Introduction to Fractions
Imagine you have one pizza and you want to share it equally between 4 friends. You cut the pizza into 4 equal slices and each person gets 1 slice. How do you write "1 slice out of 4"? You write it as 1/4 — that is a fraction! Fractions are numbers that represent parts of a whole. They help us express quantities that are not whole numbers — half a glass of water, a quarter of a cake, three-fifths of a journey. Without fractions, we would not be able to describe these everyday situations mathematically. In this topic, you will learn what fractions are, what the numerator and denominator mean, how to represent fractions visually and on a number line, and how fractions connect to division. Fractions are one of the most important topics in mathematics — they appear in cooking (half a cup of sugar), in time (quarter past 3), in sports (a batsman scoring in 3 out of 5 matches), in discounts (25% off means 1/4 off), and in science (measuring chemicals). By the end of this chapter, you will see fractions everywhere and understand exactly what they mean. Let us begin by cutting that pizza.
What is Introduction to Fractions?
A fraction is a number that represents one or more equal parts of a whole. It is written as one number over another, separated by a horizontal line:
a/b (read as "a upon b" or "a over b" or "a by b")
where:
a is the numerator — it tells you how many parts you have.
b is the denominator — it tells you how many equal parts the whole is divided into.
The horizontal line between them is called the fraction bar or vinculum.
Example: In the fraction 3/4:
Numerator = 3 (you have 3 parts)
Denominator = 4 (the whole is divided into 4 equal parts)
Read as: "three-fourths" or "three quarters"
Important Rule: The denominator can never be zero. Division by zero is undefined, so fractions like 5/0 do not exist.
Fraction as Division: Every fraction represents a division. a/b means a divided by b.
3/4 = 3 ÷ 4 = 0.75
1/2 = 1 ÷ 2 = 0.5
So a fraction is another way of writing a division that has not been carried out yet.
Visual Representation: Fractions can be shown using shapes divided into equal parts. To show 3/5, draw a rectangle, divide it into 5 equal parts, and shade 3 of them. The shaded region represents 3/5 of the whole rectangle. You can also use circles (like a pizza), number lines, or sets of objects.
Fraction on a Number Line: Fractions live between whole numbers on the number line. To plot 3/4: divide the segment between 0 and 1 into 4 equal parts. Count 3 parts from 0. That point is 3/4. To plot 5/3: since 5/3 = 1 and 2/3, it lies between 1 and 2 on the number line — divide the segment between 1 and 2 into 3 equal parts and count 2 parts from 1.
Types and Properties
Fractions come in several types based on the relationship between the numerator and denominator:
1. Proper Fractions
A fraction where the numerator is less than the denominator: numerator < denominator.
Examples: 1/2, 3/4, 5/8, 7/10, 11/15.
The value of a proper fraction is always less than 1 (it represents less than one whole).
On the number line, proper fractions lie between 0 and 1.
Visualise: if you cut a pizza into 8 slices and take 5, you have 5/8 — which is less than the full pizza.
2. Improper Fractions
A fraction where the numerator is greater than or equal to the denominator: numerator >= denominator.
Examples: 5/3, 7/4, 9/2, 12/5, 8/8.
The value of an improper fraction is 1 or greater (it represents one whole or more).
On the number line, improper fractions lie at 1 or to the right of 1.
Visualise: 5/3 means you have 5 parts when each whole has 3 parts — that is more than one whole pizza!
3. Mixed Numbers (Mixed Fractions)
A mixed number combines a whole number and a proper fraction.
Examples: 1 1/2, 2 3/4, 5 2/3.
1 1/2 means 1 whole and 1/2 more. It equals 3/2 as an improper fraction.
Every improper fraction can be written as a mixed number, and every mixed number can be written as an improper fraction.
Converting Improper Fraction to Mixed Number:
Divide the numerator by the denominator. The quotient is the whole number part, and the remainder becomes the numerator of the fractional part.
Example: 17/5 → 17 ÷ 5 = 3 remainder 2 → 3 2/5.
Converting Mixed Number to Improper Fraction:
Multiply the whole number by the denominator, add the numerator. This becomes the new numerator over the same denominator.
Example: 3 2/5 → (3 × 5 + 2) / 5 = 17/5.
4. Unit Fractions
A fraction with numerator 1.
Examples: 1/2, 1/3, 1/4, 1/5, 1/10, 1/100.
These are the simplest fractions and the building blocks of all other fractions. 3/4 = 1/4 + 1/4 + 1/4 (three unit fractions of 1/4).
5. Equivalent Fractions
Fractions that represent the same value, even though they look different.
1/2 = 2/4 = 3/6 = 4/8 = 5/10 — all of these are the same amount!
To find equivalent fractions, multiply (or divide) both the numerator and denominator by the same non-zero number.
1/2 × 3/3 = 3/6. Since we multiplied top and bottom by 3, 1/2 and 3/6 are equivalent.
6. Like and Unlike Fractions
Like fractions have the same denominator: 2/7, 3/7, 5/7 (all have denominator 7).
Unlike fractions have different denominators: 1/3, 2/5, 3/8 (denominators 3, 5 and 8 are different).
Like fractions are easy to compare and add. Unlike fractions need to be converted to like fractions first.
Solved Examples
Example 1: Example 1: Identifying numerator and denominator
Problem: For each fraction, identify the numerator and denominator: (a) 3/7, (b) 11/4, (c) 1/100, (d) 8/8.
Solution:
(a) 3/7: Numerator = 3, Denominator = 7. This means 3 parts out of 7 equal parts.
(b) 11/4: Numerator = 11, Denominator = 4. This means 11 parts when each whole has 4 parts (more than 2 wholes).
(c) 1/100: Numerator = 1, Denominator = 100. This is one part out of 100 equal parts (a very tiny fraction — 1 percent!).
(d) 8/8: Numerator = 8, Denominator = 8. This means 8 parts out of 8 — which is the complete whole = 1.
Example 2: Example 2: Representing fractions visually
Problem: Represent 2/5 using a rectangle.
Solution:
Step 1: Draw a rectangle.
Step 2: Divide it into 5 equal vertical strips (because the denominator is 5).
Step 3: Shade 2 of the 5 strips (because the numerator is 2).
Step 4: The shaded region represents 2/5 of the rectangle.
You can verify: the shaded part is less than half the rectangle (2/5 < 1/2), which looks right visually — 2 strips out of 5 is less than 2.5 strips out of 5.
Example 3: Example 3: Fraction as part of a group
Problem: There are 12 apples. 5 are red and the rest are green. What fraction of the apples are (a) red? (b) green?
Solution:
Total apples = 12. Red apples = 5. Green apples = 12 - 5 = 7.
(a) Fraction of red apples = 5/12 (5 out of 12).
(b) Fraction of green apples = 7/12 (7 out of 12).
Check: 5/12 + 7/12 = 12/12 = 1 (the whole). ✓ The red and green fractions should add up to 1 because all apples are either red or green.
Example 4: Example 4: Plotting fractions on a number line
Problem: Plot 3/4 and 5/4 on a number line.
Solution:
Step 1: Draw a number line and mark 0, 1 and 2.
Step 2: For fractions with denominator 4, divide each unit segment into 4 equal parts.
Step 3: To plot 3/4: from 0, count 3 parts to the right. Mark this point as 3/4. It lies between 0 and 1, closer to 1.
Step 4: To plot 5/4: this is an improper fraction (5 > 4), so it is beyond 1. From 0, count 5 parts (or from 1, count 1 more part). Mark this point as 5/4. It lies between 1 and 2.
Note: 5/4 = 1 1/4 as a mixed number.
Example 5: Example 5: Converting improper fraction to mixed number
Problem: Convert 23/7 to a mixed number.
Solution:
Step 1: Divide 23 by 7.
23 ÷ 7 = 3 remainder 2.
Step 2: The quotient (3) is the whole number part.
The remainder (2) is the numerator of the fraction part.
The denominator stays the same (7).
23/7 = 3 2/7
Check: 3 2/7 = (3 × 7 + 2)/7 = (21 + 2)/7 = 23/7 ✓
Example 6: Example 6: Converting mixed number to improper fraction
Problem: Convert 4 3/5 to an improper fraction.
Solution:
Step 1: Multiply the whole number by the denominator: 4 × 5 = 20.
Step 2: Add the numerator: 20 + 3 = 23.
Step 3: Write over the same denominator: 23/5.
4 3/5 = 23/5
Check: 23 ÷ 5 = 4 remainder 3 = 4 3/5 ✓
Example 7: Example 7: Finding equivalent fractions
Problem: Find three fractions equivalent to 2/3.
Solution:
Multiply both numerator and denominator by the same number:
2/3 × 2/2 = 4/6
2/3 × 3/3 = 6/9
2/3 × 4/4 = 8/12
So 2/3 = 4/6 = 6/9 = 8/12.
Verification: All these fractions simplify to 2/3. For example, 8/12: divide both by 4 → 2/3 ✓.
Example 8: Example 8: Real-world fractions — sharing equally
Problem: 3 friends share 2 rotis equally. How much roti does each person get?
Solution:
2 rotis shared among 3 people = 2 ÷ 3 = 2/3.
Each person gets 2/3 of a roti.
Visual method: Cut each roti into 3 equal parts. That gives 2 × 3 = 6 pieces total. Each person gets 6 ÷ 3 = 2 pieces. Each piece is 1/3 of a roti, so 2 pieces = 2/3 of a roti. ✓
Example 9: Example 9: Fractions in daily life
Problem: Express each as a fraction: (a) A cricket team won 7 out of 10 matches. (b) A test has 20 questions and a student answered 13 correctly. (c) 45 minutes as a fraction of an hour.
Solution:
(a) Fraction of matches won = 7/10.
(b) Fraction answered correctly = 13/20.
(c) 1 hour = 60 minutes. So 45 minutes = 45/60. Simplify by dividing both by 15: 45/60 = 3/4.
45 minutes is 3/4 of an hour.
Example 10: Example 10: Comparing unit fractions
Problem: Arrange in ascending order: 1/2, 1/5, 1/3, 1/8, 1/4.
Solution:
For unit fractions (numerator = 1), the fraction with the larger denominator is smaller. This is because the whole is divided into more parts, making each part smaller.
Think of cutting a pizza:
- Cut into 2 pieces → each piece is big (1/2).
- Cut into 8 pieces → each piece is small (1/8).
Ascending order (smallest to largest):
1/8 < 1/5 < 1/4 < 1/3 < 1/2
Answer: 1/8, 1/5, 1/4, 1/3, 1/2
Real-World Applications
Fractions are everywhere in daily life:
Cooking and Recipes: Almost every recipe uses fractions — half a cup of sugar, one-third teaspoon of salt, three-quarters of a litre of milk. Understanding fractions is essential for following recipes and adjusting quantities. If a recipe serves 4 but you want to serve 6, you need to multiply every fraction by 6/4 = 3/2.
Time: We use fractions of an hour all the time. "Quarter past 3" means 3 and 1/4 hours. "Half an hour" means 1/2 hour = 30 minutes. "Quarter to 5" means 15 minutes before 5. Digital time like 3:45 can be written as 3 3/4 hours.
Money: Prices often involve fractions. "50 paise" is 1/2 of a rupee. "25 paise" is 1/4 of a rupee. Discounts like "1/3 off" or "25% off" (which is 1/4 off) require fraction understanding to calculate savings.
Measurement: When you measure length, you often get fractions — 2 1/2 cm, 5 3/4 inches. Tailors, carpenters and engineers work with fractional measurements daily. Even your height might be 4 feet 11 1/2 inches.
Sports: A batsman's average involves fractions (total runs divided by innings). A team's win rate is a fraction (wins divided by total matches). Bowling averages, strike rates and economy rates are all fractions.
Science: In chemistry, concentrations are fractions. A "25% solution" means 1/4 of the solution is the active ingredient. In physics, fractional errors help measure accuracy. In biology, genetic probabilities use fractions (1/4 chance of a trait appearing).
Maps and Scales: A map scale of 1:100000 means 1 cm on the map represents 100000 cm (1 km) in real life. This ratio is essentially a fraction.
Key Points to Remember
- A fraction represents equal parts of a whole. It is written as a/b where a is the numerator (parts you have) and b is the denominator (total equal parts).
- The denominator can never be zero.
- A fraction is another way of writing division: a/b = a divided by b.
- Proper fraction: numerator < denominator (value less than 1). Improper fraction: numerator >= denominator (value 1 or more).
- Mixed number = whole number + proper fraction. Example: 2 3/4.
- To convert improper to mixed: divide numerator by denominator. Quotient = whole part, remainder = new numerator.
- To convert mixed to improper: (whole x denominator + numerator) / denominator.
- Equivalent fractions have the same value: 1/2 = 2/4 = 3/6. Multiply or divide both parts by the same number.
- Unit fractions have numerator 1. For unit fractions, larger denominator means smaller fraction: 1/10 < 1/5.
- Fractions can represent parts of a whole object, parts of a group, or points on a number line.
Practice Problems
- Identify the numerator and denominator of: 7/9, 15/4, 1/1000, 6/6.
- Draw a circle and shade the region representing 3/8 of the circle.
- A class has 40 students. 15 are boys. What fraction are boys? What fraction are girls?
- Plot 2/5 and 7/5 on a number line.
- Convert to mixed numbers: 19/4, 31/7, 50/9.
- Convert to improper fractions: 3 1/2, 5 2/3, 7 4/5.
- Write 3 fractions equivalent to 4/5.
- Arrange in ascending order: 1/6, 1/2, 1/4, 1/3, 1/10.
Frequently Asked Questions
Q1. What is a fraction in simple words?
A fraction is a part of a whole. When you cut a cake into 4 equal pieces and take 1 piece, you have 1/4 (one-fourth) of the cake. The bottom number (denominator) tells you how many equal parts the whole is divided into. The top number (numerator) tells you how many of those parts you have.
Q2. Why can the denominator not be zero?
The denominator tells you how many equal parts the whole is divided into. You cannot divide something into zero parts — it does not make sense. Mathematically, division by zero is undefined. So fractions like 5/0 or 3/0 are not valid numbers.
Q3. What is the difference between a proper and an improper fraction?
In a proper fraction, the numerator is smaller than the denominator (like 3/4), so the value is less than 1 — you have less than one whole. In an improper fraction, the numerator is equal to or larger than the denominator (like 5/3), so the value is 1 or more — you have one whole or more. Improper fractions can be converted to mixed numbers: 5/3 = 1 2/3.
Q4. What are equivalent fractions?
Equivalent fractions look different but represent the same amount. For example, 1/2, 2/4, 3/6 and 4/8 are all equivalent — they all represent half. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. It is like cutting the same pizza into more slices — half is still half whether you cut it into 2 pieces or 100 pieces.
Q5. How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, then add the numerator. Put this result over the same denominator. For example, 3 2/5: multiply 3 x 5 = 15, add 2 = 17, so the answer is 17/5. Check: 17 divided by 5 = 3 remainder 2 = 3 2/5.
Q6. Is 5/5 a proper or improper fraction?
5/5 is an improper fraction because the numerator (5) is equal to the denominator (5). Its value is exactly 1. Any fraction where the numerator equals the denominator equals 1: 3/3 = 1, 100/100 = 1. These are improper fractions that happen to equal a whole number.
Q7. How are fractions used in everyday life?
Fractions are used constantly: half a glass of water (1/2), quarter of an hour (1/4 = 15 minutes), 3/4 of a recipe, 2/3 of a journey, 25% discount (1/4 off), and measuring lengths like 2 1/2 cm. Sports statistics (batting average), music (time signatures like 3/4), and even phone battery (75% = 3/4 full) all use fractions.
Q8. What is a unit fraction?
A unit fraction has 1 as its numerator: 1/2, 1/3, 1/4, 1/5, 1/100 are all unit fractions. They represent exactly one equal part of the whole. Every fraction can be thought of as a sum of unit fractions: 3/4 = 1/4 + 1/4 + 1/4. Unit fractions with larger denominators are smaller: 1/100 is much smaller than 1/2.
Q9. How do you compare unit fractions?
For unit fractions (where the numerator is 1), the fraction with the smaller denominator is the larger fraction. 1/2 > 1/3 > 1/4 > 1/5 > 1/10. Think of cutting a pizza: cutting into 2 pieces gives bigger pieces than cutting into 10 pieces. So 1/2 of a pizza is bigger than 1/10 of a pizza.
Q10. Is every whole number a fraction?
Yes! Any whole number can be written as a fraction with denominator 1. For example: 5 = 5/1, 12 = 12/1, 0 = 0/1. This means whole numbers are special fractions where the denominator is 1. Every whole number is a fraction, but not every fraction is a whole number.
