+100 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)
+129899 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
