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.
