If the mean of the observations `x,2x+1,2x+5,2x+9` is 30. What is mean of last two observations?

Answer

To solve the problem step by step, we will first find the value of \( x \) and then calculate the mean of the last two observations.

### Step 1: Write down the observations
The observations given are:
1. \( x \)
2. \( 2x + 1 \)
3. \( 2x + 5 \)
4. \( 2x + 9 \)

### Step 2: Set up the equation for the mean
The mean of these observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the mean is 30 and the number of observations is 4. Thus, we can write:
\[ 30 = \frac{x + (2x + 1) + (2x + 5) + (2x + 9)}{4} \]

### Step 3: Simplify the sum of observations
Now, we simplify the sum in the numerator:
\[ x + (2x + 1) + (2x + 5) + (2x + 9) = x + 2x + 1 + 2x + 5 + 2x + 9 \]
Combining like terms:
\[ = x + 2x + 2x + 2x + 1 + 5 + 9 = 7x + 15 \]

### Step 4: Substitute the sum back into the mean equation
Now substitute the sum back into the mean equation:
\[ 30 = \frac{7x + 15}{4} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 30 \times 4 = 7x + 15 \]
\[ 120 = 7x + 15 \]

### Step 6: Solve for \( x \)
Now, isolate \( 7x \):
\[ 7x = 120 - 15 \]
\[ 7x = 105 \]
Now, divide by 7:
\[ x = \frac{105}{7} = 15 \]

### Step 7: Find the last two observations
The last two observations are:
1. \( 2x + 5 \)
2. \( 2x + 9 \)

Substituting \( x = 15 \):
- \( 2x + 5 = 2(15) + 5 = 30 + 5 = 35 \)
- \( 2x + 9 = 2(15) + 9 = 30 + 9 = 39 \)

### Step 8: Calculate the mean of the last two observations
Now, we can find the mean of the last two observations:
\[ \text{Mean} = \frac{(2x + 5) + (2x + 9)}{2} = \frac{35 + 39}{2} \]
Calculating the sum:
\[ = \frac{74}{2} = 37 \]

### Final Answer
The mean of the last two observations is **37**.