Question

find the value of r. write your answer as a decimal number.
0.2
r
0.3
r = \boxed{}

Explanation:

Step1: Determine the interval length

The distance between 0.2 and 0.3 is \(0.3 - 0.2 = 0.1\). Count the number of intervals between 0.2 and 0.3. From the number line, there are 10 intervals (since from 0.2 to 0.3, we can see the marks: let's check the positions. From 0.2 to 0.3, how many steps? Let's see, 0.2 is the first mark, then each subsequent mark. Wait, actually, from 0.2 to 0.3, the total number of intervals: let's count the number of spaces between 0.2 and 0.3. Looking at the number line, from 0.2 to 0.3, there are 10 equal intervals? Wait, no, let's check the position of \(r\). Wait, 0.2 to 0.3: let's see the number of marks. Let's list the marks: 0.2, then next marks, and then \(r\) is 2 marks before 0.3? Wait, no, let's calculate the step size. The total range from 0.2 to 0.3 is 0.1, and the number of intervals between 0.2 and 0.3: let's count the number of segments. From 0.2 to 0.3, how many vertical lines? Let's see, 0.2 is at the first line, then the next lines, and 0.3 is at the 11th line (since 0.2 is line 1, 0.3 is line 11). So the number of intervals between 0.2 and 0.3 is \(11 - 1 = 10\) intervals. So each interval is \(\frac{0.1}{10}=0.01\). Now, \(r\) is at which position? Let's see, from 0.2, how many intervals to \(r\)? Wait, 0.3 is at the 10th interval from 0.2? Wait, no, let's check the position of \(r\). Looking at the number line, \(r\) is 2 intervals before 0.3? Wait, no, let's count the number of intervals from 0.2 to \(r\). Wait, 0.2 to 0.3: total 10 intervals (each 0.01). So 0.2 + 8*0.01? Wait, no, let's see: 0.2, then each step is 0.01. Let's count the number of steps from 0.2 to \(r\). Wait, 0.3 is at the end. Let's see the position of \(r\): from 0.2, moving towards 0.3, how many steps? Let's see, the marks: 0.2, then next, next, ..., \(r\), then next, next, 0.3. Wait, 0.3 is at the 10th step from 0.2 (since 0.2 + 10*0.01 = 0.3). So \(r\) is at the 8th step? Wait, no, let's check the number of intervals between 0.2 and \(r\). Wait, maybe a better way: the number of intervals from 0.2 to \(r\): let's see, 0.2 to \(r\): how many marks? Let's count the vertical lines. From 0.2 (line 1) to \(r\) (line 9), and 0.3 is line 11? Wait, no, maybe I made a mistake. Wait, the distance between 0.2 and 0.3 is 0.1, and there are 10 equal parts (since 0.1 divided by 10 is 0.01 per part). So each small segment is 0.01. Now, \(r\) is located 2 segments before 0.3? Wait, no, 0.3 is at the end. Let's see, from 0.2 to 0.3, there are 10 segments (each 0.01). So the positions are: 0.2 (0 segments), 0.21 (1), 0.22 (2), ..., 0.29 (9), 0.3 (10). Wait, \(r\) is at 0.28? Wait, no, let's check the number line. Wait, the \(r\) is 2 marks before 0.3? Wait, 0.3 is at the last mark. Let's count the number of marks between 0.2 and 0.3. Let's see: 0.2, then the next marks, and \(r\) is the 8th mark from 0.2? Wait, no, let's do it properly. The total range from 0.2 to 0.3 is 0.1, and the number of intervals is 10 (since from 0.2 to 0.3, there are 10 equal steps). So each step is 0.01. Now, \(r\) is at the 8th step from 0.2? Wait, no, 0.2 + 8*0.01 = 0.28? Wait, no, 0.2 + 8*0.01 is 0.28, but 0.3 - 2*0.01 = 0.28. Yes, because from 0.3, moving back 2 steps (each 0.01) gives 0.3 - 0.02 = 0.28. So \(r = 0.28\). Wait, let's verify. The number of intervals between 0.2 and 0.3: 10 intervals (each 0.01). So 0.2 + 8*0.01 = 0.28? Wait, no, 0.2 to 0.3 is 10 intervals, so the positions are 0.2 (0), 0.21 (1), 0.22 (2), 0.23 (3), 0.24 (4), 0.25 (5), 0.26 (6), 0.27 (7), 0.28 (8), 0.29 (9), 0.3 (10). So \(r\) is at the 8th interval from 0.2, which is 0.28. Wait, but let's check the number line: \(r\) is 2 intervals before 0.3. 0.3 - 2*0.01 = 0.28. Yes, that's correct.

Step2: Calculate \(r\)

So each interval is \(0.01\), and \(r\) is \(0.3 - 2\times0.01 = 0.28\). Alternatively, from 0.2, the number of intervals to \(r\) is \(8\) (since 0.2 + 8*0.01 = 0.28). So \(r = 0.28\).

Answer:

\(0.28\)