Ratio of present age of 'A' to 'B', 'C' to 'D', 'E' to

Solution

Present age of 'H' = 19 - 7 = 12 years Present age of 'G' = (3/2) X 12 = 18 years Statement I: Let the present age of 'A', 'B', 'C', 'D', 'E' and 'F' be '2p', '3p', '3q', '5q', '4r' and '7r' years respectively. 2p + 4r = 28 -------- (I) 3p - 5q = 8 -------- (II) (3q + 7r) ÷ 2 = 17 3q + 7r = 34 --------- (III) On solving 3 X equation III - 5 X equation II, 3 X (3q + 7r) - 5 X (3p - 5q) = 3 X 34 - 5 X 8 Or, 9q + 21r - 15p + 25q = 102 - 40 Or, -15p + 34q + 21r = 62 -------- (IV) On solving 15 X equation I + 2 X equation IV, 15 X (2p + 4r) + 2 X (-15p + 34q + 21r) = 15 X 28 + 2 X 62 Or, 30p + 60r - 30p + 68q + 42r = 420 + 124 Or, 68q + 102r = 544 Or, 2q + 3r = 16 -------- (V) On solving 3 X equation III - 7 X equation V, We get, 3 X (3q + 7r) - 7 X (2q + 3r) = 3 X 34 - 7 X 16 Or, 9q + 21r - 14q - 21r = 102 - 112 Or, 5q = 10 Or, 'q' = 2 On putting value of 'q' in equation III, We get, 3 X 2 + 7r = 34 Or, 7r = 28 Or, 'r' = 4 On putting value of 'r' in equation I, We get, 2p + 4 X 4 = 28 Or, 2p = 12 Or, 'p' = 6 Average present age of all the given person = (2p + 3p + 3q + 5q + 4r + 7r + 18 + 12) ÷ 8 = (5 X 6 + 8 X 2 + 11 X 4 + 30) ÷ 8 = (120/8) = 15 years So, statement I is true. Statement II: Let the present age of 'A', 'B', 'C', 'D', 'E' and 'F' be '2p', '3p', '3q', '5q', '4r' and '7r' years respectively. 2p + 4r = 38 -------- (I) 3p - 5q = 24 -------- (II) (3q + 7r) ÷ 2 = 15 3q + 7r = 30 --------- (III) On solving 3 X equation III - 5 X equation II, 3 X (3q + 7r) - 5 X (3p - 5q) = 3 X 30 - 5 X 24 Or, 9q + 21r - 15p + 25q = 90 - 120 Or, -15p + 34q + 21r = -30 -------- (IV) On solving 15 X equation I + 2 X equation IV, 15 X (2p + 4r) + 2 X (-15p + 34q + 21r) = 15 X 38 + 2 X (-30) Or, 30p + 60r - 30p + 68q + 42r = 570 - 60 Or, 68q + 102r = 510 Or, 2q + 3r = 15 -------- (V) On solving 3 X equation III - 7 X equation V, We get, 3 X (3q + 7r) - 7 X (2q + 3r) = 3 X 30 - 7 X 15 Or, 9q + 21r - 14q - 21r = 90 - 105 Or, 5q = 15 Or, 'q' = 3 On putting value of 'q' in equation III, We get, 3 X 3 + 7r = 30 Or, 7r = 21 Or, 'r' = 3 On putting value of 'r' in equation I, We get, 2p + 4 X 3 = 38 Or, 2p = 26 Or, 'p' = 13 Average present age of all the given person = (2p + 3p + 3q + 5q + 4r + 7r + 18 + 12) ÷ 8 = (5 X 13 + 8 X 3 + 11 X 3 + 30) ÷ 8 = (152/8) = 19 years So, statement II is true. Statement III: Let the present age of 'A', 'B', 'C', 'D', 'E' and 'F' be '2p', '3p', '3q', '5q', '4r' and '7r' years respectively. 2p + 4r = 40 -------- (I) 3p - 5q = 14 -------- (II) (3q + 7r) ÷ 2 = 24 3q + 7r = 48 --------- (III) On solving 3 X equation III - 5 X equation II, 3 X (3q + 7r) - 5 X (3p - 5q) = 3 X 48 - 5 X 14 Or, 9q + 21r - 15p + 25q = 144 - 70 Or, -15p + 34q + 21r = 74 -------- (IV) On solving 15 X equation I + 2 X equation IV, 15 X (2p + 4r) + 2 X (-15p + 34q + 21r) = 15 X 40 + 2 X 74 Or, 30p + 60r - 30p + 68q + 42r = 600 + 148 Or, 68q + 102r = 748 Or, 2q + 3r = 22 -------- (V) On solving 3 X equation III - 7 X equation V, We get, 3 X (3q + 7r) - 7 X (2q + 3r) = 3 X 48 - 7 X 22 Or, 9q + 21r - 14q - 21r = 144 - 154 Or, 5q = 10 Or, 'q' = 2 On putting value of 'q' in equation III, We get, 3 X 2 + 7r = 48 Or, 7r = 42 Or, 'r' = 6 On putting value of 'r' in equation I, We get, 2p + 4 X 6 = 40 Or, 2p = 16 Or, 'p' = 8 Average present age of all the given person = (2p + 3p + 3q + 5q + 4r + 7r + 18 + 12) ÷ 8 = (5 X 8 + 8 X 2 + 11 X 6 + 30) ÷ 8 = (152/8) = 19 years So, statement III is false. Therefore, both statement I and II are true.