Why did everyone miss this SAT Math question?

Problem: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total?

1982 SAT Question
In the figure above, the radius of circle A is one third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?

(a) 3/2
(b) 3
(c) 6
(d) 9/2
(e) 9

Sources:

Also see the Coin Rotation Paradox

Update: some people have made really cool interactive worksheets at Geogebra. You should check these out!

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[MindYourDecisions]

Happy 4 year anniversary and 1 million views! Yes the answer really is 4, even the test makers admitted it! But it's a counter-intuitive problem. A lot of people who felt the video was mistaken now see why the video is correct, particularly after reading these links:


Sources

Coin Rotation Paradox

Geogebra demonstrations

[omarmirza]

It all depends on what you mean by "completing one revolution". If one revolution is completed when the initial point on A touching B once again touches B, which is what I originally thought, then 3 is the correct answer.

[GalileosTelescope]

The center of the small circle travels around a circle of radius R+r. Not R, as the test writers thought. So the number of rotations is 2pi(R+r)/2pi(r) = R/r + 1.

[xandercorp]

It rolls 3 times, rotates 4 times, but revolves only once. Hear me out.

A full roll is completed when an object progresses around the a surface of another, without slipping, until the contact point between the object and the surface has gone all the way around the circumference of the object and is touching again; a full rotation is completed when an object turns all the way around and is upright again with respect to the frame of reference (yes, this means that rolls = rotations on a straight surface or where the frame of reference rotates with the normal of the curve); a full revolution is completed when an object travels all the way around another object with respect to that second object (most people are familiar with the Earth rotating on it's own axis vs revolving around the Sun).

In the video example, you can see that:
1) the "bottom" of circle A, identifiable by the orientation of the text, faces the centre of circle B three times as it rolls;
2) the text is upright with respect to the camera 4 times as it rolls; and
3) circle A travels around circle B only 1 time.

Does that make sense to you?

[usdescartes]

Another way to think of this is that the center of the smaller circle is 4 units away from the center of the larger circle; thus, the center of the smaller circle travels 4 times its original circumference. Since it rolls without slipping, that also means that its edge rolls through 4 times its circumference.

Interestingly, it works out in a similar manner if the smaller circle is INSIDE the larger circle. If we think about the center of the smaller circle, it travels through a circumference of twice its radius, which means the edge travels through twice its circumference. OR we can think of it as rolling around itself three times but it travels BACKWARD one rotation because that's how an inner circle rolls. Either way, and in general, the number of rotations of an inside circle will be n – 1.

[plcamp]

“Rolls around” is actually ambiguous. One rotation could be seen as the point where the same point on the rolling coin retouches the other, in which case the answer is 3.

[SubduedRadical]

Visually mathematically:

Take Circle B's edge. Cut out all the inside circle (so you only have the rim), cut the rim in any place so you can then stretch the rim out flat on the table. This is the perimeter of Circle B. It has a value that we know, its circumference, which is s = 2*pi*r = pi*d, where pi is obviously pi, 2 is 2, r is radius, and d is diameter. Basically, we're just laying out the length of the circumference of B as a straight line on the table and measuring the total length.

Now, take Circle A's edge. Same thing, cut out so you only have a rim, snip the rim, lay the rim out flat. The total length of this circumference of A is also known and can be expressed as s = pi*d.

Now, if we put the line we now have from A next to the line for B, the line for A is 1/3rd the line for B. This means it takes THREE circumferences of A to make ONE circumference of B.

.

Mechanically (e.g. the way you rolled the circles):

The problem with your circle is, at the 1/4th point, Circle A has NOT made a full circle. You're lining it up Cartesian like using the table as a frame of reference to say that Circle A has rotated 360 degrees WITH RESPECT TO THE TABLE. But this is not WITH RESPECT TO CIRCLE B's PERIMETER.

If you notice, the STARTING orientation of Circle A with respect to Circle B's perimeter is where the "bottom" of the circle is tangent to Circle B. Look at how the A's point is pointing AWAY from the center of Circle B. That is, in order to see when Circle A has made "one full rotation with respect to Circle B's perimeter", you must rotate Circle A until the tip of the A is pointing away from the tangent point/center of Circle B. Each time the tip of A points away from Circle B, you have made one full rotation of Circle A with respect to Circle B.

Note that where you stop at the 1/4th point, the tip of the A is NOT pointing AWAY FROM Circle B. It's pointing perpendicular-ish to it.

Timestamp ~ 1:44 you can see what I'm looking at. At 1:44, notice how the tip of the A is (more or less) NOW pointing away from Circle B. So at point 1:44, you have completed one full rotation of Circle A WITH RESPECT TO Circle B.

Notice also that this is at ABOUT the 1/3rd point...

...meaning that mechanically, we've now arrived at the same solution: Circle A makes it 1/3rd of the way around Circle B when it makes one full rotation WITH RESPECT TO Circle B, ergo, it will take Circle A three full rotations WITH RESPECT TO Circle B in order to trace out the full perimeter of Circle B.

.

This is kind of some Relativity problem, so you could argue reference frames here, but the fact remains that Circle A needs to "lay out" its circumference 3 full times around Circle B's outside edge in order to cover the complete circle.

This is true both mathematically and in terms of Circle A's motion WITH RESPECT TO Circle B.

So you can argue that frame of reference can give you different answers (which is true), but 3 is _A_ correct answer, and an entirely valid answer, per Relativity.

.

That is, you can say 3 AND 4 are BOTH correct answers, but you cannot say that 4 is THE correct answer and 3 is wrong. That would make YOU wrong.

Note that I mean this with no animosity, I just dislike people making absolute statements that are not actually dictated by the situation or evidence presented. If you are given a problem that has two answers (say, the square root of 4, which can be 2 or -2), and the list of answers has one of those (say -2) but not the other. Then the correct answer should be the one answer OF THE GIVEN CHOICES that is one of the correct answers.

In this case, they presented both 3 and 4, and 3 is ONE correct answer, therefor, of the options listed, 3 would be THE correct answer, since the other four answers were incorrect.

Now, if they gave you BOTH answers (e.g. for square root of 4 they gave you -2, -1, 0, 1, and 2), then you could argue that BOTH 2 and -2 are correct, so either you pick should be counted as correct.

In which case, going back to the circles, BOTH 3 AND 4 are correct, since it depends on the frame of reference you choose, and the problem did not state a frame of reference.

Agree/disagree? I get that this is a 6 year old video, so you'll probably never see this. :)

[quistan]

However, circle A's "tread" only engaged the surface of circle B exactly 3 times.
From a topological pov, it only made 3 revolutions.

[Devastish]

I got to 3, here is how I did it:

Every rotation of the smaller circle covers a distance equal to it's circumference, therefore, I treated each circle as a straight line, and then compared them.
c= 2pi(r)
With r being the circumference of circle A
C= 2pi(R)
R=3r
C=2pi(3r)
A/B = 3

So, the A circle needs to travel 3 times it's total circumference to make it around B, which means 3 full rotations.

After watching the video and reading the comments:

My answer seems to hold true if the circles are stationary and would spin in place, such as in a gearbox, OR the B distance is a true straight line allowing A to move in a straight line.
However, they aren't stationary, and B is a circle, so that throws a wrench in it.

I'm trying to understand how A rotates 4 times as it travels around B.
My only explanation is that the center point of A is travelling a larger circle than the circumference of B as A rotates around.
This means that the distance A travels in the question is actually longer than the circumference of B.

So the real equation I should use is:
c=2pi(r)
C=2pi(3r+r)=2pi(4r)
A/B=4


Others have pointed out that there is a perspective issue at play.

A rotates 4 times, relative to the perspective of the viewer.
However, when considering the points of contact between A and B, there are only 3 rotations.
ie. the "bottom" of A is only in contact with B at 3 points.

I find this argument compelling, because from the perspective of A, it has only rotated 3 times.
If I was standing on A, looking directly at B from the start, I would only be looking directly at B 3 times during the trip.

So, I suppose it comes down to definition - what does "revolve" mean?
Is it relative to B, or is it relative to the viewer?

Personally, I would argue that the way the question is written, they meant "revolve" to be relative to B, and therefore 3 is correct.
But, it is not explicitly stated which perspective is correct, so it is still vague enough to be open to interpretation.

[SanderBuruma]

The phrasing is ambiguous. It doesn't give a frame of reference, from the perspective of the bigger circle it is 3 times. From top down it is 4 times.

[JDeWittDIY]

Both 3 and 4 are correct answers, depending on semantics. When the small circle has advanced 1/4 of the perimeter of the big circle, it has rotated/revolved one time according to our visual reference, but has not finished a full rotation in terms of where it is tangent to the big circle.

So if the frame of reference is the small circle by itself, the answer is 4.
If the frame of reference is where the two circles are tangent, the answer is 3.

[khiljaz]

The circumference that matters here is made by the path traveled by the center of coin A. So you just add the radii of both circles and solve for circumference of the new circle "C" and divide the circumference of C by A.

A = 1
B = 3
C = A + B

1*2*3.14 = 6.28
4*2*3.14 = 25.12
25.12 / 6.28 = 4

The above works for any fraction.

Any fraction with a numerator of 1 can be solved by simply adding the numerator and denominator together.

Example:
Instead of A being 1/3 of B, lets make A 1/17 of B.

A = 1
B = 17
C = 18

1*2*3.14 = 6.28
18*2*3.14 = 113.04
113.04 / 6.28 = 18

[nicholasharvey]

I'll admit I missed this one as well, but it sorta reminds me of how the Earth rotates and revolves around the Sun. It only takes 23 hours and 56 minutes for the Earth to rotate 360 degrees, but a day is 24 hours because it takes the Earth an additional 4 minutes to turn another degree or so in order for it to be facing the Sun in the exact same way, because it has advanced about one degree in its orbit.

[PatrickGSR]

I just figured it out. The actual CIRCUMFERENCE of A touches the CIRCUMFERENCE of B 3 times during its trip around. But the angular position goes through (360*4) degrees because it's a circle going around a circle. So despite the circle ending back upright after 1/4 of the trip around or 90 degrees, the original contact point between circle A and B doesn't come back around and touch B until it's 1/3 the trip around, or 120 degrees.

[_DriveTime]

3 rotations relative to its "track', but since its track is oriented as a circle 4 rotations to our eyes. Flatten the track to a straight line, and it will only be 3 rotations relative to both.

[sbelllido]

If you stretch both circumferences, their lengths are 3 to 1. If you fix the little circle and turn one time the bigger one, the little wiĺl do 3 turns.
When the little circle revolves, it do the 3 turns plus one more because the visual change. The initial contact point isn't the same at 1/4 of the turn, but the little circle center trips.

[Kirtiraj_Deshmukh]

Is anyone here after Veritasium's video?

[TheGrapeApe]

"Only using paper and pencil"
Me: *starts ripping paper into circles*

[mhoover]

The actual correct answer is 1. It rotates 4 times while revolving once around B.😀

[zrplnbz]

For your experiment with the coins at 2:42, the coin has, in actuality, only rolled half of its circumference by the time it reaches the bottom of the other coin.