To answer question number 33, we will calculate the future value of $10,000 invested at an 8% annual interest rate for 2 years using both simple interest and compound interest. Then we'll determine how much more is earned using compound interest versus simple interest.
1. **Simple Interest Calculation**:
Simple interest is calculated using the formula:
\[ I = P \times r \times t \]
where:
- \( I \) is the interest,
- \( P \) is the principal amount ($10,000),
- \( r \) is the annual interest rate (8% or 0.08 as a decimal), and
- \( t \) is the time in years (2 years).
Let's calculate the interest earned using simple interest:
\[ I = \$10,000 \times 0.08 \times 2 = \$1,600 \]
The future value (\( FV \)) using simple interest is the sum of the principal and the interest earned:
\[ FV_{simple} = P + I = \$10,000 + \$1,600 = \$11,600 \]
2. **Compound Interest Calculation**:
Compound interest is calculated using the formula:
\[ FV = P \times (1 + r)^t \]
where:
- \( FV \) is the future value,
- \( P \) is the principal amount ($10,000),
- \( r \) is the annual interest rate (8% or 0.08 as a decimal), and
- \( t \) is the time in years (2 years).
For compound interest, the interest is added to the principal at the end of each compounding period (annually in this case), and the next interest calculation is based on the new principal amount.
Let's calculate the future value using compound interest:
\[ FV_{compound} = \$10,000 \times (1 + 0.08)^2 \]
\[ FV_{compound} = \$10,000 \times (1.08)^2 \]
\[ FV_{compound} = \$10,000 \times 1.1664 \]
\[ FV_{compound} = \$11,664 \]
The difference in earnings between compound interest and simple interest:
\[ Difference = FV_{compound} - FV_{simple} \]
\[ Difference = \$11,664 - \$11,600 \]
\[ Difference = \$64 \]
So, the future value using simple interest is $11,600, while the future value using compound interest is $11,664. Therefore, using compound interest earns an additional $64 over simple interest after 2 years on the initial $10,000 investment at an 8% annual interest rate.
