On the number line above, the segment from 0 to 1 has been divided into fifths, as indicated by the large tick marks, and also into sevenths, as indicated by the small tick marks. What is the least possible distance between any two of the tick marks?

1/70
1/35
2/35
1/12
1/7

Let x = the least possible distance between any two of the tick marks.
Since the answer choices are FRACTIONS, the total distance can be ANY VALUE.
The question stem can be rephrased as follows:
If x is the least possible distance between any two of the tick marks, what FRACTION of the total distance is occupied by x?

Let the total distance = the LCM of 5 and 7 = 35, yielding the following number line:
0..........................................................................35

Since 35/7 = 5, dividing the number line into sevenths yields the following tick marks:
5, 10, 15, 20, 25 and 30.
Since 35/5 = 7, dividing the number line into fifths yields the following tick marks:
7, 14, 21, and 28.
Resulting number line:
0.....5..7...10....14.15.....20.21....25...28....30.....35

As indicated by the portions in red, x=1.
Thus:
x/total distance = 1/35.

The correct answer is B.

Another approach:

Our distance should be (x/5) - (y/7), where x and y are positive integers less than 6 and 8, respectively.

That means that our distance is

(7x/35) - (5y/35) or

(7x - 5y)/35

For A to be the answer, we'd need (7x - 5y) = .5, which is impossible, since x and y are integers.

For B to be the answer, we'd need (7x - 5y) = 1. This suggests 7x ending 1 and 5y ending in 0, or 7*3 - 5*4. So x = 3, y = 4 works, and (3/5) - (4/7) = 1/35. Since this is the smallest answer that works, we're done!