Use the compound-interest formula to find the account balance A, where P is principal, r is interest rate, n is number of compounding periods per year, t is time in years, and A is account balance.
| P | r | compounded | t |
| --------- | ------- | ---------- | - |
| $10,327 | 2.8 % | Daily | 4 |
The account balance is approximately $ [ ]. (Simplify your answer. Do not round until the final answer. Then round to two decimal places as needed.)
Answer (2)
Calculate r/n: 365 0.028 ≈ 0.00007671 .
Calculate 1 + r/n: 1 + 0.00007671 = 1.00007671 .
Calculate nt: 365 × 4 = 1460 .
Calculate A: 10327 × ( 1.00007671 ) 1460 ≈ 11550.83 .
Explanation
Understanding the Problem We are given the principal amount P = $10 , 327 , the annual interest rate r = 2.8% = 0.028 , the number of times the interest is compounded per year n = 365 (daily), and the number of years t = 4 . We need to find the account balance A using the compound interest formula.
Stating the Formula The compound interest formula is given by: A = P ( 1 + n r ) n t
Calculating r/n First, we calculate n r : n r = 365 0.028 ≈ 0.00007671
Calculating 1 + r/n Next, we calculate 1 + n r : 1 + n r = 1 + 0.00007671 = 1.00007671
Calculating nt Then, we calculate n t : n t = 365 × 4 = 1460
Calculating (1 + r/n)^(nt) Now, we calculate ( 1 + n r ) n t : ( 1 + n r ) n t = ( 1.00007671 ) 1460 ≈ 1.11858
Calculating A Finally, we calculate A : A = P × ( 1 + n r ) n t = 10327 × 1.11858 ≈ 11550.83
Final Answer Therefore, the account balance after 4 years is approximately $11550.83 .
Examples
Compound interest is a powerful concept used in many real-world financial scenarios. For example, when you deposit money into a savings account, the bank pays you interest, which is often compounded. Understanding compound interest helps you estimate how much your investments will grow over time. It's also crucial for understanding loans, mortgages, and other financial products where interest accrues over time. Knowing how to calculate compound interest allows you to make informed decisions about saving, investing, and borrowing money.
Using the compound interest formula, we calculated that the account balance after 4 years will be approximately $11,530.02. This was done by determining the interest rate per compounding period, the total number of compounding periods, and applying those values in the compound interest formula. The final amount reflects the growth of the initial principal due to daily compounding at an annual interest rate of 2.8%.
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