Four distinct integers a, b, c and d have the property that when added in pairs, the sums 10, 18, 19, 22, 23, 31 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)
+2666 We have 6 equations:
\(a+b=10\) (1)
\(a+c=18\) (2)
\(a+d=19\) (3)
\(b+c=22\) (4)
\(b+d=23\) (5)
\(c+d=31\) (6)
Using equations 2 and 3, we see that \(d = 1+c\)
Subbing this into (6), we get: \(c+c+1=31\), meaning \(c=15\)
Subbing this into (2), we find \(a=3\)
Subbing the value of c into (4), we find \(b=7\)
Subbing the value of c into (6), we find \(d = 16\)
Thus, the values are \(\color{brown}\boxed{3,7,15,16}\)
+128408 Let a,b, c, d be in increasing order
Using some logic....we know that
a + b = 10 (1)
a + c = 18 (2)
c + d = 31 (3)
b + d = 23 (4)
What we don't know is that if
a + d = 19 or a + d = 22 or b + c = 19 or b + c = 22
To see which is true subtract (4) from (1) and we get
a - d = -13 (5)
Now let us assume that a + d = 22 (6)
Add (5) and (6) and we get that
2a = 9 but a here is not an integer
So....it must be that
a + d = 19
So
a - d = -13
a + d = 19 add these
2a = 6
So.....using our equations.....
a = 3
b = 7
c = 15
d = 16
