The average of n observation is 40. If two observation of 50 and 60 are added, then the average of all the observations will be 41. What is the value of n?

Options

A. 28
B. 32
C. 42
D. 40

Answer

To solve the problem step-by-step, we can follow these steps:

### Step 1: Understand the given information
We know that the average of \( n \) observations is 40. This means that the total sum of those observations can be calculated as:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Observations} = 40n \]

**Hint:** Remember that the average is calculated by dividing the total sum by the number of observations.

### Step 2: Add the new observations
We are adding two new observations, 50 and 60. The new total sum will be:
\[ \text{New Total Sum} = 40n + 50 + 60 = 40n + 110 \]

**Hint:** When adding observations, always make sure to include all the new values in the total sum.

### Step 3: Calculate the new average
After adding the two new observations, the total number of observations becomes \( n + 2 \). The new average is given as 41. We can set up the equation:
\[ \text{New Average} = \frac{\text{New Total Sum}}{\text{Total Number of Observations}} = \frac{40n + 110}{n + 2} \]
Setting this equal to 41 gives us:
\[ \frac{40n + 110}{n + 2} = 41 \]

**Hint:** To find the new average, remember to divide the new total sum by the new total number of observations.

### Step 4: Solve the equation
Now, we can cross-multiply to eliminate the fraction:
\[ 40n + 110 = 41(n + 2) \]
Expanding the right side:
\[ 40n + 110 = 41n + 82 \]

**Hint:** When expanding, distribute the number outside the parentheses to each term inside.

### Step 5: Rearrange the equation
Now, we can rearrange the equation to isolate \( n \):
\[ 40n + 110 - 82 = 41n \]
This simplifies to:
\[ 40n + 28 = 41n \]
Subtract \( 40n \) from both sides:
\[ 28 = 41n - 40n \]
This simplifies to:
\[ 28 = n \]

**Hint:** When isolating \( n \), make sure to combine like terms correctly.

### Conclusion
The value of \( n \) is 28.

**Final Answer:** \( n = 28 \)