What is the distributive property in Class 7 Maths?

The distributive property states that a(b + c) = ab + ac. Multiplying a number by a sum equals multiplying separately by each addend and adding the results. For example, 3(x + 4) = 3x + 12. It also works with subtraction: a(b - c) = ab - ac. This is covered in NCERT Class 7 Ganita Prakash Chapter 2.

TOPIC 6 OF 31

Distributive Property and Chapter Exercises

🎓 Class 7 Mathematics CBSE Theory Ch 2 — Arithmetic Expressions ⏱ ~40 min

🌐 Language: [gtranslate]

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Removing Brackets — II: The Distributive Property

So far we have seen how to remove brackets when an expression is subtracted from a number. Now we look at a different situation — multiplying a number by an expression in brackets.

📝 Example 15 — Hotel Bill (Lhamo & Norbu)

Lhamo and Norbu each ordered one vegetable cutlet (₹43) and one rasgulla (₹24) at a hotel. The expression for each person's share is 43 + 24. What is the total they both pay together?

✦ Expression 1: 2 × (43 + 24) — multiply the total per person by 2

✦ Expression 2: 2 × 43 + 2 × 24 — pay separately for each item

✦ Key result: \(2 \times (43 + 24) = 2 \times 43 + 2 \times 24 = 86 + 48 = 134\)

📝 Example 16 — Republic Day Parade

Boy scouts march in 4 rows with 5 in each row. Girl guides march in 3 rows with 5 in each row. Total marchers = scouts + guides.

✦ Way 1: \(4 \times 5 + 3 \times 5 = 20 + 15 = 35\)

✦ Way 2: \((4 + 3) \times 5 = 7 \times 5 = 35\)

✦ Both equal 35 — the total rows times 5 is the same as adding the groups separately.

The Distributive Property^↗ — General Form

The observations from the two examples above reveal a general rule: the multiple of a sum (or difference) equals the sum (or difference) of the multiples.

Distributive Property of Multiplication over Addition/Subtraction

\(a \times (b + c) = a \times b + a \times c\)

\(a \times (b - c) = a \times b - a \times c\)

Also works with the multiplier on the right: \((b + c) \times a = b \times a + c \times a\)

Tinker the Terms II — Numbers in a Product

What happens to the value of an expression when we change a number in a product? The distributive property gives us a powerful shortcut.

🔬 Activity — Explore: Changing a Factor

Start with a known multiplication fact: 53 × 18 = 954
Now increase the first number by 10 to get 63 × 18
Think: 63 = 53 + 10, so 63 × 18 = (53 + 10) × 18 = 53×18 + 10×18
The extra amount is just 10 × 18 = 180 more than the original
Try: if the first number decreases by 3, how does the value change?

Expression	Reasoning (using distributive property)	Value
53 × 18	Given fact	954
63 × 18	(53+10)×18 = 954 + 10×18 = 954+180	1134
43 × 18	(53−10)×18 = 954 − 10×18 = 954−180	774
53 × 28	53×(18+10) = 954 + 53×10 = 954+530	1484
53 × 15	53×(18−3) = 954 − 53×3 = 954−159	795
97 × 25	(100−3)×25 = 100×25 − 3×25 = 2500−75	2425

💡 The key insight: if one number changes by k, the product changes by k × (other number).

📝 Example 17 — Using a Known Fact

Given 53 × 18 = 954, find 63 × 18 without full multiplication.

63 × 18= (53 + 10) × 18

= 53×18 + 10×18Distributive property

= 954 + 180Using the given fact + 10×18=180

✓

= 1134Answer

📝 Example 18 — Clever Multiplication: 97 × 25

97 is close to 100. Use (100 − 3) to make the calculation easy.

97 × 25= (100 − 3) × 25

= 100×25 − 3×25Distributive property

= 2500 − 75Easy calculations

✓

= 2425Answer

Try these using the same approach:

(a) 95 × 8

\(95 \times 8 = (100-5)\times 8 = 800 - 40 = \mathbf{760}\)

(b) 104 × 15

\(104 \times 15 = (100+4)\times 15 = 1500 + 60 = \mathbf{1560}\)

\(49 \times 50 = (50-1)\times 50 = 2500 - 50 = \mathbf{2450}\)

🔧 Distributive Property Explorer

Enter values for a, b, c and choose + or −. See both sides of the distributive property.

a (multiplier)

Operation

Figure it Out — Distributive Property (Pages 41–42)

Q1: Fill in the blanks / boxes so both sides are equal

Use the distributive property: \(a\times(b \pm c) = a\times b \pm a\times c\)

(a) 3 × (6 + 7) = 3 × 6 + 3 × 7

Already complete. LHS = 3×13 = 39; RHS = 18+21 = 39 ✓

(b) (8 + 3) × 4 = 8 × 4 + 3 × 4

Already complete. LHS = 11×4 = 44; RHS = 32+12 = 44 ✓

Box = +, blank = 8. 3×13 = 15+24 = 39 ✓

(d) (9 + 2) × 4 = 9 × 4 + 2 × 4

Box = +, blank = 4. 11×4 = 36+8 = 44 ✓

(e) 3 × (5 + 4) = 3×5 + 12

One possibility: blank = 5. Then 3×(5+4)=27; RHS=15+12=27 ✓

(f) (13 + 6) × 4 = 13 × 4 + 24

Blank₁ = 13, blank₂ = 6×4 = 24. 19×4=76; 52+24=76 ✓

(g) 3 × (5 + 2) = 3×5 + 3×2

Blanks = 5, 2. 3×7=21; 15+6=21 ✓

(h) (2 + 3) × 4 = 2×4 + 3×4

Blanks = 2, 3, 4. 5×4=20; 8+12=20 ✓

(i) 5 × (9 − 2) = 5×9 − 5×2

Blank = 2. 5×7=35; 45−10=35 ✓

(j) (5 − 2) × 7 = 5×7 − 2×7

Blank = 7. 3×7=21; 35−14=21 ✓

(k) 5 × (8 − 3) = 5×8 − 5×3

Box = −, blank = 3. 5×5=25; 40−15=25 ✓

(l) (8 − 3) × 7 = 8×7 − 3×7

Box = −. 5×7=35; 56−21=35 ✓

(m) 5 × (12 − 9) = 60 − 5×9

One possibility: blank = 9, second blank = 60 (= 5×12), third = 9. 5×3=15; 60−45=15 ✓

(n) (15 − 6) × 7 = 105 − 6×7

Blank₁ = 6, blank₂ = 105 (= 15×7). 9×7=63; 105−42=63 ✓

(o) 5 × (9 − 4) = 5×9 − 5×4

Blanks = 9, 4. 5×5=25; 45−20=25 ✓

(p) (17 − 9) × 7 = 17×7 − 9×7

Blanks = 17, 9, 7. 8×7=56; 119−63=56 ✓

Q2: Compare Using <, > or = (Without Evaluating)

(a) (8 − 3) × 29 __ (3 − 8) × 29

> LHS = 5×29 > 0; RHS = (−5)×29 < 0

(b) 15 + 9×18 __ (15 + 9)×18

< LHS = 15 + 162 = 177; RHS = 24×18 = 432

< LHS = 23×(17−9)=23×8; RHS = 23×(17+9)=23×26

(d) (34 − 28)×42 __ 34×42 − 28×42

= Both equal (34−28)×42 by distributive property

Q3: One way to make 14: 2 × (1 + 6) = 14. Find four more ways using a × (b + c) = 14.

(a) 7 × (1 + 1) = 14
(b) 14 × (0 + 1) = 14
(c) 1 × (7 + 7) = 14
(d) 2 × (3 + 4) = 14

Many other valid answers exist — e.g., 7×(2+0)=14, 2×(10−3)=14.

Q4: Find the sum of the dot arrangement below in at least two different ways using expressions.

Way 1 (rows): 2 rows of (6+2) = 2×8 = 16
Way 2 (columns by colour): 2×6 + 2×2 = 12+4 = 16
Both use the distributive property: 2×(6+2) = 2×6 + 2×2 = 16

Figure it Out — Application Exercises (Pages 42–44)

Q1: Write Expressions and Find Values

(a) Rahim supplies 9 kg of mangoes per day and Shyam supplies 11 kg per day to a market that runs all 7 days a week. Find the total supply in a week.

Total per day = 9 + 11 = 20 kg. Total in a week = \((9 + 11) \times 7\).
Using distributive property: \(9\times7 + 11\times7 = 63 + 77 = \mathbf{140 \text{ kg}}\)
Terms: 9×7 and 11×7

(b) Binu earns ₹20,000/month. She spends ₹5,000 on rent, ₹5,000 on food, and ₹2,000 on other expenses. How much will she save in a year?

Monthly savings = \(20000 - 5000 - 5000 - 2000 = 8000\)
Expression: \((20000 - 5000 - 5000 - 2000) \times 12\)
\(= 8000 \times 12 = \mathbf{₹96{,}000}\)

(c) A snail climbs 3 cm up a 10 cm post during the day and slips back 2 cm at night. In how many days will it reach the treat at the top?

Net progress per full day-night cycle = 3 − 2 = 1 cm.
After 7 complete cycles (7 days), the snail is at 7 cm.
On Day 8 morning it climbs 3 cm: 7 + 3 = 10 cm — reaches the top!
Answer: 8 days.
Note: The snail reaches the top during the day climb, so it does not slip back.

Q2: Melvin reads one two-page story every day except Tuesdays and Saturdays. How many stories in 8 weeks? Which expressions match?
(a) 5×2×8 (b) (7−2)×8 (c) 8×7 (d) 7×2×8 (e) 7×5−2 (f) (7+2)×8 (g) 7×8−2×8 (h) (7−5)×8

Days per week Melvin reads: 7 − 2 = 5 days.
Stories in 8 weeks = 5 × 8 = 40 stories.
Matching expressions:
(b) (7−2)×8 = 5×8 = 40 ✓
(g) 7×8 − 2×8 = 56 − 16 = 40 ✓ (distributive property!)
All other options give different values.

Q3: Find different ways to evaluate these expressions:
(a) 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10
(b) 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1

(a) Way 1 — pair consecutive terms:
(1−2)+(3−4)+(5−6)+(7−8)+(9−10) = (−1)+(−1)+(−1)+(−1)+(−1) = −5
Way 2 — separate positive and negative:
(1+3+5+7+9) − (2+4+6+8+10) = 25 − 30 = −5 ✓

(b) Way 1 — pair terms:
(1−1)+(1−1)+(1−1)+(1−1)+(1−1) = 0+0+0+0+0 = 0
Way 2 — count positives and negatives:
5 ones minus 5 ones = 5 − 5 = 0 ✓

Q4: Compare Using <, > or = (by Reasoning)

(a) 49−7+8 __ 49−7+8

= Identical expressions.

(b) 83×42−18 __ 83×40−18

> 83×42 > 83×40, same −18.

< 17×8=136 > 17×6=102, so more is subtracted on LHS.

(d) 23×48−35 __ 23×(48−35)

> LHS = 23×48−35; RHS = 23×48−23×35. Since 35 < 23×35, less is subtracted on LHS.

(e) (16−11)×12 __ −11×12+16×12

= RHS = 16×12−11×12 = (16−11)×12 by distributive property.

(f) (76−53)×88 __ 88×(53−76)

> LHS = 23×88 > 0; RHS = 88×(−23) < 0.

(g) 25×(42+16) __ 25×(43+15)

= 42+16 = 58 = 43+15, so both equal 25×58.

(h) 36×(28−16) __ 35×(27−15)

> LHS = 36×12=432; RHS = 35×12=420. First factor 36>35, same bracket value 12.

Q5: Identify Expressions Equal to the Given One (Without Computing)

(a) Given: 83 − 37 − 12
(i) 84−38−12 (ii) 84−(37+12) (iii) 83−38−13 (iv) −37+83−12

Value of given = 83−37−12 = 34.
(i) 84−38−12 = 34 ✓ — both numbers increased by 1, net change = 0
(ii) 84−(37+12) = 84−49 = 35 ✗
(iii) 83−38−13 = 32 ✗
(iv) −37+83−12 = 83−37−12 = 34 ✓ — reordering terms (commutative)
Equal expressions: (i) and (iv)

(b) Given: 93 + 37 × 44 + 76
(i) 37+93×44+76 (ii) 93+37×76+44 (iii) (93+37)×(44+76) (iv) 37×44+93+76

Terms of given expression: 93, (37×44), and 76.
(i) 37+93×44+76 — terms are 37, (93×44), 76 → different product term ✗
(ii) 93+37×76+44 — product term is 37×76 ≠ 37×44 ✗
(iii) (93+37)×(44+76) — changes structure entirely ✗
(iv) 37×44+93+76 — same three terms, just reordered ✓
Equal expression: (iv)

Bonus: Choose any number and write 10 different expressions with that value.

12+12, 4×6, 48÷2, 34−10, 3×(4+4), 5×5−1, 30−(2+4), 2×(9+3), 48−24, 100−76
Challenge: can you find 10 expressions for your own favourite number?

📋 Chapter 2 Summary — Arithmetic Expressions

An arithmetic expression is a combination of numbers and operations (+, −, ×, ÷) that has a definite value. Brackets control the order of evaluation.
Expressions can be compared through reasoning about their terms — often without fully evaluating them.
Every expression can be written as a sum of terms. A term is a part of the expression separated by + or −. Terms containing × or ÷ are evaluated as a unit first.
When terms are added in any order, the value stays the same — this is the Commutative Property^↗ of Addition.
Grouping terms differently in addition does not change the value — this is the Associative Property^↗ of Addition.
When brackets are preceded by a negative (−) sign, removing them flips all signs inside: \(a - (b + c) = a - b - c\) and \(a - (b - c) = a - b + c\)
When brackets are preceded by a positive (+) sign, removing them keeps all signs inside unchanged: \(a + (b - c) = a + b - c\)
The Distributive Property^↗: multiplying a number by a bracketed expression is the same as multiplying it by each term separately: \(a \times (b + c) = a \times b + a \times c\)
The distributive property enables powerful mental multiplication shortcuts — replacing a hard number with a nearby round number ± a small adjustment.

🛠️ Expression Engineer!

Using the four operations (+, −, ×, ÷) and brackets, a single repeated number can generate many values. Example with three 3's: (3+3)/3 = 2 3+3−3 = 3 3×3+3 = 12

Challenge 1: Four 4's (Values 1 to 20)

Using exactly four 4's with +, −, ×, ÷ and brackets, create expressions for every value from 1 to 20.

44÷44 = 1 (4÷4)+(4÷4) = 2 (4+4+4)÷4 = 3 4×(4−4)+4 = 4 4+(4×4÷4) = 8

Challenge 2: Numbers 1–5 (Values −10 to +10)

Using the digits 1, 2, 3, 4, 5 exactly once in any order, generate as many integers between −10 and +10 as you can.

1+2+3−4−5 = −3 5×(3−2)−4+1 = 2 5+4−3−2−1 = 3

Challenge 3: Digits 0–9 Make 100

Using each digit 0 through 9 exactly once, with any operations and brackets, create an expression equal to 100.

1+2+3−4+5+6+78+9 = 100 Try: 9×8+7+6−5+4−3+2−1 = ?

What other interesting puzzles can you create using repeated numbers or a fixed set of digits?

📝

Competency-Based Questions

A school is organising its Annual Sports Day. The marching contingent has 6 houses, each with boys and girls marching in separate rows. Each house sends 8 boys (in 2 rows of 4) and 5 girls (in 1 row of 5). The tuck shop sells samosas at ₹12 each and juice at ₹15 each. Every participating student gets one samosa and one juice. Use arithmetic expressions to answer the questions below.

Q1. Write an expression for the total number of students in the marching contingent from all 6 houses. L1 Remember

(A) 6 × (8 + 5)
(B) 6 × 8 + 5
(C) 6 + 8 + 5
(D) (6 + 8) × 5

Answer: (A) 6 × (8 + 5) = 6 × 13 = 78 students.
Each house contributes (8+5)=13 students, and there are 6 houses.

Q2. The expression for total tuck shop cost is 78 × (12 + 15). Using the distributive property, expand this expression and calculate the total cost. L2 Understand

78 × (12 + 15) = 78 × 12 + 78 × 15 = 936 + 1170 = ₹2,106.
The distributive property allows us to split the bracket: total samosa cost + total juice cost.

Q3. A smart student calculates 78 × 27 as (80 − 2) × 27. Write each step of this calculation and find the value. L3 Apply

Step 1: 78 × 27 = (80 − 2) × 27
Step 2: = 80 × 27 − 2 × 27 [distributive property]
Step 3: = 2160 − 54 = ₹2,106 ✓
This confirms the tuck shop cost from Q2.

Q4. Compare the two expressions 6×8 + 6×5 and 6×(8+5). Are they always equal? What property does this demonstrate? Give one more example. L4 Analyse

Yes, they are always equal — this is the Distributive Property of Multiplication over Addition: \(a\times b + a\times c = a\times(b+c)\).
Both expressions = 6×13 = 78.
Another example: 5×7 + 5×3 = 5×(7+3) = 5×10 = 50.

Q5 (HOT). A student claims: "Since 6×(8+5) = 6×8 + 6×5, then 6+(8×5) = (6+8)×(6+5) by the same logic." Is this correct? Justify with calculation. L5 Evaluate

No, this is incorrect.
LHS: 6 + (8×5) = 6 + 40 = 46
RHS: (6+8)×(6+5) = 14×11 = 154
These are NOT equal! The distributive property applies to multiplication over addition/subtraction, not to addition over multiplication. There is no general rule saying "a+(b×c) = (a+b)×(a+c)".

🔍 Assertion–Reason Questions

A: Both A and R are true; R is the correct explanation of A. B: Both A and R are true; R is NOT the correct explanation. C: A is true, R is false. D: A is false, R is true.

Assertion: 97 × 25 = 100 × 25 − 3 × 25 = 2425.
Reason: The distributive property states that \((a - b) \times c = a \times c - b \times c\).

Answer: A — Both are true and R correctly explains why 97×25 = (100−3)×25 = 100×25 − 3×25.

Assertion: \(5 \times (3 + 4) = 5 \times 3 + 4\).
Reason: When a number multiplies a bracket, it multiplies every term inside the bracket.

Answer: D — The Assertion is false: 5×(3+4)=35, but 5×3+4=19. The Reason is true (it describes the distributive property correctly), but the Assertion misapplies it by not multiplying the second term.

Assertion: \(4 \times 5 + 3 \times 5 = (4 + 3) \times 5 = 35\).
Reason: Swapping the order in which terms are added never changes the sum of an expression.

Answer: B — The Assertion is true (this is the distributive property: factoring out the common factor 5). The Reason is also true (commutative property of addition). However, R is NOT the correct explanation of A — A is explained by the distributive property, not by commutativity.

Frequently Asked Questions

What is the distributive property in algebra?

The distributive property states that multiplying a number by a sum equals the sum of individual products: a times (b plus c) equals ab plus ac. For example, 5 times (x plus 2) equals 5x plus 10. It also applies to subtraction: a times (b minus c) equals ab minus ac. NCERT Class 7 Chapter 2 covers this law.

What are like terms and how do you combine them?

Like terms have the same variable raised to the same power. For example, 3x and 7x are like terms but 3x and 3y are not. To combine like terms, add or subtract their coefficients: 3x plus 7x equals 10x. NCERT Class 7 Ganita Prakash Chapter 2 exercises require combining like terms.

How do you simplify algebraic expressions step by step?

First, remove brackets using sign rules and the distributive property. Second, identify and group like terms together. Third, combine like terms by adding or subtracting coefficients. Finally, write the simplified expression in standard form. NCERT Class 7 Chapter 2 provides structured practice.

What exercises are in Class 7 Maths Chapter 2?

Class 7 Maths Chapter 2 exercises include forming expressions from verbal statements, identifying terms and coefficients, removing brackets, applying the distributive property, combining like terms, and simplifying multi-step expressions. Problems range from basic to challenging level.

How is the distributive property used in mental maths?

The distributive property helps with mental calculations. To multiply 7 times 98, think of it as 7 times (100 minus 2) which equals 700 minus 14 equals 686. Similarly, 15 times 102 equals 15 times (100 plus 2) equals 1500 plus 30 equals 1530. NCERT Class 7 Ganita Prakash demonstrates this.

Frequently Asked Questions — Arithmetic Expressions

What is Distributive Property and Chapter Exercises in NCERT Class 7 Mathematics?

Distributive Property and Chapter Exercises is a key concept covered in NCERT Class 7 Mathematics, Chapter 2: Arithmetic Expressions. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Distributive Property and Chapter Exercises step by step?

To solve problems on Distributive Property and Chapter Exercises, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 7 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 2: Arithmetic Expressions?

The essential formulas of Chapter 2 (Arithmetic Expressions) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Distributive Property and Chapter Exercises important for the Class 7 board exam?

Distributive Property and Chapter Exercises is part of the NCERT Class 7 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Distributive Property and Chapter Exercises?

Common mistakes in Distributive Property and Chapter Exercises include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Distributive Property and Chapter Exercises?

End-of-chapter NCERT exercises for Distributive Property and Chapter Exercises cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.

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