Chapter 6: We Distribute, Yet Things Multiply

NCERT-focused learning page — concept explanations, worked examples, reasoning, MCQs and competency practice.

1. Distributive Property

The distributive property says that multiplication can be distributed over addition or subtraction: a(b+c)=ab+ac and a(b-c)=ab-ac. It is useful because a difficult multiplication can be broken into easier parts.

Example: 27×6=(20+7)×6=20×6+7×6=120+42=162.

2. Increments in Products

If one factor increases while the other remains fixed, the product increases by the fixed factor for every one-unit increase. For example, 15×20=300 and 15×21=315; the increase is 15. This is a powerful way to reason mentally instead of recalculating from the beginning.

3. Fast Multiplication Using Distribution

Numbers close to 10, 100 or 1000 can be decomposed around those convenient values. For example, 98×23=(100−2)×23=2300−46=2254.

4. Special Cases

Expressions such as (100−1)n, (100+1)n and products involving convenient numbers can be simplified quickly. The key is to preserve the complete multiplication while splitting the number.

5. Investigating Patterns

Use several examples to identify a pattern, then explain why it works generally. A pattern is not a proof by itself; mathematical reasoning must explain why the relationship continues.

6. Mind Mistake, Mend Mistake

When a distributive calculation is wrong, check whether every term inside the bracket was multiplied. For example, 4(7+3) must be 4×7+4×3, not 4×7+3.

🧠 Concept MCQs

1. 3(20+4) equals: A) 60+4 B) 60+12 C) 23×4 D) 3+24
2. 99×8 can be efficiently written as: A) (100−1)×8 B) (90+8)×8 C) 100×9 D) 98×8
3. If 12×30=360, then 12×31 is: A) 360 B) 361 C) 372 D) 390
4. Which property is used in 5(9−2)=45−10? A) Distributive B) Associative C) Identity D) Closure

Answers: 1-B, 2-A, 3-C, 4-A.

📌 Competency Case

A shopkeeper needs to calculate 48×25 mentally. Explain how you can rewrite 48 as 50−2 and obtain the answer without long multiplication.

🔥 HOTS

1. Explain why 101×37 can be calculated as 3700+37.
2. Find two different distributive methods for 49×16.
3. Create a multiplication that becomes easier by rewriting one factor as 100−n.

✍️ Exam Practice

1. State the distributive property with an example.
2. Calculate 97×24 using distribution.
3. Explain the increment pattern between 18×40 and 18×41.
4. Find and correct an incorrect distributive calculation.

🏆 Mastery Checklist

• ☐ I can use distribution over addition and subtraction.
• ☐ I can calculate strategically using convenient numbers.
• ☐ I can explain product increments.
• ☐ I can investigate and justify patterns.
• ☐ I can detect errors in a solution.

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