Calculate the difference in earnings after 2 years between a $3,000 investment at 7% simple interest and a $3,000 investment at 5% compound interest, compounded annually.

Steps to Solve

1. Calculate Simple Interest

Simple interest is calculated using the formula: $SI = P \times r \times t$, where $P$ is the principal amount, $r$ is the interest rate, and $t$ is the time in years. Given: $P = $3000$, $r = 7% = 0.07$, $t = 2$ years. $$SI = 3000 \times 0.07 \times 2$$

2. Compute Simple Interest Value

$$SI = 3000 \times 0.07 \times 2 = 420$$ So, the simple interest earned is $420.

3. Calculate Compound Interest

Compound interest is calculated using the formula: $A = P(1 + r)^t$, where $A$ is the amount after $t$ years, $P$ is the principal amount, $r$ is the interest rate, and $t$ is the time in years. The compound interest earned is then $CI = A - P$. Given: $P = $3000$, $r = 5% = 0.05$, $t = 2$ years. $$A = 3000(1 + 0.05)^2$$

4. Compute Amount After 2 Years

$$A = 3000(1.05)^2 = 3000 \times 1.1025 = 3307.50$$ So, the amount after 2 years is $3307.50.

5. Compute Compound Interest Value

$$CI = A - P = 3307.50 - 3000 = 307.50$$ So, the compound interest earned is $307.50.

6. Calculate the difference between Simple and Compound Interest

$$Difference = SI - CI = 420 - 307.50 = 112.50$$

7. Determine Which Investment Earned More and By How Much

The simple interest investment earned more than the compound interest investment by $112.50.