EXERCISE 1.5 PAGE NO: 1.25
1. Multiply:
(i) 7/11 by 5/4
Solution:
7/11 by 5/4
(7/11) × (5/4) = (7×5)/(11×4)
= 35/44
(ii) 5/7 by -3/4
Solution:
5/7 by -3/4
(5/7) × (-3/4) = (5×-3)/(7×4)
= -15/28
(iii) -2/9 by 5/11
Solution:
-2/9 by 5/11
(-2/9) × (5/11) = (-2×5)/(9×11)
= -10/99
(iv) -3/17 by -5/-4
Solution:
-3/17 by -5/-4
(-3/17) × (-5/-4) = (-3×-5)/(17×-4)
= 15/-68
= -15/68
(v) 9/-7 by 36/-11
Solution:
9/-7 by 36/-11
(9/-7) × (36/-11) = (9×36)/(-7×-11)
= 324/77
(vi) -11/13 by -21/7
Solution:
-11/13 by -21/7
(-11/13) × (-21/7) = (-11×-21)/(13×7)
= 231/91 = 33/13
(vii) -3/5 by -4/7
Solution:
-3/5 by -4/7
(-3/5) × (-4/7) = (-3×-4)/(5×7)
= 12/35
(viii) -15/11 by 7
Solution:
-15/11 by 7
(-15/11) × 7 = (-15×7)/11
= -105/11
2. Multiply:
(i) -5/17 by 51/-60
Solution:
-5/17 by 51/-60
(-5/17) × (51/-60) = (-5×51)/(17×-60)
= -255/-1020
Further can divide by 255 we get,
-255/-1020 = 1/4
(ii) -6/11 by -55/36
Solution:
-6/11 by -55/36
(-6/11) × (-55/36) = (-6×-55)/(11×36)
= 330/396
Further can divide by 66 we get,
330/396 = 5/6
(iii) -8/25 by -5/16
Solution:
-8/25 by -5/16
(-8/25) × (-5/16) = (-8×-5)/(25×16)
= 40/400
Further can divide by 40 we get,
40/400 = 1/10
(iv) 6/7 by -49/36
Solution:
6/7 by -49/36
(6/7) × (-49/36) = (6×-49)/(7×36)
= 294/252
Further can divide by 42 we get,
294/252 = -7/6
(v) 8/-9 by -7/-16
Solution:
8/-9 by -7/-16
(8/-9) × (-7/-16) = (8×-7)/(-9×-16)
= -56/144
Further can divide by 8 we get,
-56/144 = -7/18
(vi) -8/9 by 3/64
Solution:
-8/9 by 3/64
(-8/9) × (3/64) = (-8×3)/(9×64)
= -24/576
Further can divide by 24 we get,
-24/576 = -1/24
3. Simplify each of the following and express the result as a rational number in standard form:
(i) (-16/21) × (14/5)
Solution:
(-16/21) × (14/5) = (-16/3) × (2/5) (divisible by 7)
= (-16×2)/(3×5)
= -32/15
(ii) (7/6) × (-3/28)
Solution:
(7/6) × (-3/28) = (1/2) × (-1/4) (divisible by 7 and 3)
= -1/8
(iii) (-19/36) × 16
Solution:
-19/36 × 16 = (-19/9) × 4 (divisible by 4)
= (-19×4)/9 = -76/9
(iv) (-13/9) × (27/-26)
Solution:
(-13/9) × (27/-26) = (-1/1) × (3/-2) (divisible by 13 and 9)
= -3/-2 = 3/2
(v) (-9/16) × (-64/-27)
Solution:
(-9/16) × (-64/-27) = (-1/1) × (-4/-3) (divisible by 9 and 16)
= 4/-3 = -4/3
(vi) (-50/7) × (14/3)
Solution:
(-50/7) × (14/3) = (-50/1) × (2/3) (divisible by 7)
= (-50×2)/(1×3)
= -100/3
(vii) (-11/9) × (-81/-88)
Solution:
(-11/9) × (-81/-88) = (-1/1) × (-9/-8) (divisible by 11 and 9)
= (-1×-9)/(1×-8)
= 9/-8 = -9/8
(viii) (-5/9) × (72/-25)
Solution:
(-5/9) × (72/-25) = (-1/1) × (8/-5) (divisible by 5 and 9)
= (-1×8)/(1×-5)
= -8/-5 = 8/5
4. Simplify:
(i) ((25/8) × (2/5)) – ((3/5) × (-10/9))
Solution:
((25/8) × (2/5)) – ((3/5) × (-10/9)) = (25×2)/(8×5) – (3×-10)/(5×9)
= 50/40 – -30/45
= 5/4 + 2/3 (divisible by 5 and 3)
By taking LCM for 4 and 3 which is 12
= ((5×3) + (2×4))/12
= (15+8)/12
= 23/12
(ii) ((1/2) × (1/4)) + ((1/2) × 6)
Solution:
((1/2) × (1/4)) + ((1/2) × 6) = (1×1)/(2×4) + (1×3) (divisible by 2)
= 1/8 +3
By taking LCM for 8 and 1 which is 8
= ((1×1) + (3×8))/8
= (1+24)/8
= 25/8
(iii) (-5 × (2/15)) – (-6 × (2/9))
Solution:
(-5 × (2/15)) – (-6 × (2/9)) = (-1 × (2/3)) – (-2 × (2/3)) (divisible by 5 and 3)
= (-2/3) + (4/3)
Since the denominators are same we can add directly
= (-2+4)/3
= 2/3
(iv) ((-9/4) × (5/3)) + ((13/2) × (5/6))
Solution:
((-9/4) × (5/3)) + ((13/2) × (5/6)) = (-9×5)/(4×3) + (13×5)/(2×6)
= -45/12 + 65/12
Since the denominators are same we can add directly
= (-45+65)/12
= 20/12 (divisible by 2)
= 10/6 (divisible by 2)
= 5/3
(v) ((-4/3) × (12/-5)) + ((3/7) × (21/15))
Solution:
((-4/3) × (12/-5)) + ((3/7) × (21/15)) = ((-4/1) × (4/-5)) + ((1/1) × (3/5)) (divisible by 3, 7)
= (-4×4)/(1×-5) + (1×3)/(1×5)
= -16/-5 + 3/5
Since the denominators are same we can add directly
= (16+3)/5
= 19/5
(vi) ((13/5) × (8/3)) – ((-5/2) × (11/3))
Solution:
((13/5) × (8/3)) – ((-5/2) × (11/3)) = (13×8)/(5×3) – (-5×11)/(2×3)
= 104/15 + 55/6
By taking LCM for 15 and 6 which is 30
= ((104×2) + (55×5))/30
= (208+275)/30
= 483/30
(vii) ((13/7) × (11/26)) – ((-4/3) × (5/6))
Solution:
((13/7) × (11/26)) – ((-4/3) × (5/6)) = ((1/7) × (11/2)) – ((-2/3) × (5/3)) (divisible by 13, 2)
= (1×11)/(7×2) – (-2×5)/(3×3)
= 11/14 + 10/9
By taking LCM for 14 and 9 which is 126
= ((11×9) + (10×14))/126
= (99+140)/126
= 239/126
(viii) ((8/5) × (-3/2)) + ((-3/10) × (11/16))
Solution:
((8/5) × (-3/2)) + ((-3/10) × (11/16)) = ((4/5) × (-3/1)) + ((-3/10) × (11/16)) (divisible by 2)
= (4×-3)/(5×1) + (-3×11)/(10×16)
= -12/5 – 33/160
By taking LCM for 5 and 160 which is 160
= ((-12×32) – (33×1))/160
= (-384 – 33)/160
= -417/160
5. Simplify:
(i) ((3/2) × (1/6)) + ((5/3) × (7/2) – (13/8) × (4/3))
Solution:
((3/2) × (1/6)) + ((5/3) × (7/2) – (13/8) × (4/3)) =
((1/2) × (1/2)) + ((5/3) × (7/2) – (13/2) × (1/3))
(1×1)/(2×2) + (5×7)/(3×2) – (13×1)/(2×3)
1/4 + 35/6 – 13/6
By taking LCM for 4 and 6 which is 24
((1×6) + (35×4) – (13×4))/24
(6 + 140 – 52)/24
94/24
Further divide by 2 we get, 94/24 = 47/12
(ii) ((1/4) × (2/7)) – ((5/14) × (-2/3) + (3/7) × (9/2))
Solution:
((1/4) × (2/7)) – ((5/14) × (-2/3) + (3/7) × (9/2)) =
((1/2) × (1/7)) – ((5/7) × (-1/3) + (3/7) × (9/2))
(1×1)/(2×7) – (5×-1)/(7×3) + (3×9)/(7×2)
1/14 + 5/21 + 27/14
By taking LCM for 14 and 21 which is 42
((1×3) + (5×2) + (27×3))/42
(3 + 10 + 81)/42
94/42
Further divide by 2 we get, 94/42 = 47/21
(iii) ((13/9) × (-15/2)) + ((7/3) × (8/5) + (3/5) × (1/2))
Solution:
((13/3) × (-5/2)) + ((7/3) × (8/5) + (3/5) × (1/2)) =
(13×-5)/(3×2) + (7×8)/(3×5) + (3×1)/(5×2)
-65/6 + 56/15 + 3/10
By taking LCM for 6, 15 and 10 which is 30
((-65×5) + (56×2) + (3×3))/30
(-325 + 112 + 9)/30
-204/30
Further divide by 2 we get, -204/30 = -102/15
(iv) ((3/11) × (5/6)) – ((9/12) × (4/3) + (5/13) × (6/15))
Solution:
((3/11) × (5/6)) – ((9/12) × (4/3) + (5/13) × (6/15)) =
((1/11) × (5/2)) – ((1/1) × (1/1) + (1/13) × (2/1))
(1×5)/(11×2) – 1/1 + (1×2)/(13×1)
5/22 – 1/1 + 2/13
By taking LCM for 22, 1 and 13 which is 286
((5×13) – (1×286) + (2×22))/286
(65 – 286 + 44)/286
-177/286
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