Subtracting Fractions with Like Denominators - Complete Worksheet

Master subtracting fractions with like denominators through 5 detailed worksheet exercises with step-by-step solutions, visual examples, and comprehensive explanations.

Worksheet Solutions: Problems 1 to 3
1 Basic subtraction
Problem 1
Subtract:
\(\frac{7}{8} - \frac{3}{8}\)
Definition:

Fractions with like denominators: Fractions where the bottom numbers (denominators) are the same, allowing direct subtraction of the top numbers (numerators) while keeping the denominator unchanged.

Subtraction method:
  1. Confirm that both fractions have the same denominator
  2. Subtract the numerators (top numbers)
  3. Keep the denominator the same
  4. Simplify the result if possible
Original
\(\frac{7}{8} - \frac{3}{8}\)
Subtract
\(\frac{7-3}{8}\)
Result
\(\frac{4}{8}\)
Step 1: Check denominators

Both fractions have denominator 8, so they are like denominators

Step 2: Subtract numerators

\(7 - 3 = 4\)

Step 3: Keep denominator

Denominator remains 8

Step 4: Write result

\(\frac{7}{8} - \frac{3}{8} = \frac{4}{8}\)

Step 5: Simplify if possible

\(\frac{4}{8} = \frac{1}{2}\) (divide numerator and denominator by 4)

\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)
Final answer:

\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)

Applied rules:

• Like denominators: Only numerators change in subtraction

• Subtraction principle: Count how many parts are left

• Simplification: Reduce to lowest terms when possible

2 Medium difficulty
Problem 2
Subtract:
\(\frac{9}{12} - \frac{5}{12}\)
Definition:

Proper fraction: A fraction where the numerator is less than the denominator, representing a value less than 1.

Original
\(\frac{9}{12} - \frac{5}{12}\)
Subtract
\(\frac{9-5}{12}\)
Result
\(\frac{4}{12}\)
Step 1: Check denominators

Both fractions have denominator 12, so they are like denominators

Step 2: Subtract numerators

\(9 - 5 = 4\)

Step 3: Keep denominator

Denominator remains 12

Step 4: Write result

\(\frac{9}{12} - \frac{5}{12} = \frac{4}{12}\)

Step 5: Simplify if possible

\(\frac{4}{12} = \frac{1}{3}\) (divide numerator and denominator by 4)

\(\frac{9}{12} - \frac{5}{12} = \frac{1}{3}\)
Final answer:

\(\frac{9}{12} - \frac{5}{12} = \frac{1}{3}\)

Applied rules:

• Like denominators: Denominator stays the same

• Direct subtraction: Only numerators are affected

• Simplification: Always check for common factors

3 Larger denominators
Problem 3
Subtract:
\(\frac{15}{20} - \frac{7}{20}\)
Definition:

Common factors: Numbers that divide evenly into both the numerator and denominator, used for simplifying fractions to their lowest terms.

Original
\(\frac{15}{20} - \frac{7}{20}\)
Subtract
\(\frac{15-7}{20}\)
Result
\(\frac{8}{20}\)
Step 1: Check denominators

Both fractions have denominator 20, so they are like denominators

Step 2: Subtract numerators

\(15 - 7 = 8\)

Step 3: Keep denominator

Denominator remains 20

Step 4: Write result

\(\frac{15}{20} - \frac{7}{20} = \frac{8}{20}\)

Step 5: Simplify if possible

\(\frac{8}{20} = \frac{2}{5}\) (divide numerator and denominator by 4)

\(\frac{15}{20} - \frac{7}{20} = \frac{2}{5}\)
Final answer:

\(\frac{15}{20} - \frac{7}{20} = \frac{2}{5}\)

Applied rules:

• Like denominators: Denominator remains unchanged

• Subtraction: Numerator subtraction only

• Simplification: Find greatest common divisor

Like Denominator Subtraction Fundamentals
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Like Denominator Subtraction Rule
Rule 1
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Like denominators
Rule 2
\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
Like denominators addition
Rule 3
\(\frac{a}{c} = \frac{a \div d}{c \div d}\)
Simplification
Key definitions:

Denominator: The bottom number in a fraction representing equal parts of a whole

Numerator: The top number in a fraction representing selected parts

Like denominators: Same bottom numbers allowing direct operations on numerators

Simplification: Reducing a fraction to its lowest terms by dividing both numerator and denominator by their greatest common divisor

Complete methodology:
  1. Identify denominators: Confirm they are the same
  2. Subtract numerators: Perform subtraction on top numbers
  3. Keep denominator: Bottom number remains unchanged
  4. Simplify: Reduce to lowest terms if possible
Tip 1: Always check if the result can be simplified by finding the GCD of numerator and denominator.
Tip 2: Draw fraction bars to visualize the subtraction process.
Tip 3: Remember: the denominator never changes when subtracting like fractions.
Tip 4: Practice with different denominators to build fluency.
Common errors: Changing the denominator, subtracting denominators, forgetting to simplify the final answer.
Worksheet strategy: Work systematically from left to right, double-check your arithmetic.
Essential formulas:

• Subtraction: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Addition: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

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

• GCD: Greatest Common Divisor used for reducing fractions

Worksheet Solutions: Problems 4 to 5
4 Complex subtraction
Problem 4
Subtract:
\(\frac{18}{24} - \frac{10}{24}\)
Definition:

Greatest Common Divisor (GCD): The largest number that divides both the numerator and denominator without leaving a remainder, used for simplifying fractions.

Original
\(\frac{18}{24} - \frac{10}{24}\)
Subtract
\(\frac{18-10}{24}\)
Result
\(\frac{8}{24}\)
Step 1: Check denominators

Both fractions have denominator 24, so they are like denominators

Step 2: Subtract numerators

\(18 - 10 = 8\)

Step 3: Keep denominator

Denominator remains 24

Step 4: Write result

\(\frac{18}{24} - \frac{10}{24} = \frac{8}{24}\)

Step 5: Simplify if possible

Find GCD of 8 and 24: GCD(8,24) = 8

\(\frac{8}{24} = \frac{8 \div 8}{24 \div 8} = \frac{1}{3}\)

\(\frac{18}{24} - \frac{10}{24} = \frac{1}{3}\)
Final answer:

\(\frac{18}{24} - \frac{10}{24} = \frac{1}{3}\)

Applied rules:

• Like denominators: Denominator stays unchanged

• Subtraction: Only numerators are affected

• Simplification: Use GCD to reduce to lowest terms

5 Word problem application
Problem 5
Sarah ate \(\frac{5}{6}\) of a pizza. Her brother ate \(\frac{2}{6}\) of the same pizza. How much more pizza did Sarah eat than her brother?
Definition:

Word problems: Real-world scenarios requiring mathematical operations to solve, involving identification of relevant information and appropriate operation selection.

Problem
\(\frac{5}{6} - \frac{2}{6}\)
Subtract
\(\frac{5-2}{6}\)
Result
\(\frac{3}{6}\)
Step 1: Set up the subtraction

Sarah's portion - Brother's portion = \(\frac{5}{6} - \frac{2}{6}\)

Step 2: Check denominators

Both fractions have denominator 6, so they are like denominators

Step 3: Subtract numerators

\(5 - 2 = 3\)

Step 4: Keep denominator

Denominator remains 6

Step 5: Write result

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

Step 6: Simplify if possible

\(\frac{3}{6} = \frac{1}{2}\) (divide numerator and denominator by 3)

Step 7: Interpret the result

Sarah ate \(\frac{1}{2}\) more of the pizza than her brother

Sarah ate \(\frac{1}{2}\) more pizza than her brother
Final answer:

Sarah ate \(\frac{1}{2}\) more of the pizza than her brother

Applied rules:

• Problem identification: Recognize subtraction is needed

• Like denominators: Denominator remains unchanged

• Context interpretation: Answer makes sense in word problem

Subtracting Like Denominators Complete Summary
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Like Denominator Subtraction Rule
Key definitions:

Fraction: A number representing part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\)

Like denominators: Same bottom numbers allowing direct operations on numerators

Proper fraction: A fraction where numerator < denominator

Improper fraction: A fraction where numerator ≥ denominator

Complete methodology:
  1. Identify denominators: Confirm they are the same
  2. Subtract numerators: Perform subtraction on top numbers
  3. Keep denominator: Bottom number remains unchanged
  4. Simplify: Reduce to lowest terms if possible
Tip 1: Always check if the result can be simplified by finding the GCD of numerator and denominator.
Tip 2: Draw fraction bars to visualize the subtraction process.
Tip 3: Remember: the denominator never changes when subtracting like fractions.
Tip 4: Practice with different denominators to build fluency.
Common errors: Changing the denominator, subtracting denominators, forgetting to simplify the final answer.
Memory trick: "Same bottom, subtract tops" - denominator stays the same!
Essential formulas:

• Subtraction: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Addition: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

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

• GCD: Greatest Common Divisor used for reducing fractions

Visual Fraction Subtraction Guide
Fraction Visualization
Understanding how like denominator subtraction works visually:
Example: \(\frac{7}{8} - \frac{3}{8} = \frac{4}{8} = \frac{1}{2}\)

Visualization: The chart demonstrates how like denominator subtraction involves only the numerators while the denominator remains constant.

  • Start with \(\frac{7}{8}\) (shaded blue)
  • Subtract \(\frac{3}{8}\) (unshaded red)
  • Result: \(\frac{4}{8} = \frac{1}{2}\) (remaining shaded area)

Worksheet Questions & Answers

Question: Why can I only subtract the numerators when fractions have like denominators? Why doesn't the denominator change?

Answer: Think of it like this: if you have 7 apples and take away 3 apples, you have 4 apples left. The "apples" part doesn't change - only the number of apples changes.

Similarly, with \(\frac{7}{8} - \frac{3}{8}\), you're dealing with eighths - the "eighths" part is like the "apples" - it stays the same. You're just counting how many eighths you have left: 7 eighths minus 3 eighths equals 4 eighths, or \(\frac{4}{8}\).

The denominator tells us what kind of parts we're working with (eighths, sixths, etc.), while the numerator tells us how many of those parts we have. When subtracting like fractions, we're only changing how many parts we have, not what kind of parts they are.

Question: How do I know when to simplify my answer? And how do I simplify fractions?

Answer: You should always check if your answer can be simplified. A fraction can be simplified when the numerator and denominator have a common factor greater than 1.

To simplify:

  • Find the Greatest Common Divisor (GCD) of the numerator and denominator
  • Divide both the numerator and denominator by the GCD
  • Repeat until the GCD is 1

For example, with \(\frac{8}{12}\): GCD(8,12) = 4, so \(\frac{8}{12} = \frac{8÷4}{12÷4} = \frac{2}{3}\). Since GCD(2,3) = 1, \(\frac{2}{3}\) is fully simplified.

Question: What are some good ways to help students visualize like denominator subtraction?

Answer: Here are effective visualization strategies:

  • Fraction bars: Draw rectangles divided into equal parts and shade the appropriate number
  • Pie charts: Show circular representations of fractions
  • Number lines: Mark positions of fractions and show the distance between them
  • Manipulatives: Use physical fraction pieces or tiles

For \(\frac{7}{8} - \frac{3}{8}\): draw a rectangle divided into 8 equal parts, shade 7 parts, then cross out 3 of the shaded parts. Students can see that 4 parts remain shaded, showing \(\frac{4}{8}\).