(a) Yes, it is in the form p/q where p = -8, q = 13 and q ≠ 0. (b) No, denominator cannot be zero. (c) Yes, it is in the form p/q where p = -5, q = -9 and q ≠ 0. It simplifies to 5/9. (d) Yes, it can be written as 75/100 or 3/4 which is in the form p/q. (e) Yes, it is in the form p/q where p = 7, q = -8 and q ≠ 0. It is a negative rational number.(a) Numerator: -7; Denominator: 11; Type: Negative rational number. (b) Numerator: 9; Denominator: -13; Type: Negative rational number (-a/b=a/-b;equals -9/13). (c) Numerator: -8; Denominator: -15; Type: Positive rational number (equals 8/15). (d) Numerator: 7; Denominator: -8; Type: Negative rational number (equals -7/8). if numerator and denominator and type match then be consider the answer to be correct.(a) 56/-64 = -56/64 = -7/8 (dividing by HCF 8). (b) -75/125 = -3/5 (dividing by HCF 25). (c) -18/-24 = 18/24 = 3/4 (dividing by HCF 6). (d) 7/(-8) = -7/8 (negative sign should be in numerator).(a) Additive inverse: 8/11; Reciprocal: -11/8. (b) Additive inverse: -13/9; Reciprocal: 9/13. (c) Additive inverse: 1; Reciprocal: -1. (d) Additive inverse: 7/8; Reciprocal: -8/7 or 8/(-7).For reciprocal can any one of these.(a) -7/12 < -5/12 (as -7 < -5). (b) Take LCM of denominators 15 and 18, which is 90. Then -8/15 = -48/90 and -7/18 = -35/90, so -8/15 < -7/18. or cross multiply -8/15 < -7/18 is same as -8 * 18 < -7 * 15 which is -144 < -105. (c) Take LCM of denominators 9 and 11, which is 99. Then 4/9 = 44/99 and 5/11 = 45/99, so 4/9 < 5/11. or cross multiply 4/9 < 5/11 is same as 4 * 11 < 5 * 9 which is 44 < 45.(a) -7/12 + 11/15 ;Take LCM of denominators 12 and 15, which is 60. Then -7/12 = -35/60 and 11/15 = 44/60, so -7/12 + 11/15 = (-35 + 44)/60 = 9/60 = 3/20. (b) 4/9 - (-7/12) = 4/9 + 7/12 ;Take LCM of denominators 9 and 12, which is 36. Then 4/9 = 16/36 and 7/12 = 21/36, so 4/9 +7/12 = (16 + 21)/36 = 37/36. (c) -8/15 × 9/20 = (-8 × 9)/(15 × 20) .we can cancel 8 and 20(HCF is 4),9 and 15(HCF is 3). 2/5 * 3/5 = 6/25 or -8/15 × 9/20 = -72/300 = -6/25. (d) 7/(-8) + 3/4 = -7/8 + 3/4 (LCM of denominators 8 and 4, which is 8.)= -7/8 + 6/8 = -1/8.(a) -5/6 and -2/3. LCM of 6 and 3 is 6. -5/6 and -4/6. Between -5 and -4 there are no integers. So we cannot find rational numbers easily with denominator 6. Multiply both by 5 (or any number > 3). -5/6 = -25/30 and -2/3=-4/6 = -20/30. Three numbers between -25/30 and -20/30 are: -24/30, -23/30, -22/30. Here answers may change if we multiply both by other numbers like 4 or 6.Consider based on this. (b) 1/4 and 1/3. LCM of 4 and 3 is 12. 1/4 = 3/12 and 1/3 = 4/12. Between 3 and 4 there are no integers. So we cannot find rational numbers easily with denominator 12. Multiply both by 4 (or any number > 3). 3/12 = 12/48 and 4/12 = 16/48. Three numbers between 12/48 and 16/48 are: 13/48, 14/48, 15/48. Here answers may change if we multiply both by other numbers like 6 or 8.Consider based on this.(a) -11/18 ÷ 22/27 (division becomes multiplication by reciprocal) = -11/18 × 27/22 = -297/396 = -3/4. (b) (-4/9) ÷ 8/21 + 1/6 (division becomes multiplication by reciprocal) = -4/9 × 21/8 + 1/6 = -7/6 + 1/6 = -6/6 = -1. (c) 7/(-8) × (-4/5) (LCM of denominators 8 and 5, which is 40.) = -7/8 × (-4/5) = 28/40 = 7/10.(a) C gets: total chocolates(fraction=1) - (A's chocolates + B's chocolates) = 1 - (4/9 + 1/3) (LCM of denominators 9 and 3, which is 9.)= 1 - (4/9 + 3/9) = 1 - 7/9 = 2/9. (b) If 2/9 = 20 chocolates, then total = 20 × 9/2 = 90 chocolates.(a) (5/8 + 3/10) × 2/7 (LCM of denominators 8 and 10, which is 40.)= (25 + 12)/40 × 2/7 = 37/40 × 2/7 = 74/280 = 37/140. (b) 3/4 - 2/5 + 1/2 (LCM of denominators 4 and 5, which is 20.)= 15/20 - 8/20 + 10/20 = 17/20. (c) (-2/3) + (-5/6) - (-7/12) (LCM of denominators 3 and 6, which is 12.)= -8/12 - 10/12 + 7/12 = -11/12.Let the other number be x. x + 11/12 = -5/8 x = -5/8 - 11/12 (LCM of denominators 12 and 8, which is 24)= (-15 - 22)/24 = -37/24.Length of each piece =(piece of the rope)/ (number of pieces) = 7/8 ÷ 5 (reciprocal of 5 and change division to multiplication) = 7/8 × 1/5 = 7/40 meter.Next two terms: -16/243, 32/729. Pattern rule: Each term is obtained by multiplying the previous term by -2/3.Distance walked towards North = 3/4 km. Distance walked towards South = 5/6 km. Let North be positive and South be negative. Final position = 3/4 - 5/6 = (9 - 10)/12 = -1/12 km. Since the result is negative, Ravi will be 1/12 km towards south of point A.(a) Convert them to lowest terms: -7/21 = -1/3 and 3/9 = 1/3. They are not the same (opposite signs). (b) Convert them to lowest terms: -16/20 = -4/5 and 20/-25 = -20/25 = -4/5. Yes, they represent the same rational number.(a) -3/7 = -15/35 (multiply numerator and denominator by 5). (b) 4/9 = 20/45 (multiply numerator and denominator by 5). (c) -5/8 = -15/-24 (multiply numerator and denominator by 3).Let the four rational numbers be a, a+d, a+2d, a+3d. Given smallest number (a) = -7/10. Given largest number (a+3d) = 1/2. -7/10 + 3d = 1/2 3d = 1/2 + 7/10 = 5/10 + 7/10 = 12/10 = 6/5. d = (6/5) / 3 = 2/5. Second number = a + d = -7/10 + 2/5 = -7/10 + 4/10 = -3/10. Third number = a + 2d = -7/10 + 2(2/5) = -7/10 + 8/10 = 1/10. The two middle numbers are -3/10 and 1/10.(a) Commutative property of addition. (b) Associative property of multiplication. (c) Multiplicative identity property.Time taken to fill the tank = 5/4 hours. Part of tank filled in 1 hour = 1 ÷ (5/4) = 4/5. Time for which pipe is open = 3/2 hours. Part of tank filled in 3/2 hours = 4/5 × 3/2 = 12/10 = 6/5. Since 6/5 > 1, the tank will be full and overflow.(a) -3/5 + 0 = -3/5. This verifies additive identity property. (b) 7/11 × 0 = 0. This verifies zero property of multiplication. (c) -4/9 × (-9/4) = 36/36 = 1. This verifies reciprocal property (product = 1).
