Impossible Math Question from the Blue Book

Test Preparation SAT Preparation
1

<p>Practice Test 1, page 398</p>

<li><p>x f(x)</p>

<p>0 a<br>
1 24
2 b</p></li>
</ol>

<p>the table above shows some values for the function f. if f is a linear function, what is the value of a + b.</p>

<p>A) 24
B) 36
C) 48
D) 72
E) It cannot be determined from the information given.</p>

<p>For this problem, i just plugged in some numbers.</p>

<p>I made f(x) = 4x + 20 which satisfied the given function. And then plug in f(0) and f(2), but the result is not one of the choices.</p>

<p>Can someone elucidate this problem>?</p>

2

<p>Is the answer C?</p>

3

<p>ALright I checked, the answer is C</p>

<p>Use algebra to solve the problem. We know that all 3 points lie on the same line, right. So the slopes of any 2 of the points should be the same. Using the first 2 points, the slope of the line is:</p>

<p>24-a</p>

<h1>1-0</h1>

<p>24-a</p>

<p>Using the second two points, the slope is found to be:
b-24</p>

<h1>2-1</h1>

<p>b-24</p>

<p>Since these 2 expressions for the slope must be equal,</p>

<p>24-a = b-24
or
a+b=48</p>

<p>You might be prompted to choose E, because indeed you cannot find the value of a or b, but they want the value of A PLUS B, so 48, choice C.</p>

<p>I'm guessing the equation of the line would be y=x+24</p>

4

<p>why go through all that trouble when you can see that</p>

<p>0 a
1 24
2 b</p>

<p>1<em>24=24
0</em>24=0
2*24= 48!!</p>

<p>thus, answer is 48</p>

5

<p>sure it is. you just didn't add the numbers correctly.</p>

<p>f(0)=4(0)+20=20
f(2)=4(2)+20=28</p>

<p>20+28=48</p>

6

<p>Linear....</p>

<p>That means </p>

<p>a = 24 - (constant)
b = 24 + constant</p>

<p>a + b = 48</p>

7

<ul>
<li>all the #'s by 24.</li>
</ul>

8

<p>Tell me if i am wrong .</p>

<p>Linear equation means the degree of x is one , such as y= 3x +1 , y=2x-3</p>

<p>also, Linear equation is always a straight line</p>

9

<p><a href="http://talk.collegeconfidential.com/showthread.php?t=134898%5B/url%5D"&gt;http://talk.collegeconfidential.com/showthread.php?t=134898&lt;/a>
Look for
[ul]398 / 12 /[/ul].</p>

10

<p>Linear equations are of the form y = mx+b
hence at 0</p>

<p>y = 0x + b
y = a
b = a
hence, the constant term = a</p>

<p>now, if we find the slope, by finding the slope of b and 1</p>

<p>(y1-y2)/(x1-x2)=(b-24)/1</p>

<p>so, 1(m) + a = 24
m + a = 24
but m (slope) = (b-24)
so b-24 = a
a+b = 24</p>

<p>Maybe I'm dumb but I think it's a logical fallacy to assume that it's just
0<em>24
1</em>24
2*24</p>

<p>You need real math behind it</p>

11

<p>that is a logical fallacy if you think about it just like that</p>

<p>y = 0 * 24 + a = a
y = 1 * 24 + a = 24
y = 2 * 24 + a = b</p>

<p>I simply thought of it this way.</p>

<p>the difference between each term is a constant so adding the preceding term with the next term is twice the middle term since the increments cancel out.</p>

12

<p>Wow. That's very clever. I never would've seen that.</p>

13

<p>the difference between each term is a constant so adding the preceding term with the next term is twice the middle term since the increments cancel out</p>

<p>???</p>

<p>Increments cancel out from where???</p>

14

<p>
[quote]
the difference between each term is a constant so adding the preceding term with the next term is twice the middle term since the increments cancel out</p>

<p>???</p>

<p>Increments cancel out from where???

[/quote]
</p>

<p>What Sephiroth is saying is that, by definition, a linear equation is always moving at a constant rate. Thus, at x=0, the y-value of the function is a constant (call it "c") less than it is at x=1. Similarly, at x=2, the y-value of the function is the same "c" more than it is at x=1. So to find the value of a + b, which are the y-values at 0 and 2 respectively, you can add the value at 0 (24-c) and at 2 (24+c) to get 48. Or, the way he thinks of it, which is even simpler than that, is that the constant less and the constant more (the "increments") cancel out, leaving you with twice the middle number 24. So it's 48.</p>

<p>By using logic instead of algebra, he was able to cut this from 15-20 second problem to a split second problem that can be answered as fast as you can multiply a number by 2.</p>

15

<p>Lol i remember this question. Just think about it this way: a + b must always be equal. Therefore assume a is at (0,0), b is at (1,24), and therefore you see that the slope is 24/1 and that means c is (2,48). a + b = 48. You can also assume that the points are (0,24) (1,24) and (2,24) and you still get 48. Not too hard of a question if you think about it. No need to make up equations for it - just a waste of time in my opinion.</p>