Three numbers are given such that when the average of any two numbers is added to the third number, the resulting sums are 450, 420, and 400. Determine the difference between the largest and smallest numbers.

Let the numbers be A, B, and C. Given: (A + B)/2 + C = 450 → A + B + 2C = 900 (B + C)/2 + A = 420 → B + C + 2A = 840 (C + A)/2 + B = 400 → C + A + 2B = 800 Add all three equations: A + B + 2C B + C + 2A C + A + 2B = 900 + 840 + 800 = 2540 4A + 4B + 4C = 2540 A + B + C = 635 From A + B + C = 635: Use A + B + 2C = 900 (635 - C) + 2C = 900 → C = 265 Use A + C + 2B = 800 (635 - B) + 2B = 800 → B = 165 A = 635 - B - C = 635 - 165 - 265 = 205 Final values: A = 205 B = 165 C = 265 Difference between largest and smallest: 265 - 165 = 100