Question

In a list \( A = p, 24, 24, 24, 28, 20, 16 \), is “p” positive?
(1) The mean of list A is lesser than the mode of list A.
(2) The range of list A is lesser than the mode of list A.

• A. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
• B. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
• C. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When dealing with questions involving mean and mode, always express the relationships algebraically to narrow down the possible values.

Answer 1

The Correct Option is C

Step 1: Understanding Statement 1. 
Statement 1 tells us that the mean of the list is lesser than the mode. The mode of the list is 24, which appears most frequently. The mean is given by the sum of all numbers divided by 7, including \( p \). We can calculate the mean as: \[ \frac{p + 24 + 24 + 24 + 28 + 20 + 16}{7} = \frac{p + 136}{7} \] For the mean to be less than 24, we solve: \[ \frac{p + 136}{7}<24 \quad \Rightarrow \quad p + 136<168 \quad \Rightarrow \quad p<32 \] Thus, \( p \) must be less than 32 for statement 1 to be true. 
Step 2: Understanding Statement 2. 
Statement 2 tells us that the range of the list is less than the mode. The range is the difference between the largest and smallest values in the list, which are 28 and 16, respectively. Therefore, the range is \( 28 - 16 = 12 \). Since the mode is 24, the range being less than the mode implies the range is less than 24, which is true.
Step 3: Combining Statements 1 and 2. 
From statement 1, we know \( p<32 \), and from statement 2, the range condition holds. Combining both, we deduce that \( p \) must be a positive value less than 32. 
Final Answer: \[ \boxed{C} \]