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To solve this problem, let's go through each part of the question systematically.

---

### Given Information

1. **Starting Balance on May 1:** $523.31
2. **Purchase on May 6:** $118.77
3. **Payment on May 13:** -$350.00
4. **Purchase on May 19:** $32.98
5. **Purchase on May 29:** $39.90

- **Billing Date:** The 1st of each month
- **Monthly Finance Charge Rate:** 2.1%

---

### Step 1: Calculate the Average Daily Balance

To find the average daily balance, we calculate the balance for each period and multiply by the number of days in each period, then divide by the total number of days in the month (31 days for May).

#### Balance Calculations

1. **May 1 – May 5 (5 days):** $523.31
2. **May 6 – May 12 (7 days):** $523.31 + $118.77 = $642.08
3. **May 13 – May 18 (6 days):** $642.08 - $350.00 = $292.08
4. **May 19 – May 28 (10 days):** $292.08 + $32.98 = $325.06
5. **May 29 – May 31 (3 days):** $325.06 + $39.90 = $364.96

#### Calculating the Average Daily Balance

\[
\text{Average Daily Balance} = \frac{(523.31 \times 5) + (642.08 \times 7) + (292.08 \times 6) + (325.06 \times 10) + (364.96 \times 3)}{31}
\]

Breaking it down:

- $523.31 \times 5 = 2616.55$
- $642.08 \times 7 = 4494.56$
- $292.08 \times 6 = 1752.48$
- $325.06 \times 10 = 3250.60$
- $364.96 \times 3 = 1094.88$

Adding these up:

\[
2616.55 + 4494.56 + 1752.48 + 3250.60 + 1094.88 = 13209.07
\]

Now, divide by 31:

\[
\text{Average Daily Balance} = \frac{13209.07}{31} = 426.1
\]

So, the **average daily balance** is approximately **$426.10**.

---

### Step 2: Calculate the Monthly Finance Charge

The monthly finance charge is calculated based on the average daily balance and the finance charge rate (2.1%).

\[
\text{Finance Charge} = 426.10 \times \frac{2.1}{100} = 8.9481
\]

Rounding to two decimal places, the **monthly finance charge** is **$8.95**.

---

### Step 3: Calculate the Balance at the Start of the Next Month

To find the balance at the start of the next month, we add the monthly finance charge to the balance at the end of May.

- **Ending Balance on May 31:** $364.96
- **Finance Charge:** $8.95

\[
\text{Balance at the start of next month} = 364.96 + 8.95 = 373.91
\]

So, the **balance at the start of the next month** is **$373.91**.

---

### Summary of Answers

1. **Average Daily Balance:** $426.10
2. **Monthly Finance Charge:** $8.95
3. **Balance at the Start of the Next Month:** $373.91

---

Would you like further details or have any questions on these steps?

---

Here are some related questions to deepen your understanding:

1. How would the finance charge change if the finance rate was increased to 3%?
2. How does making an additional payment in the middle of the month affect the average daily balance?
3. What would be the impact of a larger purchase on the average daily balance calculation?
4. How could paying off the entire balance by May 13 affect the finance charge?
5. How does the average daily balance method differ from a simple end-of-month balance calculation?

**Tip:** Understanding the average daily balance method is key to managing credit card costs, as it directly impacts the interest you pay.

On May 1, the balance on your credit card is $523.31. On May 6, you make a purchase for $118.77. On May 13, you make a payment of $350. On May 19, you make a purchase for $32.98, and on May 29, you make a purchase for $39.90. Assume the billing date is the 1st of the month and the monthly finance charge is 2.1%.

What is the average daily balance?
What is the monthly finance charge?
What is the balance at the start of the next month?

Learn how to calculate the average daily balance, finance charge, and month-end balance for a credit card statement. With a starting balance of $523.31 and various purchases and payments throughout May, this detailed breakdown includes key finance charge calculations and tips for managing credit card costs. Suitable for high school students.

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