CBSE Class 6 Maths: Fractions Ex 7.4 - Comparing & Ordering Fractions

Understanding Proper, Improper, and Mixed Fractions

Before we dive into comparing and ordering, let's quickly review the types of fractions. Knowing these types will make it easier to handle them during comparisons.

• Proper Fraction: In a proper fraction, the numerator (the top number) is smaller than the denominator (the bottom number). These fractions always represent a value less than 1. For example, 1/2, 3/4, 5/8 are proper fractions. Think of it as a small piece of a whole pizza – you haven't eaten the entire pizza yet.

• Improper Fraction: An improper fraction has a numerator that is greater than or equal to its denominator. These fractions represent a value of 1 or more than 1. For example, 5/3, 7/4, 9/9 are improper fractions. Imagine you have more than one pizza, or you have exactly one whole pizza divided into pieces.

• Mixed Fraction: A mixed fraction is a combination of a whole number and a proper fraction. For instance, 1 1/2 (read as "one and a half") means one whole plus one half. We often convert improper fractions into mixed fractions to make them easier to understand, especially when dealing with quantities. To convert an improper fraction like 7/3 into a mixed fraction, you divide the numerator by the denominator: 7 ÷ 3 = 2 with a remainder of 1. So, 7/3 becomes 2 1/3. Similarly, to convert a mixed fraction back to improper, multiply the whole number by the denominator and add the numerator, then place it over the original denominator. For 2 1/3, it's (2 × 3 + 1) / 3 = 7/3. Understanding these conversions is key to comparing all types of fractions efficiently.

Step-by-Step Guide to Comparing Fractions

1. Step 1: Convert to Improper Fractions (if needed) — If you have any mixed fractions, convert them into improper fractions first. This makes the comparison process much simpler as you'll only be dealing with simple numerators and denominators. For example, to compare 1 1/2 and 5/3, convert 1 1/2 to 3/2.
2. Step 2: Find the Least Common Multiple (LCM) of the Denominators — To compare fractions with different denominators, we need to make their denominators the same. The easiest way to do this is to find the LCM of all the denominators. The LCM will be your new common denominator. For instance, if you're comparing 2/3 and 3/4, the denominators are 3 and 4. The LCM of 3 and 4 is 12.
3. Step 3: Convert to Equivalent Fractions with the Common Denominator — Once you have the LCM, convert each fraction into an equivalent fraction that has this LCM as its denominator. To do this, multiply both the numerator and the denominator by the same number that makes the original denominator equal to the LCM. For 2/3 and 3/4 with LCM 12: For 2/3, to get 12 in the denominator, multiply 3 by 4. So, multiply numerator 2 by 4 also: (2 × 4) / (3 × 4) = 8/12. For 3/4, to get 12 in the denominator, multiply 4 by 3. So, multiply numerator 3 by 3 also: (3 × 3) / (4 × 3) = 9/12.
4. Step 4: Compare the Numerators — Now that all your fractions have the same denominator (they are 'like fractions'), you can simply compare their numerators. The fraction with the larger numerator is the greater fraction. Following our example of 8/12 and 9/12: Since 9 > 8, it means 9/12 > 8/12. Therefore, 3/4 > 2/3.

Ordering Fractions: Ascending and Descending

• Example 1: Arrange 2/5, 1/2, and 3/10 in ascending order. Step 1: The fractions are already proper fractions. Step 2: Find the LCM of the denominators 5, 2, and 10. The multiples of 10 are 10, 20... Multiples of 5 are 5, 10... Multiples of 2 are 2, 4, 6, 8, 10... The LCM is 10. Step 3: Convert to equivalent fractions with denominator 10: 2/5 = (2 × 2) / (5 × 2) = 4/10 1/2 = (1 × 5) / (2 × 5) = 5/10 * 3/10 remains 3/10 Step 4: Compare the numerators: 3, 4, 5. In ascending order, they are 3 < 4 < 5. Step 5: Write the original fractions in ascending order: 3/10 < 2/5 < 1/2.
• Example 2: Arrange 7/3, 2 1/4, and 5/2 in descending order. Step 1: Convert the mixed fraction to an improper fraction: 2 1/4 = (2 × 4 + 1) / 4 = 9/4 The fractions are now 7/3, 9/4, and 5/2. Step 2: Find the LCM of the denominators 3, 4, and 2. Multiples of 4 are 4, 8, 12, 16... Multiples of 3 are 3, 6, 9, 12... Multiples of 2 are 2, 4, 6, 8, 10, 12... The LCM is 12. Step 3: Convert to equivalent fractions with denominator 12: 7/3 = (7 × 4) / (3 × 4) = 28/12 9/4 = (9 × 3) / (4 × 3) = 27/12 * 5/2 = (5 × 6) / (2 × 6) = 30/12 Step 4: Compare the numerators: 28, 27, 30. In descending order, they are 30 > 28 > 27. Step 5: Write the original fractions in descending order: 5/2 > 7/3 > 2 1/4.

YoLearn AI Tutor's Exam Tip: Quick Checks for Fractions

When comparing fractions, always remember these handy tips to avoid common mistakes and verify your answers:

1. Check for Common Numerators: If fractions have the same numerator, the fraction with the smaller denominator is the larger fraction. For example, 1/2 > 1/3 because a whole divided into 2 parts means larger pieces than a whole divided into 3 parts. This is a quick mental check!
2. Benchmark with 1/2 or 1: Sometimes, you can quickly estimate. Is a fraction greater than or less than 1/2? Is it greater than or less than 1? For example, 3/4 is clearly greater than 1/2, while 1/4 is less than 1/2.
3. Cross-Multiplication (for two fractions): This is a faster method for comparing just two fractions, say a/b and c/d. You compare a × d with b × c. If a × d > b × c, then a/b > c/d. If a × d < b × c, then a/b < c/d. For example, to compare 2/3 and 3/4: (2 × 4) vs (3 × 3) -> 8 vs 9. Since 8 < 9, then 2/3 < 3/4. Be careful, this only works for comparing two fractions at a time, not for ordering a list of many!
4. Always Convert Mixed to Improper: This is a golden rule. Never try to compare a mixed fraction directly with an improper or proper fraction without converting the mixed fraction first. It leads to confusion and errors.

Practice Questions with Solutions

• Q: Compare the fractions 3/5 and 2/3. Which one is greater? A: Step 1: Find the LCM of denominators 5 and 3. LCM(5, 3) = 15. Step 2: Convert to equivalent fractions: 3/5 = (3 × 3) / (5 × 3) = 9/15 2/3 = (2 × 5) / (3 × 5) = 10/15 Step 3: Compare numerators: 9 < 10. Final answer: Therefore, 2/3 is greater than 3/5 (or 3/5 < 2/3).
• Q: Arrange the following fractions in ascending order: 1/4, 5/8, 3/16. A: Step 1: All fractions are proper. Step 2: Find the LCM of denominators 4, 8, and 16. LCM(4, 8, 16) = 16. Step 3: Convert to equivalent fractions: 1/4 = (1 × 4) / (4 × 4) = 4/16 5/8 = (5 × 2) / (8 × 2) = 10/16 * 3/16 remains 3/16 Step 4: Compare numerators: 3, 4, 10. In ascending order, 3 < 4 < 10. Final answer: The ascending order is 3/16, 1/4, 5/8.
• Q: Is 1 2/5 greater than or less than 7/5? Explain. A: Step 1: Convert the mixed fraction to an improper fraction. * 1 2/5 = (1 × 5 + 2) / 5 = 7/5. Step 2: Now compare 7/5 and 7/5. Step 3: Since the numerators and denominators are identical, the fractions are equal. Final answer: 1 2/5 is equal to 7/5.
• Q: Arrange the fractions 4/3, 1 1/6, 5/2 in descending order. A: Step 1: Convert mixed fractions to improper fractions. 1 1/6 = (1 × 6 + 1) / 6 = 7/6. The fractions are now 4/3, 7/6, 5/2. Step 2: Find the LCM of the denominators 3, 6, and 2. LCM(3, 6, 2) = 6. Step 3: Convert to equivalent fractions: 4/3 = (4 × 2) / (3 × 2) = 8/6 7/6 remains 7/6 * 5/2 = (5 × 3) / (2 × 3) = 15/6 Step 4: Compare numerators: 8, 7, 15. In descending order, 15 > 8 > 7. Final answer: The descending order is 5/2, 4/3, 1 1/6.

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