Proper and Improper Fractions
You already know that a fraction like 3/4 means 3 parts out of 4 equal parts. But did you notice that sometimes the top number (numerator) is smaller than the bottom number (denominator), and sometimes it is bigger?
When the top number is smaller, we call it a proper fraction. When the top number is equal to or bigger than the bottom number, we call it an improper fraction. Knowing the difference helps you understand whether a fraction is less than 1, equal to 1, or more than 1.
In Class 6 Mathematics (NCERT), proper and improper fractions are studied in the chapter Fractions. You will also learn about mixed numbers, which are another way of writing improper fractions.
What is Proper and Improper Fractions?
Definition:
- A proper fraction is a fraction where the numerator is less than the denominator.
- An improper fraction is a fraction where the numerator is equal to or greater than the denominator.
Proper Fraction:
- Numerator < Denominator
- Value is always less than 1
- Examples: 1/2, 3/4, 5/8, 7/10, 2/5
- On a number line, proper fractions lie between 0 and 1.
Improper Fraction:
- Numerator ≥ Denominator
- Value is 1 or more
- Examples: 5/3, 7/4, 9/2, 8/8, 11/5
- On a number line, improper fractions lie at 1 or beyond 1.
Mixed Number (Mixed Fraction):
- A mixed number has a whole number part and a proper fraction part.
- Examples: 1 1/2, 2 3/4, 5 2/3
- Every improper fraction can be written as a mixed number.
- Every mixed number can be written as an improper fraction.
Proper and Improper Fractions Formula
Converting Improper Fraction to Mixed Number:
Divide the numerator by the denominator.
Quotient = whole number, Remainder = new numerator
Steps:
- Divide the numerator by the denominator.
- The quotient becomes the whole number part.
- The remainder becomes the numerator of the fraction part.
- The denominator stays the same.
Converting Mixed Number to Improper Fraction:
New Numerator = (Whole number × Denominator) + Numerator
Denominator stays the same
Steps:
- Multiply the whole number by the denominator.
- Add the numerator to the result.
- Write this sum over the original denominator.
Derivation and Proof
Understanding why these conversions work:
Why dividing gives a mixed number:
- The fraction 17/5 means 17 equal parts, each of size 1/5.
- How many whole groups of 5 can you make from 17? That is 17 ÷ 5 = 3 remainder 2.
- So you have 3 complete wholes (each made of 5 fifths) and 2 fifths left over.
- That is 3 and 2/5, written as 3 2/5.
Why multiplying gives an improper fraction:
- The mixed number 3 2/5 means 3 wholes and 2/5 more.
- Each whole has 5 fifths. So 3 wholes = 3 × 5 = 15 fifths.
- Add the extra 2 fifths: 15 + 2 = 17 fifths.
- That is 17/5.
Think of it like coins:
- If each whole = 5 one-rupee coins, then 3 wholes = 15 coins, plus 2 extra coins = 17 coins total.
- Each coin is 1/5 of a whole, so 17 coins = 17/5.
Think of it like pizza slices:
- Each pizza is cut into 4 slices. You have eaten 11 slices in total.
- How many full pizzas? 11 ÷ 4 = 2 with remainder 3.
- So you ate 2 full pizzas and 3 extra slices. That is 2 3/4 pizzas.
- As an improper fraction: 11/4.
- 2 3/4 = 11/4. Check: (2 × 4) + 3 = 11 ✓
When to use which form:
- Use mixed numbers to describe amounts in everyday language: "I drank 1 1/2 glasses of water."
- Use improper fractions for calculations: multiplying 3/2 × 4/5 is easier than multiplying 1 1/2 × 4/5.
Types and Properties
Types of fractions at a glance:
1. Proper Fractions:
- Numerator < Denominator
- Value < 1
- Examples: 1/3, 2/7, 4/9, 11/15
- You have less than one whole thing.
- Numerator ≥ Denominator
- Value ≥ 1
- Examples: 5/5 (= 1), 7/3, 12/5, 9/4
- You have one whole or more.
3. Mixed Numbers:
- Whole number + proper fraction
- Value > 1 (always)
- Examples: 1 1/4, 3 2/3, 7 5/8
- Another way of writing improper fractions.
4. Unit Fractions (special proper fractions):
- Numerator = 1
- Examples: 1/2, 1/3, 1/4, 1/10
- These are the simplest proper fractions.
Quick check:
- Is 9/11 proper or improper? Since 9 < 11, it is proper.
- Is 11/9 proper or improper? Since 11 > 9, it is improper.
- Is 7/7 proper or improper? Since 7 = 7, it is improper (equals 1).
Solved Examples
Example 1: Example 1: Identifying proper and improper fractions
Problem: Classify each fraction as proper or improper: 3/8, 11/7, 5/5, 2/9, 15/4.
Solution:
- 3/8: 3 < 8, so proper.
- 11/7: 11 > 7, so improper.
- 5/5: 5 = 5, so improper (equals 1).
- 2/9: 2 < 9, so proper.
- 15/4: 15 > 4, so improper.
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 = 4 remainder 3
Step 2: Write the mixed number:
- Whole number = 4
- Remainder = 3 (new numerator)
- Denominator stays 5
Answer: 23/5 = 4 3/5
Check: (4 × 5) + 3 = 20 + 3 = 23. So 23/5 ✓
Example 3: Example 3: Converting mixed number to improper fraction
Problem: Convert 3 2/7 to an improper fraction.
Solution:
Step 1: Multiply whole number by denominator:
- 3 × 7 = 21
Step 2: Add the numerator:
- 21 + 2 = 23
Step 3: Write over the same denominator:
- 23/7
Answer: 3 2/7 = 23/7
Check: 23 ÷ 7 = 3 remainder 2 = 3 2/7 ✓
Example 4: Example 4: Placing fractions on a number line
Problem: Place 2/5 and 7/5 on a number line. Which is proper and which is improper?
Solution:
2/5:
- 2 < 5, so it is a proper fraction.
- Its value is less than 1.
- Divide the gap between 0 and 1 into 5 equal parts. Count 2 parts from 0. That is 2/5.
7/5:
- 7 > 5, so it is an improper fraction.
- 7/5 = 1 2/5 (more than 1).
- It lies between 1 and 2 on the number line. Divide the gap between 1 and 2 into 5 parts. Count 2 parts from 1.
Answer: 2/5 is proper (between 0 and 1). 7/5 is improper (between 1 and 2).
Example 5: Example 5: Converting several improper fractions
Problem: Convert to mixed numbers: (a) 19/4, (b) 31/6, (c) 50/9.
Solution:
(a) 19/4:
- 19 ÷ 4 = 4 remainder 3
- 19/4 = 4 3/4
(b) 31/6:
- 31 ÷ 6 = 5 remainder 1
- 31/6 = 5 1/6
(c) 50/9:
- 50 ÷ 9 = 5 remainder 5
- 50/9 = 5 5/9
Example 6: Example 6: Converting several mixed numbers
Problem: Convert to improper fractions: (a) 2 3/4, (b) 5 1/3, (c) 7 5/8.
Solution:
(a) 2 3/4:
- (2 × 4) + 3 = 8 + 3 = 11
- 2 3/4 = 11/4
(b) 5 1/3:
- (5 × 3) + 1 = 15 + 1 = 16
- 5 1/3 = 16/3
(c) 7 5/8:
- (7 × 8) + 5 = 56 + 5 = 61
- 7 5/8 = 61/8
Example 7: Example 7: Real-life proper fraction
Problem: A pizza has 8 slices. Riya eats 3 slices. What fraction of the pizza did she eat? Is it proper or improper?
Solution:
Given:
- Total slices = 8
- Slices eaten = 3
Fraction eaten:
- 3/8
- 3 < 8, so this is a proper fraction.
- Riya ate less than one full pizza.
Answer: Riya ate 3/8 of the pizza. This is a proper fraction.
Example 8: Example 8: Real-life improper fraction
Problem: A cake is cut into 6 equal pieces. Amit ate 6 pieces from 2 cakes (eating a total of 10 pieces). Write this as a fraction and as a mixed number.
Solution:
Given:
- Each cake = 6 pieces
- Total pieces eaten = 10
As a fraction:
- 10/6 (10 pieces, each being 1/6 of a cake)
- 10 > 6, so this is an improper fraction.
As a mixed number:
- 10 ÷ 6 = 1 remainder 4
- 10/6 = 1 4/6 = 1 2/3 (simplify 4/6 to 2/3)
Answer: Amit ate 10/6 = 1 2/3 cakes.
Example 9: Example 9: Comparing proper and improper fractions
Problem: Which is greater: 4/5 or 7/5?
Solution:
Compare:
- 4/5 is a proper fraction (less than 1).
- 7/5 is an improper fraction (more than 1, equal to 1 2/5).
- Any improper fraction is greater than any proper fraction with the same denominator.
Answer: 7/5 > 4/5.
Example 10: Example 10: When the fraction equals 1
Problem: What type of fraction is 6/6? What is its value?
Solution:
Given:
- Numerator = 6, Denominator = 6
- Numerator = Denominator
Classification:
- Since numerator = denominator, it is an improper fraction.
- Its value = 6 ÷ 6 = 1.
Answer: 6/6 is an improper fraction equal to 1. Any fraction where the numerator equals the denominator is equal to 1.
Real-World Applications
Where do we use proper and improper fractions?
- Sharing food: If you eat 3 slices of a pizza cut into 8, you ate 3/8 (proper). If you and your friends eat from 2 pizzas and consume 11 slices (each pizza having 8), that is 11/8 (improper) = 1 3/8 pizzas.
- Measuring length: A ribbon 2 3/4 metres long is written as 11/4 metres when doing calculations. Mixed numbers are easier to picture, but improper fractions are easier to calculate with.
- Cooking: Recipes say "2 1/2 cups of flour." For doubling the recipe, converting to 5/2 and then multiplying is easier: 5/2 × 2 = 5 cups.
- Time: "One and a half hours" = 1 1/2 = 3/2 hours. Improper fractions make time calculations easier.
- Scoring in games: If a quiz has 5 questions and you answer 5 correctly, your score is 5/5 = 1 (full marks). If bonus questions take you to 7/5, that is an improper fraction meaning you scored more than the base total.
- Mathematics: When adding or multiplying fractions, it is often easier to first convert mixed numbers to improper fractions, do the calculation, then convert back.
Key Points to Remember
- Proper fraction: numerator < denominator. Value is less than 1. Examples: 1/2, 3/5, 7/10.
- Improper fraction: numerator ≥ denominator. Value is 1 or more. Examples: 5/3, 8/8, 12/7.
- Mixed number = whole number + proper fraction. Example: 2 3/4.
- To convert improper to mixed: divide numerator by denominator. Quotient is the whole part, remainder is the fraction part.
- To convert mixed to improper: (whole × denominator + numerator) / denominator.
- When the numerator equals the denominator, the fraction equals 1.
- A proper fraction lies between 0 and 1 on the number line.
- An improper fraction lies at 1 or beyond on the number line.
- For calculations (adding, subtracting, multiplying), convert mixed numbers to improper fractions first.
- For understanding the size, convert improper fractions to mixed numbers.
Practice Problems
- Classify as proper or improper: 7/12, 13/8, 6/6, 4/11, 20/9.
- Convert to mixed numbers: 17/3, 25/4, 41/7, 53/10.
- Convert to improper fractions: 4 1/5, 6 3/8, 9 2/3, 1 7/10.
- Draw a number line and mark: 3/4, 5/4, 2 1/4.
- A rope is 19/4 metres long. Write this as a mixed number.
- Which is greater: 5/6 or 8/6? Explain why.
- Write 3 proper fractions and 3 improper fractions with denominator 7.
- Raju walked 2 3/5 km. Write this distance as an improper fraction.
Frequently Asked Questions
Q1. What is a proper fraction?
A proper fraction is a fraction where the numerator (top number) is less than the denominator (bottom number). Its value is always less than 1. Examples: 2/5, 3/7, 4/9.
Q2. What is an improper fraction?
An improper fraction is a fraction where the numerator is equal to or greater than the denominator. Its value is 1 or more. Examples: 7/3, 5/5, 11/4.
Q3. What is a mixed number?
A mixed number has a whole number and a proper fraction together. Example: 3 1/2 means 3 wholes and 1/2 more. It is another way to write an improper fraction (3 1/2 = 7/2).
Q4. How do you convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same. Example: 17/5 = 3 2/5 (because 17 ÷ 5 = 3 remainder 2).
Q5. How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and write over the same denominator. Example: 4 2/3 = (4 × 3 + 2)/3 = 14/3.
Q6. Is 8/8 a proper or improper fraction?
8/8 is an improper fraction because the numerator equals the denominator. Its value is 1. Any fraction where numerator = denominator is improper and equals 1.
Q7. Can a proper fraction be greater than 1?
No. A proper fraction always has a value less than 1, because the numerator is smaller than the denominator. You have fewer parts than the total.
Q8. Why do we need to convert between improper fractions and mixed numbers?
Mixed numbers are easier to understand (2 1/2 is clearer than 5/2 for everyday use). But improper fractions are easier for calculations like addition and multiplication. So we convert between them depending on the situation.
Q9. Can a fraction have 0 as the numerator?
Yes. 0/5 = 0. It is a proper fraction (0 < 5) and its value is 0. But 5/0 is NOT allowed because the denominator cannot be 0.
Q10. Is every whole number an improper fraction?
Yes! Every whole number can be written as an improper fraction. For example, 3 = 3/1, 5 = 5/1, 7 = 7/1. In each case, the numerator is greater than the denominator (1), so it is improper.
