Math Problems

Test Preparation SAT Preparation
1

<p>1- At north High School, the ski club has 15 members and the debate club has 12 members. If a total of 11 students belong to only one of the two clubs, how many students belong to both clubs?
(A) 2
(B) 7
(C) 8
(D) 12
(E) 16</p>

<p>I understand the you have to divide 16 by 2 but why? In other words, when do I know whether I have to divide a number by 2 or not?</p>

<p>2- 6x + 3 >= a
The inequality above is true for the constant a, which of the following could be a value of x?</p>

<p>(A) a/6
(B) a/6 -1
(C) a/6 -3
(D) a-4/6
(E) a-5/6</p>

<p>I'll be grateful for your help :)</p>

2

<ol>
<li>It has to do with the fact that you’re counting the members in both clubs twice, so you have to divide by 2.</li>
</ol>

<p>For example, suppose that x students are in both clubs. Then 15-x students are only in the ski club, and 12-x students are only in the debate club. Therefore</p>

<p>(15-x) + (12-x) = 11
x = 8</p>

<p>So 8 students are in both clubs.</p>

<ol>
<li>Rearranging gives
5-a >= 6x
x <= (5-a)/6</li>
</ol>

<p>I can’t really interpret choices D or E, is D supposed to be a - (4/6) or (a-4)/6? Either way it doesn’t seem like any of the answer choices will result in a true statement for all a.</p>

3

<p>@rspence, I got it, thanks for clarifying. As for the second question, sorry for not making the choices clear enough. It’s in the form of (a-4)/6. The answer, however, is A. So how?</p>

4

<p>Choice A is not true for all a. If you substitute x = a/6 you get</p>

<p>2 - 6(a/6) + 3 >= a
5 - a >= a</p>

<p>which is not true for all a (it only holds when a <= 5/2). Of course, A <em>could</em> work if a <= 5/2, but by the same logic, all the other answer choices could work as well.</p>

5

<p>Plug in (a/6) for x. This simplifies to this: </p>

<p>6(a/6) + 3 >= a, which simplifies to:</p>

<p>a + 3 >= a. </p>

<p>Since a is a constant (a number), you know that if you add 3 to any number, the sum has to be greater than the number you started with. Exactly 3 greater. That’s why (A) is the correct answer. </p>

<p>If you go through and plug choices (B) through (E) in, you’ll get an inequality that isn’t true for ALL numbers a. It might be true if a is negative or if a is positive, but choice (A) is the only choice that yields a true inequality for ALL numbers a.</p>

6

<p>Oh wow, I think I misread and interpreted it as</p>

<p>2 - 6x + 3 >= a, instead of</p>

<p>6x + 3 >= a</p>

<p>Assuming the latter is true, A is definitely the correct answer.</p>

7

<p>@314159265, Thank you so much, so it’s a matter of plugging in choices for x. I don’t know why I missed that! </p>

<p>@rspence, lol that makes sense now.</p>

8

<p>Well, if you really wanted to, you could also solve it algebraically:</p>

<p>6x >= a - 3
x >= (a/6) - 1/2</p>

<p>Of course, a/6 is greater than (a/6) - 1/2, so we’re done.</p>