Solved Exercises on Fractions of a Whole Number in Grade 3

Master fractions of a whole number: finding parts of whole numbers, calculating fractional amounts, and understanding multiplicative relationships through these 5 detailed exercises.

Complete Summary: Fractions of a Whole Number

Definitions

Fraction of a Whole Number: A calculation that finds a part of a whole number. For example, 14 of 12 means finding one-fourth of 12.

Whole Number: Any positive integer including zero (0, 1, 2, 3, 4...).

Fraction: A number that represents a part of a whole, with a numerator and denominator.

Multiplicative Thinking: Understanding that finding a fraction of a number involves multiplication and division.

Division Concept: Finding ab of n means dividing n by b and multiplying by a.

Core Rules and Laws

Fraction of a Number Rule: ab of n = (n ÷ b) × a

Order of Operations: Divide first, then multiply (following the fraction interpretation).

Equivalent Expressions: ab of n = n × ab

Whole Equals One: nn of any number equals the whole number.

Nothing Equals Zero: 0n of any number equals 0.

Step-by-Step Method

  1. Identify the fraction and whole number: Find what fraction you need and what number you're finding it of
  2. Divide the whole number by the denominator: This finds the value of one part
  3. Multiply by the numerator: This finds the value of the required number of parts
  4. Check your answer: Verify it makes sense in the context

Examples (Simple to Advanced)

Simple: 12 of 10 = (10 ÷ 2) × 1 = 5

Intermediate: 23 of 12 = (12 ÷ 3) × 2 = 8

Advanced: 34 of 20 = (20 ÷ 4) × 3 = 15

Tips, Tricks, and Common Pitfalls

Tips: Always divide by the denominator first, then multiply by the numerator.

Tricks: Use visual models like arrays or groups to understand the concept.

Common Pitfalls: Confusing the order of operations, multiplying first instead of dividing.

Key Notes for Memorization

Memory Aid: "Divide by bottom, multiply by top" - denominator first, then numerator.

Visual Memory: Think of dividing objects into groups, then taking required groups.

Quick Check: The answer should be smaller than the original number (unless fraction > 1).

Solution: Exercises 1 to 3
1 Finding Halves
Exercise 1
What is 12 of 16?
Definition:

Finding a half: 12 of a number means dividing the number by 2.

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Method for finding fractions of whole numbers:
  1. Divide the whole number by the denominator (16 ÷ 2 = 8)
  2. Multiply the result by the numerator (8 × 1 = 8)
  3. The answer is the fraction of the whole number
Step 1: Divide by denominator: 16 ÷ 2 = 8
Step 2: Multiply by numerator: 8 × 1 = 8
Step 1: Divide by denominator

16 ÷ 2 = 8 (this gives us the value of one part)

Step 2: Multiply by numerator

8 × 1 = 8 (we want 1 part out of 2)

12 of 16 = 8
Final answer:

12 of 16 is 8

Applied rules:

• Fraction of a number rule: ab of n = (n ÷ b) × a

• Division first: Always divide by denominator before multiplying

• Visual confirmation: Half of 16 items is 8 items

2 Finding Thirds
Exercise 2
What is 23 of 15?
Definition:

Finding a fraction of a number: ab of n means dividing n by b and multiplying by a.

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Step 1: Divide by denominator: 15 ÷ 3 = 5
Step 2: Multiply by numerator: 5 × 2 = 10
Step 1: Divide by denominator

15 ÷ 3 = 5 (this gives us the value of one part)

Step 2: Multiply by numerator

5 × 2 = 10 (we want 2 parts out of 3)

Step 3: Verify the result

If 23 of 15 is 10, then 13 of 15 should be 5, and 10 + 5 = 15 ✓

23 of 15 = 10
Final answer:

23 of 15 is 10

Applied rules:

• Fraction of a number rule: ab of n = (n ÷ b) × a

• Division first: Always divide by denominator before multiplying

• Verification: Check that parts add up to the whole

3 Finding Fourths
Exercise 3
What is 34 of 20?
Definition:

Finding three-fourths: Dividing the number into 4 equal parts and taking 3 of those parts.

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Step 1: Divide by denominator: 20 ÷ 4 = 5
Step 2: Multiply by numerator: 5 × 3 = 15
Step 1: Divide by denominator

20 ÷ 4 = 5 (this gives us the value of one part)

Step 2: Multiply by numerator

5 × 3 = 15 (we want 3 parts out of 4)

Step 3: Verify the result

One-fourth of 20 is 5, so three-fourths is 3 × 5 = 15 ✓

34 of 20 = 15
Final answer:

34 of 20 is 15

Applied rules:

• Fraction of a number rule: ab of n = (n ÷ b) × a

• Division first: Always divide by denominator before multiplying

• Visual confirmation: Pattern shows 3 out of every 4 items are selected

Rules and methods, laws,...
ab of n = (n ÷ b) × a
Fraction of a Whole Number
Rule 1
12 of n = n ÷ 2
Half of a number
Rule 2
13 of n = n ÷ 3
Third of a number
Rule 3
nn of m = m
All of a number
Key definitions:

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

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

Whole number: Any positive integer including zero

Complete methodology:
  1. Identify the fraction and number: Determine what fraction and what number
  2. Divide by denominator: This finds the value of one equal part
  3. Multiply by numerator: This finds the value of required parts
  4. Verify the answer: Check if it makes sense in context
Tip 1: Always divide by the denominator first, then multiply by the numerator.
Tip 2: Use visual models to understand the concept before calculating.
Tip 3: The answer should be smaller than the original number (unless fraction > 1).
Tip 4: Check your answer by seeing if the parts add up to the whole.
Common errors: Multiplying before dividing, confusing numerator and denominator roles.
Key insights: Fractions of numbers connect multiplication and division concepts.
Formulas to know by heart:

• Fraction of a number: ab of n = (n ÷ b) × a

• Half of a number: 12 of n = n ÷ 2

• Third of a number: 13 of n = n ÷ 3

Solution: Exercises 4 to 5
4 Real World Application
Exercise 4
Sarah has 24 stickers. She gives away 14 of her stickers to her friend. How many stickers does she give away?
Definition:

Real-world fraction problems: Apply fraction calculations to solve practical problems involving quantities.

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Step 1: Divide by denominator: 24 ÷ 4 = 6
Step 2: Multiply by numerator: 6 × 1 = 6
Step 1: Identify the problem

Sarah has 24 stickers and gives away 14 of them

Step 2: Divide by denominator

24 ÷ 4 = 6 (one-fourth of 24 is 6)

Step 3: Multiply by numerator

6 × 1 = 6 (she gives away 1 part out of 4)

Step 4: Answer the question

Sarah gives away 6 stickers

14 of 24 = 6 stickers given away
Final answer:

Sarah gives away 6 stickers to her friend

Applied rules:

• Fraction of a number rule: ab of n = (n ÷ b) × a

• Real-world context: Apply mathematical operations to solve practical problems

• Verification: 6 + 18 = 24, confirming the calculation

5 Classroom Problem
Exercise 5
There are 30 students in a class. 25 of the students are wearing glasses. How many students wear glasses?
Definition:

Population fraction problems: Finding parts of a group using fractional relationships.

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Step 1: Divide by denominator: 30 ÷ 5 = 6
Step 2: Multiply by numerator: 6 × 2 = 12
Step 1: Identify the problem

30 students total, 25 wear glasses

Step 2: Divide by denominator

30 ÷ 5 = 6 (one-fifth of 30 is 6)

Step 3: Multiply by numerator

6 × 2 = 12 (two-fifths means 2 parts out of 5)

Step 4: Answer the question

12 students wear glasses

25 of 30 = 12 students wear glasses
Final answer:

12 students in the class wear glasses

Applied rules:

• Fraction of a number rule: ab of n = (n ÷ b) × a

• Group fraction problems: Apply to populations and collections

• Verification: 12 + 18 = 30, confirming the calculation

Questions & Answers

Question: Why do I divide first and then multiply when finding fractions of numbers? Can't I multiply first?

Answer: Great question! You divide first because the fraction tells you to split the number into equal parts first. Here's why:

  • Division finds the size of each part: If you want 23 of 15, first find how big one-third is: 15 ÷ 3 = 5
  • Multiplication finds the total of selected parts: Then take 2 of those parts: 5 × 2 = 10
  • Order matters: 15 ÷ 3 × 2 = 10, but 15 × 2 ÷ 3 = 10 too, but the thinking process is clearer when you divide first

Think of it as: "First, figure out how big each piece is, then count how many pieces you want."

Question: My child is having trouble with word problems involving fractions of whole numbers. What strategies can help?

Answer: Here are effective strategies for fraction word problems:

  • Circle key numbers: Have your child identify the total number and the fraction
  • Draw pictures: Visual representations help understand the problem
  • Use manipulatives: Counters, blocks, or drawings can make abstract concepts concrete
  • Follow the pattern: Always divide by denominator first, then multiply by numerator

Practice with real-life examples like sharing food, dividing toys, or splitting groups of friends to make it more relatable.

Question: How do I check if my answer is right when finding fractions of numbers?

Answer: Here are good ways to check your fraction answers:

  • Does it make sense?: Your answer should be smaller than the original number (unless the fraction is greater than 1)
  • Add the parts: If you found 25 of a number, the remaining 35 should add up to the original number
  • Reverse the operation: If 14 of 20 is 5, then 5 × 4 should equal 20
  • Estimate: Make sure your answer is reasonable compared to the original number

Always ask yourself: "Is this answer logical in the context of the problem?"