2 Circles Math Problem That Everyone Answered Incorrectly in SAT Exam

This video is adapted from MindYourDecisions. I found the question very interesting and tricky and I was able to solve it with two different approaches.
In this video I am explaining the two approaches with some graphical animation.

Problem :
The radius of circle A is one-third(1/3) the radius of circle B.
Circle A rolls around circle B one trip back to its initial position,
Question:
How many times will Circle A revolve till it reaches back to initial position?

These are your options.
(a) 3/2 (b) 3 (c) 6 (d) 9/2 (e) 9 (f) NONE of these

This question comes from 1982 S.A.T. Examination (United States)
and there were only 5 options
and all the candidates answered it incorrectly.
Reason being that the correct answer was not included in the options.
Later, 3 students claimed to the College Board, that the actual answer is 4.
And that's true.... that's why i purposely added the sixth option for you as None of these.

Please comment below your answers and suggestions. Also LIKE the video and SUBSCRIBE to my channel if you are new.

Also, please give a try to difficult puzzles and Interview Questions Answers specific to Puzzles and Riddles.

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[danbromberg]

I've seen this question posed at a few websites (including MindYourDecisions) so let me complement you on taking the time to not graphically demonstrate the problem with solution, but to also verbalize the explanation so much better than anyone else. For what it's worth, you earned yourself a bookmark (and a thumbs up) on my PC.

[ティナトピア-sj]

The problem lies in "revolve" definition being ambiguous.

Many people would interpret as "How many times does point P touches circle B" which would be three, rather than "How many times point P reaches the lowest position with respect to our eye".

First scenario refers to "revolving with respect to the big circle B" while the second scenario refers to "revolving with respect to the Euclidean coordinate system".

Indeed, as some points out, if we flatten the circle B and let A rolls on the straight line, it will revolve 3 times with unambiguous rotation. However when the surface is circular, it adds one more revolution if we view the surface from a third stationary view point, while the revolution number is still 3 from the view point of a 2D creature living in the surface, namely circle A itself.

[jayant]

Calculate Distance traveled by the small circle, not the edge of it, 3 rotation is possible if the small circle center point is on the big circle

[cedriccappelle]

is it just 3+1 because it will make a full rotation by going around the circle once? so if we have r/n, would the solution be n+1?

[jessicataylor]

I'm not sure a maths problem can be deemed tricky because everyone got it wrong on a paper that was incorrectly written...still a nice maths problem though :)

[animeverse]

No of rotation = circumference of large circle / small circle
2πr/2πr/3=2πr3/2πr = 3

[kestreladvisors]

Nope, I completely disagree.

The center point of Circle A is not the relevant point traveling around Circle B. It is the perimeter of Circle A that is traversing the perimeter of Circle B. Therefore, it is a point on the perimeter of Circle A that we must observe as Circle A travels around the perimeter of Circle B.

The focus on the center point of Circle A is relevant only for our perception of the orientation of Circle A (as "upright"), not the completion of a rotation.

To address this another way, a revolution is defined as one complete 360 degree rotation of the circle around its center of rotation (which is the center point). When a circle rolls along a flat surface, there is no revolution in the axis of rotation. However, when the circle travels along a curved surface (such as another circle), the axis of rotation, itself, is also revolving. Thus, as Circle A rolls along the curved surface of Circle B, the axis of rotation of Circle A is, itself, revolving. Therefore, even when Circle A appears "upright" again - and thus appears to have completed a 360 degree rotation - it has NOT completed a full 360 degree rotation around its axis of rotation at that point, because its axis of rotation has revolved during that segment of the trip (by ((R1 / (R1+R2)) * 360 degrees)).

To further drive home this point, consider the complementary situation (the obverse side of the coin, if you will), where Circle A is traveling along the INNER perimeter of Circle B. According to the logic presented in the video above, Circle A would complete only 2 revolutions to cover that distance. However, the inner perimeter of Circle B has EXACTLY THE SAME LENGTH as the outer perimeter of Circle B. So I ask: how much sense does it make that Circle A would have to revolve 4 times to traverse an outer perimeter having length 6pi, but only have to revolve 2 times to traverse an inner perimeter having the exact same length of 6pi? Yes, I'm waiting for an actual, logical, sensible, non-BS, legitimate response.

[christophe.gricourt]

intuition tells me this supposedly logical and mathematical demonstrations are plain wrong! if you were correct, then revolving circle A inside circle B would end up with 2 revolutions for the same length to be this is just insane, unless you consider that the "inner" perimeter of a circle is half the length of the "outer" perimeter :-) :-D please, do your math before asserting such facts, 4 is the correct answer just because it is 3 self-revolutions + 1 revolution around de circle B

[brandonservis]

Me: comes to video to scream about misspelling answer

[mikefochtman]

The way I see it, when the point in contact with the larger circle (you called it 'lower point') comes in contact with the larger circle again, the smaller circle has rotated 1 and 1/3 revolutions. Then as it continues to the next time this point touches the larger circle, the small circle has rotated 2 and 2/3 revolution. And finally when it contacts the larger circle for the third time back at the starting point, the small circle has rotated 3 and 3/3 time, or 4 times.

This particular point on the smaller circle does not draw a simple circle around the larger one, but rather a complex sort of cycloid. Which is why it's total distance traveled is larger than the circumference of the larger circle.

[alokpathak]

If we consider the radius of big circle 3
Then the radius of small circle is 1
By this circumference are 6π & 2π
According to you if small circle makes 4 rotation then distance travelled by it is 8π
But the circumference of big circle is only 6π

[mohammedzubair]

what u explained about angular rotation make sense but if we open the circumference of circle B, and make it linear, the answer we get is 3.

still confused with that.
if u could explain it...

[Roxbury_NJ]

I did it ...
and get 2,
If you put the small circle INSIDE the big circle, and go clockwise like here, then the small circle goes counterclockwise
3-1 =2
or is it...
4 - 1 is three???
I think uU can find the answer under "Smoke & Mirrors..."
🙃

or...
I mean spin the big circle and...

[jim]

Formula for small circle rotating around large circle: (R + r)/r. R = large radius, r = small radius. Here: (3 + 1)/1 = 4.
Four wasn't a choice on the subject SAT exam.

[larryb]

This is very wrong. The real answer is only 3 rotations.

You have a fault in your mathematical equations. Your first equation is the circumference of C while the second equation is 3 times the rotation of circle A, which is mathematically equal to the circumference of B. The circumference of C, which has a bigger radius by r/3, is not equal to the perimeter of B, therefore, you cannot use equation 1 (circumference of C) as equal to equation 2 (circumference of B) to get the value of N (number of rotation).
simply say: 2 X pi X (R + R/3) is not equal to 2 X pi X R☺

[abridgetool]

Rb = r
Ra = r/3
Radius of biggest circle = r+r/3 = 4r/3
Circumference of biggest circle = 2π*4r/3=8πr/3
Circumference of smallest circle =2πr/3
Number of rotation = How much times biggest circle's circumference is bigger than smallest circle = (8πr/3)/(2πr/3) = 4

[GameBoy-ftxn]

This concept is used in earth's rotation. If you study the rotation concept, you will understand this problem

[wisenheimer]

You are not solving this problem correctly.... The answer is three... When you do your demonstration, you don't take perspective into consideration... In reality, the camera should be kept horizontal to the POINT OF CONTACT... as such, as the circle travels around the larger circle, the camera will complete one revolution, if the proper perspective is maintained... You are mistakenly adding this revolution to the solution, rather than to the change in perspective...

[kestreladvisors]

And no, no, Circle A does NOT travel around imaginary Circle C; it travels around the periphery of Circle B. The periphery of Circle A travels along the periphery of Circle B, a distance of 2pr.

Imaginary Circle C does play a role in the perception of what occurs, but it creates an optical illusion which leads people to the incorrect conclusion of 4 revolutions, whereas the correct answer is actually 3 revolutions (or, more generally (r/(r/3)) revolutions.

Start with the definition of a revolution: Circle A completes a revolution once it completes a full 360 turn about its own axis of rotation. When traveling along a straight path, the orientation of Circle A's axis of rotation is fixed. However, when Circle A travels along the curved surface of Circle B, the entire system that is Circle A is rotating. When the entire system of Circle A rotates, that means that Circle A's axis of rotation itself revolves, which affects when Circle A actually completes that 360 degree turn about that revolving axis of rotation. I have provided a more thorough explanation of the illusion and the actual point of completion of revolutions in the link in my post below.

On a related note, it astounds me the lengths to which people contort logic to try to substantiate a conclusion just because someone else asserts that it is correct.

[larryharper]

Are we talking COMPLETE ROTATIONS of circle A as it travels around (and in contact with ) circle B? The answer is 3. If a vertical line is drawn through the center of circle A with an arrow at the top the base of the line will contact circle B, 3 times. If you are talking about how many times the line will return to vertical (arrow pointing up) then it is 4 times.