Question

If the mean of the numbers 3, (3p+3), 8, 14, 18, 5 and (p-2) is 9, then find their median.

• A. 6
• B. 8
• C. 9
• D. 14

Hint:

When finding the median, always remember to arrange the numbers in ascending or descending order first.
Skipping this step is a very common source of error where students select the middle term of the unsorted list.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a list of seven numbers, two of which are expressed in terms of an unknown variable \(p\).
The arithmetic mean of these seven numbers is given as 9.
We need to first find the value of \(p\) using the mean formula, then substitute it back to obtain the numerical values of all terms, and finally determine their median.

Step 2: Key Formula or Approach:
The arithmetic mean of \(N\) observations is given by:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{N} \]
The median of an ordered set of \(N\) observations (where \(N\) is odd) is:
\[ \text{Median} = \left( \frac{N + 1}{2} \right)^{\text{th}} \text{ observation} \]

Step 3: Detailed Explanation:
$\bullet$

Step 1: Set up the equation for Mean:
The seven numbers are: 3, \((3p+3)\), 8, 14, 18, 5, and \((p-2)\).
Number of terms (\(N\)) = 7.
Mean = 9.
Sum of the numbers:
\[ \text{Sum} = 3 + (3p + 3) + 8 + 14 + 18 + 5 + (p - 2) \]
Group the constant terms and the variable terms:
\[ \text{Sum} = (3p + p) + (3 + 3 + 8 + 14 + 18 + 5 - 2) \]
\[ \text{Sum} = 4p + 49 \]
Using the mean formula:
\[ 9 = \frac{4p + 49}{7} \]
\[ 4p + 49 = 63 \]
\[ 4p = 63 - 49 \]
\[ 4p = 14 \implies p = 3.5 \]
$\bullet$

Step 2: Substitute the value of p into the terms:
Now substitute \(p = 3.5\) to get all seven numbers:
First term = 3
Second term = \(3p + 3 = 3(3.5) + 3 = 10.5 + 3 = 13.5\)
Third term = 8
Fourth term = 14
Fifth term = 18
Sixth term = 5
Seventh term = \(p - 2 = 3.5 - 2 = 1.5\)
The list of numbers is: 3, 13.5, 8, 14, 18, 5, 1.5.
$\bullet$

Step 3: Sort the terms in ascending order:
Arranging the numbers from smallest to largest:
1.5, 3, 5, 8, 13.5, 14, 18.
$\bullet$

Step 4: Find the Median:
Since \(N = 7\) (which is odd), the median is the \(\left(\frac{7+1}{2}\right)^{\text{th}} = 4^{\text{th}}\) term.
Looking at our sorted list, the \(4^{\text{th}}\) term is 8.
Thus, the median is 8.

Step 4: Final Answer:
The median of the given set of numbers is 8.