Mixed Numbers
When you eat 2 full chapatis and half of a third one, you have eaten 2½ chapatis. This is neither a whole number (2) nor a simple fraction (½). It is a combination of both — a mixed number.
A mixed number (also called a mixed fraction) has a whole number part and a fractional part written together. For example, 3¼ means 3 whole parts and 1/4 of another part.
Mixed numbers are closely connected to improper fractions. An improper fraction has a numerator greater than or equal to the denominator (like 7/4). Every mixed number can be converted to an improper fraction and vice versa.
In NCERT Class 6, mixed numbers are studied in the chapter on Fractions. Learning to convert between mixed numbers and improper fractions is an important skill for adding and subtracting fractions in later chapters.
What is Mixed Numbers?
Definition: A mixed number is a number that has a whole number part and a proper fraction part.
Form:
Mixed Number = Whole Number + Proper Fraction
Examples:
- 2½ = 2 + ½ (two and a half)
- 3¼ = 3 + ¼ (three and a quarter)
- 5⅔ = 5 + ⅔ (five and two-thirds)
Key terms:
- Whole number part: The number of complete units (must be at least 1).
- Fraction part: The remaining portion, which is always a proper fraction (numerator < denominator).
- Improper fraction: A fraction where the numerator ≥ denominator (like 5/3, 7/2, 11/4).
Important:
- The fractional part of a mixed number is ALWAYS a proper fraction (less than 1).
- Every mixed number is greater than 1.
- Every mixed number has an equivalent improper fraction.
Mixed Numbers Formula
Converting Mixed Number to Improper Fraction:
Improper Fraction = (Whole × Denominator + Numerator) / Denominator
For a mixed number like a b/c:
- Improper fraction = (a × c + b) / c
Example: Convert 3¼ to an improper fraction:
- = (3 × 4 + 1) / 4 = 13/4
Converting Improper Fraction to Mixed Number:
Divide the numerator by the denominator. Quotient = whole part, Remainder = numerator of fraction part.
For an improper fraction p/q:
- Divide: p ÷ q → Quotient = a, Remainder = b
- Mixed number = a b/q
Example: Convert 17/5 to a mixed number:
- 17 ÷ 5 = Quotient 3, Remainder 2
- Mixed number = 3⅖
Types and Properties
Types of problems involving mixed numbers:
1. Converting mixed numbers to improper fractions:
- Use the formula: (whole × denominator + numerator) / denominator.
- Example: 2⅗ = (2 × 5 + 3)/5 = 13/5.
2. Converting improper fractions to mixed numbers:
- Divide numerator by denominator. Quotient is the whole part; remainder is the new numerator.
- Example: 11/4 → 11 ÷ 4 = 2 remainder 3 → 2¾.
3. Comparing mixed numbers:
- First compare the whole number parts. If equal, compare the fraction parts.
- Example: 3½ vs 3¼ → whole parts are same (3), compare ½ and ¼. Since ½ > ¼, we get 3½ > 3¼.
4. Mixed numbers on the number line:
- A mixed number lies between two consecutive whole numbers.
- Example: 2¾ lies between 2 and 3.
5. Addition and subtraction of mixed numbers:
- Convert to improper fractions first, then add/subtract, and convert back to a mixed number.
Solved Examples
Example 1: Example 1: Converting mixed number to improper fraction
Problem: Convert 4⅔ to an improper fraction.
Solution:
Given:
- Mixed number: 4⅔
- Whole part = 4, Numerator = 2, Denominator = 3
Using the formula:
- Improper fraction = (4 × 3 + 2) / 3
- = (12 + 2) / 3
- = 14/3
Answer: 4⅔ = 14/3.
Example 2: Example 2: Converting improper fraction to mixed number
Problem: Convert 23/5 to a mixed number.
Solution:
Step 1: Divide 23 by 5:
- 23 ÷ 5 = Quotient 4, Remainder 3
Step 2: Write as a mixed number:
- Whole part = 4
- Fraction part = 3/5
- Mixed number = 4⅗
Verification: 4⅗ = (4 × 5 + 3)/5 = 23/5 ✓
Answer: 23/5 = 4⅗.
Example 3: Example 3: Converting when remainder is zero
Problem: Convert 15/3 to a mixed number.
Solution:
Divide 15 by 3:
- 15 ÷ 3 = Quotient 5, Remainder 0
Since the remainder is 0:
- 15/3 is not a mixed number — it is a whole number.
- 15/3 = 5
Answer: 15/3 = 5 (a whole number, not a mixed number).
Example 4: Example 4: Identifying mixed numbers
Problem: Which of the following are mixed numbers? (a) 3/7 (b) 2⅕ (c) 9/4 (d) 6⅜ (e) 5
Solution:
- (a) 3/7 — This is a proper fraction (numerator < denominator). Not a mixed number.
- (b) 2⅕ — Has a whole part (2) and a fraction part (1/5). Yes, mixed number.
- (c) 9/4 — This is an improper fraction (numerator > denominator). Not a mixed number (but can be converted to one: 2¼).
- (d) 6⅜ — Has a whole part (6) and a fraction part (3/8). Yes, mixed number.
- (e) 5 — This is a whole number. Not a mixed number.
Answer: (b) 2⅕ and (d) 6⅜ are mixed numbers.
Example 5: Example 5: Mixed number with large whole part
Problem: Convert 12¾ to an improper fraction.
Solution:
Using the formula:
- = (12 × 4 + 3) / 4
- = (48 + 3) / 4
- = 51/4
Answer: 12¾ = 51/4.
Example 6: Example 6: Converting a large improper fraction
Problem: Convert 47/6 to a mixed number.
Solution:
Divide 47 by 6:
- 47 ÷ 6 = Quotient 7, Remainder 5
- (since 6 × 7 = 42, and 47 − 42 = 5)
Mixed number:
- = 7⅚
Verification: 7⅚ = (7 × 6 + 5)/6 = 47/6 ✓
Answer: 47/6 = 7⅚.
Example 7: Example 7: Comparing mixed numbers
Problem: Which is greater: 5⅓ or 5¼?
Solution:
Step 1: Compare whole parts:
- Both have whole part = 5. So compare the fraction parts.
Step 2: Compare ⅓ and ¼:
- Find common denominator: LCM of 3 and 4 = 12
- ⅓ = 4/12
- ¼ = 3/12
- 4/12 > 3/12
Therefore: ⅓ > ¼, so 5⅓ > 5¼.
Answer: 5⅓ is greater.
Example 8: Example 8: Mixed number on number line
Problem: Between which two whole numbers does 3⅗ lie on the number line?
Solution:
- The whole part is 3.
- The fraction part ⅗ is between 0 and 1 (it is a proper fraction).
- So 3⅗ = 3 + ⅗, which is between 3 and 4.
More precisely:
- 3⅗ = 3 + 0.6 = 3.6
- It is closer to 4 than to 3 (since 0.6 > 0.5).
Answer: 3⅗ lies between 3 and 4.
Example 9: Example 9: Real-life mixed number
Problem: A recipe needs 2½ cups of flour. Express this as an improper fraction.
Solution:
Given: 2½ cups
Using the formula:
- = (2 × 2 + 1) / 2
- = (4 + 1) / 2
- = 5/2
Meaning: 2½ cups = 5 half-cups of flour.
Answer: 2½ = 5/2.
Example 10: Example 10: Series of conversions
Problem: Convert the following improper fractions to mixed numbers: (a) 7/2 (b) 11/3 (c) 19/7 (d) 25/4
Solution:
(a) 7/2:
- 7 ÷ 2 = Quotient 3, Remainder 1
- = 3½
(b) 11/3:
- 11 ÷ 3 = Quotient 3, Remainder 2
- = 3⅔
(c) 19/7:
- 19 ÷ 7 = Quotient 2, Remainder 5
- = 2⁵⁄₇
(d) 25/4:
- 25 ÷ 4 = Quotient 6, Remainder 1
- = 6¼
Answer: (a) 3½, (b) 3⅔, (c) 2⁵⁄₇, (d) 6¼.
Real-World Applications
Real-world uses of mixed numbers:
- Cooking: Recipes use mixed numbers like 2½ cups of sugar or 1¾ teaspoons of salt. You measure whole units and then a fraction of a unit.
- Height measurement: A person's height might be 5 feet 3 inches = 5¼ feet. The mixed number shows feet and a fraction of a foot.
- Time: "2 and a half hours" is 2½ hours. Film durations, travel times, and cooking times often use mixed numbers.
- Distance: "The school is 3½ km from home." Distances are often not exact whole numbers.
- Shopping: "I bought 1¾ kg of apples." Weights at shops are usually mixed numbers.
- Construction: Builders measure lengths like 8½ feet or 3¾ inches when cutting wood or pipes.
Key Points to Remember
- A mixed number has a whole number part and a proper fraction part (e.g., 3¼).
- Every mixed number is greater than 1.
- The fraction part is always a proper fraction (numerator < denominator).
- To convert a mixed number to an improper fraction: (whole × denominator + numerator) / denominator.
- To convert an improper fraction to a mixed number: divide numerator by denominator. Quotient = whole part, remainder = fraction numerator.
- If the remainder is 0, the improper fraction equals a whole number (not a mixed number).
- A mixed number lies between two consecutive whole numbers on the number line.
- To compare mixed numbers: compare whole parts first, then fraction parts.
- Improper fractions are easier to use in calculations (addition, subtraction, multiplication). Mixed numbers are easier to understand in real life.
- Every improper fraction can be written as either a mixed number or a whole number.
Practice Problems
- Convert 5⅔ to an improper fraction.
- Convert 29/8 to a mixed number.
- Convert 7¼ to an improper fraction.
- Convert 40/9 to a mixed number.
- Which is greater: 4⅖ or 4⅜? (Hint: find common denominator for the fraction parts.)
- A rope is 6⅔ metres long. Express this as an improper fraction.
- Between which two whole numbers does 8⁷⁄₉ lie?
- Convert these improper fractions to mixed numbers: (a) 13/5 (b) 22/7 (c) 31/6
Frequently Asked Questions
Q1. What is a mixed number?
A mixed number (or mixed fraction) is a number that combines a whole number and a proper fraction. For example, 3½ has the whole part 3 and the fraction part ½. It represents 3 + ½ = 3.5.
Q2. How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and write the result over the same denominator. For 4⅔: (4 × 3 + 2)/3 = 14/3.
Q3. How do you convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The quotient becomes the whole part, and the remainder becomes the numerator of the fraction part. For 17/5: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3⅖.
Q4. Is 5/3 a mixed number?
No. 5/3 is an improper fraction (numerator > denominator). But it can be converted to the mixed number 1⅔. A mixed number must be written in the form whole + fraction, like 1⅔.
Q5. Can the fraction part of a mixed number be an improper fraction?
No. The fraction part of a mixed number must be a proper fraction (numerator < denominator). For example, 2⁷⁄₃ is not a valid mixed number. You would simplify it: 2⁷⁄₃ = 2 + 7/3 = 2 + 2⅓ = 4⅓.
Q6. Is every improper fraction a mixed number?
Not automatically. Every improper fraction can be converted to a mixed number (or a whole number if the remainder is 0). For example, 7/3 = 2⅓ (mixed number) and 6/3 = 2 (whole number).
Q7. Why do we use mixed numbers instead of improper fractions?
Mixed numbers are easier to understand in real life. Saying '2½ hours' is clearer than '5/2 hours'. However, improper fractions are easier to work with in calculations like addition and multiplication.
Q8. How do you compare two mixed numbers?
First compare the whole number parts. The one with the larger whole part is greater. If the whole parts are equal, compare the fraction parts (find a common denominator if needed). For example, 5⅓ > 5¼ because ⅓ > ¼.
Related Topics
- Proper and Improper Fractions
- Introduction to Fractions
- Addition of Fractions
- Comparing Fractions
- Equivalent Fractions
- Simplest Form of a Fraction
- Like and Unlike Fractions
- Fractions on Number Line
- Subtraction of Fractions
- Unit Fractions
- Word Problems on Fractions
- Types of Fractions
- Addition and Subtraction of Fractions
