Past Incorrect SAT Math Question?

Test Preparation
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<p>My math teacher mentioned that there has actually been an SAT math problem which was completely wrong (i.e. the correct answer was not one of the choices). He described it as seeming simple at first, but actually more complex. The problem is how many rotations does it take for a circle with radius r to rotate/spin around a circle with radius 3r. At first glance, the answer seems to be 3, but actually the answer is 4.</p>

<p>I have been trying to figure out how the problem works and so far, I have come up with something along these lines: a rotation is a straightforward concept, when it is along a straight line. If there was a line with length the circumference of the larger circle, it would obviously take 3 rotations to go along this length. But the smaller circle is going AROUND another circle, which means that what defines a rotation is altered. </p>

<p>I have no idea if this is going in the right direction, but I would appreciate any input on this problem as it has been bugging me for a while.</p>

2

<p>this problem is not incorrect, infact i remember this one, you simply need to find circumference of large circle and small circle, then divide the larger circumference by the smaller circumference and thats your answer for # of rotations</p>

3

<p>I think your math teacher is right, though I doubt the SAT folks had the more complex problem in mind.</p>

<p>Imagine that the 2 circles are drawn on a flat sheet of paper on your desk, and you are viewing from directly above. The bigger circle is 'north' of the smaller circle, and at their point of contact there is a blue dot on the large circle, and a red dot on the smaller circle.</p>

<p>Now start the smaller circle moving clockwise along the perimeter of the larger circle. Since you started with the red dot at the north (=top) of the smaller circle, a rotation is completed when this red dot is again at the top of the smaller circle, NOT WHEN THE RED DOT IS TOUCHING THE LARGER CIRCLE AGAIN. You will finish one such defined rotation when you have travelled 90 degrees (out of 360). The red dot will be at the top of the smaller circle again at 180, 270 and 360 degrees; you will have completed 4, not 3 rotations by the time you are back where you started.</p>

<p>(If you define a rotation [incorrectly, I think] as having been completed when the red dot again touches the larger circle, then you would have 3 such 'rotations' before you return to your starting point.)</p>

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<p>Christul, there are many SAT problems that have wrong proposed wrong answers. They are, however, found in the various books published by Kaplan, PR, and others abysmal SAT questions writers. </p>

<p>As far as I recall, when it comes to questions written by ETS, there have been a few cases of ambiguous questions on the PSAT. However, the only error documented on the SAT occured in 1982. Well before our time, but probably in your teacher's and Optimizer's prime time! </p>

<p>Here's a newspaper article from 1982 that discusses the same problem. </p>

<p>From the Washington Post, May 25, 1982:
College Board's Math Proved Wrong </p>

<p>Math scores are being recalculated for 300,000 students who took Scholastic Aptitude Tests across the nation May 1 because three students proved that the correct answer to one of the questions was not among the possible choices on student's answer sheets, the College Board said today. </p>

<p>"The problem came to light Friday and today we sent Mailgrams to 3,000 colleges advising them recalculated scores would go out to them within the next 10 days," said Barrie Kelly, the board's executive director of communication. </p>

<p>Daniel B. Taylor, executive vice president for operations, said as a result of the flawed question, he anticipates adjustments 10 points up or down on the tests. </p>

<p>The disputed math question shows a large circle, B, and to the left of it, a small circle, A - touching B. </p>

<p>"In the figure above," the problem states, "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?" </p>

<p>The choices given:
(a) 3/2
(b) 3
(c) 6
(d) 9/2
(e) 9 </p>

<p>"The answer to this question should have been 4, not 3," Kelly said. </p>

<p>This explanation, proving the students right, was given: </p>

<p>"The circumference of the large circle is three times the circumference of the small circle. If the small circle were to rotate among [along] a straight line segment equal in length to the circumference of the large circle, it would make three revolutions. </p>

<p>"So, the intended answer to this problem was choice (b) 3. However, the motion of the small circle is not in a straight line, but rather around a large circle. This revolving action around the large circle contributes an extra revolution as circle A rolls around circle B. Thus, the answer to this question should have been 4, not 3." </p>

<p>Now, let's look at a brain teaser that could help you get free beer through bets: Rolling one quarter around another.</p>

<p>This involves rolling one coin around another. What makes this such a goodie, of course, is that the correct answer is very counter-intuitive; plus it seems that the puzzlers who pose it are themselves not aware of all the interesting ramifications. </p>

<p>Do this: Lay 2 quarters on a flat surface so that their edges are in contact. Hold one still and roll the other one around it. How many rotations does the moving quarter make? </p>

<p>Just to make sure we're on the same wave-length and there are no quibbles, understand that there is no slippage as the rolling quarter moves; no sliding or skidding, like a well-behaved tire on a dry road. By "rotations", we mean, how many time does George Washington's head spin around by the time the quarter returns to its starting point? </p>

<p>There, do you have the answer to our "2 quarters" problem now? The moving quarter actually makes two rotations in its single trip around the stationary quarter. </p>

<p>Perhaps the above explanation from the College Board didn't make this clear. For a start, they confuse matters by their sloppy use of the word "revolve". They use it to refer to both revolution and rotation. We will not be so careless. Just remember that the earth revolves about the sun, and the earth rotates about its axis. In the college board problem, circle A rotates 4 times about its axis while it revolves 1 time around circle B. </p>

<p>The College Board's great "communicator" may also have given the impression that this unexpected extra rotation comes about because the small circle moves "around a large circle". "Largeness" has nothing to do with it. Whether the stationary circle has the diameter of a pinhead, or a button, or the equator, or the entire galaxy, the analysis is exactly the same: the rolling circle always picks up one extra rotation for each complete revolution. You calculate how many rotations it would make if the path were a straight line - and then add 1. </p>

<p>Thus it is with our "2 quarters" version of the problem. The intuitive answer (that is, the "straight line" answer) is 1. Add 1 to that to get the "circular motion" answer, 2. Pull out 2 quarters and give it a whirl - it can be quite startling the first time you see how George's head spins.</p>