Comparing Fractions Worksheets Grade 3 PDF - Step-by-Step Solutions

Master comparing fractions: equivalent fractions, visual models, number lines, and comparison strategies through these 5 detailed exercises.

Rules and Methods for Comparing Fractions
\(\frac{a}{b} \text{ vs } \frac{c}{d}\)
Comparing Two Fractions
Same Denominator
\(\frac{a}{b} \text{ vs } \frac{c}{b}\)
Compare numerators: larger numerator = larger fraction
Same Numerator
\(\frac{a}{b} \text{ vs } \frac{a}{d}\)
Compare denominators: smaller denominator = larger fraction
Different Numerators and Denominators
\(\frac{a}{b} \text{ vs } \frac{c}{d}\)
Cross multiply: compare ad vs bc
Key Definitions:

Fraction: A part of a whole represented as numerator/denominator

Numerator: Top number showing how many parts we have

Denominator: Bottom number showing total equal parts

Equivalent Fractions: Fractions with the same value but different numbers

Comparison Methods:
  1. Visual Model Method: Draw pie charts or rectangles to compare
  2. Number Line Method: Place fractions on a number line
  3. Common Denominator Method: Convert to same denominator
  4. Cross Multiplication: Compare ad vs bc
Tip 1: When denominators are the same, bigger numerator means bigger fraction.
Tip 2: When numerators are the same, smaller denominator means bigger fraction.
Tip 3: Use cross multiplication when denominators are different.
Common Mistake: Thinking larger numbers always mean larger fractions.
Memory Aid: The bigger piece of pizza is the one with the smaller denominator!
Solution: Exercises 1 to 3
1 Same Denominator
Exercise 1
Which fraction is greater: \(\frac{3}{8}\) or \(\frac{5}{8}\)?
Rule:

When denominators are the same: Compare numerators. The fraction with the larger numerator is greater.

Visual Model:

vs

\(\frac{3}{8}\) vs \(\frac{5}{8}\)

Step-by-Step Method:
  1. Both fractions have the same denominator (8)
  2. Compare numerators: 3 vs 5
  3. Since 5 > 3, then \(\frac{5}{8} > \frac{3}{8}\)
Step 1: Identify denominators

\(\frac{3}{8}\) has denominator 8, \(\frac{5}{8}\) has denominator 8

Step 2: Compare numerators

3 vs 5 → 5 is greater than 3

Step 3: Apply the rule

When denominators are the same, the fraction with the larger numerator is greater

\(\frac{5}{8} > \frac{3}{8}\)
Final answer:

\(\frac{5}{8}\) is greater than \(\frac{3}{8}\)

Applied rules:

• Same Denominator Rule: Compare numerators only

• Greater Than: 5 > 3, so \(\frac{5}{8} > \frac{3}{8}\)

• Visual Confirmation: More pieces shaded means larger fraction

2 Same Numerator
Exercise 2
Which fraction is greater: \(\frac{2}{3}\) or \(\frac{2}{5}\)?
Rule:

When numerators are the same: Compare denominators. The fraction with the smaller denominator is greater.

Visual Model:

vs

\(\frac{2}{3}\) vs \(\frac{2}{5}\)

Step-by-Step Method:
  1. Both fractions have the same numerator (2)
  2. Compare denominators: 3 vs 5
  3. Since 3 < 5, then \(\frac{2}{3} > \frac{2}{5}\)
Step 1: Identify numerators

\(\frac{2}{3}\) has numerator 2, \(\frac{2}{5}\) has numerator 2

Step 2: Compare denominators

3 vs 5 → 3 is smaller than 5

Step 3: Apply the rule

When numerators are the same, the fraction with the smaller denominator is greater

\(\frac{2}{3} > \frac{2}{5}\)
Final answer:

\(\frac{2}{3}\) is greater than \(\frac{2}{5}\)

Applied rules:

• Same Numerator Rule: Compare denominators only

• Smaller Denominator: Smaller denominator means larger pieces

• Visual Confirmation: Bigger slices mean larger fraction

3 Cross Multiplication
Exercise 3
Which fraction is greater: \(\frac{3}{4}\) or \(\frac{2}{3}\)?
Rule:

Cross Multiplication: For \(\frac{a}{b}\) vs \(\frac{c}{d}\), compare \(a \times d\) vs \(b \times c\)

Visual Model:

vs

\(\frac{3}{4}\) vs \(\frac{2}{3}\)

Step-by-Step Method:
  1. Cross multiply: \(3 \times 3 = 9\) and \(4 \times 2 = 8\)
  2. Compare results: 9 vs 8
  3. Since 9 > 8, then \(\frac{3}{4} > \frac{2}{3}\)
Step 1: Cross multiply

\(3 \times 3 = 9\) and \(4 \times 2 = 8\)

Step 2: Compare products

9 vs 8 → 9 is greater than 8

Step 3: Apply the rule

Since \(3 \times 3 > 4 \times 2\), then \(\frac{3}{4} > \frac{2}{3}\)

\(\frac{3}{4} > \frac{2}{3}\)
Final answer:

\(\frac{3}{4}\) is greater than \(\frac{2}{3}\)

Applied rules:

• Cross Multiplication: \(a \times d\) vs \(b \times c\)

• Comparison: Larger product indicates larger fraction

• Verification: Can convert to common denominator to confirm

Solution: Exercises 4 to 5
4 Equivalent Fractions
Exercise 4
Are \(\frac{4}{6}\) and \(\frac{2}{3}\) equal? Show your work.
Rule:

Equivalent Fractions: Two fractions are equivalent if their cross products are equal.

Visual Model:

=

\(\frac{4}{6}\) vs \(\frac{2}{3}\)

Step-by-Step Method:
  1. Cross multiply: \(4 \times 3 = 12\) and \(6 \times 2 = 12\)
  2. Compare results: 12 vs 12
  3. Since 12 = 12, then \(\frac{4}{6} = \frac{2}{3}\)
Step 1: Cross multiply

\(4 \times 3 = 12\) and \(6 \times 2 = 12\)

Step 2: Compare products

12 vs 12 → Products are equal

Step 3: Apply the rule

If cross products are equal, fractions are equivalent

Step 4: Alternative method - Simplify

\(\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}\)

\(\frac{4}{6} = \frac{2}{3}\)
Final answer:

Yes, \(\frac{4}{6}\) and \(\frac{2}{3}\) are equal (equivalent fractions)

Applied rules:

• Equivalence Test: Cross products must be equal

• Simplification: Divide numerator and denominator by GCD

• Verification: Both methods confirm equivalence

5 Number Line Method
Exercise 5
Order these fractions from least to greatest: \(\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{2}{3}\)
Rule:

Number Line Method: Plot fractions on a number line to compare their positions.

Number Line:

\(\frac{1}{4}\)
\(\frac{1}{2}\)
\(\frac{2}{3}\)
\(\frac{3}{4}\)
Step-by-Step Method:
  1. Convert all fractions to have common denominators or decimals
  2. Find LCD of 2, 4, 4, 3 = 12
  3. Convert: \(\frac{1}{2} = \frac{6}{12}\), \(\frac{3}{4} = \frac{9}{12}\), \(\frac{1}{4} = \frac{3}{12}\), \(\frac{2}{3} = \frac{8}{12}\)
  4. Order by numerators: 3, 6, 8, 9
Step 1: Find common denominator

LCD of 2, 4, 4, 3 is 12

Step 2: Convert fractions

\(\frac{1}{2} = \frac{6}{12}\), \(\frac{3}{4} = \frac{9}{12}\), \(\frac{1}{4} = \frac{3}{12}\), \(\frac{2}{3} = \frac{8}{12}\)

Step 3: Compare numerators

3 < 6 < 8 < 9

Step 4: Write in order

\(\frac{3}{12} < \frac{6}{12} < \frac{8}{12} < \frac{9}{12}\)

Step 5: Convert back to original fractions

\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)

\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)
Final answer:

From least to greatest: \(\frac{1}{4}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}\)

Applied rules:

• Common Denominator: Convert to same denominator for easy comparison

• Ordering: Compare numerators once denominators are equal

• Verification: Number line confirms the order visually

Questions & Answers

Question: I get confused when comparing fractions like \(\frac{1}{3}\) and \(\frac{2}{5}\). They both have small numbers, but how do I know which is bigger?

Answer: Great question! When comparing fractions with different numerators AND denominators, use cross multiplication:

  • For \(\frac{1}{3}\) vs \(\frac{2}{5}\), multiply diagonally: \(1 \times 5 = 5\) and \(3 \times 2 = 6\)
  • Compare the results: 5 vs 6
  • Since 6 > 5, then \(\frac{2}{5} > \frac{1}{3}\)

Visual way: Think of pizza slices. In \(\frac{1}{3}\), you have 1 slice out of 3 total slices. In \(\frac{2}{5}\), you have 2 slices out of 5 total slices. When you compare them visually, \(\frac{2}{5}\) represents a larger portion!

Another method is to find a common denominator. The LCD of 3 and 5 is 15: \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{2}{5} = \frac{6}{15}\). Since 6 > 5, then \(\frac{2}{5} > \frac{1}{3}\).

Question: My child is struggling with the concept that \(\frac{1}{2}\) is greater than \(\frac{1}{4}\). Even though 4 is bigger than 2, why is the fraction smaller?

Answer: This is a very common misconception! The key is to understand what the denominator represents:

  • The denominator tells us how many equal pieces the whole is divided into
  • The numerator tells us how many pieces we take
  • So \(\frac{1}{2}\) means 1 piece out of 2 total pieces (each piece is big)
  • And \(\frac{1}{4}\) means 1 piece out of 4 total pieces (each piece is small)

Real-world analogy: If you have a pizza cut into 2 pieces vs the same pizza cut into 4 pieces, taking 1 piece from the first gives you a much bigger slice than taking 1 piece from the second!

Memory trick: "The bigger the bottom number, the smaller each piece gets!" So when numerators are the same, the fraction with the smaller denominator is actually larger.

Question: How can I quickly compare fractions like \(\frac{3}{7}\) and \(\frac{4}{9}\) without drawing pictures every time?

Answer: There are several efficient methods for comparing fractions quickly:

  1. Cross Multiplication: Multiply the numerator of one fraction by the denominator of the other and compare: For \(\frac{3}{7}\) vs \(\frac{4}{9}\): \(3 \times 9 = 27\) and \(7 \times 4 = 28\). Since 28 > 27, then \(\frac{4}{9} > \frac{3}{7}\).
  2. Convert to Decimals: Divide numerator by denominator: \(\frac{3}{7} \approx 0.43\) and \(\frac{4}{9} \approx 0.44\). Since 0.44 > 0.43, then \(\frac{4}{9} > \frac{3}{7}\).
  3. Find Common Denominator: Convert both fractions to have the same denominator, then compare numerators.

Quick tip: Practice with common benchmark fractions like \(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{3}{4}\) to build intuition. Notice that \(\frac{3}{7}\) is slightly less than \(\frac{1}{2}\) (since \(\frac{3.5}{7} = \frac{1}{2}\)), while \(\frac{4}{9}\) is also slightly less than \(\frac{1}{2}\) (since \(\frac{4.5}{9} = \frac{1}{2}\)). Since 4 is closer to 4.5 than 3 is to 3.5, \(\frac{4}{9}\) is closer to \(\frac{1}{2}\) and therefore larger.