Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount at the end of \(9 \mathrm{mo}\) on an $$\$ 800\( \)deposit in a bank paying simple interest at a rate of \(6 \% /\) year.

Step-by-Step Solution

Answer

The accumulated amount at the end of 9 months on an $800 deposit in a bank paying simple interest at a rate of 6% per year is $836.

Step 1: Convert interest rate and time period

We are given the interest rate as 6% per year. To convert it into decimal form, divide by 100: \(r = \frac{6}{100} = 0.06\) The time period is given as 9 months. To convert it to years, divide by 12 (as there are 12 months in a year): \(t = \frac{9}{12} = 0.75\)

Step 2: Use the simple interest formula

Now, we will use the formula for simple interest: \(A = P(1 + rt)\) Plugging in the values, we have: \(A = 800(1 + 0.06 * 0.75)\)

Step 3: Calculate the accumulated amount

Perform the calculations: \(A = 800(1 + 0.045)\) \(A = 800(1.045)\) \(A = 836\) So the accumulated amount at the end of 9 months will be $836.

Key Concepts

Accumulated Amount

The accumulated amount refers to the total amount of money in an account after a certain period, considering the initial principal and any interest earned. It helps in understanding how much money you will have after a specified time when interest is applied. In our simple interest problem, we calculate the accumulated amount using the formula \(A = P(1 + rt)\). Here, \(A\) is the accumulated amount, \(P\) is the principal amount, \(r\) is the interest rate per period, and \(t\) is the time period in years. By substituting the values (like the principal \(\$800\) and the interest information), you can find out how much your initial deposit will grow in a given duration.

Interest Rate Conversion

Interest rates can be expressed annually, but sometimes the calculation requires the rate in a different term, like monthly. To convert an annual interest rate to a decimal form usable in formulas, divide it by 100. This converts percentages to decimals.

• Example: \(\text{Annual Rate} = 6\% \)
• Converted to Decimal: \( r = \frac{6}{100} = 0.06\)

When you need to adjust for time periods other than a year, ensure you've correctly converted the rate so that it corresponds to that time. In our exercise, the rate remains the same since the time period adjustment was directly applied to the number of months converted to years.

Interest Calculation

Calculating interest in this scenario involves using the simple interest formula: \( I = Prt \), where \(I\) is the interest earned. However, when we're asked to find the accumulated amount, we prefer using \(A = P(1 + rt)\), as it includes both the principal and interest. Simple interest means that the interest is calculated only on the initial principal, not on accumulated interest. It’s straightforward and easy to compute:

• Calculate interest earned over the specific time by multiplying the principal amount, rate, and time.
• Add this to your initial principal to find the total accumulated value.

The problem guides you through how these calculations result in discovering how much money you will finally have.

Time Period Conversion

To accurately compute interest or accumulated amounts, the time period must be in years if using an annual rate. If you have a time period in months, convert it to years by dividing by 12.

• Given: 9 Months
• Convert to Years: \( t = \frac{9}{12} = 0.75 \)

This conversion ensures that the rate and time periods align in the formula. Using compatible time units is critical to accurate interest calculations, avoiding errors from unmatched time periods. In our exercise, this conversion from months to years was essential for applying the rate correctly over the desired period.