The mean of certain number of observations is 46. If four observations whose mean is 52 are removed, the mean becomes 44.5. The original number of observations is

Options

A. 35
B. 20
C. 15
D. 12

Answer

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

### Step 1: Define the Variables
Let the original number of observations be \( n \).

### Step 2: Calculate the Total Sum of Observations
Given that the mean of the observations is 46, we can express the total sum of the observations as:
\[ \text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 46n \]

### Step 3: Calculate the Sum of the Removed Observations
We know that four observations with a mean of 52 are removed. Therefore, the total sum of these four observations is:
\[ \text{Sum of 4 Observations} = \text{Mean} \times \text{Number of Observations} = 52 \times 4 = 208 \]

### Step 4: Calculate the New Mean After Removing Observations
After removing these four observations, the new number of observations becomes \( n - 4 \), and the new mean is given as 44.5. The total sum of the remaining observations can be expressed as:
\[ \text{Total Sum of Remaining Observations} = 46n - 208 \]
The mean of the remaining observations can also be expressed as:
\[ \text{New Mean} = \frac{\text{Total Sum of Remaining Observations}}{\text{Number of Remaining Observations}} = \frac{46n - 208}{n - 4} \]
Setting this equal to the new mean:
\[ \frac{46n - 208}{n - 4} = 44.5 \]

### Step 5: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 46n - 208 = 44.5(n - 4) \]

### Step 6: Expand the Equation
Expanding the right side:
\[ 46n - 208 = 44.5n - 178 \]

### Step 7: Rearranging the Equation
Now, we will rearrange the equation to isolate \( n \):
\[ 46n - 44.5n = 208 - 178 \]
\[ 1.5n = 30 \]

### Step 8: Solve for \( n \)
Now, divide both sides by 1.5:
\[ n = \frac{30}{1.5} = 20 \]

### Conclusion
The original number of observations is \( n = 20 \).

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