How to Solve Fraction Equations: Definition, Properties, Steps, and Solved Examples

Solving fraction equations is an essential skill in mathematics that often forms the foundation for more advanced algebraic concepts. Fraction equations involve variables and constants expressed in the form of fractions. To tackle these fraction equations, it’s important to understand the underlying properties of fractions, such as the least common denominator and equivalent fractions. This guide will walk you through the definition of fraction equations, outline the properties of fractional equations, and provide clear steps for solving them. We’ll also include solved examples to illustrate the process and solidify your understanding of important terms like numerator, denominator, reciprocal, and cross-multiplication.

Definition of Fraction

A fraction is a way of expressing a part of a whole. It consists of two key components: the numerator and the denominator. The numerator is the top number, representing the number of parts you have, while the denominator is the bottom number, indicating the total number of equal parts into which the whole is divided. For example, in the fraction ¾ ​, 3 is the numerator, and 4 is the denominator, meaning the fraction represents three out of four equal parts.

Fractions can also be classified into different types based on their properties:

• Proper Fractions: The numerator is less than the denominator (e.g., 3/4).
• Improper Fractions: The numerator is greater than or equal to the denominator (e.g., 5/3​).
• Mixed Numbers: A combination of a whole number and a fraction (e.g., 2 1/2​).

Example:

Consider the fraction 7/10​. Here, 7 is the numerator, and 10 is the denominator. This fraction means that 7 out of 10 equal parts are being considered.

What is an Equation?

An equation is a mathematical statement that asserts the equality of two expressions. It consists of two parts, known as the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). An equation may include numbers, variables, constants, operators (such as addition, subtraction, multiplication, and division), and functions.

Important Terms:

• Variables: Symbols, often letters, that represent unknown values in the equation.
• Constants: Fixed values that do not change.
• Operators: Symbols that indicate mathematical operations to be performed on the values or variables.
• Solutions: Values of the variables that make the equation true when substituted into the equation.

Example:

The equation 2x+3 = 7 consists of:

• LHS: 2x+3
• RHS: 7

To solve this equation, you would find the value of xxx that makes both sides equal. In this case, x=2 is the solution because substituting 2 for x yields 2(2)+3 = 7, making both sides of the equation equal.

Also Read: Variables and Constants

Properties of Fraction Equations

Understanding the properties of fraction equations is crucial for solving them effectively. Here are the key properties:

1. Equality Property: If two fractions are equal, then the cross-products are also equal.

For example, if a/b = c/d​, then a×d = b×c.

2. Addition and Subtraction Property: You can add or subtract the same fraction (or any equivalent expression) from both sides of the equation without changing the equality. 

For example, if a/b = c/d​, then a/b + e/f = c/d + e/f.

3. Multiplication Property: You can multiply both sides of the equation by the same nonzero number (or the reciprocal of a fraction) without changing the equality. This is often used to clear fractions from an equation. 

For example, if a/b = c/d​, multiplying both sides by b×d gives a×d = c×b.

4. Division Property: You can divide both sides of the equation by the same nonzero number or fraction. However, be cautious as this can sometimes introduce extraneous solutions, especially when dealing with variables in denominators.

5. Simplification Property: You can simplify the fractions on both sides of the equation to their lowest terms, making the equation easier to solve. 

For example, 4/8 = 1/2​.

6. Reciprocal Property: If a fraction equals another fraction, their reciprocals are also equal. 

For example, if a/b = c/d​, then b/a = d/c, assuming none of the terms are zero.

Steps to Solve Fraction Equations

To solve fraction equations, follow these systematic steps:

1. Identify and Simplify: Begin by identifying the fractions in the equation and simplify them if possible. This may involve reducing fractions to their lowest terms.
2. Find a Common Denominator: If the equation involves multiple fractions, find a common denominator for all fractions to facilitate combining or comparing them.
3. Clear Fractions: Multiply every term in the equation by the common denominator to eliminate the fractions. This step turns the equation into a linear or polynomial equation, depending on the degree.
4. Combine Like Terms: Simplify the equation by combining like terms on each side of the equation.
5. Isolate the Variable: Use algebraic operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the equation.
6. Solve for the Variable: Once the variable is isolated, solve the equation to find the value of the variable.
7. Check Your Solution: Substitute the solution back into the original equation to verify that it satisfies the equation. This step is crucial, especially in equations involving variables in the denominator, to ensure no extraneous solutions are included.
8. Simplify the Solution (if necessary): Ensure that the final answer is presented in its simplest form.

Example:

Equation: x/3 + 12 = 7/6​

Step 1: Identify and Simplify: The equation has two fractions, x/3​ and 1/2​. They are already simplified.

Step 2: Find a Common Denominator: The common denominator for 3 and 2 is 6.

Step 3: Clear Fractions: Multiply each term by 6: 

6 × (x/3) + 6 × (1/2) = 6 × (7/6)

This simplifies to: 2x+3 = 7

Step 4: Combine Like Terms: There are no like terms to combine.

Step 5: Isolate the Variable: Subtract 3 from both sides: 2x = 4

Step 6: Solve for the Variable: Divide by 2: x = 2

Step 7: Check Your Solution: Substitute x = 2 back into the original equation: ⅔ + ½ = 7/6​ 

Converting to a common denominator (6): 4/6 + 3/6 = 7/6​ 

The solution checks out.

Fraction Equation Solved Examples

Here are five solved examples of fraction equations, each demonstrating the process of solving them step by step:

Example 1

Equation: 2x/5 + 13 = 7/15​

Step 1: Identify and Simplify: The fractions are already simplified.

Step 2: Find a Common Denominator: The common denominator for 5, 3, and 15 is 15.

Step 3: Clear Fractions: Multiply each term by 15: 

15 × (2x/5) + 15 × (1/3) = 15 × (7/15)

This simplifies to: 6x+5=7

Step 4: Combine Like Terms: Subtract 5 from both sides: 6x = 2

Step 5: Solve for the Variable: Divide by 6: x = 1/3​

Step 6: Check Your Solution: Substitute x = 1/3​ back into the original equation to verify.

Example 2

Equation: 3x/4 − ½ = ⅝ ​

Step 1: Identify and Simplify: The fractions are already simplified.

Step 2: Find a Common Denominator: The common denominator for 4, 2, and 8 is 8.

Step 3: Clear Fractions: Multiply each term by 8: 

8 × (3x/4) − 8 × (1/2) = 8 × (5/8)

This simplifies to: 6x−4 = 5

Step 4: Combine Like Terms: Add 4 to both sides: 6x = 9

Step 5: Solve for the Variable: Divide by 6: x = 3/2​

Step 6: Check Your Solution: Substitute x = 3/2 back into the original equation to verify.

Example 3

Equation: x/2 + x/3 = 5/6​

Step 1: Identify and Simplify: The fractions are already simplified.

Step 2: Find a Common Denominator: The common denominator for 2, 3, and 6 is 6.

Step 3: Clear Fractions: Multiply each term by 6: 

6 × (x/2) + 6 × (x/3) = 6 × (5/6)

This simplifies to: 3x+2x = 5

Step 4: Combine Like Terms: Combine the terms: 5x = 5

Step 5: Solve for the Variable: Divide by 5: x = 1

Step 6: Check Your Solution: Substitute x = 1 back into the original equation to verify.

Example 4

Equation: 4/x = 2/3​

Step 1: Identify and Simplify: The fractions are already simplified.

Step 2: Find a Common Denominator: Not applicable here as we have a fraction equated to another fraction.

Step 3: Cross Multiply: Cross-multiply to eliminate the fractions: 4×3 = 2×x

This simplifies to: 12 = 2x

Step 4: Solve for the Variable: Divide by 2: x = 6

Step 5: Check Your Solution: Substitute x = 6 back into the original equation to verify.

Example 5

Equation: (x−2)/5 = (x+1)/3

Step 1: Identify and Simplify: The fractions are already simplified.

Step 2: Cross Multiply: Cross-multiply to eliminate the fractions: 

3(x−2) = 5(x+1)

Step 3: Distribute and Simplify: 3x−6 = 5x+5

Step 4: Combine Like Terms: Subtract 3x from both sides: −6 = 2x+5

Subtract 5 from both sides: −11 = 2x

Step 5: Solve for the Variable: Divide by 2: x = −11/2​

Step 6: Check Your Solution: Substitute x = −11/2​ back into the original equation to verify.

Also Read: Heights and Distances

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