Solved Exercises on Adding Mixed Numbers with Complete Answers for Grade 4

Master adding mixed numbers with complete answers: proper fractions, regrouping, and simplification through these 3 detailed exercises.

Complete Solutions: Exercises 1 to 3
1 Basic addition
Exercise 1
Add:
\(2\frac{1}{4} + 1\frac{2}{4}\)
Definition:

Mixed number: A whole number combined with a fraction, like \(2\frac{1}{4}\)

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

\(2 + 1 = 3\)

Step 2: Add the fractions

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

Step 3: Combine the results

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

Step 4: Check if simplification is needed

\(\frac{3}{4}\) is already in lowest terms

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

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

Applied rules:

• Separate addition: Add whole numbers and fractions separately

• Like denominators: Add numerators, keep denominator

• Combination: Combine whole number and fraction results

2 Addition with regrouping
Exercise 2
Add:
\(3\frac{3}{4} + 2\frac{2}{4}\)
Definition:

Regrouping: Converting an improper fraction result into a mixed number and adding to the whole number

Expression
\(3\frac{3}{4} + 2\frac{2}{4}\)
Add Parts
\(5 + \frac{5}{4}\)
Regroup
\(6\frac{1}{4}\)
Step 1: Add the whole numbers

\(3 + 2 = 5\)

Step 2: Add the fractions

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

Step 3: Check if fraction is improper

\(\frac{5}{4} > 1\), so we need to regroup

Regrouping process:

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

Step 4: Add the regrouped amount to the whole number

\(5 + 1\frac{1}{4} = 6\frac{1}{4}\)

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

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

Applied rules:

• Separate addition: Add whole numbers and fractions separately

• Improper fraction: When numerator ≥ denominator, convert to mixed number

• Regrouping: Add the converted whole number to the existing whole number

3 Different denominators
Exercise 3
Add:
\(1\frac{1}{3} + 2\frac{1}{6}\)
Definition:

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

Expression
\(1\frac{1}{3} + 2\frac{1}{6}\)
Common Denominator
\(1\frac{2}{6} + 2\frac{1}{6}\)
Result
\(3\frac{3}{6} = 3\frac{1}{2}\)
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

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

Step 4: Add the whole numbers and fractions separately

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

Step 5: Simplify the fraction

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

Step 6: Write the final answer

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

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

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

Applied rules:

• Common denominator: Find LCD before adding fractions

• Equivalent fractions: Multiply numerator and denominator by same number

• Separate addition: Add whole numbers and fractions separately

Key Concepts and Methods
\(a\frac{b}{c} + d\frac{e}{c} = (a+d) + \frac{b+e}{c}\)
Adding 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., \(\frac{13}{4}\))

Regrouping: Converting an improper fraction result into a mixed number and adding to the whole number

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. Convert fractions: Change fractions to equivalent forms with common denominator
  3. Add whole numbers: Sum the whole number parts
  4. Add fractions: Sum the fractional parts
  5. Check for regrouping: If fraction is improper, convert to mixed number
  6. Combine results: Add any regrouped amounts to the whole number
  7. Simplify: Reduce the fraction to lowest terms
Tip 1: Always check if the fractional part of your answer is an improper fraction (numerator ≥ denominator). If so, you need to regroup.
Tip 2: When regrouping, divide the numerator by the denominator to find the whole number part.
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 regroup when fraction is improper, 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 estimation.
Key formulas and rules:

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

• Regrouping: If \(\frac{b+e}{c} > 1\), convert to mixed number and add to whole number

• 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

• Regrouping: Converting improper fraction results to mixed numbers

• Common Denominator: Required for adding fractions

Step-by-Step Process:
  1. Check if denominators are the same
  2. Find LCD if denominators differ
  3. Add whole numbers together
  4. Add fractions together
  5. Check if fraction result is improper
  6. Regroup if needed
  7. Simplify final fraction
Examples: Simple: \(2\frac{1}{4} + 1\frac{2}{4} = 3\frac{3}{4}\), Advanced: \(3\frac{3}{4} + 2\frac{2}{4} = 6\frac{1}{4}\)
Common Mistakes: Forgetting to regroup when fraction is improper, not finding common denominators, adding denominators instead of keeping them
Memory Tip: "Add whole numbers, add fractions, check for regrouping!" - Remember to always check if your fraction result needs regrouping.
Visual Examples and Properties
\(a\frac{b}{c} = \frac{ac+b}{c}\)
Mixed to Improper
Regrouping Example
\(\frac{5}{4} = 1\frac{1}{4}\)
Improper to mixed
Common Denominator
\(\frac{1}{3} = \frac{2}{6}\)
Multiply by 2/2
Simplification
\(\frac{3}{6} = \frac{1}{2}\)
Divide by 3/3
Addition 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: Add whole numbers

Sum all the whole number parts

\(a + d = (a+d)\)
Step 3: Add fractions

Sum all the fractional parts

\(\frac{b}{c} + \frac{e}{c} = \frac{b+e}{c}\)
Step 4: Check for regrouping

If fraction is improper (numerator ≥ denominator), convert to mixed number

\(\frac{5}{4} = 1\frac{1}{4}\)
Step 5: Simplify

Reduce fraction to lowest terms if possible

\(\frac{3}{6} = \frac{1}{2}\)
Visual Process
1
Check Denominators
2
Add Wholes
3
Add Fractions
4
Regroup if Needed
5
Simplify
Fraction Bar Model
Visual representation showing how mixed numbers represent parts of wholes
Practice Makes Perfect!
Frequently Asked Questions

Question: When do I need to regroup after adding mixed numbers? Can you explain this concept?

Answer: You need to regroup when the sum of the fractional parts creates an improper fraction (where the numerator is greater than or equal to the denominator). For example, in \(3\frac{3}{4} + 2\frac{2}{4}\), the fractional parts add up to \(\frac{5}{4}\), which is improper because 5 is greater than 4.

To regroup, divide the numerator by the denominator: \(5 ÷ 4 = 1\) remainder \(1\). So \(\frac{5}{4} = 1\frac{1}{4}\). Add this 1 to the whole number sum (5) to get 6, giving you the final answer of \(6\frac{1}{4}\).

Question: How do I know when to find a common denominator when adding 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 \(1\frac{1}{3} + 2\frac{1}{6}\), 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 add: \(1\frac{2}{6} + 2\frac{1}{6} = 3\frac{3}{6} = 3\frac{1}{2}\).

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.