https://file.aiquickdraw.com/mathbot/file/1728141804833jaigart1.jpg

The problem provides the data set of 12 students' marks arranged in ascending order: 

\[ 2, 3, 3, 3.4, x, x+2, 8, p, q, 8, 8, 9 \]

Given:
- The median is 6.
- The mode is 8.

### Step-by-Step Solution

#### 1. **Identify the Median:**
Since there are 12 data points, the median is the average of the 6th and 7th terms.

The 6th and 7th terms are \(x\) and \(x+2\). So:
\[
\text{Median} = \frac{x + (x + 2)}{2} = \frac{2x + 2}{2} = x + 1
\]
We are told that the median is 6, so:
\[
x + 1 = 6
\]
\[
x = 5
\]

#### 2. **Analyze the Mode:**
The mode is the value that appears most frequently, and we are told the mode is 8. So, 8 must appear at least twice in the data set. In the ordered data, we already see that 8 appears in two consecutive positions, which fits the requirement.

#### 3. **Substitute \(x = 5\):**
Substituting \(x = 5\) into the data set:
\[
2, 3, 3, 3.4, 5, 7, 8, p, q, 8, 8, 9
\]
Now we need to determine the values of \(p\) and \(q\).

#### 4. **Constraints for \(p\) and \(q\):**
The data must remain in ascending order. From the current sequence, we can see that:
- \( p \geq 8 \)
- \( q \geq 8 \)

Thus, we have two possibilities for \(p\) and \(q\):
- \( p = 8 \) and \( q = 8 \), which keeps the mode at 8.

### Final Answer:
- \( x = 5 \)
- \( p = 8 \)
- \( q = 8 \)

Let me know if you'd like further details on any part of the explanation.

Here are 5 related questions to explore:
1. What happens if we change the mode to a different value?
2. How would the solution change if there were 13 students instead of 12?
3. Can the data set have more than one mode? How would that affect the solution?
4. What is the effect on the median if the value of \(q\) is larger than 9?
5. How do we calculate the median if the number of observations is odd?

**Tip**: For a large dataset, organizing the data in ascending order simplifies the identification of the median and mode.

The data given below shows the marks of 12 students in a test, arranged in ascending order: 2, 3, 3, 3.4, x, x+2, 8, p, q, 8, 8, 9. If the given value of the median and mode is 6 and 8 respectively, find the values of x, p, q.

Learn how to find the values of x, p, and q in a dataset of 12 students' marks, with a given median of 6 and a mode of 8. This solution involves calculating the median and mode using algebraic methods. Suitable for students in Grades 10-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation