Is there a template for this problem type?

[tomslawsky]

This question type is classified as "operations with real numbers" by the OG quant supplement. In general, it goes something like this...

I am making this problem up, but it is based on one in an official guide...

For all "X", "X*" denotes the least integer greater than or equal to "X". Is Z* not equal to 0?

1) -1<Z<.01
2) (Z + 0.25)* = 1

I look at these problems, (which, I might add seem to be thinly veiled IQ questions) and I see greater than, less than, equal to, absolute value and inequalities all rolled into one and although I know if I don't overload, the question is probably pretty easy. However, my brains logic circuits all fire at once and I get frustrated. Can someone help me with a systematic way to handle these problems, the OG sucks for this explanation. I know if I just blow it off, I'm bound to see one of these on the real exam. Thank you.

[tomslawsky]

removed post and edited above

[rajkapoor]

Tom ,

I might be able to help you with OG response (the response , I must say , may be difficult to understand initally but it has an important takeaway in it )

"is Z* not Equal to 0" makes your question a lil diff from OG questions.

So we are looking for this Z* function not being equal to zero.
Lets try to answer it inversely -
Does Z* equate to 0 for all values?


so Initial rephrasing will be
does Z* = 0 for the given constraints on Z ?

further ,
taking some values of z (which could be anything , an integer or a decimal value)
and put it in this magic machine [z*] (think of Time machine or Steve Urkel's machine ) and see what comes out.
I took hints from the ranges given in the two statements and added other boundary values.

[0.5] = 1
[1]=1
[0] = 0
[-0.5] = 0
[-1.5] = -1

You can actually think through these values and their results pretty quickly.

The following is the most important derivation/rephrasing from this above exercise -
that for Z* = 0, Z must satisfy -1 < z =< 0

any value of Z beyond this range will make the machine churn out values other than 0.

Now with this info , we can go to the statements
i) -1<Z<.01
compare it to our rephrased required range of -1< z =< 0
Z is within our required range but also extends beyond 0 and thus could produce 1 when z > 0 and < 0.1
e.g [0.01 ] = 1

hence Insufficient

ii) [z+0.25]* = 1

means 0< z + 0.25 = <1 (taking cue from the previous such derivation)
-0.25 < z =<0.75
here again the value of z goes beyond our rephrased range of -1 < z =< 0


Hence insufficient


together the range of Z becomes -1 < Z <= 0.75
Again for values between -1 < Z <= 0 , Z* will be 0 , while for 0 < Z < 0.75 , the value will be 1 , hence insufficient.

AnsWer : E

Approach: find the range of Z for which the quesiton asked is satisfied and then go hunting for it in the two statements.[size=85][/size]

[RonPurewal]

1) -1<Z<.01[/quote

first of all, i'll assume that you understand the definition given in the problem. if you don't, be sure to post back and inquire.

with this one, you should just figure out what all the possibilities are. this is a pretty narrow range, so there aren't going to be a ton of possibilities.

takeaway:
IF A PROBLEM HAS ONLY A SMALL RANGE OF POSSIBILITIES, FORGET COMPLICATED THEORY; JUST CONCENTRATE ON EXHAUSTIVELY LISTING POSSIBILITIES.

if z is anywhere between 0 and 0.01 (not inclusive), then z* is 1.
if z is anywhere between -1 (not inclusive) and 0 (inclusive), then z* is 0.
if z = -1, then z* = -1.

we have possibilities that ARE and ARE NOT equal to 0, so this is INSUFFICIENT.

2) (Z + 0.25)* = 1

the key here, as in many other problems involving quantities in parentheses, is to treat the entire collective within parentheses as a unit.

takeaway:
IF THE WHOLE EXPRESSION IS ENCLOSED WITHIN PARENTHESES, AND IS OPERATED ON BY FURTHER FUNCTIONS, TREE THAT EXPRESSION AS A UNIT WHILE YOU WORK WITH THE OTHER FUNCTIONS.

if q* = 1, then we would have 0 < q < 1.
therefore,
(z + 0.25)* = 1 --> 0 < z + 0.25 < 1
--> -0.25 < z < 0.75

NOW enumerate possibilities:
if z is between -0.25 and 0 (inclusive), then z* = 0.
if z is between 0 (not inclusive) and 0.75, then z* = 1.
insufficient.


if you have the two statements together:
you still have numbers between -0.25 and 0 (which will give z* = 0).
you also still have numbers between 0 and 0.01 (which will give z* = 1).
still insufficient.

ans (e)

I look at these problems, (which, I might add seem to be thinly veiled IQ questions)

this is not my experience of what iq questions look like at all, actually, although it's admittedly been many, many years since i last looked at a real iq test.

...but that's off topic

and I see greater than, less than, equal to, absolute value and inequalities all rolled into one and although I know if I don't overload, the question is probably pretty easy. However, my brains logic circuits all fire at once and I get frustrated. Can someone help me with a systematic way to handle these problems

i would just say that you should avoid multitasking -- you should try to do exactly one thing at a time with each of these expressions or equations. hopefully, the above takeaways will help you in that effort.

remember, if you can fit each problem somehow into the takeaway template
"if i see ___________ ON ANOTHER PROBLEM, i should ____________"
that would help ENORMOUSLY in coming up with single steps in otherwise intimidating problems such as this one.