SOLUTION: When four positive integers are added three at a time, the sums are 1068, 1430. 1744, and 2013. What is the sum of the digits of the four original integers?

Question 1131957: When four positive integers are added three at a time, the sums are 1068, 1430. 1744, and 2013. What is the sum of the digits of the four original integers?
Answer by ikleyn(52834) (Show Source): You can put this solution on YOUR website!
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Let the numbers be a, b, c, and d.  Then from the condition, you have these 4 equation for 4 unknowns


a + b + c     = 1068     (1)
a + b     + d = 1430     (2)
a     + c + d = 1744     (3)
    b + c + d = 2013     (4)


Add the equation (1), (2), (3) and (4). You will get

3a + 3b + 3c + 3d = 1068 + 1430 + 1744 + 2013 = 6255.



Dividing both sides by 3, you get

a + b + c + d = 2085     (5)


Now subtract eq(1) from eq(5). You will get

d = 2085 - 1068 = 1017.


    Subtract eq(2) from eq(5). You will get

c = 2085 - 1430 =  655.


Similarly,

b = 2085 - 1744 = 341,

a = 2085 - 2013 =  72.


Now, when you have the numbers, you can easily calculate the sum of their digits.


Answer.  42.

Solved.

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