Solved Exercises on Subtracting Mixed Numbers for Grade 4

Master subtracting mixed numbers: borrowing, regrouping, and simplification through these 3 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 whole number combined with a fraction, like \(3\frac{3}{4}\)

Subtraction method:
  1. Subtract the whole numbers
  2. Subtract the fractions
  3. Combine the results
  4. Simplify if needed
Expression
\(3\frac{3}{4} - 1\frac{1}{4}\)
Subtract Parts
\(2 + \frac{2}{4}\)
Result
\(2\frac{1}{2}\)
Step 1: Subtract the whole numbers

\(3 - 1 = 2\)

Step 2: Subtract the fractions

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

Step 3: Combine the results

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

Step 4: Simplify the fraction

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

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

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

Applied rules:

• Separate subtraction: Subtract whole numbers and fractions separately

• Like denominators: Subtract numerators, keep denominator

• Simplification: Reduce fractions to lowest terms

2 Subtraction with borrowing
Exercise 2
Subtract:
\(4\frac{1}{4} - 2\frac{3}{4}\)
Definition:

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

Expression
\(4\frac{1}{4} - 2\frac{3}{4}\)
Borrowing
\(3\frac{5}{4} - 2\frac{3}{4}\)
Result
\(1\frac{2}{4} = 1\frac{1}{2}\)
Step 1: Check if borrowing is needed

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

Borrowing process:

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

Step 2: Subtract the whole numbers and fractions separately

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

Step 3: Simplify the fraction

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

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

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

Applied rules:

• Borrowing: When fractional part is smaller, borrow 1 from whole number

• Conversion: Convert borrowed 1 to fraction with same denominator

• Subtraction: Subtract whole numbers and fractions separately

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

Common denominator: Same bottom number in fractions, needed for addition/subtraction

Expression
\(5\frac{1}{3} - 2\frac{1}{6}\)
Common Denominator
\(5\frac{2}{6} - 2\frac{1}{6}\)
Result
\(3\frac{1}{6}\)
Step 1: Find the least common denominator (LCD)

The LCD of 3 and 6 is 6

Step 2: Convert fractions to equivalent fractions

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

Step 3: Rewrite the expression with common denominator

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

Step 4: Subtract the whole numbers and fractions

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

Step 5: Write the final answer

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

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

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

Applied rules:

• Common denominator: Find LCD before subtracting fractions

• Equivalent fractions: Multiply numerator and denominator by same number

• Subtraction: Subtract whole numbers and fractions separately

Key Concepts and Methods
\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
Subtracting Mixed Numbers
Key definitions:

Mixed Number: A combination of a whole number and a 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: Taking 1 from the whole number and converting it to fraction form

Common Denominator: Same bottom number in fractions, required for addition/subtraction

Complete methodology:
  1. Check denominators: If different, find the least common denominator (LCD)
  2. Compare fractions: If the top fraction is smaller, borrow from the whole number
  3. Convert if needed: Change the borrowed whole number to fraction form
  4. Subtract: Subtract whole numbers and fractions separately
  5. Simplify: Reduce the fraction to lowest terms
Tip 1: Always check if the fractional part of the first number is smaller than the second. If so, you'll need to borrow.
Tip 2: When borrowing, convert the whole number to the same denominator as the existing fraction.
Tip 3: Remember to simplify fractions to lowest terms in your final answer.
Tip 4: You can verify your answer by adding the result to the subtracted number to get back the original.
Common errors: Forgetting to borrow when needed, not finding common denominators, incorrect fraction simplification.
Practice tips: Work with visual models like fraction bars, practice converting between mixed numbers and improper fractions, check answers by addition.
Key formulas and rules:

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

• Subtraction with same denominator: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)

• Common denominator: Multiply numerator and denominator by the same number to create equivalent fractions

• Simplification: Divide numerator and denominator by their greatest common factor

Detailed Summary
Core Concepts:

• Mixed Number: A whole number combined with a fraction

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

• Common Denominator: Required for subtracting fractions

Step-by-Step Process:
  1. Check if denominators are the same
  2. Find LCD if denominators differ
  3. Compare fractional parts
  4. Borrow if needed
  5. Subtract whole numbers
  6. Subtract fractions
  7. Simplify final fraction
Examples: Simple: \(3\frac{3}{4} - 1\frac{1}{4} = 2\frac{1}{2}\), Advanced: \(4\frac{1}{4} - 2\frac{3}{4} = 1\frac{1}{2}\)
Common Mistakes: Forgetting to borrow when fractional part is smaller, not finding common denominators, subtracting denominators instead of keeping them
Memory Tip: "Check fractions first, borrow if smaller!" - Remember to always compare fractional parts before subtracting.
Visual Examples and Properties
\(a\frac{b}{c} = \frac{ac+b}{c}\)
Mixed to Improper
Borrowing Example
\(4\frac{1}{4} = 3\frac{5}{4}\)
After borrowing 1
Common Denominator
\(\frac{1}{3} = \frac{2}{6}\)
Multiply by 2/2
Simplification
\(\frac{2}{4} = \frac{1}{2}\)
Divide by 2/2
Subtraction Process
📊
Step 1: Check denominators

If denominators are different, find the LCD and convert fractions

\(\frac{1}{3} \to \frac{2}{6}\) (multiply by 2/2)
Step 2: Compare fractions

If top fraction is smaller, borrow 1 from the whole number

\(4\frac{1}{4} \to 3\frac{5}{4}\) (borrow 1 = 4/4)
Step 3: Subtract parts

Subtract whole numbers and fractions separately

\((a-d) + \frac{b-e}{c}\)
Step 4: Simplify

Reduce fraction to lowest terms if possible

\(\frac{2}{4} = \frac{1}{2}\)
Visual Process
1
Check Denominators
2
Borrow if Needed
3
Subtract Parts
4
Simplify
Fraction Bar Model
Visual representation showing how mixed numbers represent parts of wholes
Practice Makes Perfect!
Frequently Asked Questions

Question: Why do I need to borrow when subtracting mixed numbers? Can't I just subtract the fractions directly?

Answer: You need to borrow when the fractional part of the first number is smaller than the fractional part of the second number. For example, in \(4\frac{1}{4} - 2\frac{3}{4}\), you can't subtract \(\frac{3}{4}\) from \(\frac{1}{4}\) because \(\frac{1}{4}\) is smaller.

Borrowing converts one whole unit into fraction form to make the top fraction larger. In our example: \(4\frac{1}{4}\) becomes \(3\frac{5}{4}\) (since \(1 = \frac{4}{4}\), so \(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\)). Now you can subtract: \(3\frac{5}{4} - 2\frac{3}{4} = 1\frac{2}{4} = 1\frac{1}{2}\).

Question: How do I know when to find a common denominator when subtracting mixed numbers?

Answer: You must find a common denominator whenever the fractions in your mixed numbers have different denominators. You can only add or subtract fractions when they have the same denominator.

For example, in \(5\frac{1}{3} - 2\frac{1}{6}\), the denominators are 3 and 6. The least common denominator (LCD) is 6. Convert \(\frac{1}{3}\) to \(\frac{2}{6}\) by multiplying both numerator and denominator by 2. Then you can subtract: \(5\frac{2}{6} - 2\frac{1}{6} = 3\frac{1}{6}\).

Question: I sometimes get confused about simplifying fractions. How do I know if I've simplified correctly?

Answer: A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify:

  • Find the greatest common factor (GCF) of the numerator and denominator
  • Divide both the numerator and denominator by the GCF
  • Repeat until no common factors remain

For example, to simplify \(\frac{6}{8}\): The GCF of 6 and 8 is 2. Divide both by 2: \(\frac{6÷2}{8÷2} = \frac{3}{4}\). Since 3 and 4 have no common factors other than 1, \(\frac{3}{4}\) is fully simplified.