M27-11

Question

Bonnie can paint a stolen car in  x  hours, and Clyde can paint the same car in  y  hours. They start working simultaneously and independently at their respective constant rates at 9:45 am. If both  x  and  y  are odd integers, does  x  equal  y ?

(1)  x^2+y^2 \lt 12 
 
(2) Bonnie and Clyde finish painting the car together at 10:30 am

Bunuel · Accepted Answer

Official Solution:

Bonnie can paint a stolen car in  x  hours, and Clyde can paint the same car in  y  hours. They start working simultaneously and independently at their respective constant rates at 9:45 am. If both  x  and  y  are odd integers, does  x  equal  y ? 
 
When Bonnie and Clyde work together, they complete the painting of the car in  \frac{xy}{x+y}  hours (the sum of the rates equals the combined rate, which is the reciprocal of the total time:  \frac{1}{x}+\frac{1}{y}=\frac{1}{T} . From this, we can deduce that  T=\frac{xy}{x+y} ). Now, if  x=y , then the total time would be:  \frac{x^2}{2x}=\frac{x}{2} . Since  x  is odd, this time would be  \frac{odd}{2} : 0.5 hours, 1.5 hours, 2.5 hours, ...
 
(1)  x^2+y^2 \lt 12 . 
 
It's possible that  x  and  y  are odd and equal to each other if  x=y=1 , but it's also possible that  x=1  and  y=3  (or vice versa). This information is not sufficient.
 
(2) Bonnie and Clyde finish painting the car together at 10:30 am. 
 
They complete the job in  \frac{3}{4}  of an hour (45 minutes). Since this time is not  \frac{odd}{2} ,  x  and  y  cannot be equal. This information is sufficient.

Answer: B

Bunuel · Answer

"Work simultaneously and independently" means that the they work on unique parts of the job (in this case, if one person paints the left side of the car, the other person WON'T ALSO paint the left side of the car). This is important to the math because the idea is that the entire car will be painted, but it will be done in the fastest way possible (with no duplicated work). It's the type of "legalese" that has to be included in the question, otherwise the correct answer could be debated.

shasadou · Answer

"simultaneously" is fine but "independently" has confused me - why should we imply that they work on one and the same car but not on 2 identical cars simultaneously and independently?

the 1st para only sets the individual rates.

arhumsid · Answer

Given:
1. Bonnie takes x hrs to do a job
2. Clyde takes y hrs to do the same job
3. Both start independently at the same time- 9:45am
4. x and y are odd integers

to find: x=y ?

S1:  x^2+y^2 \lt 12 
Since we already know that x and y are Odd integers, there's not much left to check for in the above condition and since adding two squares will very soon pass such a small number as 12, we can do the manual work here-
x=1; y=1 => Yes (both equal and sum of squares less than 12)
x=1; y=3 => No (both unequal and still the sum of squares less than 12)

Hence NS

S2:  Bonnie and Clyde complete the painting of the car at 10:30am 
Quite straightforward.
Basically, they started at the same time and ended at the same time. Therefore, we can certainly infer that the time they took was same-  x=y 
Hence S

B

SeregaP · Answer

I approached the question this way:
1) clear

2)
Rate of Bonnie is 1/b, rate of Clyde 1/c. They work together at rate (b+c)/bc. And to paint a car it will take bc/(b+c) hours
as we know that they work for 3/4 hours, we can say bc/(b+c)=3/4. As we know b,c are odd integers, we can infer that b and c are 3 and 1, hence not the same

ulysses02 · Answer

Perhaps another approach is via inversion: set x=y and check if question stem constraint that x and y are ODD integers holds.

(1) 
Insufficient since x and y could be either same or different odd integers

(2) 
T = xy/(y+x) = 3/4 hour

Let x=y=2n+1, i.e. representing an odd integer 
Thus we have:
 
(2n+1)^2 /   = 3/4
-- cross multiply, reduce... -- 
16n^2 + 16n + 4 = 12n + 6
8n^2 + 2n - 1 = 0 
(8n + 4)(n - 1/4) = 0 
n1 = -1/2
n2 = 1/4

Plugging n1 & n2 back into x=y=2n+1 gives us x = 0 hours (invalid) or x = 1.5 hours (valid)
Therefore, if x=y, then x and y cannot both be odd integers according statement 2's parameters 
(said another way, if x and y are both odd integers, then x cannot = y)
=> Sufficient

kishor1 · Answer

Explanation of part B:

Given that both completed the work in 3/4 hours, we can write it as
1/x+1/y=1/(3/4)............(A)

Lets assume x=y ...........(B)

Using (A) and (B) we get x=3/2

However, from the question stem, x is an odd integer.

Therefore, our assumtion x=y is incorrect.

Correct answer B.

JIAA · Answer

Does " working simultaneously and independently " means working on the same job but at different parts of the job??

Also, per S2, we know both Bonnie and Clyde took 45 mins (3/4 of an hour) to complete the job.
So, x = y

Then, how come x is not equal to y??
Is it because clearly 3/4(0.75) is not an integer? 
And its stated in the Q stem that x and y are odd integers??

Can you please elaborate on this?
Thanks

Bunuel · Answer

1. Working simultaneously and independently, means that they work on different parts of the job. For example, one is painting the front and another the back of the car. This ensure that their works does not overlap and they do the job in fastest way possible.

2. When analysing the stem we got that  IF   x=y , then the total time would be:  \frac{odd}{2} : 0.5 hours, 1.5 hours, 2.5 hours, ... From (2) we got that they complete the job in  \frac{3}{4}  of an hour (45 minutes). Is  \frac{3}{4}  of an hour (45 minutes),  \frac{odd}{2}  hours: 0.5 hours, 1.5 hours, 2.5 hours, ... ? No, thus x does not equal y.

donaldmcdonald · Answer

I think the wording is bad.

In Quant every word should be crystal clear. Since we have a Verbal section full dedicated on "understanding", a Quant question (especially if it is kind of a Word Problem) should not mean different for each person. It should not test you on understanding the MEANING of a WORD, but on understanding the LOGIC BEHIND a WORD.

When you write, "simultaneosly and independently" I understood they paint separate cars. I saw the explanation above, but still, I believe in an Official question the wording would be much clear.

Bunuel · Answer

The wording is fine. Check Official Guide question: https://gmatclub.com/forum/three-machin ... 43489.html Three machines, K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. How long does it take Machine K, working alone at its constant rate, to complete the task? (1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes. (2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.

Gmat98995 · Answer

Can someone explain the second statement please? they completed the job in 3/4 hours this is understood. What does this mean since it's not odd/2 then x and y are not equal. ? Bunuel please help thanks

Bunuel · Answer

From the stem we concluded that IF x = y, then the total time would be (x hours)/2 = (odd hours)/2 (since given that x is odd). (odd hours)/2 means the total time would be 0.5 hours, 1.5 hours, 2.5 hours, ... (2) says that the total time is 3/4 hours. 3/4 hours is NOT in the form of (odd hours)/2, so x does not equal to y.

Hope it's clear.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Aakash7 · Answer

Need to find whether x=y ?

Since x and y are odd integers, So the pair (x,y) could be (1,1), (3,1), (1,3), (3,3) and so on..

S1: x^2 + y^2 < 12
All of these cases: (1,1), (3,1), (1,3) are possible. Both x=y and x != y are possible. So not sufficient

S2: They finish in 45 mins together
i.e., xy/(x+y) = 45/60 (in hours) => xy/(x+y) = 3/4
Only possible for these cases: (3,1), (1,3)
Since for both cases x != y
therefore, Sufficient

Answer is B

LUBABAYIMER · Answer

Explanation of part B:

Given that both completed the work in 3/4 hours, we can write it as
1/x+1/y=1/(3/4)............(A)

Lets assume x=y ...........(B)

Using (A) and (B) we get x=3/2

However, from the question stem, x is an odd integer.

Therefore, our assumtion x=y is incorrect.

Correct answer B.

Nipunh · Answer

I think this is a  high-quality  question and I agree with explanation.

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