# Video: Measuring Lengths Using Rulers

Liam uses two centimeter rulers to measure the length of a straight line, as shown in the diagram. He determines that the length of the line is 19.2 cm. Which of the following statements explains why this answer is incorrect? [A] The rulers are not parallel to the line. [B] The two rulers have not been placed end to end. [C] The maximum resolution of the ruler is 1 cm; thus, the length of the line should be recorded as 19 cm. [D] The second ruler has been placed the wrong way around. [E] Measure

05:08

### Video Transcript

Liam uses two centimeter rulers to measure the length of a straight line as shown in the diagram. He determines that the length of the line is 19.2 centimeters. Which of the following statements explains why this answer is incorrect? A) The rulers are not parallel to the line. B) The two rulers have not been placed end-to-end. C) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as 19 centimeters. D) The second ruler has been placed the wrong way around. E) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 20 centimeters.

Okay, so in this situation, we’ve got a line here. And Liam is trying to measure its length. He realizes that one ruler is not enough and that the line is too long for the ruler. So he has to use a second ruler. Let’s first think about the things that Liam has done correctly.

Well, we can see first of all that the zero marking on the first ruler is aligned with the start of the straight line. So that much has been done well. Secondly, we can also see that both rulers, or more specifically the edges used to do the measuring on the rulers, are parallel to the straight line. They’re all aligned in the same direction.

And coincidentally, that rules out option A. Option A says that the two rulers are not parallel to the line, where in reality they are. So this much Liam has done well. So where has it gone all wrong?

Well, let’s consider option B first of all. This option says that the two rulers have not been placed end-to-end. Well, it’s true that the two rulers have not been placed end-to-end, but that is actually a good thing. Let’s say that the two rulers had been placed end-to-end and this was the line we were trying to measure the length of.

Well then, there would be a whole chunk of this ruler and a whole chunk of this ruler, which wouldn’t be measuring anything because there are no markings on the rulers past 12 centimeters. This means that these bits of plastic on the ends of the ruler past the final marking could be arbitrarily long or short. And they wouldn’t be measuring anything. In effect, all what we’d be measuring is this length ending here and this length starting here. This whole bit would’ve been completely ignored.

So although the two rulers have not been placed end-to-end, this is a good thing. And so option B is not the correct answer to our question. Option C then, this says that the maximum resolution of the ruler is one centimeter. Well, let’s just stop there. We’ve been told that we’ve got two centimeter rulers. This means that the big markings on the rulers are every one centimeter. But we can also see that there are little markings every millimeter, every tenth of a centimeter. And so the maximum resolution of the ruler is not one centimeter. So immediately, we can cancel out this option.

Option D then, the second ruler has been placed the wrong way around. Well, that’s true. The second ruler has been placed the wrong way around. We can see that this ruler has its numbers the right way up, whereas the second ruler has its numbers upside down. So this might be the source of the problem. And in fact, that is the reason why. We’ll come back to this in a second.

However, let’s quickly make sure that option E is incorrect. Option E says that measurements using a ruler should always be rounded up. Well, why would we want to do that? Why would we want to introduce a systematic error where we always round up unless the length that we’re measuring is perfectly on a centimeter mark? This would mean that anything that’s 19.2 centimeters would be recorded as 20 centimeters and anything that’s 19.8 centimeters would also be recorded as 20 centimeters. That doesn’t make sense. So option E is wrong.

So let’s come back to option D then and work out why that’s the correct answer to our question. We can see that Liam has measured the length of the line up until this point very correctly. Liam has measured zero centimeters, one centimeters, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

But then because he’s placed the second ruler upside down, he thinks that the length of the line from here to here is seven and a bit centimeters. Specifically, that’s the second mark nearest to the seven. So he thinks that part of the line is 7.2 centimeters long.

However, what he hasn’t realized is, by placing the ruler upside down, his 12-centimeter mark now becomes the zero-centimeter mark. And that way, he can basically measure this length — that’s 12 centimeters — and this length separately. Then he can add them together to give the total length of the line.

But anyway, so placing the ruler upside down means that this becomes the zero-centimeter mark. That means that the 11-centimeter mark now becomes one, 10 becomes two, nine becomes three, eight becomes four, and seven becomes five centimeters. And so this part of the line is not 7.2 centimeters long. It’s zero, one, two, three, four, five, 5.2 centimeters long. Hence, the total length of the line is 12 centimeters from the first ruler plus 5.2 centimeters from the second ruler. And that is equal to 17.2 centimeters, not 19.2.

Now Liam could’ve equally made the second part of the measurement by placing the second ruler the correct way around — that’s the right way up — and overlapping the zero mark on the second ruler with the 12-centimeter mark on the first ruler. This way, the first part of the line already measured to be 12 centimeters can now be ignored. And anything ahead of this is an additional centimeter on top of the 12 centimeters. And two centimeters ahead means we’ve got the 12 centimeters from earlier plus two centimeters here, and so on and so forth, which in this case ends up being 5.2 centimeters more than the original 12 centimeters. So in reality, the second ruler has been placed the wrong way around. And that is the answer as to why Liam got an incorrect measurement.