Problem 109 - Elementary and Intermediate Algebra

Question

Compound Interest. If $$ 500\( is deposited in an account paying \)8.5 \%\( annual interest, compounded semiannually, how long will it take for the account to increase to $$ 800 ?\)

Step-by-Step Solution

Answer

It will take approximately 5.57 years for the account to grow to $800.

Step 1: Understand the problem

We need to find out how long it will take for a $500 deposit to grow to $800 in an account that offers 8.5% annual interest, compounded semiannually.

Step 2: Identify the compound interest formula

The formula for compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:- \(A\) is the future amount in the account,- \(P\) is the principal amount (initial deposit),- \(r\) is the annual interest rate (as a decimal),- \(n\) is the number of times interest is compounded per year,- \(t\) is the time in years.

Step 3: Substitute the known values into the formula

Given:- \(A = 800\),- \(P = 500\),- \(r = 8.5\% = 0.085\),- \(n = 2\) (since interest is compounded semiannually).Substitute these into the formula:\[ 800 = 500 \left(1 + \frac{0.085}{2}\right)^{2t} \]

Step 4: Solve for \(t\)

First, simplify \(1 + \frac{0.085}{2}\):\[ 1 + 0.0425 = 1.0425 \]Now substitute it back into the equation:\[ 800 = 500 \cdot (1.0425)^{2t} \]Divide both sides by 500:\[ \frac{800}{500} = (1.0425)^{2t} \]Simplify:\[ 1.6 = (1.0425)^{2t} \]Take the logarithm of both sides to solve for \(t\):\[ \log(1.6) = 2t \cdot \log(1.0425) \]Now solve for \(t\):\[ t = \frac{\log(1.6)}{2 \cdot \log(1.0425)} \]

Step 5: Calculate the exact value of \(t\) using a calculator

Using a calculator, find \(t\):\[ t = \frac{\log(1.6)}{2 \cdot \log(1.0425)} \approx \frac{0.2041}{2 \cdot 0.01832} \approx \frac{0.2041}{0.03664} \approx 5.57 \]Since \(t\) represents time in years, it will take approximately 5.57 years for the balance to reach $800.

Key Concepts

Compound Interest Formula

To understand compound interest, envision it as interest not only on your initial principal but also on the interest that accumulates over time. This means your money can grow faster compared to simple interest, where only the principal earns interest. The compound interest formula is given by:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• \(A\) is the future amount of money you will have.
• \(P\) is the initial principal balance. It’s the amount you start with.
• \(r\) is the annual interest rate, but it's expressed as a decimal. So, for 8.5%, it becomes 0.085.
• \(n\) denotes the number of times the interest is compounded annually. For semiannual compounding, \(n = 2\).
• \(t\) is the time the money is kept in the account, in years.

Using this formula, we can predict how much money will grow over time with varying interest rates and compounding intervals.

Annual Interest Rate Calculation

Understanding the annual interest rate is crucial for managing investments. This rate decides how much interest will apply to your principal over the course of a year.When calculating the compound interest in our scenario, the annual interest rate given is 8.5%. However, because we are dealing with semiannual compounding, this rate needs to be adjusted for the calculations. To adjust the annual rate for semiannual calculations:

• Divide the annual rate by the number of compounding periods. Here, we use 2 because it compounds every 6 months.
• This makes the semiannual rate \(\frac{8.5\%}{2} = 0.0425\) or 4.25% per period.

This adjustment ensures the interest calculations reflect the more frequent accumulation, which results in faster growth compared to yearly compounding.

Semiannual Compounding

Semiannual compounding means interest is calculated twice a year. This method is common in financial institutions because it leverages the power of compound interest more frequently. The steps to understand semiannual compounding better include:

• Recognize that each year, interest is calculated and added to the principal twice.
• The interval here is every six months. Hence, if you start with $500, you earn interest on it after the first six months.
• The new principal available for the next calculation after each six-month period is the old principal plus the accumulated interest.

Using the compound interest formula, and applying these concepts, helps ensure that each compounding period's calculations are accurate. It results in attaining the target amount faster, evident from our example where it helped reach $800 quicker than it would have with annual compounding.