How Long Does It Take to Grow $5000 to $9350 with 8.25% Annual Interest Compounded Semiannually?
Understanding Compound Interest
Compound interest is a powerful financial concept that refers to the interest calculated on the initial principal and also on the accumulated interest of previous periods. In other words, it allows your money to grow at a faster rate than simple interest, which is calculated only on the principal amount. Understanding how compound interest works can help you plan your financial goals, such as saving for college, buying a home, or retiring.Compounding Periods and Interest Rates
In our case, the interest rate is 8.25% annual interest, but the interest is compounded semiannually, meaning the interest is applied every six months. This method of compounding makes a difference in the total amount of interest earned compared to annual compounding.Problem Statement
Let's solve the problem of how long it will take for an initial investment of $5000 to grow to $9350 with an 8.25% annual interest rate compounded semiannually. Here’s the step-by-step breakdown of the process.Step 1: Determine the Growth Factor
First, calculate the growth factor, which is the total amount you want to reach divided by the initial investment: Growth Factor $9350 / $5000 1.87Step 2: Use the Compound Interest Formula
The formula for compound interest is: A P (1 r/n)^(nt) Where: - A is the amount of money accumulated after n years, including interest. - P is the principal amount (initial investment). - r is the annual interest rate (decimal). - n is the number of times that interest is compounded per year. - t is the time the money is invested or borrowed for, in years.In our scenario:
P $5000 r 8.25% or 0.0825 in decimal form n 2 (since interest is compounded semiannually) A $9350Step 3: Solve for t (Time in Years)
We need to solve for t, the time in years required for the investment to grow to $9350. The formula for t is derived from the compound interest formula and rewritten to solve for t.1 × (1 0.0825 / 2)^(N) 1.87
Where N is the number of semiannual periods. This can be further simplified to solve for N:(1 0.0825 / 2)^N 1.87
t N / 2, because N is the number of semiannual periods and we want the result in years.
Step 4: Calculate N
N log(1.87) / log(1 0.0825 / 2) Let's calculate the values:
N log(1.87) / log(1 0.04125)
N log(1.87) / log(1.04125) ≈ 15.49
N is the number of semiannual periods, so to convert this to years:
t 15.49 / 2 ≈ 7.74 years