Solved Exercises on Comparing Fractions in Grade 3

Master comparing fractions: equivalent fractions, common denominators, cross multiplication, decimal conversion, and mixed number comparison through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Equivalent fractions comparison
Exercise 1
Compare: \(\frac{2}{3}\) and \(\frac{4}{6}\)
Definition:

Equivalent fractions: Fractions that represent the same part of a whole, e.g., \(\frac{a}{b} = \frac{na}{nb}\) where n is any non-zero number

Comparison method:
  1. Find if one fraction can be simplified to match the other
  2. Or find a common denominator
  3. Compare numerators once denominators are equal
Fractions
\(\frac{2}{3}\) vs \(\frac{4}{6}\)
Simplification
\(\frac{4}{6} = \frac{2}{3}\)
Comparison
\(\frac{2}{3} = \frac{2}{3}\)
Step 1: Check if fractions are equivalent

\(\frac{4}{6}\) can be simplified by dividing numerator and denominator by their GCD (2): \(\frac{4÷2}{6÷2} = \frac{2}{3}\)

Step 2: Compare the simplified forms

\(\frac{2}{3}\) vs \(\frac{2}{3}\) → They are equal

Step 3: Verify using cross multiplication

\(2 × 6 = 12\) and \(3 × 4 = 12\) → Since both products are equal, fractions are equivalent

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

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

Applied rules:

• Equivalence: Multiplying or dividing both numerator and denominator by the same number creates an equivalent fraction

• Cross multiplication: If \(ad = bc\), then \(\frac{a}{b} = \frac{c}{d}\)

• Equality: Fractions are equal if they represent the same portion of a whole

2 Common denominator method
Exercise 2
Compare: \(\frac{3}{4}\) and \(\frac{5}{6}\)
Definition:

Common denominator: When fractions have the same denominator, the one with the larger numerator is greater

Fractions
\(\frac{3}{4}\) vs \(\frac{5}{6}\)
LCD
LCM(4,6)=12
Result
\(\frac{9}{12} < \frac{10}{12}\)
Step 1: Find the Least Common Denominator (LCD)

Denominators are 4 and 6. LCM(4,6) = 12

Step 2: Convert fractions to have LCD

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

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

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:

• Common denominator rule: When denominators are equal, compare numerators

• LCM method: Find smallest common multiple of denominators

• Conversion: Multiply both numerator and denominator by same factor

3 Cross multiplication method
Exercise 3
Compare: \(\frac{7}{8}\) and \(\frac{5}{6}\)
Definition:

Cross multiplication: For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a×d\) with \(b×c\)

Fractions
\(\frac{7}{8}\) vs \(\frac{5}{6}\)
Cross products
42 vs 40
Result
\(\frac{7}{8} > \frac{5}{6}\)
Step 1: Apply cross multiplication rule

For \(\frac{a}{b}\) vs \(\frac{c}{d}\): multiply \(a×d\) and \(b×c\)

\(\frac{7}{8}\) vs \(\frac{5}{6}\): \(7×6\) vs \(8×5\)

Step 2: Calculate cross products

\(7×6 = 42\) and \(8×5 = 40\)

Step 3: Compare results

Since \(42 > 40\), we have \(\frac{7}{8} > \frac{5}{6}\)

Step 4: Verify with common denominator

LCD of 8 and 6 is 24: \(\frac{21}{24}\) vs \(\frac{20}{24}\) → \(\frac{21}{24} > \frac{20}{24}\) ✓

\(\frac{7}{8} > \frac{5}{6}\)
Final answer:

\(\frac{7}{8} > \frac{5}{6}\)

Applied rules:

• Cross multiplication: If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\)

• Efficiency: Cross multiplication is faster than finding LCD for large denominators

• Verification: Always verify results using another method

Rules and methods, laws,...
\(\frac{a}{b} = \frac{c}{d} \iff ad = bc\)
Cross Multiplication Rule
Method 1
Common Denominator
Convert to same denominator, compare numerators
Method 2
Cross Multiplication
Multiply diagonally, compare products
Method 3
Decimal Conversion
Convert to decimals, compare numerically
Key definitions:

Fraction: A number representing 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 total equal parts in the whole

Equivalent fractions: Different fractions that represent the same value

Complete methodology:
  1. Identify fraction types: Proper, improper, mixed numbers, or whole numbers
  2. Choose comparison method: Common denominator, cross multiplication, or decimal conversion
  3. Apply the method: Follow the specific steps for chosen technique
  4. Verify result: Use alternative method to confirm accuracy
Tip 1: When denominators are small, use common denominator method.
Tip 2: Cross multiplication is fastest for larger denominators.
Tip 3: If one fraction has denominator 10, 100, or 1000, convert to decimal easily.
Tip 4: Always simplify fractions before comparing when possible.
Common errors: Forgetting to adjust both numerator and denominator when finding common denominator, confusing numerator and denominator positions in cross multiplication.
Memory aid: "Cross multiply and compare the products" - remember to multiply numerator of first by denominator of second.
Comparison rules to know:

• If denominators are equal: \(\frac{a}{c} > \frac{b}{c}\) when \(a > b\)

• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) when \(ad > bc\)

• If numerators are equal: \(\frac{a}{b} > \frac{a}{c}\) when \(b < c\)

• Equivalent fractions: \(\frac{a}{b} = \frac{ka}{kb}\) for any non-zero k

Solution: Exercises 4 to 5
4 Mixed number comparison
Exercise 4
Compare: \(1\frac{2}{3}\) and \(1\frac{3}{4}\)
Definition:

Mixed number: A number combining a whole number and a proper fraction, e.g., \(a\frac{b}{c} = \frac{ac+b}{c}\)

Mixed numbers
\(1\frac{2}{3}\) vs \(1\frac{3}{4}\)
Improper fractions
\(\frac{5}{3}\) vs \(\frac{7}{4}\)
Result
\(\frac{5}{3} < \frac{7}{4}\)
Step 1: Convert mixed numbers to improper fractions

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

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

Step 2: Find common denominator

LCD of 3 and 4 is 12

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

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

Step 3: Compare the converted fractions

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

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

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

Applied rules:

• Mixed to improper: \(a\frac{b}{c} = \frac{ac+b}{c}\)

• Comparison: Convert to same form before comparing

• Verification: Check by converting back to decimals: \(1.67 < 1.75\)

5 Decimal conversion method
Exercise 5
Compare: \(\frac{3}{8}\) and \(\frac{2}{5}\)
Definition:

Decimal conversion: Divide numerator by denominator to get decimal representation

Fractions
\(\frac{3}{8}\) vs \(\frac{2}{5}\)
Decimals
0.375 vs 0.4
Result
\(\frac{3}{8} < \frac{2}{5}\)
Step 1: Convert fractions to decimals

\(\frac{3}{8} = 3 ÷ 8 = 0.375\)

\(\frac{2}{5} = 2 ÷ 5 = 0.4\)

Step 2: Compare decimal values

Compare 0.375 and 0.400

Looking at tenths place: 3 vs 4 → Since 3 < 4, we have 0.375 < 0.4

Step 3: Verify using cross multiplication

\(3 × 5 = 15\) and \(8 × 2 = 16\)

Since \(15 < 16\), we confirm \(\frac{3}{8} < \frac{2}{5}\)

\(\frac{3}{8} < \frac{2}{5}\)
Final answer:

\(\frac{3}{8} < \frac{2}{5}\)

Applied rules:

• Division method: Divide numerator by denominator for decimal form

• Decimal comparison: Compare digits from left to right

• Alternative verification: Use cross multiplication to confirm

Comprehensive Guide: Laws, Methods, Rules, Definitions, Expressions
\(\frac{a}{b} < \frac{c}{d} \iff ad < bc\)
Fraction Comparison Rule
Key definitions:

Proper fraction: Numerator is smaller than denominator (\(\frac{a}{b}\) where \(a < b\))

Improper fraction: Numerator is greater than or equal to denominator (\(\frac{a}{b}\) where \(a ≥ b\))

Mixed number: Combination of whole number and proper fraction (\(a\frac{b}{c}\))

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Complete methodology:
  1. Identify fraction types: Determine if proper, improper, or mixed numbers
  2. Choose appropriate method: Common denominator, cross multiplication, or decimal conversion
  3. Apply the method: Follow specific steps for the chosen technique
  4. Verify the result: Use an alternative method to confirm accuracy
  5. Express final answer: Write comparison using appropriate inequality symbol
Tip 1: For quick mental comparisons, convert to decimals if denominators are factors of 10, 100, etc.
Tip 2: When comparing to 1, determine if each fraction is greater than, less than, or equal to 1.
Tip 3: If numerators are the same, the fraction with the smaller denominator is larger.
Tip 4: Practice with visual models like pie charts or number lines to build intuition.
Common errors: Confusing cross multiplication order, forgetting to adjust both parts when finding common denominators, misplacing decimal points when converting.
Exam preparation: Master cross multiplication for speed, practice converting between fraction forms, learn divisibility rules for finding LCD quickly.
Formulas and rules to know by heart:

• Common denominator: \(\frac{a}{b} = \frac{an}{bn}\) for any non-zero n

• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) if and only if \(ad > bc\)

• Mixed to improper: \(a\frac{b}{c} = \frac{ac+b}{c}\)

• Same numerator: \(\frac{a}{b} > \frac{a}{c}\) if \(b < c\)

• Same denominator: \(\frac{a}{c} > \frac{b}{c}\) if \(a > b\)

Visual Comparison: Fraction Representation
Visualizing Fraction Values
Compare the following fractions visually:
\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\)

Analysis: As the numerator and denominator increase proportionally while maintaining the relationship where numerator is one less than denominator, the value approaches 1.

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

Questions & Answers

Question: I don't understand why \(\frac{2}{3}\) and \(\frac{4}{6}\) are equal when they look different. Can you explain?

Answer: Great question! These fractions are called "equivalent fractions" because they represent the same portion of a whole:

  • Think of a pizza cut into 3 pieces. If you eat 2 pieces, you've eaten \(\frac{2}{3}\) of the pizza.
  • Now imagine the same pizza cut into 6 pieces. To eat the same amount, you'd need 4 pieces, which is \(\frac{4}{6}\) of the pizza.
  • Both scenarios represent eating the same portion of pizza!

Mathematically: \(\frac{4}{6} = \frac{4÷2}{6÷2} = \frac{2}{3}\). We divided both numerator and denominator by their common factor (2).

Rule: \(\frac{a}{b} = \frac{na}{nb}\) for any non-zero number n.

Question: When comparing \(\frac{3}{4}\) and \(\frac{5}{6}\), should I always find the least common denominator or can I use any common denominator?

Answer: You can use any common denominator, but using the LEAST common denominator (LCD) makes calculations easier:

  • Using LCD (12): \(\frac{3}{4} = \frac{9}{12}\) and \(\frac{5}{6} = \frac{10}{12}\) → Easy comparison: \(\frac{9}{12} < \frac{10}{12}\)
  • Using common denominator 24: \(\frac{3}{4} = \frac{18}{24}\) and \(\frac{5}{6} = \frac{20}{24}\) → Still correct: \(\frac{18}{24} < \frac{20}{24}\)

Using the LCD reduces the chance of calculation errors and simplifies the work. However, any common denominator will give you the correct answer.

Tip: When denominators are large, consider using cross multiplication instead of finding LCD.

Question: How can I remember which method to use for comparing fractions?

Answer: Here's a systematic approach to choose the best method:

  1. Same denominators? → Compare numerators directly (e.g., \(\frac{3}{7}\) vs \(\frac{5}{7}\) → \(\frac{5}{7} > \frac{3}{7}\))
  2. Same numerators? → Compare denominators (e.g., \(\frac{2}{3}\) vs \(\frac{2}{5}\) → \(\frac{2}{3} > \frac{2}{5}\))
  3. Small denominators? → Use common denominator method
  4. Large denominators? → Use cross multiplication
  5. One denominator is 10, 100, or 1000? → Convert to decimal

Mnemonic: "Same bottom, compare tops. Same tops, flip bottoms. Different, cross multiply!"

Practice with various problems to develop intuition for which method works best in each situation.