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We are given a group of five integers \( a \leq b \leq c \leq d \leq e \), where their average is 0, and \( e = a + 30 \). We are asked to find the largest possible median of these integers.

### Step 1: Understand the problem setup

Let the integers be \( a \), \( b \), \( c \), \( d \), and \( e \), arranged in increasing order.

- The average of these integers is 0, which gives the equation:

\[
\frac{a + b + c + d + e}{5} = 0
\]

Multiplying both sides by 5:

\[
a + b + c + d + e = 0 \tag{1}
\]

- We are also told that \( e = a + 30 \). So, substituting \( e = a + 30 \) into equation (1):

\[
a + b + c + d + (a + 30) = 0
\]

Simplifying:

\[
2a + b + c + d + 30 = 0
\]

This can be rearranged as:

\[
2a + b + c + d = -30 \tag{2}
\]

### Step 2: Maximize the median

The median of the five integers is \( c \) because the integers are arranged in increasing order.

From equation (2), we see that \( b + c + d = -30 - 2a \). To maximize \( c \), we want \( b \) and \( d \) to be as small as possible.

### Step 3: Assign values to maximize \( c \)

Let's assume \( a = -15 \), which is a reasonable value since it satisfies equation (2). Substituting \( a = -15 \) into equation (2):

\[
2(-15) + b + c + d = -30
\]

This simplifies to:

\[
-30 + b + c + d = -30
\]

Therefore:

\[
b + c + d = 0 \tag{3}
\]

Now we want to maximize \( c \), so let's choose small values for \( b \) and \( d \).

### Step 4: Test values

If we let \( b = -15 \) and \( d = 15 \), then from equation (3):

\[
-15 + c + 15 = 0
\]

This simplifies to:

\[
c = 0
\]

Thus, the largest possible value of \( c \), the median, is 0.

### Step 5: Answer

The largest possible median of these integers is \( \boxed{0} \).

Would you like more details on any part of the explanation? Here are some related questions for further exploration:

1. How does changing \( a \) affect the values of \( b \), \( c \), and \( d \)?
2. What happens if we choose different values for \( a \)?
3. Why is it necessary to minimize \( b \) and \( d \) to maximize \( c \)?
4. Could the median ever be greater than 0?
5. How would the result change if \( e = a + 40 \) instead?

**Tip**: When working with equations involving sums and averages, try isolating the target variable and adjust the other terms to maximize or minimize the desired result.

A group of five integers a ≤ b ≤ c ≤ d ≤ e has an average of 0. If e = a + 30, what is the largest possible median of these five integers?

Solve for the largest possible median of five integers a ≤ b ≤ c ≤ d ≤ e where the average is 0 and e = a + 30. Explore algebraic concepts such as averages, inequalities, and properties of medians. Suitable for high school students in Grades 10-12.

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