gmat prep number properties

[resilient]

What is the value of x/yz?


1. x=(y/2) and z=(2x/5)

2. (x/z)=(5/2) and (1/y)=(1/10)



qa is b but I put D. How is statement two insufficient?

[simplyjat]

Answer B means that the first statement is insufficient...

1. x=(y/2) and z=(2x/5)
if you substitute these values in the original expression x/yz
(y/2) / ( y * (2x/5) )
=> (y/2) / ( y * (2(y/2)/5) )
and you are left with one y in the denominator... hence the statement is insufficient...

[lunarpower]

What is the value of x/yz?


1. x=(y/2) and z=(2x/5)

2. (x/z)=(5/2) and (1/y)=(1/10)

2 approaches.

(A) SYMBOLIC: try to manipulate the expressions involving variables to get the expression you want (namely, x/yz). the best way to do this is to rearrange the expressions so that all the variables are on one side and all the numbers are on the other side.
** statement 1: rearrangement gives x/y = 1/2 and z/x = 2/5. but you can't get x/yz from these: multiplying them as is gives z/y, and taking the reciprocal of the latter one (so that x is on top, as desired) gives x^2/yz. so this is insufficient.
** statement 2: just multiplying the equations together, without rearranging anything, gives x/yz = 5/20 = 1/4. no rearrangement! aren't they nice?

(B) PLUG IN NUMBERS:
statement 1:
let x = 5
then y = 10 and z = 2
so x/yz = 1/4

let x = 10
then y = 20 and z = 4
so x/yz = 1/8

INSUFFICIENT
----
statement 2:
y has to be 10, but you can pick any x and z in the ratio 5:2.
anything you plug in will give x/yz = 1/4; try it yourself.

note that plugging in numbers is something that may be time-consuming, so it's essential that you start in on it RIGHT AWAY. you can't hem and haw for a minute and a half and THEN decide to plug in numbers, or else time will really get away from you.
you should also notice that, in statement 1, x is related to both quantities. this means that x is the best choice to pick first, from which both other variables can easily be calculated.

[p111]

What is the value of x/yz?


1. x=(y/2) and z=(2x/5)

2. (x/z)=(5/2) and (1/y)=(1/10)

2 approaches.

(A) SYMBOLIC: try to manipulate the expressions involving variables to get the expression you want (namely, x/yz). the best way to do this is to rearrange the expressions so that all the variables are on one side and all the numbers are on the other side.
** statement 1: rearrangement gives x/y = 1/2 and z/x = 2/5. but you can't get x/yz from these: multiplying them as is gives z/y, and taking the reciprocal of the latter one (so that x is on top, as desired) gives x^2/yz. so this is insufficient.

Sorry for bringing up this old topic,

hi ron/instructors,
I have a doubt :
(In orange above) how can we cancel Y in numerator with Y in denominator, if the problem doesnt states: y IS NOT= 0.
As the cancellation of Y(or for that matter any variable) involves/multiplication or division by Y on both ends.

So far as I understand GMAT has marked statements in many DS Qs as insufficient on this ground stating "we cant divide by a variable unless it is mentioned as non zero"

I observed similar issues in Q57(where m/n = 5/3 has been cross multiplied to 5n =3m) - they ignored the point that if m or n is 0 then the answer B wont be SUFFICIENT.

& Q15(in option B, they have ignored the case of x=0, which if true makes B not sufficient as it also the solution also involves division by a variable )

ARE there any particular scenarios in which we are allowed to divide/multiply variables or transfer them by cross multiplication in an equation.
OR where am i lacking in my basics ?

Please help as this is the basic & I need to know to be correct !!

[aneesh.kg]

What is the value of x/yz?


1. x=(y/2) and z=(2x/5)

2. (x/z)=(5/2) and (1/y)=(1/10)

2 approaches.

(A) SYMBOLIC: try to manipulate the expressions involving variables to get the expression you want (namely, x/yz). the best way to do this is to rearrange the expressions so that all the variables are on one side and all the numbers are on the other side.
** statement 1: rearrangement gives x/y = 1/2 and z/x = 2/5. but you can't get x/yz from these: multiplying them as is gives z/y, and taking the reciprocal of the latter one (so that x is on top, as desired) gives x^2/yz. so this is insufficient.

Sorry for bringing up this old topic,

hi ron/instructors,
I have a doubt :
(In orange above) how can we cancel Y in numerator with Y in denominator, if the problem doesnt states: y IS NOT= 0.
As the cancellation of Y(or for that matter any variable) involves/multiplication or division by Y on both ends.

So far as I understand GMAT has marked statements in many DS Qs as insufficient on this ground stating "we cant divide by a variable unless it is mentioned as non zero"

I observed similar issues in Q57(where m/n = 5/3 has been cross multiplied to 5n =3m) - they ignored the point that if m or n is 0 then the answer B wont be SUFFICIENT.

& Q15(in option B, they have ignored the case of x=0, which if true makes B not sufficient as it also the solution also involves division by a variable )

ARE there any particular scenarios in which we are allowed to divide/multiply variables or transfer them by cross multiplication in an equation.
OR where am i lacking in my basics ?

Please help as this is the basic & I need to know to be correct !!

Hi p111,

Your basics are absolutely fine.

You raise a very valid point and I agree with you.

The question must mention that x, y, z are non-zero. For e.g., x = y = z = 0 is a possible solution from Statement (1). This is not an official question and clearly, the person who made this didn't think it through.

Cheers!

[p111]

Hi Aneesh,
The stated questions are all official OG verbal second edition Questions(from the Data Sufficiency Section).
Thats actually worrying me as I want to be in sync with Gmat test makers OR I might well get stumped during the test.

I think you understand my doubt. The scenarios(x,y,z=0) will make the solution undefined. Hence answer E. (Thats the way i solved, considering all scenarios)
SO why are we not picking these values as well? The statement is meant to be true for all real numbers ~ RIGHT ?
How to decide between option D(here) & option E for such questions in exam.
Seeking lessons from your or other experts experience.

Q15 is question 15 from Data sufficiency & Q57 is question 57.

[p111]

I observed the same thing in below question:
OG-13 - PS - 41.
Is 4x +y = 8^10?
(1) x-y=9
(2) y/x=1/4

OA is D, but again what if x = 0.

How to know when to test for value 0 of a varibale..

I reall need help from Instructors.. ANYONE Please

[lunarpower]

I observed the same thing in below question:
OG-13 - PS - 41.
Is 4x +y = 8^10?
(1) x-y=9
(2) y/x=1/4

OA is D, but again what if x = 0.

remember that all GMAC problems assume that everything is limited to real numbers. so, if any numbers are excluded from the domain of a particular statement or equation, then those numbers don't need to be considered.

the key point here is that division by zero is undefined.
as a result, the equation y/x = 1/4 is NOT the same as the equation x = 4y.
in particulary, the pair (x, y) = (0, 0) is not a solution of y/x = 1/4 (because it would lead to the meaningless equation 0/0 = 1/4), but it is a perfectly good solution of x = 4y.

in the questions you've cited as "problematic" here, just about all of the statements are given as proportions, with a variable in the denominator. as soon as that happens, you know that that variable cannot be 0.

more generally, the takeaway here is that, when you are thinking about which values to consider, you should always think about values that work in the originally given version of an equation.
if the originally given version is y/x = 1/4, then ... well, that's what it is, and so (0, 0) is not a valid solution of the equation. if you cross-multiply the equation to give 4y = x, that doesn't suddenly admit values that are prohibited from the original equation; everything that was off-limits is still off-limits.

for the same reason, if i give you √x = √y, that's not the same thing as x = y; the former only admits non-negative solutions, while any value is ok in the latter.