The median of x, y, z, p, q, and r `(x gty gt zgt p gt qgt r) ` is m, then find the median of 2q, 2p, 2z, and 2y.

Options

A. `m/2`
B. m
C. `2m`
D. 2

Answer

To find the median of the numbers \(2q, 2p, 2z,\) and \(2y\), we can follow these steps:

### Step 1: Understand the given data
We are given that:
- \(x > y > z > p > q > r\)
- The median of \(x, y, z, p, q, r\) is \(m\).

### Step 2: Determine the median of \(x, y, z, p, q, r\)
Since there are 6 numbers (an even count), the median is calculated as the average of the 3rd and 4th terms when arranged in order.

From the order:
- 3rd term: \(z\)
- 4th term: \(p\)

Thus, the median \(m\) can be expressed as:
\[ m = \frac{z + p}{2} \]

### Step 3: Relate \(z + p\) to \(m\)
From the equation above, we can rearrange it to find:
\[ z + p = 2m \tag{1} \]

### Step 4: List the new numbers
We need to find the median of the numbers \(2q, 2p, 2z,\) and \(2y\).

### Step 5: Arrange \(2q, 2p, 2z,\) and \(2y\) in order
Since \(y > z > p > q\), multiplying by 2 maintains the order:
- \(2y > 2z > 2p > 2q\)

### Step 6: Find the median of \(2q, 2p, 2z,\) and \(2y\)
Now, we have 4 numbers. The median is calculated as the average of the 2nd and 3rd terms:
- 2nd term: \(2z\)
- 3rd term: \(2p\)

Thus, the median is:
\[ \text{Median} = \frac{2z + 2p}{2} = z + p \]

### Step 7: Substitute \(z + p\) using equation (1)
From equation (1), we know:
\[ z + p = 2m \]

### Final Result
Therefore, the median of \(2q, 2p, 2z,\) and \(2y\) is:
\[ \text{Median} = 2m \]