Question

If the median of the series exceeds the mean by 3, then the number by which the mode exceeds its mean is:

• A. 8
• B. 9
• C. 10
• D. 11

Hint:

When solving problems involving the empirical relationship between mean, median, and mode, always start with the formula $ \text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean} $. Use the given information to substitute and simplify the expression to find the required value.

Answer 1

The Correct Option is B

Step 1: Use the Empirical Relationship.
From the empirical relationship between the mean, median, and mode: \[ \text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean} \] Step 2: Given Information.
We are told that the median exceeds the mean by 3: \[ \text{Median} = \text{Mean} + 3 \] Step 3: Substitute into the Empirical Formula.
Substitute \( \text{Median} = \text{Mean} + 3 \) into the empirical formula: \[ \text{Mode} \approx 3 \cdot (\text{Mean} + 3) - 2 \cdot \text{Mean} \] Simplify: \[ \text{Mode} \approx 3 \cdot \text{Mean} + 9 - 2 \cdot \text{Mean} \] \[ \text{Mode} \approx \text{Mean} + 9 \] Step 4: Determine How Much the Mode Exceeds the Mean.
From the equation \( \text{Mode} \approx \text{Mean} + 9 \), we see that the mode exceeds the mean by 9. Step 5: Analyze the Options.
Option (1): 8 — Incorrect, as the mode exceeds the mean by 9.
Option (2): 9 — Correct, as this matches the calculated value.
Option (3): 10 — Incorrect, as the mode exceeds the mean by 9.
Option (4): 11 — Incorrect, as the mode exceeds the mean by 9. Step 6: Final Answer. \[ (2) 9 \]