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<p>Are there any general approach for this sort of questions. Range of x and y hath indeed become my obsession in SAT, and I usually give this up.</p>
<p>I'm imagine this like the following procedure:</p>
<p>_ _</p>
<p>Since they said 4 <= x <= 10, then there is a range of 6 value which x can receive</p>
<p>Since they said 4 < y < 10, then there is a range of 4 value which y can receive.</p>
<p>But then I get stuck...</p>
<p>You're off by just a little bit on those.</p>
<p>For the value of x, it says that x must be even--this means that x can be 4, 6, 8, or 10, for a total of 4 values.</p>
<p>For the value of y, it says 4 < y < 10--so y can be 5, 6, 7, 8, or 9, a total of 5 values.</p>
<p>So, you have that x can be any of four values and y can be any of five values. Since the value of x is completely independent of the value of y, the number of different values of the pair (x, y) is just the number of possibilities of x times the number of possibilities for y, or 4*5 = 20.</p>
<p>OK, close enough if I try to read the problem more careful.</p>
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<p>With this one, I only have to find the mid-point of the two points given, is that correct.</p>
<p>That's correct, because parabolas are symmetric with respect to the vertical line which contains the vertex point (in other words, the left side of a parabola is a mirror-image of the right side).</p>
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<p>With these sort of questions, I get a bit confused.</p>
<p>s = -2.5 (it seems so)
t = -0.75
u = 0.75
v = 2.5</p>
<p>I also notice that from s to v, it is exactly symmetrical</p>
<p>But when it comes to to choosing these options, I am dazzled...</p>
<p>I would place t around -0.25 instead of -0.75 (it's closer to zero than to -1), for a better approximation. But to be completely honest, you could use s = -2.5, t = -0.5, u = 0.5, and v = 2.5 and still get this right--approximation questions on the SAT generally have a pretty generous margin of error to leave some room for more coarse estimations (although, of course, higher precision is preferable if you have the time to figure things out).</p>
<p>Those options are absolute value expressions; all you need to do is plug in your values for each one, and odds are, one of them will be considerably larger.</p>
<p>(for ease of notation, abs(x) means the absolute value of x)</p>
<p>A) abs(-2.5 + (-0.25)) = abs(-2.75) = 2.75
B) abs(-2.5 + 2.5) = abs(0) = 0
C) abs(-2.5 - (-0.25)) = abs(-2.25) = 2.25
D) abs(-2.5 - 2.5) = abs(-5) = 5
E) abs(-2.5 + 0.75) = abs(-1.75) = 1.75</p>
<p>So answer choice D is larger than the rest.</p>
<p>How long this could take us, Scythian?</p>
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<p>Since this is a grid in problem, there could be many possible answer right?
So what I have to do is just plug in any value of x (8 to 15) and then solve the problem, is that correct?</p>
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How long this could take us, Scythian?
[/quote]
I suppose you could try to get around evaluating all five answer choices with a bit of clever reasoning, if time is a significant concern of yours. After seeing that all of the answer choices involve the absolute value function, you could determine that the greatest value of the function should involve the points with the greatest magnitude (that is, s and v), and that they should be subtracted from each other because they have opposite signs (adding them would result in a smaller value). Through that logic, you can arrive at answer choice D without many calculations.</p>
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So what I have to do is just plug in any value of x (8 to 15) and then solve the problem, is that correct?
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You mean any value of y (the question says 7 < y < 16)? That's one method of working it out. A faster way would be to realize that since 2/5 is a reduced fraction, the only fractions that can be equivalent to it are fractions of the form (2k)/(5k), where k is any positive integer (in other words, you have to multiply both the numerator and the denominator by the same value). Using this method, you can see that 2/5 = 4/10 = 6/15 = 8/20, and then you can stop there, because now the value of y is too large. The two solutions that work, then, are 4/10 and 6/15, so x can equal 4 or 6, for a total of two solutions.</p>
<p>Thanks, my eyes dash with balloon already, nothing to be done ha ha, thanks again.</p>