Question

The number x, y, z and w have an average equal to The average of x, y, and z is equal to Find value of “w”?

• A. 30
• B. 19
• C. 15
• D. 25

Hint:

To solve average problems with missing values quickly:
\[ \text{New Value} = (\text{New Count} \times \text{New Average}) - (\text{Old Count} \times \text{Old Average}) \] \[ w = (4 \times 25) - (3 \times 27) = 100 - 81 = 19. \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The arithmetic mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers in the set.
By rearranging this definition, the sum of a set of numbers is equal to their average multiplied by the number of values in the set:
\[ \text{Sum} = \text{Average} \times \text{Count} \] This relationship allows us to find missing values when averages of subsets are provided.

Step 2: Detailed Explanation:
Let us break down the problem into two parts:
1. Calculate the sum of all four numbers:
- The four numbers are $x, y, z,$ and $w$.
- Their average is given as $25$.
- The sum of these four numbers is: \[ x + y + z + w = 4 \times 25 = 100 \] 2. Calculate the sum of the first three numbers:
- The three numbers are $x, y,$ and $z$.
- Their average is given as $27$.
- The sum of these three numbers is: \[ x + y + z = 3 \times 27 = 81 \] 3. Find the value of $w$:
- We can find the value of $w$ by subtracting the sum of the first three numbers from the sum of all four numbers: \[ w = (x + y + z + w) - (x + y + z) \] \[ w = 100 - 81 \] \[ w = 19 \] Therefore, the value of $w$ is $19$.

Step 3: Final Answer:
The value of $w$ is 19.
Therefore, the correct option is (B).