Four numbers, when arranged in ascending order, are `w , x, y and z` . The average of the smallest three numbers is `25.5` while the average of the largest three numbers was `29.5`. What is the range of the data?

Options

A. 10
B. 12
C. 13
D. 11

Answer

To solve the problem step by step, we will follow the information given about the four numbers \( w, x, y, z \) arranged in ascending order.

### Step 1: Understand the averages
We know:
- The average of the smallest three numbers \( w, x, y \) is \( 25.5 \).
- The average of the largest three numbers \( x, y, z \) is \( 29.5 \).

### Step 2: Set up the equations
From the average of the smallest three numbers:
\[ \frac{w + x + y}{3} = 25.5 \]
Multiplying both sides by 3 gives:
\[ w + x + y = 76.5 \quad \text{(Equation 1)} \]

From the average of the largest three numbers:
\[ \frac{x + y + z}{3} = 29.5 \]
Multiplying both sides by 3 gives:
\[ x + y + z = 88.5 \quad \text{(Equation 2)} \]

### Step 3: Find the value of \( z \)
We can find \( z \) by subtracting Equation 1 from Equation 2:
\[ (x + y + z) - (w + x + y) = 88.5 - 76.5 \]
This simplifies to:
\[ z - w = 12 \]
Thus, we can express \( z \) in terms of \( w \):
\[ z = w + 12 \quad \text{(Equation 3)} \]

### Step 4: Calculate the range
The range of a set of numbers is calculated as the difference between the largest and smallest numbers. In this case:
\[ \text{Range} = z - w \]
From Equation 3, we have:
\[ \text{Range} = (w + 12) - w = 12 \]

### Conclusion
The range of the data is \( 12 \).