Question

the line plot above shows the distance students live (in miles) from the library. what is the difference in miles between the students who live closest and the farthest away?
a $\frac{1}{2}$ mile
b $\frac{1}{8}$ mile
c $\frac{6}{8}$ mile
d $\frac{7}{8}$ mile

Explanation:

Step1: Determine the scale of the line plot

The line plot ranges from 5 to 6, and the number of intervals between 5 and 6: Let's count the ticks. From 5 to 6, there are 8 equal intervals (since the distance between 5 and 6 is 1 mile, so each interval is $\frac{1}{8}$ mile).

Step2: Find the closest and farthest distances

• Closest: The leftmost 'x' is at 5 (since the first tick is 5, and the 'x's start at 5). Wait, no, looking at the plot: the first 'x's are at 5 (the first tick), then the next ticks are each $\frac{1}{8}$ mile apart. Wait, actually, the distance from 5: each small tick is $\frac{1}{8}$ mile. So the closest distance (minimum) is 5 miles? Wait, no, the leftmost 'x's are at 5 (the first tick), and the rightmost 'x's: let's see, the last tick is 6, and the rightmost 'x's are at 6 - $\frac{1}{8}$? Wait, no, let's re - examine. The line is from 5 to 6, divided into 8 equal parts (since 5 to 6 is 1 mile, 1 divided by 8 is $\frac{1}{8}$ per interval). So the position of a point at the $n$th tick from 5 is $5 + n\times\frac{1}{8}$.

The closest distance (minimum) is at 5 (the first tick, $n = 0$: $5+0\times\frac{1}{8}=5$). The farthest distance (maximum): let's see the rightmost 'x's. The last tick is 6, and the rightmost 'x's are at 6 - $\frac{1}{8}$? Wait, no, the number of intervals from 5 to the rightmost 'x': Let's count the ticks from 5 to the rightmost 'x'. From 5 to the rightmost 'x', how many intervals? Let's see, the rightmost 'x's are at 5 + 7\times\frac{1}{8}$? Wait, no, 5 to 6 is 8 intervals. So the rightmost 'x' is at 5 + 7\times\frac{1}{8}=5+\frac{7}{8}=\frac{40 + 7}{8}=\frac{47}{8}$, and the leftmost is at 5=\frac{40}{8}.

Step3: Calculate the difference

Difference = Farthest - Closest=\frac{47}{8}-\frac{40}{8}=\frac{7}{8}? Wait, no, wait. Wait, maybe I made a mistake. Wait, the leftmost 'x's: are they at 5 or at 5 + 0\times\frac{1}{8}? Wait, the first tick is 5, then the next tick is 5+\frac{1}{8}, then 5+\frac{2}{8},..., up to 6 (which is 5+\frac{8}{8}).

The leftmost 'x's: looking at the plot, the first 'x's are at 5 (the first tick, 5+\frac{0}{8}), and the rightmost 'x's: let's see, the rightmost 'x's are at 5+\frac{7}{8}? Wait, no, the number of 'x's on the right: the rightmost cluster of 'x's. Let's count the number of intervals from 5 to the rightmost 'x'. If we consider that the maximum value is 5+\frac{7}{8}? Wait, no, 5 to 6 is 8 intervals. So the rightmost point (farthest) is 5+\frac{7}{8}? Wait, no, 5+\frac{7}{8}=\frac{40 + 7}{8}=\frac{47}{8}=5.875, and the minimum is 5=\frac{40}{8}. Then the difference is $\frac{47}{8}-\frac{40}{8}=\frac{7}{8}$ mile.

Wait, let's check again. The distance from 5: each interval is $\frac{1}{8}$ mile. The closest distance (minimum) is 5 (at the first tick), and the farthest distance (maximum) is 5+\frac{7}{8} (since the rightmost 'x's are 7 intervals away from 5). So the difference is $(5+\frac{7}{8})-5=\frac{7}{8}$ mile.

Answer:

D. $\frac{7}{8}$ mile