Odd One Out

Let's look at the pairwise differences of the four numbers. There are six possible differences:

$\begin{aligned}(8x - 1) - (7x + 2) = x - 3 , \quad (8x - 1) - (6x - 5) = 2x - 6 , \quad (8x - 1) - (4x - 3) &= 4x + 2 \\ (7x + 2) - (6x+5) = x - 3 , \quad (7x + 2) - (4x-3) &= 3x+5 \\ (6x + 5) - (4x - 3) &= 2x + 8 \end{aligned}$

Since three of the four numbers are equal to each other, exactly three of the differences must be zero.

Let's find out when these terms are equal to zero. They require $x = 3, 3, 3, -\frac{5}{3}, -\frac{1}{2}, -4$ . When $x$ is equal to $3$ , three of the differences become zero. For other values of $x$ , either 1 or none of the differences are zero.

If we plug in $x = 3$ in the given expressions, we get the numbers as $23, 23, 23,$ and $9$ . We see that they are positive and all our conditions in the problem are satisfied. The sum of these 4 numbers is $78$ .

Since all the numbers are positive, it's obvious that $x$ needs to be positive as well. Thus we can see quite clearly that $4x-3$ is the number that isn't equal to the others, as the coefficient of $x$ is smaller than all the others and its constant term is more negative than any of the others.

One can then easily solve for $x$ by setting $8x-1=7x+2$ , which leads to the result $x=3$ .

Plugging this in yields $23+23+23+9=78$ .

First, note that $3(7x+2)=21x+6=8x-1+7x+2+6x+5$ for all x. Thus these are the equal values. From there, the value for x can be solved by equating any two of these equal binomials. $7x+2=8x-1\implies2+1=3=8x-7x=x\therefore x=3$ After this, it is simply a matter of solving for the sum of the four binomials. $8x-1+7x+2+6x+5+4x-3=25x+3=25(3)+3=75+3=78$

Case 1: The first two expressions are equal (i.e. the odd one out is #3 or #4): 8x-1=7x+2; solving for x, we get x=3, and each expression equals 23. Third expression: 6x+5=6x3+5=23 Fourth is odd one out: 4(3)-3=9 Sum of all terms = 3(23)+9=78 (which happens to be the answer).

We can check ourselves with Case 2, where one of those first two expressions IS the odd one out. Therefore the last two expressions must be equal. 6x+5=4x-3, solving for x, --> 2x = -8, x = -4, and each expression = -19 Since the problem states that the numbers are positive, this cannot be the case.

Based on the comment below, we could also see that 4x-3 has to be the odd one out for all the numbers to be positive... in which case, we knew that Case 1 would be the correct case.