The average of six observations is 12. the average decreases by 1 when a new observation is included . What is the seventh observations ?

Options

A. 1
B. 3
C. 5
D. 6

Answer

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

### Step 1: Calculate the sum of the first six observations
Given that the average of six observations is 12, we can find the sum of these observations using the formula for average:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

So, we can rearrange this to find the sum:

\[ \text{Sum of observations} = \text{Average} \times \text{Number of observations} \]

Substituting the values:

\[ \text{Sum of observations} = 12 \times 6 = 72 \]

### Step 2: Determine the new average after adding the seventh observation
When a new observation is added, the average decreases by 1. Therefore, the new average becomes:

\[ \text{New Average} = 12 - 1 = 11 \]

### Step 3: Set up the equation for the new average
Now, we can set up an equation for the new average with the seventh observation included. The total number of observations now is 7 (6 original + 1 new):

\[ \text{New Average} = \frac{\text{Sum of 6 observations} + \text{7th observation}}{7} \]

Substituting the known values:

\[ 11 = \frac{72 + x}{7} \]

Where \(x\) is the seventh observation.

### Step 4: Solve for the seventh observation
To find \(x\), we can multiply both sides of the equation by 7:

\[ 11 \times 7 = 72 + x \]

Calculating the left side:

\[ 77 = 72 + x \]

Now, isolate \(x\):

\[ x = 77 - 72 = 5 \]

### Final Answer
The seventh observation is:

\[ \boxed{5} \]

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