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VIPSolutions · Accepted Answer

## Objective

Calculate the difference in interest paid by Sabrina (using **exact interest**) and Janeen (using **ordinary interest**) for a loan of $24,000 at 8.25% from April 5, 2022, to March 9, 2023.

---

## 1. **Find Number of Days (Loan Period)**

**From:** April 5, 2022  
**To:** March 9, 2023

### Step 1: Days in 2022 (from April 5 to December 31)
- April: 30 - 5 + 1 = 26 days (including April 5)
- May: 31  
- June: 30  
- July: 31  
- August: 31  
- September: 30  
- October: 31  
- November: 30  
- December: 31

**Total for 2022:**  
= 26 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31  
= 241 days

### Step 2: Days in 2023 (from January 1 to March 9)
- January: 31  
- February: 28 (not a leap year)  
- March: 9

**Total for 2023:**  
= 31 + 28 + 9  
= 68 days

### Step 3: Total Days
Total Days = Days in 2022 + Days in 2023  
= 241 + 68  
= **309 days**

---

## 2. **Interest Formulas**

### Ordinary Interest (Banker's Rule)
$$
\text{Interest}_{\text{Ordinary}} = P \times r \times \frac{t}{360}
$$

### Exact Interest
$$
\text{Interest}_{\text{Exact}} = P \times r \times \frac{t}{365}
$$

Where:  
$P = \$24,000$  
$r = 8.25\% = .0825$  
$t = 309$ days

---

## 3. **Calculate Both Interests**

### Ordinary Interest:
$$
\text{Interest}_{\text{Ordinary}} = \$24,000 \times .0825 \times \frac{309}{360}
$$

$$
\frac{309}{360} = .8583\overline{3}
$$
$$
\$24,000 \times .0825 = \$1,980
$$
$$
\$1,980 \times .8583\overline{3} = \$1,699.50
$$

---

### Exact Interest:
$$
\text{Interest}_{\text{Exact}} = \$24,000 \times .0825 \times \frac{309}{365}
$$

$$
\frac{309}{365} = .8466
$$
$$
\$24,000 \times .0825 = \$1,980
$$
$$
\$1,980 \times .8466 = \$1,676.27
$$

---

## 4. **Difference in Interest**

$$
\text{Difference} = \text{Interest}_{\text{Ordinary}} - \text{Interest}_{\text{Exact}}
$$
$$
\text{Difference} = \$1,699.50 - \$1,676.27 = \$23.23
$$

---

## Final Answer

**The difference in interest is**  
$$
\boxed{\$23.23}
$$

Sabrina pays **\$23.23 less** in interest using exact interest compared to Janeen’s ordinary interest calculation.

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