Which expression is equivalent to the following complex fraction? :Complex fractions can be intimidating, but with a clear understanding of equivalent expressions, they become much more manageable. In this article, we will explore how to find an expression that is equivalent to a given complex fraction. By breaking down the steps and providing examples, we hope to simplify this process and boost your confidence in dealing with complex fractions.

- Understanding Complex Fractions: Complex fractions are fractions where either the numerator, denominator, or both contain fractions. They often involve operations such as addition, subtraction, multiplication, and division. Solving complex fractions requires simplifying them into an equivalent expression that is easier to work with.

- Simplifying Complex Fractions: To simplify a complex fraction, follow these steps: a) Identify the numerator and denominator. b) If the numerator or denominator contains a fraction, simplify it first. c) Multiply the numerator and denominator by the least common multiple (LCM) of the denominators involved. d) Simplify the resulting expression.
- Finding an Equivalent Expression: To find an expression equivalent to a complex fraction, you can multiply both the numerator and denominator by a suitable expression that eliminates the fraction within the complex fraction. By doing this, the resulting expression will be equivalent to the original complex fraction.
- Examples of Equivalent Expressions: Let’s explore a few examples to illustrate how to find equivalent expressions for complex fractions.

Example 1: Given complex fraction: (2/3) / (4/5)

Step 1: Identify the numerator and denominator. Numerator: 2/3 Denominator: 4/5

Step 2: Simplify the numerator and denominator. Numerator: 2/3 Denominator: 4/5

Step 3: Multiply the numerator and denominator by the LCM of the denominators. Numerator: (2/3) * 5 = 10/3 Denominator: (4/5) * 5 = 4

Step 4: Simplify the resulting expression. Equivalent expression: (10/3) / 4

Example 2:

Given complex fraction: (3/4) / [(1/2) + (1/3)] Step 1: Identify the numerator and denominator. Numerator: 3/4 Denominator: [(1/2) + (1/3)]

Step 2: Simplify the numerator and denominator. Numerator: 3/4 Denominator: (3/6) + (2/6) = 5/6

Step 3: Multiply the numerator and denominator by the LCM of the denominators. Numerator: (3/4) * 6 = 18/4 = 9/2 Denominator: (5/6) * 6 = 5

Step 4: Simplify the resulting expression. Equivalent expression: (9/2) / 5

- Frequently Asked Questions (FAQs): Q1: Can I simplify the numerator and denominator separately? A1: Yes, it is often easier to simplify the numerator and denominator before finding an equivalent expression for the complex fraction.

Q2: What if the complex fraction has multiple fractions in the numerator or denominator? A2: You can apply the same steps of simplification and finding an equivalent expression to each fraction individually.

Q3: Are there any shortcuts for finding equivalent expressions for complex fractions? A3: While there are no universal shortcuts, familiarity with common fraction operations and simplification techniques can help you streamline the process.

### Conclusion:

Solving complex fractions becomes easier when you understand how to find equivalent expressions. By following the steps outlined in this article and practicing with examples, you can confidently simplify complex fractions and find their equivalent expressions. Remember to always simplify the numerator and denominator before multiplying by the LCM to ensure accuracy. With practice, you’ll become proficient in dealing with complex fractions and conquer even the most challenging problems.

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