Fractions – Class 6 Maths Chapter 7 Notes & NCERT PDF

Class 6MathematicsChapter 7NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 7 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 7: Fractions

Introduction

Chapter 7 of Ganita Prakash builds up fractions from sharing. Shabnam and Mukta ask: if one roti is shared between 2 children, each gets ½. If it is shared among 4, each gets ¼.

More people sharing the same roti means a smaller share, so ½ > ¼.

Friends like Beni, Arvin, Jaya and Meena then explore fractional units, the number line, mixed numbers, equivalent fractions, comparing, and adding and subtracting. The chapter ends with the rich history of fractions in India.

Key Ideas

7.1 Fractional units and equal shares

When one whole is divided into equal parts, each part is a fractional unit (or unit fraction): ½, ⅓, ¼, ⅕, …, 1/100, …

Beni thinks 1/9 > 1/5 because 9 > 5. That is a common mistake. Sharing one roti among 9 people gives each person less, so 1/9 < 1/5. In the same way, 1/200 < 1/100.

Three guavas together weigh 1 kg, so each weighs about ⅓ kg.

Knowledge from the past: the Rig Veda calls ¾ tri-pada. This matches words still used today, like teen paav in Hindi and mukkaal in Tamil.

7.2 Fractional units as parts of a whole

A whole chikki is broken into pieces. The bigger piece contains 3 pieces of ¼ chikki, so it is ¾ of the chikki.

Cutting a chikki into 6 equal parts in different ways gives 1/6 pieces of different shapes, but all of them are the same size.

7.3 Measuring using fractional units

Fold a paper strip (1 unit long) in half, then in half again: ½, ¼, ⅛. Quantities are counted in fractional units: 3 halves = ½ + ½ + ½, and 5 halves = 5 times ½.

It helps to read ¾ as "3 times ¼". This shows both the fractional unit (¼) and how many of them there are (3). In ⅚, 5 is the numerator and 6 is the denominator.

7.4 Fractions on the number line

Divide the length from 0 to 1 into equal parts, for example 2, 3, 5 or 8 parts, to mark fractions. There are infinitely many fractions between 0 and 1.

7.5 Mixed fractions

In fractions less than 1, the numerator is smaller than the denominator.

In fractions greater than 1, the numerator is larger than the denominator.

7/3 = ⅓ × 7 = 1 + 1 + ⅓ = 2⅓, read "two and one third".

A mixed number has a whole part and a fractional part less than 1.

Jaya's method for turning a mixed number into a fraction: 3¾ = 1 + 1 + 1 + ¾ = 4/4 + 4/4 + 4/4 + ¾ = 15/4.

7.6 Equivalent fractions

Using paper strips and a fraction wall: ½ = 2/4 = 3/6 = 4/8 = 5/10. These are equivalent fractions, the same length written with different fractional units.

By sharing: 1 roti among 2, 2 rotis among 4 and 3 rotis among 6 all give the same share.

As facts: 1 roti among 4 children is 1 ÷ 4 = ¼, and 1 = ¼ + ¼ + ¼ + ¼ = 4 × ¼.

Comparing shares: is ½ chikki or 4/7 chikki more?

½ = 4/8.

4 chikkis shared by 7 children gives more each than 4 shared by 8, so 4/7 > ½.

To compare fractions with different denominators, make the fractional units the same. For example, ¾ = 30/40 and 7/10 = 28/40, so ¾ > 7/10. Using 20ths also works: 15/20 and 14/20.

A fraction is in its lowest terms (simplest form) when its numerator and denominator have no common factor except 1. Divide both by their highest common factor.

16/20 ÷ 4/4 = 4/5. And 36/60 = 18/30 = 9/15 = 3/5, or divide straight away by 12.

7.7 Comparing fractions

Step 1: rewrite the fractions with the same denominator (use a common multiple of the denominators).

Step 2: compare the numerators.

For example, 4/5 = 36/45 and 7/9 = 35/45, so 4/5 > 7/9. And 7/9 = 49/63 while 17/21 = 51/63, so 7/9 < 17/21.

7.8 Adding and subtracting fractions

Meena eats ½ of a chikki and her brother eats ¼. Together they eat ½ + ¼ = 2/4 + ¼ = ¾, and ¼ is left.

Same fractional unit: add the numerators. 2/5 + 1/5 = 3/5, and 4/7 + 6/7 = 10/7 = 1 3/7.

Different units: Brahmagupta's method (628 CE):

1. Find equivalent fractions with a common denominator.

2. Add (or subtract) the numerators.

3. Write the answer in lowest terms.

Examples: ¼ + ⅓ = 3/12 + 4/12 = 7/12; ⅔ + ⅕ = 10/15 + 3/15 = 13/15; ⅙ + ⅓ = 3/6 = ½.

Subtracting: 6/7 − 4/7 = 2/7; ¾ − ⅔ = 9/12 − 8/12 = 1/12.

7.9 A pinch of history

In Sanskrit a fraction was called bhinna ("broken"), or bhaga / ansha ("part").

The Bakshali manuscript (around 300 CE) wrote ½ with 1 above 2, almost as we do today. This notation was used by Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE) and Mahaviracharya (c. 850 CE).

The line between numerator and denominator was added later by the Moroccan mathematician Al-Hassar (12th century).

Egyptians and Babylonians mainly used unit fractions. Writing a fraction as a sum of different unit fractions is called an Egyptian fraction, for example 19/24 = ½ + ⅙ + ⅛.

General fractions and their arithmetic were first developed in India (even the Sulba-sutras use them). They reached Europe through the Arabs and were in general use there by about the 17th century.

Worked Examples

Write 9/2 as a mixed number. 9 halves = 4 wholes (8 halves) + ½, so 9/2 = 4½.

Compare ¾ and 7/10. A common denominator is 20: ¾ = 15/20 and 7/10 = 14/20, so ¾ > 7/10.

Reduce 36/60 to lowest terms. HCF(36, 60) = 12, and 36 ÷ 12 = 3, 60 ÷ 12 = 5, so the answer is 3/5.

Geeta buys ⅖ m of lace and Shamim buys ¾ m. Is it enough for a 1 m border? ⅖ + ¾ = 8/20 + 15/20 = 23/20 = 1 3/20 m. That is more than 1 m, so it is enough.

Jaya's school is 7/10 km away. She rides ½ km by auto. How far does she walk? 7/10 − 5/10 = 2/10 = ⅕ km.

Add 2/7 + 5/7 + 6/7. They have the same unit, so add the numerators: 13/7 = 1 6/7.

Figure it Out: Hints

To say which fraction is bigger, first ask: is the fractional unit the same? If not, make it the same.

The product of the two denominators is always a common denominator. The smallest common multiple keeps the numbers smaller.

Before writing a final answer, check whether it can be put in lowest terms.

Puzzle: two different unit fractions can never add up to 1, because ½ is the largest unit fraction and ½ + ½ = 1 uses the same one twice.

Maths Words

Fraction (भिन्न)

Fractional unit / unit fraction (इकाई भिन्न)

Numerator (अंश)

Denominator (हर)

Mixed number (मिश्र भिन्न)

Equivalent fractions (तुल्य भिन्न)

Lowest terms / simplest form (न्यूनतम रूप)

Fraction wall (भिन्न दीवार)

Number line (संख्या रेखा)

Common denominator (सार्व हर)

Ascending / descending order (आरोही / अवरोही क्रम)

Sum / difference (योग / अंतर)

Common Mistakes

Thinking a bigger denominator means a bigger fraction. In fact 1/9 < 1/5.

Adding the denominators: ½ + ¼ is not 2/6. Make the units the same first: ¾.

Comparing fractions with different denominators by looking only at the numerators.

Forgetting to simplify, for example leaving 3/6 instead of ½.

Making a mistake when converting mixed numbers: 3¾ = (3 × 4 + 3)/4 = 15/4, not 9/4.

Maths in Daily Life

Sharing rotis, chikki or juice equally.

Recipes ("½ cup sugar").

Measuring cloth and lace.

Quarter and half hours on a clock.

Words like paav, aadha, dedh and dhai are fractions we use every day.

💡 Key Learning Points

✓Understand fractional units as equal shares, and why 1/9 < 1/5
✓Mark fractions on the number line and convert between mixed numbers and fractions (e.g., 3¾ = 15/4)
✓Find equivalent fractions with a fraction wall or by sharing, and reduce fractions to lowest terms
✓Compare fractions by rewriting them with a common denominator
✓Add and subtract fractions using Brahmagupta's method, and know the Indian history of fractions

👨‍🏫 Teaching Tips

→Begin with real sharing: cut rotis or paper circles among 2, 4 and 8 students to feel why more sharers means smaller shares.
→Have every student fold paper strips into halves, quarters and eighths and build their own fraction wall.
→Ask students to collect fraction words from home languages (paav, aadha, dedh, mukkaal) and arrange them by size.
→Draw a large number line on the floor and let students stand at ½, ¾, 5/4, etc.
→Use chikki or chocolate bars to show ½ + ¼ = ¾ before writing it with numbers.
→Teach Brahmagupta's method as a three-step recipe and let pairs check each other's common denominators.
→Discuss the Bakshali manuscript and Brahmagupta's verse so students see fractions as a living part of Indian heritage.
→Set the Egyptian-fraction puzzle as an enrichment challenge for fast finishers.

📋 Assessment Questions

Can the student explain with a sharing story why 1/9 is smaller than 1/5?
Can the student convert 7/3 to a mixed number and 3¾ to a fraction?
Can the student reduce 36/60 to lowest terms and name equivalent fractions of ½?
Can the student compare ¾ and 7/10 using a common denominator?
Can the student add ⅔ + ⅕ and subtract ¾ − ⅔ correctly, simplifying the answer?

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