Solved Exercises on Subtracting Mixed Numbers in Grade 4

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

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

Mixed number: A whole number combined with a proper fraction (e.g., \(3\frac{1}{4}\))

Improper fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \(\frac{13}{4}\))

Subtraction method:
  1. Convert mixed numbers to improper fractions
  2. If necessary, borrow from the whole number part
  3. Subtract the fractions
  4. Simplify the result
Expression
\(3\frac{1}{4} - 1\frac{3}{4}\)
Convert
\(\frac{13}{4} - \frac{7}{4}\)
Result
\(\frac{6}{4} = 1\frac{1}{2}\)
Step 1: Convert mixed numbers to improper fractions

\(3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}\)

\(1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}\)

Step 2: Subtract the fractions

\(\frac{13}{4} - \frac{7}{4} = \frac{13-7}{4} = \frac{6}{4}\)

Step 3: Simplify the fraction

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

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

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

Applied rules:

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

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

• Simplification: Divide numerator and denominator by their GCD

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

Borrowing: Taking 1 from the whole number and adding it to the fraction part as the denominator

Expression
\(4\frac{1}{3} - 2\frac{2}{3}\)
Borrow
\(3\frac{4}{3} - 2\frac{2}{3}\)
Result
\(1\frac{2}{3}\)
Step 1: Check if borrowing is needed

\(\frac{1}{3} < \frac{2}{3}\), so we need to borrow from the whole number

Step 2: Borrow 1 from the whole number

\(4\frac{1}{3} = 3 + 1 + \frac{1}{3} = 3 + \frac{3}{3} + \frac{1}{3} = 3\frac{4}{3}\)

Step 3: Subtract the fractions

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

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

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

Applied rules:

• Borrowing: When fraction part is smaller, borrow 1 (which equals denominator/denominator)

• Subtraction: Subtract whole numbers and fractions separately

• Verification: Check that the result makes sense

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

Least Common Denominator (LCD): The smallest number that both denominators divide evenly into

Expression
\(5\frac{1}{2} - 2\frac{1}{3}\)
LCD
\(5\frac{3}{6} - 2\frac{2}{6}\)
Result
\(3\frac{1}{6}\)
Step 1: Find the LCD of denominators

Denominators: 2 and 3. LCD = 6

Step 2: Convert fractions to equivalent fractions with LCD

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

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

Step 3: Rewrite mixed numbers with common denominators

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

Step 4: Subtract the mixed numbers

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

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

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

Applied rules:

• LCD: Find the least common multiple of denominators

• Equivalent fractions: Multiply numerator and denominator by the same number

• Subtraction: Only possible after getting common denominators

Rules and methods, laws,...
\(a\frac{b}{c} = \frac{ac + b}{c}\)
Mixed to Improper Conversion
Rule 1
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Same denominator subtraction
Rule 2
\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
Mixed number subtraction (same denominator)
Rule 3
\(\frac{a}{b} = \frac{ac}{bc}\)
Equivalent fractions
Key definitions:

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

Improper Fraction: A fraction where numerator ≥ denominator

Borrowing: Taking 1 from the whole number and adding it as the denominator to the fraction

Complete methodology:
  1. Identify denominators: Check if they are the same or different
  2. Find LCD: If denominators differ, find the least common denominator
  3. Convert: Change fractions to equivalent forms with common denominators
  4. Check fraction parts: If the first fraction is smaller, borrow from the whole number
  5. Subtract: Subtract whole numbers and fractions separately
  6. Simplify: Reduce the fraction part to lowest terms
Tip 1: Always convert mixed numbers to improper fractions first for easier calculation.
Tip 2: When borrowing, remember that 1 whole = denominator/denominator.
Tip 3: Always simplify your final answer by reducing fractions.
Tip 4: Check your answer by adding the result back to the subtracted number.
Common errors: Forgetting to borrow when needed, not finding LCD for different denominators, not simplifying results.
Exam preparation: Practice borrowing scenarios, memorize conversion formulas, work with various denominators.
Formulas to know by heart:

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

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

• Equivalent fractions: \(\frac{a}{b} = \frac{ac}{bc}\)

• Borrowing: \(a\frac{b}{c} = (a-1)\frac{b+c}{c}\) when borrowing 1

Solution: Exercises 4 to 5
4 Complex borrowing
Exercise 4
Subtract: \(6\frac{1}{5} - 3\frac{4}{5}\)
Definition:

Complex borrowing: When the fractional part of the minuend is smaller than the subtrahend, borrowing is required

Expression
\(6\frac{1}{5} - 3\frac{4}{5}\)
Borrow
\(5\frac{6}{5} - 3\frac{4}{5}\)
Result
\(2\frac{2}{5}\)
Step 1: Compare fractional parts

\(\frac{1}{5} < \frac{4}{5}\), so borrowing is needed

Step 2: Borrow 1 from the whole number

Take 1 from 6 (making it 5) and add it to the fraction: \(\frac{1}{5} + \frac{5}{5} = \frac{6}{5}\)

So \(6\frac{1}{5} = 5\frac{6}{5}\)

Step 3: Subtract the mixed numbers

\(5\frac{6}{5} - 3\frac{4}{5} = (5-3) + (\frac{6}{5} - \frac{4}{5}) = 2 + \frac{2}{5} = 2\frac{2}{5}\)

\(6\frac{1}{5} - 3\frac{4}{5} = 2\frac{2}{5}\)
Final answer:

\(6\frac{1}{5} - 3\frac{4}{5} = 2\frac{2}{5}\)

Applied rules:

• Borrowing: When \(\frac{a}{c} < \frac{b}{c}\) in \(x\frac{a}{c} - y\frac{b}{c}\), borrow 1

• Conversion: \(1 = \frac{c}{c}\) when denominator is c

• Subtraction: Perform after borrowing to make fraction subtraction possible

5 Multiple denominators
Exercise 5
Subtract: \(7\frac{2}{3} - 4\frac{5}{6}\)
Definition:

Multiple denominators: When subtracting mixed numbers with different denominators, find LCD first

Expression
\(7\frac{2}{3} - 4\frac{5}{6}\)
LCD
\(7\frac{4}{6} - 4\frac{5}{6}\)
Borrow
\(6\frac{10}{6} - 4\frac{5}{6}\)
Result
\(2\frac{5}{6}\)
Step 1: Find the LCD of denominators

Denominators: 3 and 6. LCD = 6

Step 2: Convert fractions to equivalent fractions

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

\(\frac{5}{6}\) stays the same

Step 3: Rewrite with common denominators

\(7\frac{2}{3} = 7\frac{4}{6}\) and \(4\frac{5}{6} = 4\frac{5}{6}\)

Step 4: Compare fractional parts and borrow if needed

\(\frac{4}{6} < \frac{5}{6}\), so borrow 1 from 7

\(7\frac{4}{6} = 6\frac{4+6}{6} = 6\frac{10}{6}\)

Step 5: Subtract the mixed numbers

\(6\frac{10}{6} - 4\frac{5}{6} = (6-4) + (\frac{10}{6} - \frac{5}{6}) = 2 + \frac{5}{6} = 2\frac{5}{6}\)

\(7\frac{2}{3} - 4\frac{5}{6} = 2\frac{5}{6}\)
Final answer:

\(7\frac{2}{3} - 4\frac{5}{6} = 2\frac{5}{6}\)

Applied rules:

• LCD: Always find the least common denominator first

• Conversion: Convert to equivalent fractions with common denominators

• Borrowing: May still be needed even after finding LCD

Key Rules, Methods, and Definitions for Subtracting Mixed Numbers
\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
Mixed Number Subtraction
Key definitions:

Mixed Number: A number that combines a whole number and a proper fraction (e.g., \(3\frac{1}{4}\))

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \(\frac{13}{4}\))

Borrowing: The process of taking 1 from the whole number and adding it to the fraction part as the denominator

Least Common Denominator (LCD): The smallest number that all denominators divide evenly into

Complete methodology:
  1. Analyze the problem: Look at the denominators and fractional parts
  2. Find LCD if needed: When denominators differ, find the least common denominator
  3. Convert fractions: Change all fractions to equivalent forms with common denominators
  4. Check for borrowing: If the first fraction is smaller than the second, borrow from the whole number
  5. Perform subtraction: Subtract whole numbers and fractions separately
  6. Simplify result: Reduce fractions to lowest terms and convert improper fractions back to mixed numbers if needed
  7. Verify answer: Check by adding the result to the subtracted number
Tip 1: Always check if the fractional part of the first number is larger than the second before subtracting.
Tip 2: When borrowing, remember that 1 whole = denominator/denominator (e.g., 1 = 3/3).
Tip 3: To find LCD, list multiples of each denominator and find the smallest common one.
Tip 4: Always simplify your final answer by reducing fractions to lowest terms.
Tip 5: Verify your answer by adding the result back to the number you subtracted.
Common errors: Forgetting to borrow when needed, not finding LCD for different denominators, not simplifying results, incorrect conversion between mixed numbers and improper fractions.
Exam preparation: Practice problems with different denominators, master the borrowing technique, memorize conversion formulas, work on simplification skills.
Formulas to know by heart:

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

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

• Equivalent fractions: \(\frac{a}{b} = \frac{ac}{bc}\)

• Borrowing: \(a\frac{b}{c} = (a-1)\frac{b+c}{c}\) when borrowing 1

• Mixed number subtraction: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)

Questions & Answers

Question: I don't understand why we sometimes need to borrow when subtracting mixed numbers. Can you explain when and why this happens?

Answer: Great question! Borrowing is needed when the fractional part of the first number (minuend) is smaller than the fractional part of the second number (subtrahend). For example, in \(4\frac{1}{3} - 2\frac{2}{3}\), we have \(\frac{1}{3} < \frac{2}{3}\).

Here's why we borrow:

  • We cannot subtract a larger fraction from a smaller one
  • So we take 1 from the whole number part (4 becomes 3)
  • We add that 1 to the fraction part: \(1 = \frac{3}{3}\), so \(\frac{1}{3} + \frac{3}{3} = \frac{4}{3}\)
  • Now we have \(3\frac{4}{3} - 2\frac{2}{3}\), and we can subtract: \(\frac{4}{3} - \frac{2}{3} = \frac{2}{3}\)

Think of it like having $4.33 and needing to pay $2.67. Since 33 cents isn't enough to pay 67 cents, you "borrow" $1 from your dollar amount, turning it into $3.99 + $0.34 = $3.133, which allows you to pay the 67 cents.

Question: How do I find the least common denominator (LCD) quickly? It takes me too long to figure it out.

Answer: Here are some quick strategies to find the LCD:

  • If one denominator divides the other: The larger number is the LCD. Example: denominators 3 and 6 → LCD = 6
  • For small numbers: List multiples of the larger denominator until you find one divisible by the smaller. Example: 4 and 6 → multiples of 6: 6, 12... 12 ÷ 4 = 3, so LCD = 12
  • For prime numbers: Multiply them together. Example: denominators 3 and 5 → LCD = 15
  • Memorize common LCDs: 2 & 4 = 4, 2 & 3 = 6, 3 & 4 = 12, 4 & 6 = 12

Example: For denominators 4 and 6:

Multiples of 6: 6, 12, 18...

Does 4 divide 6? No. Does 4 divide 12? Yes! So LCD = 12.

This way you don't need to list all multiples, just check if the larger denominator's multiples are divisible by the smaller one.

Question: My child gets confused about converting between mixed numbers and improper fractions. Is there an easy way to remember the process?

Answer: Absolutely! Here's a simple memory trick for both conversions:

Mixed to Improper (M → I):

  • Remember "Multiply, Add, Keep": Multiply the whole number by the denominator, add the numerator, keep the same denominator
  • Formula: \(a\frac{b}{c} = \frac{a \times c + b}{c}\)
  • Example: \(3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}\)

Improper to Mixed (I → M):

  • Divide the numerator by the denominator
  • Quotient = whole number, remainder = new numerator, denominator stays the same
  • Example: \(\frac{17}{5}\) → 17 ÷ 5 = 3 remainder 2 → \(3\frac{2}{5}\)

You can also visualize it: \(3\frac{2}{5}\) means 3 whole items plus 2 pieces out of 5 from another item. That's 3×5=15 pieces from the wholes, plus 2 more pieces, totaling 17 pieces of size 1/5 each, or \(\frac{17}{5}\).