Question

which point on the number line is located at the number 1?
w x y z
0 \\(\frac{5}{8}\\)
\\(\bullet\\) point w
\\(\circ\\) point x
\\(\circ\\) point y
\\(\circ\\) point z

Explanation:

Step1: Determine the interval length

First, we find the distance between 0 and $\frac{5}{8}$. Let's count the number of intervals between 0 and $\frac{5}{8}$. From the number line, between 0 and $\frac{5}{8}$, there are 4 intervals (since from 0 to W is 1, W to next is 1, next to X is 1, X to $\frac{5}{8}$ is 1? Wait, no, let's look at the marks. Wait, the number line has marks: 0, then W, then two more marks, then X, then a mark, then $\frac{5}{8}$, then a mark, then Y, then Z. Wait, maybe the interval between 0 and $\frac{5}{8}$ is divided into 4 equal parts? Wait, no, let's calculate the length of each interval. Let's see, from 0 to $\frac{5}{8}$, how many units? Wait, maybe the distance from 0 to $\frac{5}{8}$ is 4 intervals? Wait, no, let's check the positions. Wait, the key is to find the value of each interval. Let's suppose that from 0 to $\frac{5}{8}$, there are 4 intervals. Wait, no, let's count the number of spaces between 0 and $\frac{5}{8}$. From 0 to W: 1 space, W to next: 1, next to X: 1, X to $\frac{5}{8}$: 1. So total 4 spaces. So each space is $\frac{5/8}{4}$? Wait, no, that can't be. Wait, maybe the number line is marked such that each interval is $\frac{1}{8}$? Wait, no, let's think again. Wait, the point $\frac{5}{8}$ is at a certain position. Let's find the value of each interval. Let's see, from 0 to W: let's say the first interval (0 to W) is length $a$, W to next is $a$, next to X is $a$, X to $\frac{5}{8}$ is $a$. So 4 intervals make $\frac{5}{8}$, so each interval $a = \frac{5/8}{4} = \frac{5}{32}$? No, that seems complicated. Wait, maybe I made a mistake. Wait, maybe the number line is divided into eighths? Wait, no, let's look at the positions. Wait, the correct approach is: we know that 1 is greater than $\frac{5}{8}$, so we need to find which point is at 1. Let's calculate the position of each point.

First, let's find the length of each segment. Let's see, from 0 to $\frac{5}{8}$, how many segments? Let's count the number of ticks between 0 and $\frac{5}{8}$. From 0 to W: 1 tick, W to next: 1, next to X: 1, X to $\frac{5}{8}$: 1. So 4 ticks, so each tick is $\frac{5/8}{4} = \frac{5}{32}$? No, that's not right. Wait, maybe the number line is marked with each interval as $\frac{1}{8}$. Wait, no, $\frac{5}{8}$ is 5 eighths. So from 0 to $\frac{5}{8}$, there are 5 intervals? No, the number line shows 0, then W, then two more, then X, then a mark, then $\frac{5}{8}$. Wait, maybe the distance from 0 to $\frac{5}{8}$ is 4 units, so each unit is $\frac{5}{32}$, but that's messy. Wait, maybe the correct way is to find the value of each point. Let's list the positions:

- Point W: 1 interval from 0. Let's say each interval is $\frac{1}{8}$. Wait, no, $\frac{5}{8}$ is 5 eighths. So if from 0 to $\frac{5}{8}$ is 5 intervals, but the number line shows 4 intervals between 0 and $\frac{5}{8}$. Wait, I think I made a mistake. Let's look at the number line again. The marks are: 0, W, (mark), (mark), X, (mark), 5/8, (mark), Y, Z. So between 0 and 5/8, there are 5 intervals? Wait, 0 to W: 1, W to next: 1, next to X: 1, X to next: 1, next to 5/8: 1. So 5 intervals. Then each interval is $\frac{5/8}{5} = \frac{1}{8}$. Ah, that makes sense! So each interval is $\frac{1}{8}$. So:

- 0 + 1*(1/8) = 1/8: W? No, that can't be. Wait, no, 5/8 is 5 eighths, so if there are 5 intervals between 0 and 5/8, each interval is 1/8. So:

- 0: 0

- W: 1*(1/8) = 1/8? No, that's not right. Wait, maybe the number line is marked with each interval as 1/4? No, 5/8 is less than 1. Wait, maybe the key is that 1 is equal to 8/8. So we need to find which point is at 8/8. Let's see the position of 5/8. Then after 5/8, how many intervals to reach 1 (8/8). The difference between 8/8 and 5/8 is 3/8. If each interval is 1/8, then 3 intervals. So from 5/8, moving 3 intervals (each 1/8) would reach 5/8 + 3*(1/8) = 8/8 = 1. Let's check the points after 5/8: 5/8, then a mark (6/8), then Y (7/8), then Z (8/8). Wait, yes! So 5/8 + 1/8 = 6/8, 6/8 + 1/8 = 7/8 (Y), 7/8 + 1/8 = 8/8 = 1 (Z). Wait, no, wait: 5/8 is at a mark, then next mark is 6/8, then Y is at 7/8, then Z is at 8/8 (which is 1). Wait, but let's check the intervals. From 0 to 5/8: how many intervals? 0, W (1/8), next (2/8), next (3/8), X (4/8), then 5/8. Wait, that makes sense. So 0, 1/8 (W), 2/8, 3/8, 4/8 (X), 5/8, 6/8, 7/8 (Y), 8/8 (Z=1). Ah! So that's the correct breakdown. So each interval is 1/8. So:

- W: 1/8

- X: 4/8 = 1/2

- Y: 7/8

- Z: 8/8 = 1

So the point at 1 is Z. Wait, but the original selected answer was W, which is wrong. Wait, let's re-express:

0 is at 0,

First mark (W): 1/8,

Second mark: 2/8,

Third mark: 3/8,

Fourth mark (X): 4/8 = 1/2,

Fifth mark: 5/8,

Sixth mark: 6/8,

Seventh mark (Y): 7/8,

Eighth mark (Z): 8/8 = 1.

Yes, so Z is at 1. Wait, but the original answer was W, which is incorrect. So the correct point is Z.

Step2: Identify the point at 1

Since each interval is 1/8, and Z is at 8/8 = 1, so the point at 1 is Z.

Answer:

Point Z