Let's break down the problem step by step:

In the class:

• 40 students total.
• 20 students like mango.
• 11 students like guava.
• 4 students dislike both fruits.

We are tasked with finding the relation between $P$ (students who like both mango and guava) and $Q = 13$.

1. Information Setup:

• Let:
  • $P$ be the number of students who like both mango and guava.
  • $M$ be the students who like only mango.
  • $G$ be the students who like only guava.

From the problem, the total is 40 students: $M + G + P + \text{students who like neither} = 40$ Since 4 students dislike both fruits, we have: $M + G + P + 4 = 40 \quad \Rightarrow \quad M + G + P = 36$

Equation 1: $M + G + P = 36$

2. Mango and Guava Groups:

• 20 students like mango (this includes those who like both): $M + P = 20$
• 11 students like guava (this includes those who like both): $G + P = 11$

3. Solving the Equations:

We now have a system of equations:

1. $M + G + P = 36$
2. $M + P = 20$
3. $G + P = 11$

Using substitution, solve for $M$ and $G$:

• From $M + P = 20$, we have $M = 20 - P$.
• From $G + P = 11$, we have $G = 11 - P$.

Substitute these into $M + G + P = 36$: $(20 - P) + (11 - P) + P = 36$ $31 - P = 36$ $P = 5$

Thus, $P = 5$.

4. Finding $Q$:

We are given that $Q = 13$, and now we have $P = 5$. Now we compare the options for the relationship between $P$ and $Q$:

• Option (a): $P = Q$ is false (since $P = 5$ and $Q = 13$).
• Option (b): $PQ = 120 - Q$ means $5 \times 13 = 120 - 13$, which is false.
• Option (c): $2P = 3Q - 2$ gives $2 \times 5 = 3 \times 13 - 2$, or $10 = 39 - 2$, which is false.
• Option (d): $P - 3 = Q + 1$ gives $5 - 3 = 13 + 1$, or $2 = 14$, which is also false.

None of the answer options seem to match based on these results. Would you like to discuss further, or review the setup to double-check the approach?

Related questions:

1. How do we determine students who like only one of the fruits?
2. How would the total number of students change if more disliked both fruits?
3. Can we derive a different relationship for $P$ and $Q$?
4. What if more categories of fruit preferences were introduced?
5. How does the problem change with varying total class sizes?

Tip:

Always check the consistency of your results with the problem's constraints, such as the total number of students.