Question

where is the blue dot on the number line? -6 -6.5 -7

Explanation:

Step1: Determine the interval length

Between -6.5 and -7, the distance is $|-6.5 - (-7)| = 0.5$. Let's count the number of divisions between -6.5 and -7. From -6.5 to -7, there are 5 small marks (since from -6.5 (1st mark after -6.5? Wait, looking at the number line: -6, then some marks, -6.5 (a mark), then marks down to -7. Wait, let's re-examine: The marks: -6 is at the top, then each mark is a division. Let's see the positions: -6, then each step down is decreasing. Let's find the value of each small division. From -6.5 to -7, the total length is 0.5, and how many intervals? Let's count the number of spaces between -6.5 and -7. Let's see: -6.5 is a mark, then the next marks: let's say from -6.5 to -7, there are 5 intervals (since from -6.5, then 1,2,3,4,5 marks to -7). So each interval is $0.5 / 5 = 0.1$. Wait, no, maybe the number of divisions: Let's check the position of the blue dot. Let's start from -6.5. The blue dot is 3 marks below -6.5? Wait, no, let's list the marks:

-6 (top), then each mark is, let's see, from -6 to -6.5: how many marks? Let's see, -6, then maybe 5 marks to -6.5? Wait, the problem's number line: -6 is at the top, then a mark, then -6.5 (a thicker mark), then marks down to -7 (thicker mark). Let's see the blue dot is 3 marks below -6.5? Wait, no, let's calculate the value.

Wait, let's find the value of each tick. Let's see between -6.5 and -7: the distance is 0.5, and if there are 5 ticks (since from -6.5 to -7, how many intervals? Let's count the number of small lines between -6.5 and -7. Let's say from -6.5 (line 1), then line 2, line 3, line 4, line 5 is -7. So that's 4 intervals? No, maybe the number of steps: Let's look at the blue dot's position. Let's see, -6.5 is at a line, then the next lines: each line is 0.1? Wait, no, let's do it properly.

Alternative approach: Let's find the coordinate of each tick. Let's assume that from -6 to -7, the total length is 1, but no, -6 to -7 is 1 unit, but -6.5 is the midpoint. Wait, the number line has -6 at the top, then -6.5, then -7 at the bottom. So between -6.5 and -7, the length is 0.5. Let's count the number of divisions between -6.5 and -7. Let's see the blue dot is 3 divisions below -6.5? Wait, no, let's look at the positions:

-6 (top)
Then some ticks, then -6.5 (a tick)
Then ticks down: let's say the ticks between -6.5 and -7: let's count the number of ticks. Let's see, -6.5, then tick 1, tick 2, tick 3, tick 4, -7. So that's 4 intervals, each of length 0.5 / 4 = 0.125? No, that can't be. Wait, maybe the number of ticks between -6.5 and -7 is 5, so each is 0.1. Wait, 0.5 divided by 5 is 0.1. So each small tick is 0.1. So from -6.5, going down (decreasing) by 0.1 each tick. So -6.5 - 0.1 = -6.6, -6.5 - 0.2 = -6.7, -6.5 - 0.3 = -6.8? Wait, no, wait, when going down the number line, the value decreases. So from -6.5, each tick down is subtracting 0.1. Wait, but -6.5 is lower than -6, so going down (more negative) means decreasing. So -6.5 is -6.5, then the next tick is -6.6, then -6.7, then -6.8, then -6.9, then -7. Wait, but the blue dot is at -6.8? Wait, no, let's check the position. Wait, the blue dot is 3 ticks below -6.5? Wait, let's count the ticks:

-6.5 (tick 1)
tick 2: -6.6
tick 3: -6.7
tick 4: -6.8
tick 5: -6.9
tick 6: -7

Wait, no, maybe the number of ticks between -6.5 and -7 is 5, so each is 0.1. So from -6.5, each tick down is 0.1 less. So the blue dot is 3 ticks below -6.5? Wait, looking at the number line, the blue dot is 3 ticks above -7? Wait, -7 is at the bottom. From -7, going up (increasing) by 0.1 each tick: -7 + 0.1 = -6.9, -7 + 0.2 = -6.8, -7 + 0.3 = -6.7. Wait, no, that's conflicting. Wait, let's use the distance. Let's see the position of the blue dot: between -6.5 and -7. Let's calculate the exact value.

Wait, let's list the marks:

- The top mark is -6.

- Then, let's say each small mark is 0.1. So from -6, each mark down is -6 - 0.1 = -6.1, -6.2, -6.3, -6.4, -6.5 (the thicker mark), then -6.6, -6.7, -6.8, -6.9, -7 (thicker mark). Wait, that makes sense. So -6.5 is at the 5th mark below -6 (since -6 - 0.5 = -6.5, so 5 marks of 0.1 each: 5*0.1=0.5). Then, below -6.5, the marks are -6.6, -6.7, -6.8, -6.9, -7. So the blue dot is at -6.8? Wait, no, looking at the number line, the blue dot is 3 marks below -6.5? Wait, -6.5 is a mark, then the next marks: -6.6 (1), -6.7 (2), -6.8 (3), -6.9 (4), -7 (5). Wait, the blue dot is at -6.8? Wait, no, maybe I miscounted. Wait, let's check the vertical position: the blue dot is 3 units below -6.5? Wait, no, let's use the formula.

Alternatively, let's find the value of each tick. The distance between -6.5 and -7 is 0.5, and there are 5 intervals (since from -6.5 to -7, how many small lines? Let's see, the number line shows -6.5, then a line, then a line, then a line, then -7. Wait, maybe 4 intervals? No, the blue dot is at the 3rd line below -6.5. Wait, maybe the correct way is:

From -6.5 to -7, the total length is 0.5. Let's count the number of segments between -6.5 and -7. If there are 5 segments, each is 0.1. So the blue dot is 3 segments below -6.5, so -6.5 - 0.3 = -6.8? Wait, no, because going down the number line (towards -7) is decreasing, so subtracting. Wait, -6.5 - 0.1 = -6.6, -6.5 - 0.2 = -6.7, -6.5 - 0.3 = -6.8. Wait, but let's check the position relative to -7. From -7, going up (increasing) by 0.1 each segment: -7 + 0.1 = -6.9, -7 + 0.2 = -6.8, -7 + 0.3 = -6.7. Wait, the blue dot is 2 segments above -7? No, that's not matching. Wait, maybe the number of segments between -6.5 and -7 is 5, so each is 0.1, and the blue dot is at -6.8? Wait, no, let's look at the number line again. The blue dot is between -6.5 and -7, closer to -7? Wait, no, let's count the ticks:

-6 (top)
-6.1
-6.2
-6.3
-6.4
-6.5 (thick)
-6.6
-6.7
-6.8 (blue dot)
-6.9
-7 (thick)

Yes, that makes sense. So the blue dot is at -6.8? Wait, no, wait, maybe I got the direction wrong. Wait, the number line is vertical, with upward being more positive (less negative) and downward being more negative. So -6 is at the top (less negative), -7 is at the bottom (more negative). So each tick down is more negative. So from -6.5 (which is -6.5), each tick down is -6.6, -6.7, -6.8, -6.9, -7. So the blue dot is at -6.8? Wait, but let's check the distance from -6.5: -6.5 to -6.8 is 0.3, and from -6.8 to -7 is 0.2. Wait, but maybe the number of ticks between -6.5 and -7 is 5, so each is 0.1, so the blue dot is 3 ticks below -6.5, so -6.5 - 0.3 = -6.8. Yes, that seems right.

Step2: Confirm the value

So each small tick is 0.1. Starting from -6.5, moving down 3 ticks (each 0.1), so -6.5 - 0.3 = -6.8. Alternatively, starting from -7, moving up 2 ticks (since -7 + 0.2 = -6.8? Wait, no, -7 to -6.9 is 0.1, -6.9 to -6.8 is 0.1, so two ticks up from -7 is -6.8. Wait, but the blue dot is 3 ticks above -7? Wait, no, let's count the ticks from -7: -7 (bottom), then -6.9 (1 tick up), -6.8 (2 ticks up), -6.7 (3 ticks up), -6.6 (4 ticks up), -6.5 (5 ticks up). Wait, the blue dot is at 2 ticks up from -7? No, that's -6.8? Wait, no, 1 tick up from -7 is -6.9, 2 ticks up is -6.8, 3 ticks up is -6.7, etc. Wait, the blue dot is at -6.8? Wait, maybe I made a mistake in counting. Let's re-express:

The number line has:

- Top: -6

- Then, each small tick is 0.1, so:

-6, -6.1, -6.2, -6.3, -6.4, -6.5 (thick tick), -6.6, -6.7, -6.8 (blue dot), -6.9, -7 (thick tick)

Yes, so the blue dot is at -6.8.

Answer:

-6.8