Three numbers are such that when the average of any two is added to the third, the results are 200, 180, and 160. What is the average of the three numbers?

The correct option is 90.

Let the three numbers be $x$, $y$, and $z$.

According to the problem, adding the average of any two numbers to the third gives the results 200, 180, and 160. We can formulate these into three equations:

$\frac{x+y}{2}+z=200$

$\frac{y+z}{2}+x=180$

$\frac{+x}{2}+y=160$

To eliminate the fractions, multiply each equation by 2:

$x+y+2z=400$

$2x+y+z=360$

$x+2y+z=320$

Now, sum all three simplified equations together:

$(x+y+2z)+(2x+y+z)+(x+2y+z)=400+360+320$

Combine the like terms:

$4x+4y+4z=1080$

Factor out 4 from the left side of the equation:

$4(x+y+z)=1080$

Divide both sides by 4 to solve for the sum of the three numbers:

$x+y+z=\frac{1080}{4}=270$

The average of the three numbers is calculated by dividing their sum by 3:

$\text{Average}=\frac{x+y+z}{3}$

Substitute the sum ($270$) into the average formula:

$\text{Average}=\frac{270}{3}=90$

Hence, the average of the three numbers is 90.