Final answer:
After creating and solving a system of algebraic equations based on the conditions of the problem, we determine that the difference between the largest and smallest original numbers is 33.
Step-by-step explanation:
The problem presents a system of equations that can be solved using algebra. Let's denote the four numbers as A, B, C, and D. The conditions given are:
- A + B + C + D = 64 (The sum of the four numbers is 64)
- A + 3 = B - 3 (The first number plus three equals the second number minus 3)
- B - 3 = 3C (Which in turn equals 3 times the third number)
- 3C = D / 3 (This, in turn, equals the fourth number divided by 3)
We can solve for one variable in terms of another and substitute into the equations step by step. Firstly, solve the second condition: A = B - 6. Substituting A in the first equation, we have:
B - 6 + B + 3C + D = 64
From the third and fourth conditions, we know B - 3 = 3C, which implies B = 3C + 3, and 3C = D / 3, which means D = 9C. Substituting these into our equation:
(3C + 3) - 6 + (3C + 3) + 3C + 9C = 64
Simplifying this gives us 3C + 3 - 6 + 3C + 3 + 3C + 9C = 64
Combining like terms, we find C = 5. Using C to solve for B, A, and D, we get B = 18, A = 12, and D = 45. The difference between the largest and smallest of the original numbers is D - A which is 45 - 12 = 33.