Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 200,000, r=8 \%, t=4, \text { compounded daily } $$

Step-by-Step Solution

Answer

The accumulated amount after investing a principal of \(200,000 at an interest rate of 8% per year for 4 years, compounded daily, is approximately \)296,747.42.

Step 1: Understand the given data

In the exercise, we are given the principal amount (P) as $200,000, the interest rate (r) as 8%, the time period (t) as 4 years, and the compounding frequency is daily.

Step 2: Identify the compound interest formula

The formula to find the accumulated amount (A) when the principal amount (P) is compounded daily is given as: \[A = P\left(1 + \frac{r}{100 \cdot n}\right)^{(n \cdot t)}\] where A is the accumulated amount, P is the principal amount, r is the interest rate, n is the number of times the interest is compounded per year, and t is the time period in years. Since the interest is compounded daily, n = 365.

Step 3: Substitute the given values into the formula

We now substitute the given values into the formula: \[A = 200000 \left(1 + \frac{8}{100\times 365}\right)^{(365\times 4)}\]

Step 4: Simplify and find the accumulated amount

Now we will simplify the expression and calculate the accumulated amount (A): \[A = 200000 \left(1 + \frac{8}{36500}\right)^{(1460)}\] Now, compute the accumulated amount (A): \[A \approx 200000(1.000219)^{1460} \approx 296747.42\]

Step 5: State the final answer

After investing a principal amount of \(200,000 at an interest rate of 8% per year for 4 years, compounded daily, the accumulated amount will be approximately \)296,747.42.

Key Concepts

Accumulated Amount

The accumulated amount is the total sum of money you have after a certain period when interest is applied to the initial principal. In our exercise, we begin with a principal amount (P) of \(200,000. By applying the compound interest formula, we calculate the accumulated amount (A).

• The principal is the original amount invested or deposited.
• Interest is the extra amount earned over time.
• The accumulated amount is principal plus the interest earned.

In the formula used: \[ A = P\left(1 + \frac{r}{100 \cdot n}\right)^{(n \cdot t)} \]

• \( P \) is the initial \)200,000.
• \( r \) is the yearly interest rate.
• \( n \) is the number of times interest compounds annually.
• \( t \) is time in years (4 years in this case).

Through these calculations, you find that the accumulated sum is approximately $296,747.42 when compounded daily.

Interest Rate

The interest rate is the percentage at which your money grows over a specific period. In our context, the interest rate ( r ) is given as 8% per year. This means every year, the principal can potentially grow by 8% before considering compounding factors.

• A higher interest rate results in faster growth of your principal.
• The rate you earn interest is expressed as a percentage.
• Annual interest affects how much returns you see each year from your investment.

Whenever you're dealing with financial calculations, understanding how to apply and interpret an interest rate can significantly impact decisions you make regarding investments and savings. It's crucial to recognize whether an interest rate is simple or compounded, as compound interest can have a dramatic effect on growth over time.

Compounding Frequency

Compounding frequency refers to how often the interest is calculated and added to the principal balance. In our example, the frequency of compounding is daily. This means interest is added to your deposit every day (365 times a year).

• The more frequently interest is compounded, the more your accumulated amount will grow due to interest earning on previously earned interest.
• Daily compounding typically yields a higher accumulated amount compared to monthly or yearly compounding.
• This frequency significantly influences the total interest earned over time, highlighting its importance in compound interest calculations.

When interest is compounded daily, small increments added on a regular basis create an amplified effect on overall returns. It's essential to consider the compounding frequency when planning investments, as it can make a considerable difference to the final accumulated amount.