Solved Exercises on Comparing Fractions with Like Numerators or Denominators in Grade 4

Master fraction comparison: rules for comparing fractions with like numerators or denominators through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Compare fractions with like denominators
Exercise 1
Compare: \(\frac{3}{8}\) and \(\frac{5}{8}\)
Definition:

Fractions with like denominators: Fractions that have the same bottom number (denominator)

Comparison method for like denominators:
  1. Compare the numerators (top numbers)
  2. The fraction with the larger numerator is greater
  3. This works because both fractions are divided into equal-sized parts
Fractions
\(\frac{3}{8} \text{ and } \frac{5}{8}\)
Numerators
\(3 \text{ and } 5\)
Comparison
\(\frac{3}{8} < \frac{5}{8}\)
Step 1: Identify the denominators

Both fractions have denominator 8, so they represent parts of the same whole divided into 8 equal pieces

Step 2: Compare the numerators

\(3 < 5\), so \(\frac{3}{8}\) represents fewer parts than \(\frac{5}{8}\)

Step 3: Make the comparison

Since 3 parts out of 8 is less than 5 parts out of 8, \(\frac{3}{8} < \frac{5}{8}\)

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

3 parts out of 8

<
\(\frac{5}{8}\)

5 parts out of 8

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

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

Applied rules:

• Like denominators rule: When denominators are the same, compare numerators directly

• Numerator comparison: Larger numerator means larger fraction

• Equal parts principle: Same denominator means same-sized parts

2 Compare fractions with like numerators
Exercise 2
Compare: \(\frac{2}{3}\) and \(\frac{2}{5}\)
Definition:

Fractions with like numerators: Fractions that have the same top number (numerator)

Fractions
\(\frac{2}{3} \text{ and } \frac{2}{5}\)
Denominators
\(3 \text{ and } 5\)
Comparison
\(\frac{2}{3} > \frac{2}{5}\)
Step 1: Identify the numerators

Both fractions have numerator 2, so we're taking the same number of parts from each whole

Step 2: Compare the denominators

Smaller denominator means larger parts (3 < 5, so thirds are bigger than fifths)

Step 3: Make the comparison

Since 2 larger parts (thirds) is more than 2 smaller parts (fifths), \(\frac{2}{3} > \frac{2}{5}\)

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

2 parts of size 1/3

>
\(\frac{2}{5}\)

2 parts of size 1/5

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

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

Applied rules:

• Like numerators rule: When numerators are the same, compare denominators inversely

• Denominator comparison: Smaller denominator means larger fraction

• Part size principle: More parts mean smaller individual parts

3 Mixed comparison scenario
Exercise 3
Compare: \(\frac{4}{7}\) and \(\frac{3}{7}\)
Definition:

Like denominators comparison: Both fractions have the same denominator, allowing direct numerator comparison

Fractions
\(\frac{4}{7} \text{ and } \frac{3}{7}\)
Numerators
\(4 \text{ and } 3\)
Result
\(\frac{4}{7} > \frac{3}{7}\)
Step 1: Check denominators

Both fractions have denominator 7, so we compare numerators directly

Step 2: Compare numerators

\(4 > 3\), so \(\frac{4}{7}\) represents more parts than \(\frac{3}{7}\)

Step 3: Determine the relationship

Since 4 parts out of 7 is more than 3 parts out of 7, \(\frac{4}{7} > \frac{3}{7}\)

\(\frac{4}{7}\)

4 parts out of 7

>
\(\frac{3}{7}\)

3 parts out of 7

\(\frac{4}{7} > \frac{3}{7}\)
Final answer:

\(\frac{4}{7} > \frac{3}{7}\)

Applied rules:

• Direct comparison: Same denominator allows direct numerator comparison

• Greater than principle: Larger numerator means larger fraction when denominators match

• Equal division: Same denominator means equal-sized portions

Rules and methods, laws,...
\(\frac{a}{c} \text{ vs } \frac{b}{c} \Rightarrow \text{Compare } a \text{ and } b\)
Compare Fractions with Like Denominators
Rule 1
\(\frac{a}{c} < \frac{b}{c} \text{ if } a < b\)
When denominators are equal, compare numerators
Rule 2
\(\frac{c}{a} > \frac{c}{b} \text{ if } a < b\)
When numerators are equal, smaller denominator is larger
Rule 3
\(\frac{a}{c} = \frac{b}{c} \text{ if } a = b\)
Fractions are equal when numerators and denominators match
Key definitions:

Numerator: The top number in a fraction representing the number of parts taken

Denominator: The bottom number in a fraction representing the total number of equal parts

Like denominators: Fractions with the same denominator (same-sized parts)

Like numerators: Fractions with the same numerator (same number of parts)

Complete methodology:
  1. Identify the pattern: Determine if fractions have like numerators or like denominators
  2. Apply the appropriate rule: Use numerator comparison for like denominators, denominator comparison for like numerators
  3. Make the comparison: Use >, <, or = symbols appropriately
  4. Verify with visualization: Draw models to confirm your reasoning
Tip 1: For like denominators, focus only on the numerators.
Tip 2: For like numerators, focus only on the denominators - smaller denominator means larger fraction.
Tip 3: Remember: more pieces of the same size is more than fewer pieces.
Tip 4: Think of pizza: 3 slices of an 8-slice pizza vs 5 slices of an 8-slice pizza.
Common errors: Confusing numerator and denominator roles, forgetting that larger denominator means smaller parts.
Memory aid: "More parts, more value" for like denominators; "Fewer divisions, bigger pieces" for like numerators.
Rules to know by heart:

• Like denominators: \(\frac{a}{c} > \frac{b}{c}\) when \(a > b\)

• Like numerators: \(\frac{c}{a} > \frac{c}{b}\) when \(a < b\)

• Visual verification: Draw models to confirm comparisons

Solution: Exercises 4 to 5
4 Compare fractions with like numerators
Exercise 4
Compare: \(\frac{3}{4}\) and \(\frac{3}{7}\)
Definition:

Like numerators comparison: Both fractions have the same numerator, so we compare denominators to determine part size

Fractions
\(\frac{3}{4} \text{ and } \frac{3}{7}\)
Denominators
\(4 \text{ and } 7\)
Comparison
\(\frac{3}{4} > \frac{3}{7}\)
Step 1: Identify the numerators

Both fractions have numerator 3, so we're taking 3 parts from each whole

Step 2: Compare the denominators

\(4 < 7\), so fourths are larger pieces than sevenths

Step 3: Make the comparison

Since 3 larger parts (fourths) is more than 3 smaller parts (sevenths), \(\frac{3}{4} > \frac{3}{7}\)

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

3 parts of size 1/4

>
\(\frac{3}{7}\)

3 parts of size 1/7

\(\frac{3}{4} > \frac{3}{7}\)
Final answer:

\(\frac{3}{4} > \frac{3}{7}\)

Applied rules:

• Like numerators rule: When numerators are the same, compare denominators inversely

• Part size principle: Smaller denominator means larger individual parts

• Equal parts count: Same numerator means same number of parts taken

5 Advanced comparison
Exercise 5
Compare: \(\frac{6}{9}\) and \(\frac{4}{9}\)
Definition:

Like denominators comparison: Both fractions have the same denominator, allowing direct comparison of numerators

Fractions
\(\frac{6}{9} \text{ and } \frac{4}{9}\)
Numerators
\(6 \text{ and } 4\)
Comparison
\(\frac{6}{9} > \frac{4}{9}\)
Step 1: Check denominators

Both fractions have denominator 9, so we compare numerators directly

Step 2: Compare numerators

\(6 > 4\), so \(\frac{6}{9}\) represents more parts than \(\frac{4}{9}\)

Step 3: Make the comparison

Since 6 parts out of 9 is more than 4 parts out of 9, \(\frac{6}{9} > \frac{4}{9}\)

\(\frac{6}{9}\)

6 parts out of 9

>
\(\frac{4}{9}\)

4 parts out of 9

\(\frac{6}{9} > \frac{4}{9}\)
Final answer:

\(\frac{6}{9} > \frac{4}{9}\)

Applied rules:

• Direct numerator comparison: Same denominator allows direct comparison

• Greater than principle: Larger numerator means larger fraction when denominators match

• Equal division: Same denominator means equal-sized portions

Complete Guide: Laws, Methods, Rules, Definitions
\(\frac{a}{c} \text{ vs } \frac{b}{c} \Rightarrow \text{Compare } a \text{ and } b\)
Like Denominators Rule
\(\frac{c}{a} \text{ vs } \frac{c}{b} \Rightarrow \text{Compare } a \text{ and } b \text{ inversely}\)
Like Numerators Rule
Key definitions:

Fraction: A number representing a part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\)

Numerator: The top number indicating how many parts are taken

Denominator: The bottom number indicating how many equal parts make up the whole

Like denominators: Fractions with identical denominators (same-sized parts)

Like numerators: Fractions with identical numerators (same number of parts)

Complete methodology:
  1. Pattern recognition: Identify whether fractions have like numerators or like denominators
  2. Rule selection: Choose the appropriate comparison rule based on the pattern
  3. Value comparison: Compare the relevant numbers (numerators or denominators)
  4. Symbol placement: Insert the correct comparison symbol (> , < , =)
  5. Verification: Use visual models or cross-multiplication to verify the result
Tip 1: For like denominators, the fraction with the larger numerator is always greater.
Tip 2: For like numerators, the fraction with the smaller denominator is always greater.
Tip 3: Remember: when the whole is divided into more parts, each part gets smaller.
Tip 4: Draw circle models or bar models to visualize the comparison.
Tip 5: Think of sharing: getting 3 pieces of a cake cut into 4 pieces vs 3 pieces of a cake cut into 7 pieces.
Tip 6: Practice with real-world examples: pizza slices, chocolate bars, etc.
Common errors: Mixing up numerator and denominator roles, forgetting that larger denominator means smaller parts, misapplying the rules.
Memory aids: "More parts, more value" for like denominators; "Fewer divisions, bigger pieces" for like numerators.
Key insight: The relationship between numerator and denominator determines the fraction's size.
Visualization: Drawing models helps solidify conceptual understanding.
Rules to know by heart:

• Like denominators rule: \(\frac{a}{c} > \frac{b}{c}\) when \(a > b\)

• Like numerators rule: \(\frac{c}{a} > \frac{c}{b}\) when \(a < b\)

• Equality rule: \(\frac{a}{b} = \frac{a}{b}\) (reflexive property)

• Transitive property: If \(\frac{a}{b} > \frac{c}{d}\) and \(\frac{c}{d} > \frac{e}{f}\), then \(\frac{a}{b} > \frac{e}{f}\)

• Antisymmetric property: If \(\frac{a}{b} > \frac{c}{d}\), then \(\frac{c}{d} < \frac{a}{b}\)

Visual Fraction Comparison

📊
Like Denominators Example

\(\frac{2}{5}\) vs \(\frac{3}{5}\): Both have 5 equal parts, so 3 parts > 2 parts

2
3
5
3
2
5
Like Numerators Example

\(\frac{2}{3}\) vs \(\frac{2}{5}\): Both take 2 parts, but 3 parts make bigger pieces than 5 parts

2
1
1
2
1
1
1
1
Comparison Workflow
1
Identify Pattern
2
Select Rule
3
Compare Values
4
Place Symbol
5
Verify
Fraction Comparison Model

Visual representation showing how the same numerator across different denominators results in different fractional values

Master These Patterns!

Questions & Answers

Question: I don't understand why \(\frac{2}{3}\) is greater than \(\frac{2}{5}\) when 5 is greater than 3. Can you explain this concept?

Answer: This is a common source of confusion! The key is to think about what the denominator represents:

  • The denominator tells us how many equal pieces the whole is divided into
  • \(\frac{2}{3}\) means 2 pieces from a whole divided into 3 equal parts
  • \(\frac{2}{5}\) means 2 pieces from a whole divided into 5 equal parts

When you divide a pizza into 3 pieces, each piece is much bigger than when you divide the same pizza into 5 pieces. So 2 big pieces (thirds) are more than 2 small pieces (fifths).

Think of it this way: If you had to share a cake with 2 friends (making 3 pieces total) versus sharing with 4 friends (making 5 pieces total), you'd get a bigger slice when there are fewer people to share with!

The rule is: when numerators are the same, the fraction with the smaller denominator is larger.

Question: When comparing \(\frac{4}{7}\) and \(\frac{6}{7}\), why does the fraction with the bigger numerator win? Doesn't having more parts make it heavier?

Answer: Great question! You're thinking about it correctly in terms of quantity. Let me clarify:

  • Both fractions have the same denominator (7), meaning both wholes are divided into the same number of equal parts
  • Each part in \(\frac{4}{7}\) is the same size as each part in \(\frac{6}{7}\)
  • So \(\frac{6}{7}\) means taking 6 parts while \(\frac{4}{7}\) means taking only 4 parts

Yes, having more parts does mean more quantity when the parts are the same size! If you have 6 pieces of the same size versus 4 pieces of that same size, you definitely have more.

Think of it like this: if you and your friend both eat slices of the same size pizza, and you eat 6 slices while your friend eats 4, you ate more pizza.

The rule is: when denominators are the same, the fraction with the larger numerator is larger.

Question: How can I remember which rule to use? I keep mixing up the like numerators and like denominators rules.

Answer: Here are some memory tricks to help you remember:

  • Like Denominators: "Same Bottom, Compare Top" - When denominators match, just look at numerators. Bigger numerator = bigger fraction.
  • Like Numerators: "Same Top, Opposite Bottom" - When numerators match, smaller denominator = bigger fraction.

Another way to remember: Think of sharing cookies

  • With like denominators (\(\frac{2}{5}\) vs \(\frac{3}{5}\)): Everyone gets the same size cookie, so more cookies (larger numerator) is better
  • With like numerators (\(\frac{2}{3}\) vs \(\frac{2}{5}\)): You get the same number of cookies (2), but the ones from 3 total are bigger than the ones from 5 total

Always ask yourself first: "Are the tops the same?" or "Are the bottoms the same?" Then apply the corresponding rule!