Solved Exercises on Multiplying Fractions by Whole Numbers in Grade 4

Master multiplying fractions by whole numbers: numerator multiplication, simplification, visual representations, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic fraction by whole number multiplication
Exercise 1
Multiply:
\(\frac{2}{5} \times 3\)
Definition:

Multiplying fraction by whole number: \(\frac{a}{b} \times n = \frac{a \times n}{b}\)

Fraction by whole number method:
  1. Write the whole number as a fraction over 1
  2. Multiply the numerators together
  3. Keep the denominator the same
  4. Simplify the result if possible
Expression
\(\frac{2}{5} \times 3\)
Convert whole number
\(\frac{2}{5} \times \frac{3}{1}\)
Result
\(\frac{6}{5}\)
Step 1: Write the whole number as a fraction

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

Step 2: Apply fraction multiplication rule

\(\frac{2}{5} \times \frac{3}{1} = \frac{2 \times 3}{5 \times 1}\)

Step 3: Multiply the numerators

\(2 \times 3 = 6\)

Step 4: Keep the denominator

The denominator remains 5

Step 5: Write the result

\(\frac{6}{5}\)

Step 6: Convert to mixed number (optional)

\(\frac{6}{5} = 1\frac{1}{5}\)

\(\frac{2}{5} \times 3 = \frac{6}{5} = 1\frac{1}{5}\)
Final answer:

\(\frac{2}{5} \times 3 = \frac{6}{5}\) or \(1\frac{1}{5}\)

Applied rules:

• Whole number as fraction: Any whole number \(n\) can be written as \(\frac{n}{1}\)

• Fraction multiplication rule: Multiply numerators by numerators, keep denominator

• Improper to mixed conversion: Divide numerator by denominator

2 Multiplication with simplification
Exercise 2
Multiply:
\(\frac{3}{8} \times 4\)
Definition:

Simplifying fractions: Divide numerator and denominator by their greatest common divisor (GCD)

Expression
\(\frac{3}{8} \times 4\)
Convert to fraction
\(\frac{3}{8} \times \frac{4}{1}\)
Simplification
\(\frac{12}{8} = \frac{3}{2}\)
Step 1: Write the whole number as a fraction

\(4 = \frac{4}{1}\)

Step 2: Apply fraction multiplication rule

\(\frac{3}{8} \times \frac{4}{1} = \frac{3 \times 4}{8 \times 1} = \frac{12}{8}\)

Step 3: Find GCD of numerator and denominator

GCD(12, 8) = 4

Step 4: Simplify by dividing by GCD

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

Step 5: Convert to mixed number (optional)

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

\(\frac{3}{8} \times 4 = \frac{3}{2} = 1\frac{1}{2}\)
Final answer:

\(\frac{3}{8} \times 4 = \frac{3}{2}\) or \(1\frac{1}{2}\)

Applied rules:

• Whole number as fraction: Write whole number as fraction over 1

• Fraction multiplication rule: Multiply numerators by numerators, keep denominator

• Simplification: Reduce to lowest terms by dividing by GCD

3 Multiplication with cross-simplification
Exercise 3
Multiply:
\(\frac{5}{6} \times 12\)
Definition:

Cross-simplification: Simplify before multiplying by finding common factors between numerator and whole number

Expression
\(\frac{5}{6} \times 12\)
Cross-simplify
\(\frac{5}{1} \times 2\)
Result
\(10\)
Step 1: Look for common factors

Denominator 6 and whole number 12 share factor 6

\(\frac{12}{6} = 2\) and \(\frac{6}{6} = 1\)

Step 2: Rewrite with simplified numbers

\(\frac{5}{6} \times 12 = \frac{5}{1} \times 2\)

Step 3: Multiply the remaining numbers

\(\frac{5}{1} \times 2 = \frac{5 \times 2}{1} = \frac{10}{1} = 10\)

Step 4: Verify the result

Without cross-simplification: \(\frac{5}{6} \times 12 = \frac{5 \times 12}{6} = \frac{60}{6} = 10\) ✓

\(\frac{5}{6} \times 12 = 10\)
Final answer:

\(\frac{5}{6} \times 12 = 10\)

Applied rules:

• Cross-simplification: Simplify before multiplying to make calculations easier

• Fraction multiplication rule: Multiply numerators by numerators, keep denominator

Rules and methods, laws,...
\(\frac{a}{b} \times n = \frac{a \times n}{b}\)
Fraction by Whole Number Rule
Rule 1
\(\frac{a}{b} \times n = \frac{an}{b}\)
Multiply numerator by whole number
Rule 2
\(n = \frac{n}{1}\)
Whole number as fraction
Rule 3
\(\frac{a}{b} = \frac{a ÷ \text{GCD}}{b ÷ \text{GCD}}\)
Simplification rule
Key definitions:

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

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

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

Improper Fraction: A fraction where numerator ≥ denominator

Mixed Number: A combination of a whole number and a proper fraction

Cross-Simplification: Simplifying before multiplication by canceling common factors

Complete methodology:
  1. Identify the fraction and whole number: Recognize the numerator, denominator, and whole number
  2. Look for cross-simplification: Check if denominator and whole number share a common factor
  3. Apply multiplication rule: Multiply the numerator by the whole number
  4. Keep the denominator: The denominator remains unchanged
  5. Simplify if needed: Find GCD and divide both numerator and denominator
  6. Convert to mixed number: If result is improper fraction (optional)
Tip 1: Always check if the final fraction can be simplified.
Tip 2: Cross-simplification before multiplying can make calculations much easier.
Tip 3: The result of multiplying a fraction by a whole number may be larger than the original fraction.
Tip 4: If the denominator divides evenly into the whole number, cross-simplification is very helpful.
Common errors: Forgetting to multiply the numerator only, not simplifying the result, misunderstanding when to convert to mixed numbers.
Memory aid: Think "multiply the top number by the whole number, keep the bottom number."
Formulas to know by heart:

• Fraction by whole number: \(\frac{a}{b} \times n = \frac{a \times n}{b}\)

• Whole number as fraction: \(n = \frac{n}{1}\)

• Simplification: \(\frac{a}{b} = \frac{a ÷ \text{GCD}}{b ÷ \text{GCD}}\)

• Improper to mixed: \(\frac{a}{b} = q\frac{r}{b}\) where \(a ÷ b = q\) remainder \(r\)

Detailed Summary for Grade 4 Students

Core Concepts: Understanding numerators, denominators, whole numbers, and cross-simplification

Key Rules: Multiply numerator by whole number, keep denominator unchanged, always simplify

Advanced Techniques: Cross-simplification, handling improper fractions, converting to mixed numbers

Problem-Solving Steps: Identify components, look for simplification, multiply, simplify, interpret result

Practical Applications: Real-world scenarios like cooking, sharing, and measurement

Memory Aids: "Top times whole, bottom stays" and "simplify first to make calculations easier"

Visual Explanation

Conceptual Understanding: Multiplying \(\frac{2}{5} \times 3\) means having 3 groups of \(\frac{2}{5}\), which totals \(\frac{6}{5}\) or 1\(\frac{1}{5}\)

Pictorial Representation: Draw 3 rectangles each divided into 5 parts, shade 2 parts in each, count total shaded parts

Solution: Exercises 4 to 5
4 Word problem application
Exercise 4
Sarah baked \(\frac{3}{4}\) of a batch of cookies yesterday. Today she wants to bake 3 times as many. How many batches of cookies will she bake today?
Definition:

Word problems: Translate the situation into a mathematical expression

Problem
Sarah has \(\frac{3}{4}\) batch, wants 3 times as many
Expression
\(3 \times \frac{3}{4}\)
Result
\(\frac{9}{4} = 2\frac{1}{4}\)
Step 1: Identify the operation

"3 times as many" means multiply 3 by \(\frac{3}{4}\)

Step 2: Set up the multiplication

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

Step 3: Apply fraction multiplication rule

\(3 \times \frac{3}{4} = \frac{3}{1} \times \frac{3}{4} = \frac{3 \times 3}{1 \times 4} = \frac{9}{4}\)

Step 4: Convert to mixed number

\(9 ÷ 4 = 2\) remainder \(1\), so \(\frac{9}{4} = 2\frac{1}{4}\)

Step 5: Answer the question

Sarah will bake \(2\frac{1}{4}\) batches of cookies today.

Sarah will bake \(2\frac{1}{4}\) batches of cookies today.
Final answer:

Sarah will bake \(2\frac{1}{4}\) batches of cookies today.

Applied rules:

• Word problem translation: "times" means multiplication

• Fraction multiplication: Multiply numerator by whole number, keep denominator

5 Complex fraction multiplication
Exercise 5
Calculate: \(\frac{7}{12} \times 24 \times \frac{2}{7}\)
Definition:

Multiplying multiple terms: Handle one multiplication at a time or look for cross-simplifications

Expression
\(\frac{7}{12} \times 24 \times \frac{2}{7}\)
Cross-simplify
\(\frac{1}{1} \times 2 \times \frac{2}{1}\)
Result
\(4\)
Step 1: Look for cross-simplification opportunities

Numerator 7 and denominator 7 share factor 7: \(\frac{7}{7} = \frac{1}{1}\)

Denominator 12 and whole number 24 share factor 12: \(\frac{24}{12} = \frac{2}{1}\)

Step 2: Rewrite with simplified terms

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

Step 3: Multiply the remaining numbers

\(\frac{1}{1} \times 2 \times \frac{2}{1} = \frac{1 \times 2 \times 2}{1 \times 1} = \frac{4}{1} = 4\)

Step 4: Verify with step-by-step multiplication

\(\frac{7}{12} \times 24 = \frac{7 \times 24}{12} = \frac{168}{12} = 14\)

Then: \(14 \times \frac{2}{7} = \frac{14 \times 2}{7} = \frac{28}{7} = 4\) ✓

\(\frac{7}{12} \times 24 \times \frac{2}{7} = 4\)
Final answer:

\(\frac{7}{12} \times 24 \times \frac{2}{7} = 4\)

Applied rules:

• Cross-simplification: Simplify before multiplying to make calculations easier

• Fraction multiplication: Multiply numerators by numerators, keep denominators

Multiplication Rules, Methods, Definitions, and Key Concepts
\(\frac{a}{b} \times n = \frac{an}{b}\)
Fraction by Whole Number Formula
Key definitions:

Numerator: The top part of a fraction that indicates how many parts are being considered

Denominator: The bottom part of a fraction that indicates the total number of equal parts

Proper Fraction: A fraction where the numerator is less than the denominator

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator

Mixed Number: A combination of a whole number and a proper fraction

GCD (Greatest Common Divisor): The largest positive integer that divides both numbers without remainder

Cross-Simplification: Simplifying before multiplication by canceling common factors

Complete methodology:
  1. Identify components: Recognize the fraction and whole number in the expression
  2. Look for cross-simplification: Check for common factors between denominator and whole number
  3. Apply multiplication rule: Multiply the numerator by the whole number
  4. Keep the denominator: The denominator remains unchanged
  5. Simplify the result: Divide numerator and denominator by their GCD
  6. Convert if necessary: Change improper fractions to mixed numbers if required
Tip 1: Cross-simplification before multiplying often makes calculations much easier.
Tip 2: When multiplying a fraction by a whole number, the result may be larger than the original fraction.
Tip 3: Always check if your final answer can be simplified further.
Tip 4: Remember that multiplying by a whole number greater than 1 makes the result larger.
Tip 5: Use visual models like area models to understand why fraction multiplication works.
Common errors: Forgetting to multiply only the numerator, not recognizing when to simplify results, not converting to mixed numbers when appropriate.
Visual representation: Imagine having multiple copies of the same fractional part.
Real-world application: Fraction by whole number multiplication is used in cooking, construction, and scaling measurements.
Formulas to know by heart:

• Basic multiplication: \(\frac{a}{b} \times n = \frac{a \times n}{b}\)

• Whole number conversion: \(n = \frac{n}{1}\)

• Simplification: \(\frac{a}{b} = \frac{a ÷ \text{GCD}}{b ÷ \text{GCD}}\)

• Mixed number conversion: \(\frac{a}{b} = q\frac{r}{b}\) where \(a ÷ b = q\) remainder \(r\)

• Multiple multiplication: \(\frac{a}{b} \times n \times \frac{c}{d} = \frac{a \times n \times c}{b \times d}\)

Detailed Summary for Grade 4 Students

Core Concepts: Understanding numerators, denominators, proper vs improper fractions, and whole numbers

Key Rules: Multiply numerator by whole number, keep denominator unchanged, always simplify

Advanced Techniques: Cross-simplification, handling multiple terms, converting mixed numbers

Problem-Solving Steps: Identify operation, apply multiplication rule, simplify, interpret result

Practical Applications: Real-world scenarios like cooking, sharing, and measurement

Memory Aids: "Top times whole, bottom stays" and "simplify first to make calculations easier"

Conceptual Understanding

Meaning of Multiplication: \(\frac{2}{3} \times 4\) means having 4 groups of \(\frac{2}{3}\), totaling \(\frac{8}{3}\)

Visual Model: Draw 4 rectangles each divided into 3 parts, shade 2 parts in each, count total shaded parts

Connection to Addition: \(\frac{2}{3} \times 4 = \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = \frac{8}{3}\)

Questions & Answers

Question: Why do we only multiply the numerator when multiplying a fraction by a whole number? Why don't we multiply the denominator too?

Answer: Great question! This happens because of what whole numbers represent:

  • When we multiply \(\frac{2}{3} \times 4\), we're essentially saying "take 4 groups of \(\frac{2}{3}\)"
  • Each group still has the same size pieces (thirds), so the denominator doesn't change
  • We only increase the number of pieces we have (the numerator)

Think of it this way: If you have 2 slices of pizza where each slice is a third of the whole pizza, and you want 4 times that amount, you'll have 8 slices of the same size (thirds). So you have \(\frac{8}{3}\) of the pizza.

The denominator tells us the size of the pieces, which stays the same when we multiply by a whole number.

Question: When should I use cross-simplification, and why is it helpful?

Answer: Cross-simplification is extremely helpful! Here's when and why to use it:

  • When to use: Whenever the denominator of the fraction and the whole number share a common factor
  • Why it helps: It reduces the size of numbers you need to multiply, making calculations easier
  • Example: In \(\frac{5}{6} \times 12\), since 6 and 12 share factor 6, you can simplify to \(\frac{5}{1} \times 2 = 10\)

Instead of calculating \(\frac{5 \times 12}{6} = \frac{60}{6} = 10\), you can do the division first: \(\frac{12}{6} = 2\), then multiply: \(5 \times 2 = 10\).

Cross-simplification is a skill that will help you throughout your math journey!

Question: How do I know if my answer should be a mixed number or an improper fraction?

Answer: The choice depends on the context of the problem:

  • Mathematical operations: Keep improper fractions for further calculations
  • Word problems: Mixed numbers are often more meaningful in real-world contexts
  • Instructions: Follow specific directions if given
  • General rule: Either form is mathematically correct

For example, if a problem asks "how many batches of cookies," the answer \(2\frac{1}{4}\) batches is more intuitive than \(\frac{9}{4}\) batches.

However, if you need to perform additional operations with the result, keeping it as \(\frac{9}{4}\) might be more convenient.

Always be able to convert between the two forms!