If average of 38 consecutive even numbers is 167, then find the sum of smallest and largest number ?

Options

A. 167
B. 334
C. 332
D. 336

Answer

To solve the problem, we need to find the sum of the smallest and largest numbers among 38 consecutive even numbers, given that their average is 167.

### Step-by-Step Solution:

1. **Understanding the Average**:
The average of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers. Here, we have 38 consecutive even numbers with an average of 167.

2. **Calculating the Total Sum**:
To find the total sum of these 38 numbers, we can use the formula:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Observations} \]
Substituting the given values:
\[ \text{Total Sum} = 167 \times 38 \]

3. **Performing the Multiplication**:
Now, we calculate \(167 \times 38\):
\[ 167 \times 38 = 6346 \]

4. **Finding the Smallest and Largest Numbers**:
Since we are dealing with consecutive even numbers, we can denote the smallest number as \(x\). The numbers can be represented as:
\[ x, x+2, x+4, \ldots, x+74 \]
Here, \(x + 74\) is the largest number in this series (since \(38 - 1 = 37\) and \(37 \times 2 = 74\)).

5. **Sum of Smallest and Largest**:
The smallest number is \(x\) and the largest number is \(x + 74\). Therefore, the sum of the smallest and largest number is:
\[ \text{Sum} = x + (x + 74) = 2x + 74 \]

6. **Using the Total Sum**:
We know that the total sum of the numbers is also equal to \(6346\). We can set up the equation:
\[ 2x + 74 = 167 \times 2 \]
Since the average of the smallest and largest number is equal to the average of all numbers, we can calculate:
\[ 167 \times 2 = 334 \]

7. **Final Calculation**:
Therefore, we have:
\[ 2x + 74 = 334 \]
To find \(2x\):
\[ 2x = 334 - 74 = 260 \]
Thus, the sum of the smallest and largest number is:
\[ \text{Sum} = 334 \]

### Conclusion:
The sum of the smallest and largest numbers among the 38 consecutive even numbers is **334**.