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}{}\\) mile

Explanation:

Step1: Determine the interval length

The number line is from 5 to 6, and there are 8 intervals (since between 5 and 6, the ticks divide it into 8 parts). So each interval is $\frac{1}{8}$ mile.

Step2: Find the closest and farthest distances

The closest distance is at 5 (which is $5 = \frac{40}{8}$) and the farthest is at the 7th interval after 5? Wait, no. Wait, the leftmost x is at 5 (which is 5.0 or $\frac{40}{8}$), and the rightmost x is at 5 + $\frac{7}{8}$? Wait, no, let's count the intervals. From 5 to 6, there are 8 marks (since between 5 and 6, the number of segments is 8). So each segment is $\frac{1}{8}$ mile. The closest distance is 5 (which is $\frac{40}{8}$) and the farthest is $5 + \frac{7}{8}$? Wait, no, the rightmost tick is 6, which is $\frac{48}{8}$. Wait, maybe I made a mistake. Wait, the line plot: the first tick is 5, then each subsequent tick is $\frac{1}{8}$ mile more. So the positions are 5, $5+\frac{1}{8}$, $5+\frac{2}{8}$, ..., 6 (which is $5+\frac{8}{8}$). Now, the leftmost x is at 5 (distance 5 miles), and the rightmost x is at $5+\frac{7}{8}$? Wait, no, the rightmost cluster is at the 7th interval? Wait, no, the rightmost tick before 6 is $5+\frac{7}{8}$, and 6 is the last tick. Wait, maybe the closest is 5 (which is $\frac{40}{8}$) and the farthest is $5+\frac{7}{8}$? Wait, no, let's check the options. Wait, the difference between the closest and farthest. Wait, maybe the closest is 5 (which is 5.0) and the farthest is $5+\frac{7}{8}$? No, wait, maybe the closest is $5$ (5.0) and the farthest is $5+\frac{7}{8}$? Wait, no, let's calculate the difference. Wait, 6 - 5 is 1, but divided into 8 parts, so each part is $\frac{1}{8}$. Wait, maybe the closest is 5 (5.0) and the farthest is $5+\frac{7}{8}$? No, wait, the leftmost x is at 5 (so distance 5), and the rightmost x is at $5+\frac{7}{8}$? Wait, no, 6 is 5 + 1, which is 5 + $\frac{8}{8}$. Wait, maybe the closest is 5 (5.0) and the farthest is $5+\frac{7}{8}$? Then the difference is $\frac{7}{8}$? But option D is $\frac{7}{8}$? Wait, but let's re-examine. Wait, the line plot: from 5 to 6, divided into 8 equal parts, so each part is $\frac{1}{8}$ mile. The leftmost x is at 5 (distance 5), and the rightmost x is at the 7th part after 5, which is $5+\frac{7}{8}$. Then the difference is $(5+\frac{7}{8}) - 5 = \frac{7}{8}$? But $\frac{7}{8}$ is an option? Wait, no, option D is $\frac{7}{8}$? Wait, the options are A: $\frac{1}{2}$, B: $\frac{1}{8}$, C: $\frac{6}{8}$, D: $\frac{7}{8}$. Wait, maybe I messed up the closest and farthest. Wait, maybe the closest is $5+\frac{0}{8}$ (5) and the farthest is $5+\frac{7}{8}$, so difference is $\frac{7}{8}$. But let's check again. Wait, the number of intervals between 5 and 6: since 5 to 6 is 1 mile, divided into 8 intervals, so each interval is $\frac{1}{8}$ mile. So the positions are 5, $5+\frac{1}{8}$, $5+\frac{2}{8}$, ..., $5+\frac{8}{8}=6$. Now, the leftmost x is at 5 (so distance 5), and the rightmost x is at $5+\frac{7}{8}$ (since the rightmost cluster is at the 7th interval before 6). Then the difference is $(5+\frac{7}{8}) - 5 = \frac{7}{8}$ mile. So the answer is D? Wait, but let's confirm. Wait, maybe the closest is $5$ (5.0) and the farthest is $5+\frac{7}{8}$, so difference is $\frac{7}{8}$. So the correct option is D. $\frac{7}{8}$ mile.

Answer:

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