Question

the blue dot is at what value on the number line?
(number line with 9, 13, and a blue dot at ?)

Explanation:

Step1: Find the interval between marks

The distance between 9 and 13 is $13 - 9 = 4$, and there are 2 intervals (from 9 to the next mark, then to 13). So each interval is $\frac{4}{2}=2$.

Step2: Calculate the value of the blue dot

From 13, we move 3 intervals to the right (since from 13 to the blue dot, count the marks: 13, then next is 15, then 17, then 19? Wait, no, let's re - check. Wait, between 9 and 13, there are two spaces (since 9, then a mark, then 13? Wait, no, the number line: let's see the positions. Let's list the marks. Let's assume the mark at 9, then the next mark is $9 + 2=11$, then $11 + 2 = 13$, then $13+2 = 15$, $15 + 2=17$, $17+2 = 19$? Wait, no, the blue dot is 3 intervals to the right of 13? Wait, no, let's count the number of intervals from 9 to 13: there are 2 intervals (because 9 to next mark: 1 interval, next mark to 13: 2nd interval). So each interval is 2. Now, from 13, how many intervals to the blue dot? Let's see: 13, then next mark (15), then next (17), then the blue dot. So that's 3 intervals? Wait, no, maybe I miscounted. Wait, the number line: let's see the positions. Let's start from 9. The marks are at 9, then after 2 units (since each interval is 2) we get 11, then 13, then 15, 17, 19. Wait, the blue dot is at the mark that is 3 intervals to the right of 13? Wait, no, let's count the number of steps from 13 to the blue dot. Let's see the number line: 9, then a mark, then 13? No, wait, the distance between 9 and 13 is 4, and there are 2 segments (so each segment is 2). So the marks are at 9, 11, 13, 15, 17, 19,... Now, the blue dot is at the mark that is 3 segments to the right of 13? Wait, no, from 13, first segment: 15, second: 17, third: 19? Wait, no, let's look at the number of marks. Let's count the intervals between 9 and the blue dot. Wait, maybe a better way: Let's find the value of each interval. The distance between 9 and 13 is 4, and there are 2 intervals (since between 9 and 13, there are two "gaps" between the vertical lines). So each interval is $4\div2 = 2$. Now, starting from 9, let's find the position of the blue dot. Let's count the number of intervals from 9 to the blue dot. Let's see: from 9, the next line is +2 (11), then +2 (13), then +2 (15), +2 (17), +2 (19). Wait, the blue dot is at the line that is 5 intervals away from 9? Wait, no, let's count the number of vertical lines. Wait, the number line has marks: let's see, the mark at 9, then the blue dot is 3 marks to the right of 9? No, this is getting confusing. Wait, another approach: the difference between 9 and 13 is 4, and there are 2 units (intervals) between them, so each unit (interval) is 2. Now, from 13, we move 3 intervals to the right: $13+2\times3=13 + 6=19$? Wait, no, wait, let's count the number of intervals from 13 to the blue dot. Let's look at the number line: after 13, the next mark is 15 (1 interval), then 17 (2 intervals), then the blue dot (3 intervals). So $13+2\times3 = 19$? Wait, but let's check again. Wait, maybe the number of intervals between 9 and 13 is 2, so each interval is 2. Then from 13, the blue dot is 3 intervals to the right? Wait, no, maybe I made a mistake in the number of intervals. Let's count the number of spaces between the vertical lines. From 9 to the next vertical line: 1 space, then to 13: 2 spaces. So 2 spaces for 4 units, so 1 space is 2 units. Now, from 13, how many spaces to the blue dot? Let's see: 13, then 1 space (15), 2 spaces (17), 3 spaces (19). So the blue dot is at 19. Wait, but let's check with another method. Let's list the values:

Mark 1: let's say the mark at 9 is the 4th mark (just a way to count). Wait, no, let's see the number of marks. Let's assume that the mark at 9 is position n, and the mark at 13 is position n + 2 (since 2 intervals). Then the blue dot is at position n+2 + 3=n + 5. The value at position n is 9, position n + 1 is 11, position n + 2 is 13, position n+3 is 15, position n + 4 is 17, position n + 5 is 19. So the blue dot is at 19.

Answer:

19