Solved Exercises on Adding and Subtracting Mixed Numbers in Grade 4

Master adding and subtracting mixed numbers: converting to improper fractions, finding common denominators, borrowing, and simplifying through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Adding mixed numbers with same denominator
Exercise 1
Add: \(2\frac{1}{4} + 1\frac{2}{4}\)
Definition:

Mixed Number: A number combining a whole number and a proper fraction (e.g., \(2\frac{1}{4}\))

Addition method for same denominators:
  1. Add the whole numbers together
  2. Add the fractions together
  3. If the fraction sum is improper, convert to mixed number and add to whole part
  4. Simplify if necessary
Expression
\(2\frac{1}{4} + 1\frac{2}{4}\)
Add wholes
\(2 + 1 = 3\)
Add fractions
\(\frac{1}{4} + \frac{2}{4} = \frac{3}{4}\)
Step 1: Separate whole numbers and fractions

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

Step 2: Add whole numbers

\(2 + 1 = 3\)

Step 3: Add fractions (same denominator)

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

Step 4: Combine results

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

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

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

Applied rules:

• Same denominators: Add numerators directly, keep denominator

• Mixed number addition: Add whole parts separately from fractional parts

• Simplification: Check if final fraction can be reduced

2 Subtracting mixed numbers with same denominator
Exercise 2
Subtract: \(3\frac{3}{5} - 1\frac{2}{5}\)
Definition:

Subtraction of mixed numbers: Subtract whole numbers and fractions separately when denominators match

Expression
\(3\frac{3}{5} - 1\frac{2}{5}\)
Subtract wholes
\(3 - 1 = 2\)
Subtract fractions
\(\frac{3}{5} - \frac{2}{5} = \frac{1}{5}\)
Step 1: Separate whole numbers and fractions

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

Step 2: Subtract whole numbers

\(3 - 1 = 2\)

Step 3: Subtract fractions (same denominator)

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

Step 4: Combine results

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

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

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

Applied rules:

• Same denominators: Subtract numerators directly, keep denominator

• Mixed number subtraction: Subtract whole parts separately from fractional parts

• Borrowing rule: If fractional part of minuend is smaller, borrow from whole number

3 Adding mixed numbers with different denominators
Exercise 3
Add: \(2\frac{1}{3} + 1\frac{1}{4}\)
Definition:

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

Expression
\(2\frac{1}{3} + 1\frac{1}{4}\)
Find LCD
\(12\)
Convert fractions
\(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)
Step 1: Find Least Common Denominator (LCD)

Denominators: 3 and 4. LCM(3,4) = 12

Step 2: Convert fractions to equivalent forms

\(\frac{1}{3} = \frac{1×4}{3×4} = \frac{4}{12}\) and \(\frac{1}{4} = \frac{1×3}{4×3} = \frac{3}{12}\)

Step 3: Add whole numbers

\(2 + 1 = 3\)

Step 4: Add converted fractions

\(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)

Step 5: Combine results

\(3 + \frac{7}{12} = 3\frac{7}{12}\)

\(2\frac{1}{3} + 1\frac{1}{4} = 3\frac{7}{12}\)
Final answer:

\(2\frac{1}{3} + 1\frac{1}{4} = 3\frac{7}{12}\)

Applied rules:

• Different denominators: Must find common denominator first

• Equivalent fractions: Multiply numerator and denominator by same number

• LCM method: Find least common multiple of denominators

Rules and methods, laws,...
\(a\frac{b}{c} = a + \frac{b}{c}\)
Mixed Number Definition
Rule 1
\(a\frac{b}{c} + d\frac{e}{c} = (a+d)\frac{b+e}{c}\)
Same denominator addition
Rule 2
\(a\frac{b}{c} - d\frac{e}{c} = (a-d)\frac{b-e}{c}\)
Same denominator subtraction
Rule 3
\(a\frac{b}{c} = \frac{ac+b}{c}\)
Convert to improper fraction
Key definitions:

Proper Fraction: Numerator < denominator (e.g., \(\frac{2}{3}\))

Improper Fraction: Numerator ≥ denominator (e.g., \(\frac{5}{3}\))

Mixed Number: Whole number + proper fraction (e.g., \(2\frac{1}{3}\))

Complete methodology:
  1. Check denominators: Same or different?
  2. For same denominators: Add/subtract whole numbers and fractions separately
  3. For different denominators: Find LCD and convert fractions
  4. Handle improper fraction results: Convert back to mixed number if needed
Tip 1: Always check if the final fraction can be simplified.
Tip 2: When subtracting and the top fraction is smaller, borrow from the whole number.
Tip 3: Convert mixed numbers to improper fractions for complex calculations.
Tip 4: Use visual models (fraction bars) to understand mixed number operations.
Common errors: Forgetting to find common denominators, miscalculating LCD, incorrect borrowing.
Key formulas: \(a\frac{b}{c} = \frac{ac+b}{c}\), LCD = LCM of denominators.
Formulas to know by heart:

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

• Improper to Mixed: Divide numerator by denominator

• LCD: Find LCM of denominators

• Borrowing: Convert 1 whole to equivalent fraction

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

Borrowing in mixed numbers: When fractional part of minuend is smaller, convert 1 whole to equivalent fraction

Expression
\(3\frac{1}{4} - 1\frac{3}{4}\)
Borrow 1
\(2\frac{5}{4}\)
Subtract
\(1\frac{2}{4}\)
Step 1: Notice fractional part issue

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

Step 2: Borrow 1 whole and convert to fraction

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

Step 3: Now perform subtraction

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

Step 4: Simplify the fraction

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

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

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

Applied rules:

• Borrowing rule: Convert 1 whole to equivalent fraction with same denominator

• Subtraction: After borrowing, subtract normally

• Simplification: Reduce fractions to lowest terms

5 Complex addition with different denominators
Exercise 5
Add: \(2\frac{2}{3} + 1\frac{3}{8}\)
Definition:

Least Common Multiple (LCM): Smallest number divisible by both denominators

Expression
\(2\frac{2}{3} + 1\frac{3}{8}\)
Find LCD
\(24\)
Convert fractions
\(\frac{16}{24} + \frac{9}{24} = \frac{25}{24}\)
Step 1: Identify denominators

Denominators are 3 and 8

Step 2: Find LCM of 3 and 8

Factors of 3: {3}, Factors of 8: {2, 2, 2}. LCM = 3 × 2³ = 24

Step 3: Convert fractions to equivalent forms

\(\frac{2}{3} = \frac{2×8}{3×8} = \frac{16}{24}\) and \(\frac{3}{8} = \frac{3×3}{8×3} = \frac{9}{24}\)

Step 4: Add whole numbers

\(2 + 1 = 3\)

Step 5: Add converted fractions

\(\frac{16}{24} + \frac{9}{24} = \frac{25}{24}\)

Step 6: Convert improper fraction to mixed number

\(\frac{25}{24} = 1\frac{1}{24}\)

Step 7: Combine with whole number sum

\(3 + 1\frac{1}{24} = 4\frac{1}{24}\)

\(2\frac{2}{3} + 1\frac{3}{8} = 4\frac{1}{24}\)
Final answer:

\(2\frac{2}{3} + 1\frac{3}{8} = 4\frac{1}{24}\)

Applied rules:

• LCM method: Find least common multiple for common denominator

• Improper fraction conversion: When fraction sum ≥ 1, convert to mixed number

• Final combination: Add converted fraction to whole number sum

Mixed Numbers Laws, Methods, Rules, and Definitions
\(a\frac{b}{c} = a + \frac{b}{c}\)
Mixed Number Structure
Key definitions:

Mixed Number: A number that combines a whole number and a proper fraction (e.g., \(3\frac{1}{2}\))

Proper Fraction: A fraction where the numerator is less than the denominator (e.g., \(\frac{2}{5}\))

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \(\frac{7}{3}\))

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

Complete methodology:
  1. Identify denominators: Check if they are the same or different
  2. Same denominators: Add/subtract whole numbers and fractions separately
  3. Different denominators: Find LCD and convert fractions to equivalent forms
  4. Handle improper results: Convert to mixed numbers if needed
  5. Simplify: Reduce fractions to lowest terms
Tip 1: When subtracting mixed numbers and the top fraction is smaller, always borrow from the whole number first.
Tip 2: To find LCD quickly, multiply the denominators together if they have no common factors.
Tip 3: Always check if your final answer can be simplified by reducing the fraction part.
Tip 4: Use visual models like fraction circles or bars to understand mixed number operations.
Common errors: Forgetting to find LCD, miscalculating when borrowing, not simplifying final answers.
Key formulas: Mixed to improper: \(a\frac{b}{c} = \frac{ac+b}{c}\), LCD = LCM of denominators.
Formulas to know by heart:

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

• Converting improper to mixed: Divide numerator by denominator

• Finding LCD: Calculate LCM of denominators

• Borrowing: Convert 1 whole to equivalent fraction with same denominator

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

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

Questions & Answers

Question: I don't understand why we need to find a common denominator when adding mixed numbers with different denominators. Why can't we just add them as they are?

Answer: Great question! Think of it this way: you can't add 3 apples and 2 oranges directly because they're different types of fruit. Similarly, fractions with different denominators represent different-sized parts:

  • \(\frac{1}{3}\) means 1 part out of 3 equal parts (each part is larger)
  • \(\frac{1}{4}\) means 1 part out of 4 equal parts (each part is smaller)

To add them, we need to make the parts the same size. When we find the common denominator (like 12), we're saying: "Let's cut everything into pieces of the same size." Then \(\frac{1}{3} = \frac{4}{12}\) and \(\frac{1}{4} = \frac{3}{12}\), so now we're adding same-sized pieces: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).

Example: \(2\frac{1}{3} + 1\frac{1}{4}\) becomes \(2\frac{4}{12} + 1\frac{3}{12} = 3\frac{7}{12}\).

Question: When I subtract mixed numbers like \(3\frac{1}{4} - 1\frac{3}{4}\), I get confused about borrowing. How do I know when to borrow and how to do it?

Answer: You need to borrow when the fractional part of the first number (minuend) is smaller than the fractional part of the second number (subtrahend). In your example:

  • You have \(\frac{1}{4}\) and need to subtract \(\frac{3}{4}\)
  • Since \(\frac{1}{4} < \frac{3}{4}\), you can't subtract directly
  • You "borrow" 1 from the whole number (3 becomes 2)
  • Convert that borrowed 1 into the same denominator: \(1 = \frac{4}{4}\)
  • Add it to the existing fraction: \(\frac{1}{4} + \frac{4}{4} = \frac{5}{4}\)

Now you have \(2\frac{5}{4} - 1\frac{3}{4} = 1\frac{2}{4} = 1\frac{1}{2}\).

Think of it like money: if you have $3.25 and want to spend $1.75, you can't take 75¢ from 25¢, so you change one dollar bill into quarters.

Question: Sometimes when I add mixed numbers, my fraction part becomes bigger than 1, like getting \(2\frac{5}{3}\). What do I do with that?

Answer: When your fraction part is an improper fraction (numerator ≥ denominator), you need to convert it to a mixed number and add it to the whole number part. For example:

  • If you get \(2\frac{5}{3}\), you notice that \(\frac{5}{3}\) is an improper fraction
  • Convert \(\frac{5}{3}\) to a mixed number: \(5 ÷ 3 = 1\) remainder \(2\), so \(\frac{5}{3} = 1\frac{2}{3}\)
  • Add this to the whole number: \(2 + 1\frac{2}{3} = 3\frac{2}{3}\)

This happens when you add fractions and their sum equals or exceeds 1 whole. It's completely normal and just requires one extra step to express the answer properly.

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