

**Tip:** Keeping a low average daily balance and making early payments can significantly reduce the finance charge on a credit card.

Find the average daily balance for the given credit card account. Assume the billing date is the same day each month, and use the actual number of days in each month. Then find the finance charge if the previous balance was $756.12 and the interest is 2.7% per month on the average daily balance. Finally, find the balance at the end of the billing cycle.

Learn how to calculate the average daily balance for a credit card account, determine the finance charge with a 2.7% interest rate, and find the final balance at the end of the billing cycle. This problem covers essential concepts in finance and interest calculation, suitable for students in Grades 9-12.

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