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The problem you're dealing with can be solved using the principle of inclusion and exclusion. Let's break it down step by step:

### Given:
- 27 students play hockey.
- 36 students play cricket.
- 13 students play both hockey and cricket.
- The number of students who play **neither** game is one less than one-third of the number of students who play hockey.

We are tasked with finding the total number of students in the class.

### Step 1: Formula for total number of students
The formula using inclusion and exclusion for students who play hockey and/or cricket is:
\[
\text{Total number of students} = (\text{Students who play hockey}) + (\text{Students who play cricket}) - (\text{Students who play both}) + (\text{Students who play neither})
\]
Let the total number of students in the class be denoted as \( N \), and let the number of students who play neither game be denoted as \( x \).

### Step 2: Condition for students playing neither game
It is given that the number of students who play neither game is one less than one-third of the number of students who play hockey:
\[
x = \frac{1}{3}(27) - 1 = 9 - 1 = 8
\]
So, 8 students play neither game.

### Step 3: Apply inclusion-exclusion principle
Now, applying the inclusion-exclusion principle, the total number of students is:
\[
N = (27 + 36 - 13) + 8 = 50 + 8 = 58
\]

Thus, the total number of students in the class is \( \boxed{58} \).

Would you like a more detailed explanation, or do you have any further questions?

### Follow-up questions:
1. How does the inclusion-exclusion principle help in calculating students involved in two sets?
2. Can you generalize the inclusion-exclusion principle to three or more sets?
3. How would the result change if the number of students playing neither game increased?
4. What if the condition for students playing neither game was different, say two less than half the number of students playing cricket?
5. How can we use Venn diagrams to visualize this problem better?

### Tip:
When solving problems with overlapping groups, always account for the overlap by subtracting the intersection—this prevents double-counting.

In a class, 27 students play hockey, 36 students play cricket, and 13 students play both games. If the number of students who play none of the games is one less than one-third of the number of students who play hockey, how many students are there in the class?

Solve a set theory problem using the inclusion-exclusion principle. The problem involves calculating the total number of students in a class where some play hockey, some play cricket, and others play both or neither. Suitable for students in Grades 8-10.

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