Question

Assertion (A) : If the Mode and Mean of a data are 12 k and 15 k, then Median of the data is 14 k.
Reason (R) : The relation between the Mean, Mode and Median of a data is : Mean = 3 Median – 2 Mode.

• A. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
• B. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

An easy way to remember the empirical formula is the "3-2-1" order of words by length:
\[ \text{Mode (4 letters)} = 3 \text{ Median (6 letters)} - 2 \text{ Mean (4 letters)} \]
This helps prevent swapping terms!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given an empirical relationship and some data values. We need to check whether the assertion and reason are true.

Step 2: Key Formula or Approach:
The correct empirical relationship between Mean, Median, and Mode is:
\[ \text{Mode} = 3 \text{ Median} - 2 \text{ Mean} \]

Step 3: Detailed Explanation:

• Evaluate Reason (R):
The stated relation in Reason is \(\text{Mean} = 3 \text{ Median} - 2 \text{ Mode}\).
Let us check the standard empirical formula:
\[ \text{Mode} = 3 \text{ Median} - 2 \text{ Mean} \]
This is different from the formula given in Reason (R). Therefore, Reason (R) is False.

• Evaluate Assertion (A):
Let \(\text{Mode} = 12k\) and \(\text{Mean} = 15k\).
Substitute these into the correct formula:
\[ 12k = 3 \text{ Median} - 2(15k) \]
\[ 12k = 3 \text{ Median} - 30k \]
\[ 3 \text{ Median} = 12k + 30k = 42k \]
\[ \text{Median} = 14k \]
This matches the value of \(14k\) given in the Assertion. Therefore, Assertion (A) is True.


Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.