<p>How many times between midnight and noon of the same day will the minute hand and the hour hand of a clock form a right angle?</p>
<p>Note:
This was question number 25. (high difficulty)</p>
<p>The answer was...
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22</p>
<p>I got 24. I don't know how they got 22. Someone explain this or tell me the book is wrong please :)</p>
<p>Well, there are two sets of such 'right-angle' times, one where the hour hand is first, and one where the minute hand is first.</p>
<p>The first set is (approx) 12:16, 1:21, ...., 11:11 and there are 12 of these.
The second set is 12:49, 1:54, ..., 11:44, and there are 12 of these.</p>
<p>So 24 looks like the right answer to me, unless there is an overlap in some of these times. You may want to write them out explicitly, to be sure.</p>
<p>You can use the method of differentiation to solve this problem.</p>
<p>The initial position of both hands are pointing to 12.</p>
<p>Let this position be the original position, i.e. 0.</p>
<p>rate of change of the minute hand = pi/30 per minute
rate of change of the hour hand = pi/360 per minute</p>
<p>when the angle between two hands are 90 degrees,
(n * rate of change of the minute hand in 1 min) - (n * rate of change of the hour hand in 1 min) = x * 90 degrees, where x can be any odd number.</p>
<p>for x = 1,</p>
<p>n = 360/22 = 16.363636</p>
<p>now divide 60*12 (i.e. 720) by 22</p>
<p>getting 32.7373737373 i.e. 360/11</p>
<p>Let n be any integer,</p>
<p>Then you can set an equation like this:</p>
<p>360/22 + 360/11n < 720</p>
<p>by solving this equation, n must be smaller than 23.</p>
<p>therefore, the answer is 22.</p>
<hr>
<p>by the way, I never imagine SAT I can set such question on differentiation.</p>
<p>Remember: SAT I math is very easy, it just making me laugh at it. A complete waste of human brain. Some cheaters might adding it by approximation without formulas and get the answer. CHEATERS! It's simply a test of IQ, not math abilities.</p>
<p>Anyway, my last SAT I math was 790 while my verbal was 480. I used to good at math but very poor in verbal expressions. Yes, I'm shamed to be a White living in Maine! And now? I'm studying pure math by myself because no good universities would admit me now, but I'm just waiting, and working hard in maths, and trying to solve hard problems for 2 years. and trying to get better result in some English exams.</p>
<p>I was careless - Neravo is right. You can verify by writing out the times, though that's tedious. There's only one '90 degree' position in 2:xx and 8:xx, all the other hours have two such positions.</p>
<p>In hindsight, a quicker way would have been to realize this beforehand. All hours have two '90 degree' positions, except for 2:xx (since the 'minute hand after' position is at 3:00) and 8:xx (since the 'minute hand before' position is at 9:00). So (12)(2) - (2 exceptions) = 24 -2 = 22.</p>
<p>Alternatively, using Neravo's logic, and re-writing his/her answer slightly:
If n is #minutes after 12:00 am, then
n(2pi/60) - n(2pi/720) = x(pi/4)
which works out to x = n(11/360)</p>
<p>If we now plug in any value for n, it shows the #times we've achieved the '90 degrees position' during that span of n minutes. There are 720 minutes between 12:00am and 12:00 pm, so</p>
<p>x = 720(11/360) = 22</p>
<p>thanks optimizerdad and neravo (good luck on your future endeavours)</p>
<p>Alright...I got this using calc also...but does anyone else have trouble believing that this was on an SAT1?</p>