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If \(\mathrm{Q}\) is an odd number and the median of \(\mathrm{Q}\) consecutive integers is \(120\), what is the largest of these integers?
Let's break down what we're given in everyday terms:
Since we have consecutive integers and Q is odd, the median (120) sits exactly in the center of our sequence.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
Let's picture this with a concrete example first. If we had 5 consecutive integers with median 120:
118, 119, 120, 121, 122
Notice that 120 is exactly in the middle, with 2 numbers to its left and 2 numbers to its right.
For any odd number Q of consecutive integers with median 120, we get this pattern:
[some numbers] ... 120 ... [some numbers]
Since Q is odd, there's exactly one number in the middle (our median 120), which means there are equal numbers of integers on both sides.
Process Skill: VISUALIZE - Using concrete examples to understand the symmetric structure
Now let's figure out how many numbers are to the right of 120 (our median).
If we have Q consecutive integers total, and one of them is the median (120), then:
Let's verify with our example: 5 consecutive integers means Q = 5
Numbers to the right of median = \(\frac{5 - 1}{2} = \frac{4}{2} = 2\)
This matches our sequence 118, 119, 120, 121, 122 where we have 2 numbers (121, 122) to the right of 120.
To find the largest integer, we need to move from the median (120) to the rightmost position.
Since there are \(\frac{\mathrm{Q} - 1}{2}\) integers to the right of 120, the largest integer is:
\(120 + \frac{\mathrm{Q} - 1}{2}\)
We can rewrite this as:
\(\frac{\mathrm{Q} - 1}{2} + 120\)
Let's verify with our example where Q = 5:
Largest = \(\frac{5 - 1}{2} + 120 = \frac{4}{2} + 120 = 2 + 120 = 122\) ✓
The largest of the Q consecutive integers is \(\frac{\mathrm{Q} - 1}{2} + 120\)
Looking at our answer choices, this matches choice (A).
Answer: (A)
1. Misunderstanding what "Q consecutive integers" means when Q is odd
Students often struggle to visualize that Q consecutive integers with an odd Q will have exactly one middle number (the median). They might incorrectly think the median falls between two numbers, leading them to set up the problem incorrectly from the start.
2. Forgetting to use the constraint that Q is odd
The problem explicitly states Q is odd, which is crucial for determining that 120 is exactly one of the integers (not an average of two middle numbers). Students might ignore this constraint and try to solve for even Q cases, leading to incorrect setup.
3. Misinterpreting the symmetric arrangement around the median
Students may not recognize that consecutive integers are symmetrically arranged around the median. They might try complex approaches involving algebraic sequences instead of using the simple fact that equal numbers of integers exist on both sides of the median.
1. Incorrect calculation of how many numbers are to the right of the median
Students might calculate this as \(\frac{\mathrm{Q}}{2}\) instead of \(\frac{\mathrm{Q} - 1}{2}\). They forget to exclude the median itself when counting the remaining numbers, leading to the wrong offset calculation.
2. Arithmetic errors when adding the offset to the median
Even with the correct offset of \(\frac{\mathrm{Q} - 1}{2}\), students may incorrectly write the final expression. They might write \(120 - \frac{\mathrm{Q} - 1}{2}\) instead of \(120 + \frac{\mathrm{Q} - 1}{2}\), confusing the direction of movement from median to largest.
3. Confusion between finding the largest versus smallest integer
Students might correctly identify that there are \(\frac{\mathrm{Q} - 1}{2}\) positions from the median but then move in the wrong direction, calculating \(120 - \frac{\mathrm{Q} - 1}{2}\) to find what they think is the largest integer.
1. Choosing algebraically equivalent but differently written expressions
Students might arrive at the correct expression \(120 + \frac{\mathrm{Q} - 1}{2}\) but fail to recognize that this equals \(\frac{\mathrm{Q} - 1}{2} + 120\) in choice (A). They might incorrectly select choice (C) \(\frac{\mathrm{Q}}{2} + 120\) thinking it's the same thing.
2. Not verifying the answer with a concrete example
Students might select an answer without checking it against a simple case like Q=5. This verification step would catch errors like choosing (B) \(\frac{\mathrm{Q}}{2} + 119\), which would give \(\frac{5}{2} + 119 = 121.5\) for our example instead of the correct 122.
Step 1: Choose a convenient value for Q
Since Q must be odd, let's choose Q = 5 (a small odd number that makes calculations manageable).
Step 2: Set up the consecutive integers
With Q = 5 consecutive integers and median = 120, we need 5 consecutive integers where the middle one is 120.
Since there are 5 integers, the middle one is the 3rd integer.
So our integers are: 118, 119, 120, 121, 122
The largest integer is 122.
Step 3: Test this result against the answer choices
Using Q = 5 and checking each answer choice:
Step 4: Verify with another value
Let's try Q = 7 to confirm our answer.
With 7 consecutive integers and median = 120: 117, 118, 119, 120, 121, 122, 123
The largest integer is 123.
Testing choice (A): \(\frac{7 - 1}{2} + 120 = \frac{6}{2} + 120 = 3 + 120 = 123\) ✓
Conclusion: The smart numbers approach confirms that answer choice (A) is correct.