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

Question

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.

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

KarishmaB · Accepted Answer

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)

darthsaini · Answer

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

bluesquare · Answer

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 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.

KarishmaB · Answer

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.

rohit8865 · Answer

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..

Mahmud6 · Answer

Would you please explain how did you get this ' a = (1 + q%)*(1-p%)a ' ?
I am in trouble to understand this.

Adarsh_24 · Answer

(1 - q%)(1 + p%) =1
Even if p and q are positive 
For example, p=100 and q=50

Not sure why the answer is C.

Kinshook · Answer

Bunuel
Please correct OA as E

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?

a > 0 ; b > 0; ; a = (1+q%)b; b = (1-p%)a
a = (1+q%)(1-p%)a
(1+q%)(1-p%) = 1
1 +q% - p% - p%q% = 1
q% - p% - p%q% = 0
p% - q% + p%q% = 0

c = a(1+p%)(1-q%) = a(1+p% - q% -p%q%) = a (1-2p%q%)

Case 1: 1 - 2p%q% = 1; p%q% = 0;
c = a; In this case p = q = 0

Case 2: 1-2p%q% > 1; p%q% <0
Not possible since p & q are of different signs

Case 3: 1-2p%q% < 1; p%q% >0
c < a
Possible

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

IMO E

bumpbot · Answer

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

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