Tricky Math Question + TestMasters Unreliability

Test Preparation SAT Preparation
1

<p>During one of my practice tests, I solved a Student Produced Response question. However, it took me a long time, and I only solved it with trail & error. Please help me find an efficient way to solve it.</p>

<p>Here it is:</p>

<p>The cost of a telephone call using long-distance carrier A is $1.00 for any time up to and including 20 minutes and $0.07 per minute thereafter. The cost using long-distance carrier B is $0.06 per minute for any amount of time. For a call that lasts t minutes, the cost using carrier A is the same as the cost using carrier B. If t is a positive integer greater than 20, what is the value of t? </p>

<p>TestMasters gave me the following solution (which is incorrect--the answer should be 40):</p>

<p>.06t + 1 = .07t
1 = .01t
100 = t</p>

<p>Therefore, 100 is the answer.</p>

<p>What the heck? Where did they pull that from? It's completely off base. Can someone please show me an efficient way of solving this problem? Thanks!</p>

2

<p>

</p>

<p>Carrier A = Carrier B
1 + .07(t - 20) = .06t
1 + 0.7t - 1.4 = 0.6t
0.7t - .4 = 0.6t
.01t = .4
t = 40</p>

<hr>

<p>Let t = 40</p>

<p>Carrier A
1 + .07(40-20)
1 + .07(20)
1 + 1.4
$2.4 <----</p>

<p>Carrier B<br>
.06 * 40
$2.4 <----</p>

<hr>

<p>You are indeed correct. Also if you plugin t = 100 you will find there is a $0.60 discrepancy between Carrier A and Carrier B.</p>

3

<p>Thanks a ton for the thorough response. I’d been struggling with that one for a while. Just wondering, how’d you do on your SAT Math? :)</p>

4

<p>I’ve got another one:</p>

<p>If n is a positive integer and 2^n +
2^n+1 = k, what is 2^n+2 in terms of k?</p>

<p>This one has really been tripping me up & the explanation in TestMasters isn’t cutting it. Help anyone?</p>

5

<p>And another:</p>

<p>On a square gameboard that is divided into n rows of n squares each, k of these squares lie along the boundary of the gameboard. Which of the following is a possible value of k?</p>

<p>A) 10
B) 25
C) 34
D) 42
E) 52</p>

<p>Please help!</p>

6

<p>2^n + 2^(n+1) = 2^n (1 + 2) = k; so 2^n = k/3; and 2^(n+2) = 2^n * 4 = 4<em>k/3.
Verify it with n=0: 1 + 2 = 3 = k; and 2^(n+2) = 4 (which is 4</em>3/3)</p>

<p>For the second problem the answer is 2n + 2 (n-2). Count the squares along the perimeter of the checkerboard. Be sure not to double count. So the answer must be writable as 4n -4 – i.e. a multiple of 4. So 52 = 13*4 is possible.</p>

7

<p>If n is a positive integer and 2^n +
2^n+1 = k, what is 2^n+2 in terms of k?</p>

<p>I assume that 2^n+1 is equal to the number 2 raised to the n+1 power (likewise for 2^n+2).</p>

<p>2^n+2^(n+1)=k</p>

<p>Factor out 2^n from each term on the left side…</p>

<p>2^n*(1+2^1)=k
2^n=k/3</p>

<p>2^(n+2)=
(2^n)<em>(2^2)=
(k/3)</em>(2^2)=
4k/3</p>

<hr>

<p>Lets look at this logically:</p>

<p>Visualize a square with a side length of, say, 5 units. This means that the area of this square is 25 units. Of these 25, how many lie on the edge.</p>

<p>A quick calculation would just be to sum the outer edges, each of 5 units each:
5*4=20</p>

<p>But, we must subtract 4 in order to account for the overlap of the corners, meaning that
4*4, or 16 squares are on the edge.</p>

<p>In order for this to be a square, its total area must be a perfect square. We can write the equation:</p>

<p>(k/4+1)^2=some perfect square</p>

<p>This perfect square must also be an integer, so we can simplify the above by saying that if k divided by 4 is an integer, k will work.</p>

<p>Of these 5 choices, only E, 52, will produce an integer when divided by 4.</p>

8

<p>Q1; See
<a href=“http://talk.collegeconfidential.com/sat-preparation/815197-math-question.html[/url]”>http://talk.collegeconfidential.com/sat-preparation/815197-math-question.html&lt;/a&gt;&lt;/p&gt;

9

<p>

I took the SAT three times: March 2010 - 800M, October 2010 - 800M, and December 2010 - 760M.</p>

<p>

</p>

<p>Total number of squares = n^2 (square gameboard that is divided into n rows)
k = 4n - 4 (4 rows - 2 b.c of overlap)
Let k = {10,25,34,42,52} </p>

<p>


10 = 4n-4
14 = 4n
n = 3.5 -----> needs to be an integer
25 = 4n-4
29 = 4n
n = 7.25 -----> needs to be an integer
34 = 4n-4
38 = 4n
n = 9.5 -----> needs to be an integer
42 = 4n-4
46 = 4n
n = 11.5 -----> needs to be an integer
52 = 4n-4
56 = 4n
n = 14 <----- Answer!

</p>

<p>Answer = E.</p>

10

<p>Thanks all for the great responses. However, I still don’t completely understand the second problem. Is there some exponent rule that I’m missing? Why can 2^n + 2^n+1 = 2^n (1 + 2)?</p>

<p>I hate to ask, but could anyone solve it with step by step explanations? I’m really lost with this one.</p>

11

<p>Oh and is that (1+2) superscripted? Or are the 1 and 2 integers?</p>

12

<p>Here is the step you missed:
You have to notice that 2^(n+1) is the same as 2 x 2^n.</p>

<p>So 2^n + 2^(n+1) can be written as 2^n + 2 x 2^n. Then, you can factor out the 2^n.</p>

<p>You get 2^n x (2 + 1)</p>

<p>and the rest is as written above.</p>

13

<p>

</p>

<p>


Let n = 1.</p>

<p>2^1 + 2^1+1 = k
2 + 4 = k
k = 6</p>

<p>2^n+2 
2^1+2
8</p>

<p>x = ratio</p>

<p>k * X = 8
6x = 8
x = 8/6
x = 4/3

</p>

<p>Adding 1 to a base 2 is the same thing as multiplying by 2. For example 2^2 = 2*2.</p>

14

<p>

I don’t understand why k = 4n


- 4

is used. I assume [4n] is 4 sides of the square to n squares, right? and what is [2 b/c of overlap?]?</p>

15

<p>The 2 was a typo. As later indicated by my work shown I met to type 4n - 4.</p>

<p>N = 1 side.
4n = 4 sides.
-4 because there will be overlap (corners).</p>

<p>A chessboard/checkerboard is 8x8.
4n - 4 = k
4(8) - 4 = k
k = 28</p>

<p><a href=“http://www.woodshayvings.com/GoodCheckerboard.jpg[/url]”>http://www.woodshayvings.com/GoodCheckerboard.jpg&lt;/a&gt;&lt;/p&gt;

<p>As you will see k = 28.</p>