Finding Common Ground - Class 7 Mathematics (CBSE)

Based on the 2026-27 Class 7 Mathematics sequence for NCERT Ganita Prakash. These notes are written for students: understand the idea first, then practise enough examples to become accurate.

1. Why this chapter matters

Finding common ground is about making numbers work together. Whether we are adding fractions, scheduling repeating events, arranging objects in equal rows, or simplifying ratios, common factors and common multiples help us choose the most efficient number.

In school tests, this chapter can appear as direct calculations, reasoning questions, short explanations, activity-based questions, and word problems. The safest preparation is not to memorise a single trick, but to know what each idea means and when to use it.

2. Core ideas

Common factors

Common factors divide two or more numbers exactly. The highest common factor is the greatest such divisor.

Common multiples

Common multiples are numbers that appear in the multiple lists of two or more numbers. The least common multiple is the smallest positive common multiple.

Where they are used

HCF is often used for grouping and simplification. LCM is often used for repeated cycles, schedules, and unlike fraction addition.

3. Rules and formulas to remember

• HCF: Greatest common factor. Largest number that divides each given number.
• LCM: Least common multiple. Smallest positive number divisible by each given number.
• Product rule for two numbers: HCF x LCM = product of numbers. Works for two positive integers.
• Fraction addition denominator: Use LCM of denominators. Keeps numbers manageable.

4. Worked examples

Example 1: Find HCF of 24 and 36.

Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36 include 1,2,3,4,6,9,12,18,36. HCF = 12.

Example 2: Find LCM of 8 and 12.

Multiples of 8: 8,16,24. Multiples of 12: 12,24. LCM = 24.

Example 3: Two bells ring every 6 and 10 minutes. When will they ring together?

LCM of 6 and 10 = 30 minutes.

Example 4: Add 5/6 + 7/9.

LCM of 6 and 9 = 18. 5/6 = 15/18, 7/9 = 14/18, sum = 29/18 = 1 11/18.

5. Activity corner

Use coloured counters to arrange 18 and 24 counters into equal groups. The largest equal group size possible for both numbers gives the HCF idea physically.

When writing an activity answer, include three things:

• What you did.
• What you observed.
• What mathematical rule or pattern the activity shows.

6. Common mistakes and how to avoid them

• Mistake: Using HCF when LCM is needed Fix: Grouping usually needs HCF; repeated timing or common denominators usually need LCM.
• Mistake: Stopping at the first common factor Fix: HCF asks for the highest common factor.
• Mistake: Listing too few multiples Fix: Continue until a common multiple appears.

7. How to write high-scoring answers

1. State the given information in mathematical form.
2. Write the rule, formula, diagram, table, or operation you are using.
3. Show every step clearly.
4. Keep units such as cm, m, rupees, degrees, or minutes where needed.
5. Check whether the answer is reasonable.

8. Practice set

1. Find HCF of 18 and 30.
2. Find LCM of 9 and 12.
3. Simplify 21/35.
4. Add 1/4 + 5/6.
5. Two lights flash every 8 s and 14 s. When together again?
6. Which is useful for making equal gift packs: HCF or LCM?

9. Answer key

1. Find HCF of 18 and 30. Answer: 6.

2. Find LCM of 9 and 12. Answer: 36.

3. Simplify 21/35. Answer: 3/5.

4. Add 1/4 + 5/6. Answer: 13/12 = 1 1/12.

5. Two lights flash every 8 s and 14 s. When together again? Answer: 56 s.

6. Which is useful for making equal gift packs: HCF or LCM? Answer: HCF.

10. Quick revision

• Main themes: factors, multiples, HCF, LCM, common denominators.
• Redo the worked examples without looking at the solutions.
• Explain the activity in your own words.
• Correct the common mistakes once before the test.
• Create one new word problem from daily life and solve it step by step.