If the mean of 11, 15, 17, y + 1 , 19 , y - 2, 3 is 14 , then the value of y is ______

Options

A. 17
B. 18
C. 9
D. 11

Answer

To find the value of \( y \) given that the mean of the numbers \( 11, 15, 17, y + 1, 19, y - 2, 3 \) is \( 14 \), we can follow these steps:

### Step 1: Write the formula for the mean
The mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers. In this case, we have 7 numbers.

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]

### Step 2: Set up the equation using the given mean
We know the mean is \( 14 \) and there are \( 7 \) observations. Therefore, we can set up the equation:

\[ 14 = \frac{11 + 15 + 17 + (y + 1) + 19 + (y - 2) + 3}{7} \]

### Step 3: Calculate the sum of the constants
First, we will calculate the sum of the constant numbers:

\[ 11 + 15 + 17 + 19 + 3 = 65 \]

Now, we include the terms involving \( y \):

\[ \text{Sum} = 65 + (y + 1) + (y - 2) = 65 + y + 1 + y - 2 = 65 + 2y - 1 = 64 + 2y \]

### Step 4: Substitute the sum back into the mean equation
Now substitute the sum back into the mean equation:

\[ 14 = \frac{64 + 2y}{7} \]

### Step 5: Multiply both sides by 7 to eliminate the fraction
To eliminate the fraction, multiply both sides by \( 7 \):

\[ 14 \times 7 = 64 + 2y \]

This simplifies to:

\[ 98 = 64 + 2y \]

### Step 6: Solve for \( 2y \)
Now, isolate \( 2y \):

\[ 2y = 98 - 64 \]

Calculating the right side gives:

\[ 2y = 34 \]

### Step 7: Solve for \( y \)
Now divide both sides by \( 2 \):

\[ y = \frac{34}{2} = 17 \]

### Final Answer
Thus, the value of \( y \) is \( 17 \).

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