Solved Exercises on Comparing Fractions with Unlike Numerators in Grade 4

Master comparing fractions with unlike numerators: finding common denominators, cross multiplication, equivalent fractions, and comparison strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic comparison with different numerators
Exercise 1
Compare: \( \frac{3}{4} \) and \( \frac{5}{6} \). Which fraction is larger?
Definition:

Fraction Comparison: To compare fractions with unlike numerators, we need to find equivalent fractions with the same denominator.

Comparison method:
  1. Find the least common denominator (LCD) of the fractions
  2. Convert each fraction to an equivalent fraction with the LCD
  3. Compare the numerators of the equivalent fractions
  4. The fraction with the larger numerator is the larger fraction
Original fractions
\( \frac{3}{4}, \frac{5}{6} \)
LCD = 12
\( \frac{9}{12}, \frac{10}{12} \)
Comparison
\( \frac{9}{12} < \frac{10}{12} \)
Step 1: Find the LCD

The denominators are 4 and 6. Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... LCD = 12

Step 2: Convert to equivalent fractions

\( \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)

\( \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \)

Step 3: Compare numerators

Since 9 < 10, we have \( \frac{9}{12} < \frac{10}{12} \)

\( \frac{3}{4} < \frac{5}{6} \)
Final answer:

\( \frac{3}{4} < \frac{5}{6} \)

Applied rules:

• Least Common Denominator: Find the smallest number that both denominators divide evenly into

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Numerator Comparison: When denominators are equal, compare numerators directly

2 Cross multiplication method
Exercise 2
Compare: \( \frac{2}{3} \) and \( \frac{4}{7} \). Which fraction is larger?
Definition:

Cross Multiplication: Multiply the numerator of one fraction by the denominator of the other and compare the products.

Original fractions
\( \frac{2}{3}, \frac{4}{7} \)
Cross products
\( 2 \times 7 = 14, 4 \times 3 = 12 \)
Comparison
\( 14 > 12 \)
Step 1: Apply cross multiplication

Multiply the numerator of the first fraction by the denominator of the second: \( 2 \times 7 = 14 \)

Multiply the numerator of the second fraction by the denominator of the first: \( 4 \times 3 = 12 \)

Step 2: Compare the products

Since 14 > 12, the first fraction is larger

Step 3: Write the comparison

\( \frac{2}{3} > \frac{4}{7} \)

\( \frac{2}{3} > \frac{4}{7} \)
Final answer:

\( \frac{2}{3} > \frac{4}{7} \)

Applied rules:

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

• Product Comparison: Larger product corresponds to the larger fraction

• Efficiency: Cross multiplication is often faster than finding LCD

3 Multiple fractions comparison
Exercise 3
Order from least to greatest: \( \frac{1}{2}, \frac{3}{5}, \frac{2}{3} \)
Definition:

Ordering Fractions: To order multiple fractions, convert them to equivalent fractions with a common denominator or use cross multiplication to compare pairs.

Original fractions
\( \frac{1}{2}, \frac{3}{5}, \frac{2}{3} \)
LCD = 30
\( \frac{15}{30}, \frac{18}{30}, \frac{20}{30} \)
Order
\( \frac{15}{30} < \frac{18}{30} < \frac{20}{30} \)
Step 1: Find the LCD for all fractions

Denominators: 2, 5, 3. LCM of 2, 5, 3 is 30

Step 2: Convert each fraction to equivalent with LCD

\( \frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30} \)

\( \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \)

\( \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \)

Step 3: Order by comparing numerators

Since 15 < 18 < 20, the order is \( \frac{15}{30} < \frac{18}{30} < \frac{20}{30} \)

Step 4: Write the final order

\( \frac{1}{2} < \frac{3}{5} < \frac{2}{3} \)

\( \frac{1}{2} < \frac{3}{5} < \frac{2}{3} \)
Final answer:

\( \frac{1}{2} < \frac{3}{5} < \frac{2}{3} \)

Applied rules:

• Multiple Fraction Ordering: Find LCD for all fractions simultaneously

• Systematic Approach: Convert all fractions before comparing

• Verification: Double-check each conversion to ensure accuracy

Rules and methods, laws,...
\( \frac{a}{b} \) vs \( \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \) vs \( \frac{c \times b}{d \times b} \)
Common Denominator Method
Method 1
Cross Multiplication
\( \frac{a}{b} \) vs \( \frac{c}{d} \) → compare \( a \times d \) vs \( c \times b \)
Method 2
Common Denominator
Convert to same denominator, then compare numerators
Method 3
Decimal Conversion
Divide numerator by denominator to get decimals
Key definitions:

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

Denominator: The bottom number in a fraction (how many equal parts the whole is divided into)

Unlike Numerators: Fractions where the top numbers are different

Equivalent Fractions: Fractions that represent the same amount but have different numerators and denominators

Complete methodology:
  1. Identify the fractions: Recognize the numerators and denominators
  2. Choose the method: Cross multiplication, common denominator, or decimal conversion
  3. Apply the method: Follow the steps of the chosen technique
  4. Compare results: Determine which fraction is larger
  5. Write the answer: Use the correct inequality symbol
Tip 1: Cross multiplication is often the fastest method for comparing two fractions.
Tip 2: When ordering multiple fractions, finding a common denominator is usually more efficient.
Tip 3: Remember that the larger the numerator (with same denominator), the larger the fraction.
Tip 4: Always verify your work by converting back to the original fractions.
Common errors: Confusing numerators with denominators, making calculation errors when finding LCD, forgetting to multiply both numerator and denominator.
Exam preparation: Practice all three methods, memorize multiplication tables, understand fraction equivalency.
Formulas to know by heart:

• Cross multiplication: \( \frac{a}{b} \) vs \( \frac{c}{d} \) → compare \( a \times d \) vs \( c \times b \)

• Common denominator: \( \frac{a}{b} = \frac{a \times n}{b \times n} \)

• Fraction comparison: If denominators are equal, compare numerators directly

Solution: Exercises 4 to 5
4 Real-world application
Exercise 4
Sarah ate \( \frac{3}{8} \) of a pizza and Tom ate \( \frac{5}{12} \) of another pizza. Who ate more pizza?
Definition:

Real-World Application: Comparing fractions helps us determine which quantity is greater in practical situations.

Fractions to compare
\( \frac{3}{8}, \frac{5}{12} \)
LCD = 24
\( \frac{9}{24}, \frac{10}{24} \)
Comparison
\( \frac{9}{24} < \frac{10}{24} \)
Step 1: Identify the fractions to compare

Sarah ate \( \frac{3}{8} \) and Tom ate \( \frac{5}{12} \)

Step 2: Find the LCD

Denominators are 8 and 12. LCM of 8 and 12 is 24

Step 3: Convert to equivalent fractions

\( \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \)

\( \frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24} \)

Step 4: Compare the numerators

Since 9 < 10, we have \( \frac{9}{24} < \frac{10}{24} \)

Step 5: Interpret the result

Tom ate more pizza since \( \frac{5}{12} > \frac{3}{8} \)

Tom ate more pizza (\( \frac{5}{12} > \frac{3}{8} \))
Final answer:

Tom ate more pizza because \( \frac{5}{12} > \frac{3}{8} \)

Applied rules:

• Problem Solving: Translate word problems into mathematical expressions

• Real-World Context: Apply fraction comparison to practical situations

• Interpretation: Connect mathematical results back to the original context

5 Complex comparison with mixed numbers
Exercise 5
Compare: \( 1\frac{2}{3} \) and \( 1\frac{3}{4} \). Which mixed number is larger?
Definition:

Mixed Numbers: Numbers that contain both a whole number and a proper fraction.

Mixed numbers
\( 1\frac{2}{3}, 1\frac{3}{4} \)
Convert to improper
\( \frac{5}{3}, \frac{7}{4} \)
LCD = 12
\( \frac{20}{12}, \frac{21}{4} \)
Step 1: Convert mixed numbers to improper fractions

\( 1\frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \)

\( 1\frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{7}{4} \)

Step 2: Find the LCD

Denominators are 3 and 4. LCM of 3 and 4 is 12

Step 3: Convert to equivalent fractions

\( \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} \)

\( \frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12} \)

Step 4: Compare the numerators

Since 20 < 21, we have \( \frac{20}{12} < \frac{21}{12} \)

Step 5: Convert back to mixed numbers if needed

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

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

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

Applied rules:

• Mixed Number Conversion: \( a\frac{b}{c} = \frac{c \times a + b}{c} \)

• Comparison Strategy: Convert mixed numbers to improper fractions for easier comparison

• Systematic Approach: Handle the whole number and fractional parts separately if needed

Fraction Comparison Laws, Methods, and Strategies
\( \frac{a}{b} \) vs \( \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \) vs \( \frac{c \times b}{d \times b} \)
Cross Multiplication Rule
Key definitions:

Like Fractions: Fractions with the same denominator (e.g., \( \frac{2}{5} \) and \( \frac{3}{5} \))

Unlike Fractions: Fractions with different denominators (e.g., \( \frac{2}{3} \) and \( \frac{4}{5} \))

Proper Fraction: A fraction where the numerator is smaller than the denominator (e.g., \( \frac{3}{4} \))

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \( \frac{5}{3} \))

Mixed Number: A combination of a whole number and a proper fraction (e.g., \( 2\frac{1}{3} \))

Complete methodology:
  1. Analyze the fractions: Look at numerators and denominators
  2. Choose the comparison method: Cross multiplication, common denominator, or decimal conversion
  3. Apply the chosen method: Follow the steps systematically
  4. Verify the result: Check calculations and make sure the answer makes sense
  5. Express the answer: Use the correct inequality symbol (<, >, or =)
Tip 1: When denominators are the same, compare numerators directly. Larger numerator = larger fraction.
Tip 2: When numerators are the same, compare denominators. Smaller denominator = larger fraction.
Tip 3: For quick mental comparisons, try to estimate if the fraction is more or less than 1/2.
Tip 4: Always double-check your LCD calculations to avoid errors.
Common errors: Forgetting to find LCD, making arithmetic mistakes, confusing numerators and denominators, not simplifying fractions before comparison.
Key notes to remember: Equivalent fractions have the same value even though they look different. The larger the denominator, the smaller each part becomes.
Fraction Comparison Laws:

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

• Reflexive Property: Any fraction is equal to itself: \( \frac{a}{b} = \frac{a}{b} \)

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

• Equivalence Property: \( \frac{a}{b} = \frac{c}{d} \) if and only if \( a \times d = b \times c \)

Visual Fraction Comparison: Understanding Size Relationships
Visualizing Fraction Sizes
Compare these fractions visually:
\( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \)
Notice how as the numerator and denominator increase proportionally, the fraction gets closer to 1.

Pattern Recognition: The chart shows how fractions with consecutive numerators and denominators approach 1 as the numbers get larger.

  • \( \frac{1}{2} = 0.5 \) (50%)
  • \( \frac{2}{3} ≈ 0.667 \) (66.7%)
  • \( \frac{3}{4} = 0.75 \) (75%)
  • \( \frac{4}{5} = 0.8 \) (80%)
  • \( \frac{5}{6} ≈ 0.833 \) (83.3%)

Questions & Answers

Question: I'm confused about when to use cross multiplication versus finding a common denominator. How do I know which method to use?

Answer: Great question! Here's how to decide:

  • Cross multiplication: Best when comparing just two fractions quickly. It's faster and doesn't require finding LCD.
  • Common denominator: Better when comparing multiple fractions or when the denominators are small numbers.

For example, comparing \( \frac{3}{4} \) and \( \frac{5}{7} \): Cross multiplication (3×7 vs 5×4) is faster.

But for ordering \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \): Finding a common denominator (12) makes it easier to order all three at once.

Both methods will give the same result - choose the one that feels more comfortable for the specific problem!

Question: My child is struggling with finding the least common denominator. Are there any tricks to make this easier?

Answer: Yes! Here are some helpful strategies:

  • Start with the larger denominator: List its multiples and see if the smaller denominator divides evenly into any of them.
  • Look for patterns: If denominators are consecutive numbers (like 3 and 4), their LCD is often their product (12).
  • Use prime factorization: Break down each denominator into prime factors and take the highest power of each prime.

Example: For denominators 6 and 8:

Multiples of 8: 8, 16, 24... Does 6 divide 8? No. Does 6 divide 16? No. Does 6 divide 24? Yes! So LCD = 24.

Practice with smaller numbers first to build confidence!

Question: How can I help my child visualize why \( \frac{3}{4} \) is larger than \( \frac{2}{3} \) without doing calculations?

Answer: Visualization is key! Try these approaches:

  • Pizza analogy: \( \frac{3}{4} \) means 3 slices out of 4 total. \( \frac{2}{3} \) means 2 slices out of 3 total. Draw both scenarios.
  • Number line: Plot both fractions on a number line from 0 to 1. \( \frac{3}{4} \) will be further to the right.
  • Think in terms of halves: \( \frac{3}{4} = 0.75 \) (more than half), \( \frac{2}{3} ≈ 0.67 \) (less than 0.75).

For \( \frac{3}{4} \) vs \( \frac{2}{3} \): Both are close to 1, but \( \frac{3}{4} \) needs only 1 piece to make a whole, while \( \frac{2}{3} \) needs 1 piece too. However, the pieces in \( \frac{3}{4} \) are smaller (fourths vs thirds), so having 3 of the smaller pieces is less than having 2 of the larger pieces. Wait - that's wrong! Let me correct: In fourths, each piece is smaller than in thirds. So having 3 of the smaller pieces vs 2 of the larger pieces needs calculation, which is why we use the methods taught!

Question: What happens when I have to compare a fraction to a whole number? Like comparing \( \frac{5}{3} \) to 2?

Answer: Great question! You can handle this in two ways:

  • Method 1: Convert the whole number to a fraction with the same denominator. \( 2 = \frac{6}{3} \). Now compare \( \frac{5}{3} \) and \( \frac{6}{3} \). Since 5 < 6, \( \frac{5}{3} < 2 \).
  • Method 2: Convert the fraction to a mixed number. \( \frac{5}{3} = 1\frac{2}{3} \). Since \( 1\frac{2}{3} < 2 \), we have \( \frac{5}{3} < 2 \).

The key is to have the same form for both numbers before comparing!

For \( \frac{5}{3} \) vs 2: Since \( \frac{5}{3} = 1\frac{2}{3} \) and 1 is less than 2, we know \( \frac{5}{3} < 2 \).

Question: Can I always use cross multiplication for comparing any two fractions?

Answer: Yes, cross multiplication works for comparing any two fractions! Here's why:

If you have \( \frac{a}{b} \) and \( \frac{c}{d} \), cross multiplication gives you \( a \times d \) and \( c \times b \).

This works because you're essentially comparing \( \frac{a \times d}{b \times d} \) and \( \frac{c \times b}{b \times d} \) without actually calculating the denominators.

Rules:

  • If \( a \times d > c \times b \), then \( \frac{a}{b} > \frac{c}{d} \)
  • If \( a \times d < c \times b \), then \( \frac{a}{b} < \frac{c}{d} \)
  • If \( a \times d = c \times b \), then \( \frac{a}{b} = \frac{c}{d} \)

Cross multiplication is especially useful when the denominators are large or when you want a quick comparison!