NCERT Solutions for Class 7 Mathematics Ganita Prakash II Chapter 3

NCERT solutions for Class 7 Mathematics Chapter Finding Common Ground – complete answers & explanations

In Class 7 Mathematics, Chapter Finding Common Ground introduces students to the essential concepts of factors, multiples, Highest Common Factor (HCF), and Lowest Common Multiple (LCM). This chapter helps students understand how numbers are related and how to solve practical problems involving everyday situations like arranging tiles, packing items, and scheduling events. The chapter explores how to efficiently find common factors and multiples using prime factorisation, making it easier to solve complex problems. Understanding these concepts improves problem-solving and logical thinking skills. This blog provides clear, reliable, and NCERT-aligned solutions that guide students through each step. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance.

What this NCERT chapter covers?

1. Understanding the concept of factors and common factors using real-life examples.
2. Learning how to find the Highest Common Factor (HCF) and applying it to solve problems.
3. Exploring prime factorisation and using it to simplify finding common factors.
4. Identifying and finding the Least Common Multiple (LCM) in practical scenarios.
5. Learning how to find the LCM using prime factorisation and the highest power rule.
6. Solving problems related to the smallest number that can be divided by multiple numbers.
7. Using prime factorisation to find all the factors of a number and applying it to real-life problems.
8. Understanding the relationship between HCF and LCM and how to use them together.
9. Solving puzzles involving common multiples and common factors, including activities like the "Jump Jackpot" game.
10. Using the LCM and HCF to optimize calculations in practical scenarios like scheduling or arranging items.
11. Applying HCF and LCM knowledge to real-world scenarios, such as food packaging, tile arrangement, and scheduling events.

How to use these NCERT solutions?

1. Begin by reading the questions carefully and attempt solving them independently.
2. After attempting the questions, check your answers using these solutions to ensure accuracy.
3. Compare your methods with the provided solutions to understand the correct approach.
4. Make sure to practice prime factorisation regularly, as it simplifies many problems.
5. Understand the concept of LCM and HCF and their applications in various situations.
6. Parents and teachers can use these solutions to guide students step by step through each question.
7. Follow the same order of questions as in the worksheet to maintain consistency.
8. Revisit questions that were difficult and reattempt them to strengthen your understanding.
9. Use these solutions for exam preparation and to ensure all concepts are thoroughly understood.

Important tips & tricks for students

1. When calculating HCF or LCM, always start by finding the prime factorisation of both numbers.
2. For HCF, use the lowest powers of common prime factors; for LCM, use the highest powers.
3. Double-check the factors and multiples to avoid confusion between HCF and LCM.
4. Prime factorisation helps in breaking down complex numbers into simpler factors for easier calculations.
5. Understand the relationship between HCF and LCM and how they can be used together in real-life problems.
6. Practice solving puzzles and real-world problems to improve reasoning skills.
7. Pay close attention to number patterns when solving word problems related to multiples.
8. Regularly revise key properties like the LCM and HCF rules to reinforce concepts.
9. Keep practicing division and multiplication involving decimals to enhance accuracy.
10. Stay consistent with exercises to improve speed and accuracy in applying the concepts.

NCERT solutions – complete answer key

3.1 The Greatest of All

Tiles should be 4 ft. in size to minimize the number of tiles required. 
Number of tiles: (12 ÷ 4) × (16 ÷ 4) = 3 × 4 = 12 tiles.

To use the fewest bags, choose the largest common factor of 84 and 108. 
Common factors of 84 and 108: 1, 2, 3, 4, 6, 12 
HCF = 12 
Answer: Each bag should be 12 kg.

Longest jump size is the HCF because it is the greatest number that divides both numbers exactly. 
(a) 14 and 30 → HCF = 2 
(b) 7 and 11 → HCF = 1 
(c) 30 and 50 → HCF = 10 
(d) 28 and 42 → HCF = 14 

Prime factorisation simplifies the process of finding common factors and the HCF.

Figure it Out 

(a) 90 = 2 × 3² × 5 
(b) 105 = 3 × 5 × 7 
(c) 132 = 2² × 3 × 11 
(d) 360 = 2³ × 3² × 5 
(e) 840 = 2³ × 3 × 5 × 7 

3.2 Least, but not Last!

The shortest toran length is 24 cm, the LCM of 6 and 8. 

The common day when free sweets will be given is the LCM of 7 and 10, which is 70 days. 
Answer: 70 days.

Figure it Out 

(a) 30, 72 → LCM = 360 
(b) 36, 54 → LCM = 108 
(c) 105, 195, 65 → LCM = 1365 
(d) 222, 370 → LCM = 1110 

3.3 Patterns, Properties, and a Pretty Procedure!

When one number is a multiple of the other, the HCF is the smaller number. 
Examples: (4, 12), (5, 20), (7, 21), (9, 27) 
General statement: If one number divides the other, the HCF is the smaller number.

Efficient Procedures for HCF and LCM

HCFs: 
For 300 and 150 → HCF = 150 
For 630 and 770 → HCF = 70

Finding LCM through Prime Factorisation: 

LCM of 96 and 360 = 1440 

Figure it Out 

(a) HCF = 10 
(b) HCF = 10 
(c) HCF = 1 
(d) HCF = 2 
(e) HCF = 81 

3.4 Property Involving both the HCF and the LCM

Product of two numbers = HCF × LCM. 
For example: 
LCM of 105 and 95 = 1995. 

Figure it Out 

(a) LCM = 315 
(b) LCM = 96800 
(c) LCM = 41070 

Product of two numbers is always a common multiple. If HCF = 1, then the LCM is the smallest common multiple.

Figure it Out 

(a) Answer: 6 positions 
(b) Answer: 7 × 7 × 8 = LCM = 56 

4.4 Look Before You Leap!

In 100 calendar years, the number of days is 36,525. 
In 100 years, Earth will need 36,524.22 days to complete its revolutions.

Figure it Out 

1. Answers may vary. 
2. Answers may vary. 
3. Answers may vary. 
4. Answers may vary. 
5. Answers may vary. 
6. The number of cows = 105 
7. Answer: 105 cows 
8. Cube sizes = 6 cm, 3 cm, 2 cm

Why NCERT solutions help students?

NCERT solutions help students build a clear understanding of how to apply HCF and LCM in real-life situations. They provide the correct methods for solving problems and improve mathematical accuracy. By practicing these solutions, students gain confidence and become better problem-solvers in mathematics.

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