Solved Exercises on Subtracting Mixed Numbers in Grade 4

Master subtracting mixed numbers: converting to improper fractions, borrowing when needed, and simplifying results through these 5 detailed exercises.

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

Mixed Number: A number consisting of a whole number and a proper fraction (e.g., \(3\frac{3}{4}\))

Subtraction method:
  1. Subtract the whole numbers
  2. Subtract the fractions
  3. If the fraction part of the first number is smaller, borrow from the whole number
  4. Simplify the result if possible
Original Expression
\(3\frac{3}{4} - 1\frac{1}{4}\)
Separate Parts
\((3 - 1) + (\frac{3}{4} - \frac{1}{4})\)
Final Result
\(2\frac{2}{4} = 2\frac{1}{2}\)
Step 1: Subtract whole numbers

\(3 - 1 = 2\)

Step 2: Subtract fractions (same denominator)

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

Step 3: Simplify the fraction

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

Step 4: Combine results

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

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

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

Applied rules:

• Same Denominator: When denominators are equal, subtract numerators only

• Whole Numbers: Subtract separately from fractions

• Combining: Combine whole number and fraction parts

2 Borrowing needed
Exercise 2
Subtract: \(2\frac{1}{4} - 1\frac{3}{4}\)
Definition:

Borrowing: Taking 1 whole from the whole number part and converting it to fraction form

Original Expression
\(2\frac{1}{4} - 1\frac{3}{4}\)
Borrow from 2
\(1\frac{5}{4} - 1\frac{3}{4}\)
Subtract Parts
\((1-1) + (\frac{5}{4}-\frac{3}{4})\)
Final Result
\(\frac{2}{4} = \frac{1}{2}\)
Step 1: Check if borrowing is needed

\(\frac{1}{4} < \frac{3}{4}\), so we need to borrow from the whole number

Step 2: Borrow 1 from the whole number

Take 1 from 2, leaving 1. Convert 1 to \(\frac{4}{4}\)

\(2\frac{1}{4} = 1 + \frac{4}{4} + \frac{1}{4} = 1\frac{5}{4}\)

Step 3: Now subtract with sufficient fraction

\(1\frac{5}{4} - 1\frac{3}{4} = (1-1) + (\frac{5}{4}-\frac{3}{4}) = 0 + \frac{2}{4}\)

Step 4: Simplify the fraction

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

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

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

Applied rules:

• Borrowing: When fraction part is too small, borrow 1 whole

• Conversion: 1 whole = denominator in fraction form

• Subtraction: Now perform subtraction with sufficient fraction

3 Different denominators
Exercise 3
Subtract: \(3\frac{1}{3} - 1\frac{1}{6}\)
Definition:

Common Denominator: The least common multiple of denominators

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

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

Step 2: Convert fractions to LCD

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

Step 3: Rewrite with common denominator

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

Step 4: Subtract whole numbers and fractions

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

Step 5: Combine results

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

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

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

Applied rules:

• Find LCD: Lowest common denominator for subtraction

• Equivalent Fractions: Convert to same denominator

• Subtraction: Subtract whole numbers and fractions separately

Rules and methods, laws,...
\(a\frac{b}{c} - d\frac{e}{f} = (a-d) + \left(\frac{b}{c} - \frac{e}{f}\right)\)
Mixed Number Subtraction Formula
Rule 1
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Same Denominator Subtraction
Rule 2
\(a\frac{b}{c} = \frac{ac + b}{c}\)
Mixed to Improper Conversion
Rule 3
\(1 = \frac{c}{c}\)
Borrowing Rule
Key definitions:

Mixed Number: A combination of a whole number and a proper fraction

Improper Fraction: A fraction where numerator ≥ denominator

Common Denominator: Same denominator for fraction subtraction

Borrowing: Taking 1 whole and converting to fraction form when needed

Complete methodology:
  1. Check denominators: If same, proceed directly
  2. Find LCD: If different, find lowest common denominator
  3. Convert fractions: Change to equivalent fractions with LCD
  4. Check fraction subtraction: If first fraction < second fraction, borrow
  5. Subtract separately: Subtract whole numbers and fractions independently
  6. Simplify: Reduce final fraction if possible
Tip 1: Always check if the fraction part of the first number is larger than the second before subtracting.
Tip 2: When borrowing, remember that 1 whole equals the denominator in fraction form.
Tip 3: Use fraction bars or circles to visualize the subtraction process.
Tip 4: Practice finding LCD by listing multiples of each denominator.
Common errors: Forgetting to borrow, subtracting denominators, not finding LCD, not simplifying results.
Exam preparation: Master LCD finding, practice borrowing, review conversion between mixed and improper fractions.
Formulas to know by heart:

• Mixed to Improper: \(a\frac{b}{c} = \frac{ac + b}{c}\)

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

• Borrowing: When needed, convert 1 whole to \(\frac{c}{c}\)

• Subtraction Steps: Separate whole and fraction parts, then combine

Exercise with Visualization: Mixed Number Operations
Exercise 6: Mixed Number Comparison
Consider the following mixed number subtractions:
\(f_1(x) = x - 1\frac{1}{2}\)
\(f_2(x) = x - 2\frac{1}{4}\)
\(f_3(x) = x - 1\frac{3}{4}\)

Analysis: The chart shows how mixed numbers behave differently when subtracted from variables.

  • \(f_1(x) = x - 1\frac{1}{2} = x - 1.5\) (Decimal equivalent)
  • \(f_2(x) = x - 2\frac{1}{4} = x - 2.25\) (Mixed number subtraction)
  • \(f_3(x) = x - 1\frac{3}{4} = x - 1.75\) (Fraction simplification)
Solution: Exercises 4 to 5
4 Complex borrowing
Exercise 4
Subtract: \(5\frac{1}{8} - 2\frac{5}{8}\)
Definition:

Complex Borrowing: When the fraction part of the first number is smaller than the second

Original Expression
\(5\frac{1}{8} - 2\frac{5}{8}\)
Borrow from 5
\(4\frac{9}{8} - 2\frac{5}{8}\)
Subtract Parts
\((4-2) + (\frac{9}{8}-\frac{5}{8})\)
Final Result
\(2\frac{4}{8} = 2\frac{1}{2}\)
Step 1: Check if borrowing is needed

\(\frac{1}{8} < \frac{5}{8}\), so we need to borrow from the whole number

Step 2: Borrow 1 from the whole number

Take 1 from 5, leaving 4. Convert 1 to \(\frac{8}{8}\)

\(5\frac{1}{8} = 4 + \frac{8}{8} + \frac{1}{8} = 4\frac{9}{8}\)

Step 3: Now subtract with sufficient fraction

\(4\frac{9}{8} - 2\frac{5}{8} = (4-2) + (\frac{9}{8}-\frac{5}{8}) = 2 + \frac{4}{8}\)

Step 4: Simplify the fraction

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

Step 5: Combine results

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

\(5\frac{1}{8} - 2\frac{5}{8} = 2\frac{1}{2}\)
Final answer:

\(5\frac{1}{8} - 2\frac{5}{8} = 2\frac{1}{2}\)

Applied rules:

• Need to Borrow: When \(\frac{1}{8} < \frac{5}{8}\), borrowing is required

• Conversion: 1 whole = \(\frac{8}{8}\) in this case

• Subtraction: Now perform subtraction with sufficient fraction

5 Multiple mixed numbers
Exercise 5
Subtract: \(4\frac{1}{2} - 1\frac{1}{3} - 1\frac{1}{6}\)
Definition:

Multiple Subtraction: Subtracting more than two mixed numbers requires common denominator for all

Original Expression
\(4\frac{1}{2} - 1\frac{1}{3} - 1\frac{1}{6}\)
Find LCD
\(LCD(2,3,6) = 6\)
Convert All
\(4\frac{3}{6} - 1\frac{2}{6} - 1\frac{1}{6}\)
Final Result
\(2\frac{0}{6} = 2\)
Step 1: Find LCD for all denominators (2, 3, 6)

Multiples of 2: 2, 4, 6, 8...
Multiples of 3: 3, 6, 9...
Multiples of 6: 6, 12...
LCD = 6

Step 2: Convert all fractions to LCD

\(\frac{1}{2} = \frac{3}{6}\) (multiply by \(\frac{3}{3}\))
\(\frac{1}{3} = \frac{2}{6}\) (multiply by \(\frac{2}{2}\))
\(\frac{1}{6} = \frac{1}{6}\) (already has LCD)

Step 3: Rewrite with common denominator

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

Step 4: Subtract all whole numbers

\(4 - 1 - 1 = 2\)

Step 5: Subtract all fractions

\(\frac{3}{6} - \frac{2}{6} - \frac{1}{6} = \frac{3-2-1}{6} = \frac{0}{6} = 0\)

Step 6: Combine results

\(2 + 0 = 2\)

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

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

Applied rules:

• Multiple LCD: Find LCD for all denominators simultaneously

• Sequential Subtraction: Subtract all whole numbers, then all fractions

• Zero Fraction: When fraction result is 0, only whole number remains

Detailed Summary: Subtracting Mixed Numbers
\(a\frac{b}{c} - d\frac{e}{f} = (a-d) + \left(\frac{bf}{cf} - \frac{ec}{cf}\right)\)
General Subtraction Formula
Key definitions:

Mixed Number: A number expressed as a whole number plus a proper fraction

Proper Fraction: A fraction where numerator < denominator

Improper Fraction: A fraction where numerator ≥ denominator

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

Borrowing: Taking 1 whole from the whole number part and converting to fraction form

Complete methodology:
  1. Analyze the problem: Identify whole numbers and fractional parts
  2. Check denominators: Determine if LCD is needed
  3. Find LCD: Calculate the least common denominator
  4. Convert fractions: Change to equivalent fractions with LCD
  5. Check fraction subtraction: If first fraction < second fraction, borrow
  6. Subtract separately: Subtract whole numbers and fractions independently
  7. Simplify: Reduce final fraction if possible
Tip 1: When subtracting multiple mixed numbers, convert all fractions to the same denominator first.
Tip 2: To find LCD quickly, list multiples of the largest denominator until you find one divisible by others.
Tip 3: Always simplify your final answer by reducing fractions to lowest terms.
Tip 4: Check your work by converting to improper fractions and back.
Common errors: Forgetting to borrow when needed, subtracting denominators instead of numerators, not finding LCD, not converting improper fraction results.
Memory aids: "Same denominator, subtract numerators only" and "Check if first fraction is big enough before subtracting."
Formulas to know by heart:

• Mixed to Improper: \(a\frac{b}{c} = \frac{ac + b}{c}\)

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

• Mixed Number Subtraction: \(a\frac{b}{c} - d\frac{e}{c} = (a-d)\frac{b-e}{c}\) (if b≥e)

• Borrowing: When b

Exercise with Visualization: Mixed Number Operations
Exercise 6: Mixed Number Comparison
Consider the following mixed number subtractions:
\(f_1(x) = x - 1\frac{1}{2}\)
\(f_2(x) = x - 2\frac{1}{4}\)
\(f_3(x) = x - 1\frac{3}{4}\)

Analysis: The chart shows how mixed numbers behave differently when subtracted from variables.

  • \(f_1(x) = x - 1\frac{1}{2} = x - 1.5\) (Decimal equivalent)
  • \(f_2(x) = x - 2\frac{1}{4} = x - 2.25\) (Mixed number subtraction)
  • \(f_3(x) = x - 1\frac{3}{4} = x - 1.75\) (Fraction simplification)

Questions & Answers

Question: When do I need to borrow when subtracting mixed numbers? I'm confused about when this is necessary.

Answer: You need to borrow when the fraction part of the first mixed number is smaller than the fraction part of the second mixed number:

  • You cannot subtract a larger fraction from a smaller fraction (e.g., \(\frac{1}{4} - \frac{3}{4}\) is impossible without borrowing).
  • Example: In \(2\frac{1}{4} - 1\frac{3}{4}\), since \(\frac{1}{4} < \frac{3}{4}\), you need to borrow.
  • To borrow: Take 1 whole from the whole number part and convert it to fraction form with the same denominator.

Think of it like money: If you have $2.25 and want to spend $1.75, you need to break the $2 bill into smaller pieces to make the transaction work.

After borrowing: \(2\frac{1}{4}\) becomes \(1\frac{5}{4}\) (since 1 whole = \(\frac{4}{4}\)), so now you can subtract: \(1\frac{5}{4} - 1\frac{3}{4} = \frac{2}{4} = \frac{1}{2}\).

Question: How do I know what to convert the whole number to when borrowing? Like in \(3\frac{1}{5} - 1\frac{3}{5}\), what do I change the 3 to?

Answer: When borrowing, convert 1 whole to a fraction with the same denominator as your fractional part:

  • 1 whole always equals the denominator in fraction form: \(1 = \frac{5}{5}\), \(1 = \frac{3}{3}\), \(1 = \frac{8}{8}\), etc.
  • So in \(3\frac{1}{5}\), when you borrow 1 from the 3, you convert it to \(\frac{5}{5}\).
  • The 3 becomes 2, and you add \(\frac{5}{5}\) to \(\frac{1}{5}\) to get \(\frac{6}{5}\).

Example: \(3\frac{1}{5} - 1\frac{3}{5}\)

  • Borrow 1 from 3: \(3\frac{1}{5} = 2 + 1 + \frac{1}{5} = 2 + \frac{5}{5} + \frac{1}{5} = 2\frac{6}{5}\)
  • Now subtract: \(2\frac{6}{5} - 1\frac{3}{5} = (2-1) + (\frac{6}{5}-\frac{3}{5}) = 1 + \frac{3}{5} = 1\frac{3}{5}\)

Remember: 1 whole = denominator in fraction form!

Question: I have trouble finding the LCD quickly. Is there a faster method than listing multiples?

Answer: Yes! Here are some 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.
  • Multiply and reduce: 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 can I check if my answer is correct when subtracting mixed numbers?

Answer: Here are effective verification methods:

  • Estimation: Round mixed numbers to nearest whole numbers and subtract. Your answer should be close to this estimate.
  • Addition check: Add your answer back to the subtracted number. You should get the original number.
  • Convert to improper fractions: Convert all mixed numbers to improper fractions, subtract, then convert back.

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

  • Estimation: \(4 - 1 = 3\), so answer should be close to 2.5
  • Addition check: \(2\frac{1}{2} + 1\frac{1}{4} = 3\frac{3}{4}\) ✓
  • Improper fractions: \(\frac{15}{4} - \frac{5}{4} = \frac{10}{4} = 2\frac{1}{2}\) ✓

These methods help catch common errors!

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

Answer: Here's a systematic approach for multiple mixed number 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. Subtract whole numbers: Perform all whole number subtractions
  4. Subtract fractions: Perform all fraction subtractions
  5. Combine results: Add the whole number result to the fraction result
  6. Simplify: Reduce final fraction if possible

Example: \(4\frac{1}{2} - 1\frac{1}{3} - 1\frac{1}{6}\)

  • LCD(2,3,6) = 6
  • Convert: \(4\frac{3}{6} - 1\frac{2}{6} - 1\frac{1}{6}\)
  • Whole numbers: \(4 - 1 - 1 = 2\)
  • Fractions: \(\frac{3}{6} - \frac{2}{6} - \frac{1}{6} = \frac{0}{6} = 0\)
  • Final answer: \(2 + 0 = 2\)

Work systematically to avoid missing any terms!