Types of Fractions

Class 6Fractions

In the previous topic, you learned what fractions are — numbers that represent parts of a whole, written as a/b with a numerator and a denominator. Now it is time to explore the different types of fractions in detail. Just as there are different types of numbers (natural, whole, integers), there are different types of fractions, and each type has its own special properties. Knowing the type of fraction you are working with is important because it affects how you compare, add, subtract, and simplify fractions. In this chapter, you will learn about proper fractions, improper fractions, mixed fractions, like fractions, unlike fractions, equivalent fractions, and unit fractions. You will understand what makes each type unique, how to identify them, and how to convert between them. Think of this as learning to classify animals into groups — once you know whether a fraction is proper or improper, like or unlike, you know exactly how to handle it. This knowledge is essential for the operations on fractions (addition, subtraction, multiplication, division) that you will study next. Let us dive deep into each type.

What is Types of Fractions?

Here are the main types of fractions you need to know:

1. Proper Fraction: A fraction where the numerator is less than the denominator.
Examples: 1/2, 3/5, 7/10, 11/15, 99/100.
Value: Always less than 1.
Visual: Less than one complete shape is shaded.

2. Improper Fraction: A fraction where the numerator is greater than or equal to the denominator.
Examples: 5/3, 7/4, 9/2, 12/5, 8/8, 15/15.
Value: Equal to 1 or greater than 1.
Visual: One complete shape (or more) is shaded.

3. Mixed Fraction (Mixed Number): A combination of a whole number and a proper fraction.
Examples: 1 1/2, 2 3/4, 5 2/7, 10 1/3.
Value: Always greater than 1 (since it has a whole number part plus a fraction).
Every improper fraction can be written as a mixed number and vice versa.

4. Like Fractions: Fractions that have the same denominator.
Examples: 2/7, 3/7, 5/7 (all have denominator 7). Or 1/10, 3/10, 9/10 (all have denominator 10).
Like fractions are easy to compare and add because they have the same "size" of parts.

5. Unlike Fractions: Fractions that have different denominators.
Examples: 1/3, 2/5, 4/7 (denominators 3, 5 and 7 are all different).
To compare or add unlike fractions, you must first convert them to like fractions by finding a common denominator.

6. Equivalent Fractions: Fractions that have different numerators and denominators but represent the same value.
Examples: 1/2 = 2/4 = 3/6 = 4/8 = 50/100.
You get equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.

7. Unit Fraction: A fraction with numerator equal to 1.
Examples: 1/2, 1/3, 1/4, 1/5, 1/100.
A unit fraction represents exactly one part of the whole. Any fraction a/b can be written as a sum of 'a' unit fractions of 1/b.

8. Decimal Fraction: A fraction whose denominator is a power of 10 (10, 100, 1000, etc.).
Examples: 3/10, 17/100, 9/1000.
These fractions are easily written as decimals: 3/10 = 0.3, 17/100 = 0.17, 9/1000 = 0.009.

Types and Properties

Let us understand each type more deeply with properties and comparisons:

1. Proper vs Improper Fractions — How to Tell Them Apart
The quick test is simple: compare the numerator and denominator.
- If numerator < denominator → proper (value < 1).
- If numerator >= denominator → improper (value >= 1).

Examples of proper fractions with real-world meaning:
- You ate 2 slices of a 6-slice pizza → 2/6 (proper).
- A student scored 45 out of 50 → 45/50 (proper).
- You have walked 3 km of a 5 km path → 3/5 (proper).

Examples of improper fractions with real-world meaning:
- You need 5 quarter-cups of flour → 5/4 cups (more than one cup).
- You have 7 halves of a chocolate bar → 7/2 bars (3.5 bars).
- 10/10 of the work is done → the work is complete (= 1).

2. Converting Between Improper Fractions and Mixed Numbers

Improper → Mixed: Divide numerator by denominator.
17/5 → 17 ÷ 5 = 3 remainder 2 → 3 2/5.
Think of it as: how many whole pizzas (of 5 slices each) can you make from 17 slices? 3 whole pizzas with 2 slices left over.

Mixed → Improper: Multiply whole number by denominator, add numerator.
3 2/5 → (3 × 5 + 2) / 5 = 17/5.
Think of it as: 3 whole pizzas of 5 slices = 15 slices, plus 2 more = 17 slices, each slice is 1/5.

3. Like vs Unlike Fractions — Why It Matters

Like fractions (same denominator) are easy to work with:
- Comparing: just compare numerators. 3/7 < 5/7 because 3 < 5.
- Adding: just add numerators. 2/7 + 3/7 = 5/7.
- Subtracting: just subtract numerators. 5/7 - 2/7 = 3/7.

Unlike fractions (different denominators) need extra work:
- First find the LCM of the denominators (Least Common Multiple).
- Convert each fraction to an equivalent fraction with that LCM as denominator.
- Now they are like fractions and you can compare/add/subtract easily.
Example: compare 2/3 and 3/4. LCM of 3 and 4 = 12. So 2/3 = 8/12 and 3/4 = 9/12. Since 8/12 < 9/12, we know 2/3 < 3/4.

4. Equivalent Fractions — The Big Idea

A fraction does not change its value if you multiply or divide both numerator and denominator by the same non-zero number. This is because you are multiplying by 1 (in disguise): 2/2 = 1, 3/3 = 1, etc.

2/3 × 4/4 = 8/12 → 2/3 and 8/12 are equivalent.
18/24 ÷ 6/6 = 3/4 → 18/24 and 3/4 are equivalent.

Simplest form (lowest terms): A fraction is in its simplest form when the numerator and denominator have no common factor other than 1 (their HCF = 1).
8/12 → HCF(8,12) = 4 → divide both by 4 → 2/3 (simplest form).
15/25 → HCF(15,25) = 5 → divide both by 5 → 3/5 (simplest form).

5. Unit Fractions — Special Properties

Unit fractions decrease as the denominator increases: 1/2 > 1/3 > 1/4 > 1/5 > ...
Think of sharing a pizza: sharing among 2 people gives bigger pieces than sharing among 5 people.

Every fraction is a multiple of a unit fraction: 3/4 = 3 × (1/4). So 3/4 is three copies of the unit fraction 1/4.

Ancient Egyptians used only unit fractions (called Egyptian fractions). They wrote 3/4 as 1/2 + 1/4. This is a fun historical fact!

Solved Examples

Example 1: Example 1: Classifying fractions as proper or improper

Problem: Classify each as proper or improper: (a) 4/9, (b) 12/7, (c) 5/5, (d) 1/100, (e) 101/100.

Solution:
(a) 4/9: 4 < 9, so it is a proper fraction (value less than 1).

(b) 12/7: 12 > 7, so it is an improper fraction (value greater than 1).

(c) 5/5: 5 = 5, so it is an improper fraction (value equals 1).

(d) 1/100: 1 < 100, so it is a proper fraction (a tiny fraction, equal to 0.01).

(e) 101/100: 101 > 100, so it is an improper fraction (just barely greater than 1, equal to 1.01).

Example 2: Example 2: Converting improper fractions to mixed numbers

Problem: Convert to mixed numbers: (a) 11/4, (b) 25/6, (c) 40/7.

Solution:
(a) 11/4: 11 ÷ 4 = 2 remainder 3. So 11/4 = 2 3/4.

(b) 25/6: 25 ÷ 6 = 4 remainder 1. So 25/6 = 4 1/6.

(c) 40/7: 40 ÷ 7 = 5 remainder 5. So 40/7 = 5 5/7.

Check (b): 4 1/6 = (4 × 6 + 1)/6 = 25/6 ✓

Example 3: Example 3: Converting mixed numbers to improper fractions

Problem: Convert to improper fractions: (a) 3 1/4, (b) 7 2/3, (c) 1 5/8.

Solution:
(a) 3 1/4: (3 × 4 + 1)/4 = 13/4. So 3 1/4 = 13/4.

(b) 7 2/3: (7 × 3 + 2)/3 = 23/3. So 7 2/3 = 23/3.

(c) 1 5/8: (1 × 8 + 5)/8 = 13/8. So 1 5/8 = 13/8.

Example 4: Example 4: Identifying like and unlike fractions

Problem: Which are like fractions and which are unlike? (a) 2/5, 4/5, 7/5. (b) 1/3, 1/4, 1/5. (c) 3/11, 8/11, 1/11.

Solution:
(a) 2/5, 4/5, 7/5: All have denominator 5. These are like fractions.

(b) 1/3, 1/4, 1/5: Denominators are 3, 4, 5 — all different. These are unlike fractions. (Note: even though the numerators are all 1, it is the denominators that must match for like fractions.)

(c) 3/11, 8/11, 1/11: All have denominator 11. These are like fractions.

Example 5: Example 5: Finding equivalent fractions

Problem: (a) Find 3 fractions equivalent to 3/4. (b) Check if 6/10 and 9/15 are equivalent.

Solution:
(a) Multiply numerator and denominator by 2, 3, and 5:
3/4 × 2/2 = 6/8
3/4 × 3/3 = 9/12
3/4 × 5/5 = 15/20
So 3/4 = 6/8 = 9/12 = 15/20.

(b) Simplify both fractions to their simplest form:
6/10: HCF(6,10) = 2 → 6/10 = 3/5.
9/15: HCF(9,15) = 3 → 9/15 = 3/5.
Both simplify to 3/5, so yes, they are equivalent.

Alternative method — cross multiply: 6 × 15 = 90 and 10 × 9 = 90. Since the cross products are equal, the fractions are equivalent. ✓

Example 6: Example 6: Reducing fractions to simplest form

Problem: Reduce to simplest form: (a) 12/18, (b) 35/50, (c) 24/36.

Solution:
(a) 12/18: HCF(12, 18) = 6. Divide both by 6: 12/18 = 2/3.

(b) 35/50: HCF(35, 50) = 5. Divide both by 5: 35/50 = 7/10.

(c) 24/36: HCF(24, 36) = 12. Divide both by 12: 24/36 = 2/3.

Note: 12/18 and 24/36 both reduce to 2/3, so they are equivalent fractions!

Example 7: Example 7: Comparing like fractions

Problem: Arrange in ascending order: 7/9, 2/9, 5/9, 1/9, 8/9.

Solution:
These are all like fractions (denominator = 9). For like fractions, the one with the smaller numerator is the smaller fraction.

Numerators in ascending order: 1, 2, 5, 7, 8.

Ascending order: 1/9, 2/9, 5/9, 7/9, 8/9.

Think of it as pizza slices: 1 slice out of 9 is less than 8 slices out of 9.

Example 8: Example 8: Comparing unlike fractions

Problem: Which is greater: 2/3 or 5/8?

Solution:
These are unlike fractions (denominators 3 and 8 are different). Convert to like fractions using LCM.

LCM of 3 and 8 = 24.

2/3 = (2 × 8) / (3 × 8) = 16/24.
5/8 = (5 × 3) / (8 × 3) = 15/24.

Now compare: 16/24 > 15/24.

Therefore, 2/3 > 5/8.

Example 9: Example 9: Pizza sharing problem — identifying fraction types

Problem: 3 pizzas are shared equally among 4 friends. (a) What fraction does each person get? (b) Is this a proper or improper fraction? (c) Write it as a mixed number.

Solution:
(a) 3 pizzas ÷ 4 friends = 3/4 pizza each.

(b) 3/4: numerator (3) < denominator (4), so this is a proper fraction.

(c) Since 3/4 is already less than 1, it does not need to be written as a mixed number. It is simply 3/4.

But what if 5 pizzas were shared among 4 friends? Then each person gets 5/4 (improper), which equals 1 1/4 pizzas (mixed number).

Example 10: Example 10: Finding the fraction that does not belong

Problem: One of these fractions is NOT equivalent to the others. Find it: 2/3, 4/6, 6/9, 8/11, 10/15.

Solution:
Simplify each fraction:
2/3 = 2/3 ✓
4/6 = 2/3 ✓ (divided by 2)
6/9 = 2/3 ✓ (divided by 3)
8/11 = 8/11 ✗ (cannot be simplified to 2/3)
10/15 = 2/3 ✓ (divided by 5)

Answer: 8/11 is not equivalent to the others. All the others are equivalent to 2/3.

Real-World Applications

Understanding the types of fractions helps in many practical situations:

Cooking and Baking: Recipes use proper fractions (1/2 cup, 3/4 teaspoon) and mixed numbers (1 1/2 cups, 2 3/4 tablespoons). When you double a recipe, 3/4 becomes 6/4 (improper), which you convert to 1 1/2. Knowing how to switch between fraction types is essential in the kitchen.

Measurement and Construction: Carpenters measure in fractions of an inch (3/8 inch, 5/16 inch). They need to compare unlike fractions to find the right drill bit or nail size. Equivalent fractions help: 4/8 inch = 1/2 inch, so they can use either measurement interchangeably.

Shopping and Discounts: A 25% discount means 1/4 off. A 33% discount is approximately 1/3 off. Comparing which discount is better requires comparing unlike fractions. Is 2/5 off better than 3/8 off? You need equivalent fractions with a common denominator to decide.

Time Management: We describe time using different fractions: half an hour (1/2 hour = 30 min), quarter hour (1/4 hour = 15 min), three-quarters of an hour (3/4 hour = 45 min). Converting between these fractions and minutes is a daily skill.

Academic Scoring: Test scores are fractions — 17/20, 85/100, 42/50. To compare scores from tests with different totals, you convert to equivalent fractions (or percentages, which are fractions with denominator 100). Is 17/20 better than 42/50? Convert: 17/20 = 85/100 and 42/50 = 84/100. Yes, 17/20 is slightly better!

Data and Statistics: Surveys express results as fractions: "3 out of 5 students prefer maths" (3/5). Comparing survey results requires converting unlike fractions to like fractions or decimals.

Key Points to Remember

Proper fraction: numerator < denominator, value < 1. Example: 3/5.
Improper fraction: numerator >= denominator, value >= 1. Example: 7/4.
Mixed number: whole number + proper fraction. Example: 2 3/4. Every improper fraction has a mixed number form.
Like fractions: same denominator (easy to compare and add). Unlike fractions: different denominators (need common denominator first).
Equivalent fractions: different-looking fractions with the same value. Created by multiplying or dividing both parts by the same number.
Simplest form: numerator and denominator share no common factor other than 1. Divide both by their HCF to simplify.
Unit fraction: numerator = 1. For unit fractions, larger denominator = smaller fraction.
Decimal fractions have denominators of 10, 100, 1000, etc., and convert directly to decimals.
To compare unlike fractions: find LCM of denominators, convert to equivalent like fractions, then compare numerators.
Cross multiplication is a quick test for equivalence: a/b = c/d if and only if a x d = b x c.

Practice Problems

Classify as proper, improper or mixed: 5/8, 13/4, 2 1/5, 9/9, 100/99, 1/1000.
Convert to mixed numbers: 17/3, 29/8, 45/7, 100/9.
Convert to improper fractions: 2 5/6, 4 1/3, 6 7/8, 10 2/5.
Find 4 equivalent fractions for 5/6.
Reduce to simplest form: 16/24, 25/35, 42/56, 60/100.
Identify like and unlike fractions: {3/8, 5/8, 1/8}, {2/3, 4/5, 1/6}, {7/12, 11/12}.
Compare: 3/4 and 5/7. (Hint: find a common denominator.)
Which of these are equivalent to 2/5: 4/10, 6/14, 8/20, 10/25?

Frequently Asked Questions

Q1. What is the difference between proper and improper fractions?

In a proper fraction, the top number (numerator) is smaller than the bottom number (denominator), so the value is less than 1 — like 3/4 (three-quarters of a pizza). In an improper fraction, the top number is equal to or bigger than the bottom number, so the value is 1 or more — like 5/3 (more than one whole pizza). Improper fractions can be rewritten as mixed numbers.

Q2. How do you convert an improper fraction to a mixed number?

Divide the numerator by the denominator. The quotient becomes the whole number part, and the remainder becomes the new numerator (the denominator stays the same). For example, 19/4: 19 divided by 4 = 4 remainder 3, so 19/4 = 4 3/4.

Q3. What are like fractions?

Like fractions are fractions that have the same denominator. Examples: 2/9, 5/9, 7/9 (all have denominator 9). Like fractions are easy to compare — just look at the numerators. They are also easy to add and subtract — just add or subtract the numerators and keep the denominator.

Q4. How do you make unlike fractions into like fractions?

Find the LCM (Least Common Multiple) of the denominators. Then convert each fraction to an equivalent fraction with that LCM as the new denominator. For example, to make 1/3 and 2/5 into like fractions: LCM of 3 and 5 = 15. So 1/3 = 5/15 and 2/5 = 6/15. Now they are like fractions with denominator 15.

Q5. How do you know if two fractions are equivalent?

Two fractions a/b and c/d are equivalent if they simplify to the same simplest form. A quick test is cross multiplication: if a times d equals b times c, they are equivalent. For example, 3/4 and 9/12: 3 x 12 = 36 and 4 x 9 = 36. Equal, so they are equivalent.

Q6. What is the simplest form of a fraction?

A fraction is in its simplest form (also called lowest terms) when the numerator and denominator have no common factor other than 1. To simplify, divide both by their HCF (Highest Common Factor). For example, 12/18: HCF of 12 and 18 is 6, so 12/18 = 2/3 (simplest form).

Q7. Why are unit fractions important?

Unit fractions (like 1/2, 1/3, 1/4) are the building blocks of all fractions. Any fraction a/b equals a copies of 1/b. For example, 3/5 = 1/5 + 1/5 + 1/5 (three copies of one-fifth). Understanding unit fractions helps you visualise what fractions really mean.

Q8. Is 7/7 proper or improper?

7/7 is an improper fraction because the numerator (7) equals the denominator (7). Its value is exactly 1. Similarly, 3/3, 10/10, and 100/100 are all improper fractions equal to 1.

Q9. Can a mixed number have an improper fractional part?

No. A properly written mixed number always has a whole number plus a proper fraction (where the numerator is less than the denominator). If you get something like 3 5/4, you must simplify: 5/4 = 1 1/4, so 3 5/4 = 4 1/4. The fractional part must always be proper.

Q10. What is a decimal fraction?

A decimal fraction is a fraction whose denominator is a power of 10 (10, 100, 1000, etc.). Examples: 7/10, 23/100, 456/1000. These are special because they can be directly written as decimals: 7/10 = 0.7, 23/100 = 0.23, 456/1000 = 0.456. Percentages are also decimal fractions: 75% = 75/100.