Question

which number line shows the fraction \\(\frac{2}{5}\\)?

Explanation:

Step1: Understand the fraction

The fraction is $\frac{2}{5}$, which is between 0 and 1 (since the numerator 2 is less than the denominator 5, so $\frac{2}{5}<1$). So we can eliminate the number lines that have endpoints at 2 (first and fourth options) because $\frac{2}{5}$ is less than 1.

Step2: Analyze the remaining number lines (second and third)

For a number line from 0 to 1, to represent $\frac{2}{5}$, we need to divide the segment from 0 to 1 into 5 equal parts (since the denominator is 5) and then mark the point that is 2 parts away from 0.

Let's assume the number of divisions between 0 and 1:
- In the second option: Let's count the number of intervals between 0 and 1. If we look, the dot is at a position closer to 1, maybe 4/5 or something, not 2/5.
- In the third option: Let's count the intervals. If we divide 0 - 1 into 5 equal parts, the 2nd part (since we start from 0) would be at $\frac{2}{5}$. Let's check the number of ticks. From 0 to 1, if there are 5 intervals (so 6 ticks including 0 and 1), then the 2nd tick after 0 (excluding 0) would be at $\frac{2}{5}$. Wait, actually, when we divide 0 - 1 into n equal parts, the k-th part (starting from 0) is at $\frac{k}{n}$. So for $\frac{2}{5}$, we need 5 parts, so the 2nd mark (after 0) should be at $\frac{2}{5}$. Let's check the third number line: from 0 to 1, how many segments? Let's count the ticks. From 0, then some ticks, then the dot. If the number of segments between 0 and the dot is 2 when divided into 5 equal parts, then that's $\frac{2}{5}$. Wait, maybe I miscounted earlier. Wait, the second number line: from 0 to 1, how many intervals? Let's see, the dot is at a position that's 4/5 maybe? Wait, no, let's re - evaluate.

Wait, the key is: $\frac{2}{5}$ is less than $\frac{1}{2}$ (since $\frac{2}{5}=0.4$ and $\frac{1}{2}=0.5$). So the dot should be before the mid - point (0.5) of 0 - 1.

Looking at the second number line: the dot is after the mid - point (closer to 1), so that's more than 0.5, so not $\frac{2}{5}$.

Looking at the third number line: the dot is before the mid - point, and if we divide 0 - 1 into 5 equal parts, the 2nd part is at 0.4 ($\frac{2}{5}$). So the third number line (the one with 0 to 1, and the dot at the 2nd part when divided into 5 parts) is the correct one. Wait, but maybe the options are labeled differently. Wait, the original problem's options: let's assume the third option (the one with 0 to 1, and the dot at the 4th tick? No, wait, maybe I made a mistake. Wait, let's think again.

Wait, the fraction $\frac{2}{5}$: 0 < $\frac{2}{5}$ < 1. So we can eliminate the first and fourth (end at 2). Now between second and third (end at 1).

For the second number line: from 0 to 1, how many intervals? Let's count the number of small segments. If there are 8 segments (so 9 ticks), then the dot is at the 7th tick (since from 0, 0,1,2,3,4,5,6,7,1). Then the position is $\frac{7}{8}\approx0.875$, which is not $\frac{2}{5}$.

For the third number line: from 0 to 1, how many segments? Let's count the ticks. 0, then some ticks, then the dot, then more ticks to 1. If we have 5 segments (so 6 ticks: 0,1,2,3,4,5,1? No, maybe 0 to 1 with 5 intervals, so 6 ticks. So the ticks are at 0, $\frac{1}{5}$, $\frac{2}{5}$, $\frac{3}{5}$, $\frac{4}{5}$, 1. So the dot at $\frac{2}{5}$ would be at the 2nd tick after 0 (the first tick after 0 is $\frac{1}{5}$, the second is $\frac{2}{5}$). So the third number line (the one with 0 to 1, and the dot at the 2nd interval mark) is the correct one. Wait, maybe the options are:

First option: 0 to 2, dot near 1 (but 0 - 2, so scale is different)

Second option: 0 to 1, dot near 1 (like 4/5)

Third option: 0 to 1, dot at 2/5

Fourth option: 0 to 2, dot near 1 (scale 0 - 2)

So the correct number line is the third one (the one with 0 to 1, and the dot at the position corresponding to 2/5). Wait, but maybe the labels of the options are different. Wait, the user's image: the third option (the one with 0, then some ticks, then the dot, then more ticks to 1) is the one that shows 2/5.

Answer:

The third number line (the one with 0 and 1 as endpoints, and the dot at the position representing $\frac{2}{5}$)