If 5 is added to each and every item of a data, then the A.M. is

Options

A. 5 times to the first A.M.
B. increased by 5 to the first A.M.
C. equal to the first A.M.
D. decreased by 5 to the first A.M.

Answer

To solve the problem, we need to determine how the arithmetic mean (A.M.) of a dataset changes when a constant value (in this case, 5) is added to each item in the dataset.

### Step-by-Step Solution:

1. **Define the Arithmetic Mean (A.M.):**
The arithmetic mean of a dataset consisting of \( n \) observations \( a_1, a_2, a_3, \ldots, a_n \) is given by the formula:
\[ \text{A.M.} = \frac{a_1 + a_2 + a_3 + \ldots + a_n}{n} \]

2. **Consider the New Dataset:**
If we add 5 to each observation, the new observations become \( a_1 + 5, a_2 + 5, a_3 + 5, \ldots, a_n + 5 \).

3. **Calculate the New A.M.:**
The new arithmetic mean (let's denote it as A.M.') is calculated as follows:
\[ \text{A.M.'} = \frac{(a_1 + 5) + (a_2 + 5) + (a_3 + 5) + \ldots + (a_n + 5)}{n} \]

4. **Simplify the New A.M.:**
We can simplify the numerator:
\[ \text{A.M.'} = \frac{(a_1 + a_2 + a_3 + \ldots + a_n) + 5n}{n} \]
This can be rewritten as:
\[ \text{A.M.'} = \frac{(a_1 + a_2 + a_3 + \ldots + a_n)}{n} + \frac{5n}{n} \]
\[ \text{A.M.'} = \text{A.M.} + 5 \]

5. **Conclusion:**
Therefore, when 5 is added to each item of the data, the new arithmetic mean is equal to the original arithmetic mean plus 5:
\[ \text{A.M.'} = \text{A.M.} + 5 \]

### Final Answer:
The new A.M. is increased by 5 compared to the original A.M.