How to Get Equivalent Fractions - Detailed Exercises for 4th Graders

Master equivalent fractions: learn how to multiply or divide numerators and denominators by the same number to find equivalent fractions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic equivalent fraction
Exercise 1
Find an equivalent fraction to 1/2 by multiplying both numerator and denominator by 3.
Definition:

Equivalent fractions: Fractions that represent the same amount even though they have different numerators and denominators.

Method for finding equivalent fractions:
  1. Multiply both numerator and denominator by the same non-zero number
  2. Or divide both numerator and denominator by the same non-zero number
  3. The value of the fraction stays the same
Original fraction
\( \frac{1}{2} \)
Multiply by 3
\( \frac{1 \times 3}{2 \times 3} \)
Result
\( \frac{3}{6} \)
Step 1: Identify the original fraction

We start with \( \frac{1}{2} \)

Step 2: Multiply both parts by the same number

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

Step 3: Calculate the result

\( \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \)

Step 4: Verify equivalence

\( \frac{3}{6} = \frac{1}{2} \) because \( 3 \div 3 = 1 \) and \( 6 \div 3 = 2 \)

\( \frac{1}{2} = \frac{3}{6} \)
Final answer:

\( \frac{1}{2} = \frac{3}{6} \)

Applied rules:

• Equivalent fractions rule: Multiply or divide both numerator and denominator by the same number

• Fraction equality: Two fractions are equal if their cross products are equal

• Visual representation: Both fractions represent the same portion of a whole

2 Multiplying by larger numbers
Exercise 2
Find an equivalent fraction to 2/3 by multiplying both numerator and denominator by 4.
Definition:

Cross multiplication: To verify equivalence, multiply diagonally: \( \frac{a}{b} = \frac{c}{d} \) if \( a \times d = b \times c \)

Original fraction
\( \frac{2}{3} \)
Multiply by 4
\( \frac{2 \times 4}{3 \times 4} \)
Result
\( \frac{8}{12} \)
Step 1: Identify the original fraction

We start with \( \frac{2}{3} \)

Step 2: Multiply both numerator and denominator by 4

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

Step 3: Calculate each part

Numerator: \( 2 \times 4 = 8 \)

Denominator: \( 3 \times 4 = 12 \)

Step 4: Write the equivalent fraction

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

Step 5: Verify using cross multiplication

\( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \), so they are equivalent

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

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

Applied rules:

• Equivalent fractions rule: Multiply both parts by the same number

• Cross multiplication: Verify equivalence by checking if cross products are equal

• Visual verification: Both fractions represent the same portion

3 Dividing to find equivalent fractions
Exercise 3
Find an equivalent fraction to 8/12 by dividing both numerator and denominator by their greatest common divisor.
Definition:

Greatest Common Divisor (GCD): The largest number that divides both the numerator and denominator evenly.

Original fraction
\( \frac{8}{12} \)
Find GCD
GCD(8,12)=4
Divide by GCD
\( \frac{8÷4}{12÷4} \)
Result
\( \frac{2}{3} \)
Step 1: Identify the original fraction

We start with \( \frac{8}{12} \)

Step 2: Find the GCD of numerator and denominator

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

Step 3: Divide both parts by the GCD

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

Step 4: Verify the result

\( \frac{2}{3} = \frac{8}{12} \) because \( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \)

Step 5: Check if fraction is in simplest form

GCD of 2 and 3 is 1, so \( \frac{2}{3} \) is in simplest form

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

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

Applied rules:

• Equivalent fractions rule: Divide both parts by the same number

• GCD simplification: Divide by the greatest common divisor to get simplest form

• Simplified fraction: When GCD of numerator and denominator is 1

Rules, Methods, and Key Concepts
\( \frac{a}{b} = \frac{a \times n}{b \times n} \)
Creating Equivalent Fractions
Key definitions:

Numerator: The top number in a fraction (represents parts taken)

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

Equivalent fractions: Fractions that represent the same value

Complete methodology:
  1. Identify the original fraction
  2. Decide whether to multiply or divide both parts by the same number
  3. Perform the operation on both numerator and denominator
  4. Verify the result using cross multiplication
Tip 1: Always multiply or divide BOTH numerator AND denominator by the same number.
Tip 2: To simplify a fraction, divide by the GCD of numerator and denominator.
Tip 3: Cross multiplication helps verify if two fractions are equivalent.
Tip 4: Draw pictures or use fraction bars to visualize equivalent fractions.
Common errors: Forgetting to change both numerator and denominator, multiplying only one part, confusing addition/subtraction with multiplication/division.
Memory aid: "Whatever you do to the top, do to the bottom" - keeps the fraction equivalent.
Essential rules to remember:

• 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 ≠ 0

• Cross multiplication: \( \frac{a}{b} = \frac{c}{d} \) if \( a \times d = b \times c \)

• Simplest form: When GCD of numerator and denominator is 1

\( \frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c \)
Cross Multiplication Rule
Solution: Exercises 4 to 5
4 Complex equivalent fractions
Exercise 4
Find three equivalent fractions to 3/4 by multiplying by 2, 3, and 5 respectively.
Definition:

Multiple equivalent fractions: One fraction can have many equivalent forms by multiplying by different numbers.

Original
\( \frac{3}{4} \)
×2
\( \frac{6}{8} \)
×3
\( \frac{9}{12} \)
×5
\( \frac{15}{20} \)
Step 1: Start with the original fraction

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

Step 2: Multiply by 2

\( \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \)

Step 3: Multiply by 3

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

Step 4: Multiply by 5

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

Step 5: Verify all are equivalent

Cross multiplication: 3×8=24, 4×6=24 → ✓

Cross multiplication: 3×12=36, 4×9=36 → ✓

Cross multiplication: 3×20=60, 4×15=60 → ✓

\( \frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{15}{20} \)
Final answer:

\( \frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{15}{20} \)

Applied rules:

• Equivalent fractions rule: Multiply both parts by the same number

• Multiple equivalents: One fraction has infinite equivalent forms

• Verification: Use cross multiplication to confirm equivalence

5 Real-world application
Exercise 5
Sarah ate 2/3 of her pizza. Her brother ate 4/6 of his pizza. Did they eat the same amount? Show your work.
Definition:

Real-world comparison: Use equivalent fractions to compare different measurements or portions.

Sarah's portion
\( \frac{2}{3} \)
Brother's portion
\( \frac{4}{6} \)
Comparison
Are they equal?
Step 1: Identify the fractions to compare

Sarah: \( \frac{2}{3} \), Brother: \( \frac{4}{6} \)

Step 2: Find equivalent forms to compare

Method 1: Simplify \( \frac{4}{6} \)

GCD of 4 and 6 is 2

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

Step 3: Alternative method - cross multiplication

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

\( 2 \times 6 = 12 \) and \( 3 \times 4 = 12 \)

Since 12 = 12, the fractions are equivalent

Step 4: Alternative method - convert to same denominator

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

This matches the brother's portion

Step 5: State the conclusion

Yes, Sarah and her brother ate the same amount of pizza

Yes, \( \frac{2}{3} = \frac{4}{6} \), so they ate the same amount.
Final answer:

Yes, Sarah and her brother ate the same amount of pizza because \( \frac{2}{3} = \frac{4}{6} \).

Applied rules:

• Comparison method: Convert fractions to equivalent forms for comparison

• Cross multiplication: Use to verify equivalence quickly

• Real-world application: Equivalent fractions help compare different measurements

Complete Guide: Laws, Methods, Rules, and Definitions
\( \frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a ÷ m}{b ÷ m} \)
Equivalent Fractions Formula
Key definitions:

Equivalent fractions: Different fractions that represent the same value or portion of a whole

Numerator: The number above the fraction bar (parts taken)

Denominator: The number below the fraction bar (total parts)

GCD (Greatest Common Divisor): Largest number that divides both numerator and denominator

Simplest form: A fraction where the GCD of numerator and denominator is 1

Complete methodology for equivalent fractions:
  1. Identify the original fraction and determine what you want to achieve
  2. Choose the operation: Multiply to find equivalent fractions, divide to simplify
  3. Apply the same operation to both numerator and denominator
  4. Verify your result using cross multiplication
  5. Check if in simplest form when simplifying
Memory trick: "Whatever you do to the top, do to the bottom" - maintains equivalence.
Visual method: Draw pie charts or fraction bars to see equivalence visually.
Quick check: Use cross multiplication to verify equivalence quickly.
Simplification: Always divide by the GCD to get the simplest form.
Pattern recognition: Look for multiplication relationships between numerators/denominators.
Common errors: Forgetting to apply operation to both parts, confusing operations with addition/subtraction, not recognizing equivalent forms.
Key insight: Equivalent fractions represent the same portion of a whole, just divided differently.
Practical tip: In word problems, convert to equivalent fractions to make comparisons easier.
Essential formulas and rules:

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

• Simplification rule: \( \frac{a}{b} = \frac{a ÷ m}{b ÷ m} \) where m ≠ 0

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

• Simplification process: Divide by GCD of numerator and denominator

• Identity property: \( \frac{a}{b} = \frac{a}{b} \) (every fraction is equivalent to itself)

\( \frac{a}{b} = \frac{c}{d} \iff ad = bc \)
Cross Product Property
Problem-solving strategies:
  1. Read carefully: Identify what equivalent fractions are needed
  2. Plan: Decide whether to multiply or divide based on the goal
  3. Execute: Apply the operation to both numerator and denominator
  4. Verify: Use cross multiplication to confirm your answer
  5. Check: Ensure the fraction is in the required form (simplest, etc.)

Questions & Answers

Question: I don't understand why 1/2 and 2/4 are the same. They look different with different numbers!

Answer: Great question! Think of it like pizza slices:

  • If you have 1 slice out of 2 total slices, that's half the pizza
  • If you cut each of those 2 slices in half, you now have 4 pieces total
  • So the 1 slice you originally had is now 2 smaller slices out of 4 total

Both situations represent the SAME AMOUNT of pizza - half of it! The numbers are different, but the actual amount is the same.

Using the formula: \( \frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \)

Cross check: \( 1 \times 4 = 4 \) and \( 2 \times 2 = 4 \), so they're equivalent!

Question: My child gets confused when they need to find an equivalent fraction. What's the easiest way to explain the process?

Answer: Here's a simple explanation for your child:

  1. Start with your fraction (like 3/4)
  2. Pick a number to multiply by (like 2)
  3. Multiply TOP and BOTTOM by that number: \( \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \)
  4. Check: Does 3×8 equal 4×6? Yes! 24 = 24 ✓

Memory trick: "What you do to the top, do to the bottom!"

Use visual aids like fraction circles or bars to show that both fractions represent the same amount.

Question: Can you find equivalent fractions by adding or subtracting? Why only multiplication and division?

Answer: That's a very insightful question! Let me show you why:

Adding doesn't work: Take \( \frac{1}{2} \) and add 1 to top and bottom: \( \frac{1+1}{2+1} = \frac{2}{3} \)

Is \( \frac{1}{2} = \frac{2}{3} \)? Cross multiply: 1×3 = 3, 2×2 = 4. Since 3 ≠ 4, they're NOT equivalent!

Subtracting doesn't work: \( \frac{3}{4} \) minus 1: \( \frac{3-1}{4-1} = \frac{2}{3} \)

3×3 = 9, 4×2 = 8. Since 9 ≠ 8, they're NOT equivalent!

Why multiplication works: When you multiply both parts by the same number, you're essentially multiplying by 1: \( \frac{n}{n} = 1 \)

So \( \frac{a}{b} \times \frac{n}{n} = \frac{a \times n}{b \times n} \), which equals the original fraction since we multiplied by 1!