Fill In The Blank To Make The Equation True:\[$\frac{1}{12} = \frac{3}{\square}\$\]

Introduction

In mathematics, equations are a fundamental concept that help us understand and describe the world around us. Solving equations is a crucial skill that is used in various fields, including science, engineering, and finance. In this article, we will focus on solving a simple equation that involves fractions. We will use the equation $\frac{1}{12} = \frac{3}{\square}$ as an example and provide a step-by-step guide on how to solve it.

Understanding the Equation

The given equation is $\frac{1}{12} = \frac{3}{\square}$. This equation states that the ratio of 1 to 12 is equal to the ratio of 3 to an unknown number, denoted by $\square$. Our goal is to find the value of the unknown number.

Step 1: Identify the Unknown Number

The unknown number is represented by the symbol $\square$. To solve the equation, we need to find the value of this number.

Step 2: Cross-Multiply

To solve the equation, we can use the method of cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In this case, we multiply 1 by $\square$ and 12 by 3.

$\frac{1}{12} = \frac{3}{\square}$

$1 \times \square = 3 \times 12$

$\square = 36$

Step 3: Check the Solution

Now that we have found the value of the unknown number, we need to check if it satisfies the original equation. We can do this by substituting the value of $\square$ back into the original equation.

$\frac{1}{12} = \frac{3}{36}$

To check if the equation is true, we can simplify the right-hand side by dividing 3 by 36.

$\frac{3}{36} = \frac{1}{12}$

Since the two fractions are equal, we can conclude that the value of $\square$ is indeed 36.

Conclusion

In this article, we solved the equation $\frac{1}{12} = \frac{3}{\square}$ using the method of cross-multiplication. We found that the value of the unknown number is 36, and we verified that this value satisfies the original equation. This example illustrates the importance of solving equations in mathematics and provides a step-by-step guide on how to solve a simple equation involving fractions.

Tips and Variations

• To solve equations involving fractions, it is essential to use the method of cross-multiplication.
• When solving equations, it is crucial to check the solution to ensure that it satisfies the original equation.
• Equations can be solved using various methods, including cross-multiplication, addition, and subtraction.
• Solving equations is a fundamental skill that is used in various fields, including science, engineering, and finance.

Real-World Applications

Solving equations is a crucial skill that has numerous real-world applications. Here are a few examples:

• Finance: Solving equations is used in finance to calculate interest rates, investment returns, and other financial metrics.
• Science: Solving equations is used in science to model real-world phenomena, such as the motion of objects and the behavior of chemical reactions.
• Engineering: Solving equations is used in engineering to design and optimize systems, such as bridges and buildings.
• Computer Science: Solving equations is used in computer science to develop algorithms and solve problems in fields such as machine learning and data analysis.

Common Mistakes

When solving equations, it is essential to avoid common mistakes. Here are a few examples:

• Incorrect Cross-Multiplication: When cross-multiplying, it is essential to multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
• Incorrect Simplification: When simplifying fractions, it is essential to divide the numerator and denominator by their greatest common divisor.
• Incorrect Solution: When solving equations, it is essential to check the solution to ensure that it satisfies the original equation.

Conclusion

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It is a fundamental concept in mathematics that helps us understand and describe the world around us.

Q: What is the difference between an equation and an expression?

A: An expression is a mathematical statement that contains variables, constants, and mathematical operations. An equation, on the other hand, is a statement that two expressions are equal.

Q: How do I solve an equation?

A: To solve an equation, you need to isolate the variable on one side of the equation. This can be done using various methods, including addition, subtraction, multiplication, and division.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: To simplify an equation, you need to combine like terms and eliminate any unnecessary operations. This can be done by adding or subtracting the same value to both sides of the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. This can be done using various methods, including addition, subtraction, multiplication, and division.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you two solutions for the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to use various methods, including substitution, elimination, and graphing. The goal is to find the values of the variables that satisfy all the equations in the system.

Q: What is the difference between a linear system and a nonlinear system?

A: A linear system is a system of equations in which the highest power of the variable is 1. A nonlinear system, on the other hand, is a system of equations in which the highest power of the variable is greater than 1.

Q: How do I solve a linear system?

A: To solve a linear system, you need to use various methods, including substitution, elimination, and graphing. The goal is to find the values of the variables that satisfy all the equations in the system.

Q: What is the difference between a homogeneous system and a nonhomogeneous system?

A: A homogeneous system is a system of equations in which all the equations are equal to zero. A nonhomogeneous system, on the other hand, is a system of equations in which not all the equations are equal to zero.

Q: How do I solve a homogeneous system?

A: To solve a homogeneous system, you need to use various methods, including substitution, elimination, and graphing. The goal is to find the values of the variables that satisfy all the equations in the system.

Conclusion

In conclusion, solving equations is a fundamental skill that is used in various fields, including science, engineering, and finance. By following the step-by-step guide provided in this article, you can learn how to solve simple equations involving fractions. Remember to use the method of cross-multiplication, check the solution, and avoid common mistakes. With practice and patience, you can become proficient in solving equations and apply this skill to real-world problems.