9. Sean plans to buy a new tablet for $$ \$ 315 $$. The store offers him an interest- free payment plan that allows him to pay in monthly installments of $$ \$ 25 $$. a) How much will Sean still owe after one payment? b) Explain in words how the amount owed depends on the number of payments made. Then write an equation for calculating the amount owed $$ a $$ for any number of payments $$ \boldsymbol{n} $$. After three payments? - payments?

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Let's tackle each part of Sean's tablet payment problem step by step.

---

### **Problem Overview**
- **Total Cost of Tablet:** \$315
- **Monthly Payment:** \$25
- **Interest Rate:** 0% (Interest-free payment plan)

---

### **a) Amount Owed After One Payment**

**Calculation:**
1. **Initial Amount Owed:** \$315
2. **Payment Made:** \$25

$$
\text{Amount Owed} = \text{Initial Amount} - \text{Payment}
$$
$$
\text{Amount Owed} = \$315 - \$25 = \$290
$$

**Answer:** After one payment, Sean still owes **\$290**.

---

### **b) Relationship Between Amount Owed and Number of Payments**

**Explanation:**
- **Linear Relationship:** The amount Sean owes decreases by a fixed amount (\$25) each month as he makes payments.
- **Dependence on Number of Payments:** For every additional payment $$ n $$, the total amount owed decreases by $$ 25n $$ dollars.

**Mathematical Equation:**
$$
a = 315 - 25n
$$
- **Where:**
  - $$ a $$ = Amount owed after $$ n $$ payments
  - $$ n $$ = Number of monthly payments made

**Interpretation:**
- **$$ n = 0 $$:** Before any payments, $$ a = 315 $$ (the full price).
- **$$ n = 1 $$:** After one payment, $$ a = 315 - 25 = 290 $$.
- **And so on.**

---

### **Additional Question: Amount Owed After Three Payments**

**Using the Equation:**
$$
a = 315 - 25n
$$
$$
\text{For } n = 3:
$$
$$
a = 315 - 25 \times 3 = 315 - 75 = 240
$$

**Answer:** After three payments, Sean still owes **\$240**.

---

### **Summary**

- **After One Payment:** \$290 owed
- **After Three Payments:** \$240 owed
- **General Formula:** $$ a = 315 - 25n $$

This formula allows Sean to calculate the remaining amount owed for any number of payments he makes.
After one payment, Sean owes \$290. The amount owed decreases by \$25 for each payment. The equation to calculate the amount owed $$ a $$ after $$ n $$ payments is $$ a = 315 - 25n $$. After three payments, Sean owes \$240.

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