Kuta Software – Infinite Precalculus
Compound Interest
1) Brenda invests $4,848 in a savings account with a fixed annual interest rate of 5% compounded 2 times per year. What will the account balance be after 6 years?
The correct answer and explanation is :
To solve this problem, we can use the formula for compound interest, which is given by:
[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
]
Where:
- ( A ) is the amount of money accumulated after ( t ) years, including interest.
- ( P ) is the principal amount (initial investment).
- ( r ) is the annual interest rate (as a decimal).
- ( n ) is the number of times the interest is compounded per year.
- ( t ) is the time the money is invested or borrowed for, in years.
Given:
- ( P = 4848 ) (initial investment).
- ( r = 5\% = 0.05 ) (annual interest rate).
- ( n = 2 ) (compounded twice per year).
- ( t = 6 ) years.
Step 1: Plug in the values into the formula.
[
A = 4848 \left( 1 + \frac{0.05}{2} \right)^{2 \times 6}
]
[
A = 4848 \left( 1 + 0.025 \right)^{12}
]
[
A = 4848 \left( 1.025 \right)^{12}
]
Step 2: Calculate ( (1.025)^{12} ).
Using a calculator, we find:
[
(1.025)^{12} \approx 1.795856
]
Step 3: Multiply the result by the principal.
[
A = 4848 \times 1.795856 \approx 8706.60
]
So, the account balance after 6 years is approximately $8,706.60.
Explanation:
This problem involves calculating compound interest, which is the process of earning interest on both the initial investment (the principal) and the interest that has already been added to the account. In this case, the interest is compounded semi-annually (twice per year), meaning that the interest is applied twice each year, rather than just once. This leads to a higher overall balance due to the interest being applied to a larger amount each time. The compound interest formula takes into account both the rate of interest and the frequency of compounding. By raising the term ( (1 + \frac{r}{n}) ) to the power of ( nt ), the formula calculates how much the investment will grow over the specified period.