Exactly 18 months ago management in a certain retail electronics store began making monthly observations of the percentage of shoppers...

1. Translate the problem requirements: Convert the timeline information into clear mathematical relationships - we need to find when observations began relative to key reference points, and understand that showrooming percentage increases linearly by \(\mathrm{0.5\%}\) each month.
2. Establish the timeline and reference points: Determine the relationship between "18 months ago when observations began," "10 months ago when \(\mathrm{p = 10.5\%}\)," and "x months after observations began."
3. Calculate the initial percentage: Use the known data point (10 months ago, \(\mathrm{p = 10.5\%}\)) and the monthly increase rate to find what the percentage was when observations began.
4. Construct the linear equation: Combine the initial value with the monthly increase rate to form the equation \(\mathrm{p = 0.5x + [initial\,value]}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is really asking in plain English. We have a store that started tracking showrooming behavior exactly 18 months ago. Every month, they noticed that \(\mathrm{0.5\%}\) more customers were showrooming compared to the previous month. This is a steady, consistent increase - like climbing stairs where each step is exactly the same height.

We're told that 10 months ago, exactly \(\mathrm{10.5\%}\) of customers were showrooming. We need to find an equation where 'x' represents the number of months after observations began, and 'p' represents the percentage of customers showrooming at that time.

Process Skill: TRANSLATE - Converting the timeline language into mathematical relationships is crucial here

The key mathematical relationship is: p increases by \(\mathrm{0.5\%}\) each month, so this is a linear equation with slope \(\mathrm{0.5}\).

2. Establish the timeline and reference points

Now let's map out the timeline clearly. Think of it like this:

• 18 months ago: Observations began (this is our starting point, \(\mathrm{x = 0}\))
• 10 months ago: We know \(\mathrm{p = 10.5\%}\)
• Today: 18 months have passed since observations began

The crucial insight is figuring out what 'x' value corresponds to "10 months ago." Since observations began 18 months ago, and we're looking at a point 10 months ago, we need to count forward from the start:
18 months ago → 10 months ago represents 8 months after observations began.

So when \(\mathrm{x = 8}\), we know that \(\mathrm{p = 10.5\%}\).

3. Calculate the initial percentage

Now we can work backwards to find what the percentage was when observations began (when \(\mathrm{x = 0}\)).

Since the percentage increases by \(\mathrm{0.5\%}\) each month, and we know that after 8 months (\(\mathrm{x = 8}\)) the percentage was \(\mathrm{10.5\%}\), we can calculate:

• At \(\mathrm{x = 8}\): \(\mathrm{p = 10.5\%}\)
• At \(\mathrm{x = 7}\): \(\mathrm{p = 10.5\% - 0.5\% = 10.0\%}\)
• At \(\mathrm{x = 6}\): \(\mathrm{p = 10.0\% - 0.5\% = 9.5\%}\)
• And so on...

Or more directly: At \(\mathrm{x = 0}\), \(\mathrm{p = 10.5\% - (8 \times 0.5\%) = 10.5\% - 4\% = 6.5\%}\)

So when observations began, \(\mathrm{6.5\%}\) of customers were showrooming.

4. Construct the linear equation

Now we have everything we need for our linear equation:

• Starting value (when \(\mathrm{x = 0}\)): \(\mathrm{p = 6.5\%}\)
• Rate of increase: \(\mathrm{0.5\%}\) per month

This gives us the equation: \(\mathrm{p = 0.5x + 6.5}\)

Let's verify this works with our known data point:

• When \(\mathrm{x = 8}\): \(\mathrm{p = 0.5(8) + 6.5 = 4 + 6.5 = 10.5\%}\) ✓

This matches what we were told about 10 months ago (\(\mathrm{x = 8}\)).

Final Answer

The equation that most accurately models the findings is \(\mathrm{p = 0.5x + 6.5}\).

This corresponds to answer choice C.

Common Faltering Points

Errors while devising the approach

1. Timeline confusion and reference point misunderstanding

Students often struggle with the multiple time references in this problem. The question mentions "18 months ago" (when observations began), "10 months ago" (when \(\mathrm{p = 10.5\%}\)), and uses "x months after observations began" as the variable. Students frequently confuse what x represents - some think \(\mathrm{x = 10}\) corresponds to "10 months ago" rather than correctly identifying that "10 months ago" means \(\mathrm{x = 8}\) (since \(\mathrm{18 - 10 = 8}\) months after observations began).

2. Misunderstanding the linear relationship structure

While students typically recognize this is a linear equation with slope \(\mathrm{0.5}\), they often get confused about which point to use as their reference. Some students try to use "10 months ago" as their starting point (\(\mathrm{x = 0}\)) instead of recognizing that \(\mathrm{x = 0}\) represents when observations began 18 months ago. This leads them to incorrectly think the y-intercept should be \(\mathrm{10.5}\).

3. Incorrectly interpreting the rate of change direction

Some students misunderstand whether the \(\mathrm{0.5\%}\) represents an increase or decrease each month, or they get confused about the direction when working backwards from the known data point to find the initial value.

Errors while executing the approach

1. Arithmetic errors in timeline calculation

Even when students understand the approach, they often make simple arithmetic mistakes when calculating that "10 months ago" corresponds to \(\mathrm{x = 8}\). Common errors include: \(\mathrm{10 + 8 = 18}\) (incorrectly adding instead of subtracting), or miscalculating \(\mathrm{18 - 10 = 8}\).

2. Wrong direction calculation when finding initial value

When working backwards to find the initial percentage (when \(\mathrm{x = 0}\)), students sometimes add \(\mathrm{0.5\%}\) per month instead of subtracting, or they calculate the wrong number of steps. For example, calculating \(\mathrm{10.5\% + (8 \times 0.5\%)}\) instead of \(\mathrm{10.5\% - (8 \times 0.5\%)}\), leading to an incorrect y-intercept of \(\mathrm{14.5\%}\).

3. Verification calculation mistakes

Students may arrive at the correct equation but make arithmetic errors when verifying their answer. For instance, when checking \(\mathrm{p = 0.5(8) + 6.5}\), they might calculate \(\mathrm{0.5 \times 8 = 3}\) instead of \(\mathrm{4}\), leading them to doubt their correct equation.

Errors while selecting the answer

No likely faltering points - once students have correctly calculated that the y-intercept is \(\mathrm{6.5}\) and the slope is \(\mathrm{0.5}\), the equation \(\mathrm{p = 0.5x + 6.5}\) directly corresponds to answer choice C without additional interpretation needed.