Comparing Fractions with Like Denominators and Numerators: Complete Worksheet

Master comparing fractions with like denominators and numerators through 5 detailed exercises with step-by-step solutions, rules, and visual aids.

Worksheet: Exercises 1 to 3

Instructions:

Compare the fractions using the appropriate rule. Remember:

  • When denominators are the same, compare numerators
  • When numerators are the same, compare denominators (smaller denominator = larger fraction)
  • Use >, <, or = to complete each comparison
1 Like Denominators
Problem 1
Compare: \( \frac{4}{9} \) and \( \frac{7}{9} \)
Definition:

Like denominators: Fractions with the same bottom number (denominator)

Comparing like denominators: Only compare the numerators

Like Denominator Method:
  1. Check that denominators are the same
  2. Compare the numerators
  3. The fraction with the larger numerator is greater
Original Fractions
\( \frac{4}{9}, \frac{7}{9} \)
Same Denominator
9 = 9
Compare Numerators
4 < 7
Step 1: Verify denominators are the same

Both fractions have denominator 9

So we can directly compare numerators

Step 2: Compare numerators

Numerator of first fraction: 4

Numerator of second fraction: 7

Since 4 < 7

Step 3: Write the comparison

Since 4 < 7 and denominators are the same, then \( \frac{4}{9} < \frac{7}{9} \)

Answer: \( \frac{4}{9} < \frac{7}{9} \)
Applied rules:

• Like Denominator Rule: When denominators are equal, compare numerators

• Comparison: Larger numerator means larger fraction

• Simplification: No conversion needed when denominators match

2 Like Numerators
Problem 2
Compare: \( \frac{3}{5} \) and \( \frac{3}{8} \)
Definition:

Like numerators: Fractions with the same top number (numerator)

Comparing like numerators: Compare denominators (smaller denominator = larger fraction)

Original Fractions
\( \frac{3}{5}, \frac{3}{8} \)
Same Numerator
3 = 3
Compare Denominators
5 < 8
Step 1: Verify numerators are the same

Both fractions have numerator 3

So we compare denominators

Step 2: Compare denominators

Denominator of first fraction: 5

Denominator of second fraction: 8

Since 5 < 8

Step 3: Apply the rule for like numerators

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

So \( \frac{3}{5} > \frac{3}{8} \)

Answer: \( \frac{3}{5} > \frac{3}{8} \)
Applied rules:

• Like Numerator Rule: When numerators are equal, smaller denominator means larger fraction

• Reasoning: More pieces of the same size = larger amount

• Comparison: \( \frac{n}{a} > \frac{n}{b} \) when \( a < b \)

3 Mixed Comparison
Problem 3
Order from least to greatest: \( \frac{2}{7}, \frac{5}{7}, \frac{2}{9}, \frac{2}{5} \)
Definition:

Ordering fractions: Arranging fractions from smallest to largest (or vice versa)

Original Fractions
\( \frac{2}{7}, \frac{5}{7}, \frac{2}{9}, \frac{2}{5} \)
Group by Type
Like num: 2/7,2/9,2/5; Like den: 2/7,5/7
Ordered Result
\( \frac{2}{9} < \frac{2}{7} < \frac{2}{5} < \frac{5}{7} \)
Step 1: Identify patterns

\( \frac{2}{7} \) and \( \frac{5}{7} \) have like denominators

\( \frac{2}{7}, \frac{2}{9}, \frac{2}{5} \) have like numerators

Step 2: Compare like denominators

\( \frac{2}{7} \) vs \( \frac{5}{7} \): Since 2 < 5, then \( \frac{2}{7} < \frac{5}{7} \)

Step 3: Compare like numerators

\( \frac{2}{7}, \frac{2}{9}, \frac{2}{5} \): Smaller denominator = larger fraction

Since 5 < 7 < 9, then \( \frac{2}{5} > \frac{2}{7} > \frac{2}{9} \)

Step 4: Combine comparisons

From like numerators: \( \frac{2}{9} < \frac{2}{7} < \frac{2}{5} \)

From like denominators: \( \frac{2}{7} < \frac{5}{7} \)

So: \( \frac{2}{9} < \frac{2}{7} < \frac{2}{5} < \frac{5}{7} \)

Answer: \( \frac{2}{9} < \frac{2}{7} < \frac{2}{5} < \frac{5}{7} \)
Applied rules:

• Like denominator rule: Compare numerators

• Like numerator rule: Compare denominators (inversely)

• Transitive property: Combine multiple comparisons

Worksheet Rules and Methods
\( \frac{a}{n} \text{ vs } \frac{b}{n} \Rightarrow \text{Compare } a \text{ vs } b \)
Like Denominator Rule
\( \frac{n}{a} \text{ vs } \frac{n}{b} \Rightarrow \text{Compare } a \text{ vs } b \text{ (inverted)} \)
Like Numerator Rule
Rule 1
Like Denominators
Compare numerators only
Rule 2
Like Numerators
Compare denominators inversely
Rule 3
Mixed Comparison
Identify patterns first
Key definitions:

Like Denominators: Fractions with the same denominator (bottom number)

Like Numerators: Fractions with the same numerator (top number)

Numerator: The top number in a fraction (how many parts)

Denominator: The bottom number in a fraction (how many total parts)

Comparison: Determining which fraction is greater, lesser, or equal

Worksheet methodology:
  1. Identify patterns: Look for like denominators or like numerators
  2. Apply appropriate rule: Use like denominator or like numerator rule
  3. Compare systematically: Follow the specific comparison method
  4. Order if needed: Arrange from least to greatest or vice versa
  5. Verify result: Double-check using alternative methods
Tip 1: Always check if denominators or numerators are the same before comparing.
Tip 2: When numerators are the same, smaller denominator means larger fraction.
Tip 3: Use visual models to understand why these rules work.
Tip 4: Remember: same pieces, more pieces = larger amount (like denominators).
Common errors: Forgetting to verify like denominators/numerators, confusing the rules, ordering incorrectly.
Key insight: Like denominators = same-sized pieces; like numerators = same number of pieces.
Rules to remember:

• Like Denominator Rule: \( \frac{a}{n} > \frac{b}{n} \) if \( a > b \)

• Like Numerator Rule: \( \frac{n}{a} > \frac{n}{b} \) if \( a < b \)

• Equality: \( \frac{a}{n} = \frac{b}{n} \) if \( a = b \)

• Ordering: Arrange fractions by following the appropriate rule

Worksheet: Exercises 4 to 5
4 Word Problem
Problem 4
Sarah read \( \frac{5}{8} \) of her book and John read \( \frac{3}{8} \) of the same book. Who read more?
Definition:

Word problem: Real-world scenario requiring mathematical solution

Given Fractions
\( \frac{5}{8}, \frac{3}{8} \)
Same Denominator
8 = 8
Compare Numerators
5 > 3
Step 1: Identify the fractions to compare

Sarah read: \( \frac{5}{8} \)

John read: \( \frac{3}{8} \)

Step 2: Verify denominators are the same

Both fractions have denominator 8

So we can compare numerators directly

Step 3: Compare numerators

Sarah's numerator: 5

John's numerator: 3

Since 5 > 3

Step 4: Interpret the result

Since \( \frac{5}{8} > \frac{3}{8} \), Sarah read more of the book than John

Answer: Sarah read more because \( \frac{5}{8} > \frac{3}{8} \)
Applied rules:

• Real-world application: Apply mathematical rules to practical situations

• Comparison: Use like denominator rule for direct comparison

• Context interpretation: Translate mathematical result back to the word problem

5 Advanced Comparison
Problem 5
Compare: \( \frac{7}{12} - \frac{2}{12} \) and \( \frac{4}{9} \)
Definition:

Multi-step comparison: Problems involving operations before comparison

Original Expression
\( \frac{7}{12} - \frac{2}{12} \) vs \( \frac{4}{9} \)
Simplify Left Side
\( \frac{5}{12} \) vs \( \frac{4}{9} \)
Result
Convert to common denominator
Step 1: Simplify the left side

\( \frac{7}{12} - \frac{2}{12} = \frac{7-2}{12} = \frac{5}{12} \)

Step 2: Now compare \( \frac{5}{12} \) and \( \frac{4}{9} \)

These fractions have neither like denominators nor like numerators

Step 3: Find common denominator

LCD of 12 and 9 is 36

\( \frac{5}{12} = \frac{15}{36} \) and \( \frac{4}{9} = \frac{16}{36} \)

Step 4: Compare with like denominators

Since 15 < 16, then \( \frac{15}{36} < \frac{16}{36} \)

Step 5: Final answer

Therefore \( \frac{7}{12} - \frac{2}{12} < \frac{4}{9} \)

Answer: \( \frac{7}{12} - \frac{2}{12} < \frac{4}{9} \)
Applied rules:

• Fraction subtraction: When denominators are the same, subtract numerators

• Common denominator: Convert fractions to have the same denominator

• Multi-step problems: Solve step by step, simplifying as needed

Worksheet Summary: Comparing Fractions with Like Denominators and Numerators
\( \frac{a}{n} \text{ vs } \frac{b}{n} \Rightarrow \text{Compare } a \text{ vs } b \)
Like Denominator Rule
\( \frac{n}{a} \text{ vs } \frac{n}{b} \Rightarrow \text{Compare } a \text{ vs } b \text{ (inverted)} \)
Like Numerator Rule

Key Concepts Summary:

  • Like denominators: Compare numerators directly
  • Like numerators: Compare denominators (smaller denominator = larger fraction)
  • Neither like: Find common denominator or convert to decimals
  • Ordering: Arrange fractions by applying the appropriate rule
Key definitions:

Like Denominators: Fractions with the same denominator (bottom number)

Like Numerators: Fractions with the same numerator (top number)

Numerator: The top number in a fraction indicating how many parts

Denominator: The bottom number in a fraction indicating total parts

Comparison: Determining which fraction is greater, lesser, or equal

Ordering: Arranging fractions from least to greatest or greatest to least

Complete methodology:
  1. Pattern identification: Look for like denominators or like numerators
  2. Rule application: Apply the appropriate comparison rule
  3. Systematic comparison: Follow the specific method for each case
  4. Verification: Double-check using alternative methods when possible
  5. Context application: Apply rules to word problems and real scenarios
Tip 1: Always verify denominators or numerators are equal before applying rules.
Tip 2: For like numerators, think "more pieces of the same size" = larger amount.
Tip 3: Use visual models (pies, bars) to reinforce understanding.
Tip 4: Remember: same bottom, compare tops; same top, smaller bottom wins.
Common errors: Forgetting to check for like denominators/numerators, applying wrong rules, ordering incorrectly.
Memory aid: "Same bottom, compare tops; same top, smaller bottom wins."
Essential formulas and rules:

• Like Denominator Rule: \( \frac{a}{n} > \frac{b}{n} \) if and only if \( a > b \)

• Like Numerator Rule: \( \frac{n}{a} > \frac{n}{b} \) if and only if \( a < b \)

• Equality: \( \frac{a}{n} = \frac{b}{n} \) if and only if \( a = b \)

• Ordering: To order fractions with like denominators, order their numerators

• Ordering like numerators: To order, order denominators in reverse (smallest first)

Quick Reference Guide

📋
Like Denominators

Example: \( \frac{3}{7} \) vs \( \frac{5}{7} \)

Same bottom, compare tops

3 < 5, so \( \frac{3}{7} < \frac{5}{7} \)

Like Numerators

Example: \( \frac{4}{5} \) vs \( \frac{4}{7} \)

Same top, smaller bottom wins

5 < 7, so \( \frac{4}{5} > \frac{4}{7} \)

Neither Like

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

Find common denominator

Convert and compare

Worksheet Questions & Answers

Question: On the worksheet, I sometimes see fractions that don't have like denominators OR like numerators. What do I do then?

Answer: Good question! When fractions don't have like denominators or numerators, you need to make them comparable:

  • Find a common denominator: Convert both fractions to have the same denominator
  • Example: Compare \( \frac{2}{3} \) and \( \frac{3}{4} \)
  • Find LCD (Least Common Denominator): 12
  • Convert: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{3}{4} = \frac{9}{12} \)
  • Compare: Since 8 < 9, then \( \frac{8}{12} < \frac{9}{12} \), so \( \frac{2}{3} < \frac{3}{4} \)

This is more work than like denominators/numerators, but it always works!

Question: Why does a smaller denominator make a fraction larger when numerators are the same? That seems backwards!

Answer: It might seem backwards, but think of it this way:

  • \( \frac{3}{4} \) means 3 pieces out of 4 total pieces
  • \( \frac{3}{8} \) means 3 pieces out of 8 total pieces
  • The pieces in \( \frac{3}{4} \) are larger because the whole is divided into fewer pieces
  • So 3 larger pieces (\( \frac{3}{4} \)) is more than 3 smaller pieces (\( \frac{3}{8} \))

Think of pizza: 3 slices of a pizza cut into 4 pieces is more than 3 slices of a pizza cut into 8 pieces!

Question: How do I order a set of fractions that has both like denominators and like numerators mixed together?

Answer: Break it down step by step:

  • First: Group fractions by pattern (like denominators together, like numerators together)
  • Second: Order each group using the appropriate rule
  • Third: Compare across groups to finalize the order

Example: Order \( \frac{2}{5}, \frac{3}{5}, \frac{2}{7}, \frac{2}{3} \)

  • Like denominators: \( \frac{2}{5} < \frac{3}{5} \)
  • Like numerators: \( \frac{2}{7} < \frac{2}{5} < \frac{2}{3} \) (since 7 > 5 > 3)
  • Combine: \( \frac{2}{7} < \frac{2}{5} < \frac{3}{5} < \frac{2}{3} \)

Take it one step at a time!