Solved Exercises on Review of Fractions in Grade 4

Master fractions: identifying, comparing, and understanding fraction concepts through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying fractions
Exercise 1
What fraction of the circle is shaded?
Definition:

Fraction: A number that represents part of a whole. Written as a/b where a is the numerator and b is the denominator.

Method for identifying fractions:
  1. Count the total number of equal parts in the whole
  2. Count the number of shaded (or selected) parts
  3. Write the fraction as: shaded parts / total parts
  4. Simplify if possible
Total Parts
4 equal parts
Shaded Parts
3 parts
Fraction
3/4
Step 1: Count total equal parts

The circle is divided into 4 equal parts (quarters)

Step 2: Count shaded parts

3 out of 4 parts are shaded yellow

Step 3: Write the fraction

Shaded parts / Total parts = 3/4

Step 4: Check if fraction can be simplified

GCD of 3 and 4 is 1, so 3/4 is already in simplest form

Fraction shaded: 3/4
Final answer:

The fraction of the circle that is shaded is 3/4.

Applied rules:

• Equal Parts: All parts must be equal in size for valid fraction representation

• Numerator: Top number represents the counted parts

• Denominator: Bottom number represents the total number of parts

2 Equivalent fractions
Exercise 2
Show that 2/3 and 4/6 are equivalent fractions.
Definition:

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

Fractions
2/3 and 4/6
Method
Cross multiply
Result
Equivalent
Step 1: Cross multiply to compare

For 2/3 and 4/6: (2 × 6) vs (3 × 4)

2 × 6 = 12

3 × 4 = 12

Since 12 = 12, the fractions are equivalent

Step 2: Alternative method - Simplify

4/6 = (4÷2)/(6÷2) = 2/3

Since 4/6 simplifies to 2/3, they are equivalent

Step 3: Visual confirmation

2/3 means 2 out of 3 equal parts

4/6 means 4 out of 6 equal parts

When 6 parts are grouped into 3 sets of 2, 4 parts represent 2 of the 3 sets

2/3 = 4/6 (equivalent fractions)
Final answer:

2/3 and 4/6 are equivalent fractions because they represent the same value.

Applied rules:

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

• Simplification: Reduce to lowest terms to compare

• Visual Verification: Draw models to confirm equivalence

3 Comparing fractions
Exercise 3
Compare 3/5 and 2/3. Which is greater?
Definition:

Comparing Fractions: Determining which fraction represents a larger or smaller portion of the whole.

Fractions
3/5 vs 2/3
Method
Find common denominator
Result
2/3 > 3/5
Step 1: Find a common denominator

Denominators are 5 and 3

LCM of 5 and 3 is 15

Step 2: Convert fractions to equivalent forms

3/5 = (3×3)/(5×3) = 9/15

2/3 = (2×5)/(3×5) = 10/15

Step 3: Compare the numerators

9/15 vs 10/15

Since 10 > 9, then 10/15 > 9/15

Therefore 2/3 > 3/5

2/3 > 3/5
Final answer:

2/3 is greater than 3/5.

Applied rules:

• Common Denominator: Convert fractions to same denominator for comparison

• Numerator Comparison: With same denominator, compare numerators

• Alternative Method: Cross multiply to compare (a/b vs c/d → a×d vs b×c)

Key Concepts: Fraction Fundamentals and Methods
Fraction = Numerator / Denominator
Basic Fraction Structure
Concept 1
Numerator: Parts taken
Top number in fraction
Concept 2
Denominator: Total parts
Bottom number in fraction
Concept 3
Equivalent: Same value
Different numerators/denominators
Key definitions:

Fraction: A number representing part of a whole, written as a/b where a is the numerator and b is the denominator

Numerator: The top number indicating how many parts are taken

Denominator: The bottom number indicating how many equal parts make up the whole

Equivalent Fractions: Fractions that represent the same value despite having different numerators and denominators

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

Complete methodology:
  1. Identify the whole: Determine what represents the complete unit
  2. Count equal parts: Ensure the whole is divided into equal sections
  3. Count selected parts: Identify the relevant portions
  4. Write the fraction: Place selected parts over total parts
  5. Simplify if needed: Reduce to lowest terms by dividing by GCD
Tip 1: The bigger the denominator, the smaller each part becomes.
Tip 2: Fractions with same denominator can be compared by their numerators.
Tip 3: To find equivalent fractions, multiply or divide both numerator and denominator by the same number.
Tip 4: Always check if a fraction can be simplified by finding common factors.
Common errors: Miscounting parts, forgetting to simplify, confusing numerator with denominator.
Memory aids: "Numerator" starts with "N" like "Number of pieces taken".
Essential fraction facts:

• Any number divided by itself equals 1 (n/n = 1)

• Zero divided by any number equals 0 (0/n = 0)

• You cannot divide by zero (n/0 is undefined)

• Proper fractions have numerator less than denominator

• Improper fractions have numerator greater than or equal to denominator

Fraction Visualization: Understanding Relationships
Exercise 6: Fraction Comparison
Visualize and compare different fractions:
1/2, 1/3, 1/4, 2/3, 3/4
Understand how numerators and denominators affect size

Analysis: The chart shows how different fractions compare in size.

  • As denominator increases (with same numerator), fraction gets smaller
  • As numerator increases (with same denominator), fraction gets larger
  • 3/4 is the largest among these fractions
  • 1/4 is the smallest among these fractions

Questions & Answers

Question: Why do we need equivalent fractions? Don't they just make things more complicated?

Answer: Actually, equivalent fractions make many operations easier!

Uses of equivalent fractions:

  • Adding/subtracting fractions: Need common denominators
  • Comparing fractions: Easier when denominators are the same
  • Simplifying calculations: Sometimes working with smaller numbers is easier
  • Real-world applications: Recipes, measurements, scaling

Example: To add 1/2 + 1/4, you convert 1/2 to 2/4, then add: 2/4 + 1/4 = 3/4

Equivalent fractions help us express the same value in different forms, making mathematical operations more manageable!

Question: How can I tell if a fraction is in its simplest form?

Answer: A fraction is in simplest form when the numerator and denominator have no common factors except 1.

Steps to check:

  1. Find the factors of the numerator
  2. Find the factors of the denominator
  3. See if they share any common factors besides 1
  4. If yes, divide both by their Greatest Common Divisor (GCD)

Example 1: 6/8 → Factors of 6: 1,2,3,6; Factors of 8: 1,2,4,8 → Common factor: 2 → 6÷2/8÷2 = 3/4

Example 2: 5/7 → Factors of 5: 1,5; Factors of 7: 1,7 → Only common factor: 1 → Already in simplest form

Practice recognizing common factors to become faster at simplifying fractions!

Question: What's the easiest way to compare fractions with different denominators?

Answer: There are two main methods, and you can choose the one that feels easier:

Method 1: Common Denominator

  • Find the Least Common Multiple (LCM) of the denominators
  • Convert both fractions to have this common denominator
  • Compare the numerators

Method 2: Cross Multiplication

  • For a/b vs c/d, compare a×d vs b×c
  • If a×d > b×c, then a/b > c/d
  • If a×d < b×c, then a/b < c/d

Example: Compare 2/3 and 3/4

Cross multiply: 2×4 = 8 vs 3×3 = 9 → Since 8 < 9, then 2/3 < 3/4

Both methods work perfectly, so pick the one you prefer!

Question: Why can't the denominator be zero? What happens if it is?

Answer: Division by zero is undefined in mathematics, and here's why:

Intuitive explanation:

  • Think of division as sharing: 10 ÷ 2 means "share 10 items among 2 people"
  • But 10 ÷ 0 means "share 10 items among 0 people" - this makes no sense!
  • How many items does each person get when there are no people?

Mathematical reasoning:

  • If 10 ÷ 0 = x, then 0 × x should equal 10
  • But 0 × anything = 0, never 10
  • So no value of x satisfies this equation

In fraction terms, having 0 in the denominator means trying to divide something into 0 parts, which is impossible. That's why fractions with denominator 0 are undefined!

Question: How do fractions help us in real life?

Answer: Fractions are everywhere in daily life! Here are some examples:

  • Cooking: Recipe measurements (1/2 cup, 3/4 teaspoon)
  • Shopping: Sale prices (1/2 off, 25% = 1/4 off)
  • Time: Half hour (1/2 hour), quarter past (1/4 hour)
  • Construction: Measurements (3/4 inch, 1/8 meter)
  • Grades: Test scores (18/20 correct answers)

Fractions help us measure, divide, compare, and understand portions of whole things. They're essential for precise measurement and fair sharing of resources.

Understanding fractions also prepares you for more advanced math topics like decimals, percentages, ratios, and proportions!