Comparing Fractions

Definition: Comparing fractions means determining which fraction is greater, smaller, or if two fractions are equal.

Key terms:

• Like fractions: Fractions with the same denominator (e.g., 2/7 and 5/7).
• Unlike fractions: Fractions with different denominators (e.g., 3/4 and 2/5).
• Unit fractions: Fractions with numerator 1 (e.g., 1/2, 1/3, 1/4).

Basic rules:

• For like fractions (same denominator): the fraction with the larger numerator is greater. Example: 5/8 > 3/8.
• For unit fractions (numerator = 1): the fraction with the smaller denominator is greater. Example: 1/3 > 1/5 (because thirds are bigger than fifths).
• For unlike fractions: you must convert them to like fractions or use cross multiplication.

Understanding why these methods work:

Like fractions (same denominator):

When two fractions have the same denominator, the pieces are the same size. So you just count how many pieces you have — the one with more pieces is bigger.

• 3/7 means 3 pieces, each of size 1/7.
• 5/7 means 5 pieces, each of size 1/7.
• 5 pieces > 3 pieces, so 5/7 > 3/7.

Cross multiplication — why it works:

To compare a/b and c/d, we can make them into like fractions with denominator b x d:

• a/b = (a x d) / (b x d)
• c/d = (c x b) / (b x d)

Now both have the same denominator (b x d), so we just compare numerators: a x d vs c x b. This is exactly cross multiplication.

LCM method — step by step:

Compare 3/4 and 5/6:

• LCM of 4 and 6 = 12
• 3/4 = (3 x 3)/(4 x 3) = 9/12
• 5/6 = (5 x 2)/(6 x 2) = 10/12
• 9/12 vs 10/12 → 9 < 10 → 3/4 < 5/6

Unit fractions:

For fractions with numerator 1, the bigger the denominator, the smaller the fraction:

• 1/2 means one out of 2 parts (a large piece)
• 1/10 means one out of 10 parts (a tiny piece)
• So 1/2 > 1/3 > 1/4 > 1/5 > ... > 1/10 > ...

Types of fraction comparison problems:

Type 1: Like fractions (same denominator)

• Compare numerators directly.
• Example: 4/9 ___ 7/9 → 4 < 7 → 4/9 < 7/9

Type 2: Like numerators (same numerator)

• The fraction with the smaller denominator is greater.
• Example: 3/5 ___ 3/8 → 5 < 8 → 3/5 > 3/8

Type 3: Unlike fractions

• Use cross multiplication or LCM method.
• Example: 2/3 ___ 3/5 → cross multiply: 2x5=10, 3x3=9 → 10 > 9 → 2/3 > 3/5

Type 4: Comparing unit fractions

• Smaller denominator = larger fraction.
• Example: 1/4 > 1/6

Type 5: Arranging fractions in order

• Convert all to like fractions using LCM, then arrange.

Type 6: Comparing with 1/2 or 1

• Check if a fraction is greater than, less than, or equal to 1/2.
• Rule: a/b > 1/2 when 2a > b.

Comparing fractions in daily life:

• Sharing food: If you get 3/4 of a cake and your friend gets 2/3, who gets more? Comparing fractions answers this.
• Discounts: A shop offers 1/3 off vs another offering 1/4 off. Which discount is bigger? 1/3 > 1/4, so the first shop gives a bigger discount.
• Recipes: A recipe needs 2/3 cup of sugar. You have 3/4 cup. Do you have enough? Since 3/4 > 2/3, yes you do.
• Time: You spend 3/5 of the day studying and 1/4 playing. Which takes more time? 3/5 > 1/4.
• Sports: A batsman scored runs in 2/3 of his matches. Another scored in 3/5 of his matches. Who was more consistent? Compare 2/3 and 3/5.
• Measurements: Comparing fractional measurements (3/4 inch vs 5/8 inch) in woodworking or sewing.