Question

If the average of two numbers is 3y and one of the numbers is y + z, what is the other number, in terms of y and z?

• A. 5y + z
• B. 4y - z
• C. 3y + z
• D. 5y - z
• E. y + z

Hint:

A quick way to think about averages is in terms of balance. If the average is \(3y\), the sum must be \(2 \times 3y = 6y\). If you have one part (\(y+z\)), the other part must be whatever is needed to reach the total sum: \( (y+z) + \text{?} = 6y\). Solving for the unknown gives \(6y - (y+z)\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The average (or arithmetic mean) of a set of numbers is their sum divided by the count of the numbers. This problem can be solved by using the definition of an average to find the sum of the two numbers, and then subtracting the known number to find the unknown one.
Step 2: Key Formula or Approach:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} \] From this, we can derive:
\[ \text{Sum of numbers} = \text{Average} \times \text{Count of numbers} \] Step 3: Detailed Explanation:
Let the two numbers be \(N_1\) and \(N_2\).
We are given that their average is \(3y\). Since there are two numbers, the count is 2.
Using the formula for the sum:
\[ \text{Sum} = N_1 + N_2 = 3y \times 2 = 6y \] We are given that one of the numbers is \(y+z\). Let's set \(N_1 = y+z\).
We need to find the other number, \(N_2\). We can rearrange the sum equation:
\[ N_2 = \text{Sum} - N_1 \] Substitute the values we know:
\[ N_2 = 6y - (y+z) \] It is important to use parentheses to ensure we subtract the entire expression. Distribute the negative sign:
\[ N_2 = 6y - y - z \] Combine the like terms (the \(y\) terms):
\[ N_2 = 5y - z \] Step 4: Final Answer:
The other number is \(5y - z\).