Each of the 110 residents in Dashville is either a rancher or

Each of the 110 residents in Dashville is either a rancher or an tractor-driver, but not both. How many residents of Dashville are ranchers?

Since each of the 110 residents in Dashville is either a rancher or a tractor-driver, but not both, we have r + t = 110, where r is the number of ranchers and t is the number of tractor-drivers.

(1) There are more than 81 ranchers who reside in Dashville

This implies that r > 81. However, the exact number of ranchers is still unclear. Not sufficient.

(2) The number of ranchers who reside in Dashville is less than three times the number of tractor-drivers who reside in Dashville

This implies r < 3t. Substituting t = 110 - r, we get r < 3(110 - r), which simplifies to r < 82.5. Still, the exact number of ranchers is unclear. Not sufficient.

(1)+(2) Combining the above, we get 81 < r < 82.5. Therefore, r must be 82. Sufficient.

Answer: C.

Hope it's clear.

Let Rangers be X and Tractor drivers be Y.
X+Y=110
We need to find x.

Statement 1: x > 81. Thus x can take any values such as 82,82,83, etc. Thus Insufficient.
Statement 2: x < 3y. There can multiple combinations of numbers which can satisfy this equation and still the sum of x & y equals 110. Thus Insufficient.

Combined 1 & 2: By addition the equations,
\( x- 81 > 0\)
\(-x+3y > 0\)

\(3y -81 > 0\)
\( y>27\)

Thus we now use x>81 and y>27 and find that only one solution satisfies the equations i.e., x=82 and y =28. Thus sufficient.

Option C is correct.