Question

Ramesh analysed the monthly salary figures of five vice presidents of his company. All the salary figures are integers. The mean and the median salary figures are 5 lakh, and the only mode is 8 lakh. Which of the options below is the sum (in lakh) of the highest and the lowest salaries?

• A. 9
• B. 10
• C. 11
• D. 12
• E. None of the above

Hint:

- Always apply conditions in order: mean \(\Rightarrow\) sum, median \(\Rightarrow\) middle value, and mode \(\Rightarrow\) frequency.
- Check integer constraints carefully to avoid invalid salary values.

Answer 1

The Correct Option is A

Step 1: Represent the salaries.
Let the five salaries (in lakhs) in ascending order be: \(a \leq b \leq c \leq d \leq e\).
Step 2: Use the mean condition.
The mean salary is given as 5 lakh. So, the total of all salaries is:
\[ a + b + c + d + e = 5 \times 5 = 25 \]
Step 3: Use the median condition.
Since there are 5 numbers, the median is the third one, i.e., \(c = 5\).
Step 4: Use the mode condition.
The only mode is 8 lakh. Hence, 8 must appear at least twice and more frequently than any other number. Thus, \(d = 8\) and \(e = 8\).
Step 5: Substitute values.
So far, we have \(a + b + 5 + 8 + 8 = 25 \Rightarrow a + b = 4\).
Step 6: Find the lowest and highest salaries.
The lowest salary is \(a\), the highest is \(e = 8\). Their sum is:
\[ a + e = a + 8 \]
Since \(a + b = 4\) and both are integers with \(a \leq b\), possible pairs are \((0,4), (1,3), (2,2)\). The only feasible non-negative salary pair with median \(5\) is \((1,3)\). Thus, \(a = 1\).
So, \(a + e = 1 + 8 = 9\).
\[ \boxed{9} \]