Question

Suppose a, b and c are three real numbers such that Max(a, b, c) + Min(a, b, c) = 15, and Median(a, b, c) - Mean(a, b, c) = 2. Then the median of a, b and c is

• A. 11
• B. 9.5
• C. 10.5
• D. 10

Hint:

When a problem involves Min, Median, and Max, it's almost always helpful to first assume an order for the variables (e.g., \(a \le b \le c\)). This allows you to replace the functional notation (Min, Max, etc.) with the variables themselves, making algebraic manipulation much easier.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem deals with the statistical measures of a set of three numbers: minimum, maximum, median, and mean. We need to use the relationships given to find the value of the median.
Step 2: Key Formula or Approach:
1. Let the three numbers be ordered as \( x_1 \le x_2 \le x_3 \). 2. By definition: Min = \(x_1\), Median = \(x_2\), Max = \(x_3\). 3. The mean is \( \frac{x_1 + x_2 + x_3}{3} \). 4. Set up a system of equations based on the given information and solve for the median (\(x_2\)).
Step 3: Detailed Explanation:
Let's order the numbers as \( x_1 \le x_2 \le x_3 \). The given conditions can be written as: Equation 1: \( \text{Max} + \text{Min} = 15 \implies x_3 + x_1 = 15 \) Equation 2: \( \text{Median} - \text{Mean} = 2 \implies x_2 - \frac{x_1 + x_2 + x_3}{3} = 2 \) We need to find the value of the median, which is \( x_2 \). Let's work with Equation 2: \[ x_2 - \frac{x_1 + x_2 + x_3}{3} = 2 \] Multiply the entire equation by 3 to eliminate the fraction: \[ 3x_2 - (x_1 + x_2 + x_3) = 6 \] Distribute the negative sign: \[ 3x_2 - x_1 - x_2 - x_3 = 6 \] Combine the terms with \(x_2\): \[ 2x_2 - x_1 - x_3 = 6 \] Factor out the negative sign: \[ 2x_2 - (x_1 + x_3) = 6 \] Now we have a simplified equation relating the three numbers. We can use Equation 1, which states \( x_1 + x_3 = 15 \). Substitute this value into our simplified equation: \[ 2x_2 - (15) = 6 \] Now, solve for \(x_2\): \[ 2x_2 = 6 + 15 \] \[ 2x_2 = 21 \] \[ x_2 = \frac{21}{2} = 10.5 \] Step 4: Final Answer:
The median of a, b and c is 10.5.