The average of x, y and z is 45. x is as much more than the average as y is less than the average. Find the value of z.

Options

A. (a) 45
B. (b) 25
C. (c) 35
D. (d) 15

Answer

To solve the problem step by step, we will follow the information given in the question and derive the value of \( z \).

### Step 1: Understand the average
The average of \( x, y, z \) is given as 45. The formula for the average of three numbers is:
\[ \text{Average} = \frac{x + y + z}{3} \]
Given that the average is 45, we can set up the equation:
\[ \frac{x + y + z}{3} = 45 \]

### Step 2: Multiply to eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 3:
\[ x + y + z = 135 \]

### Step 3: Define the difference from the average
We know that \( x \) is as much more than the average (45) as \( y \) is less than the average. Let’s denote the difference from the average as \( a \):
\[ x = 45 + a \quad \text{(since \( x \) is more than the average)} \]
\[ y = 45 - a \quad \text{(since \( y \) is less than the average)} \]

### Step 4: Substitute \( x \) and \( y \) in the equation
Now, substitute the expressions for \( x \) and \( y \) into the equation \( x + y + z = 135 \):
\[ (45 + a) + (45 - a) + z = 135 \]

### Step 5: Simplify the equation
When we simplify the equation, the \( a \) terms cancel out:
\[ 45 + a + 45 - a + z = 135 \]
\[ 90 + z = 135 \]

### Step 6: Solve for \( z \)
Now, isolate \( z \) by subtracting 90 from both sides:
\[ z = 135 - 90 \]
\[ z = 45 \]

### Conclusion
The value of \( z \) is \( 45 \).

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