Video: Using Two Rulers to Measure Length

Anthony uses two centimetre rulers to measure the length of a straight line, as shown in the diagram. He determines that the length of the line is 17.1 cm. Which of the following statements correctly explains why this answer is incorrect? [A] The ruler is not parallel to the line. [B] The maximum resolution of the ruler is 1 cm. Thus, the length of the line should be recorded as 17 cm. [C] Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 18 cm. [D] He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually longer than 17.1 cm. [E] He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually shorter than 17.1 cm.

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Video Transcript

Anthony uses two centimetre rulers to measure the length of a straight line, as shown in the diagram. He determines that the length of the line is 17.1 centimetres. Which of the following statements correctly explains why this answer is incorrect? A) The ruler is not parallel to the line. B) The maximum resolution of the ruler is one centimetre. Thus, the length of the line should be recorded as 17 centimetres. C) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 18 centimetres. D) He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually longer than 17.1 centimetres. E) He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually shorter than 17.1 centimetres.

Okay, so in this question, Anthony’s trying to measure the length of this line starting here and ending here. He’s had to use two rulers to measure the length of this line, as we can see, because just one would’ve been too short. And he claims that the length of the line is 17.1 centimetres. We need to work out why this answer is not correct. So let’s start by looking at option A as a possible reason for this.

Option A says that the ruler is not parallel to the line. Well clearly, this is not true. In fact, both of the rulers that have been used in this measurement are parallel to the line. We can see that the straight line is in this direction. And the edges of the rulers used to do the measuring are also aligned in the same direction. They are parallel. And, therefore, we can rule out option A.

Moving on to option B then, this one says the maximum resolution of the ruler is one centimetre. Thus, the length of the line should be recorded as 17 centimetres. Well, in the question, we’ve been told that Anthony uses two centimetre rulers. This means that these two rulers are both identical. And a large markings that they have on them are every centimetre. However, we can see that the ruler has small markings as well, every tenth of a centimetre or, in other words, every millimetre. Therefore, the maximum resolution of each one of these rulers is not one centimetre. It’s actually one millimetre. And hence, option B is incorrect as well.

Option C then, measurements using a ruler should always be rounded up. Well, this is definitely, definitely a bad idea. We should never do this. In fact, we should always round correctly because otherwise we’re introducing a systematic error into our measurement. The only time we wouldn’t have to round up is if the end of the line fell exactly on a centimetre marking. However, in this case, we can see that our line ends on the 5.1-centimetre marking. And if we were to round that up, then we would round that up to six centimetres.

And even if the end of the line was somewhere here, like 5.8 centimetres, we would still round that up to six centimetres. So 5.1 rounds up to six and 5.8 rounds up to six. How does that make sense? We’re introducing a systematic error into our measurement by always rounding up. We’re always measuring lengths unless they fall exactly on a centimetre marking as longer than they actually are. And hence, option C is out of the question as well.

Let’s look at option D then. Option D says that he has positioned the second ruler at the end of the first one. And there is a space in between. Now, it’s worth noting that option E starts exactly the same way. It says that he has positioned the second ruler at the end of the first one. And there is a space in between. So they’re both saying that this is the reason why his measurement is incorrect. And that’s true he has placed the ruler end-to-end. And there’s also a little gap in between these two rulers. And that actually is a reason as to why he’s measured the length of the line incorrectly.

Now, the reason for this is that Anthony’s correctly measured from here all the way to 12 centimetres and all the way from this zero-centimetre mark to the 5.1-centimetre mark. However, this part of the line is not being measured at all because there are no markings after the 12-centimetre mark on the first ruler or before the zero-centimetre mark on the second ruler. Therefore, these bits of plastic on the ends of the ruler could be arbitrarily long. And that entire portion of line is just not being measured at all.

So this is the reason why Anthony should not have placed the rulers end-to-end or had a gap between the two. But then what is the consequence of this? Option D says that because of this, the line is actually longer than 17.1 centimetres, whereas option E says that the line is actually shorter than 17.1 centimetres. So which is it? Well, as we’ve said already, Anthony has correctly measured this part of the line and this part of the line. However, he’s failed to account for this part which means that the line must be longer than 17.1 centimetres.

Another way to think about this is that the second ruler, this one on the right, should be moved to the left so that the zero marking on this ruler aligns with the 12-centimetre marking on the first ruler. That way, the first 12 centimetres of the line that have been measured are all well and good. And then anything greater than that is measured on the second ruler. One centimetre ahead from the 12-centimetre mark would mean 12 plus one centimetres. Two centimetres ahead would be 12 plus two, and so on and so forth.

But then if we have to move the second ruler to the left in order to move the zero marking onto the 12 centimetre marking on the first ruler, then, of course, the whole ruler is going to move left. And, therefore, the reading of the end of the line that was 5.1 centimetres is now going to be somewhere along here, maybe 5.4, 5.5, 5.6, whatever it is. And that shows that the length of the line is actually greater than 17.1 centimetres because the total length of the line is the 12 centimetres from the first ruler plus the 5 point whatever centimetres that’s greater than 5.1 centimetres on the second ruler.

Hence, option E, which says that the line is actually shorter, is out of the question. So at this point, we found our final answer. The reason that Anthony measured the length of the line incorrectly is because he positioned the second ruler at the end of the first one. And there was a space in between. Thus, the line is actually longer than 17.1 centimetres.

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