Solved Exercises on Multiplying Fractions in Grade 4

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

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

Multiplying fractions: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)

Fraction 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 if it can be simplified

GCD of 8 and 15 is 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:

• Fraction multiplication rule: Multiply numerators and denominators separately

• Simplification: Reduce to lowest terms by dividing by GCD

• Product property: Result is always a fraction

2 Fraction multiplication with whole number
Exercise 2
Multiply: \(3 \times \frac{2}{7}\)
Definition:

Whole number as fraction: \(3 = \frac{3}{1}\), so \(3 \times \frac{2}{7} = \frac{3}{1} \times \frac{2}{7}\)

Expression
\(3 \times \frac{2}{7}\)
Conversion
\(\frac{3}{1} \times \frac{2}{7}\)
Result
\(\frac{6}{7}\)
Step 1: Convert whole number to fraction

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

Step 2: Apply fraction multiplication rule

\(\frac{3}{1} \times \frac{2}{7} = \frac{3 \times 2}{1 \times 7}\)

Step 3: Calculate products

\(3 \times 2 = 6\) and \(1 \times 7 = 7\)

Step 4: Write final answer

\(\frac{6}{7}\) is already in simplest form

\(3 \times \frac{2}{7} = \frac{6}{7}\)
Final answer:

\(3 \times \frac{2}{7} = \frac{6}{7}\)

Applied rules:

• Whole number conversion: Any whole number can be written as a fraction over 1

• Fraction multiplication: Same rule applies regardless of type of multiplier

• Result interpretation: Product of whole number and fraction is a fraction

3 Fraction multiplication with simplification
Exercise 3
Multiply: \(\frac{3}{4} \times \frac{8}{9}\)
Definition:

Simplification before multiplication: Cancel common factors between numerators and denominators before multiplying

Expression
\(\frac{3}{4} \times \frac{8}{9}\)
Simplification
\(\frac{1}{1} \times \frac{2}{3}\)
Result
\(\frac{2}{3}\)
Step 1: Look for common factors

Numerator 3 and denominator 9 share factor 3: \(\frac{3}{9} = \frac{1}{3}\)

Numerator 8 and denominator 4 share factor 4: \(\frac{8}{4} = \frac{2}{1}\)

Step 2: Cancel common factors

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

Step 3: Multiply simplified fractions

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

Step 4: Verify result

\(\frac{2}{3}\) is in simplest form since GCD(2,3) = 1

\(\frac{3}{4} \times \frac{8}{9} = \frac{2}{3}\)
Final answer:

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

Applied rules:

• Cross-simplification: Cancel common factors before multiplication to avoid large numbers

• Factor identification: Find GCD of numerators and denominators across fractions

• Simplification benefit: Makes calculations easier and results cleaner

Rules and methods, laws,...
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Fraction Multiplication Rule
Rule 1
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Multiply numerators and denominators
Rule 2
\(n \times \frac{a}{b} = \frac{n \times a}{b}\)
Whole number times fraction
Rule 3
\(\frac{a}{b} \times \frac{b}{a} = 1\)
Reciprocal multiplication
Key definitions:

Numerator: Top number in a fraction representing parts taken

Denominator: Bottom number in a fraction representing total equal parts

Simplification: Reducing a fraction to its lowest terms by dividing by GCD

Complete methodology:
  1. Identify the fractions: Recognize numerators and denominators
  2. Apply multiplication rule: Multiply numerators together, denominators together
  3. Simplify if possible: Cancel common factors before or after multiplication
  4. Verify the result: Check if fraction is in simplest form
Tip 1: Always simplify before multiplying when possible to make calculations easier.
Tip 2: Cross-cancel common factors between any numerator and any denominator.
Tip 3: The product of two proper fractions is always smaller than both original fractions.
Common errors: Adding numerators and denominators instead of multiplying, forgetting to simplify, misidentifying numerators vs denominators.
Key insight: Multiplying fractions means taking a part of a part - the result gets smaller.
Essential formulas to remember:

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

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

• Reciprocal property: \(\frac{a}{b} \times \frac{b}{a} = 1\)

• Identity property: \(\frac{a}{b} \times 1 = \frac{a}{b}\)

Solution: Exercises 4 to 5
4 Mixed fraction multiplication
Exercise 4
Multiply: \(\frac{2}{3} \times \frac{9}{10} \times \frac{5}{6}\)
Definition:

Multiplying multiple fractions: Multiply all numerators together and all denominators together

Expression
\(\frac{2}{3} \times \frac{9}{10} \times \frac{5}{6}\)
Multiplication
\(\frac{90}{180}\)
Simplification
\(\frac{1}{2}\)
Step 1: Multiply all numerators

\(2 \times 9 \times 5 = 90\)

Step 2: Multiply all denominators

\(3 \times 10 \times 6 = 180\)

Step 3: Write intermediate result

\(\frac{90}{180}\)

Step 4: Simplify the fraction

GCD of 90 and 180 is 90: \(\frac{90 \div 90}{180 \div 90} = \frac{1}{2}\)

\(\frac{2}{3} \times \frac{9}{10} \times \frac{5}{6} = \frac{1}{2}\)
Final answer:

\(\frac{2}{3} \times \frac{9}{10} \times \frac{5}{6} = \frac{1}{2}\)

Applied rules:

• Multiple fraction multiplication: Extend basic rule to multiple fractions

• Sequential multiplication: Can multiply two at a time or all together

• Simplification priority: Always reduce to simplest form

5 Word problem with fraction multiplication
Exercise 5
Sarah ate \(\frac{3}{4}\) of her pizza. Then she ate \(\frac{2}{3}\) of what was left. What fraction of the whole pizza did she eat in total?
Definition:

Word problem interpretation: First find remaining pizza, then calculate fraction of remaining portion eaten

Initial eating
\(\frac{3}{4}\)
Remaining
\(\frac{1}{4}\)
Additional eating
\(\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}\)
Total eaten
\(\frac{3}{4} + \frac{1}{6} = \frac{11}{12}\)
Step 1: Calculate remaining pizza after first eating

Total pizza = 1, eaten = \(\frac{3}{4}\), remaining = \(1 - \frac{3}{4} = \frac{1}{4}\)

Step 2: Calculate amount of remaining pizza eaten

\(\frac{2}{3}\) of \(\frac{1}{4}\) = \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}\)

Step 3: Calculate total amount eaten

First eating + Second eating = \(\frac{3}{4} + \frac{1}{6}\)

Step 4: Add the fractions

LCM of 4 and 6 is 12: \(\frac{3}{4} = \frac{9}{12}\), \(\frac{1}{6} = \frac{2}{12}\)

\(\frac{9}{12} + \frac{2}{12} = \frac{11}{12}\)

Sarah ate \(\frac{11}{12}\) of the whole pizza
Final answer:

Sarah ate \(\frac{11}{12}\) of the whole pizza

Applied rules:

• Problem decomposition: Break complex problems into simpler steps

• Fraction operations: Combine multiplication and addition appropriately

• Real-world context: Interpret mathematical results in practical situations

Complete Guide to Fraction Multiplication Rules and Methods
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Basic Fraction Multiplication
Key definitions:

Proper fraction: Numerator is less than denominator (\(\frac{2}{3}\))

Improper fraction: Numerator is greater than or equal to denominator (\(\frac{5}{3}\))

Mixed number: Combination of whole number and fraction (\(1\frac{1}{2}\))

Equivalent fractions: Different fractions representing the same value (\(\frac{1}{2} = \frac{2}{4}\))

Complete methodology:
  1. Identify fractions: Recognize all numerators and denominators
  2. Look for simplification opportunities: Check for common factors across fractions
  3. Apply multiplication rule: Multiply numerators together and denominators together
  4. Simplify result: Reduce to lowest terms if possible
  5. Verify answer: Check reasonableness and accuracy
Tip 1: Cross-cancel before multiplying to avoid large numbers: \(\frac{6}{8} \times \frac{4}{9} = \frac{6}{8} \times \frac{4}{9} = \frac{3}{4} \times \frac{1}{3} = \frac{1}{4}\)
Tip 2: When multiplying by a whole number, write it as a fraction over 1: \(5 = \frac{5}{1}\)
Tip 3: The product of two proper fractions is always smaller than both original fractions.
Tip 4: To check your answer, divide the product by one factor to get the other.
Commutative property: \(\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}\)
Associative property: \((\frac{a}{b} \times \frac{c}{d}) \times \frac{e}{f} = \frac{a}{b} \times (\frac{c}{d} \times \frac{e}{f})\)
Distributive property: \(\frac{a}{b} \times (\frac{c}{d} + \frac{e}{f}) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f}\)
Formulas to know by heart:

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

• With whole number: \(n \times \frac{a}{b} = \frac{na}{b}\)

• Reciprocal: \(\frac{a}{b} \times \frac{b}{a} = 1\)

• Identity: \(\frac{a}{b} \times 1 = \frac{a}{b}\)

• Zero property: \(\frac{a}{b} \times 0 = 0\)

Visualizing Fraction Multiplication: Area Models
Understanding Fraction Products
Visual representation of fraction multiplication using area models:
\(\frac{2}{3} \times \frac{3}{4}\) as overlapping rectangles

Concept: Fraction multiplication can be visualized as finding the area of overlapping sections.

  • \(\frac{2}{3} \times \frac{3}{4}\) means taking 2/3 of 3/4 of a whole
  • This is equivalent to \(\frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}\)
  • Area model helps understand why the result is smaller than original fractions

Questions & Answers

Question: Why does multiplying two fractions give a smaller number? It seems counterintuitive compared to regular multiplication.

Answer: Great observation! When you multiply two fractions between 0 and 1, you're essentially taking a part of a part:

  • If you have half a pizza (\(\frac{1}{2}\)) and take half of that (\(\frac{1}{2}\)), you end up with \(\frac{1}{4}\) of the whole pizza
  • Think of fraction multiplication as "taking a fraction OF another fraction"
  • So \(\frac{2}{3} \times \frac{3}{4}\) means "take \(\frac{2}{3}\) of \(\frac{3}{4}\)"

This is different from multiplying whole numbers where you're combining quantities. With fractions, you're finding portions of portions, which naturally results in a smaller quantity.

Note: This only applies when both fractions are between 0 and 1. If one fraction is improper (>1), the result could be larger.

Question: When should I simplify fractions before multiplying versus after multiplying? Is there a rule?

Answer: Always simplify BEFORE multiplying when possible! This is called cross-cancellation:

  • Before multiplying \(\frac{4}{6} \times \frac{3}{8}\), notice that 4 and 8 share factor 4, and 3 and 6 share factor 3
  • You can cancel: \(\frac{4}{6} \times \frac{3}{8} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
  • This avoids working with large numbers like \(\frac{12}{48}\) that need further simplification

Cross-cancellation is preferred because:

  • It reduces calculation errors
  • It prevents dealing with unnecessarily large numbers
  • It often gives the answer in simplest form automatically

The rule is: Look for common factors between any numerator and any denominator before multiplying!

Question: How do I multiply a mixed number by a fraction? Do I convert the mixed number first?

Answer: Yes, always convert mixed numbers to improper fractions before multiplying! Here's the process:

To multiply \(2\frac{1}{3} \times \frac{3}{4}\):

  1. Convert mixed number to improper fraction: \(2\frac{1}{3} = \frac{7}{3}\)
  2. Now multiply: \(\frac{7}{3} \times \frac{3}{4}\)
  3. Apply fraction multiplication: \(\frac{7 \times 3}{3 \times 4} = \frac{21}{12}\)
  4. Simplify: \(\frac{21}{12} = \frac{7}{4} = 1\frac{3}{4}\)

Alternative method: You can also multiply the whole number part and fractional part separately, but converting to improper fraction is usually more straightforward.

Remember: \(a\frac{b}{c} = \frac{a \times c + b}{c}\) for conversion!

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

Answer: Here are several verification methods for fraction multiplication:

  1. Reverse division: If \(\frac{a}{b} \times \frac{c}{d} = \frac{e}{f}\), then \(\frac{e}{f} \div \frac{c}{d}\) should equal \(\frac{a}{b}\)
  2. Decimal conversion: Convert fractions to decimals and multiply: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\) → 0.5 × 0.333... ≈ 0.166..., which equals 1/6
  3. Reasonableness check: The product of two proper fractions should be smaller than both original fractions

For example, checking \(\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}\):

  • \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{2}{3}\) ✓
  • \(\frac{1}{2} = 0.5\), which is indeed smaller than both \(\frac{2}{3} ≈ 0.67\) and \(\frac{3}{4} = 0.75\) ✓

These checks help catch calculation errors and build confidence in your results!

Question: What is the best way to visualize fraction multiplication? I learn better with pictures.

Answer: Visual models make fraction multiplication much clearer! Here are the most effective approaches:

  1. Area model: Draw a rectangle and divide it according to the fractions. For \(\frac{2}{3} \times \frac{3}{4}\):
    • Divide rectangle into 3 columns, shade 2 (representing \(\frac{2}{3}\))
    • Divide the same rectangle into 4 rows, shade 3 (representing \(\frac{3}{4}\))
    • The overlapping shaded area represents the product
  2. Number line: Show jumps representing the fraction of a distance
  3. Pie charts: Shade portions of circles to represent fractions

The area model is particularly powerful because it shows why we multiply numerators together and denominators together - you're finding the overlap of two separate partitions of the same whole.

Try drawing these models yourself - they make the abstract concept very concrete!