The average of 5 consecutive numbers is A If the next 3 numbers are also taken, then what will be the new average?

Options

A. A+1
B. A+1.5
C. A +2
D. A +2.5

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the average of 5 consecutive numbers
Let’s denote the 5 consecutive numbers as \( n, n+1, n+2, n+3, n+4 \). The average of these numbers can be calculated as follows:

\[ \text{Average} = \frac{n + (n+1) + (n+2) + (n+3) + (n+4)}{5} \]

This simplifies to:

\[ \text{Average} = \frac{5n + 10}{5} = n + 2 \]

Given that this average is \( A \), we can express \( n \) in terms of \( A \):

\[ n + 2 = A \implies n = A - 2 \]

### Step 2: Identify the next 3 consecutive numbers
The next 3 consecutive numbers after our initial 5 are \( n+5, n+6, n+7 \). Substituting \( n \) from the previous step, we have:

\[ n+5 = (A - 2) + 5 = A + 3 \]
\[ n+6 = (A - 2) + 6 = A + 4 \]
\[ n+7 = (A - 2) + 7 = A + 5 \]

### Step 3: Calculate the new average
Now we have a total of 8 numbers: \( n, n+1, n+2, n+3, n+4, n+5, n+6, n+7 \). The sum of these numbers is:

\[ \text{Sum} = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) + (n+6) + (n+7) \]

This can be simplified as:

\[ \text{Sum} = 5n + 10 + (n+5) + (n+6) + (n+7) = 5n + 10 + 3n + 18 = 8n + 28 \]

Substituting \( n = A - 2 \):

\[ \text{Sum} = 8(A - 2) + 28 = 8A - 16 + 28 = 8A + 12 \]

Now, the new average is:

\[ \text{New Average} = \frac{\text{Sum}}{8} = \frac{8A + 12}{8} = A + 1.5 \]

### Conclusion
Thus, the new average when the next 3 numbers are included is:

\[ \text{New Average} = A + 1.5 \]