Compare These Fractions. Choose The Sign That Makes The Statement True.$\frac{8}{12} \, ? \, \frac{3}{12}$A. $\ \textless \ $B. $\ \textgreater \ $C. $=$

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Introduction

Comparing fractions is an essential skill in mathematics, and it's crucial to understand how to make the right choice when comparing two or more fractions. In this article, we will explore the concept of comparing fractions and provide a step-by-step guide on how to choose the sign that makes the statement true.

What are Fractions?

A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator (the top number) and a denominator (the bottom number). For example, the fraction 3/4 can be read as "three-fourths" or "three out of four."

Comparing Fractions: A Step-by-Step Guide

To compare fractions, we need to follow a simple step-by-step guide:

  1. Check if the denominators are the same: If the denominators are the same, we can compare the numerators directly. If the denominators are different, we need to find a common denominator.
  2. Find a common denominator: If the denominators are different, we need to find a common denominator. We can do this by finding the least common multiple (LCM) of the two denominators.
  3. Compare the numerators: Once we have a common denominator, we can compare the numerators directly.
  4. Choose the sign: Based on the comparison, we can choose the sign that makes the statement true.

Example 1: Comparing 8/12 and 3/12

Let's compare the fractions 8/12 and 3/12.

  • Check if the denominators are the same: Yes, the denominators are the same (12).
  • Compare the numerators: 8 is greater than 3.
  • Choose the sign: Since 8 is greater than 3, the statement is true, and we choose the sign >.

Example 2: Comparing 1/4 and 1/2

Let's compare the fractions 1/4 and 1/2.

  • Check if the denominators are the same: No, the denominators are different (4 and 2).
  • Find a common denominator: The LCM of 4 and 2 is 4.
  • Compare the numerators: 1 is less than 2.
  • Choose the sign: Since 1 is less than 2, the statement is true, and we choose the sign <.

Example 3: Comparing 3/8 and 2/8

Let's compare the fractions 3/8 and 2/8.

  • Check if the denominators are the same: Yes, the denominators are the same (8).
  • Compare the numerators: 3 is greater than 2.
  • Choose the sign: Since 3 is greater than 2, the statement is true, and we choose the sign >.

Conclusion

Comparing fractions is a simple process that requires attention to detail and a step-by-step approach. By following the guide outlined in this article, you can make the right choice when comparing two or more fractions. Remember to check if the denominators are the same, find a common denominator if necessary, compare the numerators, and choose the sign that makes the statement true.

Common Mistakes to Avoid

When comparing fractions, it's essential to avoid common mistakes. Here are a few to watch out for:

  • Not checking if the denominators are the same: If the denominators are different, you need to find a common denominator before comparing the numerators.
  • Not finding a common denominator: If the denominators are different, you need to find a common denominator before comparing the numerators.
  • Comparing the denominators instead of the numerators: Remember to compare the numerators, not the denominators.

Practice Exercises

To practice comparing fractions, try the following exercises:

  1. Compare the fractions 5/6 and 3/6.
  2. Compare the fractions 2/3 and 1/3.
  3. Compare the fractions 7/8 and 5/8.

Answer Key

  1. (5 is greater than 3)

  2. (2 is greater than 1)

  3. (7 is greater than 5)

Conclusion

Q: What is the first step in comparing fractions?

A: The first step in comparing fractions is to check if the denominators are the same. If the denominators are the same, we can compare the numerators directly. If the denominators are different, we need to find a common denominator.

Q: How do I find a common denominator?

A: To find a common denominator, we need to find the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators can divide into evenly.

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

A: The least common multiple (LCM) is the smallest number that both denominators can divide into evenly. For example, the LCM of 4 and 6 is 12.

Q: How do I compare fractions with different denominators?

A: To compare fractions with different denominators, we need to find a common denominator and then compare the numerators. We can do this by multiplying the numerator and denominator of each fraction by the same number, which will give us a common denominator.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number in a fraction, and it represents the number of equal parts we have. The denominator is the bottom number in a fraction, and it represents the total number of parts.

Q: Can I compare fractions with different numerators and denominators?

A: Yes, you can compare fractions with different numerators and denominators. However, you need to find a common denominator first, and then compare the numerators.

Q: How do I know which sign to use when comparing fractions?

A: When comparing fractions, you need to use the following signs:

  • (greater than) if the numerator of the first fraction is greater than the numerator of the second fraction

  • < (less than) if the numerator of the first fraction is less than the numerator of the second fraction
  • = (equal to) if the numerators of both fractions are equal

Q: Can I compare fractions with decimals?

A: Yes, you can compare fractions with decimals. However, you need to convert the decimal to a fraction first, and then compare the fractions.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you need to follow these steps:

  1. Write the decimal as a fraction by placing the decimal part over the place value of the last digit.
  2. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction evenly.

Q: Can I compare fractions with mixed numbers?

A: Yes, you can compare fractions with mixed numbers. However, you need to convert the mixed number to an improper fraction first, and then compare the fractions.

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

A: To convert a mixed number to an improper fraction, you need to follow these steps:

  1. Multiply the whole number part by the denominator.
  2. Add the product to the numerator.
  3. Write the result as an improper fraction.

Q: What is an improper fraction?

A: An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: Can I compare fractions with negative numbers?

A: Yes, you can compare fractions with negative numbers. However, you need to follow the same rules as comparing fractions with positive numbers.

Q: How do I compare fractions with negative numbers?

A: To compare fractions with negative numbers, you need to follow these steps:

  1. Determine the sign of the numerator and denominator.
  2. Compare the absolute values of the numerators and denominators.
  3. Use the following signs:

  • (greater than) if the numerator of the first fraction is greater than the numerator of the second fraction

  • < (less than) if the numerator of the first fraction is less than the numerator of the second fraction
  • = (equal to) if the numerators of both fractions are equal

Conclusion

Comparing fractions is a fundamental skill in mathematics that requires attention to detail and a step-by-step approach. By following the guide outlined in this article, you can make the right choice when comparing two or more fractions. Remember to check if the denominators are the same, find a common denominator if necessary, compare the numerators, and choose the sign that makes the statement true.