The mean of a certain number of observations is 35. What is the new value of the mean if each observation is :
increased by 7.

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Understand the given information
We know that the mean of a certain number of observations is 35.

### Step 2: Define variables
Let:
- \( S \) = sum of the observations
- \( N \) = number of observations

### Step 3: Write the formula for mean
The mean \( M \) is given by the formula:
\[ M = \frac{S}{N} \]
Given that the mean \( M = 35 \), we can write:
\[ 35 = \frac{S}{N} \]

### Step 4: Express the sum of observations in terms of \( N \)
From the equation above, we can rearrange it to find the sum of observations \( S \):
\[ S = 35N \]

### Step 5: Determine the new sum after increasing each observation
If each observation is increased by 7, then the new sum \( S' \) can be calculated as follows:
- Each observation \( x_i \) becomes \( x_i + 7 \)
- Therefore, the new sum \( S' \) is:
\[ S' = (x_1 + 7) + (x_2 + 7) + (x_3 + 7) + ... + (x_N + 7) \]
This can be simplified to:
\[ S' = (x_1 + x_2 + x_3 + ... + x_N) + 7N \]
Substituting \( S \) for the sum of the original observations:
\[ S' = S + 7N \]
Now substituting \( S = 35N \):
\[ S' = 35N + 7N = 42N \]

### Step 6: Calculate the new mean
The new mean \( M' \) is given by:
\[ M' = \frac{S'}{N} \]
Substituting \( S' = 42N \):
\[ M' = \frac{42N}{N} = 42 \]

### Conclusion
The new mean after increasing each observation by 7 is:
\[ \boxed{42} \]