If the average of seven consecutive even numbers is 62, then the one-fourth of twice of total of first and sixth number is

Options

A. 60
B. 62
C. None of these
D. 61

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

**Step 1: Understand the Problem**
We are given that the average of seven consecutive even numbers is 62. We need to find one-fourth of twice the total of the first and sixth numbers.

**Step 2: Set Up the Consecutive Even Numbers**
Let the seven consecutive even numbers be:
- First number: \( x - 6 \)
- Second number: \( x - 4 \)
- Third number: \( x - 2 \)
- Fourth number: \( x \)
- Fifth number: \( x + 2 \)
- Sixth number: \( x + 4 \)
- Seventh number: \( x + 6 \)

Here, \( x \) is the middle number.

**Step 3: Calculate the Average**
The average of these seven numbers is given by:
\[ \text{Average} = \frac{\text{Sum of all numbers}}{7} \]
The sum of the numbers is:
\[ (x - 6) + (x - 4) + (x - 2) + x + (x + 2) + (x + 4) + (x + 6) = 7x \]
Thus, the average is:
\[ \frac{7x}{7} = x \]
Since the average is given as 62, we have:
\[ x = 62 \]

**Step 4: Find the First and Sixth Numbers**
Now we can find the first and sixth numbers:
- First number: \( x - 6 = 62 - 6 = 56 \)
- Sixth number: \( x + 4 = 62 + 4 = 66 \)

**Step 5: Calculate the Total of First and Sixth Numbers**
Now we calculate the total of the first and sixth numbers:
\[ \text{Total} = 56 + 66 = 122 \]

**Step 6: Calculate One-Fourth of Twice the Total**
Now we need to find one-fourth of twice this total:
\[ \text{Twice the Total} = 2 \times 122 = 244 \]
\[ \text{One-fourth of Twice the Total} = \frac{244}{4} = 61 \]

**Final Answer:**
Thus, the answer is \( 61 \).

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