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Cook Security Systems has a line of credit, which charges an
annual percentage rate of prime rate plus . The starting balance on
October 1 was .
On October 4 they made a payment of . On October 13 the business
borrowed , and on October 19 they borrowed . If the current
prime rate is , what is the new balance (in )? (Round your answer to
the nearest cent.)
Ln 2, Col 1

Ask by Bird Barrett. in the United States

Mar 26,2025

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File Edit Format View Help
Cook Security Systems has a line of credit, which charges an
annual percentage rate of prime rate plus . The starting balance on
October 1 was .
On October 4 they made a payment of . On October 13 the business
borrowed , and on October 19 they borrowed . If the current
prime rate is , what is the new balance (in )? (Round your answer to
the nearest cent.)
Ln 2, Col 1

Ask by Bird Barrett. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

We are told that Cook Security Systems has a line of credit with

$$
\text{APR} = \text{prime rate} + 4\%
$$

and the current prime rate is $$6\%$$. Thus the effective annual rate is

$$
6\% + 4\% = 10\%.
$$

Interest on a line of credit is typically computed daily using the daily rate

$$
r_\text{daily} = \frac{10\%}{365} = \frac{0.10}{365}.
$$

The transactions during October are as follows:

1. The balance on October 1 is $$\$9,300$$.
2. On October 4 a payment of $$\$1,800$$ is made.
3. On October 13 the business borrows $$\$2,900$$.
4. On October 19 the business borrows $$\$4,400$$.

We assume that the interest is computed on the balance that exists during each time segment, and then the interest is added at the end of the period. We can break the month into periods during which the balance is constant.

1. **October 1 to October 3 (3 days):**  
   The balance is $$\$9,300$$.  
   Interest for this period (Period 1) is:
   $$
   I_1 = 9300 \times \frac{0.10}{365} \times 3.
   $$

2. **October 4 to October 12 (9 days):**  
   The payment on October 4 reduces the balance:
   $$
   9300 - 1800 = 7500.
   $$
   Interest for this period (Period 2) is:
   $$
   I_2 = 7500 \times \frac{0.10}{365} \times 9.
   $$

3. **October 13 to October 18 (6 days):**  
   On October 13, the business borrows $$\$2,900$$ so that:
   $$
   7500 + 2900 = 10400.
   $$
   Interest for this period (Period 3) is:
   $$
   I_3 = 10400 \times \frac{0.10}{365} \times 6.
   $$

4. **October 19 to October 31 (13 days):**  
   On October 19, an additional $$\$4,400$$ is borrowed; the new balance becomes:
   $$
   10400 + 4400 = 14800.
   $$
   Interest for this period (Period 4) is:
   $$
   I_4 = 14800 \times \frac{0.10}{365} \times 13.
   $$

Now we compute the interest for each period using the daily rate

$$
\frac{0.10}{365} \approx 0.0002739726.
$$

- **Period 1:**
  $$
  I_1 = 9300 \times 0.0002739726 \times 3 \approx 9300 \times 0.0008219178 \approx 7.64.
  $$

- **Period 2:**
  $$
  I_2 = 7500 \times 0.0002739726 \times 9 \approx 7500 \times 0.0024657534 \approx 18.49.
  $$

- **Period 3:**
  $$
  I_3 = 10400 \times 0.0002739726 \times 6 \approx 10400 \times 0.0016438356 \approx 17.09.
  $$

- **Period 4:**
  $$
  I_4 = 14800 \times 0.0002739726 \times 13 \approx 14800 \times 0.0035616438 \approx 52.68.
  $$

Now add all the interest charges:

$$
I_\text{total} = I_1 + I_2 + I_3 + I_4 \approx 7.64 + 18.49 + 17.09 + 52.68 \approx 95.91.
$$

At the end of the month, the principal balance (after all transactions) is

$$
\text{Principal} = 14800.
$$

Thus, the new balance including accrued interest is:

$$
\text{New Balance} = 14800 + 95.91 \approx 14895.91.
$$

Rounded to the nearest cent, the new balance is $$\$14,895.91$$.
The new balance is \$14,895.91.

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