Which Expression Is Equivalent To The Following Complex Fraction 1-1/X/2

Which Expression Is Equivalent To The Following Complex Fraction 1-1/X/2

<strong>Simplify the Complex Fraction: 1-1/x/2

Are you struggling to simplify complex fractions like 1-1/x/2? Don’t worry, you’re not alone. Many students find these expressions challenging, but with the right approach, they can be simplified with ease.

The Solution

To simplify 1-1/x/2, we can start by rewriting the fraction as follows:

1 - 1/x / 2
= 1 - 2/x * 1/2
= 1 - 1/x
= x-1/x

Therefore, the expression 1-1/x/2 is equivalent to x-1/x.

Summary

In this post, we discussed how to simplify the complex fraction 1-1/x/2. By rewriting the fraction and using simple algebraic operations, we were able to determine that its equivalent expression is x-1/x. This simplification technique can be applied to various complex fractions, making it a valuable tool for understanding and solving algebraic equations. Remember, practice is key to mastering these concepts.

Simplifying Complex Fractions: An Exploration of (1 – 1/x)/2

Introduction

In algebra, complex fractions arise when a fraction contains another fraction in its numerator or denominator. Simplifying such fractions requires careful manipulation to obtain an equivalent expression that is easier to work with. This article delves into the process of simplifying complex fractions, focusing specifically on the expression (1 – 1/x)/2.

Definition of a Complex Fraction

A complex fraction is a fraction that has another fraction in its numerator or denominator. It can be expressed in the form a/b, where a and b are fractions.

Equivalents of (1 – 1/x)/2

To simplify the complex fraction (1 – 1/x)/2, we can apply the following steps:

**<center><img src="https://tse1.mm.bing.net/th?q=Simplify%20a%20complex%20fraction" alt="Simplify a complex fraction" title="Simplify a complex fraction"></center><br>**
  • Step 1: Multiply numerator and denominator by 2x.
= (2x)(1 - 1/x)/2(2x)
= (2x - 2)/4x
  • Step 2: Expand the numerator.
= (2x - 2)/4x
= x - 1/2x

Therefore, the expression (1 – 1/x)/2 is equivalent to x – 1/2x.

Transition to Properties of Equivalent Fractions

Two fractions are equivalent if they represent the same numerical value. This means that they can be used interchangeably in any expression or equation without altering the result.

Applications of Simplification

Simplifying complex fractions has practical applications in various fields:

  • Mathematics: Simplifying complex fractions simplifies calculations and equations, making them easier to solve.
  • Science: In physics and engineering, simplified fractions facilitate the analysis of data and the derivation of equations.
  • Finance: Simplifying fractions is crucial for calculating interest rates, ratios, and other financial metrics.

How to Simplify Different Types of Complex Fractions

Complex fractions can vary in complexity. Here are some additional examples:

  • Simplifying a fraction with a fraction in the numerator:
(a/b)/(c/d) = (a/b) * (d/c)
  • Simplifying a fraction with a fraction in the denominator:
(e/f)/(g/h) = (e/f) * (h/g)
  • Simplifying a fraction with fractions in both the numerator and denominator:
((a/b)/(c/d))/(e/f) = ((a/b)/(c/d)) * (f/e)

Transition to Examples

The following examples illustrate the process of simplifying complex fractions:

  • Example 1: Simplifying (3/4)/(1/2)
(3/4)/(1/2) = (3/4) * (2/1) = 3/2
  • Example 2: Simplifying (2/(x+1))/(x-1)
(2/(x+1))/(x-1) = (2/(x+1)) * (1/(x-1)) = 2/((x+1)(x-1))

Using Properties to Simplify Complex Fractions

Properties of fractions can be leveraged to simplify complex fractions:

  • Multiplication property: Multiplying or dividing the numerator or denominator by the same non-zero number does not change the value of the fraction.
  • Addition and subtraction property: Adding or subtracting the same number to or from the numerator and denominator does not change the value of the fraction.
  • Reciprocal property: The reciprocal of a fraction is the fraction with the numerator and denominator switched.

Transition to Conclusion

Simplifying complex fractions is an essential skill for working with rational expressions. By understanding the concepts and applying the appropriate techniques, complex fractions can be transformed into simpler and more manageable expressions.

Conclusion

In conclusion, the expression (1 – 1/x)/2 can be simplified to x – 1/2x. This process involves multiplying both the numerator and denominator by 2x and then expanding the numerator. The simplified expression is more straightforward to use in mathematical operations and equations.

FAQs

  1. What is a complex fraction?
  • A complex fraction is a fraction that contains another fraction in its numerator or denominator.
  1. How do you simplify a complex fraction?
  • You can simplify a complex fraction by multiplying both the numerator and denominator by the least common multiple of the denominators of the individual fractions.
  1. When is it necessary to simplify complex fractions?
  • Simplifying complex fractions is necessary when you need to simplify calculations, solve equations, or perform mathematical operations.
  1. Can complex fractions be simplified in different ways?
  • Yes, there are different methods for simplifying complex fractions, but they all involve multiplying both the numerator and denominator by an appropriate value.
  1. Are complex fractions important in mathematics?
  • Yes, complex fractions are important in mathematics as they are used in various areas, including algebra, calculus, and statistics.

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