Fraction Numerator Denominator
Fraction Numerator Denominator
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How To Compare Fractions With The Same Numerator Easily?

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Comparing fractions might seem daunting, but with the right approach, it can become a breeze. Are you struggling to compare fractions with the same numerator? This article on COMPARE.EDU.VN provides a comprehensive guide on how to easily compare these fractions. We’ll break down the concept, explore different methods, and provide practical examples to help you master this essential skill. Learn about fraction comparison, like numerators, and fraction ordering for effective math problem-solving.

1. Understanding Fractions: Numerators and Denominators

Before diving into comparing fractions with the same numerator, let’s quickly recap what numerators and denominators are.

  • Numerator: The top number in a fraction, representing the number of parts you have.
  • Denominator: The bottom number in a fraction, representing the total number of equal parts the whole is divided into.

For example, in the fraction $frac{3}{5}$, 3 is the numerator and 5 is the denominator. This means we have 3 parts out of a total of 5.

2. What Are Fractions With The Same Numerator?

Fractions with the same numerator, also known as like numerators, are fractions where the top number (numerator) is the same, but the bottom number (denominator) differs.

For example, $frac{5}{7}$ and $frac{5}{9}$ are fractions with the same numerator (5).

3. The Key to Comparing: Focus on the Denominator

When comparing fractions with the same numerator, the trick is to focus on the denominators. The fraction with the smaller denominator is actually the larger fraction. This might seem counterintuitive at first, but let’s understand why.

Imagine you have a pizza.

  • If you divide it into 4 slices and take 1 slice ($frac{1}{4}$), you get a larger slice than if you divide it into 8 slices and take 1 slice ($frac{1}{8}$).

  • Because the numerator is the same (1 slice), the size of the slice depends on how many total slices (denominator) there are. Fewer slices mean each slice is bigger.

4. Comparing Fractions With The Same Numerator: Step-by-Step

Here’s a simple step-by-step guide to comparing fractions with the same numerator:

  1. Identify the Numerators: Ensure that the numerators of the fractions you are comparing are indeed the same.
  2. Compare the Denominators: Look at the denominators of the fractions.
  3. Determine the Larger Fraction: The fraction with the smaller denominator is the larger fraction.
  4. Use Inequality Symbols: Use the “greater than” (>) or “less than” (<) symbols to show the comparison.

Example:

Compare $frac{3}{5}$ and $frac{3}{7}$.

  1. Numerators: Both fractions have a numerator of 3.
  2. Denominators: The denominators are 5 and 7.
  3. Larger Fraction: Since 5 is smaller than 7, $frac{3}{5}$ is larger than $frac{3}{7}$.
  4. Inequality Symbol: $frac{3}{5} > frac{3}{7}$

5. Visual Aids: Using Models to Understand

Visual models can be incredibly helpful for understanding why the fraction with the smaller denominator is larger.

  • Circle Models: Draw two circles of the same size. Divide one into the number of parts represented by the first fraction’s denominator and shade the number of parts represented by the numerator. Repeat for the second fraction. Visually compare the shaded areas.

  • Bar Models: Draw two bars of the same length. Divide one into the number of parts represented by the first fraction’s denominator and shade the number of parts represented by the numerator. Repeat for the second fraction. Visually compare the shaded lengths.

By using visual models, you can clearly see that when the numerators are the same, the fraction with the smaller denominator represents a larger portion of the whole.

6. Real-Life Examples

Let’s look at some real-life examples to solidify your understanding.

  • Sharing a Cake: You and your friend both have the same amount of cake left. You cut your remaining cake into 2 pieces, while your friend cuts theirs into 4 pieces. You each eat one piece. Who ate more cake? You did, because $frac{1}{2} > frac{1}{4}$.

  • Running a Race: Two runners, Alex and Ben, both run for the same amount of time. Alex runs $frac{2}{5}$ of the total distance, while Ben runs $frac{2}{7}$ of the total distance. Who ran further? Alex did, because $frac{2}{5} > frac{2}{7}$.

7. Ordering Fractions With The Same Numerator

Once you can compare two fractions with the same numerator, you can easily order a series of such fractions.

7.1 Ascending Order

Ascending order means arranging the fractions from smallest to largest. When fractions have the same numerator, the fraction with the largest denominator is the smallest.

Example: Arrange the following fractions in ascending order: $frac{1}{33}, frac{1}{45}, frac{1}{27}, frac{1}{19}$

  1. Identify the Numerators: All fractions have a numerator of 1.
  2. Compare the Denominators: The denominators are 33, 45, 27, and 19.
  3. Arrange in Ascending Order: Remember, the largest denominator corresponds to the smallest fraction. Therefore, the ascending order is: $frac{1}{45} < frac{1}{33} < frac{1}{27} < frac{1}{19}$

7.2 Descending Order

Descending order means arranging the fractions from largest to smallest. When fractions have the same numerator, the fraction with the smallest denominator is the largest.

Example: Arrange the following fractions in descending order: $frac{1}{33}, frac{1}{45}, frac{1}{27}, frac{1}{19}$

  1. Identify the Numerators: All fractions have a numerator of 1.
  2. Compare the Denominators: The denominators are 33, 45, 27, and 19.
  3. Arrange in Descending Order: Remember, the smallest denominator corresponds to the largest fraction. Therefore, the descending order is: $frac{1}{19} > frac{1}{27} > frac{1}{33} > frac{1}{45}$

8. Addition and Subtraction With The Same Numerator

Adding or subtracting fractions with the same numerator but different denominators requires finding a common denominator first. Here’s how:

  1. Find the Least Common Multiple (LCM): Determine the LCM of the denominators. This will be your common denominator.
  2. Convert the Fractions: Convert each fraction to an equivalent fraction with the LCM as the denominator.
  3. Add or Subtract the Numerators: Once the fractions have the same denominator, you can add or subtract the numerators.
  4. Simplify: Simplify the resulting fraction if possible.

Example: Addition

Add $frac{2}{3}$ and $frac{2}{5}$

  1. LCM: The LCM of 3 and 5 is 15.
  2. Convert:
    • $frac{2}{3} = frac{2 times 5}{3 times 5} = frac{10}{15}$
    • $frac{2}{5} = frac{2 times 3}{5 times 3} = frac{6}{15}$
  3. Add: $frac{10}{15} + frac{6}{15} = frac{16}{15}$

Example: Subtraction

Subtract $frac{2}{3}$ and $frac{2}{5}$

  1. LCM: The LCM of 3 and 5 is 15.
  2. Convert:
    • $frac{2}{3} = frac{2 times 5}{3 times 5} = frac{10}{15}$
    • $frac{2}{5} = frac{2 times 3}{5 times 3} = frac{6}{15}$
  3. Subtract: $frac{10}{15} – frac{6}{15} = frac{4}{15}$

9. How to Make Numerators The Same

Sometimes, you might encounter fractions that don’t have the same numerator. In such cases, you can make the numerators the same by finding the LCM of the numerators and then multiplying each fraction by the appropriate factor.

Example: Compare $frac{3}{4}$ and $frac{9}{11}$

  1. Find the LCM: The LCM of 3 and 9 is 9.
  2. Convert the Fractions: Convert $frac{3}{4}$ to an equivalent fraction with a numerator of 9.
    • $frac{3}{4} = frac{3 times 3}{4 times 3} = frac{9}{12}$
  3. Compare: Now you can compare $frac{9}{12}$ and $frac{9}{11}$. Since 12 > 11, $frac{9}{12} < frac{9}{11}$, which means $frac{3}{4} < frac{9}{11}$.

10. Common Mistakes to Avoid

  • Thinking Bigger Denominator Means Bigger Fraction: Remember, with the same numerators, the smaller denominator means the bigger fraction.
  • Forgetting to Find a Common Denominator for Addition/Subtraction: You can’t add or subtract fractions unless they have the same denominator.
  • Not Simplifying Fractions: Always simplify your answer to its simplest form.

11. Solved Examples

Let’s go through some more solved examples to reinforce your understanding.

Example 1: Write the following fractions in descending order: $frac{7}{20}, frac{7}{9}, frac{7}{11}, frac{7}{19}, frac{7}{25}$

Solution:

Since the numerators are the same, we compare the denominators. The smaller the denominator, the greater the fraction.

Ordering the denominators from smallest to largest: 9 < 11 < 19 < 20 < 25

Therefore, the fractions in descending order are: $frac{7}{9} > frac{7}{11} > frac{7}{19} > frac{7}{20} > frac{7}{25}$

Example 2: Find the fractions with the same numerator from the following group: $frac{3}{5}, frac{3}{10}, frac{1}{6}, frac{3}{8}, frac{3}{19}, frac{8}{13}$

Solution:

The fractions with the same numerator are: $frac{3}{5}, frac{3}{10}, frac{3}{8}, frac{3}{19}$

Example 3: Add $frac{1}{3} + frac{1}{5} + frac{1}{9}$

Solution:

  1. Find the LCM: The LCM of 3, 5, and 9 is 45.
  2. Convert the Fractions:
    • $frac{1}{3} = frac{1 times 15}{3 times 15} = frac{15}{45}$
    • $frac{1}{5} = frac{1 times 9}{5 times 9} = frac{9}{45}$
    • $frac{1}{9} = frac{1 times 5}{9 times 5} = frac{5}{45}$
  3. Add: $frac{15}{45} + frac{9}{45} + frac{5}{45} = frac{29}{45}$

Example 4: Write the following fractions in ascending order: $frac{135}{178}, frac{135}{199}, frac{135}{101}, frac{135}{119}, frac{135}{229}$

Solution:

Since the numerators are the same, we compare the denominators. The greater the denominator, the smaller the fraction.

Ordering the denominators from largest to smallest: 229 > 199 > 178 > 119 > 101

Therefore, the fractions in ascending order are: $frac{135}{229} < frac{135}{199} < frac{135}{178} < frac{135}{119} < frac{135}{101}$

12. Practice Problems

Test your knowledge with these practice problems:

  1. Which of the following is true?
    • a) $frac{13}{35} > frac{13}{34}$
    • b) $frac{15}{17} > frac{15}{13}$
    • c) $frac{11}{34} > frac{11}{49}$
    • d) $frac{21}{34} < frac{21}{49}$
  2. Which sign will come in between $frac{2}{7}$ and $frac{4}{13}$?
    • a) >
    • b) <
    • c) =
    • d) None of these
  3. What will be the fraction $frac{1}{3} – frac{1}{4}$ and $frac{1}{5} – frac{1}{6}$ called?
    • a) Improper Fraction
    • b) Equivalent Fraction
    • c) Like Fraction
    • d) Fraction with Same Numerators

13. Frequently Asked Questions (FAQs)

Q1: What is the difference between like numerators and like denominators?

Fractions with like numerators have the same number on top but different numbers on the bottom, such as $frac{3}{5}$ and $frac{3}{7}$. Like denominators have the same number on the bottom, such as $frac{2}{9}$ and $frac{5}{9}$.

Q2: Are fractions with the same numerator called like fractions?

No, fractions with the same denominator are called like fractions. Fractions with the same numerator are simply called fractions with the same numerator or like numerators.

Q3: How do you compare fractions with the same numerator?

When fractions have the same numerator, compare their denominators. The fraction with the smaller denominator is the larger fraction. For example, $frac{4}{5} > frac{4}{7}$ because 5 is smaller than 7.

14. Conclusion

Comparing fractions with the same numerator becomes easy when you remember to focus on the denominators. The smaller the denominator, the larger the fraction. By using visual aids, real-life examples, and practicing regularly, you can master this skill and confidently compare fractions.

Ready to explore more comparison techniques? Visit COMPARE.EDU.VN for comprehensive guides and tools that simplify decision-making. Whether it’s comparing fractions or evaluating products, we provide the resources you need.

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