<p>Here is an interesting problem I ran into while reviewing for Math II Subject Sat. I think there is a mistake in the explanation guide; however, before I decide so, I want to get some input from board members ;) Here it goes:</p>
<p>The postal rate for first class mail is 41 cents for the first ounce or portion thereof and 17 cents for each additional ounce or portion thereof up to 3.5 ounce. The cost in cents of first class postage for a letter weighing N ounces (N<=3.5) is
(A) 41 + [N-1]<em>17
(B) [N-41]</em>17
(C) 41 + [N]<em>17
(D) 1 + [N]</em>17
(E) none of the above</p>
<p>Please provide the answer and your explanation. :)</p>
<p>Sorry, I should probably clarify that [N] - is an integer function (the so-called step function), i.e., it equals N if N is an integer and equals the largest integer smaller than N if N is a fraction.</p>
<p>The reason A will not work is that if you try a small fraction, say 0.5 ounce. 41 + [0.5-1]<em>17 = 41 + [-0.5]</em>17= 41 + (-1)+17 = 24 cents, whereas it should have been 41 cents for the first ounce. ;)</p>
<p>Oh, didn’t notice the brackets were square xD thanks!
Yeah, I thought there’d be a problem with values smaller than one…</p>
<p>I think my reasoning with B and D still holds, though, so it’s down to C or E.
Tried 1.2 with C, just randomly, that doesn’t seem to work.
So… E.</p>
<p>The answer is E, but C comes very close; in fact, in real life, C is the function the post office would probably use for the indicated pricing scheme.</p>
<p>Could you specify what is floor() ? Also, the explanation guide of Barrons Math SAT II says the answer is C, but it doesnt work(Anwser C fails when you use values of 1<=N<2, for instance N=1.2 gives you 41+1*17=58 whereas it should be 41). So, I guess I officially found a mistake in a book most highly regarded on this board I am so disappointed because of this mistake since the book seemed perfect thus far (I found mistakes in PR and other SAT books in the past)</p>
<p>P.S. I answered it as E when I was doing the practice test and then went to check the answers. Thats how it all started.</p>
<p>You’re right, of course. For any integer value of N, the function in (C) is wrong. However, for the vast majority of numbers, it is right.</p>
<p>The reason it’s wrong is that it was approached from a “floor” function angle. The floor function maps real numbers to the greatest integer less than the real number; in symbols, floor(x) = , using your notation. The ceiling function maps real numbers to the smallest integer greater than the real number; in symbols using the same notation, ceil(x) = +1.</p>
<p>The function I give is correct… since it is not one of the choices, the correct answer is E, as you indicated. In that respect, you have found a mistake in the book. I would let them know about it.</p>
<p>Do you understand why MY function is correct? The best way to approach problems like these is to construct your own function, verify it is correct, and then see which answer matches it. Testing the given functions out is, generally, a terrible way to go about it.</p>
<p>Yes. I understand what your function is now. The reason I knew pretty fast that none of the answers A-D were correct while taking the test is that I have done a similar problem in the past except that the values of the argument were positive integers greater than one. It was similar to the given problem except I didnt have to mess with step functions like ceil and floor ;)</p>
<p>Btw, speaking of your function, f(N) = 58 + floor(N-1)<em>17. What if we try a 1 ounce letter? f(1)=58 + floor(1-1)</em>17 = 58+floor(0)*17. Isnt floor(0) = 0? Doesnt this give us f(1)=58, which is different from 41? </p>
<p>P.S. I tried graphing the function y=int(x) on my TI 89 and it shows that y(0)=-1 (this fits your function nicely), which kinda baffles me since 0 is an integer. Shouldnt the function be y(0)=0? :D</p>
<p>I just traced that function near the x=0 and found that as x approaches x=0 from the right, the function equals zero. However, as it approaches x=0 from the left (negative numbers), function equals -1. It seems to me that your function works well with the interval (0,1) but it has a problem that I specified above at x=1. Any thoughts? Is my reasoning correct?</p>
<p>P.S. Btw, I am already off the original problem since we decided that E is the right answer. I am now trying to find the proper function!!!</p>
<p>It’s not true that ceil(x) = floor(x) + 1, technically. That is only true when x is not an integer. naturally, when x is an integer, ceil(x) = floor(x). Trust me, my function is right… but the one with the ceil function, not the extension with the floor function. My “floor” version suffers from the same problem as the authors’.</p>
I am so disappointed because of this mistake since the book seemed perfect thus far (I found mistakes in PR and other SAT books in the past)</p>
lol</p>