Problem 12 - Prealgebra

Question

A These problems are similar to the examples found in this section. They should be set up and solved in the same way. (Problems 1-12 involve simple interest.) Savings Account \(\$ 1,800\) is deposited in a savings account that pays \(6 \%\). If the money is withdrawn at the end of 30 days, how much interest is earned?

Step-by-Step Solution

Answer

The interest earned is approximately \$8.88.

Step 1: Understand Simple Interest Formula

The simple interest formula is given by \( I = P \times r \times t \). Here, \( I \) is the interest earned, \( P \) is the principal amount (\$1,800), \( r \) is the interest rate per year (as a decimal), and \( t \) is the time in years.

Step 2: Convert Interest Rate to Decimal

The interest rate given is \(6\%\). To convert this percentage into a decimal for the formula, divide by 100: \( r = \frac{6}{100} = 0.06 \).

Step 3: Convert Time from Days to Years

The time given is 30 days. Since the interest rate is annual, we need the time in years. There are 365 days in a year, so the time in years is \( t = \frac{30}{365} \).

Step 4: Substitute Values into the Formula

Substitute \( P = 1800 \), \( r = 0.06 \), and \( t = \frac{30}{365} \) into the formula: \[ I = 1800 \times 0.06 \times \frac{30}{365} \].

Step 5: Calculate the Simple Interest

Solve the equation: \[ I = 1800 \times 0.06 \times \frac{30}{365} \approx 8.877 \]. Rounded to the nearest cent, the interest is approximately \$8.88.

Key Concepts

Interest Rate

The interest rate is the percentage of the principal amount that a lender charges for the use of its money. In the context of simple interest, the interest rate is applied over a period of one year. For your calculations, you need to express this rate in decimal form. To convert a percentage into a decimal, divide the percentage by 100. For example, an interest rate of 6% becomes 0.06 when converted to a decimal. This decimal form is crucial as it simplifies the use of the formula for simple interest.

Principal Amount

The principal amount is the original sum of money that is invested or loaned. It serves as the baseline for calculating interest. In the exercise, the principal amount is given as $1,800. Understanding the principal amount is key. It dictates the scale of the interest earned or owed. Without knowing the principal, you cannot apply the simple interest formula effectively. This amount remains unchanged over the specified time for simple interest calculations, ensuring ease of use in computations.

Time Conversion

When working with interest rates, the time period used must align with the time unit of the interest rate. Interest rates are usually annual, but time might be given in other units, like days, months, or years. For accurate calculations, you need to convert any given time to years. In the example exercise, 30 days must be converted into years by dividing by 365, as there are 365 days in one year. This conversion is necessary to match the annual nature of the interest rate.

Interest Calculation

Calculating simple interest involves using the formula:

• \( I = P \times r \times t \)

Here:

• \( I \) is the interest,
• \( P \) is the principal (\(1,800),
• \( r \) is the interest rate in decimal (0.06), and
• \( t \) is the time in years (\( \frac{30}{365} \)).

You plug these values into the formula to find \( I \). Multiplying them will give you the interest earned over the specified period. For our example, the interest, when rounded, amounts to approximately \)8.88. This simple approach provides clarity and ease in understanding how money grows over time with simple interest.