The sum of a number and its reciprocal is 3 41 80 . Find the number.

Let the required number be x . The sum of a number and its reciprocal is 3 41 80 Then, x + 1 x = 281 80 ⇒ x 2 + 1 x = 281 80 ⇒ 80 x 2 + 1 = 281 x ⇒ 80 x 2 + 80 = 281 x So, the quadratic equation formed is: ⇒ 80 x 2 - 281 x + 80 = 0 Splitting the middle terms, we get, ⇒ 80 x 2 - 256 x - 25 x + 80 = 0 ⇒ 16 x 5 x - 16 - 5 5 x - 16 = 0 ⇒ 5 x - 16 ( 16 x - 5 ) = 0 ⇒ 5 x - 16 = 0 or 16 x - 5 = 0 ⇒ 5 x = 16 or 16 x = 5 ⇒ x = 16 5 or x = 5 16 Hence, the required number is 16 5 or 5 16

The sum of a number and its reciprocal is 3 41 80 . Find the number.

Let the required number be x . The sum of a number and its reciprocal is 3 41 80 Then, x + 1 x = 281 80 ⇒ x 2 + 1 x = 281 80 ⇒ 80 x 2 + 1 = 281 x ⇒ 80 x 2 + 80 = 281 x So, the quadratic equation formed is: ⇒ 80 x 2 - 281 x + 80 = 0 Splitting the middle terms, we get, ⇒ 80 x 2 - 256 x - 25 x + 80 = 0 ⇒ 16 x 5 x - 16 - 5 5 x - 16 = 0 ⇒ 5 x - 16 ( 16 x - 5 ) = 0 ⇒ 5 x - 16 = 0 or 16 x - 5 = 0 ⇒ 5 x = 16 or 16 x = 5 ⇒ x = 16 5 or x = 5 16 Hence, the required number is 16 5 or 5 16

The sum of two numbers is 16 . The sum of their reciprocals is 1 3 . Find the numbers.

Let the required numbers be x and 16 - x It is given that the sum of their reciprocals is 1 3 Then, 1 x + 1 16 - x = 1 3 ⇒ 16 - x + x x 16 - x = 1 3 ⇒ 16 16 x - x 2 = 1 3 ⇒ 16 x - x 2 = 48 So, the quadratic equation formed is, ⇒ x 2 - 16 x + 48 = 0 On splitting the middle terms, we get, ⇒ x 2 - 12 x - 4 x + 48 = 0 ⇒ x x - 12 - 4 x - 12 = 0 ⇒ x - 12 x - 4 = 0 ⇒ x - 12 = 0 or x - 4 = 0 ⇒ x = 12 or x = 4 When x = 12 , then the required numbers are 12 and 4 When x = 4 , then the required numbers are 4 and 12 Hence, the required numbers are 4 and 12