Subtracting Mixed Numbers Practice 12-9 for Grade 4

Practice set 12-9 covering definitions, rules, step-by-step methods, examples, and exercises for subtracting mixed numbers.

Lesson Content: Subtracting Mixed Numbers Practice 12-9
1 Introduction
Definition:

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

Proper Fraction: A fraction where the numerator is less than the denominator (e.g., \(\frac{3}{4}\)).

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 adding it to the fraction part as the denominator.

Key Rule:

When subtracting mixed numbers, subtract the whole numbers and fractions separately. If the fraction part of the minuend is smaller than the subtrahend, borrow from the whole number.

\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\) (when \(b \geq e\))
Subtraction of Mixed Numbers
Simple Example

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

Why This Works:

Think of having 5 whole pizzas and 3 slices, and you eat 2 whole pizzas and 1 slice. You keep 3 whole pizzas and 2 slices.

2 Step-by-Step Process
Steps to Subtract Mixed Numbers:
  1. Check denominators: Make sure both fractions have the same denominator
  2. Compare fractions: If the first fraction is smaller than the second, borrow from the whole number
  3. Borrow if needed: Take 1 from the whole number and add it to the fraction
  4. Subtract whole numbers: Subtract the whole number parts
  5. Subtract fractions: Subtract the fractional parts
  6. Simplify: Reduce the fraction to its simplest form
Step-by-Step Example

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

Step 1: Check denominators

Both fractions have denominator 3, so they have like denominators ✓

Step 2: Compare fractions

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

Step 3: Borrow from the whole number

Take 1 from 4 (making it 3) and add it to the fraction: \(\frac{1}{3} + \frac{3}{3} = \frac{4}{3}\)

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

Step 4: Subtract whole numbers

\(3 - 2 = 1\)

Step 5: Subtract fractions

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

Step 6: Write the answer

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

3 More Examples
Example 1: No Borrowing Needed

\(7\frac{5}{6} - 3\frac{2}{6} = (7-3) + (\frac{5}{6} - \frac{2}{6}) = 4 + \frac{3}{6} = 4\frac{1}{2}\)

Example 2: Borrowing Needed

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

Since \(\frac{1}{4} < \frac{3}{4}\), we need to borrow:

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

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

Example 3: Different Denominators

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

First, find LCD: 2 and 3 have LCD = 6

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

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

Tip 1: Always check if the first fraction is smaller than the second before subtracting.
Tip 2: When borrowing, remember that 1 whole = denominator/denominator.
Definitions, Rules, and Key Concepts
Key Definitions:

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

Minuend: The number being subtracted from (the first number in a subtraction problem).

Subtrahend: The number being subtracted (the second number in a subtraction problem).

Difference: The result of a subtraction problem.

Like Denominators: Fractions that have the same denominator.

Unlike Denominators: Fractions that have different denominators.

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

Core Rules:

• Subtraction Rule: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\) (when \(b \geq e\))

• Borrowing Rule: When \(b < e\) in \(a\frac{b}{c} - d\frac{e}{c}\), borrow 1 from \(a\)

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

• Simplification: Always reduce fractions to their lowest terms

\(a\frac{b}{c} = \frac{ac + b}{c}\)
Mixed to Improper Conversion
Visual Model: Imagine you have 4 pizzas cut into 3 pieces each. If you have 4\(\frac{1}{3}\) of these pizzas, that's 4 whole pizzas plus 1 piece out of 3 from another pizza. This equals 13 pieces of size 1/3 each.
Real-world Connection: Subtracting mixed numbers is used in cooking (measuring ingredients), construction (measuring lengths), and many other practical applications.
Memory Aid: "Borrow when small from big" - if the first fraction is smaller than the second, you need to borrow from the whole number.
Check Your Work: Add your answer back to the subtrahend to see if you get the minuend.
Practice Problems and Solutions
Problem Set 1
Problem 1

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

Solution:

Both fractions have the same denominator (5), and \(\frac{3}{5} > \frac{1}{5}\), so no borrowing needed.

Subtract whole numbers: \(5 - 2 = 3\)

Subtract fractions: \(\frac{3}{5} - \frac{1}{5} = \frac{2}{5}\)

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

Problem 2

\(7\frac{2}{3} - 4\frac{1}{3} = ?\)

Solution:

Both fractions have the same denominator (3), and \(\frac{2}{3} > \frac{1}{3}\), so no borrowing needed.

Subtract whole numbers: \(7 - 4 = 3\)

Subtract fractions: \(\frac{2}{3} - \frac{1}{3} = \frac{1}{3}\)

\(7\frac{2}{3} - 4\frac{1}{3} = 3\frac{1}{3}\)

Problem Set 2
Problem 3

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

Solution:

Both fractions have the same denominator (4), but \(\frac{1}{4} < \frac{3}{4}\), so borrowing is needed.

Borrow 1 from 6: \(6\frac{1}{4} = 5\frac{5}{4}\) (since \(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\))

Subtract whole numbers: \(5 - 3 = 2\)

Subtract fractions: \(\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}\)

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

Problem 4

\(8\frac{1}{6} - 5\frac{4}{6} = ?\)

Solution:

Both fractions have the same denominator (6), but \(\frac{1}{6} < \frac{4}{6}\), so borrowing is needed.

Borrow 1 from 8: \(8\frac{1}{6} = 7\frac{7}{6}\) (since \(\frac{1}{6} + \frac{6}{6} = \frac{7}{6}\))

Subtract whole numbers: \(7 - 5 = 2\)

Subtract fractions: \(\frac{7}{6} - \frac{4}{6} = \frac{3}{6} = \frac{1}{2}\)

\(8\frac{1}{6} - 5\frac{4}{6} = 2\frac{1}{2}\)

Problem Set 3
Problem 5

\(9\frac{1}{2} - 6\frac{1}{3} = ?\)

Solution:

Denominators are different (2 and 3), so find LCD first.

LCD of 2 and 3 is 6.

Convert: \(9\frac{1}{2} = 9\frac{3}{6}\) and \(6\frac{1}{3} = 6\frac{2}{6}\)

Now subtract: \(9\frac{3}{6} - 6\frac{2}{6} = (9-6) + (\frac{3}{6} - \frac{2}{6}) = 3 + \frac{1}{6} = 3\frac{1}{6}\)

Advanced Concepts and Applications
\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\) (when \(b \geq e\))
Subtraction of Mixed Numbers
Advanced Definitions:

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

Greatest Common Factor (GCF): The largest number that divides two or more numbers evenly.

Simplifying Fractions: Reducing a fraction to its lowest terms by dividing both numerator and denominator by their GCF.

Converting Mixed Numbers to Improper Fractions:

To convert a mixed number to an improper fraction:

  1. Multiply the whole number by the denominator
  2. Add the numerator
  3. Place the result over the original denominator

Example: \(3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}\)

Tip 1: When subtracting mixed numbers with different denominators, always find the LCD first.
Tip 2: Always simplify your final answer by reducing fractions to lowest terms.
Tip 3: Check if your answer makes sense by estimating the result.
Tip 4: Remember that borrowing changes the whole number by 1 and adds the denominator to the numerator of the fraction part.
Common Mistakes: Forgetting to borrow when needed, not finding LCD for different denominators, not simplifying the final answer, subtracting the wrong order (subtrahend minus minuend instead of minuend minus subtrahend).
Application: Subtracting mixed numbers is used in cooking (adjusting recipes), construction (measuring materials), crafts (cutting fabric), and many other practical situations where measurements need to be compared or reduced.
Important Reminders:

• Always check if borrowing is needed before subtracting fractions

• Find LCD when denominators are different

• Simplify your final answer when possible

• Verify your answer by adding it back to the subtrahend

Questions & Answers

Question: Why do I need to borrow when subtracting mixed numbers? It seems confusing.

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

Think of it like this: You have $4.33 and need 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.

Similarly, in \(4\frac{1}{3}\), we borrow 1 whole (which equals \(\frac{3}{3}\)) and add it to \(\frac{1}{3}\), giving us \(\frac{4}{3}\). Now we can subtract: \(\frac{4}{3} - \frac{2}{3} = \frac{2}{3}\).

Question: How can I help my child understand when to find the LCD for subtracting mixed numbers?

Answer: Teach your child to first look at the denominators of the fractions in the mixed numbers. If the denominators are different, they must find the LCD before subtracting.

Here's a simple approach:

1. Look at the denominators (bottom numbers) of both fractions

2. If they're the same, you can subtract directly

3. If they're different, find the LCD

4. Convert both fractions to equivalent fractions with the LCD

5. Then subtract

Example: In \(3\frac{1}{4} - 1\frac{1}{3}\), denominators are 4 and 3, so LCD = 12. Convert to \(3\frac{3}{12} - 1\frac{4}{12}\).

Question: What are some good visual models to help students understand subtracting mixed numbers?

Answer: Visual models are excellent for understanding mixed number subtraction! Here are effective approaches:

Circle Models (Pizza/Cake): Draw circles divided into equal parts. Show the first number with shaded parts, then "erase" the parts being subtracted.

Rectangle Models (Chocolate Bar): Draw rectangles divided into equal sections. Color the parts representing the first number, then cross out the parts being subtracted.

Number Lines: Mark the starting mixed number, then move backward by the amount of the second number.

Pattern Blocks: Use manipulatives where hexagons represent wholes, triangles represent sixths, etc.

For \(4\frac{1}{3} - 2\frac{2}{3}\), show 4 whole shapes plus 1 part out of 3, then remove 2 whole shapes plus 2 parts out of 3 (with borrowing needed).