Generating Equivalent Fractions Using Multiplication

Complete lesson on generating equivalent fractions through multiplication, with examples, exercises, and solutions for 4th grade students.

Lesson Examples: Problems 1 to 3
1 Basic Multiplication
Example 1
Find an equivalent fraction to \(\frac{2}{3}\) by multiplying both numerator and denominator by 4.
Definition:

Equivalent fractions: Fractions that represent the same value or the same portion of a whole, even though they may look different.

Multiplication Method Steps:
  1. Identify the original fraction
  2. Choose a number to multiply both numerator and denominator by
  3. Multiply both parts by the same number
  4. Verify the result using cross multiplication
Original Fraction
\(\frac{2}{3}\)
Multiply by 4
\(\frac{2 \times 4}{3 \times 4}\)
Result
\(\frac{8}{12}\)
Step 1: Identify the original fraction

Original fraction: \(\frac{2}{3}\)

Step 2: Choose a number to multiply by

We'll multiply both parts by 4

Step 3: Multiply both numerator and denominator

Numerator: \(2 \times 4 = 8\)

Denominator: \(3 \times 4 = 12\)

Step 4: Write the equivalent fraction

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

Step 5: Verify using cross multiplication

\(2 \times 12 = 24\) and \(3 \times 8 = 24\)

Since both products are equal, the fractions are equivalent

\(\frac{2}{3} = \frac{8}{12}\)
Final answer:

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

Applied rules:

• Multiplication Property: Multiply both numerator and denominator by the same non-zero number

• Identity Property: Multiplying by \(\frac{n}{n} = 1\) doesn't change the value

• Cross Multiplication Test: \(a/b = c/d\) if \(a \times d = b \times c\)

2 Intermediate Multiplication
Example 2
Find an equivalent fraction to \(\frac{3}{7}\) by multiplying both parts by 5.
Definition:

Generating equivalent fractions: Creating new fractions that have the same value as the original by multiplying both the numerator and denominator by the same non-zero number.

Original Fraction
\(\frac{3}{7}\)
Multiply by 5
\(\frac{3 \times 5}{7 \times 5}\)
Result
\(\frac{15}{35}\)
Step 1: Identify the original fraction

Original fraction: \(\frac{3}{7}\)

Step 2: Choose the multiplier

We'll multiply both parts by 5

Step 3: Multiply the numerator

Numerator: \(3 \times 5 = 15\)

Step 4: Multiply the denominator

Denominator: \(7 \times 5 = 35\)

Step 5: Write the equivalent fraction

\(\frac{3}{7} = \frac{15}{35}\)

Step 6: Verify using cross multiplication

\(3 \times 35 = 105\) and \(7 \times 15 = 105\)

Since both products are equal, the fractions are equivalent

\(\frac{3}{7} = \frac{15}{35}\)
Final answer:

\(\frac{3}{7} = \frac{15}{35}\)

Applied rules:

• Multiplication Property: Multiply both numerator and denominator by the same non-zero number

• Value Preservation: The fraction's value remains unchanged

• Verification: Use cross multiplication to confirm equivalence

3 Multiple Multiplications
Example 3
Generate three equivalent fractions for \(\frac{1}{4}\) using different multipliers.
Definition:

Multiple equivalent fractions: There are infinitely many equivalent fractions for any given fraction, generated through various multiplication operations.

Original
\(\frac{1}{4}\)
×2
\(\frac{2}{8}\)
×3
\(\frac{3}{12}\)
×4
\(\frac{4}{16}\)
Step 1: Start with the original fraction

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

Step 2: Multiply by \(\frac{2}{2}\) to get first equivalent

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

Step 3: Multiply by \(\frac{3}{3}\) to get second equivalent

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

Step 4: Multiply by \(\frac{4}{4}\) to get third equivalent

\(\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}\)

Step 5: Verify one of the equivalences

Check \(\frac{1}{4} = \frac{3}{12}\): \(1 \times 12 = 12\) and \(4 \times 3 = 12\) ✓

Step 6: Conclusion

Any number can be used as a multiplier to generate equivalent fractions

\(\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\)
Final answer:

Three equivalent fractions: \(\frac{2}{8}, \frac{3}{12}, \frac{4}{16}\)

Applied rules:

• Infinite Equivalents: Any fraction has infinitely many equivalent fractions

• Operation Consistency: Always apply the same operation to both numerator and denominator

• Verification: Use cross multiplication to confirm equivalences

Multiplication Method Concepts and Rules
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Multiplication Property of Equivalent Fractions
Multiplication Rule
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Multiply both parts by same number
Identity Property
\(\frac{a}{b} \times \frac{n}{n} = \frac{a}{b}\)
Multiplying by 1 in disguise
Cross Product
\(\frac{a}{b} = \frac{c}{d}\) if \(a \times d = b \times c\)
Test for equivalence
Key definitions:

Equivalent fractions: Fractions that represent the same value or the same portion of a whole.

Numerator: Top number in a fraction, representing the number of parts taken.

Denominator: Bottom number in a fraction, representing the total number of equal parts.

Multiplication method: Technique for generating equivalent fractions by multiplying both numerator and denominator by the same non-zero number.

Multiplication Method Steps:
  1. Identify the original fraction you want to create an equivalent for
  2. Choose a number to multiply both numerator and denominator by
  3. Multiply both parts of the fraction by the chosen number
  4. Verify equivalence using cross multiplication
Tip 1: Always multiply both numerator AND denominator by the same number to maintain equivalence.
Tip 2: Think of multiplying by \(\frac{n}{n}\) as multiplying by 1, which doesn't change the value.
Tip 3: Cross multiplication is the fastest way to check if two fractions are equivalent.
Tip 4: Any non-zero number can be used as a multiplier to generate equivalent fractions.
Common errors: Forgetting to multiply both parts, multiplying numerator and denominator by different numbers, confusing equivalent with equal.
Key insight: Equivalent fractions represent the same point on a number line and the same area in visual models.
Essential formulas:

• Generating equivalent fractions: \(\frac{a}{b} = \frac{a \times n}{b \times n}\)

• Testing equivalence: \(\frac{a}{b} = \frac{c}{d}\) if \(a \times d = b \times c\)

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

Lesson Practice: Problems 4 to 5
4 Finding Missing Numerators
Example 4
Find the missing number: \(\frac{2}{5} = \frac{?}{20}\)
Definition:

Finding missing parts: Use the relationship between equivalent fractions to determine unknown numerators or denominators.

Given
\(\frac{2}{5} = \frac{?}{20}\)
Find multiplier
\(5 \times ? = 20\)
Apply to numerator
\(2 \times 4 = 8\)
Result
\(\frac{8}{20}\)
Step 1: Determine how the denominator changed

From 5 to 20: \(20 \div 5 = 4\)

So we multiplied the denominator by 4

Step 2: Apply the same multiplier to the numerator

Original numerator: 2

New numerator: \(2 \times 4 = 8\)

Step 3: Write the complete equivalent fraction

\(\frac{2}{5} = \frac{8}{20}\)

Step 4: Verify using cross multiplication

\(2 \times 20 = 40\) and \(5 \times 8 = 40\)

Since both products are equal, the fractions are equivalent

\(\frac{2}{5} = \frac{8}{20}\)
Final answer:

The missing number is 8, so \(\frac{2}{5} = \frac{8}{20}\)

Applied rules:

• Consistency Rule: Whatever operation is done to the denominator must also be done to the numerator

• Multiplier Identification: Find the factor by which one denominator was multiplied to get the other

• Cross Multiplication Check: Verify the result by cross multiplying

5 Word Problem Application
Example 5
A pizza was cut into 8 slices and 3 slices were eaten. Express this as a fraction. Then find an equivalent fraction by multiplying both parts by 3.
Definition:

Word problem application: Converting real-world situations into mathematical fractions and then applying multiplication to generate equivalent fractions.

Word Problem
3 out of 8
Write Fraction
\(\frac{3}{8}\)
Multiply by 3
\(\frac{9}{24}\)
Step 1: Translate the word problem to a fraction

3 slices eaten out of 8 total slices

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

Step 2: Multiply both parts by 3

Numerator: \(3 \times 3 = 9\)

Denominator: \(8 \times 3 = 24\)

Step 3: Write the equivalent fraction

\(\frac{3}{8} = \frac{9}{24}\)

Step 4: Verify the equivalence

Cross multiplication: \(3 \times 24 = 72\) and \(8 \times 9 = 72\) ✓

Step 5: Interpret in context

\(\frac{3}{8}\) and \(\frac{9}{24}\) represent the same amount of pizza eaten

\(\frac{3}{8} = \frac{9}{24}\)
Final answer:

\(\frac{3}{8} = \frac{9}{24}\) of the pizza was eaten

Applied rules:

• Problem Translation: Convert real-world ratios to mathematical fractions

• Multiplication Application: Use multiplication method to generate equivalent fractions

• Contextual Interpretation: Understand the meaning of the equivalent fraction

Detailed Summary: Equivalent Fractions Using Multiplication
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Core Multiplication Principle
Key Definitions:

Equivalent fractions: Different fractions that represent the same value or the same portion of a whole. They occupy the same position on a number line.

Multiplication method: A technique for generating equivalent fractions by multiplying both the numerator and denominator by the same non-zero number.

Numerator: The top number in a fraction; tells how many parts are being considered.

Denominator: The bottom number in a fraction; tells how many equal parts the whole is divided into.

Multiplication Method Steps:
  1. Identify the original fraction you want to create an equivalent for
  2. Choose a number to multiply both numerator and denominator by
  3. Multiply both parts of the fraction by the chosen number
  4. Verify equivalence using cross multiplication
Tip 1: Always multiply both numerator AND denominator by the same number to maintain equivalence.
Tip 2: Think of multiplying by \(\frac{n}{n}\) as multiplying by 1, which doesn't change the value.
Tip 3: Cross multiplication is the most reliable way to check if two fractions are equivalent: if \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\).
Tip 4: Any non-zero number can be used as a multiplier to generate equivalent fractions.
Tip 5: The multiplication method increases the size of the numbers but keeps the value the same.
Core Rule: The multiplication property states that \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0. This preserves the fraction's value while changing its appearance.
Practical Application: The multiplication method is essential for finding common denominators when adding or subtracting fractions with different denominators.
Essential Rules and Formulas:

• Multiplication Property: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0

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

• Equivalence Test: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a \times d = b \times c\)

Simple Example

Find an equivalent fraction to \(\frac{1}{2}\) by multiplying by 3:

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

Verification: \(1 \times 6 = 6\) and \(2 \times 3 = 6\) ✓

Intermediate Example

Find an equivalent fraction to \(\frac{4}{5}\) by multiplying by 7:

\(\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}\)

Verification: \(4 \times 35 = 140\) and \(5 \times 28 = 140\) ✓

Advanced Example

Find the missing number: \(\frac{5}{8} = \frac{?}{40}\)

From 8 to 40: \(40 \div 8 = 5\)

Apply to numerator: \(5 \times 5 = 25\)

\(\frac{5}{8} = \frac{25}{40}\)

Verification: \(5 \times 40 = 200\) and \(8 \times 25 = 200\) ✓

Key Takeaways:
  • Equivalent fractions have the same value but different appearances
  • Multiplication method creates larger equivalent fractions
  • Always multiply both numerator and denominator by the same number
  • Use cross multiplication to verify equivalences
  • Any non-zero number can be used as a multiplier
  • The value of the fraction remains unchanged during multiplication
  • There are infinitely many equivalent fractions for any given fraction
Practice Problems
Practice Problems

Solve the following problems using the multiplication method:

  1. Find an equivalent fraction to \(\frac{3}{4}\) by multiplying both parts by 6.
  2. Find an equivalent fraction to \(\frac{2}{7}\) by multiplying both parts by 9.
  3. Find the missing number: \(\frac{4}{9} = \frac{?}{27}\)
  4. List three equivalent fractions for \(\frac{5}{6}\) using different multipliers.
  5. A recipe calls for \(\frac{1}{3}\) cup of sugar. Find an equivalent fraction by multiplying both parts by 4.
Answer Key:

1. \(\frac{3}{4} = \frac{18}{24}\) (multiply both by 6)

2. \(\frac{2}{7} = \frac{18}{63}\) (multiply both by 9)

3. The missing number is 12, so \(\frac{4}{9} = \frac{12}{27}\)

4. Three equivalent fractions: \(\frac{10}{12}, \frac{15}{18}, \frac{20}{24}\) (multiplied by 2, 3, and 4 respectively)

5. \(\frac{1}{3} = \frac{4}{12}\) (multiply both by 4)