Which fraction lies between 2/3 and 6/7? a. 3/5 b. 4/8 c. 3/7 d. 5/6

One way we could do it is to find a common multiple for all of them. 
A common number of 3 and 7 is 21. 
2/3 x 7 = 14/21. 6/7 x 3 = 18/21. We need to then find a number that is in between 14/21 and 18/21. Let's start with another common denominator, 3/7. Multiplying this we find that is 9/21, which is no where near the 14/21 that we need. We can eliminate c.
6 is double of 3, which means that we can do that one next.
Unfortunately, 6 does not line up with 21 so we need to go a bit farther up to 42. This is only double of what we had before, so we can multiply this easily!
2/3 becomes 28/42, and 6/7 becomes 36/42. 5/6 x 7 = 35/42 which is between our goal. We could stop here (which is where I am going to) or we could continue going and double checking everything (recommended).

With that being said, the answer is D. 5/6.