Proper and Improper Fractions
Proper and improper fractions are two categories of fractions based on the relationship between the numerator and denominator. Understanding this classification helps you compare fractions, convert between forms, and solve problems accurately.
When you cut a pizza into 4 equal slices and take 3, you have 3/4 — a proper fraction. But if you have 2 full pizzas and take 5 slices of a 4-slice pizza, you have 5/4 — an improper fraction.
What is Proper and Improper Fractions - Class 4 Maths (Fractions)?
Proper Fraction: A fraction where the numerator is less than the denominator. Its value is always less than 1.
Examples: 1/2, 3/4, 5/8, 7/10
Improper Fraction: A fraction where the numerator is greater than or equal to the denominator. Its value is 1 or more.
Examples: 5/3, 7/4, 9/2, 6/6
Proper: Numerator < Denominator (value < 1)
Improper: Numerator ≥ Denominator (value ≥ 1)
Types and Properties
Comparison table:
| Feature | Proper Fraction | Improper Fraction |
|---|---|---|
| Numerator vs Denominator | Numerator < Denominator | Numerator ≥ Denominator |
| Value | Less than 1 | 1 or greater |
| On number line | Between 0 and 1 | At 1 or beyond |
| Example | 3/5 | 7/5 |
Mixed number: An improper fraction can be written as a mixed number (whole number + proper fraction). Example: 7/4 = 1 3/4.
Solved Examples
Example 1: Example 1: Identify the Type
Problem: Classify these fractions as proper or improper: 3/7, 9/4, 5/5, 2/9.
Solution:
3/7: 3 < 7 → Proper
9/4: 9 > 4 → Improper
5/5: 5 = 5 → Improper (value = 1)
2/9: 2 < 9 → Proper
Example 2: Example 2: Convert Improper to Mixed
Problem: Convert 11/4 to a mixed number.
Solution:
Step 1: Divide 11 by 4: 11 ÷ 4 = 2 remainder 3.
Step 2: Write as: whole number = 2, fraction = 3/4.
Answer: 11/4 = 2 3/4
Example 3: Example 3: Convert Mixed to Improper
Problem: Convert 3 2/5 to an improper fraction.
Solution:
Step 1: Multiply whole number by denominator: 3 x 5 = 15.
Step 2: Add the numerator: 15 + 2 = 17.
Step 3: Keep the same denominator: 17/5.
Answer: 3 2/5 = 17/5
Example 4: Example 4: Comparing Proper Fractions
Problem: Which is greater: 5/8 or 3/8?
Solution:
Both have the same denominator (8). Compare numerators: 5 > 3.
Answer: 5/8 > 3/8
Example 5: Example 5: Word Problem
Problem: Ria ate 3/4 of a chapati and Neha ate 5/4 of a chapati. Who ate more than one chapati?
Solution:
3/4: numerator (3) < denominator (4) → proper fraction → less than 1 chapati.
5/4: numerator (5) > denominator (4) → improper fraction → more than 1 chapati.
Answer: Neha ate more than one chapati.
Example 6: Example 6: Value of Improper Fraction
Problem: What is the value of 8/8?
Solution:
8/8 = 8 ÷ 8 = 1
When the numerator equals the denominator, the fraction equals 1. It is classified as improper.
Answer: 1
Example 7: Example 7: Listing Proper Fractions
Problem: Write all proper fractions with denominator 6.
Solution:
The numerator must be less than 6: 1, 2, 3, 4, 5.
Answer: 1/6, 2/6, 3/6, 4/6, 5/6
Example 8: Example 8: Converting Multiple Fractions
Problem: Convert 13/5 and 2 1/3 to the other form.
Solution:
13/5: 13 ÷ 5 = 2 remainder 3 → 2 3/5
2 1/3: (2 x 3) + 1 = 7 → 7/3
Example 9: Example 9: True or False
Problem: True or False: All improper fractions are greater than 1.
Solution:
When numerator = denominator (e.g., 4/4), the value is exactly 1, not greater than 1.
Answer: False. Improper fractions are greater than or equal to 1.
Example 10: Example 10: Word Problem — Cricket Overs
Problem: A bowler bowled 17/6 overs. Express this as a mixed number. (6 balls = 1 over)
Solution:
17 ÷ 6 = 2 remainder 5.
Answer: The bowler bowled 2 5/6 overs (2 complete overs and 5 balls).
Key Points to Remember
- Proper fraction: numerator < denominator; value < 1.
- Improper fraction: numerator ≥ denominator; value ≥ 1.
- Every improper fraction can be converted to a mixed number by dividing.
- Every mixed number can be converted to an improper fraction by multiplying and adding.
- When numerator = denominator (e.g., 5/5), the value is exactly 1.
- On a number line, proper fractions lie between 0 and 1; improper fractions lie at 1 or beyond.
Practice Problems
- Classify as proper or improper: 7/3, 4/9, 6/6, 1/5, 10/7.
- Convert 15/4 to a mixed number.
- Convert 4 3/8 to an improper fraction.
- Write all proper fractions with denominator 5.
- Dev drank 7/3 glasses of milk. How many full glasses did he drink?
- Which is greater: 9/10 or 11/10?
- True or False: 3/3 is a proper fraction.
Frequently Asked Questions
Q1. What is a proper fraction?
A proper fraction is one where the numerator is smaller than the denominator. Its value is always less than 1. Examples: 2/5, 3/7, 4/9.
Q2. What is an improper fraction?
An improper fraction is one where the numerator is greater than or equal to the denominator. Its value is 1 or more. Examples: 5/3, 8/5, 6/6.
Q3. How do you convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The quotient is the whole number part, and the remainder over the denominator is the fractional part. For example, 11/3: 11 / 3 = 3 remainder 2, so 11/3 = 3 2/3.
Q4. How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator and add the numerator. Place this over the original denominator. For example, 2 3/4: (2 x 4) + 3 = 11, so 2 3/4 = 11/4.
Q5. Is 5/5 proper or improper?
5/5 is an improper fraction because the numerator equals the denominator. Its value is 1.
Q6. Can a proper fraction ever equal 1?
No. A proper fraction is always less than 1 because the numerator is strictly less than the denominator.
Q7. Why do we need improper fractions?
Improper fractions are useful in calculations, especially in addition and subtraction of fractions. They are easier to work with than mixed numbers during computation.
Q8. Are proper and improper fractions part of NCERT Class 4?
Yes. Identifying, converting, and comparing proper and improper fractions is part of the CBSE/NCERT Class 4 Maths curriculum under the fractions chapter.
Related Topics
- Fractions (Grade 4)
- Mixed Numbers and Improper Fractions
- Equivalent Fractions (Grade 4)
- Simplifying Fractions
- Comparing Fractions (Grade 4)
- Ordering Fractions
- Addition of Fractions (Grade 4)
- Subtraction of Fractions (Grade 4)
- Fraction Word Problems (Grade 4)
- Fractions on a Number Line (Grade 4)
- Fraction of a Number (Grade 4)
