The mean of the set of 5 numbers {42, 3, 11, 27, x} is 24, and the median of the set of 4 numbers {53, 8, 29,y} is 38. If it can be determined, which of the following values is equal to x - y?

Options

A. `-38`
B. `-10`
C. `10`
D. `38`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) based on the given conditions regarding the mean and median.

### Step 1: Find the value of \( x \)

We know that the mean of the set of numbers \( \{42, 3, 11, 27, x\} \) is 24. The formula for the mean is:

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

Substituting the values we have:

\[ 24 = \frac{42 + 3 + 11 + 27 + x}{5} \]

### Step 2: Calculate the sum of the known numbers

First, calculate the sum of the known numbers:

\[ 42 + 3 + 11 + 27 = 83 \]

### Step 3: Substitute the sum back into the mean equation

Now substitute this sum back into the mean equation:

\[ 24 = \frac{83 + x}{5} \]

### Step 4: Multiply both sides by 5 to eliminate the fraction

\[ 120 = 83 + x \]

### Step 5: Solve for \( x \)

Now, isolate \( x \):

\[ x = 120 - 83 = 37 \]

### Step 6: Find the value of \( y \)

Next, we need to find \( y \) from the set \( \{53, 8, 29, y\} \) where the median is 38. Since there are four numbers, the median will be the average of the two middle numbers when arranged in ascending order.

### Step 7: Arrange the numbers in ascending order

We need to arrange \( 8, 29, 53, y \). The median is given as 38, which means:

\[ \text{Median} = \frac{\text{2nd number} + \text{3rd number}}{2} = 38 \]

### Step 8: Identify the positions of \( y \)

Assuming \( y < 29 \), the numbers in order would be \( y, 8, 29, 53 \). The median would then be:

\[ \text{Median} = \frac{8 + 29}{2} = 18.5 \quad (\text{not valid}) \]

Assuming \( 29 < y < 53 \), the order would be \( 8, 29, y, 53 \). The median would be:

\[ \text{Median} = \frac{29 + y}{2} = 38 \]

### Step 9: Solve for \( y \)

Now, solve for \( y \):

\[ 29 + y = 76 \quad (\text{since } 38 \times 2 = 76) \]
\[ y = 76 - 29 = 47 \]

### Step 10: Calculate \( x - y \)

Now that we have both \( x \) and \( y \):

\[ x - y = 37 - 47 = -10 \]

### Final Answer

Thus, the value of \( x - y \) is:

\[ \boxed{-10} \]

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