M37-89

Question

A merchant wants to increase the price of an item so that both the percent of increase and the resulting price in cents are positive integers. If the price of the item is $$2.25$, what is the least percent increase the merchant can apply? ($$1 = 100$ cents)

A. $2\%$
B. $4\%$
C. $10\%$
D. $33\%$
E. $100\%$

Show Answer

Official Answer

B

A.  2\% 
B.  4\% 
C.  10\% 
D.  33\% 
E.  100\%

Bunuel · Answer

Official Solution:

A merchant wants to increase the price of an item so that both the percent of increase and the resulting price in cents are positive integers. If the price of the item is $$2.25$, what is the least percent increase the merchant can apply? ($$1 = 100$ cents)

A. $2\%$
B. $4\%$
C. $10\%$
D. $33\%$
E. $100\%$

Show Answer

Official Answer

B

A.  2\% 
B.  4\% 
C.  10\% 
D.  33\% 
E.  100\%

Let the percent increase is  x\% .
 
We need to find the least positive  integer   x  such that  225*\frac{x}{100}  is an integer. This is because the initial price (225 cents) is already an integer, and for the resulting price to also be an integer, the change in price ( 225*\frac{x}{100} ) must be an integer too.
 
 225*\frac{x}{100}=integer ;
 
 x*\frac{9}{4}=integer  
 
The least positive  integer  value of  x  is 4.

Answer: B

Engineer1 · Answer

Bunuel  I do not understand this question.
Does "resulting price in cents" mean "2.25 + x% increase" or just the "x% increase" is an integer? (the question mentions cents, so looks like maybe the percent increase?).
Moreover, can you elaborate on the solution?

I did this way:

2% of 2.25 = 0.045. Therefore, 2.25 + 0.045 not an integer
4% of 2.25 = 0.9. Therefore, 2.25 + 0.09 is also not an integer

And these questions are not saved on my error log? :O

Thanks!

Bunuel · Answer

HERE  you can find alternative explanations. Hope those will help.

Sudhanwa23 · Answer

Hi,
I have a doubt. Since the question says, what is the least percentage increase, shouldn't we be looking for least x such that 225 * (1+x/100) is a positive integer? Someone please clarify.. I feel like I am going wrong somewhere

Bunuel · Answer

In my solution, I used 225 * (x/100), which represents  the amount of the percent increase . Since the initial price, 225, is an integer, ensuring that the amount of the percent increase is an integer will make the resulting price an integer as well. Therefore, for simplicity, I focused directly on 225 * (x/100) to find the least percent increase that results in an integer price.

gmatbd · Answer

I don’t quite agree with the solution. This is a poorly drafted question which isn't clear. What do we want to be the integer: The resulting price or the absolute value of change in price?

Bunuel · Answer

The question is clear and not poorly drafted. It explicitly states:

“...so that both the percent of increase and the resulting price in cents are positive integers.”

That clearly means the  final price , after applying the percent increase, must be a whole number in cents.

That said, since the initial price (225 cents) is already an integer, and the resulting price must also be an integer, the change in price will automatically be an integer too — but that’s just a consequence of the setup, not an extra requirement.

The only thing you're solving for is the smallest integer x such that:

225 * (1 + x/100)  is an integer — which means  225 * (x/100)  must be an integer.

There’s nothing unclear about that. Misreading the question doesn’t make it badly written.

Please read the question and the solution more carefully, and also go through the full discussion. Your doubt has already been addressed earlier in the thread  HERE .

M37-89

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