Question

The median of 10, 15, x and y is 18.5, and x < y.

Column A	Column B
\(x\)	22

Answer 1

Step 1: Understanding the Concept:
This question involves the statistical concept of a median. We need to use the definition of the median for an even-sized data set to find the value of one of the variables.
Step 2: Key Formula or Approach:
The median of a set of numbers is the middle value when the numbers are arranged in order. For a set with an even number of elements (like the 4 numbers here), the median is the average of the two middle elements.
Step 3: Detailed Explanation:
The data set is {10, 15, x, y}. We are given that \(x < y\).
To find the median, we first need to order the numbers. We know \(10 < 15\). We need to figure out where \(x\) and \(y\) fit in this order.
Let's consider the possible ordered arrangements of the four numbers. The two middle numbers must average to 18.5.
- If the two middle numbers were 10 and 15, the median would be \((10+15)/2 = 12.5\), which is not 18.5. This is not valid.
- One of the two middle numbers must be 15, and the other must be \(x\).
- Using the order 10, 15, x, y (valid as \(15 < x\)), the two middle numbers are 15 and \(x\).
Solve for \(x\) using the median formula:
\[ \frac{15 + x}{2} = 18.5 \]
Multiply both sides by 2:
\[ 15 + x = 37 \]
Subtract 15 from both sides:
\[ x = 22 \]
This satisfies \(15 < 22\) and requires \(y > 22\).
Step 4: Final Answer:
Column A has the value \(x=22\). Column B has the value 22. The two quantities are equal.