M14-16

Question

A stationery store sells pens and pencils. If the price of each item in cents has a positive integer value, how much do 5 pencils cost?

(1) 5 pencils and 3 pens cost 30 cents.
 
(2) 4 pencils and 4 pens cost 32 cents.

Bunuel · Accepted Answer

Official Solution:

A stationery store sells pens and pencils. If the price of each item in cents has a positive integer value, how much do 5 pencils cost? 
 
(1) 5 pencils and 3 pens cost 30 cents.
 
This implies  5x+3y=30 . This equation has only one positive integer solution:  x=3  and  y=5 , so the price of 5 pencils is  5*3=15  cents. Sufficient.
 
(2) 4 pencils and 4 pens cost 32 cents. 
 
This implies  4x+4y=32 , which gives  x+y=8 . This equation has more than one positive integer solution, for example:  x=1  and  y=7  or  x=3  and  y=5 . Not sufficient.

Answer: A

minwoswoh · Answer

Hi Bunuel,

I think this is a great question. I falled for the trap and picked C here. At a very first glance I read this question as:

- Question stem: 5x = ?
- Statement 1: 5x + 3y = 30. I thought that since I had 1 equation and 2 variables I couldn´t solve for this.
- Statement 2: x + y = 8. Again, 1 equation and 2 variables.

So I thought the answer was C.

I understand that what makes Statement 1 sufficient is that the question stem gives us an "Integer Constraint"  (“…if the price of each item in cents has a positive integer value…”)

But my question is: how will I realize about this in the next similar problem I have and prevent from falling for this trap again? Should I pay close attention to whether the question stem mentions this "Integer Constraint" (such as in the portion of the question mentioned above)?

Bunuel · Answer

This is a C-Trap question. "C trap" question is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer. Check for more here: c-trap-questions-177044.html Hope it helps.

rmadmit · Answer

Very tricky one indeed. One more thing which needs to be highlighted is the fact that there could be another solution to the equation 5x+3y=30 which is x=0 and y=10 but since we are given in the question that the number of pens and pencils are positive integer values, this solution is not possible.

manlog · Answer

IMO you should be always cautious on similar tasks (number of cakes/pencils/animals). They actually follow a similar pattern, so just solve as much of them as possible. And alway pay attention to INTEGER CONSTRAINT.

bailz314 · Answer

How can I get fast to X=3 and Y=5? should I just plug in numbers? is this the best way to do it?

Bunuel · Answer

Yes, testing values and common sense is pretty much the way to solve.

amanvermagmat · Answer

Hi

Let me see if I can help a bit.

we have 5x + 3y = 30 and we know both x and y have to be positive integers. 
5x is obviously a multiple of 5, and on the right side 30 is also a multiple of 5. So, by simple logic, '3y' also must be a multiple of 5.
For '3y' to be a multiple of 5, 'y' must be a multiple of 5.

so we put y=5, and get 5x + 3*5 = 30. This gives us x=3
If we put y=10, we get x =0 which is not possible.

There is another way. 5x + 3y = 30 and we know both x and y have to be positive integers. 
3y is obviously a multiple of 3, and on the right side 30 is also a multiple of 3. So, by simple logic, '5x' also must be a multiple of 3.
For '5x' to be a multiple of 3, 'x' must be a multiple of 3.

so we put x=3, and get 5*3 + 3y = 30. This gives us y=5
If we put x=6, we get y =0 which is not possible

This method sometimes helps.

swethalagisetti · Answer

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

OhMy · Answer

Really interesting question, involving diophantine equations.
for such equations ( ax+by=c ), the first answer has to be found somehow, maybe by guessing.
The other answers can be obtained through
 x=x_0+\frac{b}{d}*n 
 y=y_0-\frac{a}{d}*n 
where  d=GCF(a,b)

Statement (1):   5x+3y=30 
Figure out the first solution through testing. One of the obvious answers would be the point
 (3,5) 
other solutions would be  (3+3n, 5-5n)  -> so we can find all solutions, like with this pattern:
 . 
 . 
 . 
 (9,-5) 
 (6,0) 
 (3,5) 
 (0,10) 
 (-3,15) 
 . 
 . 
 . 
but since the question stem states that only  positive  integers are allowed, only the solution  (3,5)  is valid. (See attached image)
 Sufficient.

Statement (2):   4x+4y=32 
Guess one first solution:  (4,4)  -> and find other solutions. Which would look like this:
 . 
 . 
 . 
 (-1,9) 
 (0,8) 
 (1,7) 
 (2,6) 
 (3,5) 
 (4,4) 
 (5,3) 
 (6,2) 
 (7,1) 
 (8,0) 
 (9,-1) 
 . 
 . 
 . 
Even though only positive integers are allowed, there are still 7 solutions to this equation  (1,7)  to (7,1) .
 Not Sufficient.

Together, they would be clearly sufficient, since 2 equations and 2 variables will give a definitive solution.
However, the GMAT throws this curve ball at you, and disguises that Statement 1 actually has only 1  positive integer solution  and is therefore sufficient on its own.
Hope this helps.  Answer A .

theGisforGorilla · Answer

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

BottomJee · Answer

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

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.

mellowmev · Answer

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

HenryHai · Answer

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

MedhanshMahto · Answer

I like the solution - it’s helpful.

kartik134 · Answer

Do we need to assume that prices of all pens and all pencils is same?

Bunuel · Answer

______________________
Yes.

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