How much is $1000 worth at the end of 2 years, if the interest rate of 6% is compounded daily?
A $1060.00
B $1127.49
C $1182.43
D $1083.27
E $1105.16
F $1183.54
The Correct Answer and Explanation is :
To determine how much $1000 is worth at the end of 2 years with an interest rate of 6% compounded daily, we use the compound interest formula:
[
A = P \left(1 + \frac{r}{n}\right)^{nt}
]
Where:
- (A) is the amount of money accumulated after interest (including principal).
- (P) is the principal amount (initial investment), which is $1000.
- (r) is the annual interest rate (decimal), which is 6% or 0.06.
- (n) is the number of times interest is compounded per year, which is 365 for daily compounding.
- (t) is the time the money is invested or borrowed for, in years, which is 2 years.
Step-by-step Calculation:
- Identify the values:
- (P = 1000)
- (r = 0.06)
- (n = 365)
- (t = 2)
- Substitute these values into the formula:
[
A = 1000 \left(1 + \frac{0.06}{365}\right)^{365 \times 2}
]
- Simplify the equation:
- First, calculate ( \frac{0.06}{365} \approx 0.000164384 )
- Then, calculate ( 365 \times 2 = 730 )
So the equation becomes:
[
A = 1000 \left(1 + 0.000164384\right)^{730}
]
- Evaluate the expression:
- ( 1 + 0.000164384 = 1.000164384 )
- Raise this to the power of 730: ( (1.000164384)^{730} \approx 1.12749 )
- Multiply by the principal:
[
A = 1000 \times 1.12749 = 1127.49
]
Thus, the amount after 2 years is $1127.49.
Answer:
The correct answer is B) $1127.49.
Explanation:
When interest is compounded daily, the money earns interest each day, which results in slightly higher growth compared to annual compounding. In this case, the 6% annual interest rate is split into daily increments, and the number of compounding periods (365 per year) plays a significant role in increasing the final amount. Over the 2-year period, this compounding effect results in an accumulation of $1127.49, showing the power of compound interest over time.