To find out how many people ate eggs, multiple 1/7 by 42, which becomes 1/7 times 42/1. After multiplying, you get 42/7, which is 6, so 6 people ate eggs.
2/3 of the people ate both eggs and pancakes, so you multiply 2/3 by 42/1 which then becomes 84/3. So, 28 people ate both eggs and pancakes.
6 people ate eggs and 28 ate eggs and pancakes so if you add that up, that 6 + 28 = 34 people who did not eat pancakes only. 42 (total amount of people) minus 38 (total who did not eat pancakes only) becomes 4, so your answer is 4 people ate pancakes only.