Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount at the end of \(8 \mathrm{mo}\) on a $$\$ 1200$$ bank deposit paying simple interest at a rate of \(7 \% /\) year.

Step-by-Step Solution

Answer

The accumulated amount at the end of 8 months for a $1200 deposit with a simple interest rate of 7% per year is $1256.

Step 1: Convert Months to Years

We will convert the time (T) from months to years; \[ T = \frac{8 \mathrm{mo}}{12 \mathrm{mo/yr}} \] \(T = \frac{2}{3} \mathrm{yr} \)

Step 2: Calculate Simple Interest

Now, we will put the given values into the formula for simple interest to calculate the interest earned ( I ): \[ I = P * R * T \] \[ I = 1200 * 0.07 * \frac{2}{3} \] \[ I = 56 \]

Step 3: Calculate Accumulated Amount

Finally, add the interest earned (I) to the principal amount (P) to find the accumulated amount (A) at the end of the 8 months: \[ A = P + I \] \[ A = 1200 + 56 \] \[ A = $ 1256 \] The accumulated amount at the end of 8 months for a \(1200 deposit with a simple interest rate of 7% per year is \)1256.

Key Concepts

Accumulated Amount

When we're dealing with bank deposits and simple interest, it's essential to understand what the accumulated amount means. Simply put, the accumulated amount is the total sum of money you have after earning interest on your initial deposit (the principal). To calculate the accumulated amount, you need to add the interest earned to the principal.

In our example, the principal is $1200. The interest earned over 8 months is calculated as $56. So, the accumulated amount is found by adding these together:

• Principal = $1200
• Interest Earned = $56
• Accumulated Amount = Principal + Interest = $1200 + $56 = $1256

Understanding this concept helps you see how your money grows when it's deposited in a bank account with interest.

Interest Rate

The interest rate is a crucial factor that determines how much interest you earn on your deposited money. In simple interest calculations, the interest rate is typically expressed as a percentage per year, also known as the annual interest rate.

In the exercise example, we have an interest rate of 7% per year. This means for every $100 you deposit, you earn $7 as interest, provided it's held for a full year. However, if the deposit term is shorter, like our 8-month example, the actual interest earned will be proportionately less than the annual rate due to the shorter time frame.

• Interest Rate = 7% per year
• Shows the percentage growth of your money in one year

To use the interest rate in calculations, convert it to a decimal by dividing by 100 (7% becomes 0.07). This step is crucial for plugging into the simple interest formula, making it an integral component in calculating interest earned.

Time Conversion

To accurately calculate simple interest, converting time into the correct unit is necessary when the interest rate is annual. This conversion ensures that time frames match up with the rate's time scale.

In our scenario, the deposit duration is 8 months, but the interest rate is per year. So, we need to convert months into years for compatibility. Here's how:

• Time Conversion Formula: \( T = \frac{\text{months}}{12} \)
• For 8 months: \( T = \frac{8}{12} = \frac{2}{3} \text{ years} \)

This conversion process is vital as it aligns the interest calculation with the yearly rate. Without it, the interest calculation would not correctly represent the interest earned over the specified period.