When 4 positive integers are added three at a time, the sums are 1068,1430,1744 and 2013. The sum of the digits of the four original integers is

Question

Plz it’s math homework!! Need help thx so much!

Answer 1

Let the four positive integers be:  a, b, c, d

 

  a  +  b  +  c            =  1068

  a  +  b         +   d    =  1430

  a          +  c  +  d    =  1744

          b  +  c  +  d    =  2013

------------------------------------ [add equations]

 3a + 3b + 3c + 3d   =  6255

 

3(a+b+c+d) = 6255

  a+b+c+d = 2085

 

 

  a+b+c+d  =  2085

  a+b+c      =  1068

 ------------------------   [elimination; subtract equations]

               d  = 1017

 

  a + b + c + d = 2085

  a + b       + d = 1430

------------------------    [elimination; subtract equations]

              c        = 655

 

   a + b + c + d = 2085

   a       + c + d = 1744

 ---------------------------    [elimination; subtract equations]

         b              =  341

 

   a + b + c + d = 2085
         b + c + d = 2013
--------------------------- [elimination; subtract equations]

    a                  =    72

 

 

."The sum of the digits of the four original integers is"

  The sum of the digits of 1017. 655. 351 and 72 is:

               1+0+1+7+6+5+5+3+5+1+7+2 = 43

Answer 2

a+b+c   =k₁

a+b    +d=k₂

a  +c + d=k₃

   b+c+d=k₄

where the k values are as you stated. Solve this system.

You could use determinants (Cramer's rule) or, preferably, the Reduced Row Echelon Function on a calculator.

Then, when you have the answers, add up all of their digits, if that is what you want.

 

Answer 3

There is a shortcut here that prevents you having to solve the system of linear equations:

 

Let a,b,c, and d be the four positive integers described in the problem.  We can build the four relationships mentioned above

 

a+b+c=1068

a+c+d=1430

a+b+d=1744

b+c+d=2013

 

We don't have to find a,b,c, and d individually, we just need to determine a+b+c+d.

 

Notice if we add the four equations together we have

 

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

 

3(a+b+c+d)=6255

a+b+c+d=6255/3

a+b+c+d=2065

 

Voila'

 

Good Luck

 

Cheers

 

-Chris