Robots X, Y, and Z each assemble components at their

Robots X, Y, and Z each assemble components at their respective constant rates. If r(x) is the ratio of robot X's constant rate to robot Z's constant rate and r(y) is the ratio of robot Y's constant rate to robot Z's constant rate, is robot Z's constant rate the greatest of the three?

Let the rates of robots X, Y, and Z be x, y, and z respectively. Given: \(r_x=\frac{x}{z}\) and \(r_y=\frac{y}{z}\). Question is \(z>x\) and \(z>y\)?

(1) \(r_x<r_y\) --> \(\frac{x}{z}<\frac{y}{z}\) --> \(x<y\). Not sufficient.

(2) \(r_y<1\) --> \(\frac{y}{z}<1\) --> \(y<z\). Not sufficient.

(1)+(2) As \(x<y\) and \(y<z\) then \(x<y<z\). Sufficient.

Answer: C.
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Does this equation work when plugging in numbers, opposed to looking at pure variables?

Sure. Lets say the rates are X = 5, Y = 6 and Z = 7.

Y/Z = 6/7 -> as stated in II

X/Y = 5/6 and Y/Z = 6/7 -> 5/6< 6/7 as stated in I


Cheers

Bunuel,

Please help me clarify. The question says "Robots X, Y, and Z each assemble components at their respective constant rates. If r(x) is the ratio of robot X's constant rate to robot Z's constant rate", so if it is rates, why is X's constant rate not 1/X (which is the rate of completing one unit of work, and Z's rate would therefore be 1/Z. Thus r(x) would be 1/X : 1/Z? What am I misunderstanding here?

Because we denoted rates by x , y, and z: let the rates of robots X, Y, and Z be x, y, and z respectively.
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Hello Everyone,
In OG 13, the question uses "rx" and "ry" in the question stem but "r_x (x in suffix)" and "r_y(y in suffix)".
Are these typos?
Or am I supposed to guess that rx and ry of question stem has been converted to r_x and r_y in the two given options?
TIA,

It's a typo. x and y must be indexes in both stem and the statements.
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Thank you for your reply Bunuel.
Also thank you for pointing out the error/typo in Diagnostic Test Q 5 (Cylindrical tank contains 36PI f3 of water...)

At lest for the second question (Cylindrical Tank...) we will never know if its a typo or the guys who had this question in their real GMAT were unfortunate!

Thanks Anyway!

Assume the individual rates be X, Y and Z respectively for the robots.
Given: X/Z = \(r_x\) and Y/Z = \(r_y\)
Required: Is Z the greatest?

Statement 1: \(r_x < r_y\)
Hence X/Z < Y/Z
Or, X < Y - (i)
We do not have any information about Z.
INSUFFICIENT

Statement 2: \(r_y < 1\)
Or Y/Z < 1
Hence Y < Z - (ii)
We do not know anything about X.
INSUFFICIENT

Combining both statements:
From (i) and (ii), we know that
X < Y and Y < Z
Hence X < Y < Z

SUFFICIENT

Correct Option: C

Fist of all we need to remember that rate can never be negative.
this makes the question super easy to deal

(1) \(r_x<r_y\)

meaning

\(\frac{x}{z}<\frac{y}{z}\) {since neither x,y,z are rates and rate cannot be negative; therefore we can remove z from both side it easily without worrying about the sign}

so

\(x<y\)

NO info about z

INSUFFICINET

(2) \(r_y<1\)

meaning\(\frac{y}{z}\) is less than 1

No info about rate of x or rate of z

INSUFFICIENT

merge both statements

\(\frac{x}{z}<\frac{y}{z}<1\)

multiply each term with z

\(z*\frac{x}{z}<z*\frac{y}{z}<z*1\)

x<y<z

Therefore Z is greatest

SUFFICIENT
ANSWER IS C

Hi Bunuel, I prefectly understad your solution, but why I can't reach the same conclusion if I use the notation 1/x, 1/y, 1/z to denote the 3 rates? After all the rate is output/time, therfore it should work also in this way...

Thanks

EDIT: sorry I wrote something stuopid, it works fine also using the notation 1/x etc

Bunuel, why are we not taking the 1/x, 1/y format? shouldnt r(x) be (1/x)/1/z = z/x? and r(y) in the same manner? Thank you.

In the solution I denoted rates by x , y, and z because we need to find the ratio of rates and it makes more sense to do this way. We could denoted the times of robots X, Y, and Z by x , y, and z and in this case, since the rate is reciprocal of time, the rates would be 1/x. 1/y, and 1/z but the way I did is better for this problem.
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I understand Bunuel, thank you. Please review if the following solution is correct too -

If we use 1/x, 1/y and 1/z, we get rx = z/x and ry as z/y. To have the greatest rate, the denominator should be lowest; so z will have to be lower than x and y.

Statement 1 - rx<ry that means z/x<z/y => y<x. Not sufficieent.

Statement 2 - ry<1=> z/y<1 =>z<y => z<y. Not sufficient.

Statement 1+2 - z<y<x - sufficient. C.

We are given that rx = the ratio of robot X’s constant rate to robot Z’s constant rate.

If we let A = the rate of robot X and C = the rate of robot Z, we can say:

A/C = rx

We are also given that ry = the ratio of robot Y’s constant rate to robot Z’s constant rate. If we let B = the rate of robot Y, we can say:

B/C = ry

We need to determine whether C is greater than both A and B.

Statement One Alone:

rx < ry

Statement one tells us that A/C < B/C. We can multiply both sides by C and obtain:

A < B

Thus, the rate of robot X is less than the rate of robot Y. However, we still do not know whether the rate of robot Z is greater than the rate of either robot X or robot Y. Statement one is not sufficient to answer the question.

Statement Two Alone:

ry < 1

Since ry < 1, B/C < 1 or B < C.

Thus, the rate of robot Z is greater than the rate of robot Y. However, we still do not know whether the rate of robot Z is greater than the rate of robot X. Statement two is not sufficient to answer the question.

Statements One and Two Together:

From statements one and two we know that the rate of robot X is less than the rate of robot Y and that the rate of robot Z is greater than the rate of robot Y. Thus, if the rate of robot Y is less than the rate of robot Z, then the rate of robot X must also be less than the rate of robot Z. Therefore, the rate of robot Z must be the greatest.

Answer: C
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