16. Higher Order Thinking You have two different savings accounts. For Account A, the interest earned after 18 months is $$ \$ 12.00 $$. For Account B, the interest earned after 27 months is $$ \$ 27.00 $$. a. If the interest rate is $$ 3.2 \% $$ for Account $$ A $$, how much is the principal? b. If the interest rate is $$ 2.4 \% $$ for Account B, how much is the principal? c. Which account earned you the most interest in the first year? Explain.

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Let's tackle each part of the problem step by step.

### **a. Calculating the Principal for Account A**

**Given:**
- **Interest (I):** \$12.00
- **Interest Rate (r):** 3.2% per annum
- **Time (t):** 18 months

First, convert the time from months to years:
$$
t = \frac{18 \text{ months}}{12 \text{ months/year}} = 1.5 \text{ years}
$$

**Using the Simple Interest Formula:**
$$
I = P \times r \times t
$$
Where:
- $$ I $$ = Interest earned
- $$ P $$ = Principal amount
- $$ r $$ = Annual interest rate (in decimal)
- $$ t $$ = Time in years

Rearranging the formula to find $$ P $$:
$$
P = \frac{I}{r \times t}
$$

**Plugging in the values:**
$$
P = \frac{12}{0.032 \times 1.5} = \frac{12}{0.048} = 250
$$

**Answer:**
The principal for **Account A** is **\$250.00**.

---

### **b. Calculating the Principal for Account B**

**Given:**
- **Interest (I):** \$27.00
- **Interest Rate (r):** 2.4% per annum
- **Time (t):** 27 months

First, convert the time from months to years:
$$
t = \frac{27 \text{ months}}{12 \text{ months/year}} = 2.25 \text{ years}
$$

**Using the Simple Interest Formula:**
$$
I = P \times r \times t
$$

Rearranging to find $$ P $$:
$$
P = \frac{I}{r \times t}
$$

**Plugging in the values:**
$$
P = \frac{27}{0.024 \times 2.25} = \frac{27}{0.054} = 500
$$

**Answer:**
The principal for **Account B** is **\$500.00**.

---

### **c. Comparing Interest Earned in the First Year**

To determine which account earned more interest in the first year, calculate the interest for each account over 1 year.

**For Account A:**
- **Principal (P):** \$250.00
- **Interest Rate (r):** 3.2% per annum
- **Time (t):** 1 year

$$
I = 250 \times 0.032 \times 1 = \$8.00
$$

**For Account B:**
- **Principal (P):** \$500.00
- **Interest Rate (r):** 2.4% per annum
- **Time (t):** 1 year

$$
I = 500 \times 0.024 \times 1 = \$12.00
$$

**Conclusion:**
**Account B** earned **\$12.00** in one year, whereas **Account A** earned **\$8.00**. Therefore, **Account B** earned more interest in the first year.
**a. Account A Principal:** \$250.00 **b. Account B Principal:** \$500.00 **c. Account B earned more interest in the first year.**

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