Divide And Reduce The Answer To Lowest Terms: ${ 4 \frac{2}{3} + 8 \frac{1}{6} = }$A. { \frac{4}{7}$}$B. { \frac{5}{6}$}$C. { \frac{1}{2}$}$D. { \frac{3}{4}$}$

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Introduction

When dealing with mixed numbers and fractions, it's essential to understand how to simplify them to their lowest terms. This involves adding or subtracting fractions with different denominators, which can be a challenging task. In this article, we'll explore the process of dividing and reducing fractions to their simplest form, using the example of adding two mixed numbers: 423+8164 \frac{2}{3} + 8 \frac{1}{6}.

Understanding Mixed Numbers and Fractions

Before we dive into the example, let's review the basics of mixed numbers and fractions. A mixed number is a combination of a whole number and a fraction, written in the form abca \frac{b}{c}. For example, 4234 \frac{2}{3} is a mixed number where a=4a = 4, b=2b = 2, and c=3c = 3. A fraction, on the other hand, is a part of a whole, written in the form ab\frac{a}{b}.

Adding Mixed Numbers with Different Denominators

To add mixed numbers with different denominators, we need to follow a step-by-step process:

  1. Convert the mixed numbers to improper fractions: We'll convert both 4234 \frac{2}{3} and 8168 \frac{1}{6} to improper fractions.
  2. Find the least common multiple (LCM) of the denominators: We'll find the LCM of the denominators of the two improper fractions.
  3. Add the fractions: We'll add the two improper fractions using the LCM as the common denominator.
  4. Simplify the result: We'll simplify the resulting fraction to its lowest terms.

Step 1: Convert the Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we'll multiply the whole number by the denominator and add the numerator.

  • 423=(4×3)+23=12+23=1434 \frac{2}{3} = \frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}
  • 816=(8×6)+16=48+16=4968 \frac{1}{6} = \frac{(8 \times 6) + 1}{6} = \frac{48 + 1}{6} = \frac{49}{6}

Step 2: Find the Least Common Multiple (LCM) of the Denominators

To find the LCM of two numbers, we'll list the multiples of each number and find the smallest multiple they have in common.

  • The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
  • The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
  • The smallest multiple they have in common is 6.

Step 3: Add the Fractions

Now that we have the LCM, we can add the two fractions using the LCM as the common denominator.

  • 143+496=14×23×2+496=286+496=28+496=776\frac{14}{3} + \frac{49}{6} = \frac{14 \times 2}{3 \times 2} + \frac{49}{6} = \frac{28}{6} + \frac{49}{6} = \frac{28 + 49}{6} = \frac{77}{6}

Step 4: Simplify the Result

To simplify the resulting fraction, we'll divide both the numerator and the denominator by their greatest common divisor (GCD).

  • The GCD of 77 and 6 is 1.
  • 776=77÷16÷1=776\frac{77}{6} = \frac{77 \div 1}{6 \div 1} = \frac{77}{6}

Conclusion

In this article, we've explored the process of dividing and reducing fractions to their simplest form. We've used the example of adding two mixed numbers: 423+8164 \frac{2}{3} + 8 \frac{1}{6}. By converting the mixed numbers to improper fractions, finding the LCM of the denominators, adding the fractions, and simplifying the result, we've arrived at the final answer: 776\frac{77}{6}.

Answer

The correct answer is:

776\boxed{\frac{77}{6}}

Discussion

This problem requires a deep understanding of fractions, mixed numbers, and the process of adding and simplifying fractions. It's essential to follow the step-by-step process and to simplify the result to its lowest terms.

Common Mistakes

When adding mixed numbers with different denominators, it's common to make mistakes by:

  • Not converting the mixed numbers to improper fractions
  • Not finding the LCM of the denominators
  • Not adding the fractions correctly
  • Not simplifying the result to its lowest terms

Tips and Tricks

To simplify the process of adding mixed numbers with different denominators, you can:

  • Use a calculator to find the LCM of the denominators
  • Use a chart or table to list the multiples of each number
  • Practice, practice, practice! The more you practice, the more comfortable you'll become with adding and simplifying fractions.

Final Thoughts

Q: What is the first step in adding mixed numbers with different denominators?

A: The first step is to convert the mixed numbers to improper fractions. This involves multiplying the whole number by the denominator and adding the numerator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you'll multiply the whole number by the denominator and add the numerator. For example, 423=(4×3)+23=12+23=1434 \frac{2}{3} = \frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}.

Q: What is the least common multiple (LCM) of two numbers?

A: The LCM of two numbers is the smallest multiple they have in common. To find the LCM, you can list the multiples of each number and find the smallest multiple they have in common.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple they have in common. For example, the multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... and the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ... The smallest multiple they have in common is 6.

Q: What is the next step after finding the LCM?

A: After finding the LCM, you'll add the two fractions using the LCM as the common denominator. This involves multiplying the numerator and denominator of each fraction by the necessary factor to make the denominators equal.

Q: How do I add the fractions?

A: To add the fractions, you'll multiply the numerator and denominator of each fraction by the necessary factor to make the denominators equal. For example, 143+496=14×23×2+496=286+496=28+496=776\frac{14}{3} + \frac{49}{6} = \frac{14 \times 2}{3 \times 2} + \frac{49}{6} = \frac{28}{6} + \frac{49}{6} = \frac{28 + 49}{6} = \frac{77}{6}.

Q: What is the final step in adding mixed numbers with different denominators?

A: The final step is to simplify the resulting fraction to its lowest terms. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: How do I simplify the resulting fraction?

A: To simplify the resulting fraction, you'll divide both the numerator and the denominator by their GCD. For example, 776=77÷16÷1=776\frac{77}{6} = \frac{77 \div 1}{6 \div 1} = \frac{77}{6}.

Q: What are some common mistakes to avoid when adding mixed numbers with different denominators?

A: Some common mistakes to avoid when adding mixed numbers with different denominators include:

  • Not converting the mixed numbers to improper fractions
  • Not finding the LCM of the denominators
  • Not adding the fractions correctly
  • Not simplifying the result to its lowest terms

Q: What are some tips and tricks for adding mixed numbers with different denominators?

A: Some tips and tricks for adding mixed numbers with different denominators include:

  • Using a calculator to find the LCM of the denominators
  • Using a chart or table to list the multiples of each number
  • Practicing, practicing, practicing! The more you practice, the more comfortable you'll become with adding and simplifying fractions.

Q: What is the final answer to the problem $4 \frac{2}{3} + 8 \frac{1}{6} = ?

A: The final answer is 776\boxed{\frac{77}{6}}.

Conclusion

Adding mixed numbers with different denominators can be a challenging task, but with practice and patience, you can master it. Remember to follow the step-by-step process, simplify the result to its lowest terms, and to use a calculator or chart to find the LCM of the denominators. With these tips and tricks, you'll be well on your way to becoming a master of fractions!