Choose The Symbol That Correctly Compares These Mixed Numbers:${ 6 \frac{5}{7} \quad ? \quad 6 \frac{3}{8} }$A. ${ = }$ B. ${ \ \textgreater \ }$ C. ${ \ \textless \ }$

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Introduction

When comparing mixed numbers, it's essential to understand the concept of equivalent ratios and how to convert them into a comparable format. In this article, we will explore the process of comparing mixed numbers, focusing on the given problem: ${ 6 \frac{5}{7} \quad ? \quad 6 \frac{3}{8} }$. We will examine the different comparison symbols and determine which one correctly represents the relationship between these two mixed numbers.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form abca \frac{b}{c}, where aa is the whole number part, and bc\frac{b}{c} is the fractional part. For example, 6576 \frac{5}{7} is a mixed number with a whole number part of 6 and a fractional part of 57\frac{5}{7}.

Comparing Mixed Numbers: A Step-by-Step Approach

To compare mixed numbers, we need to convert them into a comparable format. One way to do this is to convert the mixed numbers into improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator. The result is then written as the new numerator over the denominator.

Example 1: Converting 6576 \frac{5}{7} to an Improper Fraction

  • Multiply the whole number part (6) by the denominator (7): 6×7=426 \times 7 = 42
  • Add the numerator (5) to the result: 42+5=4742 + 5 = 47
  • Write the result as the new numerator over the denominator: 477\frac{47}{7}

Example 2: Converting 6386 \frac{3}{8} to an Improper Fraction

  • Multiply the whole number part (6) by the denominator (8): 6×8=486 \times 8 = 48
  • Add the numerator (3) to the result: 48+3=5148 + 3 = 51
  • Write the result as the new numerator over the denominator: 518\frac{51}{8}

Comparing Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can compare them. To do this, we need to find a common denominator for the two fractions.

Finding a Common Denominator

A common denominator is the smallest multiple that both denominators can divide into evenly. In this case, the denominators are 7 and 8. The least common multiple (LCM) of 7 and 8 is 56.

Converting Fractions to Have a Common Denominator

To convert the fractions to have a common denominator, we multiply the numerator and denominator of each fraction by the necessary factor.

Example 1: Converting 477\frac{47}{7} to have a denominator of 56

  • Multiply the numerator (47) by the factor needed to get to 56: 47×8=37647 \times 8 = 376
  • Multiply the denominator (7) by the same factor: 7×8=567 \times 8 = 56
  • Write the result as the new fraction: 37656\frac{376}{56}

Example 2: Converting 518\frac{51}{8} to have a denominator of 56

  • Multiply the numerator (51) by the factor needed to get to 56: 51×7=35751 \times 7 = 357
  • Multiply the denominator (8) by the same factor: 8×7=568 \times 7 = 56
  • Write the result as the new fraction: 35756\frac{357}{56}

Comparing the Fractions

Now that we have converted the fractions to have a common denominator, we can compare them. To do this, we compare the numerators.

Comparing the Numerators

The numerator of the first fraction is 376, and the numerator of the second fraction is 357. Since 376 is greater than 357, the first fraction is greater than the second fraction.

Conclusion

In conclusion, the correct comparison symbol for the given mixed numbers is ${ \ \textgreater \ }$. This is because the improper fraction equivalent of 6576 \frac{5}{7} is greater than the improper fraction equivalent of 6386 \frac{3}{8}.

Answer

The correct answer is B. ${ \ \textgreater \ }$

Final Thoughts

Introduction

Comparing mixed numbers can be a complex task, but with the right approach, it can be made much easier. In this article, we will address some of the most frequently asked questions about comparing mixed numbers.

Q: What is the first step in comparing mixed numbers?

A: The first step in comparing mixed numbers is to convert them into improper fractions. This involves multiplying the whole number part 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, follow these steps:

  1. Multiply the whole number part by the denominator.
  2. Add the numerator to the result.
  3. Write the result as the new numerator over the denominator.

Example: Converting 6576 \frac{5}{7} to an Improper Fraction

  • Multiply the whole number part (6) by the denominator (7): 6×7=426 \times 7 = 42
  • Add the numerator (5) to the result: 42+5=4742 + 5 = 47
  • Write the result as the new numerator over the denominator: 477\frac{47}{7}

Q: What is the next step in comparing mixed numbers?

A: The next step in comparing mixed numbers is to find a common denominator for the two fractions. This involves finding the least common multiple (LCM) of the two denominators.

Q: How do I find a common denominator?

A: To find a common denominator, follow these steps:

  1. List the multiples of each denominator.
  2. Identify the smallest multiple that both denominators have in common.
  3. This is the least common multiple (LCM).

Example: Finding the LCM of 7 and 8

  • List the multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56
  • List the multiples of 8: 8, 16, 24, 32, 40, 48, 56
  • Identify the smallest multiple that both denominators have in common: 56

Q: How do I convert fractions to have a common denominator?

A: To convert fractions to have a common denominator, follow these steps:

  1. Multiply the numerator and denominator of each fraction by the necessary factor.
  2. Write the result as the new fraction.

Example: Converting 477\frac{47}{7} to have a denominator of 56

  • Multiply the numerator (47) by the factor needed to get to 56: 47×8=37647 \times 8 = 376
  • Multiply the denominator (7) by the same factor: 7×8=567 \times 8 = 56
  • Write the result as the new fraction: 37656\frac{376}{56}

Q: How do I compare the fractions?

A: To compare the fractions, compare the numerators. The fraction with the greater numerator is greater than the other fraction.

Example: Comparing 37656\frac{376}{56} and 35756\frac{357}{56}

  • Compare the numerators: 376 is greater than 357
  • The fraction 37656\frac{376}{56} is greater than the fraction 35756\frac{357}{56}

Q: What is the final step in comparing mixed numbers?

A: The final step in comparing mixed numbers is to determine which mixed number is greater or lesser based on the comparison of the improper fractions.

Conclusion

Comparing mixed numbers can be a challenging task, but by following the steps outlined in this article, you can confidently compare mixed numbers and determine which one is greater or lesser. Remember to convert the mixed numbers to improper fractions, find a common denominator, and compare the numerators to determine the correct comparison.

Additional Resources

For more information on comparing mixed numbers, check out the following resources:

Final Thoughts

Comparing mixed numbers is an essential skill in mathematics, and with practice, you can become proficient in comparing mixed numbers. Remember to take your time, follow the steps outlined in this article, and practice comparing mixed numbers to become a master of this skill.