What's the fastest way to do these math problems?

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1

<p>I got these right, I just want to know if there's a more efficient way to solve them. </p>

<p>**1. **The following points are on line f
(4, 5) (8, a) (12, 9)</p>

<p>The following are on line g
(5, 4) (10, b) (20, 7)</p>

<p>What is a - b?</p>

<p>What I did was notice that a is just the average of 9 and 5 (7) because its x coordinate is the average of 4 and 12.</p>

<p>Getting b was trickier. What I did was say there's some point (15, y) on g. Then you can get a system of equations using the fact that b is the average of 4 and y and that y is the average of b and 7. You can solve the system, get b = 5, so a - b = 7 - 5 = 2.</p>

<p>What's a faster way? I guess you could get the point slope equation for g and then just plug in 10, but that seems too lengthy.</p>

<p>2. If AB = 5 and BC = 6, what could be the length of AC?</p>

<p>I imagined a triangle ABC. AB + BC > AC, so AC < 11. The only answer choice less than 11 is 10, so that's it. </p>

<p>Any other way to do this?</p>

2

<ol>
<li><p>You were on the way to the fast answer! You had already found ‘a’ the fast way, by observing that it had to be the midpoint of 5 and 9 because 8 is the midpoint of 4 and 12. Then you noticed that given the x coordinates 5, 10 and 20, it is useful to think of the point at x = 15. 10 is not the midpoint – it is 1/3 of the the way between 5 and 20. So the y-coordinate will be 1/3 of the way between 4 and 7 – the difference between 4 and 7 is three, one third of that is 1, so add 1 to 4 and get 5. (This kind of reasoning only works for linear functions.)</p></li>
<li><p>Your way is exactly right – can’t think of anything quicker.</p></li>
</ol>

3

<ol>
<li>Is the only method to do it. Known as cauchy-schwarz inequality in general. :)</li>
</ol>

4

<p>for 1. Just equate the slope for points on the same line to get the values of a and b</p>

5

<p>Thanks for both your help guys. Funny how you forget stuff from algebra 1 ><</p>

6

<p>Instead of creating a new thread, I’ll just post more questions here.</p>

<p>3. If (r+t)/(r-t) = 5/2, what is the value of r/t?</p>

<p>The way I did it was cross multiply to get 2(r+t) = 5(r-t), and then solve for r/t. However, there must be a faster way. </p>

<p>**4. ** In the xy plane, the graph of y = -x^2 + 9 intersects line l at (p, 5) and (t, -7). What is the least possible value of the slope of l?</p>

<p>slope of line = 12/(p-t)</p>

<p>From the graph, the possible values of p are -2 and 2. The possible values of t are -4 and 4. Using those values to get possibilities for p-t, we get 2, -6, 6, and -2. The possible slopes are hence 6, -2, 2, and -6, -6 being the smallest.</p>

<p>Is there a better way?</p>

7

<p>3 is pretty quick already. 2r+2t=5r-5t so 7t=3r so r/t=7/3. I don’t really see a faster way. </p>

<p>4 also seems like the fastest way possible. If you were really short on time, you could figure that the least slope (or greatest negative slope) will occur when the x values are closest to each other, hence making -2/-4 and 2/4 the only possibilities that will work, and thus quickly realizing that the slopes of these points will be + or - 6, hence -6. But that is prone to some human errors. When I decide to do a problem a shortcut way (on any math test), I’ll mark it, and if (usually when) I have time at the end, I’ll come back to it and solve out all the math just to cover my bases.</p>

8

<p>Your method for 3 is direct. You might pick up a tiny bit of speed by dividing by t after cross multiplying to get 2(r/t+1) = 5(r/t-1), so 3(r/t) = 7, r/t = 7/3. (Probably no better; perhaps attractive for style points.)</p>

9

<p>Your method for 4 is also direct. </p>

<p>You might sort things out a little more quickly if you realize that y = -x^2 + 9 specifies a concave downward parabola and that the y values of 5 and -7 specify horizontal lines at 5 and -7. The 4 resulting intersections define a trapezoid with the short base on top. Your 4 possible slopes are either side of the trapezoid or either diagonal. With this in mind, the right side should jump out as having the least slope.</p>

<p>You could translate the parabola to place its vertex at (0,0) by subtracting 9 from the parabola and lowering lines by 9 so the intersections would be (p, -4) and (t, -16). The advantage of this method is that we can “read” p and t very quickly as +/- 2 and +/- 4. Since we know we want the right side of the trapezoid slope, delta x will be 2 and delta y will be -12 for slope = -6. (Potential for error here, but perhaps thinking about the problem this way is helpful.)</p>

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<p>

</p>

<p>That’s how I did it on the test, but reviewing my answers I did my other method.</p>

<p>Thanks to all for helping.</p>

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<p>IDK if you’ve done componendo and dividendo in ratios and proportions but:
if a/b = c/d
then
a+b /a-b = c+d / c-d</p>

<p>Using this:</p>

<ol>
<li>If (r+t)/(r-t) = 5/2, what is the value of r/t?</li>
</ol>

<p>apply C &D:
2r/2t = 5+2/5-2 = 7/3</p>

<p>:D</p>

12

<p>For 3 –</p>

<p>You could also just mess around until you find numbers that work…</p>

<p>For example, I tried r=4, s=1 so that r+s=5 but then r-s = 3, not 2.
Then I tried r=3, s=2 so r+s=5 but now r-s=1, not 2. </p>

<p>So now I know to try r=3.5, s=1.5. Sure enough, r+s=5 and r-s=2.</p>

<p>Then you just divide to get r/s=3.5/1.5=7/3 or 2.333</p>

13

<p>^ I tried that, but when I realized they were decimal values I tried another method.</p>

<p>Sorry that this is a lot of problems, but it’ll chase away boredom for a bit :)</p>

<p>5. The numbers 6, 9, 8, 4, and 8 are arranged into a column on the left. The numbers 4, 7, 5, 13, and 6 are arranged into a column on the right. The average of the numbers in each column is k. If 6, 8, and 4 are moved from the left to right column, which of the following combination of numbers can then be moved from the right to the left column so that k remains the average of the numbers in each column?</p>

<p>A. 6, 13
B. 4, 5, 6
C. 4, 7, 5
D. 4, 7, 6
E. 7, 5, 6</p>

<p>I quickly found that the sum of each column was 35. Using the grouping method, I eliminated A and assumed that three numbers had to be moved. The sum of the two numbers remaining in the left column was 17, so the three numbers that were to be moved from right to left had to add up to 18, or E. This only works because the same number of numbers would end up on both sides.</p>

<p>6. In the figure here <a href=“ImageShack - Best place for all of your image hosting and image sharing needs”>ImageShack - Best place for all of your image hosting and image sharing needs; line l is parallel to line m. If v = 2w, which of the following must be equal to q?</p>

<p>A. v + t
B. v - t
C. t
D. 2v
E. s + t</p>

<p>v + w = 180, so 2w + w = 180 and w = 60. Then v = 120, q = 120, t = 60, s = 60. q being 120, we can eliminate v + t (180), v - t (60), t (60), and 2v (240) leaving answer E.</p>

<p>Another quick question: considering that v = 2w, wouldn’t the value of w ALWAYS be 60 (since v + w = 180 = 3w)? </p>

<p>7. How many positive four digit integers have 1 as their first digit and 2 or 5 as their last digit?</p>

<p>A. 144
B. 180
C. 200
D. 300
E. 720</p>

<p>I’ve never been good at these types of problems, so I got stuck. I realize that the list of numbers is all with the form 1 _ _ 5 and 1 _ _ 2 where _ is any number from 0 to 9, but I’m not sure how to find out how many four digit integers have that form.</p>

<p>8. In the figure <a href=“ImageShack - Best place for all of your image hosting and image sharing needs”>ImageShack - Best place for all of your image hosting and image sharing needs; a square with sides of length 6 units is divided into 9 squares. What is the area of the circle (not shown) that passes through the points A, B, C, and D which are the centers of the four corner squares.</p>

<p>The radius of the circle extends from the center of the big square to the center of one of the corner squares, meaning 2 * diagonal of one small square. Diagonal of one small square is 2/sqrt(2), so twice that is 4/sqrt(2) = radius of circle. pi (4/sqrt(2))^2 = 8pi.</p>

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<ol>
<li>How many positive four digit integers have 1 as their first digit and 2 or 5 as their last digit?</li>
</ol>

<p><1> _ _ <2 or 5></p>

<p>the 2 spots in between can be filled in 10<em>10 ways (each 0-9) and multiply that by 2 since we have 2 or 5 in the end.
so 100</em>2 = 200</p>

15

<ol>
<li>The diameter would be twice of the small square’s length (2 units) so the radius would be 2 units. And the area = pi.r^2 = 4pi</li>
</ol>

<hr>

<p>The radius of the circle extends from the center of the big square to the center of one of the corner squares, meaning 2 * diagonal of one small square.*********
It will be 1 not 2… Check again. Two half diagonals are added up.</p>

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<ol>
<li>Yes v=2w will mean w = 60 degrees</li>
</ol>

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<ol>
<li>The numbers 6, 9, 8, 4, and 8 are arranged into a column on the left. The numbers 4, 7, 5, 13, and 6 are arranged into a column on the right. The average of the numbers in each column is k. If 6, 8, and 4 are moved from the left to right column, which of the following combination of numbers can then be moved from the right to the left column so that k remains the average of the numbers in each column?</li>
</ol>

<p>A. 6, 13
B. 4, 5, 6
C. 4, 7, 5
D. 4, 7, 6
E. 7, 5, 6</p>

<p>You need to move three numbers that total the same as 6, 8 and 4.
ie. 18 which can be made by 6, 5, 7 :)</p>

18

<p>

</p>

<p>Yeah, I meant 2 * half diagonal. </p>

<p>But that being so, the radius would then be 2sqrt(2). Area = pi (2sqrt(2))^2 = 8pi. I’m not sure what you did.</p>

19

<p>yes, you’re right. :slight_smile: Sorry. I got too caught by that 2*half statement and forgot what i wrote. 8 pi is the correct answer.</p>

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<p>It’s no problem. Thanks a lot for helping with these problems!</p>