Consider the sets S, T and U, where S = {31, 14, 64, 22, 43, 66}, T = {x, 31,...

Visualization

Let's create a table to track our sets and their properties:

Calculate S's Properties

First, let's find the mean of S:

• Sum = 31 + 14 + 64 + 22 + 43 + 66 = 240
• Mean of S = \(240 \div 6 = 40\)

For the median, we need to sort S:

• S sorted = {14, 22, 31, 43, 64, 66}
• With 6 elements (even count), median = \((31 + 43) \div 2 = 37\)

Phase 2: Understanding the Question

We need to find:

• Value x such that T has mean = \(40 - 5 = 35\)
• Value y such that U has median = \(37 - 4 = 33\)

Key insight: Both T and U have 7 elements (the 6 from S plus one additional element).

Phase 3: Finding the Answer

Finding x (for set T)

T has 7 elements with sum = 240 + x

• Mean of T = \((240 + x) \div 7 = 35\)
• \(240 + x = 245\)
• \(x = 5\) ✓

Finding y (for set U)

U has 7 elements. When sorted, the median is the 4th element (middle of 7).
We need the 4th element to equal 33.

Let's determine where y must be positioned:

• If y ≤ 14: sorted U = {y, 14, 22, 31, 43, 64, 66} → 4th element = 31 ✗
• If 14 < y ≤ 22: sorted U = {14, y, 22, 31, 43, 64, 66} → 4th element = 31 ✗
• If 22 < y ≤ 31: sorted U = {14, 22, y, 31, 43, 64, 66} → 4th element = 31 ✗
• If 31 < y < 43: sorted U = {14, 22, 31, y, 43, 64, 66} → 4th element = y = 33 ✓

Since we need the 4th element to be 33, and this only happens when y itself is the 4th element, \(y = 33\).

Phase 4: Solution

Our final answers are:

• X = 5
• Y = 33

Both values are in our answer choices and satisfy all the given conditions.