The average of four numbers is 45. If 4 is added to the first number, the second number is multiplied by 2, 4 is subtracted from the third number, and the fourth number is divided by 2, then all the resulting values become equal. Find the difference between the second-highest number and the smallest number.

ATQ, Let the common resulting value be y. When 4 is added to the first number, the result is y. So, first number = y − 4 When the second number is multiplied by 2, the result is y. So, second number = y/2 When 4 is subtracted from the third number, the result is y. So, third number = y + 4 When the fourth number is divided by 2, the result is y. So, fourth number = 2y Average of four numbers = 45 So, total of four numbers = 45 × 4 = 180 Now add the four numbers: (y − 4) + y/2 + (y + 4) + 2y = 180 (2y) + 2y + y/2 = 180 4y + y/2 = 180 (8y + y)/2 = 180 9y/2 = 180 9y = 180 × 2 9y = 360 y = 360/9 y = 40 First number = 40 − 4 = 36 Second number = 40/2 = 20 Third number = 40 + 4 = 44 Fourth number = 2 × 40 = 80 Numbers are 36, 20, 44 and 80 Smallest number = 20 Second-highest number = 44 Required difference = 44 − 20 = 24