Problem 58 - Algebra for College Students

Question

A certificate of deposit for 1 yr pays $25.50 simple interest on a principal of 3400. What is the interest rate being paid on this deposit?

Step-by-Step Solution

Answer

0.75%

Step 1 - Identify the Given Values

Identify the given values in the problem: the principal (P) is 3400, the interest (I) is 25.50, and the time (T) is 1 year.

Step 2 - Recall the Simple Interest Formula

Recall the formula for simple interest: \[ I = P \times R \times T \] where I is the interest, P is the principal, R is the interest rate, and T is the time.

Step 3 - Rearrange the Formula to Solve for the Interest Rate

Rearrange the formula to solve for the interest rate (R): \[ R = \frac{I}{P \times T} \]

Step 4 - Substitute the Given Values into the Formula

Substitute the given values into the formula: \[ R = \frac{25.50}{3400 \times 1} \]

Step 5 - Calculate the Interest Rate

Calculate the interest rate: \[ R = \frac{25.50}{3400} = 0.0075 \]

Step 6 - Convert the Interest Rate to a Percentage

Convert the decimal interest rate to a percentage by multiplying by 100: \[ R = 0.0075 \times 100 = 0.75\text{%} \]

Key Concepts

Simple Interest Formula

Simple interest is a way to calculate the interest you earn on a principal amount over a period of time. The formula for simple interest is quite straightforward and easy to remember: \[ I = P \times R \times T \] Here,

• \( I \) stands for the interest earned
• \( P \) is the principal amount, which is the initial sum of money that you invested or loaned
• \( R \) represents the interest rate, which is the percentage of the principal that you earn as interest each year
• \( T \) is the time period for which the money is invested or loaned, usually expressed in years

Using this formula, you can quickly determine how much interest you would earn for a given principal, interest rate, and time period. For example, if you know \( P \), \( R \), and \( T \), you can calculate \( I \) directly by plugging in these values. If you need to find any other variable, you can rearrange the formula accordingly.

Interest Rate Calculation

Calculating the interest rate helps you understand how much return you are getting on your investment or loan. Let’s break down the steps we used in the original exercise to find the interest rate on a certificate of deposit.First, recall the formula for simple interest: \[ I = P \times R \times T \]In the given problem, we already know the principal \( P = 3400 \), interest \( I = 25.50 \), and time \( T = 1 \) year. To find the interest rate \( R \), we need to rearrange the formula. The rearranged formula to find \( R \) is: \[ R = \frac{I}{P \times T} \]Next, we substitute the known values into this formula:\[ R = \frac{25.50}{3400 \times 1} \]After performing the division, we get:\[ R = \frac{25.50}{3400} = 0.0075 \]The result is in decimal form. To convert it into a percentage, multiply by 100:\[ R = 0.0075 \times 100 = 0.75\% \]Now we know the interest rate is 0.75%. Calculating interest rates like this helps you evaluate different investment opportunities or understand your loan agreements better.

Principal and Interest

Principal and interest are two fundamental concepts in finance. Understanding them is crucial for making informed financial decisions. 1. **Principal**:The principal is the initial amount of money you invest or borrow. For example, in our exercise, the principal amount is \( 3400 \). This is the sum you start with. When you invest money, the principal is what earns interest. When you borrow money, the principal is the amount you owe and on which you pay interest.2. **Interest**:Interest is the reward for lending your money or the cost of borrowing money, usually expressed as a percentage of the principal. In simple interest calculations, it is straightforward to calculate based on the three key variables: principal, interest rate, and time.In our previous example, the interest earned on a \( 3400 \) principal over 1 year at an interest rate of \( 0.75\% \) is \( 25.50 \). Understanding the relationship between these concepts with calculations helps you manage investments and loans more effectively.