Math Olympiad | A Nice Rational Equation | 90% Failed to solve!

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In first sight I understood that X should be common

mohdjahangir

(x/18) + (x/54) + (x/108) + (x/180) + (x/270) + (x/378) = 100 → from 18 to get 54, you multiply by 3 → x/18 = 3x/54
(3x/54) + (x/54) + (x/108) + (x/180) + (x/270) + (x/378) = 100 → you know that: (3x/54) + (x/54) = 4x/54
(4x/54) + (x/108) + (x/180) + (x/270) + (x/378) = 100 → from 54 to get 108, you multiply by 2 → 4x/54 = 8x/108
(8x/108) + (x/108) + (x/180) + (x/270) + (x/378) = 100 → you know that: (8x/108) + (x/108) = 9x/108
(9x/108) + (x/180) + (x/270) + (x/378) = 100 → you know that: (9x/108) = (9x)/(9 * 12) = x/12
(x/12) + (x/180) + (x/270) + (x/378) = 100 → from 12 to get 180, you multiply by 15 → x/12 = 15x/180
(15x/180) + (x/180) + (x/270) + (x/378) = 100 → you know that: (15x/180) + (x/180) = 16x/180
(16x/180) + (x/270) + (x/378) = 100 → you know that: (16x/180) = (2 * 8x)/(2 * 90) = 8x/90
(8x/90) + (x/270) + (x/378) = 100 → from 90 to get 270, you multiply by 3 → 8x/90 = 24x/270
(24x/270) + (x/270) + (x/378) = 100 → you know that: (24x/270) + (x/270) = 25x/270
(25x/270) + (x/378) = 100 → you know that: (25x/270) = (5 * 5x)/(5 * 54) = 5x/54
(5x/54) + (x/378) = 100 → from 54 to get 378, you multiply by 7 → 5x/54 = 35x/378
(35x/378) + (x/378) = 100 → you know that: (35x/378) + (x/378) = 36x/378
36x/378 = 100
(6 * 6x)/(6 * 63) = 100
6x/63 = 100
(3 * 2x)/(3 * 21) = 100
2x/21 = 100 → you multiply by 21 both sides
(2x/21) * 21 = 100 * 21
2x = 2100 → you divide by 2 both sides
x = 1050

key_board_x

सर जी नमस्कार।
ऐसे भी तो सरलता से हल किया जा सकता है कि स्टेप में ही x/18को कॉमन लिया जाए, दुसरे स्टेप में 1/21को छोड़ कर 60लघुत्तम ले-ले और अंतिम एक -दो स्टेज में हम उत्तर प्राप्त कर लेंगे। यानी केवल पांच स्टेप में हल हो जायेगा।

rameshchandrajoshi

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