We have already studied that multiplication distributes over addition on whole numbers. Let us check the property for integers.

**Distributive
Property of Multiplication over Addition**

We have already studied that multiplication distributes
over addition on whole numbers. Let us check the property for integers.

Take for example (−2)×(4 + 5) = [(−2)× 4] + [(−2)×
5]

LHS = (−2)×(4 + 5)

= (−2)× 9

= (−18)

= −18

RHS = [(−2)× 4] + [(−2)× 5]

= (−8 )+(−10)

= − 8 − 10

= − 18

From the above, we can observe that (−2)×(4 +
5) = [(−2)× 4] + [(−2)× 5]

Hence, "distributive
property of multiplication over addition" is true for integers.

Therefore, for any three integers
*a, b, c ;* *a × *(*b+c*)* = *(*a
× b*)* + *(*a×c*)

** **

**Try these**

**1. Find the values of the
following and check for equality:**

**(i) (–6) × (4+(–5)) and ((–6) ×4)
+ ((–6)×(–5))**

(–6) × [4 + (–5)] = (–6) × (–l) = 6

[(–6) × 4] + [(–6 × (–5)] = –24 + 30 = 6

(–6) × [(4 + (–5)] = [(–6) × 4) + (–6 × (–5)]

They are equal

**(ii) (–3)×[2+(–8)] and
[(–3)×2]+[(–3)×8]**

(–3) × [2 + (–8)] = –3 × –6 = 18

[(–3) × 2] + [(–3) × 8] = –6 – 24 = –30

(–3) × [2+(–8)] ≠ [(–3 × 2] + [(–3) × 8]

They are not equal

**2. Prove the following:**

**(i) (–5) × [(–76)+8] = [(–5)
×(–76)] + [(–5)×8]**

(–5) × [(–76) + 8] = (–5)
× (–68) = 340

[(–5) × (–76)] + [(–5) × 8] = 380 + (–40) = 340

(–5) × [(–76) + 8] = [(–5) × (–76)] + [(–5) × 8]

Hence it is proved

**(ii) 42 × [7+(–3)] = (42 ×7) +
[42×(–3)]**

42 × [7 + (–3)] = 42 × 4 = 168

(42 × 7) + [42 × (–3)] = 294 + (–126) = 168

42 × [7 + (–3)] = (42 × 7) + [42 × (–3)]

Hence it is proved

**(iii) (–3) × [(–4)+(–5)] = ((–3)
× (–4)) + [(–3)×(–5)]**

(–3) × [–4 + (–5)] = –3 × –9 = 27

[(–3) × (–4)] + [(–3) × (–5)] = 12+ 15 = 27

(–3) × [–4 + (–5)] = [(–3) × (–4)] + [(–3) × (–5)]

Hence it is proved

**(iv) 103 × 25 = (100+3) × 25 = (100×25) +(3×25)**

103 × 25 = 2575

(100 + 3) × 25 = 103 × 25 = 2575

(100 × 25) + (3 × 25) = 2500 + 75 = 2575

103 × 25 = (100 + 3) × 25 = (100 × 25) + (3 × 25)

Hence it is proved

** **

__Example 1.20__

Prove that (–7) × (+8) is an integer and
mention the property.

**Solution**

(–7) × (+8) = (–56)

Hence, –56 is an integer.

Therefore, (–7) × (+8) is closed under
multiplicaton.

** **

__Example 1.21__

Are (–42) × (–7) and (–7) × (–42) equal?
Mention the property.

**Solution**

Consider, (–42) × (–7),

(–42) × (–7) = +294

Consider, (–7) × (–42),

(–7) × (–42) = +294

Therefore, (–42) × (–7) and (–7) × (–42) are
equal.

It is commutative.

** **

__Example 1.22__

Prove that [(–2) × 3] ×(–4) = (–2) ×[3×(–4)].

**Solution**

In the first case (–2) and (3) are grouped
together and in the second case (3) and (–4) are grouped together

L.H.S = [(–2) × 3] ×(–4)

= (–6) × (–4) = 24

R.H.S =
(–2) ×[3×(–4)]

= (–2) × (–12) = 24

Therefore, L.H.S. = R.H.S.

[(–2) × 3] ×(–4) = (–2) ×[3×(–4)]

Hence it is proved.

** **

__Example 1.23__

Are (–81) × [5×(–2)] and [(–81) × 5]×(–2)
equal? Mention the property.

*Solution*

Consider, (–81) × [5×(–2)],

(–81) × [5×(–2)] = (–81) × (–10) = 810

Consider, [(–81) × 5]×(–2),

[(–81) × 5]×(–2) = (–405) × (–2) = 810

Therefore, (–81) × [5×(–2)] and [(–81) ×
5]×(–2) are equal.

It is associative.

** **

__Example 1.24__

Are 3 × [(–4)+6] and [3 ×(–4)]+(3×6) equal?
Mention the property.

*Solution*

Consider, 3 × [(–4)+6],

3×[(–4)+6]=3×2=6

Consider, [3 ×(–4)]+[3×6],

[3 ×(–4)]+[3×6] = –12+18 = 6

Therefore, 3 × [(–4)+6] and [3 ×(–4)]+3×6 are
equal.

It is the distributive property of
multiplication over addition.

Tags : Number System | Term 1 Chapter 1 | 7th Maths , 7th Maths : Term 1 Unit 1 : Number System

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7th Maths : Term 1 Unit 1 : Number System : Distributive Property of Multiplication over Addition | Number System | Term 1 Chapter 1 | 7th Maths

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