Let's work through the given problem step-by-step.

Problem Analysis

We have five numbers, and we are given the following:

1. The mean (average) of the five numbers is 50.
2. Two of the numbers are given as 35 and 55.
3. The other three numbers are $x$, $x + 10$, and $2x - 10$.

We need to find:

1. The value of $x$ (Part a).
2. The median of the five numbers (Part b).

Step 1: Set Up the Mean Equation

Since the mean of the five numbers is 50, the sum of these five numbers should be: $\text{Sum of five numbers} = 5 \times 50 = 250$

The five numbers are: $35, \quad 55, \quad x, \quad x + 10, \quad \text{and} \quad 2x - 10$

Thus, the sum of these five numbers can be represented as: $35 + 55 + x + (x + 10) + (2x - 10) = 250$

Step 2: Simplify and Solve for $x$

Combine all terms on the left side: $35 + 55 + x + x + 10 + 2x - 10 = 250$ $90 + 4x = 250$

Now, isolate $x$ by solving the equation: $4x = 250 - 90$ $4x = 160$ $x = 40$

So, the value of $x$ is 40.

Step 3: Find the Median

Now that we know $x = 40$, we can substitute it back to find the values of all five numbers:

1. $x = 40$
2. $x + 10 = 40 + 10 = 50$
3. $2x - 10 = 2(40) - 10 = 80 - 10 = 70$

Thus, the five numbers are: $35, \quad 55, \quad 40, \quad 50, \quad 70$

To find the median, we need to order these numbers: $35, \quad 40, \quad 50, \quad 55, \quad 70$

The median is the middle value, which is 50.

Final Answers

(a) $x = 40$

(b) The median is 50.

Would you like further details on any of these steps?

Related Questions:

1. How would the solution change if the mean were 60 instead of 50?
2. What if the given numbers were changed to different values while keeping the mean the same?
3. How do we find the median in a set with an even number of terms?
4. What would happen if one of the expressions involving $x$ were nonlinear, such as $x^2$ or $\sqrt{x}$?
5. Can you solve for $x$ if you only know four of the numbers and the mean?

Tip:

When given a mean and a set of values with variables, use the formula for the mean (sum = mean × count) to set up an equation to solve for unknowns.