Consider a set of integers w, x, y, and z, where w is less than or equal to x, which is less than or equal to y, and so on. W is 9 and z is 20, and w + x + y = 36. Find the median of the set 

We are given the set of integers \( w, x, y, z \), where \( w \leq x \leq y \leq z \). The following conditions are provided:

- \( w = 9 \)
- \( z = 20 \)
- \( w + x + y = 36 \)

We need to determine the median of the set \( \{ w, x, y, z \} \).

### Step 1: Solve for \( x \) and \( y \)
We are told that \( w + x + y = 36 \), and since \( w = 9 \), we can substitute this into the equation:

\[
9 + x + y = 36
\]

Solving for \( x + y \):

\[
x + y = 36 - 9 = 27
\]

### Step 2: Analyze the arrangement
We are also given that \( w \leq x \leq y \leq z \). Thus, we know the following:

- \( w = 9 \)
- \( z = 20 \)
- \( x + y = 27 \)
- \( x \leq y \)

Since \( x + y = 27 \), the possible values of \( x \) and \( y \) that satisfy this inequality are \( x = 13 \) and \( y = 14 \) (since \( x \leq y \)).

### Step 3: Determine the set and median
The set of integers is now \( \{ w, x, y, z \} = \{ 9, 13, 14, 20 \} \). The median of a set of four numbers is the average of the two middle numbers. In this case, the two middle numbers are 13 and 14.

Thus, the median is:

\[
\text{Median} = \frac{13 + 14}{2} = \frac{27}{2} = 13.5
\]

### Final Answer:
The median of the set is \( 13.5 \).

Do you want any further details or clarifications?

### 5 Related Questions:
1. How do we calculate the median of an odd-numbered set of integers?
2. Can the median ever be a value that is not an element of the set?
3. What is the mode of the set \( \{9, 13, 14, 20\} \)?
4. How do you find the range of a set of numbers?
5. How do the properties of inequality affect the arrangement of numbers in a set?

### Tip:
For an even-numbered set, always remember that the median is the average of the two middle numbers after sorting.

Consider a set of integers w, x, y, and z, where w is less than or equal to x, which is less than or equal to y, and so on. W is 9 and z is 20, and w + x + y = 36. Find the median of the set.

Learn how to find the median of a set of integers w, x, y, z under given conditions. This problem explores key algebraic and statistical concepts, including solving for unknowns and calculating the median in a four-element set. Suitable for students in Grades 7-9.

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