The real number R is expressed as the sum of three numbers so that the median of the three numbers...

Phase 1: Owning the Dataset

Visualization

Let's arrange the three numbers on a number line, calling them a, b, and c in ascending order:

a -----(+x)-----> b -----(+y)-----> c
(least)        (median)        (greatest)

This visual shows:

• a is the least of the three numbers
• b is the median (middle value)
• c is the greatest
• The median is x greater than the least: \(\mathrm{b = a + x}\)
• The median is y less than the greatest: \(\mathrm{b = c - y}\)

Key Relationships

From our visualization:

1. \(\mathrm{b = a + x}\) (median is x greater than least)
2. \(\mathrm{b = c - y}\) (median is y less than greatest)
3. \(\mathrm{a + b + c = R}\) (the three numbers sum to R)

Phase 2: Understanding the Question

We need to find algebraic expressions for:

• The least of the three numbers (a)
• The greatest of the three numbers (c)

Both expressions should be in terms of R, x, and y.

Phase 3: Finding the Answer

1. Express c in terms of b
   From \(\mathrm{b = c - y}\), we get:
   \(\mathrm{c = b + y}\)
2. Express c in terms of a
   Since \(\mathrm{b = a + x}\), we can substitute:
   \(\mathrm{c = (a + x) + y = a + x + y}\)
3. Use the sum condition
   Substituting our expressions into \(\mathrm{a + b + c = R}\):
   \(\mathrm{a + (a + x) + (a + x + y) = R}\)
   \(\mathrm{a + a + x + a + x + y = R}\)
   \(\mathrm{3a + 2x + y = R}\)
   
   Solving for a:
   \(\mathrm{3a = R - 2x - y}\)
   \(\mathrm{a = \frac{R - 2x - y}{3}}\)
4. Find c
   Using \(\mathrm{c = a + x + y}\):
   \(\mathrm{c = \frac{R - 2x - y}{3} + x + y}\)
   
   To simplify, we need a common denominator:
   \(\mathrm{c = \frac{R - 2x - y}{3} + \frac{3x}{3} + \frac{3y}{3}}\)
   \(\mathrm{c = \frac{R - 2x - y + 3x + 3y}{3}}\)
   \(\mathrm{c = \frac{R + x + 2y}{3}}\)

Verification

Let's verify our answers sum to R:

• \(\mathrm{a = \frac{R - 2x - y}{3}}\)
• \(\mathrm{b = a + x = \frac{R - 2x - y}{3} + x = \frac{R + x - y}{3}}\)
• \(\mathrm{c = \frac{R + x + 2y}{3}}\)

Sum: \(\mathrm{\frac{R - 2x - y + R + x - y + R + x + 2y}{3} = \frac{3R}{3} = R}\) ✓

Phase 4: Solution

Final Answer:

• Least: \(\mathrm{\frac{R - 2x - y}{3}}\)
• Greatest: \(\mathrm{\frac{R + x + 2y}{3}}\)

These expressions correctly represent the least and greatest of the three numbers that sum to R with the given median relationships.