The mean of 6,10,11,x,12,y is 10. Also y is 7 more than x. Find the value of x and y.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given the conditions about their mean and their relationship. Let's break it down step by step.

### Step 1: Set up the equation for the mean
The mean of the numbers \( 6, 10, 11, x, 12, y \) is given as 10. The formula for the mean is:

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

In this case, the number of observations is 6. Therefore, we can write:

\[ 10 = \frac{6 + 10 + 11 + x + 12 + y}{6} \]

### Step 2: Calculate the sum of the known numbers
Now, let's calculate the sum of the known numbers:

\[ 6 + 10 + 11 + 12 = 39 \]

### Step 3: Substitute the sum into the mean equation
Now we can substitute this sum back into the mean equation:

\[ 10 = \frac{39 + x + y}{6} \]

### Step 4: Multiply both sides by 6
To eliminate the fraction, multiply both sides by 6:

\[ 60 = 39 + x + y \]

### Step 5: Rearrange the equation
Now, rearranging gives us:

\[ x + y = 60 - 39 \]
\[ x + y = 21 \]

### Step 6: Use the relationship between \( x \) and \( y \)
We are also given that \( y \) is 7 more than \( x \):

\[ y = x + 7 \]

### Step 7: Substitute \( y \) in the equation
Now, substitute \( y \) in the equation \( x + y = 21 \):

\[ x + (x + 7) = 21 \]

### Step 8: Simplify the equation
This simplifies to:

\[ 2x + 7 = 21 \]

### Step 9: Solve for \( x \)
Now, subtract 7 from both sides:

\[ 2x = 21 - 7 \]
\[ 2x = 14 \]

Now, divide by 2:

\[ x = \frac{14}{2} = 7 \]

### Step 10: Find \( y \)
Now that we have \( x \), we can find \( y \):

\[ y = x + 7 = 7 + 7 = 14 \]

### Final Answer
Thus, the values are:

\[ x = 7 \quad \text{and} \quad y = 14 \]