In a survey of 120 8th graders, 84 liked math, 73 liked science and 23 liked math but not science. how many students disliked both subjects

24  
We can divide the 8th graders into 4 groups. They are: 
1. Like both math and science. 
2. Like only math. 
3. Like only science. 
4. Don't like either math or science.  
The sum of the people in all 4 groups will add up to 120. So let's see how many are in each group. 
We've been given 23 as the number of people who like only math (group 2).
And since we know that 84 people like math, that means that 84 - 23 = 61 people are in group 1. And since we know that 73 people like science, that means that 73 - 61 = 12 are in group 3. So of the 4 groups, we now know. 
1. Like both math and science (61 people).
 2. Like only math (23 people).
 3. Like only science (12 people).
 4. Don't like either math or science (unknown).
 
 So for the people left in group 4 (don't like either subject) that will be 120 - 61 - 23 - 12 = 24.

The mode of the numbers is 29.