Solved Exercises on Multiplying Fractions by Fractions in Grade 4

Master multiplying fractions by 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 the numerators together
  2. Multiply the denominators together
  3. Simplify the result if possible
Expression
\(\frac{2}{3} \times \frac{4}{5}\)
Multiplication
\(\frac{2 \times 4}{3 \times 5}\)
Result
\(\frac{8}{15}\)
Step 1: Multiply the numerators

\(2 \times 4 = 8\)

Step 2: Multiply the denominators

\(3 \times 5 = 15\)

Step 3: Write the result

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

Step 4: Check if simplification is needed

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 by numerators, denominators by denominators

• Simplification: Reduce to lowest terms if GCD > 1

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

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

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

\(3 \times 2 = 6\)

Step 2: Multiply the denominators

\(4 \times 6 = 24\)

Step 3: Write the intermediate result

\(\frac{6}{24}\)

Step 4: Find GCD of numerator and denominator

GCD(6, 24) = 6

Step 5: Simplify by dividing by GCD

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

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

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

Applied rules:

• Fraction multiplication rule: Multiply numerators by numerators, denominators by denominators

• Simplification: Reduce to lowest terms by dividing by GCD

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

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

Expression
\(\frac{2}{3} \times 4\)
Convert to fraction
\(\frac{2}{3} \times \frac{4}{1}\)
Result
\(\frac{8}{3}\)
Step 1: Convert whole number to fraction

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

Step 2: Apply fraction multiplication rule

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

Step 3: Calculate numerator and denominator

Numerator: \(2 \times 4 = 8\), Denominator: \(3 \times 1 = 3\)

Step 4: Write the result

\(\frac{8}{3}\)

Step 5: Convert to mixed number (optional)

\(\frac{8}{3} = 2\frac{2}{3}\)

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

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

Applied rules:

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

• Fraction multiplication rule: Multiply numerators by numerators, denominators by denominators

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

Complete methodology:
  1. Identify the fractions: Recognize the numerators and denominators
  2. Apply multiplication rule: Multiply numerators together, multiply denominators together
  3. Simplify if needed: Find GCD and divide both numerator and denominator
  4. Convert to mixed number: If result is improper fraction (optional)
Tip 1: Always check if the final fraction can be simplified.
Tip 2: To multiply a fraction by a whole number, write the whole number as a fraction over 1.
Tip 3: Cross-simplification before multiplying can make calculations easier.
Tip 4: The result of multiplying two fractions is usually smaller than both original fractions.
Common errors: Adding numerators and denominators instead of multiplying, forgetting to simplify the result.
Memory aid: Think "multiply across" - numerator × numerator, denominator × denominator.
Formulas to know by heart:

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

• 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 \(r\)

Solution: Exercises 4 to 5
4 Cross-simplification
Exercise 4
Multiply:
\(\frac{4}{9} \times \frac{3}{8}\)
Definition:

Cross-simplification: Simplify before multiplying by finding common factors between numerators and denominators

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

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

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

Step 2: Simplify the fractions

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

Step 3: Multiply the simplified fractions

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

Step 4: Verify the result

Without cross-simplification: \(\frac{4 \times 3}{9 \times 8} = \frac{12}{72} = \frac{1}{6}\) ✓

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

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

Applied rules:

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

• Fraction multiplication rule: Multiply numerators by numerators, denominators by denominators

5 Complex fraction multiplication
Exercise 5
Multiply:
\(\frac{5}{6} \times \frac{3}{10} \times \frac{4}{7}\)
Definition:

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

Expression
\(\frac{5}{6} \times \frac{3}{10} \times \frac{4}{7}\)
All numerators
\(5 \times 3 \times 4\)
All denominators
\(6 \times 10 \times 7\)
Result
\(\frac{60}{420} = \frac{1}{7}\)
Step 1: Multiply all numerators together

\(5 \times 3 \times 4 = 60\)

Step 2: Multiply all denominators together

\(6 \times 10 \times 7 = 420\)

Step 3: Write the result

\(\frac{60}{420}\)

Step 4: Find GCD of numerator and denominator

GCD(60, 420) = 60

Step 5: Simplify by dividing by GCD

\(\frac{60 ÷ 60}{420 ÷ 60} = \frac{1}{7}\)

\(\frac{5}{6} \times \frac{3}{10} \times \frac{4}{7} = \frac{1}{7}\)
Final answer:

\(\frac{5}{6} \times \frac{3}{10} \times \frac{4}{7} = \frac{1}{7}\)

Applied rules:

• Multiple fraction multiplication: Multiply all numerators together and all denominators together

• Simplification: Reduce to lowest terms by dividing by GCD

Multiplication Rules, Methods, Definitions, and Key Concepts
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Fraction Multiplication 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 between any numerator and any denominator

Complete methodology:
  1. Identify fractions: Recognize numerators and denominators in each fraction
  2. Look for simplification opportunities: Check for common factors between any numerator and any denominator
  3. Apply multiplication rule: Multiply all numerators together, multiply all denominators together
  4. Simplify the result: Divide numerator and denominator by their GCD
  5. 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 fractions, the result is usually smaller than the original fractions.
Tip 3: Always check if your final answer can be simplified further.
Tip 4: Remember that multiplying by a fraction less than 1 makes the result smaller.
Tip 5: Use visual models like area models to understand why fraction multiplication works.
Common errors: Adding numerators and denominators instead of multiplying, forgetting to simplify results, not recognizing when to convert to mixed numbers.
Visual representation: Imagine a rectangle divided into parts to represent fraction multiplication.
Real-world application: Fraction multiplication is used in cooking, construction, and scaling measurements.
Formulas to know by heart:

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

• 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 fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} \times \frac{e}{f} = \frac{a \times c \times e}{b \times d \times f}\)

Questions & Answers

Question: Why do we multiply the numerators together and the denominators together when multiplying fractions? It seems different from adding fractions.

Answer: Great question! The difference comes from what multiplication versus addition means for fractions:

  • When you add fractions, you're combining parts of the same whole, so you need the same denominator (same-sized pieces) to add the numerators (number of pieces).
  • When you multiply fractions, you're finding a part of a part. For example, \(\frac{2}{3} \times \frac{3}{4}\) means "what is \(\frac{2}{3}\) of \(\frac{3}{4}\)?"

Think of it visually: if you have \(\frac{3}{4}\) of a pizza and you eat \(\frac{2}{3}\) of that portion, you're taking 2 out of every 3 parts from the \(\frac{3}{4}\) portion. This creates 6 small pieces out of a total of 12 possible pieces, giving you \(\frac{6}{12} = \frac{1}{2}\) of the whole pizza.

So multiplication combines both the "part" aspect (numerators) and the "whole" aspect (denominators) simultaneously.

Question: When should I use cross-simplification, and why does it work?

Answer: Cross-simplification is a helpful technique you should use whenever possible! Here's when and why:

  • When to use: Before multiplying, check if any numerator has a common factor with any denominator.
  • Why it works: Since multiplication is commutative, you can rearrange the factors. For example, in \(\frac{4}{9} \times \frac{3}{8}\), you can rewrite as \(\frac{4 \times 3}{9 \times 8}\) and then rearrange to \(\frac{4}{8} \times \frac{3}{9}\).

This simplifies your multiplication significantly. Instead of calculating \(\frac{12}{72}\), you work with \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\), which is much easier and reduces the chance of computational errors.

Always look for cross-simplification opportunities - it's a valuable skill that will help in more advanced math too!

Question: I noticed that when I multiply fractions, the answer is sometimes smaller than both of the original fractions. Why does this happen?

Answer: That's an excellent observation! This happens because multiplication by a fraction less than 1 is essentially scaling down. Here's why:

  • When you multiply by a fraction less than 1 (like \(\frac{1}{2}\)), you're taking only part of the original amount.
  • For example, \(\frac{1}{2} \times \frac{1}{3}\) means "half of one-third", which is clearly smaller than both \(\frac{1}{2}\) and \(\frac{1}{3}\).
  • This is different from multiplying by whole numbers greater than 1, which increases the value.

Think of it this way: if you have half a cake and someone takes half of that, they get only a quarter of the original cake - much less than the half you started with. This is the nature of multiplying fractions less than 1!

However, if you multiply by a fraction greater than 1 (improper fraction), the result could be larger than one or both of the original fractions.