NCERT-aligned: The current Grade 7 Ganita Prakash Part I builds on prior fraction knowledge and focuses on multiplication and division of fractions, including the methods of the Indian mathematician Brahmagupta and problem-solving contexts. citeturn1search0turn1view0

🌟 Chapter at a Glance

1. 🍕 Fractions Represent Quantities

A fraction a/b represents a quantity made from equal parts. The denominator tells how many equal parts make one whole, while the numerator tells how many of those parts are being considered.

Fractions can represent parts of a whole, points on a number line, ratios and quantities produced by division. Equivalent fractions have the same value: 1/2 = 2/4 = 4/8.

2. ✖️ Multiplying Fractions

For positive fractions, multiply numerators together and denominators together:

a/b × c/d = ac/bd

Example: 2/3 × 5/7 = 10/21.

What does multiplication mean?

When we find 3/4 of 20, we mean 3/4 × 20 = 15. Here multiplication is not simply repeated whole-number addition; it can mean taking a fractional part of a quantity.

3. ✂️ Simplify Before Multiplying

Cancellation can make calculations easier. For 6/15 × 10/21, cancel common factors across numerator and denominator before multiplying. This reduces the arithmetic and lowers the chance of error.

Important: cancellation is division by a common non-zero factor; it does not mean deleting digits.

4. ➗ Dividing Fractions

Division asks how many groups of one quantity fit into another. For non-zero fractions, dividing by a fraction is equivalent to multiplying by its reciprocal:

a/b ÷ c/d = a/b × d/c

Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.

5. 🔄 Reciprocal

The reciprocal of a non-zero fraction a/b is b/a. Their product is 1:

a/b × b/a = 1

For example, the reciprocal of 7/9 is 9/7. The reciprocal of 5 is 1/5.

6. 🧠 Why Does “Invert and Multiply” Work?

Dividing by c/d asks for a number x such that (c/d) × x equals the original quantity. Since d/c is the reciprocal of c/d, multiplying by d/c reverses the effect:

c/d × d/c = 1.

This gives a reason for the rule instead of treating it as a trick to memorise.

7. 🥛 Mixed Numbers and Improper Fractions

A mixed number such as 2 1/3 contains 2 wholes and 1/3. To convert it to an improper fraction:

(2 × 3 + 1)/3 = 7/3.

To convert 7/3 back: 7 ÷ 3 gives 2 wholes and remainder 1, so 7/3 = 2 1/3.

8. 🇮🇳 Brahmagupta and Fraction Operations

Brahmagupta was an Indian mathematician associated with the 7th century. His work included rules for arithmetic with fractions. Studying these methods helps us see that mathematical ideas have developed through many cultures and historical traditions.

9. 🧮 Worked Examples

1. 2/5 × 15: 30/5 = 6.
2. 3/8 × 4/9: 12/72 = 1/6.
3. 5/6 ÷ 10/9: 5/6 × 9/10 = 45/60 = 3/4.
4. 2 1/4 × 4/9: 9/4 × 4/9 = 1.
5. 7/8 ÷ 7/16: 7/8 × 16/7 = 2.

🎯 Think & Explore

1. Why is 1/2 × 1/2 smaller than 1/2?
2. Can multiplying two positive fractions give an answer greater than both fractions? Find examples.
3. Explain why division by a fraction smaller than 1 can make a number larger.
4. Use a number line or area model to explain 2/3 × 3/4.

🧩 Interactive MCQs

1. 3/5 × 10 = A. 2 B. 6 C. 15 D. 30
2. The reciprocal of 8/11 is: A. 8/11 B. 11/8 C. −11/8 D. 1/11
3. 2/3 ÷ 4/5 equals: A. 8/15 B. 5/6 C. 6/5 D. 2/3
4. 2 1/2 as an improper fraction is: A. 5/2 B. 4/2 C. 3/2 D. 7/2
5. Which statement is always true for non-zero a/b? A. a/b + b/a = 1 B. a/b × b/a = 1 C. a/b ÷ b/a = 1 D. a/b = b/a

Answers: 1-B, 2-B, 3-B, 4-A, 5-B

📌 Case-Based Challenge — The School Garden

A school uses 3/5 of a rectangular garden for vegetables. Of the vegetable area, 2/3 is used for leafy vegetables.

1. What fraction of the whole garden is used for leafy vegetables?
2. If the whole garden is 900 m², what area is used for leafy vegetables?
3. The school has 3/4 kg of seeds and packs them equally into bags of 1/8 kg. How many bags can be filled?
4. Explain why the second calculation uses division.

🔥 HOTS / Competition Corner

1. Find a fraction x such that 3/4 × x = 9/16.
2. Without fully calculating, decide whether 7/8 ÷ 14/15 is greater or less than 1. Explain.
3. A student says 2/3 ÷ 4/5 = 8/15 because “division means multiply numerators and denominators.” Find and explain the error.
4. Create a real-life problem whose solution is 5/6 ÷ 1/3.
5. Find two different pairs of fractions whose product is 1/2.

📝 Chapter Worksheet

🏆 Mastery Checklist

• ☐ I can multiply fractions accurately.
• ☐ I can simplify before and after multiplication.
• ☐ I understand the reciprocal.
• ☐ I can divide fractions and explain why the reciprocal is used.
• ☐ I can convert mixed numbers and improper fractions.
• ☐ I can apply fraction operations to real situations.
• ☐ I can explain a method rather than only memorising it.

🔁 One-Minute Revision

Multiply: numerator × numerator and denominator × denominator. Divide: multiply by the reciprocal of the divisor. Reciprocal: reverse numerator and denominator for a non-zero fraction. Mixed number: whole × denominator + numerator over denominator. Always simplify and check whether the answer makes sense.

🏠 Class 7 Maths Main Page | ← Chapter 7