To determine which investment will earn the most money, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per unit \( t \).
- \( t \) is the time the money is invested for, in years.
We will calculate the total amount \( A \) for each investment.
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Investment 1: $2,800 at 6% for 2 years compounded monthly.
- \( P = 2800 \)
- \( r = 0.06 \)
- \( n = 12 \)
- \( t = 2 \) \[ A = 2800 \left(1 + \frac{0.06}{12}\right)^{12 \times 2} \] \[ A = 2800 \left(1 + 0.005\right)^{24} \] \[ A = 2800 \left(1.005\right)^{24} \] \[ A \approx 2800 \times 1.12749 \approx 3,159.97 \]
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Investment 2: $1,200 at 2% for 8 years compounded monthly.
- \( P = 1200 \)
- \( r = 0.02 \)
- \( n = 12 \)
- \( t = 8 \) \[ A = 1200 \left(1 + \frac{0.02}{12}\right)^{12 \times 8} \] \[ A = 1200 \left(1 + 0.0016667\right)^{96} \] \[ A = 1200 \left(1.0016667\right)^{96} \] \[ A \approx 1200 \times 1.173 \approx 1,407.60 \]
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Investment 3: $4,200 at 12% for 4 years compounded monthly.
- \( P = 4200 \)
- \( r = 0.12 \)
- \( n = 12 \)
- \( t = 4 \) \[ A = 4200 \left(1 + \frac{0.12}{12}\right)^{12 \times 4} \] \[ A = 4200 \left(1 + 0.01\right)^{48} \] \[ A = 4200 \left(1.01\right)^{48} \] \[ A \approx 4200 \times 1.644 \approx 6,905.55 \]
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Investment 4: $2,400 at 4% for 12 years compounded monthly.
- \( P = 2400 \)
- \( r = 0.04 \)
- \( n = 12 \)
- \( t = 12 \) \[ A = 2400 \left(1 + \frac{0.04}{12}\right)^{12 \times 12} \] \[ A = 2400 \left(1 + 0.0033333\right)^{144} \] \[ A = 2400 \left(1.0033333\right)^{144} \] \[ A \approx 2400 \times 1.490 \approx 3,576.00 \]
Now, let's summarize the computed amounts:
- Investment 1: $3,159.97
- Investment 2: $1,407.60
- Investment 3: $6,905.55
- Investment 4: $3,576.00
From the calculations, the investment that earns the most amount of money is Investment 3: $4,200 at 12% interest compounded monthly over 4 years.