<p>I’ll go top to bottom</p>
<p>1) Just know that all the expressions are supposed to be integers in consecutive order, after that, it’s all about plugging in your own numbers.</p>
<p>So, say we want the order to be 1, 2, 3, 4 (we won’t even need the 4, because by the 3rd expression we should be able to do w * y, and we’ll even have x). Plug in some random numbers for v and w to make them equal to 4 so that we can do 4/4 for it to equal 1. Let’s choose v = 1, w = 3. Simple. Now, for the next one. Make w + x, aka 3 + x should equal to 8 so we can divide it by 4 to get 2 (still remember, we do this because of consecutive order). Obviously, x is 5. Now, for the next one, y must be 7 to get 12/4 = 3. </p>
<p>We have our x, w, and y; those are all we need. w = 3, x = 5, y = 7.</p>
<p>wy = 21
(x^2) - 4 = (5^2) - 4 = 21
Answer is E.</p>
<p>2) So I don’t know any cool tricks to answer this, but, when I was answering it, I found a pattern:</p>
<p>Starting from 3, if you divide 90/3 you get 30. If you minus it by 1, to get 29, you can add 29, 30, and 31 to get 90. So, it works for 3. Now the pattern starts here: If you divide 90 by 4, you get 22.5, and if you subtract by 1.5 (up .5 from the 1 from before), you get 21, and if you add 21, 22, 23, 24, you get 90. </p>
<p>If you do the same thing for 5, 90/5 = 18, and increase 1.5 to 2 and then subtract by it, you get 16. Add 16, 17, 18, 19, 20, and you get 90. Everything breaks down for 6, and that’s why 6 is the answer. If you think that doesn’t make any sense, go check it for 9: increasing by .5 up to 9, you get 4. 90/9 = 10, minus that by 4 and you get 6. 6+7+9+10+11+12+13+14 = 90, so…yea</p>
<p>3) Well, for this, you’ve got to do circumference measurements. Don’t think about just 1 clock, but rather make 2 circles, 1 which has point A on it, and one with point B. Basically, the circle with point B (the hour hand one) will have half a radius.</p>
<p>2pr = C. If you want, put some random number for R for point A, and divide it by 2 for Point B, but that isn’t exactly necessary. Instead, for point B, just divide 2pr/2, to get just pr. The actual key thing here is to think about the distance they’d travel. To make it easy for ourselves, let’s just say in 3 hours the minute hand will travel 1/4 of the clock, and, in 9 hours, the hour hand will travel 3/4. If you basically do 1/4 * 2pr (for the minute hand), you get .5pr. If you do 3/4 * pr, you get .75pr. If you do the ratios, you can bring .75 up to 3, and .5 up to 2, making it 2:3.</p>
<p>Ok, it may seem like a lot of words, but, when actually doing the calculations and thinking of it, it’ll go MUCH faster. Also, I’d like someone else to check my work.</p>
<p>4) Since all of their legs are exactly the same lengths (they’re all diagonals of the faces), all of them must be the same angles, aka 60.</p>
<p>5) First, figure out how many minutes are in 7.5 hours. It’s 450 minutes. Now figure out what 1 painter can do in 450 minutes. Well, 90 minutes (which is how much it takes 1 painter to do 2 rooms) goes into 450 five times, so he can do 10 rooms (2 rooms*5) in 450 minutes. In that case, using the same reasoning, we just need 4 more painters to do 50 rooms (4 more painters can do a total of 40 rooms, and add that with the original, and we get 50).</p>
<p>6) Ok, so I might be wrong here, but I’ll say what I thought of so far:</p>
<p>If g(k) = g(2k-3), you can think of g(2k-3) as g(x), and so g(2k-3) = 2(2k-3)-3, which is also 4k-9.
So, that means g(k) = 4k-9. g(4k) = 4(4k)-9 = 16k-9.
Still, I’m not so sure about it, so please someone check for me.</p>
<p>7) Yes, I think you’re right. 4x3x2x1 = 24</p>
<p>8) You’ve got the right idea, considering the marks you’ve put.</p>
<p>If the 60 side = 9, then 9 = x * sqrt(3), square everything to get 81=3(x^2). Divide by 3 to get 27=x^2, or sqrt(27) = x. Since the 90 side = 2x = 2<em>sqrt(27), do 2x + 2x = 4x AKA 4</em>sqrt(27). If you use a calculator, you can see that 12<em>sqrt(3) is equal to 4</em>sqrt(27). Sorry if this a little messy, but this technique works perfectly fine for me, although I’m sure there’s better.</p>
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