Problem 23 - Algebra and Trigonometry

Question

Investments What annual rate of interest would you have to earn on an investment of \(\$ 3500\) to ensure receiving \(\$ 262.50\) interest after 1 year?

Step-by-Step Solution

Answer

The annual interest rate is 7.5%.

Step 1: Define the Problem

To find the annual rate of interest, we know the principal amount (\( P = \\( 3500 \)), the interest earned (\( I = \\) 262.50 \)), and the time period, which is 1 year.

Step 2: Understand the Formula

We use the formula for simple interest: \[ I = P \times r \times t \]where \( I \) is the interest, \( P \) is the principal, \( r \) is the rate of interest, and \( t \) is the time in years. Since \( t = 1 \) year, the equation simplifies to \( I = P \times r \).

Step 3: Substitute the Known Values

Insert the known values into the simplified formula: \[ 262.50 = 3500 \times r \]

Step 4: Solve for the Interest Rate

Rearrange the formula to solve for \( r \): \[ r = \frac{262.50}{3500} \]Calculate to find \( r \): \[ r = \frac{262.50}{3500} \approx 0.075 \]

Step 5: Convert Rate to Percentage

Convert the decimal answer to a percentage by multiplying by 100: \[ 0.075 \times 100 = 7.5\% \]

Key Concepts

Understanding Interest Rates

Interest rates are a key factor in investment and savings calculations. They represent the cost of borrowing money or the return on investment. When talking about simple interest, the interest rate is expressed as a percentage of the principal amount and determines how much interest an investment will earn over a period, typically a year.
For example, in the exercise, we needed to find out what interest rate would yield $262.50 from a $3,500 investment in one year. By understanding how interest rates work, we can easily calculate the returns on different investments.

The Principal Amount

The principal amount is the initial sum of money put into an investment or loan. It's the amount on which interest is calculated before any interest is added. In the case of the exercise, the principal amount was $3,500.
It's essential to identify the principal precisely, as it forms the foundation for calculating interest. Once you know the principal, you can effectively gauge how different interest rates will affect your returns.

Calculating Annual Interest

Annual interest refers to the money earned or paid on a principal amount over a year. It is usually calculated using the formula for simple interest: \[ I = P \times r \times t \]where:

• I is the interest earned or paid,
• P is the principal amount,
• r is the annual interest rate,
• t is the time, in years.

In this particular exercise, the time (t) is one year, which simplifies the equation to:\[ I = P \times r \]This equation allows us to determine the interest amount easily when we know the principal and rate, or vice versa. Using this, we can quickly see what an investment will yield over its time period.

Mastering Investment Calculations

Investment calculations are crucial for anyone looking to save or grow their money over time. By mastering these calculations, you can make informed decisions about where to allocate your resources for the best returns.
To determine an interest rate, like in the exercise, we rearrange the simple interest formula to solve for the rate:\[ r = \frac{I}{P} \]For our exercise, this translates to:\[ r = \frac{262.50}{3500} \approx 0.075 \]Converting this to a percentage, by multiplying by 100, we get a 7.5% annual interest rate. This highlights how simple math can provide insights into financial decisions, ensuring that your money works as hard as possible for you.