Solved Exercises on Multiplying Fractions in Grade 4

Master multiplying fractions: numerator multiplication, denominator multiplication, simplification, visual representations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic fraction multiplication
Exercise 1
Multiply and simplify:
\(\frac{2}{3} \times \frac{4}{5}\)
Definition:

Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)

Multiplication method:
  1. Multiply numerators together
  2. Multiply denominators together
  3. Simplify the resulting fraction if possible
Expression
\(\frac{2}{3} \times \frac{4}{5}\)
Multiplication
\(\frac{8}{15}\)
Step 1: Multiply numerators

\(2 \times 4 = 8\)

Step 2: Multiply denominators

\(3 \times 5 = 15\)

Step 3: Write the result

\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)

Step 4: Check for simplification

GCD(8, 15) = 1, so \(\frac{8}{15}\) is already in simplest form

\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
Final answer:

\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)

Applied rules:

• Numerator multiplication: Multiply top numbers together

• Denominator multiplication: Multiply bottom numbers together

• Simplification: Reduce fraction to lowest terms if possible

2 Multiplication with simplification
Exercise 2
Multiply and simplify:
\(\frac{3}{4} \times \frac{2}{6}\)
Definition:

Fraction simplification: Reduce by dividing numerator and denominator by their GCD

Expression
\(\frac{3}{4} \times \frac{2}{6}\)
Multiplication
\(\frac{6}{24}\)
Simplification
\(\frac{1}{4}\)
Step 1: Multiply numerators

\(3 \times 2 = 6\)

Step 2: Multiply denominators

\(4 \times 6 = 24\)

Step 3: Write the result

\(\frac{3}{4} \times \frac{2}{6} = \frac{6}{24}\)

Step 4: Find GCD of 6 and 24

Factors of 6: 1, 2, 3, 6

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

GCD(6, 24) = 6

Step 5: Simplify the fraction

\(\frac{6 \div 6}{24 \div 6} = \frac{1}{4}\)

\(\frac{3}{4} \times \frac{2}{6} = \frac{1}{4}\)
Final answer:

\(\frac{3}{4} \times \frac{2}{6} = \frac{1}{4}\)

Applied rules:

• Multiplication: Numerator × numerator over denominator × denominator

• Simplification: Divide by greatest common divisor

• Reduction: Always express fractions in simplest form

3 Multiplication with whole number
Exercise 3
Multiply and simplify:
\(3 \times \frac{2}{5}\)
Definition:

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

Expression
\(3 \times \frac{2}{5}\)
Conversion
\(\frac{3}{1} \times \frac{2}{5}\)
Result
\(\frac{6}{5}\)
Step 1: Convert whole number to fraction

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

Step 2: Apply multiplication rule

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

Step 3: Calculate products

Numerator: \(3 \times 2 = 6\)

Denominator: \(1 \times 5 = 5\)

Step 4: Write the result

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

Step 5: Check for simplification

GCD(6, 5) = 1, so \(\frac{6}{5}\) is already in simplest form

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

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

Applied rules:

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

• Multiplication: Apply same rule for fraction × fraction

• Simplification: Reduce if possible

Rules and methods, laws,...
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Fraction Multiplication
Rule 1
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Multiply numerators and denominators
Rule 2
\(n \times \frac{a}{b} = \frac{n \times a}{b}\)
Multiply whole number by numerator
Rule 3
\(\frac{a}{b} \times \frac{b}{a} = 1\)
Multiplicative inverse property
Key definitions:

Numerator: Top number in a fraction representing parts taken

Denominator: Bottom number in a fraction representing total equal parts

Simplification: Reducing fraction to lowest terms using GCD

Complete methodology:
  1. Identify the fractions: Recognize numerators and denominators
  2. Multiply numerators: Multiply all top numbers together
  3. Multiply denominators: Multiply all bottom numbers together
  4. Simplify: Reduce to lowest terms if possible
Tip 1: Always simplify your final answer to lowest terms.
Tip 2: When multiplying by a whole number, write it as a fraction over 1.
Tip 3: You can simplify before multiplying to make calculations easier.
Tip 4: Cross-cancel common factors before multiplying to avoid large numbers.
Common errors: Adding instead of multiplying, forgetting to simplify, mixing up numerators and denominators.
Memory aids: "Top times top, bottom times bottom" - always multiply across, not up and down.
Formulas to know by heart:

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Whole number × fraction: \(n \times \frac{a}{b} = \frac{na}{b}\)

• Simplification: Divide numerator and denominator by GCD

• Cross-cancellation: Cancel common factors before multiplying

Solution: Exercises 4 to 5
4 Multiplication with simplification before multiplying
Exercise 4
Multiply and simplify:
\(\frac{4}{9} \times \frac{3}{8}\)
Definition:

Cross-cancellation: Simplify by canceling common factors before multiplication

Expression
\(\frac{4}{9} \times \frac{3}{8}\)
Cross-cancellation
\(\frac{1}{3} \times \frac{1}{2}\)
Result
\(\frac{1}{6}\)
Step 1: Identify common factors

Numerator 4 and denominator 8 share factor 4

Numerator 3 and denominator 9 share factor 3

Step 2: Cancel common factors

\(\frac{4}{9} \times \frac{3}{8} = \frac{4 \div 4}{9} \times \frac{3}{8 \div 4} = \frac{1}{9} \times \frac{3}{2}\)

Then: \(\frac{1}{9 \div 3} \times \frac{3 \div 3}{2} = \frac{1}{3} \times \frac{1}{2}\)

Step 3: Multiply simplified fractions

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

Step 4: Verify result

Original method: \(\frac{4 \times 3}{9 \times 8} = \frac{12}{72} = \frac{1}{6}\) ✓

\(\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}\)
Final answer:

\(\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}\)

Applied rules:

• Cross-cancellation: Cancel common factors before multiplying

• Simplification: Makes calculations easier

• Verification: Check result using original method

5 Mixed operations with fractions
Exercise 5
Solve:
\(\frac{2}{3} \times \frac{5}{6} + \frac{1}{9}\)
Definition:

Order of operations: Perform multiplication before addition

Expression
\(\frac{2}{3} \times \frac{5}{6} + \frac{1}{9}\)
Multiplication
\(\frac{10}{18} + \frac{1}{9}\)
Simplification
\(\frac{2}{3}\)
Step 1: Perform multiplication first

\(\frac{2}{3} \times \frac{5}{6} = \frac{2 \times 5}{3 \times 6} = \frac{10}{18}\)

Step 2: Simplify the multiplication result

\(\frac{10}{18} = \frac{10 \div 2}{18 \div 2} = \frac{5}{9}\)

Step 3: Add fractions

\(\frac{5}{9} + \frac{1}{9} = \frac{5 + 1}{9} = \frac{6}{9}\)

Step 4: Simplify final result

\(\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}\)

\(\frac{2}{3} \times \frac{5}{6} + \frac{1}{9} = \frac{2}{3}\)
Final answer:

\(\frac{2}{3} \times \frac{5}{6} + \frac{1}{9} = \frac{2}{3}\)

Applied rules:

• Order of operations: Multiplication before addition

• Fraction multiplication: Numerator × numerator over denominator × denominator

• Fraction addition: Same denominator, add numerators

Key Concepts: Fraction Multiplication Laws, Methods, and Definitions
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Basic Multiplication Rule
Key definitions:

Fraction: Represents part of a whole, written as \(\frac{\text{numerator}}{\text{denominator}}\)

Numerator: Number of parts being considered (top number)

Denominator: Total number of equal parts (bottom number)

Product: Result of multiplication operation

Complete methodology:
  1. Analyze the fractions: Identify numerators and denominators
  2. Apply multiplication rule: Multiply numerators together, denominators together
  3. Simplify if possible: Reduce to lowest terms using GCD
  4. Verify result: Check by cross-multiplication or alternative method
Tip 1: Cross-cancel common factors before multiplying to simplify calculations.
Tip 2: Remember: "multiply straight across" - top × top, bottom × bottom.
Tip 3: When multiplying by a whole number, put it over 1.
Tip 4: Always express final answers in simplest form.
Common errors: Adding numerators and denominators separately, forgetting to simplify, mixing up multiplication and addition rules.
Exam preparation: Practice with various fraction sizes, master simplification, understand order of operations.
Formulas to know by heart:

• Basic multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Whole number multiplication: \(n \times \frac{a}{b} = \frac{na}{b}\)

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

• Cross-cancellation: Cancel common factors diagonally before multiplying

Visual Representation: Fraction Multiplication Concepts
Understanding Fraction Multiplication
Visual representations of fraction multiplication:
Area model showing \(\frac{2}{3} \times \frac{3}{4}\)
Number line representation
Pie chart comparison

Analysis: The chart shows how fraction multiplication creates smaller parts of the whole.

  • \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) (Area model)
  • Fraction multiplication results in a smaller portion of the original unit
  • Visual models help understand why multiplication makes fractions smaller

Questions & Answers

Question: I don't understand why when we multiply fractions, the result gets smaller. Shouldn't multiplication make things bigger?

Answer: Great observation! This is a common misconception. When we multiply whole numbers, the result is usually larger because we're adding groups of items. But with fractions, we're taking a part of a part:

  • Think of \(\frac{1}{2} \times \frac{1}{3}\) as "half of one-third"
  • We take one-third and then take half of that amount
  • This gives us \(\frac{1}{6}\), which is smaller than both original fractions

In general, when you multiply two fractions between 0 and 1, the result is smaller than both original fractions. It's like taking a portion of a portion, which naturally gives you less than you started with.

Question: When do I need to simplify fractions after multiplying? Is it always necessary?

Answer: Yes, it's standard mathematical practice to express fractions in their simplest form. Here's why and when:

  • Always simplify your final answer unless specifically told otherwise
  • Simplified fractions are easier to work with in further calculations
  • Teachers typically expect answers in lowest terms

To simplify, find the Greatest Common Divisor (GCD) of numerator and denominator, then divide both by the GCD. For example, \(\frac{8}{12}\) becomes \(\frac{2}{3}\) because GCD(8,12) = 4.

Tip: You can also simplify BEFORE multiplying by canceling common factors between any numerator and any denominator!

Question: I have trouble remembering whether to multiply the top numbers together or the bottom numbers together. Is there a way to remember?

Answer: Here are some helpful memory aids for fraction multiplication:

  • "Top times top, bottom times bottom" - multiply across the same positions
  • Think of it as "numerators together, denominators together"
  • Visual: Draw arrows from numerator to numerator and denominator to denominator

The rule is: \(\frac{\text{numerator}_1}{\text{denominator}_1} \times \frac{\text{numerator}_2}{\text{denominator}_2} = \frac{\text{numerator}_1 \times \text{numerator}_2}{\text{denominator}_1 \times \text{denominator}_2}\)

Remember: Unlike addition/subtraction, you don't need common denominators for multiplication!

Question: How can I check if my fraction multiplication is correct? Sometimes I'm not sure about my answer.

Answer: Here are several methods to verify your fraction multiplication:

Method 1: Reverse calculation - If \(\frac{a}{b} \times \frac{c}{d} = \frac{e}{f}\), then \(\frac{e}{f} \div \frac{c}{d}\) should equal \(\frac{a}{b}\)

Method 2: Decimal conversion - Convert fractions to decimals and multiply to see if results match

Method 3: Estimate - Make sure your answer is reasonable (should be smaller than original fractions if both are less than 1)

Method 4: Factor check - Verify that the numerator and denominator of your result have no common factors (already simplified)

Example: For \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\), check: \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}\) ✓

Question: What happens when I multiply a fraction by a mixed number? Do I follow the same rules?

Answer: When multiplying a fraction by a mixed number, you must first convert the mixed number to an improper fraction:

  1. Convert mixed number to improper fraction: \(2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}\)
  2. Then multiply fractions using the standard rule: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
  3. Finally, simplify if needed

Example: \(\frac{2}{5} \times 1\frac{3}{4} = \frac{2}{5} \times \frac{7}{4} = \frac{14}{20} = \frac{7}{10}\)

Remember: You cannot multiply the whole number part and fractional part separately - you must convert to improper fraction first!