Convert the decimals 0 . 9 ¯ , 14 . 5 ¯ and 1 . 2 4 ¯ to rational form. Can you find any easy method other than formal method?

Let x = 0 . 9 ¯ ⇒ x = 0 . 999 . . . . _________ i The periodicity of the decimal is one. Multiplying both sides of equation i by 10 , we get, 10 × x = 10 × 0 . 9999 . . . . . . . . ⇒ 10 x = 9 . 99 . . . . . . _________ ii Subtracting equation i from ii 10 x - x = 9 . 99 . . . . . - 0 . 99 . . . . . . ⇒ 9 x = 9 ⇒ x = 9 9 ⇒ x = 1 The rational form of 0 . 9 ¯ is 1 . Let y = 14 . 5 ¯ ⇒ y = 14 . 555 . . . . . . . . ________ iii The periodicity of the decimal is one. Multiplying both sides of equation iii by 10 , we get, 10 × y = 10 × 14 . 55555 . . . . . . . . ⇒ 10 y = 145 . 555555 . . . . . . _________ iv Subtracting equation iii from iv 10 y - y = 145 . 5555 . . . . . - 14 . 5555 . . . . . . ⇒ 9 y = 131 ⇒ y = 131 9 The rational form of 14 . 5 ¯ is 131 9 . Let m = 1 . 2 4 ¯ ⇒ m = 1 . 2444 . . . . ________ v The periodicity of the decimal is one. Multiplying both sides of equation v by 10 , we get, 10 × m = 10 × 1 . 24 . . . . . . . . ⇒ 10 m = 12 . 4444 . . . . . . _________ vi Subtracting equation v from vi 10 m - m = 12 . 4444 . . . . . - 1 . 24444 . . . . . . ⇒ 9 m = 11 . 2 ⇒ m = 11 . 2 9 ⇒ m = 11 . 2 × 10 9 × 10 ⇒ m = 112 90 ⇒ m = 56 45 The rational form of 1 . 2 4 ¯ is 56 45 . No, there is no other easier method other than the formal method.