Solved Exercises on Subtracting Mixed Numbers with Regrouping in Grade 4

Master subtracting mixed numbers with regrouping: borrowing from whole numbers, finding common denominators, and simplifying through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic regrouping with same denominator
Exercise 1
Subtract:
\(3\frac{1}{4} - 1\frac{3}{4}\)
Definition:

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

Regrouping: Borrowing 1 whole unit from the whole number to add to the fractional part

Subtraction method with regrouping:
  1. Check if the top fraction is smaller than the bottom fraction
  2. If yes, borrow 1 from the whole number and add it to the fraction
  3. Subtract the whole numbers
  4. Subtract the fractions
  5. Simplify if needed
Original
\(3\frac{1}{4} - 1\frac{3}{4}\)
Regroup
\(2\frac{5}{4} - 1\frac{3}{4}\)
Result
\(1\frac{2}{4} = 1\frac{1}{2}\)
Step 1: Compare fractions

\(\frac{1}{4} < \frac{3}{4}\), so we need to regroup

Step 2: Regroup the first mixed number

Borrow 1 from 3: \(3 = 2 + 1\)
Convert 1 to fourths: \(1 = \frac{4}{4}\)
Add to existing fraction: \(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\)
So \(3\frac{1}{4} = 2\frac{5}{4}\)

Step 3: Subtract the whole numbers

\(2 - 1 = 1\)

Step 4: Subtract the fractions

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

Step 5: Simplify the fraction

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

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

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

Applied rules:

• Regrouping necessity: When top fraction < bottom fraction

• Borrowing: Take 1 whole unit and convert to fractional form

• Same denominators: Subtract numerators, keep denominator

2 Regrouping with different denominators
Exercise 2
Subtract:
\(4\frac{1}{3} - 2\frac{2}{5}\)
Definition:

Common denominator: A shared multiple of the denominators

LCM: Least Common Multiple - smallest number both denominators divide evenly into

Original
\(4\frac{1}{3} - 2\frac{2}{5}\)
Find LCD
LCD = 15
Compare
\(\frac{5}{15} < \frac{6}{15}\)
Result
\(1\frac{14}{15}\)
Step 1: Find LCD for denominators 3 and 5

Factors: 3=3, 5=5
LCD = 3×5 = 15

Step 2: Convert fractions to equivalent forms

\(\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\)
\(\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}\)

Step 3: Rewrite problem with common denominators

\(4\frac{5}{15} - 2\frac{6}{15}\)

Step 4: Compare fractions

\(\frac{5}{15} < \frac{6}{15}\), so we need to regroup

Step 5: Regroup the first mixed number

Borrow 1 from 4: \(4 = 3 + 1\)
Convert 1 to fifteenths: \(1 = \frac{15}{15}\)
Add to existing fraction: \(\frac{5}{15} + \frac{15}{15} = \frac{20}{15}\)
So \(4\frac{5}{15} = 3\frac{20}{15}\)

Step 6: Subtract the whole numbers

\(3 - 2 = 1\)

Step 7: Subtract the fractions

\(\frac{20}{15} - \frac{6}{15} = \frac{14}{15}\)

Step 8: Write final answer

\(1\frac{14}{15}\)

\(4\frac{1}{3} - 2\frac{2}{5} = 1\frac{14}{15}\)
Final answer:

\(4\frac{1}{3} - 2\frac{2}{5} = 1\frac{14}{15}\)

Applied rules:

• Find LCD: Before comparing fractions with different denominators

• Equivalent fractions: Multiply numerator and denominator by same number

• Regrouping: When top fraction < bottom fraction

3 Regrouping with whole number zero
Exercise 3
Subtract:
\(3 - 1\frac{2}{3}\)
Definition:

Whole number as mixed number: Any whole number can be written as a mixed number with 0 fractional part

Zero fractional part: \(3 = 3\frac{0}{3}\) when working with thirds

Original
\(3 - 1\frac{2}{3}\)
Rewrite
\(2\frac{3}{3} - 1\frac{2}{3}\)
Subtract
\(1\frac{1}{3}\)
Step 1: Rewrite whole number as mixed number

\(3 = 3\frac{0}{3}\) (using denominator 3)

Step 2: Compare fractions

\(\frac{0}{3} < \frac{2}{3}\), so we need to regroup

Step 3: Regroup the first mixed number

Borrow 1 from 3: \(3 = 2 + 1\)
Convert 1 to thirds: \(1 = \frac{3}{3}\)
Add to existing fraction: \(\frac{0}{3} + \frac{3}{3} = \frac{3}{3}\)
So \(3\frac{0}{3} = 2\frac{3}{3}\)

Step 4: Subtract the whole numbers

\(2 - 1 = 1\)

Step 5: Subtract the fractions

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

Step 6: Write final answer

\(1\frac{1}{3}\)

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

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

Applied rules:

• Whole to mixed: Any whole number can be expressed as mixed number with 0 fraction

• Regrouping: Even when starting with whole number

• Subtraction: Follow same process after regrouping

Rules and methods, laws,...
\(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \left(\frac{b-e}{c}\right)\) if \(b ≥ e\)
Mixed Number Subtraction
Rule 1
Regroup when needed
Top fraction < bottom fraction
Rule 2
Find common denominator
Before comparing fractions
Rule 3
Borrow 1 whole unit
Convert to fractional form
Key definitions:

Mixed number: Combination of a whole number and a proper fraction

Proper fraction: Fraction where numerator < denominator

Common denominator: Same denominator for both fractions

Regrouping: Borrowing 1 whole unit from whole number to add to fraction

LCM (Least Common Multiple): Smallest number both denominators divide evenly into

Complete methodology:
  1. Identify components: Separate whole numbers and fractions in each mixed number
  2. Find LCD: If denominators differ, find least common denominator
  3. Convert fractions: Change fractions to equivalent forms with common denominator
  4. Compare fractions: Check if top fraction ≥ bottom fraction
  5. Regroup if needed: Borrow 1 from whole number and add to fraction
  6. Subtract whole numbers: After possible regrouping
  7. Subtract fractions: Subtract numerators, keep denominator
  8. Simplify: Reduce fraction if possible
Tip 1: Always compare fractions BEFORE subtracting to see if regrouping is needed.
Tip 2: When regrouping, take 1 from the whole number and convert it to the same fractional parts.
Tip 3: To convert 1 whole to fractional form, use the denominator of the fraction part.
Tip 4: Remember that 1 whole = denominator/denominator (e.g., 1 = 4/4 when working with fourths).
Common errors: Forgetting to regroup when needed, subtracting denominators instead of just numerators, not finding common denominators before comparing fractions.
Memory aid: "CFR" - Compare fractions first, Regroup if needed, then subtract.
Solution: Exercises 4 to 5
4 Complex regrouping with multiple denominators
Exercise 4
Subtract:
\(5\frac{1}{6} - 2\frac{3}{4}\)
Definition:

Least Common Denominator (LCD): Smallest number both denominators divide evenly into

Original
\(5\frac{1}{6} - 2\frac{3}{4}\)
LCD
LCD = 12
Convert
\(\frac{2}{12} < \frac{9}{12}\)
Result
\(2\frac{7}{12}\)
Step 1: Find LCD for denominators 6 and 4

Factors: 6=2×3, 4=2²
LCD = 2²×3 = 12

Step 2: Convert fractions to equivalent forms

\(\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)
\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

Step 3: Rewrite problem with common denominators

\(5\frac{2}{12} - 2\frac{9}{12}\)

Step 4: Compare fractions

\(\frac{2}{12} < \frac{9}{12}\), so we need to regroup

Step 5: Regroup the first mixed number

Borrow 1 from 5: \(5 = 4 + 1\)
Convert 1 to twelfths: \(1 = \frac{12}{12}\)
Add to existing fraction: \(\frac{2}{12} + \frac{12}{12} = \frac{14}{12}\)
So \(5\frac{2}{12} = 4\frac{14}{12}\)

Step 6: Subtract the whole numbers

\(4 - 2 = 2\)

Step 7: Subtract the fractions

\(\frac{14}{12} - \frac{9}{12} = \frac{5}{12}\)

Step 8: Write final answer

\(2\frac{5}{12}\)

\(5\frac{1}{6} - 2\frac{3}{4} = 2\frac{5}{12}\)
Final answer:

\(5\frac{1}{6} - 2\frac{3}{4} = 2\frac{5}{12}\)

Applied rules:

• Find LCD: Always before comparing fractions with different denominators

• Equivalent fractions: Multiply numerator and denominator by same number

• Regrouping necessity: When top fraction < bottom fraction after conversion

• Subtraction: Subtract numerators only, keep denominator

5 Regrouping with mixed number result
Exercise 5
Subtract:
\(6\frac{1}{8} - 3\frac{5}{8}\)
Definition:

Proper fraction: A fraction where the numerator is less than the denominator

Original
\(6\frac{1}{8} - 3\frac{5}{8}\)
Regroup
\(5\frac{9}{8} - 3\frac{5}{8}\)
Subtract
\(2\frac{4}{8} = 2\frac{1}{2}\)
Step 1: Compare fractions

\(\frac{1}{8} < \frac{5}{8}\), so we need to regroup

Step 2: Regroup the first mixed number

Borrow 1 from 6: \(6 = 5 + 1\)
Convert 1 to eighths: \(1 = \frac{8}{8}\)
Add to existing fraction: \(\frac{1}{8} + \frac{8}{8} = \frac{9}{8}\)
So \(6\frac{1}{8} = 5\frac{9}{8}\)

Step 3: Subtract the whole numbers

\(5 - 3 = 2\)

Step 4: Subtract the fractions

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

Step 5: Simplify the fraction

\(\frac{4}{8} = \frac{1}{2}\) (divide both by GCD = 4)

Step 6: Write final answer

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

\(6\frac{1}{8} - 3\frac{5}{8} = 2\frac{1}{2}\)
Final answer:

\(6\frac{1}{8} - 3\frac{5}{8} = 2\frac{1}{2}\)

Applied rules:

• Regrouping: When top fraction < bottom fraction, even with same denominators

• Conversion: 1 whole = denominator/denominator

• Simplification: Reduce fraction to lowest terms by dividing by GCD

Complete Summary: Subtracting Mixed Numbers with Regrouping
\(a\frac{b}{c} - d\frac{e}{c} = (a-1-d) + \left(\frac{c+b-e}{c}\right)\) if \(b < e\)
Regrouping Formula
Key definitions:

Mixed number: A number combining 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., \(\frac{3}{4}\))

Common denominator: The same denominator shared by two or more fractions

Least Common Denominator (LCD): The smallest positive integer that is a multiple of all denominators

Regrouping: The process of borrowing 1 whole unit from the whole number and converting it to fractional form

LCM (Least Common Multiple): The smallest number that both denominators divide evenly into

Complete methodology:
  1. Analyze the problem: Identify all mixed numbers and their components
  2. Find common denominator: Determine LCD for all fraction denominators
  3. Convert fractions: Change all fractions to equivalent forms with the common denominator
  4. Compare fractions: Check if the top fraction is smaller than the bottom fraction
  5. Regroup if necessary: Borrow 1 from the whole number and add it to the fraction part
  6. Subtract whole numbers: After possible regrouping
  7. Subtract fractions: Subtract numerators while keeping the common denominator
  8. Simplify the result: Reduce the fraction to lowest terms if possible
  9. Write the final answer: In mixed number form unless otherwise specified
Tip 1: Always compare the fractions BEFORE subtracting to determine if regrouping is needed!
Tip 2: When regrouping, remember that 1 whole = denominator/denominator (e.g., 1 = 8/8 when working with eighths).
Tip 3: To find LCD, list multiples of the larger denominator until you find one that the smaller denominator divides into evenly.
Tip 4: After regrouping, the whole number decreases by 1, but the fraction increases by the denominator amount.
Tip 5: Always simplify your final fraction by dividing both numerator and denominator by their greatest common divisor.
Common errors: Forgetting to regroup when the top fraction is smaller, subtracting denominators instead of just numerators, not finding common denominators before comparing fractions, not simplifying the final fraction.
Memory aids: "CFR" - Compare fractions first, Regroup if needed, then subtract. Or "BORROW" - Borrow One from the whole number, Regroup Over the fraction, Work the subtraction.
Essential formulas and rules:

• Regrouping: \(a\frac{b}{c} = (a-1)\frac{c+b}{c}\) when \(b < e\) in subtraction

• Finding LCD: Find LCM of all denominators

• Subtracting fractions: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\) (same denominator)

• Equivalent fractions: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) for any non-zero n

• Simplification: Divide numerator and denominator by their GCD

Visual Representation: Mixed Number Subtraction
Understanding Mixed Number Subtraction
Visual representation of mixed number subtraction using area models and number lines:
\(3\frac{1}{4} - 1\frac{3}{4} = 1\frac{1}{2}\)
\(4\frac{1}{3} - 2\frac{2}{5} = 1\frac{14}{15}\)
\(3 - 1\frac{2}{3} = 1\frac{1}{3}\)

Analysis: The chart shows how mixed number subtraction works by separating whole numbers and fractions.

  • \(3\frac{1}{4} - 1\frac{3}{4} = 2\frac{5}{4} - 1\frac{3}{4} = 1\frac{2}{4} = 1\frac{1}{2}\)
  • \(4\frac{1}{3} - 2\frac{2}{5} = 4\frac{5}{15} - 2\frac{6}{15} = 3\frac{20}{15} - 2\frac{6}{15} = 1\frac{14}{15}\)
  • \(3 - 1\frac{2}{3} = 2\frac{3}{3} - 1\frac{2}{3} = 1\frac{1}{3}\)

Questions & Answers

Question: Why do I need to regroup when subtracting mixed numbers? It seems like extra work!

Answer: Great question! Think of regrouping like exchanging money. If you have $3.25 and want to spend $1.75, you can't take 75 cents away from 25 cents, right? You need to change one dollar bill into four quarters first.

In fractions, when you have \(\frac{1}{4} - \frac{3}{4}\), you can't take away more than you have! So you need to borrow 1 whole unit and convert it to the same fractional parts.

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

  • Can't do \(\frac{1}{4} - \frac{3}{4}\) because \(\frac{1}{4} < \frac{3}{4}\)
  • So we borrow 1 from 3: \(3 = 2 + 1\)
  • Convert 1 to fourths: \(1 = \frac{4}{4}\)
  • Add to existing fraction: \(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\)
  • Now we can do \(\frac{5}{4} - \frac{3}{4} = \frac{2}{4}\)

It's like borrowing from your whole number to "pay" for the subtraction!

Question: When I regroup, I don't understand why the whole number goes down by 1 but the fraction goes up. Isn't that strange?

Answer: Actually, it's perfectly logical! You're not changing the total amount, just moving it around. It's like having 3 apples and 1 orange, and then deciding to trade 1 apple for 4 oranges. You still have the same amount of fruit, just distributed differently.

When you regroup \(3\frac{1}{4}\) to \(2\frac{5}{4}\):

  • You take 1 whole unit from the 3 (so 3 becomes 2)
  • You convert that 1 whole to fractional parts (1 = \(\frac{4}{4}\))
  • You add those fractional parts to the existing fraction (\(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\))

The total amount stays the same: \(3\frac{1}{4} = \frac{13}{4}\) and \(2\frac{5}{4} = \frac{13}{4}\)

You're just rearranging the same quantity to make subtraction possible!

Question: My child is having trouble knowing when to regroup. Are there any quick checks?

Answer: Yes! Here's a simple 3-step check system:

  1. Line up the fractions: Look at the fraction parts after converting to common denominators
  2. Compare: Ask "Is the top fraction smaller than the bottom fraction?"
  3. Regroup if YES: If the top fraction is smaller, you MUST regroup

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

  • Find LCD: 15
  • Convert: \(4\frac{5}{15} - 2\frac{6}{15}\)
  • Compare: Is \(\frac{5}{15} < \frac{6}{15}\)? YES!
  • Regroup: \(4\frac{5}{15} = 3\frac{20}{15}\)

A helpful memory trick: "Small minus big? Borrow from your friend!"