Find $\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right$\].

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Introduction

Fractions are an essential part of mathematics, and they can be used to represent a wide range of mathematical concepts. However, when we encounter complex fractions, it can be challenging to solve them. In this article, we will focus on finding the value of the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right). We will break down the problem into smaller steps and provide a clear explanation of each step.

Understanding the Problem

The given problem involves adding and subtracting fractions. To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: None in this problem.
  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.

Step 1: Evaluate the Expression Inside the Parentheses

To evaluate the expression inside the parentheses, we need to find a common denominator for the fractions 56\frac{5}{6} and 29\frac{2}{9}. The least common multiple (LCM) of 6 and 9 is 18.

# Finding the Common Denominator

## **Step 1: Find the Least Common Multiple (LCM)**

To find the LCM of 6 and 9, we can list the multiples of each number:

Multiples of 6: 6, 12, 18, 24, ...
Multiples of 9: 9, 18, 27, 36, ...

## **Step 2: Identify the Least Common Multiple**

The smallest number that appears in both lists is 18. Therefore, the LCM of 6 and 9 is 18.

## **Step 3: Rewrite the Fractions with the Common Denominator**

To rewrite the fractions with the common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factor:

$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$

$\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

## **Step 4: Subtract the Fractions**

Now that we have the fractions with the common denominator, we can subtract them:

$\frac{15}{18} - \frac{4}{18} = \frac{15 - 4}{18} = \frac{11}{18}$

Step 2: Add the Fractions

Now that we have evaluated the expression inside the parentheses, we can add the fractions:

712+1118\frac{7}{12} + \frac{11}{18}

To add these fractions, we need to find a common denominator. The LCM of 12 and 18 is 36.

# Finding the Common Denominator

## **Step 1: Find the Least Common Multiple (LCM)**

To find the LCM of 12 and 18, we can list the multiples of each number:

Multiples of 12: 12, 24, 36, 48, ...
Multiples of 18: 18, 36, 54, 72, ...

## **Step 2: Identify the Least Common Multiple**

The smallest number that appears in both lists is 36. Therefore, the LCM of 12 and 18 is 36.

## **Step 3: Rewrite the Fractions with the Common Denominator**

To rewrite the fractions with the common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factor:

$\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}$

$\frac{11}{18} = \frac{11 \times 2}{18 \times 2} = \frac{22}{36}$

## **Step 4: Add the Fractions**

Now that we have the fractions with the common denominator, we can add them:

$\frac{21}{36} + \frac{22}{36} = \frac{21 + 22}{36} = \frac{43}{36}$

Conclusion

In this article, we have solved the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right). We broke down the problem into smaller steps and provided a clear explanation of each step. We found the common denominator for the fractions inside the parentheses and then added the fractions. The final answer is 4336\frac{43}{36}.

Introduction

Fractions are an essential part of mathematics, and they can be used to represent a wide range of mathematical concepts. However, when we encounter complex fractions, it can be challenging to solve them. In this article, we will focus on finding the value of the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right). We will break down the problem into smaller steps and provide a clear explanation of each step.

Understanding the Problem

The given problem involves adding and subtracting fractions. To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: None in this problem.
  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.

Step 1: Evaluate the Expression Inside the Parentheses

To evaluate the expression inside the parentheses, we need to find a common denominator for the fractions 56\frac{5}{6} and 29\frac{2}{9}. The least common multiple (LCM) of 6 and 9 is 18.

# Finding the Common Denominator

## **Step 1: Find the Least Common Multiple (LCM)**

To find the LCM of 6 and 9, we can list the multiples of each number:

Multiples of 6: 6, 12, 18, 24, ...
Multiples of 9: 9, 18, 27, 36, ...

## **Step 2: Identify the Least Common Multiple**

The smallest number that appears in both lists is 18. Therefore, the LCM of 6 and 9 is 18.

## **Step 3: Rewrite the Fractions with the Common Denominator**

To rewrite the fractions with the common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factor:

$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$

$\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

## **Step 4: Subtract the Fractions**

Now that we have the fractions with the common denominator, we can subtract them:

$\frac{15}{18} - \frac{4}{18} = \frac{15 - 4}{18} = \frac{11}{18}$

Step 2: Add the Fractions

Now that we have evaluated the expression inside the parentheses, we can add the fractions:

712+1118\frac{7}{12} + \frac{11}{18}

To add these fractions, we need to find a common denominator. The LCM of 12 and 18 is 36.

# Finding the Common Denominator

## **Step 1: Find the Least Common Multiple (LCM)**

To find the LCM of 12 and 18, we can list the multiples of each number:

Multiples of 12: 12, 24, 36, 48, ...
Multiples of 18: 18, 36, 54, 72, ...

## **Step 2: Identify the Least Common Multiple**

The smallest number that appears in both lists is 36. Therefore, the LCM of 12 and 18 is 36.

## **Step 3: Rewrite the Fractions with the Common Denominator**

To rewrite the fractions with the common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factor:

$\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}$

$\frac{11}{18} = \frac{11 \times 2}{18 \times 2} = \frac{22}{36}$

## **Step 4: Add the Fractions**

Now that we have the fractions with the common denominator, we can add them:

$\frac{21}{36} + \frac{22}{36} = \frac{21 + 22}{36} = \frac{43}{36}$

Conclusion

In this article, we have solved the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right). We broke down the problem into smaller steps and provided a clear explanation of each step. We found the common denominator for the fractions inside the parentheses and then added the fractions. The final answer is 4336\frac{43}{36}.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  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 the least common multiple (LCM) of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists. Alternatively, you can use the following formula:

LCM(a, b) = (a Γ— b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. You can do this by listing the multiples of each denominator and finding the smallest number that appears in both lists. Alternatively, you can use the following formula:

LCM(a, b) = (a Γ— b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the final answer to the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right)?

A: The final answer to the complex fraction 712+(56βˆ’29)\frac{7}{12}+\left(\frac{5}{6}-\frac{2}{9}\right) is 4336\frac{43}{36}.