1. Add, each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
(∵ Denominators are same, ∴ LCM = 8)
(ii) (-8)/13 and (-4)/13
(∵ LCM of 13 and 13 = 13)
(iii) 6/11 and (-9)/11
(∵ Denominator are same, ∴ LCM = 11)
(iv) 5/(-26) and 8/39
= (-15 + 16)/78 (∵ LCM of 26 and 39 = 78)
(v) 5/(-6) and 2/3
(∵ LCM of 6 and 3 = 6)
(vi) (-2) and 2/5
(vii) 9/-(4) and (-3)/8
(∵ LCM of 4 and 8 = 8)
(viii) 7/(-18) and 8/27 2. Evaluate:
= (-32 – 15)/36 (∵ LCM of 9 and 12 = 36) = (-47)/36
= (0 × 7)/(1 × 7) – (2 × 1)/(7 × 1) (∵ LCM of 0 and 7 = 7) = (0 – 2)/7 = -2/7
(viii) 2 + (-3)/5 = 2/1 – 3/5 (∵ LCM of 1 and 5 = 5) = (2 × 5)/(1 × 5) – (3 × 1)/(5 × 1) = (10 – 3)/5 = 7/5 = 1 2/5 (ix) 4/(-9) + 1 = (-4)/9 + 1/1 (∵ LCM of 9 and 1 = 9) = (-4 × 1)/(9 × 1) + (1 × 9)/(1 × 9) = (-4 + 9)/9 = 5/9 3. Evaluate : (i) 3/7 + (-4/9) + (-11/7) + 7/9 (ii) 2/3 + -4/5 + 1/3 + 2/5 (iii) 4/7 + 0 + (-8)/9 + (-13)/7 + 17/9 (iv) 3/8 + (-5)/12 + 3/7 + 3/12 + (-5)/8 + (-2)/7 Solution (i) 3/7 + (-4)/9 + (-11)/7 + 7/9 = {3/7 + (-11)/7} + {(-4)/9 + 7/9) = (3 – 11)/7 + (-4 + 7)/9 = (-8)/7 + 3/9 = (-8)/7 + 1/3 ∴ LCM of 3 and 2 = 3 × 7 = 21 = {(-8 × 3)/(7 × 3) + (1 × 7)/(3 × 7)} (∵ LCM of 7 and 3 = 21) = (-24 + 7)/21 = (-17)/21 (ii) 2/3 + (-4)/5 + 1/3 + 2/5 (2/3 + 1/3) + (-4/5 + 2/5) = (2 + 1)/3 + (-4 + 2)/5 = 3/3 + (-2/5) ∴ LCM of 3 and 5 = 3 × 5 = 15 = (3 × 5)/(3 × 5) + (-2 × 3)/(5 × 3) (∵ LCM of 3 and 5 = 15) = (15 – 6)/15 = 9/15 = 3/5
(iv) 3/8 + (-5)/12 + 3/7 + 3/12 + (-5)/8 + (-2)/7 = (3/8 – 5/8) + {(-5)/12 + 3/12} + (3/7 – 2/7) = (-2)/8 – 2/12 + 1/7 = (-1)/4 – 1/6 + 1/7 ∴ LCM of 4, 6 and 7 = 2 × 2 × 3 × 7 = 84 = (-1 × 21)/(1 × 14)/(6 × 14) + (1 × 12)/(7 × 12) (∵ LCM of 4, 6 and 7 = 84) = (-21 – 14 + 12)/84 = (-35 + 12)/84 = (-23)/844. For each pair of rational numbers, verify commutative property of addition of rational numbers: (i) -8/7 and 5/14 (ii) 5/9 and 5/-12 (iii) -4/5 and -13/-15 (iv) 2/-5 and 11/-15 (v) 3 and -2/7 (vi) -2 and 3/-5 Solution (i) (-8)/7 and 5/14 To show that : - (-8)/7 + 5/14 = 5/14 + (-8)/7 ∵ (-8)/7 + 5/14 ∴ LCM of 2 and 7 = 14 = (-8 × 2)/(7 × 2) + (5 × 1)/(14 × 1) = (- 16 + 5)/14 = -(11)/14 And, 5/14 + (-8)/7 = {(5 × 1)/(14 × 1) + (-8 × 2)/(7 × 2)} = (5 – 16)/14 = (-11)/14 ∴ (-8)/7 + 5/14 = 5/14 + (-8)/7 This verifies the commutative property for the addition of rational numbers. (ii) 5/9 and 5/(-12) To show that : 5/9 + 5/(-12) = 5/(-12) + 5/9 ∵ 5/9 + 5/(-12) ∴ LCM of 9 and 12 = 2 × 2 × 3 × 3 = 36 = (5 × 4)/(9 × 4) – (5 × 3)/(12 × 3) = (20 – 15)/36 = 5/36 And, 5/(-12) + 5/9 = {(5 × 3)/(-12 × 3) + (5 × 4)/(12 × 3)} = (-15 + 20)/36 = 5/36 ∴ 5/9 + 5/(-12) = 5/(-12) + 5/9 This verifies the commutative property for the addition of rational numbers.(iii) (-4/5) and (-13/-15) To show that : (-4/5) and (-13/-15) = (-13/-15) + (-4)/5 ∵ (-4)/5 + 13/15 ∴ LCM of 5 and 15 = 5 × 3 = 15 = (-4 × 3)/(5 × 3) + (13 × 1)/(15 × 1) = (-12 + 13)/15 = 1/15 And, 13/15 + (-4)/5 = {(13 × 1)/(15 × 1) + (-4 × 3)/(5 × 3)} = (13 – 12)/15 = 1/15 ∴ (-4)/5 + (-13/-15) = (-13/-15) + (-4)/5 This verifies the commutative property for the addition of rational numbers.(iv) 2/(-5) and 11/(-15) Show that: 2/(-5) + 11/(-15) = 11/(-15) + 2/(-5) = 2/(-5) + 11/(-15) ∴ LCM of 5 and 15 = 15 = (-2 × 3)/(5 × 3) – (11 × 1)/(15 × 1) = (-6 – 11)/15 = (-17)/15 And, 11/(-15) + 2/(-5) = (-11 × 1)/(15 × 1) – (2 × 3)/(5 × 3) = (- 11 – 6)/15 = (-17)/15 ∴ 2/(-5) + 11/(-15) = (11/-15) + 2/(-5) This verifies the commutative property for the addition of rational numbers.(v) 3 and (-2)/7 Show that : 3/1 + (-2)/7 = (-2)/7 + 3/1 = 3/1 + (-2)/7 (∵ LCM of 1 and 7 = 7) = (3 × 7)/(1 × 7) - (2 × 1)/(7 × 1) = (21 – 2)/7 = 19/7 And, (-2)/7 + 3/1 = {(-2 × 1)/(7 × 1) + (3 × 7)/(1 × 7)} = (-2 + 21)/7 = 19/7 ∴ 3/1 + (-2)/7 = (-2)/7 + 3/1 This verifies the commutative property for the addition of rational numbers.
= (-2)/1 + (-3)/5 (∵ LCM of 1 and 5 = 5) = {(-2 × 5)/(1 × 5) + (-3 × 1)/(5 × 1)} = (-10 – 3)/5 = (-13)/5 And, (-3)/5 + (-2)/1 = {(-3 × 1)/(5 × 1) + (-2 × 5)/(1 × 5)} = (-3 – 10)/5 = (-13)/5 ∴ (-2)/1 + (-3)/5 = (-3)/5 + (-2)/1 This verifies the commutative property for the addition of rational numbers.5. For each set of rational numbers, given below, verify the associative property of addition of rational numbers. (i) 1/2, 2/3 and -1/6 (ii) -2/5, 4/15 and -7/10 (iii) -7/9, 2/-3 and -5/18 (iv) -1, 5/6 and -2/3 Solution (i) 1/2, 2/3 and (-1)/6 Show that : ½ + {2/3 + (-1)/6} = (1/2 + 2/3) + (-1)/6 ∵ ½ + {2/3 + (-1)/6} ∴ LCM of 3 and 6 = 6 = ½ + {(2 × 2)/(3 × 2) + (-1 × 1)/(6 × 1)} = ½ + (4/6 – 1/6) = ½ + {(4 – 1)/6} = ½ + (3/6) = (1 × 3)/(2 × 3) + (3 × 1)/(6 × 1) (∵ LCM of 2 and 6 = 3) = (3 + 3)/6 = 6/6 = 1 And, (1/2 + 2/3) + (-1/6) ∴ LCM of 2 and 3 = 6 = {(1 × 3)/(2 × 3) + (2 × 2)/(3 × 2) + (-1)/6} = {(3 + 4)/6 + (-1)/6} ∴ ½ + {2/3 + (-1)/6} = (1/2 + 2/3) + (-1)/6 This verifies associative property of the addition of rational numbers.(ii) (-2/5), (4/15) and (-7/10) Show that: (-2)/5 + {4/15 + (-7)/10} = (-2/5 + 4/15) + (-7)/10 ∵ (-2)/5 + {4/15 + (-7)/10} ∴ LCM of 15, 10 = 2 × 3 × 5 = 30 = (-2)/5 + {(4 × 2)/(15 × 2) + (-7 × 3)/(10 × 3)} (∵ LCM of 15 and 10 = 30) = (-2)/5 + {(8 – 21)/30} = (-2)/5 – 13/30 = {(-2 × 6)/(5 × 6) – (13 × 1)/(30 × i)} = (-12 – 13)/30 = (-25)/30 = (-5)/6 And, {(-2)/5 + 4/15} + (-7)/10 ∴ LCM of 5 and 15 = 3 × 5 = 15 = {(-2 × 3)/(5 × 3) + (4 × 1)/(15 × 1) + (-7)/10 ∴ LCM of 5 and 15 = 15 = (-6 + 4)/15 + (-7)/10 = (-2)/15 + (-7/10) = (-2 × 2)/(15 × 2) – (7 × 3)/(10 × 3) = (-4)/30 – 21/30 = (-25)/30 = (-5)/6 ∴ (-2)/5 + {4/15 + (-7)/10} = {(-2)/5 + 4/15 + (-7)/10} This verifies associative property of the addition of rational numbers.(iii) (-7)/9, 2/(-3) and (-5)/18 Show that : (-7)/9 + {2/(-3) + (-15)/18} = {(-7)/9 + 2/(-3)} + (-5)/18 ∵ (-7)/9 + {2/(-3) + (-5)/18} ∴ LCM of 3 and 18 = 2 × 3 × 3 = 18 = (-7)/9 + {(-2 × 6)/(3 × 6) + (-5 × 1)/(18 × 1)} (∵ LCM of 3 and 18 = 18) = (-7)/9 + {(-12 – 5)/18} = (-7)/9 + (-17)/18 = (-7 × 2)/(9 × 2) – (17 × 1)/(18 × 1) (∵ LCM of 9 and 18 = 18) = (- 14 – 17)/18 = (-31)/18 And, (-7)/9 + 2/(-3)} + (-5)/18 ∴ LCM of 3 and 9 = 3 = {(-7 × 1)/(9 × 1) + (-2 × 3)/(3 × 3) + (-5)/18 (∵ LCM = 9 and 3 = 9) = {(-7 – 6)/9 + (-5)/18} = (-13)/9 + (-5)/18 = (-13 × 2)/(9 × 2) + (-5 × 1)/(18 × 1) = (-26 – 5)/18 = (-31)/18 ∴ (-7)/9 + {2/(-3) + (-15)/18} = {(-7)/9 + 2/(-3)} + (-5)/18} This verifies associative property of the addition of rational numbers.(iv) (-1), 5/6 and (-2)/3 Show that: This verifies associative property of the addition of rational numbers. (-1)/1 + {5/6 + (-2)/3} = {(-1)/1 + 5/6} + (-2)/3 ∵ (-1)/1 + (5/6 + (-2)/3} ∴ LCM of 6 and 3 = 6 = (-1)/1 + {(5 × 1)/(6 × 1)+ (-2 × 2)/(2 × 2)} (∵ LCM of 6 and 3 = 6) = (-1)/1 + {(5 - 4)/6} = {(-1)/1 + 1/6} = {(-1 × 6)/(1 × 6) + (1 × 1)/(6 × 1)} (∵ LCM of 1 and 6 = 1) = (-6 + 1)/6 = (-5)/6 And, {(-1)/1 + 5/6} + (-2)/3 = {(-1 × 6)/(1 × 6) + (5 × 1)/(6 × 1) + (-2)/3 (∵ LCM of 1 and 6 = 6) = (-6 + 5)/6 + (-2)/3 = (-1)/6 + (-2)/3 = {(-1 × 1)/(6 × 1) + (-2 × 2)/(3 × 2)} (∵ LCM of 6 and 3 = 6) = (-1 – 4)/6 = (-5)/6 ∴ (-1)/1 + {(5/6 + (-2)/3} = (-1)/1 + 5/6} + (-2)/36. Write the additive inverse (negative) of: (i) -3/8 (ii) 4/-9 (iii) -7/5 (iv) -4/-13 (v) 0 (vi) -2 (vii) 1 (viii) -1/3 (ix) -3/1 Solution (i) The additive inverse of (-3)/8 = (3/8) (ii) The additive inverse of 4/(-9) = 4/9 (iii) The additive inverse of (-7)/5 = 7/5 (iv) The additive inverse of (-4/-13) or (4/13) = - 4/13 (v) The additive inverse of 0 = 0 (vi) The additive inverse of – 2 = 2 (vii) The additive inverse of 1 = (-1) (viii) The additive inverse of - 1/3 = 1/3 (ix) The additive inverse of (-3)/1 = 3 7. Fill in the blanks: (i) Additive inverse of -5/-12 = _______ (ii) -5/-12 + its additive inverse = _______ (iii) If a/b is additive inverse of -c/d, then -c/d is additive inverse of _______ Solution (i) Additive inverse of (-5)/(-12) = -5/12 (ii) (-5)/(-12) + its additive inverse = (-5)/(-15) + (- 5/15) = 0. (iii) If a/b is additive inverse of (-c)/d, then (-c)/d is additive inverse of a/b. Also, a/b + (-c)/d – (-c)/d + a/b = 0 8. State true of false: (i) 7/9 = (7 + 5)/(9 + 5) (ii) 7/9 = (7 – 5)/(9 – 5) (iii) 7/9 = (7 × 5)/(9 × 5) (iv) 7/9 = (7 + 5)/(9 + 5) (v) (-5)/(-12) is a negative rational number (vi) (-13)/25 is smaller than (-25)/13. Solution (i) False (ii) False (iii) True (iv) True (v) False (vi) False 1. Evaluate: = (2 × 5)/(3 × 5) – (4 × 3)/(5 × 3) (∵ LCM of 3 and 5 = 15) = (10 – 12)/15 = (-2)/15(ii) (-4)/9 – 2/(-3) = (-4 × 1)/(9 × 1) – (-2 × 3)/(3 × 3)(∵ LCM of 3 and 9 = 9)= (-4 + 6)/9 = 2/9
(vi) 5/21 – (-13)/42 ∴ LCM of 21, 42 = 2 × 3 × 7 = 42= (5 × 2)/(21 × 2) – (-13 × 1)/(42 × 1)(∵ LCM of 21 and 42 = 42)= (10 + 13)/42 = 23/422. Subtract: (i) 5/8 from -3/8 (ii) -8/11 from 4/11 (iii) 4/9 from -5/9 (iv) 1/4 from -3/8 (v) -5/8 from -13/16 (vi) (9/22 from 5/33 Solution (i) 5/8 from (-3)/8= (-3)/8 – 5/8= (-3 × 1)/(8 × 1) – (5 × 1)/(8 × 1)= (-3 – 5)/8= (-8)/8 = -1
(iii) 4/9 and (-5)/9 = (-5)/9 – 4/9= (-5 – 4)/9= (-9)/9 = -1
= (-3)/8 – ¼ (∵ LCM of 8 and 4 = 8) = (-3 × 1)/(8 × 1) – (1 × 2)/(4 × 2)= (-3 – 2)/8= (-5)/8
(vi) (-9)/22 from 5/33 ∴ LCM of 22 and 33 = 2 × 3 × 11 = 66= 5/33 – (-9)/22= (5 × 22)/(33 × 2) + (9 × 3)/(22 × 3) (∵ LCM of 22 and 33 = 66)= (10 + 27)/66= 37/663. The sum of two rational number is 9/20. If one of them is 2/5, find the other. (∵ LCM of 20 and 5 = 20) 4 The sum of the two rational numbers is -2/3. If one of them is -8/5, find the other. 5 The sum of the two rational numbers is -6. If the one of them is -8/5, fins the other. 6. Which rational number should be added to -7/8 to get 5/9? Required rational number = 5/9 – (-7)/8= 5/9 + 7/8∴ LCM of 9 and 8 = 2 × 2 × 2 × 3 × 3 = 72= (5 × 8)/(9 × 8) + (7 × 9)/(8 × 9)(∵ LCM of 9 and 8 = 72)= 40/72 + 63/72= (40/72 + 63/72= (40 + 63)/72= 103/72= 1.31/72 7. Which rational number should be added to -5/9 to get -2/3? 8. Which rational number should be added to -5/6 to get 4/9 ? 9. (i) What should be subtracted from (-2) to get 3/8 (ii)What should be added to -2 to get 3/8 Solution (i) Set the required number be = xAccording to the condition,-2 – x = 3/8⇒ -x = 3/8 + 2⇒ -x = (3 + 16)/8⇒ x = (-19)/8∴ The required number = (-19)/8 (ii) Let the required number be = x According to the question,-2 + x = 3/8⇒ x = 3/8 + 2⇒ x = 3/8 + 2⇒ x = (3 + 16)/8 = 19/8 = 2 3/8∴ The required number = 19/8 = 2 3/810. Evaluate: (i) 3/7 + (-4)/9 – (-11)/7 – 7/9 (ii) 2/3 + (-4)/5 – 1/3 – 2/5 (iii) 4/7 – (-8)/9 – (-13)/7 + 17/9 Solution (i) 3/7 + (-4)/9 – (-11)/7 – 7/9= {3/7 – (-11)/7} + {(-4)/9 – 7/9}= (3/7 + 11/7) + {(-4)/9 – 7/9}= 14/7 + (-11)/9= 2 – 11/9 = (2 × 9 – 11)/9 = 7/9 (ii) 2/3 + (-4)/5 – 1/3 – 2/5 = (2/3 – 1/3) + {(-4)/5 – 2/5}= 1/3 + (-6)/5= 1/3 – 6/5= (1 × 5 – 6 × 3)/15 (∵ LCM of 3 and 5 = 15) = (5 – 18)/15 = - 13/15 (iii) 4/7 – (-8)/9 – (-13)/7 + 17/9 = {4/7 – (-13)/7} - {(-8)/9 – 17/9)= (4/7 + 13/7) – {(-8)/9 – 17/9}= 17/7 – (-25)/9= 17/7 + 25/9 (∵ LCM of 7 and 9 = 63) = (17 × 9 + 25 × 7)/63= (153 + 175)/63 1. Evaluate: = 12/5 = 2 2/5 (ii) 7/6 × (-18)/91 = {7 × (-18)}/(6 × 91)= {1 × (-3)}/(1 × 13) = -3/13 (iii) (-125)/72 × 9/(-5) = {(-125) × 9}/{72 × (-5)} = 3 1/8 (iv) (-11)/9 × (-51)/(-44) = {(-11) × (-51)}/{9 × (-44)} = {1 × (-51)}/(9 × 4) = (-17)/12 (v) (-16)/5 × 20/8 2. Multiply:
= {2 × (-14)}/(7 × 9) = (-4)/9 = {(-7) × 1}/(2 × 1) = 3 ½ = {9 × (-9)/{(-7) × 7} = 1 32/49 = 28/75 = {1 × (-7)}/{(-2) × 1} = 3 ½ 3. Evaluate: (i) (2/-3 × 5/4) + (5/9 × 3/-10) (ii) (2 × 1/4) – (-18/7 × -7/15) (iii) (-5 × 2/15) – (-6 × 2/9) (iv) (8/5 × -3/2) + (-3)/10 × 9/16) Solution (i){2/(-3) × 5/4} + {5/9 × 3/(-10)}= (2 × 5)/{(-3) × 4} + (5 × 3)/{9 × (-10)}= (1 × 5)/{(-3) × 2} + (1 × 1)/{3 × (-2)}= (-5)/6 + (-1)/6= (- 5 – 1)/6 = (-6)/6 (ii) (2 × 1/4) – {(-18)/7 × (-7)/15}= (2 × 1)/(1 × 4) – {(-18) × (-7)}/(7 × 15)= (1 × 1)/(1 × 2) – {(-18) × (-1)}/(1 × 15)= ½ - 18/15 ∴ LCM of 2 and 15 is 2 × 3 × 5 = 30= (1 × 15)/(2 × 15) – (18 × 2)/(15 × 2)(∵ LCM of 2 and 15 = 30) = (15 – 36)/30 = -7/10 = (-2 + 4)/3 = 2/3 (iv) {8/5 × (-3)/2} + {(-3)/10 × 9/16} = {8 × (-3)}/(5 × 2) + {(-3) × 9}/(10 × 16)= {4 × (-3)}/(5 × 1) + {(-3) × 9}/(10 × 16)= (-12)/5 + (-27)/160∴ LCM of 5 and 160 = 160= {(-12) × 32}/(5 × 32) + {(-27 × 1)/(160 × 1)}= (-384 – 27)/160= (-411)/1604. Multiply each rational number, given below, by one (1): (i) 7/(-5) (ii) (-3)/(-4) (iii) 0 (iv) (-8)/13 (v) (-6)/(-7) Solution (i) 7/(-5) = 7/(-5) × 1 (ii) (-3)/(-4) (iv) (-8)/13 (v) (-6)/(-7) = 6/7 = (-2)/45 = (-2)/45 (ii) 5/(-3) and 3/(-11) = (5 × 13)/{(-3) × (-11)} = 65/33 = (13 × 5)/{(-3) × (-11)} = 65/33
= {1 × (-8)}/(1 × 3) = (-8)/3 = (-8)/3
= {0 × (-12)}(1 × 17) = 0 = {(-12) × 0}/(17 × 1) = 0
= 1 2/15 7. Find the reciprocal (multiplicative inverse) of (i) 3/5 × 2/3 (ii) -8/3 × 13/-7 (iii) -3/5 × -1/13 Solution (i) 3/5 × 2/3 = (3 × 2)/(5 × 3) = 5/2 = {(-8) × 13}/(3 × (-7)} = 21/104 = {(-3) × (-1)}/(5 × 13) = 21 2/3 8. Verify that (x + y) × z = x × z + y × z, if (i) x = 4/5, y = -2/3 and z = -4 (ii) x = 2, y = 4/5 and z = 3/-10 Solution (i) x = 4/5, y = (-2)/3 and z = -4Using, (x + y) × z = x × z + y × z⇒ [{4/5 + (-2)/3} × -4] = [4/5 × (-4) + (-2)/3 × (-4)]⇒ {(4 × 3)/(5 × 3) - (2 × 5)/(3 × 5)} × (-4) = {(-16)/5 × 8/3)⇒ {(12 – 10)/15 × (-4)} = (- 48 + 40)/15 = (-8)/15 = (-8)/15 (ii) x = 2, y = 4/5 and z = 3/(-10)Using, (x + y) × z = x × z + y × z⇒ (2/1 + 4/5) × 3/(-10) = 2 × 3/(-10) + 4/5 × 3/(-10)⇒ {(2 × 5)/(1 × 5) + (4 × 1)/(5 × 1)} × 3/(-10) = 3/(-5) + 6/(-25)⇒ {(10 + 4)/5} × 3/(-10) = {(-3 × 5)/(5 × 5) + {(-6) × 1}/(5 × 5)⇒ 14/5 × 3/(-10) = (-15 – 6)/25⇒ (-21)/25 = (-21)/25Hence, proved. 9. Verify that x × (y – z) = x × y – x × z, if (i) x = 4/5, y = - 7/4 and z = 3 (ii) x = ¾ , y = 8/9 and z = -5 Solution (i) x = 4/5, y = -7/4 and z = 3Using, x × (y – z) = x × y – x × z⇒ 4/5 × {(-7)/4 – 3} = {4/5 × (-7)/4 – 4/5 × 3}⇒ 4/5 {(-7 × 1 – 3 × 4)/4} = (-7)/5 – 12/5⇒ {4/5 × (-7 – 12)/4} = (-7 – 12)/5 ⇒ 4/5 × (-19)/4 = -19/5 ⇒ -19/5 = -19/5 10. Name the multiplication property of rational numbers shown below: (i) 3/5 × -8/9 = -8/9 × 3/5 (ii) -3/4 × (5/7 × -8/15) = (-3/4 × 5/7) × -8/15 (iii) 4/5 × {3/-8 + (-4)/7} = (4/5 × 3/-8) + 4/5 × -4/7 (iv) -7/5 × 5/-7 = 1 (v) 8/-9 × 1 = 1 × 8/-9 = 8/-9 (vi) -3/4 × 0 = 0 Solution(i) Commutativity property.(ii) Associativity property(iii) Distributivity property(iv) Existence of inverse.(v) Existence of identity.(vi) Existence of inverse. 11. Fill in the blanks: (i) The product of two of positive rational numbers is always ………. (ii) The product of two negative rational numbers is always………… (iii) If two rational numbers have opposite signs then their product is always……. (iv) The reciprocal of a positive rational number is ……… and the reciprocal of a negative rational number is……… (v) Rational number 0 has ……..reciprocal. (vi) The product of a rational number and its reciprocal is……… (vii) The numbers …… and …….. are their own reciprocals. (viii) If m is reciprocal of n, then the reciprocal of n is……… Solution(i) The product of two of positive rational numbers is always positive.(ii) The product of two negative rational numbers is always positive.(iii) If two rational numbers have opposite signs then their product is always negative.(iv) The reciprocal of a positive rational number is positive and the reciprocal of a negative rational number is negative.(v) Rational number 0 has no reciprocal.(vi) The product of a rational number and its reciprocal is 1.(vii) The numbers 1 and -1 are their own reciprocals. (viii) If m is reciprocal of n, then the reciprocal of n is m. 1. Evaluate: (ii) 3 ÷ 3/5 = 3 × 5/3 = 5 (iii) - 5/12 ÷ 1/16 = -5/12 × 16/1= (-5 × 4)/(3 × 1) = -5 5/3 (iv) –21/16 ÷ (-7)/8 = - 21/16 × 8/(-7)= (3 × 1)/(2 × 1)= 3/2 = 1 ½ (v) 0 ÷ (-4/7) = 0 × (-7/4) = 0 (vi) 8/(-5) ÷ 24/25 = 8/(-5) × 25/24= (2 × 5)/(-1 × 6)= (1 × 5)/(-1 × 3) = (-5)/3 (vii) – ¾ ÷ (-9) = -¾ × 1/(-9) = 1/12 (viii) ¾ ÷ (- 5/12) = ¾ × (- 12/5)= (3 × 3)/(1 × (-5)} = -9/5 (ix) -5 ÷ (- 10/11) = (-5) × 11/(-10)= (1 × 11)/(1 × 2) = 11/2 = 5 ½(x) (-7)/11 ÷ (-3)/44 = {(-7)/11 × 44/(-3)}= {(-7) × 4}/{1 × (-3)} = 9 1/3 2. Divide : (i) 3 by 1/3 (ii) -2 by - ½ (iii) 0 by 7/-9 (iv) -5/8 by ¼ (v) -¾ by -9/16 Solution (i) 3 by 1/3 = 3 ÷ 1/3 = 9 (ii) (-2) by (- ½) = (-2) ÷ (- ½)= (-2) × 2/(-1) = 4 (iii) 0 by 7/(-9) = 0 ÷ 7/(-9)= 0 × (-9)/7 = 0 (iv) (-5)/8 by ¼ = (-5)/8 ÷ ¼= (-5)/8 × 4/1= {(-5) × 1}/(2 × 1) = (-5)/2 (v) – ¾ by - 9/16 = (-4)/(-3) = 4/3 = 1 1/3 3. The product of two rational numbers is -2. If one of them is 4/7, find the other. Question 4: The product of two numbers is (-4)/9. If one of them is (-2)/27, find the other. Solution 4: ∵ The product of two numbers is = - 4/9And, one of them is = (-2)/27∵ The other number = - 4/9 ÷ (-2)/27= -4/9 × 27/(-2)= (2 × 3)/(1 × 1)= 6 5. m and n are two rational numbers such that m × n = -25)/9. (i) if m = 5/3, find n, (ii) if n = -10/9, find m. Solution∵ m × n = -25/9 (i) m = 5/3∴ 5/3 × n = (-25)/9n = (-25)/9 × 3/5n = (-5 × 1)/(3 × 5) = (-5)/3 (ii) m × -10/9 = (-25)/9m = {(-25)/9 × 9/(-10)} m = (5 × 1)/(1 × 2) = 2 ½ 6. By what number must -3/4 be multiplied so that the product is -9/16? 7. By what number should (-8)/13 be multiplied to get 16?
Given, Cost of 3 2/5 or 17/5 metre cloth or = ₹88 ½ = ₹177/2∴ Cost of one metre cloth = 177/2 ÷ 17/5= (177/2 × 5/17) = ₹ 885/34 = ₹26 1/34 10. Divide the sum of 3/7 and -5/14 by -1/2.
(i) m = 2/3 and n = 3/2 (ii) m = ¾ and n = 4/3 (iii) m = 4/5 and n = -3/10 Solution (i) m = 2/3 and n = 3/2Using formula (m + n) ÷ (m – n)= (2/3 + 3/2) ÷ (2/3 – 3/2)= {(2 × 2)/(3 × 2) + (3 × 3)/(2 × 3)} ÷ {(2 × 2)/(3 × 2) – (3 × 3)/(2 × 3)}(∵ LCM of 3 and 2 = 6)= {(4 + 9)/6} ÷ {(4 – 9)/6}= 13/6 ÷ (-5)/6= 13/6 × 6/(-5) = -13/5
13. Divide the sum of 5/8 and (-11)/12 by the difference of 3/7 and 5/14 = -4 1/12 or 4 1/12 1. Draw a number line and mark ¾, 7/4, -3/4 and -7/4 on it. 2. On a number line mark the points 2/3, -8/3, 7/3, -)/3 and -2. SolutionDraw a number line as shown below: 3. Insert one rational number between (ii) 3.5 and 5 4. Insert two rational numbers between (i) 6 and 7 (ii) 4.8 and 6 (iii) 2.7 and 6.3 Solution (i) 6 and 7Given numbers = 6 and 7= 6, (6 + 7)/2, 7(Inserting one rational number between 6 and 7)= 6, 13/2, 7= 6, 6.5, 7= 6, (6 + 6.5)/2, 6.5, 7= 6, 6.25, 6.5, 7∴ Required rational numbers between 6 and 7 are = 6.25 and 6.5 (ii) 4.8 and 6 Given numbers = 4.8 and 6= 4.8, (4.8 + 6)/2, 6= 4.8, 5.4, 6(Insert one rational number 4.8 and 6)= 4.8, (4.8 + 5.4)/2, 5.4, 6= 4.8, 5.1, 5.4, 6∴ Required rational numbers between 4.8 and 6 are = 5.1 and 5.4(iii) 2.7 and 6.3 Given numbers = 2.7 and 6.3= 2.7, (2.7 + 6.3)/2, 6.3= 2.7, 4.5, 6.3= 2.7, 4.5, (4.5 + 6.3)/2 , 4.5, 6.3= 2.7, 4.5, 5.4, 6.3∴ Required rational numbers between 2.7 and 6.3 are 4.5 and 5.45:.Insert three rational numbers between (i) 3 and 4 (ii) 10 and 12 Solution (i) 3 and 4Given numbers = 3 and 4= 3, (3 + 4)/2, 4= 3, 3.5, 4= 3, (3 + 3.5)/2, 3.5, (3.5 + 4)/2, 4= 3, 3.25, 3.5, 3.75, 4Required rational numbers between 3 and 4 are = 3.25, 3.5 and 3.75 (ii) 10 and 12 Given numbers = 10 and 12= 10, (10 + 12)/2, 12= 10, 11, 12= 10, (10 + 11)/2, 11, (11 + 12)/2, 2= 10. 10.5, 11, 11.5, 12Required rational numbers between 10 and 12 are = 10.5, 11, 11.56. Insert five rational numbers between 3/5 and 2/5 7. Insert six rational numbers between 5/6 and 8/9 8. Insert seven rational numbers between 2 and 3. = 2 1/8, 2 ¼, 2 3/8, 2 ½, 2 5/8, 2 ¾ and 2 7/8 |