Solved Exercises on Understanding Equivalent Fractions for Grade 3

Master equivalent fractions: finding and recognizing equivalent fractions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic equivalent fraction
Exercise 1
Show that 1/2 and 2/4 are equivalent fractions. Draw pictures to explain why they are equivalent.
Definition:

Equivalent fractions: Fractions that represent the same amount or value, even though they may look different.

Equivalent fraction method:
  1. Draw two identical shapes (rectangles, circles, etc.)
  2. Divide the first shape according to the first fraction
  3. Divide the second shape according to the second fraction
  4. Shade the appropriate parts in each shape
  5. Compare the shaded areas to see if they're the same size
Fractions
1/2, 2/4
Multiplication
1×2/2×2 = 2/4
Verification
Both represent same area
Step 1: Draw two identical rectangles

Draw two rectangles of the same size

Step 2: Divide the first rectangle into 2 equal parts

Shade 1 out of 2 parts (1/2)

Step 3: Divide the second rectangle into 4 equal parts

Shade 2 out of 4 parts (2/4)

Step 4: Compare the shaded areas

Both shaded areas cover the same amount of space

1/2 and 2/4 are equivalent fractions
Final answer:

1/2 = 2/4 because they represent the same amount of space.

Mathematical proof: 1/2 × 2/2 = 2/4 (multiplying numerator and denominator by the same number)

Applied rules:

• Multiplication by 1: Multiplying by 2/2 is the same as multiplying by 1

• Equal Area: Equivalent fractions represent equal portions of the same whole

• Proportional Relationship: The numerator and denominator change by the same factor

2 Pizza equivalent fraction
Exercise 2
A pizza is cut into 6 equal slices. If you eat 3 slices, what fraction of the pizza did you eat? Find an equivalent fraction with a smaller denominator.
Definition:

Simplification: Reducing a fraction to its simplest form by dividing both numerator and denominator by their greatest common factor.

Eaten slices
3
Total slices
6
Initial fraction
3/6
Equivalent fraction
1/2
Step 1: Identify the fraction eaten

You ate 3 slices out of 6 total slices

Fraction eaten = 3/6 of the pizza

Step 2: Find an equivalent fraction

Divide both numerator and denominator by their common factor (3)

3 ÷ 3 = 1, 6 ÷ 3 = 2

So 3/6 = 1/2

Step 3: Verify the equivalence

3/6 means 3 out of 6 parts

1/2 means 1 out of 2 parts

Both represent the same amount: half the pizza

You ate 3/6 (or 1/2) of the pizza
Final answer:

You ate 3/6 of the pizza.

An equivalent fraction is 1/2 (with a smaller denominator).

Applied rules:

• Simplification: Divide both numerator and denominator by the same number

• Common Factors: Use the greatest common factor to reduce to lowest terms

• Preservation of Value: The fraction's value remains unchanged

3 Visual equivalent fraction
Exercise 3
Are 2/3 and 4/6 equivalent fractions? Use multiplication to show how to get from 2/3 to 4/6.
Definition:

Creating equivalent fractions: Multiplying both the numerator and denominator by the same non-zero number.

Starting fraction
2/3
Multiplier
× 2/2
Result
4/6
Verification
2×2/3×2 = 4/6
Step 1: Identify the starting fraction

We start with 2/3

Step 2: Find what to multiply by

To get from 2 to 4: 2 × 2 = 4

To get from 3 to 6: 3 × 2 = 6

So multiply by 2/2

Step 3: Perform the multiplication

2/3 × 2/2 = (2×2)/(3×2) = 4/6

Step 4: Conclusion

Since we multiplied both numerator and denominator by the same number, 2/3 = 4/6

Yes, 2/3 and 4/6 are equivalent fractions
Final answer:

Yes, 2/3 and 4/6 are equivalent fractions.

2/3 × 2/2 = 4/6

Both fractions represent the same amount.

Applied rules:

• Multiplication Method: Multiply both numerator and denominator by the same number

• Identity Property: Multiplying by 1 (like 2/2) doesn't change the value

• Proportional Scaling: Both parts of the fraction must scale by the same factor

Understanding Equivalent Fractions: Complete Guide
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Equivalent Fraction Formula
Multiplication
× n/n
Same number top/bottom
Division
÷ gcf/gcf
Reduce to lowest terms
Verification
Cross multiply
ad = bc if a/b = c/d
Key definitions:

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

Simplification: Reducing a fraction to its lowest terms by dividing both numerator and denominator by their greatest common factor.

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

Lowest Terms: A fraction where the numerator and denominator have no common factors other than 1.

Cross Multiplication: A method to verify if two fractions are equivalent by multiplying diagonally.

Equivalent Fraction Process:
  1. Identify the Original Fraction: Determine the fraction you're starting with
  2. Choose a Multiplier: Select a non-zero number to multiply by
  3. Multiply Both Parts: Multiply both numerator and denominator by the same number
  4. Verify Equivalence: Check that the new fraction represents the same value
  5. Simplify if Needed: Reduce to lowest terms when required
Tip 1: Always multiply or divide both numerator and denominator by the same number.
Tip 2: Use visual models like fraction bars or circles to understand equivalence.
Tip 3: Cross multiply to verify: if a/b = c/d, then a×d = b×c.
Tip 4: The numerator and denominator must change by the same proportional factor.
Tip 5: Equivalent fractions occupy the same position on a number line.
Common errors: Forgetting to multiply both parts, not using the same multiplier, not simplifying to lowest terms.
Success strategies: Visual models, cross multiplication verification, systematic multiplication/division.
Equivalent Fraction Rules:

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

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

• Identity Property: Multiplying by 1 (like 2/2, 3/3) doesn't change the value

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

Solution: Exercises 4 to 5
4 Numerical equivalent fraction
Exercise 4
Find two equivalent fractions for 3/5. Then verify that they are equivalent using cross multiplication.
Definition:

Cross multiplication: A method to verify equivalence by multiplying the numerator of one fraction by the denominator of the other.

Original fraction
3/5
First equivalent
6/10
Second equivalent
9/15
Verification
Cross multiply
Step 1: Find first equivalent fraction

Multiply 3/5 by 2/2: 3×2/5×2 = 6/10

Step 2: Find second equivalent fraction

Multiply 3/5 by 3/3: 3×3/5×3 = 9/15

Step 3: Verify first equivalence (3/5 = 6/10)

Cross multiply: 3×10 = 30, 5×6 = 30

Since 30 = 30, 3/5 = 6/10 ✓

Step 4: Verify second equivalence (3/5 = 9/15)

Cross multiply: 3×15 = 45, 5×9 = 45

Since 45 = 45, 3/5 = 9/15 ✓

6/10 and 9/15 are equivalent to 3/5
Final answer:

Two equivalent fractions for 3/5 are: 6/10 and 9/15

Verification: 3/5 = 6/10 because 3×10 = 5×6 = 30

Verification: 3/5 = 9/15 because 3×15 = 5×9 = 45

Applied rules:

• Multiple Equivalents: A fraction has infinitely many equivalent fractions

• Cross Multiplication: Reliable method to verify equivalence

• Systematic Generation: Multiply by 2/2, 3/3, 4/4, etc. to generate equivalents

5 Finding missing number
Exercise 5
Find the missing number: 2/3 = ?/9. Explain how you found your answer.
Definition:

Missing number problem: A problem where one part of an equivalent fraction equation is unknown.

Known fraction
2/3
Target denominator
9
Multiplier
3
Missing numerator
6
Step 1: Identify what changed

Denominator changed from 3 to 9

3 × ? = 9, so 9 ÷ 3 = 3

The denominator was multiplied by 3

Step 2: Apply the same change to the numerator

Since denominator was multiplied by 3, multiply numerator by 3

2 × 3 = 6

Step 3: Verify the answer

2/3 = 6/9

Cross multiply: 2×9 = 18, 3×6 = 18

Since 18 = 18, the fractions are equivalent ✓

2/3 = 6/9
Final answer:

The missing number is 6.

2/3 = 6/9

I found this by determining that the denominator was multiplied by 3 (3×3=9), so I multiplied the numerator by 3 (2×3=6).

Applied rules:

• Proportional Change: What you do to the denominator, you do to the numerator

• Division to Find Multiplier: Use division to find the scaling factor

• Verification: Always verify with cross multiplication

Advanced Equivalent Fraction Concepts
\(\frac{a}{b} = \frac{c}{d} \iff ad = bc\)
Cross Multiplication Rule
Advanced definitions:

Unit Rate: A rate with a denominator of 1, often found by simplifying equivalent fractions.

Proportion: An equation stating that two ratios are equal (a/b = c/d).

Common Denominator: A shared denominator that allows for fraction comparison.

Decimal Equivalence: Equivalent fractions expressed as decimal numbers.

Percentage Equivalence: Equivalent fractions expressed as percentages.

Advanced Equivalent Fraction Methodology:
  1. Pattern Recognition: Identify multiplication or division patterns between numerators and denominators
  2. Cross Multiplication: Use cross multiplication to verify equivalences
  3. Simplification: Reduce fractions to lowest terms to identify equivalences
  4. Common Denominator: Convert fractions to have the same denominator for comparison
  5. Verification: Check answers using multiple methods
  6. Real-world Application: Apply concepts to practical situations
Tip 1: Always check that both numerator and denominator are multiplied/divided by the same number.
Tip 2: Use cross multiplication to quickly verify if fractions are equivalent.
Tip 3: The simplest form of a fraction is unique.
Tip 4: Equivalent fractions represent the same point on a number line.
Tip 5: Practice with visual models to strengthen conceptual understanding.
Advanced errors: Forgetting to apply operations to both numerator and denominator, not simplifying completely, making calculation errors.
Advanced strategies: Systematic approach, multiple verification methods, visual confirmation.
Advanced Fraction Properties:

• Reflexive Property: Every fraction is equivalent to itself (a/b = a/b)

• Symmetric Property: If a/b = c/d, then c/d = a/b

• Transitive Property: If a/b = c/d and c/d = e/f, then a/b = e/f

• Uniqueness of Simplest Form: Every fraction has exactly one equivalent in lowest terms

• Proportional Relationships: Equivalent fractions maintain the same ratio

Questions & Answers

Question: How do I know if two fractions are equivalent? They look different but I think they might be the same.

Answer: Great question! Here are ways to check if fractions are equivalent:

  • Cross multiply: If a/b = c/d, then a×d should equal b×c
  • Reduce both fractions: Simplify to lowest terms and see if they're the same
  • Visualize: Draw pictures to see if they represent the same amount

Example: Are 2/4 and 3/6 equivalent?

Cross multiply: 2×6 = 12, 4×3 = 12. Since 12 = 12, they are equivalent!

Question: My child is struggling with the concept of equivalent fractions. How can I help them understand better?

Answer: Here are some helpful strategies:

  • Use physical objects like pizza slices, chocolate bars, or paper folding
  • Draw pictures showing the same amount divided differently
  • Practice with fraction bars or circles that can be compared visually
  • Start with simple examples like 1/2 = 2/4 before moving to more complex ones
  • Emphasize that equivalent fractions represent the same amount of space

Hands-on activities make abstract concepts more concrete!

Question: Can I multiply the top and bottom by different numbers to get an equivalent fraction?

Answer: No, you must multiply both the numerator (top) and denominator (bottom) by the same number to get an equivalent fraction!

  • Correct: 1/2 × 3/3 = 3/6 (multiplied both by 3)
  • Incorrect: 1/2 × 3/4 = 3/8 (multiplied by different numbers)

When you multiply by the same number top and bottom, you're essentially multiplying by 1 (like 3/3 = 1), which doesn't change the value.

Multiplying by different numbers changes the value of the fraction.