Evaluate :
(i) 427 × 8 + 2 × 427
(ii) 394 × 12 + 394 × (-2)
(iii) 558 × 27 + 3 × 558
Answer
Using the distributive property a × b + a × c = a × (b + c) :
(i) Solving,
427 × 8 + 2 × 427
= 427 × (8 + 2)
= 427 × 10
= 4270.
∴ 427 × 8 + 2 × 427 = 4270.
(ii) Solving,
394 × 12 + 394 × (-2)
= 394 × {12 + (-2)}
= 394 × 10
= 3940.
∴ 394 × 12 + 394 × (-2) = 3940.
(iii) Solving,
558 × 27 + 3 × 558
= 558 × (27 + 3)
= 558 × 30
= 16740.
∴ 558 × 27 + 3 × 558 = 16740.
Evaluate :
(i) 673 × 9 + 673
(ii) 1925 × 101 - 1925
Answer
(i) Solving,
673 × 9 + 673
= 673 × (9 + 1)
= 673 × 10
= 6730.
∴ 673 × 9 + 673 = 6730.
(ii) Solving,
1925 × 101 - 1925
= 1925 × (101 - 1)
= 1925 × 100
= 192500.
∴ 1925 × 101 - 1925 = 192500.
Verify :
(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)
(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19
(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7
(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)
Answer
(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)
L.H.S. = 37 × {8 + (-3)} = 37 × 5 = 185
R.H.S. = 37 × 8 + 37 × (-3) = 296 + (-111) = 296 - 111 = 185
∴ L.H.S. = R.H.S.
Hence, verified.
(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19
L.H.S. = (-82) × {(-4) + 19} = (-82) × 15 = -1230
R.H.S. = (-82) × (-4) + (-82) × 19 = 328 + (-1558) = 328 - 1558 = -1230
∴ L.H.S. = R.H.S.
Hence, verified.
(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7
L.H.S. = {7 - (-7)} × 7 = (7 + 7) × 7 = 14 × 7 = 98
R.H.S. = 7 × 7 - (-7) × 7 = 49 - (-49) = 49 + 49 = 98
∴ L.H.S. = R.H.S.
Hence, verified.
(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)
L.H.S. = {(-15) - 8} × (-6) = (-23) × (-6) = 138
R.H.S. = (-15) × (-6) - 8 × (-6) = 90 - (-48) = 90 + 48 = 138
∴ L.H.S. = R.H.S.
Hence, verified.
Evaluate :
(i) 15 × 8
(ii) 15 × (-8)
(iii) (-15) × 8
(iv) (-15) × -8
Answer
(i) 15 × 8 = 120.
(ii) 15 × (-8) = -120.
(iii) (-15) × 8 = -120.
(iv) (-15) × (-8) = 120.
Evaluate :
(i) 4 × 6 × 8
(ii) 4 × 6 × (-8)
(iii) 4 × (-6) × 8
(iv) (-4) × 6 × 8
(v) 4 × (-6) × (-8)
(vi) (-4) × (-6) × 8
(vii) (-4) × 6 × (-8)
(viii) (-4) × (-6) × (-8)
Answer
(i) 4 × 6 × 8 = 24 × 8 = 192.
(ii) 4 × 6 × (-8) = 24 × (-8) = -192.
(iii) 4 × (-6) × 8 = (-24) × 8 = -192.
(iv) (-4) × 6 × 8 = (-24) × 8 = -192.
(v) 4 × (-6) × (-8) = (-24) × (-8) = 192.
(vi) (-4) × (-6) × 8 = 24 × 8 = 192.
(vii) (-4) × 6 × (-8) = (-24) × (-8) = 192.
(viii) (-4) × (-6) × (-8) = 24 × (-8) = -192.
Evaluate :
(i) 2 × 4 × 6 × 8
(ii) 2 × (-4) × 6 × 8
(iii) (-2) × 4 × (-6) × 8
(iv) (-2) × (-4) × 6 × (-8)
(v) (-2) × (-4) × (-6) × (-8)
Answer
(i) 2 × 4 × 6 × 8 = 8 × 48 = 384.
(ii) 2 × (-4) × 6 × 8 = (-8) × 48 = -384.
(iii) (-2) × 4 × (-6) × 8 = (-8) × (-48) = 384.
(iv) (-2) × (-4) × 6 × (-8) = 8 × (-48) = -384.
(v) (-2) × (-4) × (-6) × (-8) = 8 × 48 = 384.
Determine the integer whose product with '-1' is :
(i) -47
(ii) 63
(iii) -1
(iv) 0
Answer
We will use the property:
(-1) x a = -a
Since multiplying by -1 changes only the sign and not the value of an integer, the required integer has the same value as the given product but with the opposite sign .
(i) Product = -47
(-1) x 47 = -47
Hence, required integer = 47.
(ii) Product = 63
(-1) x -63 = 63
Hence, required integer = -63.
(iii) Product = -1
(-1) x 1 = -1
Hence, required integer = 1.
(iv) Product = 0
(-1) x 0 = 0
Hence, required integer = 0.
Eighteen integers are multiplied together. What will be the sign of their product, if :
(i) 15 of them are negative and 3 are positive ?
(ii) 12 of them are negative and 6 are positive ?
(iii) 9 of them are positive and the remaining are negative ?
(iv) all are negative ?
Answer
The sign of the product depends on the number of negative integers. If the number of negative integers is even, the product is positive; if it is odd, the product is negative.
(i) Number of negative integers = 15, which is odd.
∴ The product will be negative.
(ii) Number of negative integers = 12, which is even.
∴ The product will be positive.
(iii) Number of negative integers = 18 - 9 = 9, which is odd.
∴ The product will be negative.
(iv) Number of negative integers = 18, which is even.
∴ The product will be positive.
Find which is greater ?
(i) (8 + 10) × 15 or 8 + 10 × 15
(ii) 12 × (6 - 8) or 12 × 6 - 8
(iii) {(-3) - 4} × (-5) or (-3) - 4 × (-5)
Answer
(i) (8 + 10) × 15 = 18 × 15 = 270
8 + 10 × 15 = 8 + 150 = 158
Since 270 > 158,
∴ (8 + 10) × 15 is greater.
(ii) 12 × (6 - 8) = 12 × (-2) = -24
12 × 6 - 8 = 72 - 8 = 64
Since 64 > -24,
∴ 12 × 6 - 8 is greater.
(iii) {(-3) - 4} × (-5) = (-7) × (-5) = 35
(-3) - 4 × (-5) = -3 - (-20) = -3 + 20 = 17
Since 35 > 17,
∴ {(-3) - 4} × (-5) is greater.
State true or false :
(i) product of two different integers can be zero.
(ii) product of 120 negative integers and 121 positive integers is negative.
(iii) a × (b + c) = a × b + c
(iv) (b - c) × a = b - c × a.
Answer
(i) True. If one of the two different integers is 0, their product is 0. For example, 0 × 5 = 0.
(ii) False. The number of negative integers is 120, which is even, so the product is positive (positive integers do not change the sign).
(iii) False. By the distributive property, a × (b + c) = a × b + a × c, not a × b + c.
(iv) False. By the distributive property, (b - c) × a = b × a - c × a, not b - c × a.