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Tom's Personal Loan: Option B



Background Info: Tom finds a second personal loan option. This loan would also require him to repay the principal in one lump sum after three years.



Loan Option B:

Principal: \$9,000

Type of Interest: Compound Interest

Interest Rate: 8%

Rate of Accrual: Once per year



Use the formula for annual compound interest:

$A=P\left(1+\frac{r}{n}\right)^{n t}$



Remember, $A$ refers to the total amount owed. Calculate the total amount that Tom would repay.


A\$10,337
B\$11,337
C\$12,337
D\$13,337

Verified Answer :

  • Identify the principal \(P = \$9,000\), annual interest rate \(r = 0.08\), frequency \(n = 1\), and time \(t = 3\) years.
  • Substitute these into the formula \(A = P(1 + \frac{r}{n})^{nt}\), resulting in \(A = 9,000(1.08)^3\).
  • Calculate the total repayment amount: \(A = 11,337.408\).
  • Conclude that the total repayment is \(\boxed{\$11,337}\).

Explanation

  1. Identifying Loan Parameters

To determine the total amount Tom would repay for his loan, we identify the key financial parameters given:

  • The principal amount (P) is \(\$9,000\).
  • The annual interest rate (r) is \(8\%\), which as a decimal is \(0.08\).
  • The interest is compounded once per year, meaning the rate of accrual (n) is \(1\).
  • The duration of the loan (t) is \(3\) years.

We use the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) represents the total amount owed after the given time period.

  1. Setting up the Equation

Substituting the identified values into the compound interest formula:
\[A = 9,000 \left(1 + \frac{0.08}{1}\right)^{1 \times 3}\]
Simplifying the expression inside the parentheses:
\[1 + 0.08 = 1.08\]
Calculating the exponent:
\[1 \times 3 = 3\]
Now the equation becomes:
\[A = 9,000 \times (1.08)^3\]

  1. Calculating the Final Amount

We calculate the value of \((1.08)^3\):
\[1.08 \times 1.08 = 1.1664\]
\[1.1664 \times 1.08 = 1.259712\]
Multiplying this by the principal amount \(P = 9,000\):
\[A = 9,000 \times 1.259712 = 11,337.408\]
Rounding to the nearest dollar, the total amount Tom would repay is \(\$11,337\). Comparing this result to the given options, we find it matches the second option. Thus, the total amount is \(\boxed{\$11,337}\).

Examples

Understanding compound interest is essential for managing personal finances. For example, when you take out a loan or invest money in a savings account, interest is often compounded annually or monthly. By using the compound interest formula, you can calculate the total growth of your debt or savings over time, helping you make informed decisions about your financial future. This mathematical approach allows you to see clearly how even a small interest rate can impact the final balance significantly over several years.

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