Average Calculation: x, y, and z

We are given two pieces of information about the averages:

• The average of $x$ and $y$ is 30.
• The average of $x$, $y$, and $z$ is 40.

We need to find the average of $x$, $y$, and $2z$.

Deriving Sums from Averages

From the first statement, the average of $x$ and $y$ is 30. This means:

$ \frac{x + y}{2} = 30 $

Multiplying both sides by 2 gives the sum of $x$ and $y$:

$ x + y = 2 \times 30 = 60 $

From the second statement, the average of $x$, $y$, and $z$ is 40. This means:

$ \frac{x + y + z}{3} = 40 $

Multiplying both sides by 3 gives the sum of $x$, $y$, and $z$:

$ x + y + z = 3 \times 40 = 120 $

Calculating the Value of z

Now we can find the value of $z$ by substituting the sum of $x$ and $y$ (which is 60) into the second sum equation:

$ (x + y) + z = 120 $

$ 60 + z = 120 $

Subtracting 60 from both sides gives:

$ z = 120 - 60 = 60 $

Finding the Required Average

We need to calculate the average of $x$, $y$, and $2z$. First, let's find the sum:

$ \text{Sum} = x + y + 2z $

We know $x + y = 60$ and $z = 60$. Substituting these values:

$ \text{Sum} = 60 + 2 \times (60) $

$ \text{Sum} = 60 + 120 = 180 $

The average is the sum divided by the count of numbers (which is 3):

$ \text{Average} = \frac{\text{Sum}}{3} = \frac{180}{3} = 60 $

Therefore, the average of $x$, $y$, and $2z$ is 60.