Loading...
Exactly 18 months ago management in a certain retail electronics store began making monthly observations of the percentage of shoppers who appear to be showrooming—examining a product while in the store and then buying the product online from another store. The results of the observations were that the percentage of customers who appeared to be showrooming increased by \(0.5\%\) each month. If \(\mathrm{p}\) is the percent of customers who appeared to be showrooming in their store \(\mathrm{x}\) months after the monthly observations began and 10 months ago \(\mathrm{p}\) was equal to \(10.5\%\), which of the following equations most accurately models the findings of the management team for the past 18 months?
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 \(\mathrm{0.5}\).
Now let's map out the timeline clearly. Think of it like this:
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 \(\mathrm{p = 10.5\%}\).
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:
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.
Now we have everything we need for our linear equation:
This gives us the equation: \(\mathrm{p = 0.5x + 6.5}\)
Let's verify this works with our known data point:
This matches what we were told about 10 months ago (\(\mathrm{x = 8}\)).
The equation that most accurately models the findings is \(\mathrm{p = 0.5x + 6.5}\).
This corresponds to answer choice C.
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).
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 \(\mathrm{10.5}\).
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.
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 \(\mathrm{18 - 10 = 8}\).
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 \(\mathrm{14.5\%}\).
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.
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.