Video: Measuring Lengths Using Rulers

Sophia uses a centimeter ruler to measure the length of a straight line, as shown in the diagram. She determines that the length of the line is 9.4 cm. Which of the following statements explains why the answer is incorrect? [A] The 0 marker of the ruler was not positioned at the start of the line; thus, the line is actually shorter than 9.4 cm. [B] The 0 marker of the ruler was not positioned at the start of the line; thus, the line is actually longer than 9.4 cm. [C] The ruler is not parallel to the line. [D] The maximum resolution of the ruler is 1 cm; thus, the length of the line should be recorded as 9 cm. [E] Measurements using a ruler should always be rounded up; thus, the length of the line should be recorded as 10 cm.

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Video Transcript

Sophia uses a centimeter ruler to measure the length of a straight line as shown in the diagram. She determines that the length of the line is 9.4 centimeters. Which of the following statements explains why the answer is incorrect? A) The zero marker of the ruler was not positioned at the start of the line. Thus, the line is actually shorter than 9.4 centimeters. B) The zero marker of the ruler was not positioned at the start of the line. Thus, the line is actually longer than 9.4 centimeters. C) The ruler is not parallel to the line. D) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as nine centimeters. E) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 10 centimeters.

Okay, so first things first, here’s the line that we’re trying to measure. We can see that the line is pointing in this direction, and so is the edge of the ruler that we use to make measurements. They’re both aligned. And therefore, the line and the ruler are in fact parallel. This means that option C is immediately ruled out because they say that the ruler is not parallel to the line.

So Sophia has done this correctly. She has aligned the ruler with the straight line. Now let’s consider options D and E before we do anything else. Looking at option D, this says that the maximum resolution of the ruler is one centimeter. Well, this is clearly not true. Of course, there are markings every centimeter, but there are also smaller markings every millimeter. And so the maximum resolution of the ruler is not one centimeter. It’s actually one millimeter. So option D is not our answer either.

Moving on to option E, this one says that measurements using a ruler should always be rounded up. Well, that is definitely bad practice. We should not be doing this, because if we’re rounding everything up, then unless some length that we’re measuring is perfectly on a centimeter mark, we would always end up rounding it up. And so we would always measure it to be longer than it actually is.

The idea with rounding though is that, on average, half of the time we would round up and half of the time we would round down. This way, any errors in the rounding up would be canceled out by the errors in the rounding down. That’s if we’re measuring different lengths of course. But if we’re always rounding up, then we’re introducing a systematic error. And so option E is out of the question as well.

So let’s now look at options A and B. They start off very similarly. Both of them say that the zero marker of the ruler was not positioned at the start of the line. Option B says exactly the same thing. And that, in fact, is true. Here’s the zero marker of the line, and here is the start of the line. So this is going to result in an error in the measurement. But what is the consequence of this?

Well, option A says that the line is actually shorter than 9.4 centimeters. That’s shorter than what Sophia measured it to be, which happens to be a reading here at 9.4 centimeters on the ruler, whereas option B says that the line is actually longer than 9.4 centimeters. So which is it?

Well, to answer this question, we need to imagine that we correctly placed a ruler next to the line. We need to ensure that the zero mark — that’s this mark over here — meets up with the start of the line. To do this, we would either have to move the ruler left or move the straight line to the right. Either way, we would have to do this until this end of the line moved to the zero mark. And we can imagine this as just moving the straight line to the right.

But then in that situation, this end of the line moves to the right as well. And so once we’ve aligned the line correctly, the actual reading is going to be larger than 9.4 centimeters. It’s gonna be somewhere closer to 10 centimeters. And hence, the line is actually longer than 9.4 centimeters. That is what option B says, and so option A is incorrect.

Hence, we found our answer to the question. The length of the line as measured by Sophia is incorrect because the zero marker of the ruler was not positioned at the start of the line and thus the line is actually longer than 9.4 centimeters.

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