https://file.aiquickdraw.com/mathbot/file/1726938865856y4xc232i.jpg

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.

In a class of 40 students, 20 like mango, 11 like guava, and 4 dislike both. What is the relationship between P (students who like both) and Q = 13?

Learn how to solve a set theory problem involving 40 students, with 20 liking mango, 11 liking guava, and 4 disliking both. This problem includes solving systems of linear equations to find the number of students who like both fruits. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation