Four numbers when arranged in ascending order are w, x,y and z. The average of the smallest three numbers is 22, whereas the average of the largest three numbers is 28. What is the range of the data?
(a)16
(b)17
(c)18
(d)19

Options

A. 16
B. 17
C. 18
D. 19

Answer

To solve the problem step by step, we will first define the variables and then use the information given to find the range of the numbers.

### Step 1: Define the Variables
Let the four numbers be arranged in ascending order as follows:
- \( w \) (smallest)
- \( x \)
- \( y \)
- \( z \) (largest)

### Step 2: Calculate the Sum of the Smallest Three Numbers
We know that the average of the smallest three numbers \( w, x, y \) is 22. The formula for average is given by:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Thus, we can express the sum of the smallest three numbers:

\[ \frac{w + x + y}{3} = 22 \]

Multiplying both sides by 3 gives:

\[ w + x + y = 66 \quad \text{(Equation 1)} \]

### Step 3: Calculate the Sum of the Largest Three Numbers
Next, we know that the average of the largest three numbers \( x, y, z \) is 28. Using the same average formula, we have:

\[ \frac{x + y + z}{3} = 28 \]

Multiplying both sides by 3 gives:

\[ x + y + z = 84 \quad \text{(Equation 2)} \]

### Step 4: Find the Value of \( z \)
Now, we can find \( z \) by subtracting Equation 1 from Equation 2:

\[ (x + y + z) - (w + x + y) = 84 - 66 \]

This simplifies to:

\[ z - w = 18 \]

### Step 5: Calculate the Range
The range of a set of numbers is defined as the difference between the largest and smallest numbers. Therefore, the range can be calculated as:

\[ \text{Range} = z - w \]

From the previous step, we found that:

\[ z - w = 18 \]

Thus, the range of the data is:

\[ \text{Range} = 18 \]

### Conclusion
The range of the data is **18**, which corresponds to option (c).

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