Jane can paint the wall in J hours, and Bill can paint the

Question

Jane can paint the wall in J hours, and Bill can paint the same wall in B hours. They begin at noon together. If J and B are both even numbers is J=B?

(1) Jane and Bill finish at 4:48 p.m.
(2) (J+B)^2=400

Bunuel · Accepted Answer

Nice solutions atish and sriharimurthy, +1 to both of you.

Though it can be done easier.

Jane can paint the wall in J hours, and Bill can paint the same wall in B hours. They begin at noon together. If J and B are both even numbers is J=B?

Jane and Bill working together will paint the wall in  T=\frac{JB}{J+B}  hours. Now suppose that  J=B  -->  T=\frac{J^2}{2J}=\frac{J}{2} , as  J  and  B  are even  J=2n  -->  T=\frac{2n}{2}=n , as  n  is an integer, working together Jane and Bill will paint the wall in whole number of hours, meaning that in any case  T  must be an integer.

(1) They finish painting in 4 hours and 48 minutes,  T  is not an integer, -->  J  and  B  are not equal. Sufficient.

(2)  J+B=20 , we can even not consider this one, clearly insufficient.  J  and  B  can be  10  and  10  or  12  and  8 .

Answer: A.

sriharimurthy · Answer

Question Stem : In one hour Jane can paint  \frac{1}{J}  of the wall and Bill can paint  \frac{1}{B}  of the wall.
 Together, in one hour, they can paint  \frac{1}{J} + \frac{1}{B}  of the wall.
Starting time : 12:00 p.m.
If J and B are even, can we tell if J = B ?

St. (1) : Finish time : 4:48 p.m.
 This tells us that when they worked together, they finished painting the wall in 288 minutes or  \frac{288}{60}  hours.
Therefore, in one hour, working together, they finished painting  \frac{60}{288}  of the wall.

Therefore we get the equation :  \frac{1}{J} + \frac{1}{B}  =  \frac{60}{288}

This can be reduced to  \frac{J+B}{J*B} = \frac{5}{24}

Now let us assume that A and B are equal.

Therefore,  \frac{J+J}{J*J} = \frac{5}{24}  --->  \frac{2*J}{J^2} = \frac{5}{24}  --->  J = \frac{48}{5}

Thus it is obvious that if J is equal to B, in order to satisfy statement 2, J and B cannot be even. Since we are given that J and B have to be even, it is possible to conclude that J is not equal to B.

Hence this statement is Sufficient.

St. (2) :  (J + B)^2 = 400 
 This one is easy to discard by just plugging in some values.
The statement holds true for J = B = 10 as well as for J = 12 and B = 8.
Since it gives contradicting solutions for the question stem (J=B in case and J#B in the other), this statement is Insufficient.

Answer : A

atish · Answer

If both J & B work they will finish the job in J*B/(J+B) hrs.
Also given both J&B are even.

Lets start with statement B
(J+B)^2 = 400
If J & B were equal then J=B=10
They will finish the work in 5 hrs i.e 5 PM, statement A is different hence we know J & B are not equal. Sufficient but before we jump to C lets look at statement A alone

J & B finish the job by 4:48p.m they take 4 48/60 hrs = 4 4/5 = 24/5

So we know J*B/(J+B) = 24/5
since J and B are even they can't be fractions, hence 
J*B has to be a multiple of 24 
J+B has to be a multiple of 5
since J & B are even J+B can only be multiples of 10.
Lets assume J=B and start plugging in values
when J*B/(J+B) = 48/10
J+B = 10, J=B=5 so J*B = 25 too small
when J*B/(J+B) = 96/20
J+B = 10, J=B=10 so J*B = 100 too big
when J*B/(J+B) = 144/30
J+B = 30, J=B=15 so J*B = 225 too big

J*B values keep increasing and will never equal the value derived by assuming j&B are equal.

Hence answer should be A.

atish · Answer

very elegant solution Bunuel. And thanks for the awesome question. +1

kartik222 · Answer

@bunel,

I have a question regarding option a) 
4hr 48 mins if you look at it from hours prespective then yes, T is not an integer BUT:
4 hr 48 mins == 288 mins. Now, 288 is a integer.

thoughts?

I feel the question is not clear on this!

-Kartik

Bunuel · Answer

J and B are given in hours and we are told that both are even numbers, so the number of  hours  they need together to paint (T) must be even number too.

Hope it's clear.

RyanCJW · Answer

(1) They're going to finish at 4:48, I broke this down into a fraction so I could try to make the even numbers match. The ratio that could hit this is in two minute intervals, you'll hit the hour but also the 48 minute mark on the final hour).

For each hour you have 30 units - so 144 units altogether (30 for the 12:00 hour, 30 for 1:00, 30 for 2:00, 30 for 3:00 and 24 for the 48 minutes in the 4:00 hour - each two minute span in the time period). This means that together they're painting 30/144 wall per hour. 30/2 is 15 - therefore these two numbers are not equal if J and B are even numbers.

A is sufficient.

(2) (J+B)^2=400 
J+B= 20

There aren't any limiting parameters here and no finish time. If you decide to try it anyway - make it fit the rules of the problem.

10+10=20

Both parties are working at 1/10 wall per hour. Which means that together they're working at 2/10 wall per hour. It will take them 5 hours to paint the wall.

But since there's nothing that limits you, find another set of even numbers that aren't equal.

16+4=20. One party is working at 1/16 wall per hour, the other is working at 1/4 wall per hour. Together they're working at 5/16 wall per hour.It will take them 3 1/5 hours to paint the wall. These numbers give you an answer as well, this option isn't sufficient enough to give you an answer.

B is insufficient. The answer is (A).

mau5 · Answer

We are supposed to find out whether J=B? Assume that it is. Thus, the question can be interpreted as this : If Jane/Bill working alone take J=B hrs, how much time will they take if they work together? The time taken will be exactly half of what they took individually--> J/2 = H/2. Now as J=H=even=2k, thus the new time taken(in hours) must be an integral value(k). However, from F.S 1, we can see that the hours taken for them is not an an integral value. Thus  J
eq{H} .Sufficient.

From F.S 2, we know that (J+B) = 20. Thus, J=B=10 is a valid solution; and so is J=14,B=6. Insufficient.

A.

sheolokesh · Answer

I have a clarification. Both the stmts J and B are both even numbers and Jane and Bill finish at 4:48 p.m. are given in the question, I mean the whole question, where the 2 information that we need to use are contradicting each other.. ie., lets keep it aside that if J=B.. But the info Jane and Bill finish at 4:48 p.m that we need to use says that J and B are not integers, contradicting J and B are both even numbers. Do we get these kind of self contradicting questions?

EMPOWERgmatRichC · Answer

Hi sheolokesh,

This DS question is a bit more 'layered' than most DS questions. The two Facts do NOT contradict though. Here's why:

From the prompt, we know that J and B are both EVEN INTEGERS. We're asked if J and B are equal. This is a YES/NO question.

Fact 1 tells us that it takes the two people 4 4/5 hours to paint the wall together. There are only 2 possible pairs of even integers that will lead to THAT result:

6 and 24
8 and 12

In both situations, the answer to the question is NO, so Fact 1 is SUFFICIENT.

Fact 2 tells us that B+J = 20

So we could have....
8 and 12 ...and the answer to the question is NO
10 and 10 ...and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Final Answer:  A

GMAT assassins aren't born, they're made,
Rich

lexxus · Answer

Looking at the first statement:

Ra = 1/a=1/(2k) 
Rb=1/b=1/(2n)
convert 4:48 to hours: 4 + 48/60 = 4 + ⅘ = 20/5 + ⅘ = 24/5
24/5 (1/2k + 1/2n) = 1
24/5 (1/2) (1/k + 1/n) = 1
12/5 (k+n)/(kn) = 1
(k+n)/(kn) = 5/12 
k+n=5
kn=12

However, there are no such positive integers k and n. Is the problem ill-defined or I miss something?

EMPOWERgmatRichC · Answer

Hi lexxus,

This DS question is a bit more 'layered' than most DS questions. From the prompt, we know that J and B are both EVEN INTEGERS. We're asked if J and B are equal. This is a YES/NO question.

Fact 1 tells us that it takes the two people 4 4/5 hours to paint the wall together. There are only 2 possible pairs of even integers that will lead to THAT result:

6 and 24
8 and 12

In both situations, the answer to the question is NO, so Fact 1 is SUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

Rishovnits · Answer

Hi Bunuel,

Correct me if I am wrong.

Before jumping into the individual statements, it has been found that "T" must be an integer.
Now in statement 2: 1st case :10 and 10 gives me T as an integer but 2nd case: 12 and 8 {(1/12+1/8)*T =1} does not give T as an integer. So is it not that J and B have to be 10 and 10. Please give your suggestion.

Thank you.

Bunuel · Answer

From the stem we got that T is an integer  IF  J = B, not that T is an integer in all cases.

Sparta_750 · Answer

Bunuel

In statement 2,

(J+B)^2= 400

So, J^2 + B^2 + 2JB = 400

We want J & B to be even and equal, so there are only limited even numbers we have to check i.e

It does not hold true for any even number such that J=B

Thus would not the answer choice become D instead of B?

Bunuel · Answer

(J+B)^2= 400 has many even solutions for J and B. For example,  J  and  B  can be  10  and  10  or  12  and  8 .

Kritisood · Answer

Hi  Bunuel  does this mean that J and B would always be integers since its given that they are even numbers? are even numbers always = even integers?

Bunuel · Answer

______________________________
Only integers can be even or odd...

vaibhav1221 · Answer

The question asks whether a=b. If this is true, then the time they take working together must be exactly half of the time they take working alone.
So, we need to figure out whether the time they take working together is
 \frac{(a+b)}{2}  ....  (1)

Statement A says that they finish the work at 4:48 and according to the question they started at noon. This means that they took 4 hours and 48 minutes to finish the work. Now, if we assume that this is exactly half of the time a and b would've taken to finish the work independently or that a=b, then this statement doesn't hold because a and b are *even integers*. Doubling 4 hours 48 minutes would give us a time that is in fraction hours. This is not consistent with what we are given. So, Option A is sufficient to answer the question and the answer is No;  a≠b .

Statement B says that a+b=20. This alone is insufficient.

If we were to combine information in A and B, then we could assume that a=10, b=10 (again, assuming a=b). Till here, it is consistent with what is given in the question - a and b are even integers. So, if A can finish the work in 10 hours and B can finish the work in 10 hours, then together they would take 5 hours because of  (1) . But Statement A says that they took 4 hours and 48 minutes. Hence,  a≠b .

C is a good trap if you don't think deep enough.

Jane can paint the wall in J hours, and Bill can paint the

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GMAT Data Sufficiency (DS) Questions

Jane can paint the wall in J hours, and Bill can paint the

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