The average of the first and the second of three numbers is 10 more than the average of the second and the third of these numbers. What is the difference between the first and the third of these three numbers?

Answer

To solve the problem step by step, let's define the three numbers as follows:

Let:
- \( x \) = the first number
- \( y \) = the second number
- \( z \) = the third number

### Step 1: Set up the equation based on the average of the first two numbers
The average of the first and second numbers \( x \) and \( y \) can be expressed as:
\[ \text{Average of } x \text{ and } y = \frac{x + y}{2} \]

### Step 2: Set up the equation based on the average of the second and third numbers
The average of the second and third numbers \( y \) and \( z \) can be expressed as:
\[ \text{Average of } y \text{ and } z = \frac{y + z}{2} \]

### Step 3: Write the equation based on the problem statement
According to the problem, the average of the first and second numbers is 10 more than the average of the second and third numbers:
\[ \frac{x + y}{2} = \frac{y + z}{2} + 10 \]

### Step 4: Eliminate the fractions by multiplying through by 2
To eliminate the fractions, multiply the entire equation by 2:
\[ x + y = y + z + 20 \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ x + y - y = z + 20 \]
This simplifies to:
\[ x = z + 20 \]

### Step 6: Find the difference between the first and third numbers
We need to find the difference between the first number \( x \) and the third number \( z \):
\[ x - z = 20 \]

### Conclusion
Thus, the difference between the first and the third numbers is:
\[ \boxed{20} \]