Generate Equivalent Fractions to Get Like Units

Lesson Examples: Problems 1 to 3

1 Adding Fractions with Different Denominators

Example 1

Find equivalent fractions to get like units for \(\frac{1}{3} + \frac{1}{6}\).

Definition:

Like units: Fractions that have the same denominator, allowing them to be easily compared, added, or subtracted.

Method to Find Like Units:

Identify the different denominators
Find the least common multiple (LCM) of the denominators
Convert each fraction to an equivalent fraction with the LCM as denominator
Add or compare the fractions with like units

Original Fractions

\(\frac{1}{3}, \frac{1}{6}\)

Find LCM

LCM(3,6) = 6

Convert

\(\frac{2}{6}, \frac{1}{6}\)

Add

\(\frac{3}{6}\)

Step 1: Identify the denominators

The denominators are 3 and 6

Step 2: Find the least common multiple (LCM)

LCM of 3 and 6 is 6 (since 6 is a multiple of 3)

Step 3: Convert \(\frac{1}{3}\) to equivalent fraction with denominator 6

To get from 3 to 6, multiply by 2

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

Step 4: Keep \(\frac{1}{6}\) as is (already has denominator 6)

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

Step 5: Add the fractions with like units

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

Step 6: Simplify if possible

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

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

Final answer:

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

Applied rules:

• Common Denominator: Fractions must have the same denominator to be added

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Adding Like Fractions: Add numerators, keep denominator the same

2 Comparing Fractions with Different Denominators

Example 2

Compare \(\frac{2}{5}\) and \(\frac{3}{7}\) by generating equivalent fractions with like units.

Definition:

Comparing fractions: Determining which fraction is greater, less than, or equal to another by converting them to have the same denominator.

Original Fractions

\(\frac{2}{5}, \frac{3}{7}\)

Find LCM

LCM(5,7) = 35

Convert

\(\frac{14}{35}, \frac{15}{35}\)

Compare

\(\frac{14}{35} < \frac{15}{35}\)

Step 1: Identify the denominators

The denominators are 5 and 7

Step 2: Find the least common multiple (LCM)

LCM of 5 and 7 is 35 (since 5 and 7 are both prime)

Step 3: Convert \(\frac{2}{5}\) to equivalent fraction with denominator 35

To get from 5 to 35, multiply by 7

\(\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}\)

Step 4: Convert \(\frac{3}{7}\) to equivalent fraction with denominator 35

To get from 7 to 35, multiply by 5

\(\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}\)

Step 5: Compare the fractions with like units

Since 14 < 15, \(\frac{14}{35} < \frac{15}{35}\)

Step 6: Write the comparison with original fractions

\(\frac{2}{5} < \frac{3}{7}\)

Final answer:

\(\frac{2}{5} < \frac{3}{7}\)

Applied rules:

• Common Denominator: Fractions must have the same denominator to be compared

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Comparing Like Fractions: Compare numerators when denominators are the same

3 Subtracting Fractions with Different Denominators

Example 3

Find equivalent fractions to get like units for \(\frac{3}{4} - \frac{1}{6}\).

Definition:

Subtracting fractions: To subtract fractions with different denominators, convert them to equivalent fractions with the same denominator.

Original Fractions

\(\frac{3}{4}, \frac{1}{6}\)

Find LCM

LCM(4,6) = 12

Convert

\(\frac{9}{12}, \frac{2}{12}\)

Subtract

\(\frac{7}{12}\)

Step 1: Identify the denominators

The denominators are 4 and 6

Step 2: Find the least common multiple (LCM)

LCM of 4 and 6 is 12 (multiples of 4: 4, 8, 12...; multiples of 6: 6, 12...)

Step 3: Convert \(\frac{3}{4}\) to equivalent fraction with denominator 12

To get from 4 to 12, multiply by 3

\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

Step 4: Convert \(\frac{1}{6}\) to equivalent fraction with denominator 12

To get from 6 to 12, multiply by 2

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

Step 5: Subtract the fractions with like units

\(\frac{9}{12} - \frac{2}{12} = \frac{7}{12}\)

Step 6: Check if the result can be simplified

Factors of 7: 1, 7

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

Only common factor is 1, so \(\frac{7}{12}\) is in simplest form

\(\frac{3}{4} - \frac{1}{6} = \frac{7}{12}\)

Final answer:

\(\frac{3}{4} - \frac{1}{6} = \frac{7}{12}\)

Applied rules:

• Common Denominator: Fractions must have the same denominator to be subtracted

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Subtracting Like Fractions: Subtract numerators, keep denominator the same

Like Units Concepts and Methods

\(\frac{a}{b} = \frac{a \times n}{b \times n}\)

Generating Equivalent Fractions to Get Like Units

Multiplication Rule

\(\frac{a}{b} = \frac{a \times n}{b \times n}\)

Generate equivalent fractions

Addition

\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

Add fractions with like units

Comparison

\(\frac{a}{c} \lt \frac{b}{c}\) if \(a \lt b\)

Compare fractions with like units

Key definitions:

Like units: Fractions that have the same denominator, allowing them to be easily compared, added, or subtracted.

Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

Equivalent fractions: Fractions that represent the same value but have different numerators and denominators.

Common denominator: The same denominator shared by fractions to allow for comparison or arithmetic operations.

Steps to Generate Like Units:

Identify the different denominators in the fractions
Find the least common multiple (LCM) of the denominators
Convert each fraction to an equivalent fraction with the LCM as denominator
Perform the required operation (addition, subtraction, or comparison)

Tip 1: To find the LCM, list multiples of each denominator until you find the smallest common multiple.

Tip 2: When denominators have no common factors (are relatively prime), their LCM is their product.

Tip 3: Always multiply both numerator AND denominator by the same number to maintain equivalence.

Tip 4: When comparing fractions with like units, compare the numerators only.

Common errors: Forgetting to multiply the numerator when changing the denominator, not finding the correct LCM, comparing numerators without ensuring like units.

Key insight: Like units allow direct comparison of numerators since the "size" of the pieces being compared is the same.

Essential formulas:

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

• Adding like fractions: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Subtracting like fractions: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Comparing like fractions: \(\frac{a}{c} < \frac{b}{c}\) if \(a < b\)

Lesson Practice: Problems 4 to 5

4 Adding Fractions with Like Units

Example 4

Find equivalent fractions to get like units for \(\frac{2}{3} + \frac{1}{4}\).

Definition:

Adding fractions with different denominators: Requires finding equivalent fractions with the same denominator before performing addition.

Original Fractions

\(\frac{2}{3}, \frac{1}{4}\)

Find LCM

LCM(3,4) = 12

Convert

\(\frac{8}{12}, \frac{3}{12}\)

Add

\(\frac{11}{12}\)

Step 1: Identify the denominators

The denominators are 3 and 4

Step 2: Find the least common multiple (LCM)

LCM of 3 and 4 is 12 (since 3×4=12 and they have no common factors)

Step 3: Convert \(\frac{2}{3}\) to equivalent fraction with denominator 12

To get from 3 to 12, multiply by 4

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

Step 4: Convert \(\frac{1}{4}\) to equivalent fraction with denominator 12

To get from 4 to 12, multiply by 3

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

Step 5: Add the fractions with like units

\(\frac{8}{12} + \frac{3}{12} = \frac{11}{12}\)

Step 6: Check if the result can be simplified

Factors of 11: 1, 11 (prime number)

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

Only common factor is 1, so \(\frac{11}{12}\) is in simplest form

\(\frac{2}{3} + \frac{1}{4} = \frac{11}{12}\)

Final answer:

\(\frac{2}{3} + \frac{1}{4} = \frac{11}{12}\)

Applied rules:

• Common Denominator: Fractions must have the same denominator to be added

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Adding Like Fractions: Add numerators, keep denominator the same

5 Real-World Application

Example 5

Sarah ate \(\frac{1}{2}\) of a pizza and Tom ate \(\frac{1}{3}\) of the same pizza. Who ate more pizza?

Definition:

Real-world fraction comparison: Converting fractions to like units to solve practical problems involving comparison of amounts.

Original Fractions

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

Find LCM

LCM(2,3) = 6

Convert

\(\frac{3}{6}, \frac{2}{6}\)

Compare

\(\frac{3}{6} > \frac{2}{6}\)

Step 1: Identify the fractions to compare

Sarah: \(\frac{1}{2}\), Tom: \(\frac{1}{3}\)

Step 2: Find the least common multiple (LCM)

LCM of 2 and 3 is 6 (since 2×3=6 and they have no common factors)

Step 3: Convert \(\frac{1}{2}\) to equivalent fraction with denominator 6

To get from 2 to 6, multiply by 3

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

Step 4: Convert \(\frac{1}{3}\) to equivalent fraction with denominator 6

To get from 3 to 6, multiply by 2

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

Step 5: Compare the fractions with like units

Since 3 > 2, \(\frac{3}{6} > \frac{2}{6}\)

Step 6: Answer the question in context

Since \(\frac{3}{6} > \frac{2}{6}\), Sarah ate more pizza than Tom

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

Final answer:

Sarah ate more pizza because \(\frac{1}{2} > \frac{1}{3}\)

Applied rules:

• Common Denominator: Fractions must have the same denominator to be compared

• Equivalent Fractions: Multiply both numerator and denominator by the same number

• Comparing Like Fractions: Compare numerators when denominators are the same

Detailed Summary: Equivalent Fractions to Get Like Units

\(\frac{a}{b} = \frac{a \times n}{b \times n}\)

Core Principle for Like Units

Key Definitions:

Like units: Fractions that have the same denominator, allowing them to be easily compared, added, or subtracted. They represent parts of the same whole divided into the same number of equal parts.

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers. Used to find the common denominator.

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.

Common denominator: The same denominator shared by two or more fractions, necessary for addition, subtraction, or comparison.

Steps to Generate Like Units:

Identify the different denominators in the fractions you need to work with
Find the least common multiple (LCM) of all the denominators
Convert each fraction to an equivalent fraction with the LCM as denominator
Perform the required operation (addition, subtraction, or comparison)

Tip 1: To find the LCM, list multiples of each denominator until you find the smallest common multiple.

Tip 2: When denominators have no common factors (are relatively prime), their LCM is their product.

Tip 3: Always multiply both numerator AND denominator by the same number to maintain equivalence.

Tip 4: When comparing fractions with like units, compare the numerators only.

Tip 5: After finding like units, check if the final answer can be simplified to lowest terms.

Core Rule: The fundamental principle is that \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0. This allows us to create equivalent fractions with desired denominators.

Practical Application: Getting like units is essential for fraction operations: addition, subtraction, and comparison. Without like units, these operations cannot be performed directly.

Essential Rules and Formulas:

• Generating Equivalent Fractions: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0

• Adding Like Fractions: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Subtracting Like Fractions: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Comparing Like Fractions: \(\frac{a}{c} < \frac{b}{c}\) if and only if \(a < b\)

Simple Example

Compare \(\frac{1}{2}\) and \(\frac{2}{3}\) by getting like units:

LCM of 2 and 3 is 6

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

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

Since 3 < 4, \(\frac{3}{6} < \frac{4}{6}\), so \(\frac{1}{2} < \frac{2}{3}\)

Intermediate Example

Add \(\frac{3}{8} + \frac{1}{4}\) by getting like units:

LCM of 8 and 4 is 8

\(\frac{3}{8} = \frac{3}{8}\) (already has denominator 8)

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

\(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\)

Advanced Example

Subtract \(\frac{5}{6} - \frac{2}{9}\) by getting like units:

LCM of 6 and 9 is 18

\(\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}\)

\(\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}\)

\(\frac{15}{18} - \frac{4}{18} = \frac{11}{18}\)

Key Takeaways:

Like units are fractions with the same denominator
Getting like units requires finding equivalent fractions
Use the LCM of denominators to find the common denominator
Always multiply both numerator and denominator by the same number
With like units, compare or operate on numerators only
Check if the final answer can be simplified
Like units are essential for fraction addition, subtraction, and comparison

Practice Problems

Solve the following problems by generating equivalent fractions to get like units:

Compare \(\frac{3}{4}\) and \(\frac{5}{6}\)
Add \(\frac{2}{5} + \frac{1}{3}\)
Subtract \(\frac{7}{8} - \frac{1}{4}\)
Compare \(\frac{2}{7}\) and \(\frac{3}{10}\)
Add \(\frac{1}{6} + \frac{1}{9}\)

Answer Key:

1. \(\frac{3}{4} < \frac{5}{6}\) (LCM=12: \(\frac{9}{12} < \frac{10}{12}\))

2. \(\frac{2}{5} + \frac{1}{3} = \frac{11}{15}\) (LCM=15: \(\frac{6}{15} + \frac{5}{15} = \frac{11}{15}\))

3. \(\frac{7}{8} - \frac{1}{4} = \frac{5}{8}\) (LCM=8: \(\frac{7}{8} - \frac{2}{8} = \frac{5}{8}\))

4. \(\frac{2}{7} > \frac{3}{10}\) (LCM=70: \(\frac{20}{70} > \frac{21}{70}\) → Actually \(\frac{2}{7} < \frac{3}{10}\) since \(\frac{20}{70} < \frac{21}{70}\))

5. \(\frac{1}{6} + \frac{1}{9} = \frac{5}{18}\) (LCM=18: \(\frac{3}{18} + \frac{2}{18} = \frac{5}{18}\))