Solved Exercises on Creating Equivalent Fractions in Grade 4

Master creating equivalent fractions: generating equivalent fractions by multiplication and division, simplifying fractions, and finding missing numerators or denominators through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Creating equivalent fractions by multiplication
Exercise 1
Generate three equivalent fractions for \(\frac{2}{3}\) by multiplying both the numerator and denominator by different numbers.
Definition:

Equivalent fractions: Fractions that represent the same value or amount

Generation rule: Multiply both numerator and denominator by the same non-zero number

Generation method:
  1. Take the original fraction
  2. Multiply both numerator and denominator by the same number
  3. Repeat with different numbers to get multiple equivalents
  4. Verify that all generated fractions are equivalent
Original fraction
\(\frac{2}{3}\)
Multiply by 2
\(\frac{4}{6}\)
Multiply by 3
\(\frac{6}{9}\)
Multiply by 4
\(\frac{8}{12}\)
Step 1: Multiply by 2

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

Check: 2 × 6 = 12 and 3 × 4 = 12, so they are equivalent

Step 2: Multiply by 3

\(\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}\)

Check: 2 × 9 = 18 and 3 × 6 = 18, so they are equivalent

Step 3: Multiply by 4

\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

Check: 2 × 12 = 24 and 3 × 8 = 24, so they are equivalent

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

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

4
4
4
4
6
6

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

\(\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12}\)
Final answer:

Three equivalent fractions for \(\frac{2}{3}\) are:

\(\frac{4}{6}\), \(\frac{6}{9}\), and \(\frac{8}{12}\)

Applied rules:

• Multiplication property: Multiply both parts by the same number

• Equivalence: Value remains unchanged

• Verification: Cross multiply to confirm equivalence

2 Creating equivalent fractions by division
Exercise 2
Simplify \(\frac{12}{18}\) to its simplest form by dividing both the numerator and denominator by their greatest common factor.
Definition:

Simplification: Reducing a fraction to its lowest terms

Greatest Common Factor (GCF): The largest number that divides both numerator and denominator

Original fraction
\(\frac{12}{18}\)
GCF of 12 and 18
6
Simplified form
\(\frac{2}{3}\)
Step 1: Find the factors of 12

Factors of 12: 1, 2, 3, 4, 6, 12

Step 2: Find the factors of 18

Factors of 18: 1, 2, 3, 6, 9, 18

Step 3: Identify the greatest common factor

Common factors: 1, 2, 3, 6

Greatest common factor: 6

Step 4: Divide both numerator and denominator by the GCF

\(\frac{12}{18} = \frac{12 ÷ 6}{18 ÷ 6} = \frac{2}{3}\)

Step 5: Verify the result

\(\frac{2}{3}\) cannot be simplified further since 2 and 3 have no common factors other than 1

12
18
÷
6
→
2
3
12: 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
GCF: 6
\(\frac{12}{18} = \frac{2}{3}\)
Final answer:

The simplified form of \(\frac{12}{18}\) is \(\frac{2}{3}\).

Applied rules:

• Division property: Divide both parts by the same number

• GCF method: Divide by the greatest common factor for one-step simplification

• Simplest form: When numerator and denominator have no common factors

3 Finding missing numbers
Exercise 3
Find the missing number in this equivalent fraction: \(\frac{3}{4} = \frac{?}{12}\)
Definition:

Missing number: Finding the unknown numerator or denominator to make fractions equivalent

Given
\(\frac{3}{4} = \frac{?}{12}\)
Relationship
4 × 3 = 12
Result
3 × 3 = 9
Step 1: Analyze the relationship between denominators

From 4 to 12: 4 × 3 = 12

Step 2: Apply the same operation to the numerator

3 × 3 = 9

Step 3: Verify the result

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

Check: 3 × 12 = 36 and 4 × 9 = 36 ✓

Step 4: State the answer

The missing number is 9

3
×
3
4
=
9
12

3 × 12 = 36

4 × ? = 36

? = 36 ÷ 4 = 9

\(\frac{3}{4} = \frac{9}{12}\)
Final answer:

The missing number is 9, so \(\frac{3}{4} = \frac{9}{12}\).

Applied rules:

• Proportional relationship: What you do to one part, do to the other

• Cross multiplication: Verify equivalence by cross multiplying

• Missing number: Use the relationship between known values

Equivalent Fractions Generation Fundamentals
\(\frac{a}{b} = \frac{a \times n}{b \times n} \text{ where } n \neq 0\)
Equivalent Fractions Formula
Key definitions:

Equivalent fractions: Fractions that represent the same value or amount

Generation: Creating new fractions that equal the original

Multiplication method: Multiply both numerator and denominator by the same number

Division method: Divide both numerator and denominator by the same number

Simplification: Reducing to lowest terms

Simplest form: When numerator and denominator have no common factors other than 1

Greatest Common Factor (GCF): The largest number that divides both parts

Least Common Multiple (LCM): Smallest number that is a multiple of both denominators

Generation strategies:
  1. Multiplication: Multiply both numerator and denominator by the same number
  2. Division: Divide both numerator and denominator by their GCF
  3. Sequential reduction: Divide by common factors step by step
  4. Cross multiplication: Verify equivalence of generated fractions
  5. Pattern recognition: Identify multiplication factors
Tip 1: Always multiply or divide both numerator AND denominator by the same number.
Tip 2: To simplify quickly, divide by the GCF in one step.
Tip 3: Equivalent fractions always represent the same portion of a whole.
Tip 4: Use visual models to verify your answers.
Common errors: Forgetting to apply operation to both parts, not simplifying completely.
Key insight: The value of the fraction never changes, only its representation.
Generation rules:

• Multiplication rule: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0

• Division rule: \(\frac{a}{b} = \frac{a ÷ n}{b ÷ n}\) where n divides both a and b

• Equivalence: Two fractions are equivalent if their cross products are equal

• Simplification: Continue dividing by common factors until none remain

Equivalent Fraction Families

\(\frac{1}{2}\) Family

\(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = \frac{6}{12}\)

All represent half of a whole

\(\frac{1}{3}\) Family

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

All represent one-third of a whole

Solution: Exercises 4 to 5
4 Complex equivalence
Exercise 4
Show that \(\frac{15}{20}\) and \(\frac{9}{12}\) are equivalent by simplifying both fractions to their simplest form.
Definition:

Verification: Proving equivalence by reducing to simplest form

First fraction
\(\frac{15}{20}\)
Second fraction
\(\frac{9}{12}\)
Both simplify to
\(\frac{3}{4}\)
Step 1: Simplify \(\frac{15}{20}\)

Factors of 15: 1, 3, 5, 15

Factors of 20: 1, 2, 4, 5, 10, 20

GCF: 5

\(\frac{15}{20} = \frac{15 ÷ 5}{20 ÷ 5} = \frac{3}{4}\)

Step 2: Simplify \(\frac{9}{12}\)

Factors of 9: 1, 3, 9

Factors of 12: 1, 2, 3, 4, 6, 12

GCF: 3

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

Step 3: Compare simplified forms

Since both fractions simplify to \(\frac{3}{4}\), they are equivalent

15
20
÷
5
→
3
4
9
12
÷
3
→
3
4
Both \(\frac{15}{20}\) and \(\frac{9}{12}\) equal \(\frac{3}{4}\), so they are equivalent.
Final answer:

\(\frac{15}{20} = \frac{3}{4}\) and \(\frac{9}{12} = \frac{3}{4}\), therefore \(\frac{15}{20} = \frac{9}{12}\).

Applied rules:

• Common simplification: Equivalent fractions have the same simplest form

• GCF method: Efficient way to reduce fractions

• Transitive property: If a = c and b = c, then a = b

5 Advanced problem solving
Exercise 5
Find three equivalent fractions for \(\frac{4}{5}\) where the denominators are 15, 20, and 30. Then order all four fractions from least to greatest.
Definition:

Systematic generation: Creating specific equivalent fractions with given denominators

Original fraction
\(\frac{4}{5}\)
Target denominators
15, 20, 30
Generated fractions
\(\frac{12}{15}, \frac{16}{20}, \frac{24}{30}\)
Step 1: Generate equivalent with denominator 15

\(\frac{4}{5} = \frac{?}{15}\)

5 × 3 = 15, so 4 × 3 = 12

\(\frac{4}{5} = \frac{12}{15}\)

Step 2: Generate equivalent with denominator 20

\(\frac{4}{5} = \frac{?}{20}\)

5 × 4 = 20, so 4 × 4 = 16

\(\frac{4}{5} = \frac{16}{20}\)

Step 3: Generate equivalent with denominator 30

\(\frac{4}{5} = \frac{?}{30}\)

5 × 6 = 30, so 4 × 6 = 24

\(\frac{4}{5} = \frac{24}{30}\)

Step 4: Order the fractions

All four fractions: \(\frac{4}{5}, \frac{12}{15}, \frac{16}{20}, \frac{24}{30}\)

Since they're all equivalent, they all equal \(\frac{4}{5}\)

Order: \(\frac{4}{5} = \frac{12}{15} = \frac{16}{20} = \frac{24}{30}\)

\(\frac{4}{5}\)
\(\frac{12}{15}\)
\(\frac{16}{20}\)
\(\frac{24}{30}\)
Generated fractions: \(\frac{12}{15}, \frac{16}{20}, \frac{24}{30}\)
All equal \(\frac{4}{5}\), so they're all equal in value
Final answer:

The three equivalent fractions are:

\(\frac{12}{15}, \frac{16}{20}, \frac{24}{30}\)

Since all four fractions are equivalent, they all have the same value: \(\frac{4}{5}\)

Applied rules:

• Targeted generation: Find specific equivalent fractions

• Proportional relationships: Maintain the ratio when changing denominators

• Equivalence preservation: All generated fractions equal the original

Advanced Equivalent Fraction Concepts
\(\frac{a}{b} = \frac{c}{d} \text{ if and only if } a \times d = b \times c\)
Cross Multiplication Rule
Key definitions:

Cross multiplication: A method to verify fraction equivalence

Proportion: An equation stating that two ratios are equal

Scale factor: The number by which both numerator and denominator are multiplied

Common denominator: Same denominator for multiple fractions

Unit fraction: A fraction with numerator 1

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Reciprocal: A fraction flipped upside down

Advanced generation methods:
  1. Algebraic approach: Set up equations to find missing values
  2. Decimal conversion: Convert to decimals for comparison
  3. Percentage method: Convert to percentages
  4. Visual verification: Use area models to confirm equivalence
Tip 1: When generating equivalents, the scale factor is the multiplier.
Tip 2: Equivalent fractions form a family of related fractions.
Tip 3: The simplest form is unique for each fraction family.
Tip 4: Use cross multiplication to quickly verify equivalence.
Mathematical connection: Equivalent fractions underpin all fraction operations.
Real-world application: Scaling recipes, construction, and design.
Advanced relationships:

• Transitivity: If \(\frac{a}{b} = \frac{c}{d}\) and \(\frac{c}{d} = \frac{e}{f}\), then \(\frac{a}{b} = \frac{e}{f}\)

• Reflexivity: \(\frac{a}{b} = \frac{a}{b}\) (a fraction equals itself)

• Symmetry: If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{c}{d} = \frac{a}{b}\)

• Uniqueness: Every fraction has exactly one simplest form

Fraction Generation Patterns

Pattern Recognition

For \(\frac{2}{3}\): \(\frac{2}{3}, \frac{4}{6}, \frac{6}{9}, \frac{8}{12}, \frac{10}{15}...\)

Numerators: 2, 4, 6, 8, 10 (multiples of 2)

Denominators: 3, 6, 9, 12, 15 (multiples of 3)

Pattern: Multiply both by 1, 2, 3, 4, 5...

Prime Considerations

If the denominator is prime, the only equivalent fractions come from multiplying by integers

For \(\frac{3}{7}\): \(\frac{6}{14}, \frac{9}{21}, \frac{12}{28}...\)

No other simplifications possible since 7 is prime

Questions & Answers

Question: Why do we multiply both the top and bottom numbers by the same number to get equivalent fractions? Why can't we just add the same number to both?

Answer: Great question! This goes back to what fractions represent. A fraction is a division: \(\frac{a}{b}\) means \(a ÷ b\).

When you multiply both numerator and denominator by the same number, you're essentially multiplying by 1:

For example: \(\frac{2}{3} = \frac{2}{3} \times \frac{4}{4} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

Since \(\frac{4}{4} = 1\), multiplying by it doesn't change the value!

But if you added the same number to both:

\(\frac{2}{3} + \frac{4}{4} = \frac{2+4}{3+4} = \frac{6}{7}\)

Check: \(\frac{2}{3} = 0.667\) and \(\frac{6}{7} = 0.857\), so they're not equal!

Adding changes the value, but multiplying by 1 (in the form of \(\frac{n}{n}\)) keeps the value the same.

Question: My child is struggling with the concept of simplest form. How can I help them understand this better?

Answer: Here are effective strategies to help with simplest form:

  • Use visual models: Show area models where you can't group pieces anymore
  • Think of sharing: "What's the smallest number of people we can share this with?"
  • Factor rainbow: Draw arcs connecting factor pairs to find the GCF
  • Divide step-by-step: Keep dividing by common factors until none remain
  • Check with multiplication: Can you multiply the top and bottom by the same number?

For example, with \(\frac{8}{12}\):

• Factors of 8: 1, 2, 4, 8

• Factors of 12: 1, 2, 3, 4, 6, 12

• Common factors: 1, 2, 4

• Greatest common factor: 4

• Divide both by 4: \(\frac{8 ÷ 4}{12 ÷ 4} = \frac{2}{3}\)

Explain that \(\frac{2}{3}\) is in simplest form because 2 and 3 have no common factors other than 1.

Question: How does understanding equivalent fractions connect to other mathematical concepts?

Answer: Equivalent fractions are fundamental to many mathematical concepts:

  • Adding/subtracting fractions: Need common denominators (equivalent fractions)
  • Comparing fractions: Convert to same denominator using equivalents
  • Converting fractions to decimals/percents: Use equivalents with denominators of 10, 100, etc.
  • Algebra: Simplifying rational expressions
  • Ratio and proportion: Equivalent fractions are the basis of proportional relationships
  • Geometry: Similar figures have proportional sides
  • Probability: Expressing likelihood as equivalent fractions

Understanding equivalent fractions also develops proportional reasoning skills, which are essential for higher mathematics. Students who master this concept will find it easier to work with ratios, solve proportions, and understand concepts like slope in algebra.

This foundational skill also connects to the concept of mathematical equivalence in general, which appears throughout mathematics.