NCERT Solutions for Class 7 Mathematics Chapter 3 Finding Common Ground
NCERT Solutions for Class 7 Mathematics Chapter 3 Finding Common Ground
NCERT Solutions for Class 7 Mathematics Chapter Finding Common Ground
This worksheet provides complete and accurate NCERT Solutions for Class 7 Mathematics Chapter Finding Common Ground. This worksheet helps students understand important concepts like factors, Highest Common Factor (HCF), and Least Common Multiple (LCM) through simple real-life situations such as tiles, bags, and number games. The chapter is important because it builds strong number sense and helps students solve practical problems involving division, grouping, and common multiples. All solutions are strictly aligned with NCERT structure and order, ensuring reliable learning and exam preparation.
Chapter summary: stories, poems & themes
This chapter includes real-life situations like arranging tiles, packing bags, and number-based games such as Jump Jackpot. It is an activity-based chapter where students learn how to find common factors and multiples through observation and reasoning. The main focus is on understanding how numbers relate to each other using factors, prime factorisation, HCF, and LCM. The chapter is concept-driven and uses practical examples to make learning meaningful.
What this NCERT chapter covers?
• Understanding factors and common factors
• Concept of Highest Common Factor (HCF)
• Prime numbers and prime factorisation
• Finding factors using prime factorisation
• Finding HCF using prime factorisation
• Concept of Least Common Multiple (LCM)
• Finding LCM using prime factorisation
How to use these NCERT solutions?
Students should first try solving all questions independently and then verify their answers using this worksheet. Parents and teachers can guide students step-by-step using the provided explanations. These solutions follow the exact NCERT sequence, ensuring that no concept or question is missed. This helps in better understanding, structured revision, and exam readiness.
Student tips & learning tricks
Students should clearly understand the difference between factors and multiples. A common mistake is confusing HCF with LCM, so always check whether the question is asking for the greatest common factor or the smallest common multiple. While using prime factorisation, write steps carefully to avoid missing any prime factor. Practice regularly to improve speed and accuracy.
Why NCERT solutions are important?
NCERT Solutions are important because they strictly follow the CBSE syllabus and textbook. They help students develop strong fundamentals and improve logical thinking. These solutions also help in building confidence and preparing effectively for exams. Practicing NCERT-based questions ensures clarity in concepts and better performance.
Complete answer key – NCERT solutions
3.1 The greatest of all
1. The tile side must divide both 12 ft and 16 ft
HCF = 4 ft
Number of tiles = (12 ÷ 4) × (16 ÷ 4) = 3 × 4 = 12 tiles
2. To use the fewest bags, choose the largest common factor
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12
Answer: Each bag should be 12 kg
3. To land on both numbers, we find the HCF
(a) 14 and 30 → HCF = 2
(b) 7 and 11 → HCF = 1
(c) 30 and 50 → HCF = 10
(d) 28 and 42 → HCF = 14
4. Yes
The longest jump size is the HCF because it divides both numbers exactly
5. Yes
The process can be simplified using prime factorisation
Factors of a number using prime factorisation
6. Yes, because 840 ÷ 14 = 60
7. Yes, because 840 ÷ 8 = 105
8. No, because 840 ÷ 27 is not a whole number
9. Yes, by using prime factorisation we can form all factors
10. Prime factorisation: 225 = 3 × 3 × 5 × 5
11. Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, 225
12. Yes, all factors are included
Figure it out
1. (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
Finding the HCF of numbers using prime factorisation
13. 45 = 3 × 3 × 5
75 = 3 × 5 × 5
Common factors: 1, 3, 5, 15
HCF = 15
14. 112 = 2 × 2 × 2 × 2 × 7
84 = 2 × 2 × 3 × 7
Common factors: 1, 2, 4, 7, 14, 28
HCF = 28
15. 96 = 2 × 2 × 2 × 2 × 2 × 3
275 = 5 × 5 × 11
Common factors: 1
HCF = 1
Figure it out
16. (a) Common factors: 1, 2, 5, 10 → HCF = 10
(b) Common factors: 1, 2, 5, 10 → HCF = 10
(c) Common factors: 1 → HCF = 1
(d) Common factors: 1, 2 → HCF = 2
(e) Common factors: 1, 3, 9, 27, 81 → HCF = 81
Further problems
17. (a) 24, 180 → HCF = 12
(b) 42, 75, 24 → HCF = 3
(c) 240, 378 → HCF = 6
(d) 400, 2500 → HCF = 100
(e) 300, 800 → HCF = 100
18. No
The numbers share common prime factors
3.2 Least, but not last
19. The shortest toran length is 24 cm
It is the LCM of 6 and 8
20. Free sweets are given every 7 days and Kabamai visits every 10 days
LCM = 70 days
21. Yes
2 × 3 × 5 × 7 = 210 is a common multiple
22. First common multiples
(a) 4 and 6 → LCM = 12
(b) 7 and 11 → LCM = 77
(c) 14 and 30 → LCM = 210
(d) 15 and 55 → LCM = 165
23. Yes
The first common occurrence is the LCM
24. Method
Prime factorise both numbers
Take all prime factors with highest powers
Multiply to get LCM
25. Example
12 = 2² × 3
18 = 2 × 3²
LCM = 36
26. Yes
210 is the LCM
27. It is the LCM of 14 and 35
28. LCM of 96 and 360 = 1440
Master Class 7 Mathematics concepts like HCF and LCM with clear NCERT Solutions for better understanding and exam success