If the average of x and y is 50, and the aveage of y and z is 80, what is the value of `z-x`?

Answer

To solve the problem step by step, we will use the information given about the averages of x, y, and z.

### Step 1: Set up the equations based on the averages.
Given:
1. The average of x and y is 50.
2. The average of y and z is 80.

From these statements, we can write the following equations:

\[ \frac{x + y}{2} = 50 \]

\[ \frac{y + z}{2} = 80 \]

### Step 2: Multiply both equations by 2 to eliminate the fractions.
For the first equation:

\[ x + y = 2 \times 50 = 100 \]

For the second equation:

\[ y + z = 2 \times 80 = 160 \]

### Step 3: Rearrange both equations to express y in terms of x and z.
From the first equation:

\[ y = 100 - x \quad \text{(Equation 1)} \]

From the second equation:

\[ y = 160 - z \quad \text{(Equation 2)} \]

### Step 4: Set the two expressions for y equal to each other.
Since both expressions equal y, we can set them equal:

\[ 100 - x = 160 - z \]

### Step 5: Rearrange the equation to find z - x.
Rearranging gives us:

\[ z - x = 160 - 100 \]

### Step 6: Simplify the right side.
Calculating the right side:

\[ z - x = 60 \]

### Final Answer:
Thus, the value of \( z - x \) is \( 60 \).

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