To calculate Emma's ending balance after depositing $1200 into an account with a 3% annual interest rate compounded monthly, 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 year.
- \( t \) is the number of years the money is invested for.
For Emma's account:
- \( P = 1200 \)
- \( r = 0.03 \) (3%)
- \( n = 12 \) (compounded monthly)
- \( t = 1 \)
Plugging in the values:
\[ A = 1200 \left(1 + \frac{0.03}{12}\right)^{12 \times 1} \]
Calculating step-by-step:
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Calculate \( \frac{0.03}{12} \): \[ \frac{0.03}{12} = 0.0025 \]
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Add 1 to the interest rate: \[ 1 + 0.0025 = 1.0025 \]
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Calculate \( 12 \times 1 \): \[ 12 \times 1 = 12 \]
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Raise \( 1.0025 \) to the power of 12: \[ (1.0025)^{12} \approx 1.0304 \]
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Multiply by the principal amount: \[ A \approx 1200 \times 1.0304 \approx 1236.48 \]
Thus, Emma's ending balance after one year is approximately $1,236.48. From your options, the closest is $1,236.50.
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