Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 500\) at \(4 \%\) for 1 year

Short answer

Answer: The simple interest earned on the investment is $20, and the final balance after 1 year is $520.

Step-by-step solution

Convert the interest rate to a decimal

Since the interest rate is given as a percentage (4%), we need to convert it to a decimal by dividing by 100. So, \(\displaystyle \frac{4}{100} = 0.04\). Hence, our annual interest rate (\(r\)) is 0.04.

Input values into the simple interest formula

We will now input the values into the simple interest formula \(I = Prt\). Our principal (\(P\)) is \(500, our annual interest rate (\)r\() is \)0.04, and our number of years (\(t\)) is 1.

Calculate the simple interest

Plug the values into the formula and perform the calculation. \(I = Prt = (500)(0.04)(1) = 20\) Hence, the simple interest earned on the investment is $20.

Calculate the final balance

Now that we know the simple interest earned, we can find the final balance by adding the simple interest to the initial investment (\(P\)). \(\displaystyle Final \ Balance = P + I = 500+20 = \$520\) So, the final balance after 1 year is $520.

Key concepts

Simple Interest Formula

Understanding the simple interest formula is essential when dealing with basic financial calculations. The formula is expressed as \( I = Prt \), where \( I \) stands for the interest earned, \( P \) is the principal amount (the initial sum of money), \( r \) is the annual interest rate in decimal form, and \( t \) is the time the money is invested or borrowed for, in years.

When you plug in the values from the exercise, you can see how each part of the formula corresponds to a specific piece of information you would have: the principal ($500), the interest rate (converted to 0.04), and the time period (1 year). Multiplying these values together gives you the interest earned over the specified time without considering any compounding. In essence, it tells you how much money you can expect to make from your investment solely based on the time and rate offered.

Converting Percentage to Decimal

To use an interest rate in the simple interest formula, you must first convert it from a percentage to a decimal. This is a crucial step for any interest rate calculation.

To convert a percentage to a decimal, you simply divide it by 100. For example, a 4% interest rate becomes 0.04 after conversion (\(4 \div 100 = 0.04\)). This is because a percent represents a part out of 100. So by dividing by 100, you're turning the 'part out of 100' into a 'part of 1', which is what a decimal represents. Being comfortable with converting percentages to decimals is important not just for calculating simple interest, but for all sorts of financial calculations and beyond.

Interest Rate Calculation

Calculating the interest rate, when not readily provided, can be approached by rearranging the simple interest formula to solve for \( r \). The formula transformed would be \( r = \frac{I}{Pt} \). However, in our given problem, the interest rate is directly provided as 4%, which we converted to a decimal before using it in our formula.

Understanding how the interest rate affects your earnings is crucial. A higher rate means higher interest earned, while a longer time period also contributes to a greater return. If you know the interest earned and the time, but not the rate, you can find it using the rearranged formula. The interest rate is the percentage of the principal that one earns (or owes) per year. Being able to calculate it can help you compare different financial products or loans, and make informed decisions about where to invest or borrow money.