Fractions — Types, Operations & Solved Examples

Imagine you have one roti and you want to share it equally between two people. Each person gets half the roti. That “half” is a fraction — a way of showing a part of a whole thing. We use fractions every single day without even realising it.

What Is a Fraction?

A fraction tells us how many parts of a whole we have. When we cut something into equal pieces and take some of those pieces, we write it as a fraction.

A fraction has two parts separated by a line:

The numerator (top number) — how many parts we have
The denominator (bottom number) — how many equal parts the whole is divided into

Fraction = Numerator / Denominator

Example: 3/8 means we have 3 parts out of 8 equal parts

Think of a watermelon cut into 8 equal slices. If you eat 3 slices, you have eaten 3/8 of the watermelon. Your friend has 5/8 left.

The denominator can never be zero. We cannot divide something into zero parts — that makes no sense. Always check that your denominator is a non-zero number.

Types of Fractions

Proper Fractions

A proper fraction has its numerator smaller than its denominator. The value of a proper fraction is always less than 1.

Examples: 1/2, 3/4, 5/7, 2/9

If you eat 2 pieces from a pizza cut into 6 slices, you have eaten 2/6 of the pizza. Since 2 is less than 6, this is a proper fraction.

Improper Fractions

An improper fraction has its numerator equal to or greater than its denominator. The value is always 1 or more than 1.

Examples: 7/3, 5/5, 9/4, 11/6

Suppose you need 7 half-rotis for a big family lunch. We write this as 7/2. Since 7 is bigger than 2, it is an improper fraction.

Mixed Fractions (Mixed Numbers)

A mixed fraction is a combination of a whole number and a proper fraction. We use it to show improper fractions in a friendlier way.

Step 1: Divide numerator by denominator Step 2: Quotient = whole number part Step 3: Remainder = new numerator (denominator stays same)

Example: 7/3 7 ÷ 3 = 2 remainder 1 So 7/3 = 2 and 1/3 (written as 2⅓)

If you have 7/3, it means 2 whole parts and 1/3 more. Think of it as having 2 full glasses of juice and one-third of another glass.

To convert a mixed fraction back to an improper fraction: multiply the whole number by the denominator, then add the numerator. Example: 2⅓ = (2 × 3 + 1)/3 = 7/3

Equivalent Fractions

Equivalent fractions look different but they are equal in value. Cutting a chocolate bar into 2 pieces and taking 1 is the same as cutting it into 4 pieces and taking 2. Both are half the chocolate.

1/2 = 2/4 = 3/6 = 4/8 — all of these are the same amount.

How to find equivalent fractions:

Multiply or divide both the numerator and denominator by the same number (but never by zero).

1/2 × 3/3 = 3/6 ✓ 3/6 ÷ 3/3 = 1/2 ✓

Rule: Multiply or divide numerator and denominator by the same number

Simplest form (lowest terms): A fraction is in its simplest form when the numerator and denominator have no common factor other than 1. To simplify, divide both by their HCF (Highest Common Factor).

Example: Simplify 12/16 HCF of 12 and 16 is 4 12 ÷ 4 = 3, and 16 ÷ 4 = 4 So 12/16 = 3/4 in simplest form.

Comparing Fractions

Comparing Like Fractions (Same Denominator)

When the denominators are the same, just compare the numerators. The fraction with the bigger numerator is greater.

3/7 vs 5/7 → since 5 > 3, we get 5/7 > 3/7

Think of two friends sharing a birthday cake cut into 7 pieces. The one who gets 5 pieces has more cake than the one who gets 3 pieces.

Comparing Unlike Fractions (Different Denominators)

When the denominators are different, we first find the LCM (Least Common Multiple) of the denominators, then convert both fractions to equivalent fractions with that LCM as the denominator.

Example: Compare 3/4 and 2/3

Step 1: Find LCM of 4 and 3 → LCM = 12

Step 2: Convert both fractions:

3/4 = 9/12 (multiply top and bottom by 3)
2/3 = 8/12 (multiply top and bottom by 4)

Step 3: Compare → 9/12 > 8/12, so 3/4 > 2/3

In exams, always write out all the steps when comparing unlike fractions. Finding LCM and showing the conversion will get you full marks.

Adding Fractions

Adding Like Fractions

When the denominators are the same, add only the numerators. The denominator stays the same.

a/c + b/c = (a + b)/c

Example: 2/7 + 3/7 = (2 + 3)/7 = 5/7

Priya ate 2/8 of a mango and her brother ate 3/8. Together they ate 2/8 + 3/8 = 5/8 of the mango.

Adding Unlike Fractions

When the denominators are different, we must first make them the same by finding the LCM.

Example: Add 1/3 + 1/4

Step 1: LCM of 3 and 4 = 12

Step 2: Convert:

1/3 = 4/12
1/4 = 3/12

Step 3: Add → 4/12 + 3/12 = 7/12

Step 1: Find LCM of denominators Step 2: Convert each fraction (keep value the same) Step 3: Add the numerators Step 4: Simplify if needed

Subtracting Fractions

The rules are exactly the same as addition — same denominator means subtract the numerators; different denominators means find LCM first.

Like fractions: 5/9 − 2/9 = 3/9 = 1/3

Unlike fractions: 3/4 − 1/6

LCM of 4 and 6 = 12
3/4 = 9/12, and 1/6 = 2/12
9/12 − 2/12 = 7/12

Ravi had 3/4 of a bottle of lemonade. He drank 1/6 of the bottle. He is left with 7/12 of the bottle.

Word Problems with Fractions

Learning to solve word problems helps us use fractions in real life. Here’s a method that always works:

Step 1: Read the problem carefully and identify what fraction operation is needed. Step 2: Write the fractions clearly. Step 3: Solve step by step. Step 4: Check if the answer makes sense.

Problem: A farmer has a field. He planted wheat in 2/5 of the field and rice in 1/5 of the field. What fraction of the field is used for crops?

Solution: 2/5 + 1/5 = 3/5 of the field is used for crops. The remaining 2/5 is unused.

Problem: Seema finished 3/8 of her homework before dinner and 2/8 after dinner. What fraction is still left?

Total done = 3/8 + 2/8 = 5/8 Fraction left = 1 − 5/8 = 8/8 − 5/8 = 3/8

5 Common Mistakes to Avoid

Mistake 1: Adding denominators Wrong: 1/3 + 1/4 = 2/7 Right: 1/3 + 1/4 = 4/12 + 3/12 = 7/12 Never add the denominators together!

Mistake 2: Forgetting to simplify When the question asks for the answer in simplest form, always divide by the HCF. 6/8 should be simplified to 3/4.

Mistake 3: Mixing up numerator and denominator In 3/7, the numerator is 3 (top) and the denominator is 7 (bottom). The denominator tells us the total number of equal parts.

Mistake 4: Making errors when converting mixed to improper fractions For 3⅖: multiply 3 × 5 = 15, then add 2 = 17. So 3⅖ = 17/5. Many students write 35/5 by just placing the digits next to each other.

Mistake 5: Using wrong LCM when adding unlike fractions Always double-check your LCM. The LCM of 4 and 6 is 12, not 24. Using a larger common multiple gives a correct answer but needs more simplification.

Practice Questions

Q1. Write three equivalent fractions for 2/3.

Multiply numerator and denominator by 2, 3, and 4:

2/3 × 2/2 = 4/6
2/3 × 3/3 = 6/9
2/3 × 4/4 = 8/12

Answer: 4/6, 6/9, and 8/12 are three equivalent fractions of 2/3.

Q2. Is 5/9 a proper or improper fraction? What about 11/7?

5/9 is a proper fraction because 5 (numerator) < 9 (denominator). 11/7 is an improper fraction because 11 (numerator) > 7 (denominator).

Q3. Convert 17/4 into a mixed fraction.

Divide 17 by 4: 17 ÷ 4 = 4 remainder 1 So 17/4 = 4¼ (4 and one-quarter)

Q4. Which is greater: 5/6 or 7/9?

LCM of 6 and 9 = 18 5/6 = 15/18 7/9 = 14/18 Since 15 > 14, we get 5/6 > 7/9. Answer: 5/6 is greater.

Q5. Add 3/8 + 5/8 and simplify if possible.

Same denominators, so add numerators: 3/8 + 5/8 = 8/8 = 1 HCF of 8 and 8 is 8, so 8/8 = 1 (a whole number). Answer: 1

Q6. Subtract 2/5 from 4/5.

4/5 − 2/5 = (4 − 2)/5 = 2/5 Answer: 2/5

Q7. Add 1/2 + 1/6.

LCM of 2 and 6 = 6 1/2 = 3/6 1/6 = 1/6 3/6 + 1/6 = 4/6 = 2/3 (simplified by dividing by HCF = 2) Answer: 2/3

Q8. Aarav ate 3/7 of a pizza. His sister ate 2/7. What fraction is left?

Total eaten = 3/7 + 2/7 = 5/7 Fraction left = 1 − 5/7 = 7/7 − 5/7 = 2/7 Answer: 2/7 of the pizza is left.

Frequently Asked Questions

Q: Can a fraction equal a whole number?

Yes! When the numerator is a multiple of the denominator, the fraction equals a whole number. For example, 8/4 = 2, and 15/5 = 3. Also 8/8 = 1.

Q: What is a unit fraction?

A unit fraction has 1 as the numerator. Examples: 1/2, 1/3, 1/7. These are the building blocks of all fractions — any fraction is a multiple of a unit fraction.

Q: Are 2/4 and 1/2 the same?

Yes, they are equivalent fractions. They represent the same amount. Half a pizza is the same whether you cut it into 2 equal parts and take 1, or cut it into 4 equal parts and take 2.

Q: When do we use fractions in daily life?

We use fractions all the time — sharing food, telling time (quarter past 3 means 3¼), measuring ingredients while cooking (½ cup of sugar), and even when we say something is “half-price.”

Q: What if the numerator is 0?

A fraction with 0 as the numerator equals 0. For example, 0/5 = 0. We have zero parts out of 5, which means nothing.

Q: How is a fraction different from a ratio?

A fraction shows a part of a whole. A ratio compares two separate quantities. But they are written the same way, so context matters. 3/5 as a fraction means 3 parts out of 5 equal parts of one thing. 3:5 as a ratio means comparing 3 of one thing with 5 of another thing.