Solved Exercises on Subtracting Fractions with Different Denominators in Grade 4

Master subtracting fractions with different denominators: finding common denominators, creating equivalent fractions, and simplifying results through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic subtraction
Exercise 1
Subtract: \(\frac{3}{4} - \frac{1}{2}\)
Definition:

Equivalent Fractions: Fractions that represent the same value (e.g., \(\frac{1}{2} = \frac{2}{4}\))

Subtraction method:
  1. Find the Least Common Denominator (LCD) of both fractions
  2. Convert each fraction to an equivalent fraction with the LCD
  3. Subtract the numerators while keeping the common denominator
  4. Simplify the result if possible
Original Expression
\(\frac{3}{4} - \frac{1}{2}\)
Find LCD
\(LCD(4,2) = 4\)
Convert Fractions
\(\frac{3}{4} - \frac{2}{4}\)
Final Result
\(\frac{1}{4}\)
Step 1: Find the LCD of denominators 4 and 2

Multiples of 4: 4, 8, 12...
Multiples of 2: 2, 4, 6, 8...
LCD = 4

Step 2: Convert \(\frac{1}{2}\) to equivalent fraction with denominator 4

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

Step 3: Subtract numerators with common denominator

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

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

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

Applied rules:

• Common Denominator: Must have same denominator to subtract fractions

• Equivalent Fractions: Multiply numerator and denominator by same number

• Subtraction: Only subtract numerators, keep denominator unchanged

2 Larger denominators
Exercise 2
Subtract: \(\frac{5}{6} - \frac{1}{3}\)
Definition:

Least Common Multiple (LCM): The smallest positive integer that is divisible by both numbers

Original Expression
\(\frac{5}{6} - \frac{1}{3}\)
Find LCD
\(LCD(6,3) = 6\)
Convert Fractions
\(\frac{5}{6} - \frac{2}{6}\)
Final Result
\(\frac{3}{6} = \frac{1}{2}\)
Step 1: Find the LCD of denominators 6 and 3

Multiples of 6: 6, 12, 18...
Multiples of 3: 3, 6, 9, 12...
LCD = 6

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

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

Step 3: Subtract numerators with common denominator

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

Step 4: Simplify the fraction

\(\frac{3}{6} = \frac{1}{2}\) (divide both by GCD of 3)

\(\frac{5}{6} - \frac{1}{3} = \frac{1}{2}\)
Final answer:

\(\frac{5}{6} - \frac{1}{3} = \frac{1}{2}\)

Applied rules:

• LCM Finding: List multiples to find smallest common multiple

• Equivalent Conversion: Multiply numerator and denominator by same factor

• Simplification: Divide numerator and denominator by their GCD

3 Borrowing needed
Exercise 3
Subtract: \(\frac{1}{3} - \frac{1}{4}\)
Definition:

Positive Result: When subtracting fractions, ensure the result makes sense (positive values in context)

Original Expression
\(\frac{1}{3} - \frac{1}{4}\)
Find LCD
\(LCD(3,4) = 12\)
Convert Fractions
\(\frac{4}{12} - \frac{3}{12}\)
Final Result
\(\frac{1}{12}\)
Step 1: Find the LCD of denominators 3 and 4

Multiples of 3: 3, 6, 9, 12...
Multiples of 4: 4, 8, 12, 16...
LCD = 12

Step 2: Convert both fractions to equivalent with denominator 12

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

Step 3: Subtract numerators with common denominator

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

\(\frac{1}{3} - \frac{1}{4} = \frac{1}{12}\)
Final answer:

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

Applied rules:

• Compare Values: \(\frac{1}{3} > \frac{1}{4}\) so result should be positive

• Common Denominator: Essential for fraction comparison/subtraction

• Subtraction Process: Numerator subtraction with same denominator

Rules and methods, laws,...
\(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)
General Subtraction Formula
Rule 1
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Same Denominator Subtraction
Rule 2
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Equivalent Fraction Rule
Rule 3
\(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)
Cross-Multiplication Method
Key definitions:

Denominator: The bottom number in a fraction (shows how many equal parts the whole is divided into)

Numerator: The top number in a fraction (shows how many parts we have)

Common Denominator: The same denominator for both fractions

Equivalent Fractions: Fractions that represent the same value

Complete methodology:
  1. Identify denominators: Look at the denominators of both fractions
  2. Find LCD: Calculate the least common denominator
  3. Convert fractions: Change both fractions to equivalent forms with LCD
  4. Subtract numerators: Keep denominator the same
  5. Simplify: Reduce the result to lowest terms
Tip 1: Always check that your answer is positive when subtracting a smaller fraction from a larger one.
Tip 2: To find LCD quickly, start with the larger denominator and see if it's divisible by the smaller.
Tip 3: Draw fraction bars to visualize the subtraction process.
Tip 4: Remember to multiply both numerator and denominator by the same number when creating equivalent fractions.
Common errors: Forgetting to find LCD, subtracting denominators, not converting to equivalent fractions.
Exam preparation: Practice LCD finding, master equivalent fraction creation, review simplification.
Formulas to know by heart:

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

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

• Cross Multiplication: \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)

• Simplification: Divide by GCD of numerator and denominator

Exercise with Visualization: Fraction Subtraction
Exercise 6: Fraction Comparison
Consider the following fraction subtractions:
\(f_1(x) = \frac{3}{4} - x\)
\(f_2(x) = \frac{5}{6} - x\)
\(f_3(x) = \frac{2}{3} - x\)

Analysis: The chart shows how different fractions behave when subtracted from constants.

  • \(f_1(x) = \frac{3}{4} - x = 0.75 - x\) (Decimal equivalent)
  • \(f_2(x) = \frac{5}{6} - x = 0.833... - x\) (Fraction to decimal)
  • \(f_3(x) = \frac{2}{3} - x = 0.666... - x\) (Repeating decimal)
Solution: Exercises 4 to 5
4 Complex denominators
Exercise 4
Subtract: \(\frac{7}{8} - \frac{2}{3}\)
Definition:

Least Common Denominator (LCD): Smallest number divisible by both denominators

Original Expression
\(\frac{7}{8} - \frac{2}{3}\)
Find LCD
\(LCD(8,3) = 24\)
Convert Fractions
\(\frac{21}{24} - \frac{16}{24}\)
Final Result
\(\frac{5}{24}\)
Step 1: Find the LCD of denominators 8 and 3

Multiples of 8: 8, 16, 24, 32...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...
LCD = 24

Step 2: Convert \(\frac{7}{8}\) to equivalent fraction with denominator 24

\(\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}\)

Step 3: Convert \(\frac{2}{3}\) to equivalent fraction with denominator 24

\(\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\)

Step 4: Subtract numerators with common denominator

\(\frac{21}{24} - \frac{16}{24} = \frac{21-16}{24} = \frac{5}{24}\)

Step 5: Check if the result can be simplified

GCD(5,24) = 1, so \(\frac{5}{24}\) is already in simplest form

\(\frac{7}{8} - \frac{2}{3} = \frac{5}{24}\)
Final answer:

\(\frac{7}{8} - \frac{2}{3} = \frac{5}{24}\)

Applied rules:

• LCM Finding: List multiples systematically to find LCD

• Conversion: Multiply both numerator and denominator by same factor

• Simplification Check: Verify if GCD of numerator and denominator is 1

5 Multiple operations
Exercise 5
Subtract: \(\frac{3}{4} - \frac{1}{6} - \frac{1}{8}\)
Definition:

Multiple Subtraction: When subtracting more than two fractions, find LCD for all denominators

Original Expression
\(\frac{3}{4} - \frac{1}{6} - \frac{1}{8}\)
Find LCD
\(LCD(4,6,8) = 24\)
Convert All
\(\frac{18}{24} - \frac{4}{24} - \frac{3}{24}\)
Final Result
\(\frac{11}{24}\)
Step 1: Find LCD for all denominators (4, 6, 8)

Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Multiples of 8: 8, 16, 24...
LCD = 24

Step 2: Convert all fractions to LCD

\(\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}\)
\(\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}\)
\(\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}\)

Step 3: Perform the subtraction in order

\(\frac{18}{24} - \frac{4}{24} - \frac{3}{24} = \frac{18-4-3}{24} = \frac{11}{24}\)

Step 4: Check if the result can be simplified

GCD(11,24) = 1, so \(\frac{11}{24}\) is already in simplest form

\(\frac{3}{4} - \frac{1}{6} - \frac{1}{8} = \frac{11}{24}\)
Final answer:

\(\frac{3}{4} - \frac{1}{6} - \frac{1}{8} = \frac{11}{24}\)

Applied rules:

• Multiple LCD: Find LCD for all denominators simultaneously

• Sequential Operations: Perform operations from left to right

• Simplification: Check for common factors in numerator and denominator

Detailed Summary: Subtracting Fractions with Different Denominators
\(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)
Cross-Multiplication Formula
Key definitions:

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

Denominator: The bottom number indicating how many equal parts the whole is divided into

Numerator: The top number indicating how many parts we have

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

Least Common Denominator (LCD): The smallest common multiple of the denominators

Complete methodology:
  1. Analyze the fractions: Identify numerators and denominators
  2. Check denominators: Determine if LCD is needed
  3. Find LCD: Calculate the least common denominator
  4. Create equivalents: Convert both fractions to equivalent forms with LCD
  5. Subtract numerators: Keep denominator the same
  6. Simplify result: Reduce to lowest terms if possible
  7. Verify: Check if answer makes logical sense
Tip 1: When subtracting fractions, the result is positive only when the first fraction is larger than the second.
Tip 2: To find LCD quickly, try multiplying the denominators together, then divide by their GCD.
Tip 3: Draw visual models (like pie charts or fraction bars) to understand the subtraction conceptually.
Tip 4: Always verify your answer by adding the result back to the subtracted fraction to see if you get the original.
Common errors: Forgetting to find LCD, subtracting denominators, not converting to equivalent fractions, incorrect simplification.
Memory aids: "Same bottom, subtract tops" (only for same denominators) and "Find the bottom that fits both."
Formulas to know by heart:

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

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

• Cross Multiplication: \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)

• Simplification: \(\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}\)

Exercise with Visualization: Fraction Subtraction
Exercise 6: Fraction Comparison
Consider the following fraction subtractions:
\(f_1(x) = \frac{3}{4} - x\)
\(f_2(x) = \frac{5}{6} - x\)
\(f_3(x) = \frac{2}{3} - x\)

Analysis: The chart shows how different fractions behave when subtracted from constants.

  • \(f_1(x) = \frac{3}{4} - x = 0.75 - x\) (Decimal equivalent)
  • \(f_2(x) = \frac{5}{6} - x = 0.833... - x\) (Fraction to decimal)
  • \(f_3(x) = \frac{2}{3} - x = 0.666... - x\) (Repeating decimal)

Questions & Answers

Question: Why do I need to find a common denominator when subtracting fractions? Can't I just subtract the numerators and denominators separately?

Answer: Great question! You cannot subtract numerators and denominators separately because fractions represent parts of a whole, and those parts must be the same size:

  • Think of fractions like slices of pizza. \(\frac{1}{4}\) represents 1 slice of a pizza cut into 4 pieces, while \(\frac{1}{3}\) represents 1 slice of a pizza cut into 3 pieces. The slices are different sizes!
  • You can only subtract slices that are the same size. So \(\frac{3}{4} - \frac{1}{4}\) works because both fractions refer to quarters of the same whole.
  • When denominators are different, you need to cut both pizzas into the same number of pieces (find LCD) so the slices become the same size.

Example: \(\frac{1}{2} - \frac{1}{4}\) cannot be calculated as \(\frac{1-1}{2-4} = \frac{0}{-2}\) because that doesn't make sense. Instead, convert to \(\frac{2}{4} - \frac{1}{4} = \frac{1}{4}\).

Question: I have trouble finding the LCD quickly. Is there an easier method than listing all the multiples?

Answer: Yes! Here are several efficient methods for finding LCD:

  • Start with larger denominator: Begin with the larger of the denominators and check if it's divisible by the smaller one.
  • Prime factorization: Break each denominator into prime factors and take the highest power of each prime.
  • Product method: Multiply the denominators together, then divide by their GCD.

Examples:

  • LCD(4,6): Start with 6, check if divisible by 4? No. Try 6×2=12, is 12÷4=3? Yes! LCD=12
  • LCD(3,5): Since 3 and 5 are both prime, LCD=3×5=15
  • LCD(6,9): Prime factors: 6=2×3, 9=3². Take highest powers: 2¹×3²=18

Practice with small numbers first to build confidence!

Question: How do I know if my answer is correct when subtracting fractions?

Answer: Here are effective verification methods:

  • Visual estimation: Compare the sizes of the fractions. The result should make logical sense.
  • Addition check: Add your answer back to the subtracted fraction. You should get the original fraction.
  • Convert to decimals: Convert all fractions to decimals and perform the operation to verify.

Example: \(\frac{3}{4} - \frac{1}{2} = \frac{1}{4}\)

  • Estimation: \(\frac{3}{4} > \frac{1}{2}\), so result should be positive ✓
  • Addition check: \(\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\) ✓
  • Decimal check: 0.75 - 0.50 = 0.25 = \(\frac{1}{4}\) ✓

These methods help catch common errors!

Question: What happens if I subtract a larger fraction from a smaller fraction? Will I get a negative answer?

Answer: Yes, when you subtract a larger fraction from a smaller fraction, you will get a negative result:

  • This is mathematically correct and represents a value less than zero.
  • For example: \(\frac{1}{4} - \frac{1}{2} = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4}\)
  • Negative fractions mean "owe" or "less than zero" in practical applications.

However, in grade 4, you typically work with positive results. If you encounter a situation where the first fraction is smaller, it might indicate you need to reverse the problem or reconsider the context.

Example: If you have \(\frac{1}{4}\) of a pizza and eat \(\frac{1}{2}\) of a pizza, you can't do that with just \(\frac{1}{4}\) pizza. This is why the answer is negative.

Question: What is the best strategy when subtracting more than two fractions at once?

Answer: Here's a systematic approach for multiple fraction subtraction:

  1. Find common denominator: Calculate LCD for ALL denominators at once
  2. Convert all fractions: Change each fraction to equivalent with the LCD
  3. Perform operations: Subtract numerators from left to right
  4. Keep denominator: Use the common denominator throughout
  5. Simplify result: Reduce to lowest terms if possible

Example: \(\frac{3}{4} - \frac{1}{6} - \frac{1}{8}\)

  • LCD(4,6,8) = 24
  • Convert: \(\frac{18}{24} - \frac{4}{24} - \frac{3}{24}\)
  • Subtract: \(\frac{18-4-3}{24} = \frac{11}{24}\)

Work systematically to avoid missing any terms or making calculation errors!