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Four distinct integers a, b, c and d have the property that when added in pairs, the sums 16, 19, 20, 21, 22, and 25 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)

 Jan 3, 2020
 #1
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The four integers are 6, 10, 12, 13.

 Jan 3, 2020
 #3
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This is impossible guest.....

 

10 + 13   =   23

 

But  23  isn't one of the pair  sums....

 

 

 

cool cool cool

CPhill  Jan 3, 2020
 #2
avatar+107098 
+1

Four distinct integers a, b, c and d have the property that when added in pairs, the sums 16, 19, 20, 21, 22, and 25 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)

 

Let the integers, in increasing order, be

a, b , c , d

 

We know the the  sum of the  smallest two must  be  the least sum...so...

a + b   = 16     (1)

 

And we know that the sum   of the  largest two must be the  greatest sum....so....

c + d  =  25      (2)

 

And it must be that  

b + d  =   22      (3)

 

Subtracting  (3)  from (2)    we have that

c - b  =   3       so

b + 3   = c

 

So

a + b + c + d  = 41

a + b + (b + 3) + d = 41

a + 2b + d  = 38

a + d  =   38  - 2b      (4)    

 

 

But   a + d   must be even   because  the right side of  (4)  is even

So  either

a + d  = 20       or      a  +  d   =  22  

 

If  a + d   = 20 , then  by (4)  ....... b  = 9     c    =   12      d  = 13     and.....by (1)..... a  = 7

 

This  must be true

 

a = 7     b   = 9      c   =  12       d   = 13

 

Check

a + b = 16

a + c = 19

a + d  = 20

b + c  = 21

b + d =  22

c + d =  25

 

 

cool cool cool

 Jan 3, 2020

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