Solve for the missing item in the following. Note: Do not round intermediate calculations. Round your answer to the nearest cent.
| Principal | Interest rate | Time | Simple interest |
| --------- | ------------- | ------ | --------------- |
| | 6.50 % | 4 1/4 years | $ |
Answer (1)
The formula for simple interest is I = P × r × t , where I is the simple interest, P is the principal, r is the interest rate, and t is the time.
Convert the interest rate to decimal form: r = 6.50% = 0.065 .
Convert the time to decimal form: t = 4 4 1 = 4.25 years.
Solve for the principal: P = r × t I = 0.065 × 4.25 I = 0.27625 I .
Without the value of I , the principal is: P = 0.27625 I
Explanation
Understanding the Problem and Formula We are given the interest rate, time, and simple interest, and we need to find the principal. Let's denote the principal by P , the interest rate by r , the time by t , and the simple interest by I . The formula for simple interest is: I = P × r × t We need to rearrange this formula to solve for P :
P = r × t I We are given that the interest rate is 6.50% , which can be written as a decimal by dividing by 100: r = 100 6.50 = 0.065 The time is given as 4 4 1 years, which can be written as a decimal as: t = 4 + 4 1 = 4.25 years We are not given the simple interest I , so we cannot find the principal. Let's assume the simple interest is given as $ I . Then we can calculate the principal as: P = 0.065 × 4.25 I = 0.27625 I For example, if the simple interest is $100 , then the principal would be: P = 0.27625 100 ≈ 362.03 So, if the simple interest is $ I , the principal is 0.27625 I .
Calculations and Solving for Principal Let's assume the simple interest is $ I . We have the formula for the principal: P = r × t I We found that r = 0.065 and t = 4.25 . Therefore, r × t = 0.065 × 4.25 = 0.27625 . So, P = 0.27625 I To find the principal, we need the value of the simple interest I . Without this value, we can only express the principal in terms of I .
Final Formula for Principal Since the problem is missing the value of the simple interest, we cannot find a numerical value for the principal. However, we have derived the formula for the principal in terms of the simple interest I :
P = 0.27625 I If we assume the simple interest is $ I , then the principal is 0.27625 I .
Examples
Understanding simple interest is crucial in personal finance. For instance, if you deposit money into a savings account, the bank pays you interest. Knowing how to calculate the principal, interest rate, and time helps you estimate your earnings. For example, if you deposit $1000 at a 5% interest rate for 2 years, you can calculate the simple interest earned as I = 1000 × 0.05 × 2 = $100 . This helps you understand how your savings grow over time.
