The sum of the digits of a 3-digit number is three times the sum of the digits of a 2-digit number. When the 2-digit number is subtracted from the 3-digit number it gave another 2-digit number of which its first digit is three times the first digit of the first 2-digit number. If the last digit of the 3-digit number equals the last digit of the second 2-digit number, and three times of the sum of the first and second digit of the 3-digit number equals the first digit of the second 2-digit number.
Find the 3-digit number if two times the first digit of the first 2-digit number equals the last digit of the 3-digit number and the sum of the first and second digit of the 3-digit number equals the first digit of the first 2-digit number and also two times the sum of the first and second digit of the 3-digit number equals the last digit of the second 2-digit number. Also the sum of all the digits of the three different number is 27. And the second 2-digit number is greater than the first 2-digit by 66. Finally when the digits of the second 2-digit number are interchanged it is 39 greater than the first 2-digit number.
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Let's break down the problem into smaller parts and identify the given information and the unknowns.
Given information:
1. The sum of all the digits of the three different numbers is 27.
2. The second 2-digit number is greater than the first 2-digit number by 66.
3. When the digits of the second 2-digit number are interchanged, it is 39 greater than the first 2-digit number.
Unknowns:
1. First 2-digit number (let's call it AB, where A represents the tens digit and B represents the ones digit)
2. Second 2-digit number (let's call it CD, where C represents the tens digit and D represents the ones digit)
3. 3-digit number (let's call it XYZ, where X represents the hundreds digit, Y represents the tens digit, and Z represents the ones digit)
Now, let's set up the equations based on the given information:
Equations based on the sum of the digits:
1. A + B + C + D + X + Y + Z = 27
Equation based on the difference between the second and first 2-digit numbers:
2. CD - AB = 66
Equation based on the difference when the digits of the second 2-digit number are interchanged:
3. 10D + C - (10B + A) = 39
Equations based on the given conditions about the 3-digit number:
4. A + B + C = 3(D + X + Y + Z)
5. (100X + 10Y + Z) - (10A + B) = (10C + D)
6. 3(A + B) = C
These are the equations that represent the given information. You can solve this system of equations to find the values of A, B, C, D, X, Y, and Z.