12.) Calculate:a) { \frac{3}{4} + \frac{1}{3} - \left(1 - \frac{1}{2}\right)$}$b) { \frac{5}{6} + \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{3}\right)$} C ) \[ C) \[ C ) \[ \frac{15}{16} - \frac{1}{2} - \left(\frac{7}{8} -

by ADMIN 223 views

Introduction

In mathematics, fractions are a fundamental concept that is used to represent a part of a whole. When dealing with complex fractions, it can be challenging to simplify them. In this article, we will explore how to simplify complex fractions using the order of operations and basic fraction rules.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions within its numerator or denominator. For example, the expression 34+13−(1−12)\frac{3}{4} + \frac{1}{3} - \left(1 - \frac{1}{2}\right) is a complex fraction because it contains multiple fractions and a subtraction operation within the parentheses.

Simplifying Complex Fractions: A Step-by-Step Guide

To simplify complex fractions, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions within the parentheses first.
  2. Exponents: Evaluate any exponents next (none in this case).
  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.

Example 1: Simplifying 34+13−(1−12)\frac{3}{4} + \frac{1}{3} - \left(1 - \frac{1}{2}\right)

To simplify this complex fraction, we need to follow the order of operations:

  1. Evaluate the expression within the parentheses: 1−12=121 - \frac{1}{2} = \frac{1}{2}
  2. Rewrite the expression: 34+13−12\frac{3}{4} + \frac{1}{3} - \frac{1}{2}
  3. Find a common denominator for the fractions: The least common multiple of 4, 3, and 2 is 12.
  4. Rewrite the fractions with the common denominator: 912+412−612\frac{9}{12} + \frac{4}{12} - \frac{6}{12}
  5. Add and subtract the fractions: 912+412−612=712\frac{9}{12} + \frac{4}{12} - \frac{6}{12} = \frac{7}{12}

Example 2: Simplifying 56+12−(12−13)\frac{5}{6} + \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{3}\right)

To simplify this complex fraction, we need to follow the order of operations:

  1. Evaluate the expression within the parentheses: 12−13=36−26=16\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
  2. Rewrite the expression: 56+12−16\frac{5}{6} + \frac{1}{2} - \frac{1}{6}
  3. Find a common denominator for the fractions: The least common multiple of 6 and 2 is 6.
  4. Rewrite the fractions with the common denominator: 56+36−16\frac{5}{6} + \frac{3}{6} - \frac{1}{6}
  5. Add and subtract the fractions: 56+36−16=76\frac{5}{6} + \frac{3}{6} - \frac{1}{6} = \frac{7}{6}

Example 3: Simplifying 1516−12−(78−38)\frac{15}{16} - \frac{1}{2} - \left(\frac{7}{8} - \frac{3}{8}\right)

To simplify this complex fraction, we need to follow the order of operations:

  1. Evaluate the expression within the parentheses: 78−38=48=12\frac{7}{8} - \frac{3}{8} = \frac{4}{8} = \frac{1}{2}
  2. Rewrite the expression: 1516−12−12\frac{15}{16} - \frac{1}{2} - \frac{1}{2}
  3. Find a common denominator for the fractions: The least common multiple of 16 and 2 is 16.
  4. Rewrite the fractions with the common denominator: 1516−816−816\frac{15}{16} - \frac{8}{16} - \frac{8}{16}
  5. Add and subtract the fractions: 1516−816−816=1516−1616=−116\frac{15}{16} - \frac{8}{16} - \frac{8}{16} = \frac{15}{16} - \frac{16}{16} = -\frac{1}{16}

Conclusion

Introduction

In our previous article, we explored how to simplify complex fractions using the order of operations and basic fraction rules. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions within its numerator or denominator.

Q: Why do I need to simplify complex fractions?

A: Simplifying complex fractions is important because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to evaluate and compare fractions
  • Avoid errors and confusion when working with fractions

Q: What is the order of operations for simplifying complex fractions?

A: The order of operations for simplifying complex fractions is:

  1. Parentheses: Evaluate any expressions within the parentheses first.
  2. Exponents: Evaluate any exponents next (none in this case).
  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 find a common denominator for fractions?

A: To find a common denominator for fractions, you need to find the least common multiple (LCM) of the denominators. For example, if you have fractions with denominators 4 and 6, the LCM is 12.

Q: Can I simplify complex fractions with different denominators?

A: Yes, you can simplify complex fractions with different denominators by finding a common denominator and rewriting the fractions with that denominator.

Q: What if I have a fraction with a negative exponent?

A: If you have a fraction with a negative exponent, you can rewrite it as a fraction with a positive exponent by taking the reciprocal of the fraction. For example, 1x−2=x2\frac{1}{x^{-2}} = x^2.

Q: Can I simplify complex fractions with variables?

A: Yes, you can simplify complex fractions with variables by following the same steps as you would with numerical fractions.

Q: What if I have a complex fraction with multiple levels of parentheses?

A: If you have a complex fraction with multiple levels of parentheses, you can simplify it by following the order of operations and working from the innermost parentheses outwards.

Q: Are there any shortcuts for simplifying complex fractions?

A: While there are no shortcuts for simplifying complex fractions, you can use the following tips to make the process easier:

  • Use a calculator to find the LCM of the denominators
  • Use a table or chart to keep track of the fractions and their denominators
  • Break down the complex fraction into smaller parts and simplify each part separately

Conclusion

Simplifying complex fractions requires following the order of operations and using basic fraction rules. By answering some of the most frequently asked questions about simplifying complex fractions, we hope to have provided you with a better understanding of this important math concept. With practice and patience, you can become proficient in simplifying complex fractions and tackle even the most challenging math problems.