Shown Below Are The Steps A Student Took To Solve The Simple Interest Formula A = P ( 1 + R T A = P(1 + Rt A = P ( 1 + R T ] For R R R .${ \begin{align} A &= P(1 + Rt) \ A &= P + Rt \ A - P &= Rt \ r &= \frac{A - P}{t} \end{align} }$Explain The

Mar 12, 2025 by ADMIN 248 views

**Solving for the Rate of Interest in the Simple Interest Formula**

Introduction

The simple interest formula is a fundamental concept in mathematics, particularly in finance and economics. It is used to calculate the interest earned on a principal amount over a specific period of time. The formula is given by $A = P(1 + rt)$ , where $A$ is the amount of money accumulated after $t$ years, including interest, $P$ is the principal amount, $r$ is the rate of interest, and $t$ is the time in years. In this article, we will discuss the steps a student took to solve for the rate of interest $r$ in the simple interest formula.

Step 1: Isolate the Variable

The first step in solving for $r$ is to isolate the variable on one side of the equation. In this case, we want to isolate $r$ . To do this, we need to get rid of the term $P$ on the right-hand side of the equation. We can do this by subtracting $P$ from both sides of the equation.

A - P &= P + rt - P \\
A - P &= rt

Step 2: Factor Out the Common Term

The next step is to factor out the common term $t$ from the right-hand side of the equation. This will allow us to isolate $r$ .

A - P &= rt \\
A - P &= t \cdot r

Step 3: Divide Both Sides by t

Now that we have factored out the common term $t$ , we can divide both sides of the equation by $t$ to isolate $r$ .

\frac{A - P}{t} &= r

Conclusion

In this article, we have discussed the steps a student took to solve for the rate of interest $r$ in the simple interest formula. By isolating the variable, factoring out the common term, and dividing both sides by $t$ , we were able to solve for $r$ . The final formula for solving for $r$ is given by $r = \frac{A - P}{t}$ . This formula is a fundamental concept in mathematics and is used extensively in finance and economics.

Real-World Applications

The simple interest formula has many real-world applications. For example, it can be used to calculate the interest earned on a savings account or a loan. It can also be used to determine the rate of interest on a investment. In addition, the simple interest formula can be used to calculate the interest earned on a bond or a stock.

Example Problems

Here are a few example problems that illustrate how to use the simple interest formula to solve for $r$ .

Example 1

Suppose we want to calculate the interest earned on a principal amount of $1000 over a period of 5 years at a rate of 5% per year. Using the simple interest formula, we can calculate the interest earned as follows:

A &= P(1 + rt) \\
A &= 1000(1 + 0.05 \cdot 5) \\
A &= 1000(1 + 0.25) \\
A &= 1000 \cdot 1.25 \\
A &= 1250

Example 2

Suppose we want to determine the rate of interest on a investment that earns $500 in interest over a period of 10 years on a principal amount of $1000. Using the simple interest formula, we can calculate the rate of interest as follows:

r &= \frac{A - P}{t} \\
r &= \frac{500 - 1000}{10} \\
r &= \frac{-500}{10} \\
r &= -0.05 \\
r &= -5\%

Conclusion

Q: What is the simple interest formula?

A: The simple interest formula is a mathematical formula used to calculate the interest earned on a principal amount over a specific period of time. The formula is given by $A = P(1 + rt)$ , where $A$ is the amount of money accumulated after $t$ years, including interest, $P$ is the principal amount, $r$ is the rate of interest, and $t$ is the time in years.

Q: How do I calculate the interest earned using the simple interest formula?

A: To calculate the interest earned using the simple interest formula, you need to plug in the values of $P$ , $r$ , and $t$ into the formula. The formula is $A = P(1 + rt)$ . You can then calculate the interest earned by subtracting the principal amount from the total amount.

Q: What is the rate of interest (r) in the simple interest formula?

A: The rate of interest (r) is the percentage of the principal amount that is earned as interest over a specific period of time. It is usually expressed as a decimal value.

Q: How do I solve for the rate of interest (r) in the simple interest formula?

A: To solve for the rate of interest (r) in the simple interest formula, you need to isolate the variable r on one side of the equation. You can do this by subtracting P from both sides of the equation, factoring out the common term t, and then dividing both sides by t.

Q: What is the difference between simple interest and compound interest?

A: Simple interest is calculated only on the principal amount, while compound interest is calculated on both the principal amount and any accrued interest.

Q: When is the simple interest formula used?

A: The simple interest formula is used in a variety of situations, including calculating the interest earned on a savings account or a loan, determining the rate of interest on an investment, and calculating the interest earned on a bond or a stock.

Q: What are some real-world applications of the simple interest formula?

A: Some real-world applications of the simple interest formula include:

Calculating the interest earned on a savings account or a loan
Determining the rate of interest on an investment
Calculating the interest earned on a bond or a stock
Calculating the interest earned on a mortgage or a car loan

Q: How do I use the simple interest formula to calculate the interest earned on a savings account?

A: To use the simple interest formula to calculate the interest earned on a savings account, you need to plug in the values of P, r, and t into the formula. The formula is $A = P(1 + rt)$ . You can then calculate the interest earned by subtracting the principal amount from the total amount.

Q: How do I use the simple interest formula to determine the rate of interest on an investment?

A: To use the simple interest formula to determine the rate of interest on an investment, you need to plug in the values of A, P, and t into the formula. The formula is $r = \frac{A - P}{t}$ . You can then calculate the rate of interest by dividing the difference between the total amount and the principal amount by the time period.

Conclusion

In conclusion, the simple interest formula is a fundamental concept in mathematics that is used extensively in finance and economics. By understanding the simple interest formula and how to use it, you can calculate the interest earned on a savings account or a loan, determine the rate of interest on an investment, and calculate the interest earned on a bond or a stock.