Solved Exercises on Finding Equivalent Fractions in Grade 4

Master finding equivalent fractions: methods, properties, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding equivalent by multiplying
Exercise 1
Find two equivalent fractions for 2/5 by multiplying numerator and denominator by the same number.
Definition:

Equivalent Fractions: Fractions that represent the same value or amount, created by multiplying or dividing both numerator and denominator by the same non-zero number.

Method for finding equivalents by multiplication:
  1. Take the original fraction
  2. Multiply both numerator and denominator by the same number
  3. Simplify if needed
  4. Verify using cross multiplication
Original
2/5
Multiply by 2
4/10
Multiply by 3
6/15
2/5
→
4/10
→
6/15
Step 1: Multiply numerator and denominator by 2

2/5 = (2×2)/(5×2) = 4/10

Check: 2×10 = 20 and 5×4 = 20 → Equivalent ✓

Step 2: Multiply numerator and denominator by 3

2/5 = (2×3)/(5×3) = 6/15

Check: 2×15 = 30 and 5×6 = 30 → Equivalent ✓

Step 3: Verify all fractions are equivalent

2/5 = 4/10 = 6/15 (all represent the same value)

Equivalent fractions: 4/10 and 6/15
Final answer:

Two equivalent fractions for 2/5 are: 4/10 and 6/15

Applied rules:

• Multiplication Property: Multiply both numerator and denominator by same non-zero number

• Cross Multiplication Check: For a/b = c/d, verify a×d = b×c

• Equivalence Preservation: Value remains unchanged when multiplying by n/n

2 Finding equivalent by dividing
Exercise 2
Find an equivalent fraction for 12/18 by dividing both numerator and denominator by their GCD.
Definition:

Simplification: Finding an equivalent fraction in its simplest form by dividing both numerator and denominator by their Greatest Common Divisor (GCD).

Original
12/18
GCD
6
Simplified
2/3
12/18
→
6/9
→
2/3
Step 1: Find the GCD of 12 and 18

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

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

Common factors: 1, 2, 3, 6

Greatest Common Factor: 6

Step 2: Divide both numerator and denominator by GCD

12 ÷ 6 = 2

18 ÷ 6 = 3

12/18 = 2/3

Step 3: Verify the fractions are equivalent

12 × 3 = 36 and 18 × 2 = 36 → Equivalent ✓

12/18 = 2/3 (simplified form)
Final answer:

The equivalent fraction for 12/18 in simplest form is 2/3.

Applied rules:

• Division Property: Divide both numerator and denominator by same non-zero number

• GCD Method: Use Greatest Common Divisor for complete simplification

• Simplification: Result is in lowest terms when GCD is used

3 Finding missing numerator
Exercise 3
Find the missing numerator: 3/4 = ?/12
Definition:

Finding Missing Values: Use the principle that equivalent fractions have proportional numerators and denominators.

Given
3/4 = ?/12
Method
Cross multiply
Answer
9
Step 1: Set up the equation using cross multiplication

If 3/4 = x/12, then 3 × 12 = 4 × x

36 = 4x

Step 2: Solve for the missing numerator

36 = 4x

x = 36 ÷ 4 = 9

Step 3: Verify the answer

3/4 = 9/12

Check: 3 × 12 = 36 and 4 × 9 = 36 → Equivalent ✓

Step 4: Alternative method - Find the multiplier

Denominator: 4 → 12 means multiply by 3

So numerator: 3 × 3 = 9

3/4 = 9/12

3/4 = 9/12
Final answer:

The missing numerator is 9, so 3/4 = 9/12.

Applied rules:

• Cross Multiplication: For a/b = c/d, then a×d = b×c

• Proportional Relationship: Equivalent fractions maintain the same ratio

• Multiplier Method: Find what multiplier converts one denominator to another

Key Concepts: Equivalent Fractions Properties and Methods
a/b = (a×n)/(b×n) = (a÷d)/(b÷d) where n≠0, d≠0
Equivalent Fractions Formula
Property 1
Multiplication: a/b = (a×n)/(b×n)
Multiply both by same number
Property 2
Division: a/b = (a÷d)/(b÷d)
Divide both by same number
Property 3
Cross Product: a/b = c/d → a×d = b×c
Check equivalence
Key definitions:

Equivalent Fractions: Fractions that represent the same value or portion of a whole, even though they have different numerators and denominators

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

Simplified Form: A fraction where the numerator and denominator have no common factors except 1

Common Factor: A number that divides both the numerator and denominator without remainder

Complete methodology:
  1. Identify the operation: Determine if you need to multiply or divide
  2. Find the multiplier/divisor: Use given information to find the number to apply
  3. Apply to both parts: Perform the operation on both numerator and denominator
  4. Verify equivalence: Use cross multiplication to confirm
Tip 1: Always multiply or divide both numerator AND denominator by the same number.
Tip 2: Use cross multiplication to check if fractions are equivalent: a/b = c/d means a×d = b×c.
Tip 3: To simplify completely, divide by the Greatest Common Divisor (GCD).
Tip 4: When finding missing values, set up a proportion and solve using cross multiplication.
Common errors: Forgetting to apply operation to both numerator and denominator, using different multipliers.
Memory aids: "Whatever you do to the top, you do to the bottom" to maintain equality.
Essential equivalent fractions facts:

• Multiplying by n/n (where n≠0) always produces an equivalent fraction

• Dividing by a common factor always produces an equivalent fraction

• Every fraction has infinitely many equivalent fractions

• The simplest form is unique for each fraction

• Equivalent fractions occupy the same position on a number line

Equivalent Fractions Visualization: Family Tree
Exercise 6: Fraction Families
Visualize the equivalent fractions family for 1/2:
1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 6/12
See how they all represent the same value

Analysis: The chart shows how different fractions can represent the same value.

  • All equivalent fractions for 1/2 have the same decimal value (0.5)
  • Each equivalent is created by multiplying numerator and denominator by the same number
  • They all represent exactly half of a whole
  • Simplification reduces to the simplest form (1/2)

Questions & Answers

Question: Why do equivalent fractions have the same value even though they look different?

Answer: Think of it like cutting a pizza differently but eating the same amount!

Example: 1/2 of a pizza cut into 2 pieces = 2/4 of a pizza cut into 4 pieces

Mathematical explanation:

  • When you multiply both numerator and denominator by the same number n, you're essentially multiplying by n/n = 1
  • Multiplying by 1 doesn't change the value
  • So 2/5 × 2/2 = 4/10, but the value remains the same

It's like having different names for the same amount: "one half," "two quarters," and "four eighths" all mean the same thing!

Question: How do I find the GCD (Greatest Common Divisor) quickly?

Answer: Here are efficient methods to find the GCD:

Method 1: Factor Lists

  1. List all factors of each number
  2. Identify the common factors
  3. Pick the largest common factor

Method 2: Prime Factorization

  1. Break each number into prime factors
  2. Identify common prime factors
  3. Multiply the common prime factors

Method 3: Euclidean Algorithm (Advanced)

  1. Divide the larger by the smaller
  2. Replace the larger with the smaller and the smaller with the remainder
  3. Repeat until remainder is 0

For 4th grade, the factor lists method is most accessible and reliable!

Question: How can I check if my equivalent fraction is correct?

Answer: There are several ways to verify your equivalent fractions:

Method 1: Cross Multiplication

  • For a/b = c/d, check if a×d = b×c
  • Example: Is 2/3 = 4/6? → 2×6 = 12 and 3×4 = 12 → Equal ✓

Method 2: Decimal Conversion

  • Convert both fractions to decimals
  • If decimals are equal, fractions are equivalent
  • Example: 2/3 ≈ 0.667 and 4/6 ≈ 0.667 → Equivalent ✓

Method 3: Simplification

  • Reduce both fractions to simplest form
  • If results are identical, fractions are equivalent
  • Example: 4/6 = 2/3, and 2/3 = 2/3 → Equivalent ✓

Cross multiplication is usually the quickest and most reliable method!

Question: How many equivalent fractions does a fraction have?

Answer: Every fraction has infinitely many equivalent fractions!

Why infinite?

  • You can multiply both numerator and denominator by any non-zero number
  • There are infinitely many non-zero numbers to choose from
  • For example, with 1/2: 2/4, 3/6, 4/8, 5/10, 6/12, 7/14... and so on

Example sequence for 2/3:

  • 2/3 = 4/6 (×2)
  • 2/3 = 6/9 (×3)
  • 2/3 = 8/12 (×4)
  • 2/3 = 10/15 (×5)
  • And so on indefinitely...

However, each fraction has only ONE simplest form (lowest terms).

Question: When would I use equivalent fractions in real life?

Answer: Equivalent fractions are used in many practical situations:

  • Cooking: Converting recipe measurements (1/2 cup = 2/4 cup = 4/8 cup)
  • Construction: Measuring materials (1/2 inch = 2/4 inch = 4/8 inch)
  • Money: Understanding that 50 cents = 1/2 dollar = 5/10 dollar
  • Time: 30 minutes = 1/2 hour = 3/6 hour = 6/12 hour
  • Shopping: Comparing unit prices and discounts

Practical examples:

  • If a recipe calls for 2/3 cup and you only have a 1/3 cup measure, you need 2 scoops (2/3 = 2×1/3)
  • When comparing prices, converting to equivalent fractions helps determine the better deal
  • Understanding that 3/4 full is the same as 6/8 full when filling containers

Equivalent fractions help us work with different measurements that represent the same quantity!