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The positive number a is q percent greater than the positive number b,

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The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 11 May 2015, 07:18
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The positive number a is q percent greater than the positive number b, which is p percent less than a itself. If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?

I. c > a
II. c = a
III. c < a

A) I only
B) II only
C) III only
D) I and II only
E) II and III only

Kudos for a correct solution.

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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 12 May 2015, 23:29
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Bunuel wrote:
The positive number a is q percent greater than the positive number b, which is p percent less than a itself. If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?

I. c > a
II. c = a
III. c < a

A) I only
B) II only
C) III only
D) I and II only
E) II and III only

Kudos for a correct solution.


a = (1 + q%)b
b = (1-p%)a
a = (1 + q%)*(1-p%)a
So (1 + q%)(1 - p%) = 1 = No change
One easy solution to this would be q = 0 and p = 0. In that case, a = b = c. So statement II can hold.

Assuming p and q are not 0,
(1 + q%)(1 - p%) = 1 = No change
So if you increase something by q% (it becomes bigger) and then decrease it by p% (now you will need to decrease it by a lesser %), there is no change.
Such as, you increase 100 by 25% and it becomes 125. Then you decrease 125 by only 20% and it comes back to 100. (This should remind you of cost price, sale price, profit and margin). So q would be a higher percentage than p.
Now if you increase something by p% (the lower %) and then decrease by q% (the higher %), the value you will obtain will certainly be lower than original.
Say you increase a (100) by p% (20%) to get 120 and then decrease it by q% (25%), you will get c (90). Hence c < a is possible too.

Answer (E)
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 11 May 2015, 07:55
Let q = 10% then p = 100/11 %
let b = 100 then a = 110
after increasing a by p and decreasing by q we get c = 108
therefore c<a
C it is
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 11 May 2015, 21:50
Quote:
The positive number a is q percent greater than the positive number b, which is p percent less than a itself.

I assume this means that a > b, and thus q and p are positive.
Because b is positive, p < 100.

1) a = (1+q)b = b+bq
2) b = (1-p)a = a - ap
plug 2 into 1:
a = a - ap + (a - ap)q = a - ap + aq - apq
subtract a from both sides and rearrange the right side:
0 = aq - ap - apq

Quote:
If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?

Because c is a positive number, q < 100.

a*(1+p)*(1-q)=c
(a+ap)*(1-q)=c
a+ap-aq-apq=c

subtract a from both sides (because we want to see if c - a > 0, = 0 or < 0):

ap - aq - apq = c - a

from above:
0 = aq - ap - apq or ap = aq - apq
plug that in:
aq - apq -aq - apq = c-a
aq - aq = 0
-apq - apq = -2apq

so c - a = -2apq, because a, p and q are all positive, c - a = -2 * positive number, which is < 0. This is C.

NOTE: if p and q are allowed to be 0 or negative, then the answer changes. If p and q are allowed to be 0, so a = b, then II can be correct. For example, a = 10, b = 10, p = 0, q = 0.
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 11 May 2015, 22:03
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darthsaini wrote:
Let q = 10% then p = 100/11 %
let b = 100 then a = 110
after increasing a by p and decreasing by q we get c = 108
therefore c<a
C it is


A hint here: the question is - "could be true"
Plugging in values gave you that one of the three statements (c < a) could be true. But what can you say about the other two statements? By taking a single set of values, how can you prove that they cannot be true?
Suggest you to think logically.
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 27 May 2016, 19:24
VeritasPrepKarishma wrote:
Bunuel wrote:
The positive number a is q percent greater than the positive number b, which is p percent less than a itself. If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?

I. c > a
II. c = a
III. c < a

A) I only
B) II only
C) III only
D) I and II only
E) II and III only

Kudos for a correct solution.


a = (1 + q%)b
b = (1-p%)a
a = (1 + q%)*(1-p%)a
So (1 + q%)(1 - p%) = 1 = No change
One easy solution to this would be q = 0 and p = 0. In that case, a = b = c. So statement II can hold.

Assuming p and q are not 0,
(1 + q%)(1 - p%) = 1 = No change
So if you increase something by q% (it becomes bigger) and then decrease it by p% (now you will need to decrease it by a lesser %), there is no change.
Such as, you increase 100 by 25% and it becomes 125. Then you decrease 125 by only 20% and it comes back to 100. (This should remind you of cost price, sale price, profit and margin). So q would be a higher percentage than p.
Now if you increase something by p% (the lower %) and then decrease by q% (the higher %), the value you will obtain will certainly be lower than original.
Say you increase a (100) by p% (20%) to get 120 and then decrease it by q% (25%), you will get c (90). Hence c < a is possible too.

Answer (E)


completely stumped
could not think of p=q=0 could be one of solution.
question should mention either of p or q is also positive number.then the ans will be C otherwise its E

thanks karishma..
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 28 Jan 2017, 08:18
VeritasPrepKarishma wrote:
Bunuel wrote:
The positive number a is q percent greater than the positive number b, which is p percent less than a itself. If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?

I. c > a
II. c = a
III. c < a

A) I only
B) II only
C) III only
D) I and II only
E) II and III only

Kudos for a correct solution.


a = (1 + q%)b
b = (1-p%)a
a = (1 + q%)*(1-p%)a
So (1 + q%)(1 - p%) = 1 = No change
One easy solution to this would be q = 0 and p = 0. In that case, a = b = c. So statement II can hold.

Assuming p and q are not 0,
(1 + q%)(1 - p%) = 1 = No change
So if you increase something by q% (it becomes bigger) and then decrease it by p% (now you will need to decrease it by a lesser %), there is no change.
Such as, you increase 100 by 25% and it becomes 125. Then you decrease 125 by only 20% and it comes back to 100. (This should remind you of cost price, sale price, profit and margin). So q would be a higher percentage than p.
Now if you increase something by p% (the lower %) and then decrease by q% (the higher %), the value you will obtain will certainly be lower than original.
Say you increase a (100) by p% (20%) to get 120 and then decrease it by q% (25%), you will get c (90). Hence c < a is possible too.

Answer (E)



Would you please explain how did you get this 'a = (1 + q%)*(1-p%)a' ?
I am in trouble to understand this.
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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New post 29 Jan 2017, 04:49
Hi Bunuel,

Correct answer in the question is C; however the explanation given by VeritasPrepKarishma says that the answer is E. Please clarify which one is correct. If it is C then why we neglected the case when p=0, q=0
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Re: The positive number a is q percent greater than the positive number b,  [#permalink]

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