Problem 52 - Prealgebra

Question

Use any of the methods developed in this chapter to solve them. Rate In early January, your bank sent out a form called a 1099 -INT, which summarizes the amount of interest you have received on a savings account for the previous year. If you received \(\$ 72\) interest for the year on an account in which you started with \(\$ 1,200,\) determine the annual interest rate paid by your bank.

Step-by-Step Solution

Answer

The annual interest rate is 6\%.

Step 1: Understand the Given Information

We know the interest received is \(72, the principal amount (initial deposit) is \)1,200, and we need to find the interest rate. The formula to calculate simple interest is \( I = P \times r \times t \) where \( I \) is the interest, \( P \) is the principal, \( r \) is the rate, and \( t \) is the time in years. Since the time is one year, \( t = 1 \).

Step 2: Use the Simple Interest Formula

We substitute the known values into the simple interest formula. The equation becomes \( 72 = 1,200 \times r \times 1 \). This simplifies to \( 72 = 1,200r \).

Step 3: Solve for the Interest Rate

To find \( r \), we need to solve the equation \( 72 = 1,200r \). Divide both sides by 1,200 to isolate \( r \). Thus, \( r = \frac{72}{1,200} \).

Step 4: Calculate the Interest Rate

Perform the division to find \( r \): \( r = \frac{72}{1,200} = 0.06 \). Therefore, the interest rate is 0.06.

Step 5: Convert to Percentage

To express \( r \) as a percentage, multiply by 100. So, \( 0.06 \times 100 = 6\% \). Therefore, the annual interest rate is 6\%.

Key Concepts

Understanding Annual Interest Rate

The annual interest rate is a crucial component of simple interest calculations. It represents the percentage of the principal amount you earn over one year. When you deposit money into a savings account, the bank pays you for keeping your money with them. This payment is calculated using the annual interest rate.

To find the annual interest rate in simple interest problems, you use the formula:

• Simple Interest Formula: \( I = P \times r \times t \)
• Where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years.

In our example, the interest earned is 72 dollars, and the principal is 1,200 dollars. Simplifying 72 = 1,200 \( r \) gives us the interest rate \( r \) as 0.06.

Converting this decimal to a percentage involves multiplying by 100, turning 0.06 into a 6% annual interest rate. This percentage helps compare with rates from different banks or financial products.

Interest Calculation Basics

Interest calculation helps understand how much extra money you earn or owe. Specifically, with simple interest, the amount you gain is proportional to the principal amount, the interest rate, and the period of investment.

Here's a quick breakdown of the steps:

• Identify all components, which include the principal amount (initial investment), rate (as a percentage), and time (generally in years).
• Use the formula to calculate the interest: \( I = P \times r \times t \).
• In our scenario, time is one year, so the equation simplifies to \( I = P \times r \).

Performing these calculations accurately is fundamental to understanding basic financial transactions. It ensures you know what to expect from your investments or savings over time.

Building Financial Literacy through Simple Interest

Financial literacy is the ability to understand and effectively use financial skills, including managing money, budgeting, and investing. Knowing how to calculate simple interest is a foundational skill that contributes significantly to financial literacy.

Why is this important?

• Empowers informed decision-making: Understanding interest rates helps assess savings products and loans.
• Helps in financial planning: By knowing how much interest you will earn or pay, you can better plan your finances.
• Boosts confidence: With a firm grasp of financial concepts, you approach financial matters with assurance.

Interest calculations are part of everyday financial activities. From determining how much extra money you will receive from your bank deposit over a set period to calculating the cost of financing a car, understanding simple interest is crucial. Building these skills improves your financial well-being.