The median of nine positive integers is x and the range is 8 more than the median. If the greatest...

1. Translate the problem requirements: We have 9 positive integers arranged in order. The middle value (5th position) is \(\mathrm{x}\), the difference between largest and smallest is \(\mathrm{(x + 8)}\), and the largest value is 15. We need to find an expression for the smallest value.
2. Apply the range definition to find the smallest integer: Since range equals largest minus smallest, and range = \(\mathrm{x + 8}\), we can set up the equation to solve for the smallest value.
3. Substitute known values and solve: Use the fact that the largest integer is 15 to find the expression for the smallest integer in terms of \(\mathrm{x}\).
4. Verify the solution makes sense: Check that our answer produces positive integers and maintains the proper ordering.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday terms:

We have 9 positive integers lined up from smallest to largest. Think of them like this:

[1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th]

The median is the middle number when we arrange all numbers in order. Since we have 9 numbers, the median is the 5th number, which equals \(\mathrm{x}\).

The range tells us how spread out our numbers are. It's simply the biggest number minus the smallest number. We're told this range is "8 more than the median," so the range = \(\mathrm{x + 8}\).

We also know the biggest number (the 9th number) is 15.

We need to find what the smallest number (the 1st number) equals.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships

2. Apply the range definition to find the smallest integer

Now let's use the definition of range in plain English:

Range = Largest number - Smallest number

We know:

• Range = \(\mathrm{x + 8}\) (given in the problem)
• Largest number = 15 (given in the problem)
• Smallest number = ? (this is what we're looking for)

So we can write:

\(\mathrm{x + 8 = 15 - \text{Smallest number}}\)

This is our key equation that will help us find the smallest number.

3. Substitute known values and solve

Let's solve for the smallest number step by step:

Starting with: \(\mathrm{x + 8 = 15 - \text{Smallest number}}\)

To isolate "Smallest number," we rearrange:

Smallest number = \(\mathrm{15 - (x + 8)}\)

Smallest number = \(\mathrm{15 - x - 8}\)

Smallest number = \(\mathrm{7 - x}\)

So the smallest integer is represented by the expression \(\mathrm{(7 - x)}\).

4. Verify the solution makes sense

Let's check our answer with a concrete example:

Suppose \(\mathrm{x = 10}\) (so the median is 10)

• Range would be \(\mathrm{x + 8 = 10 + 8 = 18}\)
• Largest number is 15 (given)
• Smallest number would be \(\mathrm{7 - x = 7 - 10 = -3}\)

Wait! The problem states we have positive integers, but our example gives -3. This means \(\mathrm{x}\) must be small enough that \(\mathrm{7 - x}\) stays positive. So \(\mathrm{x}\) must be less than 7.

Let's try \(\mathrm{x = 5}\):

• Range = \(\mathrm{5 + 8 = 13}\)
• Largest = 15, Smallest = \(\mathrm{7 - 5 = 2}\)
• Check: \(\mathrm{15 - 2 = 13}\) ✓
• All numbers are positive ✓

Our formula works! The smallest integer is \(\mathrm{7 - x}\).

Final Answer: C. \(\mathrm{7 - x}\)

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "median" means for 9 numbers

Students often confuse the median position. With 9 numbers, some students think the median is the average of the 4th and 5th numbers (like they would do with an even number of data points), rather than simply the 5th number in the ordered list.

2. Incorrectly interpreting "8 more than the median"

The phrase "the range is 8 more than the median" can be misread as "the range is 8 times the median" or "the median is 8 more than the range." Students need to carefully parse that range = median + 8, which means range = \(\mathrm{x + 8}\).

3. Overlooking the constraint that all integers must be positive

Students often ignore the word "positive" in the problem statement when setting up their approach. This constraint is crucial because it limits the possible values of \(\mathrm{x}\) and affects which answer makes mathematical sense.

Errors while executing the approach

1. Sign errors when rearranging the range equation

When solving \(\mathrm{x + 8 = 15 - \text{(smallest number)}}\) for the smallest number, students commonly make sign errors. They might write: smallest = \(\mathrm{15 - x + 8}\) instead of smallest = \(\mathrm{15 - x - 8}\), leading to \(\mathrm{23 - x}\) instead of \(\mathrm{7 - x}\).

2. Confusing which number is largest vs. smallest in the range formula

Students sometimes write Range = Smallest - Largest instead of Range = Largest - Smallest, which completely reverses their equation setup and leads to incorrect algebraic manipulation.

Errors while selecting the answer

1. Picking \(\mathrm{x - 7}\) instead of \(\mathrm{7 - x}\) due to order confusion

Even when students correctly solve for \(\mathrm{7 - x}\), they might select answer choice D \(\mathrm{(x - 7)}\) because the numbers "7" and "x" appear in their work. They don't carefully check that the order matters: \(\mathrm{7 - x \neq x - 7}\).

2. Not verifying that their chosen answer yields positive integers

Students might select an answer choice without testing whether it actually satisfies the "positive integers" constraint. For example, they could pick an expression that would make some numbers negative, violating the problem's conditions.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for the median

Let's choose \(\mathrm{x = 7}\) as our median. This is a strategic choice because:

• It's a positive integer that will keep our calculations simple
• It ensures all our integers remain positive (since the smallest will be \(\mathrm{7 - x = 0}\), but we'll verify this works)
• It's small enough to make the arithmetic manageable

Step 2: Calculate the range and smallest integer

Given: Range = \(\mathrm{x + 8 = 7 + 8 = 15}\)

Since Range = Largest - Smallest, and Largest = 15:

\(\mathrm{15 = 15 - \text{Smallest}}\)

Smallest = \(\mathrm{15 - 15 = 0}\)

But wait - we need positive integers! Let's try \(\mathrm{x = 8}\).

Step 3: Recalculate with \(\mathrm{x = 8}\)

Range = \(\mathrm{x + 8 = 8 + 8 = 16}\)

Since Range = Largest - Smallest, and Largest = 15:

\(\mathrm{16 = 15 - \text{Smallest}}\)

Smallest = \(\mathrm{15 - 16 = -1}\)

This gives a negative number, which contradicts our requirement for positive integers.

Step 4: Try \(\mathrm{x = 9}\)

Range = \(\mathrm{x + 8 = 9 + 8 = 17}\)

\(\mathrm{17 = 15 - \text{Smallest}}\)

Smallest = \(\mathrm{15 - 17 = -2}\)

Still negative. Let's work backwards from our constraint.

Step 5: Work backwards to find valid \(\mathrm{x}\)

For the smallest integer to be positive, we need:

Smallest = \(\mathrm{15 - (x + 8) > 0}\)

\(\mathrm{15 - x - 8 > 0}\)

\(\mathrm{7 - x > 0}\)

\(\mathrm{x < 7}\)

Let's try \(\mathrm{x = 6}\):

Range = \(\mathrm{6 + 8 = 14}\)

Smallest = \(\mathrm{15 - 14 = 1}\) ✓ (positive!)

Step 6: Verify our pattern

With \(\mathrm{x = 6}\): Smallest = \(\mathrm{7 - 6 = 1}\) ✓

Let's try \(\mathrm{x = 5}\): Smallest = \(\mathrm{15 - (5 + 8) = 15 - 13 = 2}\)

Using our formula: \(\mathrm{7 - 5 = 2}\) ✓

Step 7: Confirm the pattern

In every case, Smallest = \(\mathrm{7 - x}\), which matches answer choice C.