Distributive Property of Multiplication over Addition

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.