COMPOUND INTEREST

Use I = (P × (1 + R/N)NT) – P and A = (P × (1 + R/N)NT)  to solve these compound interest problems. Remember that financial answers should be rounded to the nearest hundredth.

1.   Suzanne is looking at taking out a personal loan. Opportunity Loans is offering her $1600 at 3.45% for 1 year, with interest compounding monthly. General Loans is offering her $1600 at 4.2% for 1 year, with interest compounding quarterly. Assuming she makes no payments until the 1 year is up, how much interest will have accrued on each loan? On which loan will she earn the least amount of interest?

Opportunity Loans = $Answer 1 Question 4
in accrued interest.

General Loans = $Answer 2 Question 4
in accrued interest.

The loan from Answer 3 Question 4
Opportunity Loans
will earn the least amount of interest.

2.     General. Gavin wants to put $3,475 into a savings account when his daughter is born. Examine the account options below to determine which will have a higher accrued value at the end of 5 years.

Community Bank

1.90%

Compounds daily

Capital Bank

2.05%

Compounds monthly

The loan from Community Bank will accrue $Answer 4 Question 4

The loan from Capital Bank will accrue Answer 5 Question 4
.

The loan from Answer 6 Question 4
Capital Bank
will have a higher accrued value after 5 years.

Let's solve each problem step by step using the formulas provided.

### **Problem 1: Suzanne's Personal Loan**

**Formulas:**
- **Interest Accrued**: $ I = P \times \left(1 + \frac{R}{N}\right)^{NT} - P $
- **Accrued Amount**: $ A = P \times \left(1 + \frac{R}{N}\right)^{NT} $

Where:
- $ P $ = Principal amount (initial loan amount)
- $ R $ = Annual interest rate (as a decimal)
- $ N $ = Number of compounding periods per year
- $ T $ = Time the money is invested or borrowed for (in years)

#### **Opportunity Loans:**
- $ P = 1600 $ 
- $ R = 3.45\% = 0.0345 $
- $ N = 12 $ (monthly compounding)
- $ T = 1 $ year

Using the interest formula:
\[
I = 1600 \times \left(1 + \frac{0.0345}{12}\right)^{12 \times 1} - 1600
\]

Let's calculate this:

\[
I = 1600 \times \left(1 + 0.002875\right)^{12} - 1600
\]
\[
I = 1600 \times \left(1.035106\right) - 1600
\]
\[
I = 1600 \times 1.035106 - 1600
\]
\[
I = 1656.17 - 1600
\]
\[
I = 56.17
\]

**Opportunity Loans Interest:** \$56.17

#### **General Loans:**
- $ P = 1600 $ 
- $ R = 4.2\% = 0.042 $
- $ N = 4 $ (quarterly compounding)
- $ T = 1 $ year

Using the interest formula:
\[
I = 1600 \times \left(1 + \frac{0.042}{4}\right)^{4 \times 1} - 1600
\]

Let's calculate this:

\[
I = 1600 \times \left(1 + 0.0105\right)^{4} - 1600
\]
\[
I = 1600 \times \left(1.042832\right) - 1600
\]
\[
I = 1668.53 - 1600
\]
\[
I = 68.53
\]

**General Loans Interest:** \$68.53

**Comparison:**
- **Least Interest**: Suzanne will earn the least interest with **Opportunity Loans** (\$56.17).

### **Problem 2: Gavin's Savings Account**

#### **Community Bank:**
- $ P = 3475 $ 
- $ R = 1.90\% = 0.019 $
- $ N = 365 $ (daily compounding)
- $ T = 5 $ years

Using the accrued amount formula:
\[
A = 3475 \times \left(1 + \frac{0.019}{365}\right)^{365 \times 5}
\]

Let's calculate this:

\[
A = 3475 \times \left(1 + 0.00005205\right)^{1825}
\]
\[
A = 3475 \times \left(1.098388\right)
\]
\[
A = 3818.15
\]

**Community Bank Accrued Value:** \$3818.15

#### **Capital Bank:**
- $ P = 3475 $ 
- $ R = 2.05\% = 0.0205 $
- $ N = 12 $ (monthly compounding)
- $ T = 5 $ years

Using the accrued amount formula:
\[
A = 3475 \times \left(1 + \frac{0.0205}{12}\right)^{12 \times 5}
\]

Let's calculate this:

\[
A = 3475 \times \left(1 + 0.00170833\right)^{60}
\]
\[
A = 3475 \times \left(1.106507\right)
\]
\[
A = 3844.63
\]

**Capital Bank Accrued Value:** \$3844.63

**Comparison:**
- **Higher Accrued Value**: The Capital Bank account will have the higher accrued value after 5 years (\$3844.63).

### **Summary:**

1. **Opportunity Loans Interest Accrued:** \$56.17
2. **General Loans Interest Accrued:** \$68.53
3. **Least Interest Loan:** Opportunity Loans
4. **Community Bank Accrued Value:** \$3818.15
5. **Capital Bank Accrued Value:** \$3844.63
6. **Higher Accrued Value:** Capital Bank

Would you like more details or have any questions?

**Related Questions:**

1. How does increasing the compounding frequency affect the interest accrued on a loan?
2. What is the impact of a longer loan term on the total interest accrued?
3. How do different interest rates affect the decision to choose between two savings accounts?
4. How would semi-annual compounding compare to monthly compounding for these loans?
5. What happens to the accrued interest if Suzanne decides to pay off the loan early?

**Tip:** When comparing financial options, always consider both the interest rate and the compounding frequency, as these can significantly affect the final amount of interest or accrued value.

Learn how to solve compound interest problems using formulas. Calculate the interest accrued on personal loans and the accrued value of savings accounts with different compounding frequencies and interest rates. Suitable for high school students studying financial mathematics.

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