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Chapter 1: Problem 28

Use the appropriate formula to solve each problem. Simple Interest If you borrow 100 dollars and pay back 105 dollars at the end of one month, then what is the simple annual interest rate?

Short Answer

Expert verified
The simple annual interest rate is 60%.

Step by step solution

01

- Understand the Simple Interest Formula

The simple interest formula is given by \( I = P \times R \times T \), where \( I \) is the interest, \( P \) is the principal amount, \( R \) is the rate of interest per time period, and \( T \) is the time the money is borrowed for.
02

- Identify Known Values

In this problem, the principal \( P \) is 100 dollars, the total amount paid back is 105 dollars, and since the loan period is one month, the time \( T \) is \( \frac{1}{12} \) years. The interest \( I \) can be calculated as \( 105 - 100 = 5 \) dollars.
03

- Rearrange the Formula to Solve for the Interest Rate

Rearrange the simple interest formula to solve for the interest rate \( R \): \[ R = \frac{I}{P \times T} \]
04

- Substitute the Known Values into the Formula

Substitute \( I = 5 \), \( P = 100 \), and \( T = \frac{1}{12} \) into the formula: \[ R = \frac{5}{100 \times \frac{1}{12}} \]
05

- Calculate the Simple Annual Interest Rate

Calculate the value: \[ R = \frac{5}{100 \times \frac{1}{12}} = \frac{5 \times 12}{100} = \frac{60}{100} = 0.60 \] Thus, the simple annual interest rate is 60%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Interest Formula
In finance and lending, the simple interest formula is your best friend for quick calculations. The formula is expressed as: \( I = P \times R \times T \), where:
  • I is the Interest
  • P is the Principal amount
  • R is the Rate of interest per time period
  • T is the Time period
The formula is used to calculate the interest accrued on a principal over a specified period. Simple interest is a straightforward method to understand how your investment or loan grows over time.
Principal Amount
The principal amount, denoted by \( P \), is the initial sum of money borrowed or invested. It is the starting point in our interest calculations. For example, if you take out a loan of \(100, the principal amount is \)100. Understanding the principal is crucial because it directly impacts the interest calculation. The more significant the principal, the higher the calculated interest, given the same rate and time period.
Loan Period Calculation
The loan period, represented by \( T \), is the time frame over which the loan is taken or the investment is made. It is crucial to express this period in the same time unit used for the interest rate. Often, the time period is converted into years for standardization. For instance, if a loan is taken for one month, it translates to \( \frac{1}{12} \) of a year. This conversion ensures consistency and accuracy in our calculations.
Rate of Interest
The rate of interest, denoted as \( R \), is the percentage charged on the principal for a specific time period. This rate can differ based on the lender, borrower鈥檚 creditworthiness, and other factors. For simple interest calculations, the rate is typically expressed as an annual percentage. In our example, if you need to find the annual rate from a monthly context, the formula rearrangement helps you calculate it correctly. Knowing the rate of interest is vital for understanding how much extra you will pay or earn over the loan or investment period.
Step-by-Step Solution
Let's tie everything together with a step-by-step approach:
Step 1 - Understand the Simple Interest Formula: \( I = P \times R \times T \)
Step 2 - Identify Known Values: \( P = 100 \), \( I = 5 \), \( T = \frac{1}{12} \)
Step 3 - Rearrange the formula to solve for \( R \): \[ R = \frac{I}{P \times T} \]
Step 4 - Substitute and solve: \[ R = \frac{5}{100 \times \frac{1}{12}} = \frac{5 \times 12}{100} \]
Step 5 - Calculate: \[ R = \frac{60}{100} = 0.60 \], so the annual interest rate is 60%. This methodical approach ensures you tackle each part of the problem systematically. By breaking it down step-by-step, you can simplify complex calculations.

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