3, 7, 9, 14, x
The numbers in the list above are ordered from least to greatest. If the average (arithmetic mean) is 2 greater than the median, what is the value of x?
The numbers in the list above are ordered from least to greatest. If the average (arithmetic mean) is 2 greater than the median, what is the value of x?
Show Hint
- 22
- 20
- 17
- 16
- 15
The Correct Option is A
Solution and Explanation
This problem connects the concepts of average (mean) and median for a small, ordered set of numbers. We need to use the given relationship between them to find the unknown value, x.
Step 2: Detailed Explanation:
1. Find the median. - The list has 5 numbers and is already ordered from least to greatest. The median is the middle value. - The middle value is the 3rd number in the list, which is 9. - So, the Median = 9.
2. Find the average (mean). - The problem states that the average is 2 greater than the median. - Average = Median + 2 = \(9 + 2 = 11\).
3. Use the average to find x. - The formula for the average is: Average = (Sum of numbers) / (Count of numbers). - We know the average is 11 and there are 5 numbers in the list. \[ 11 = \frac{3 + 7 + 9 + 14 + x}{5} \] - To solve for x, first multiply both sides by 5: \[ 11 \times 5 = 3 + 7 + 9 + 14 + x \] \[ 55 = 33 + x \] - Now, isolate x by subtracting 33 from both sides: \[ x = 55 - 33 = 22 \] 4. Check the condition. The problem states the list is ordered from least to greatest. Our resulting list is 3, 7, 9, 14, 22. Since \(22 \textgreater 14\), this condition is met.
Step 3: Final Answer:
The value of x is 22.
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