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805+ (Hard)|   Algebra|      
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santorasantu
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If a number N is decreased by p percent and then the resulting value 01 Oct 2015, 06:32
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If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.
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C
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If a number N is decreased by p percent and then the resulting value 11 Jan 2017, 01:13
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Hi iMyself,

Condition 1)

N * ( 1 - p/100 ) ( 1 + q / 100 ) = N
( 1 - p / 100 ) ( 1 + q / 100 ) = 1
( 100 - p ) ( 100 + q ) = 10,000 = 2^4 * 5^4

We have two cases satisfying the first condition.

case 1: 100 - p = 2^4, 100 + q = 5^4
100 - p = 16, 100 + q = 625
p = 84, q = 525

case 2: 100 - p = 5^2, 100 + q = 2^4 * 5^2
100 - p = 25, 100 + q = 400
p = 75, q = 300

Not Sufficient


Condition 2)

case 1: 100 - p = 2^4, 100 + q = 5^4
p = 84, q = 525

case 2: 100 - p = 2^4 * 5, 100 + q = 5^3
100 - p = 80, 100 + q = 125
p = 20, q = 25

Not Sufficient


Considering both conditions together, the only case for them is 100 - p = 2^4 and 100 + q = 5^4., since prime factors 2 and 5 cannot be together in 100 - p or 100 + q and p must be less than or equal to 100.
Then 100 - p = 16 and 100 + q = 625

p = 84 and q = 525 is the unique solution

The correct answer is C.

Happy Studying!
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If a number N is decreased by p percent and then the resulting value 16 Nov 2015, 04:40
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Harley1980:

Can you provide some insight on this one.
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If a number N is decreased by p percent and then the resulting value 16 Nov 2015, 11:06
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If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.
Show more

N decreased by p Percent

i.e. N becomes-----> N*[1-(p/100)]

Resulting value increased by q Percent

i.e. N*[1-(p/100)] becomes-----> N*[1-(p/100)]*[1+(q/100)]

Now, N*[1-(p/100)]*[1+(q/100)] = N

i.e. (100-p)*(100+q) = 100*100 = \(2^4*5^4\)

Statement 1: p is not a multiple of 10.
Case 1: p = 75 and q = 300
Case 2: p = 95 and q = 1900
NOT SUFFICIENT

Statement 2: q is not a multiple of 10.
Case 1: q = 25 and p = 20
Case 2: q = 525 and p = 84
NOT SUFFICIENT

Combining the two statements
Only possible values are
q = 525 and p = 84
SUFFICIENT
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If a number N is decreased by p percent and then the resulting value 17 Nov 2015, 01:51
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Hi GMATinsight,

When combined 1 & 2, I found difficult to get the answer and in the real test it will be waste of time. Do you have any short cut to know there is only one unique answer such that the answer will be C?

Thanks
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If a number N is decreased by p percent and then the resulting value 17 Nov 2015, 09:36
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Hi GMATinsight,

When combined 1 & 2, I found difficult to get the answer and in the real test it will be waste of time. Do you have any short cut to know there is only one unique answer such that the answer will be C?

Thanks
Show more

i.e. (100-p)*(100+q) = 100*100 = \(2^4∗5^4\)

Combining the two statements
Since and p and q are both NON-MULTIPLE of 10 so (100-p) and (100+q) also will be NON-MULTIPLE of 10

i.e. the values of (100-p) and (100+q) will include either powers of only 2 (i.e. 2 or 2^2 or 2^3 or 2^4) or powers of only 5 (i.e. 5 or 5^2 or 5^3 or 5^4)

Also, Note that p can NOT be greater than 100 because (100-p) must be POSITIVE

Only possible values of p are
q = 525 and p = 84
(100-p)*(100+q) = 16*625 = 100*100
SUFFICIENT

I Hope this helps!!!
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If a number N is decreased by p percent and then the resulting value 17 Nov 2015, 10:38
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.


If we modify the question by multiplying both sides by 100 and dividing by n, we get (100-p)(100+q)=10,000
There are 2 variables (p,q) and one equation (100-p)(100+q)=10,000. There are 2 more equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, p=98, q=4,900/ p=84, q=525. This is insufficient, as there is no unique answer.
For condition 2, p=20, q=25/ p=84, q=525. This is also insufficient for the same reason.
Looking at the condition together, however, we get p=84, q=525, which is a unique answer.
The answer is therefore (C).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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If a number N is decreased by p percent and then the resulting value 17 Nov 2015, 10:56
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MathRevolution
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.


If we modify the question by multiplying both sides by 100 and dividing by n, we get (100-p)(100+q)=10,000
There are 2 variables (p,q) and one equation (100-p)(100+q)=10,000. There are 2 more equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, p=98, q=4,900/ p=84, q=525. This is insufficient, as there is no unique answer.
For condition 2, p=20, q=25/ p=84, q=525. This is also insufficient for the same reason.
Looking at the condition together, however, we get p=84, q=525, which is a unique answer.
The answer is therefore (C).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
Show more


how would one think of such possible answers during the test? what level hardness question is this?
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If a number N is decreased by p percent and then the resulting value 17 Nov 2015, 11:00
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Hi GMATinsight,

When combined 1 & 2, I found difficult to get the answer and in the real test it will be waste of time. Do you have any short cut to know there is only one unique answer such that the answer will be C?

Thanks

i.e. (100-p)*(100+q) = 100*100 = \(2^4∗5^4\)

Combining the two statements
Since and p and q are both NON-MULTIPLE of 10 so (100-p) and (100+q) also will be NON-MULTIPLE of 10

i.e. the values of (100-p) and (100+q) will include either powers of only 2 (i.e. 2 or 2^2 or 2^3 or 2^4) or powers of only 5 (i.e. 5 or 5^2 or 5^3 or 5^4)

Also, Note that p can NOT be greater than 100 because (100-p) must be POSITIVE

Only possible values of p are
q = 525 and p = 84
(100-p)*(100+q) = 16*625 = 100*100
SUFFICIENT

I Hope this helps!!!
Show more


on simplifying using another method, i arrive at
100(q-p) = pq

any help from here?
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If a number N is decreased by p percent and then the resulting value 17 May 2016, 14:13
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santorasantu


Now, N*[1-(p/100)]*[1+(q/100)] = N

i.e. (100-p)*(100+q) = 100*100 = \(2^4*5^4\)

Show more


I cannot find an algebraic way to arrive at this equation. How is this determined?
I can get to { [1-(p/100)]*[1+(q/100)] = 1 }.
However, when multiplying both sides by 100, I get { (100-p)*(100+q) = 100 }..
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If a number N is decreased by p percent and then the resulting value 06 Dec 2016, 13:47
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MathRevolution
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.


If we modify the question by multiplying both sides by 100 and dividing by n, we get (100-p)(100+q)=10,000
There are 2 variables (p,q) and one equation (100-p)(100+q)=10,000. There are 2 more equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, p=98, q=4,900/ p=84, q=525. This is insufficient, as there is no unique answer.
For condition 2, p=20, q=25/ p=84, q=525. This is also insufficient for the same reason.
Looking at the condition together, however, we get p=84, q=525, which is a unique answer.
The answer is therefore (C).
Show more
The original question stem says:
N*(100-p / 100)*(100+q / 100) = N
can we really divide the equation by N?
There is no indication in the question that ''N is not zero''. So, why do we divide the equation by N?
Thanks...
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If a number N is decreased by p percent and then the resulting value 13 Dec 2016, 04:51
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iMyself
MathRevolution
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.


If we modify the question by multiplying both sides by 100 and dividing by n, we get (100-p)(100+q)=10,000
There are 2 variables (p,q) and one equation (100-p)(100+q)=10,000. There are 2 more equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, p=98, q=4,900/ p=84, q=525. This is insufficient, as there is no unique answer.
For condition 2, p=20, q=25/ p=84, q=525. This is also insufficient for the same reason.
Looking at the condition together, however, we get p=84, q=525, which is a unique answer.
The answer is therefore (C).
The original question stem says:
N*(100-p / 100)*(100+q / 100) = N
can we really divide the equation by N?
There is no indication in the question that ''N is not zero''. So, why do we divide the equation by N?
Thanks...
Show more

Hi iMyself,

Because there is a condition that n is not 0.

Happy Studying!
Math Revolution
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If a number N is decreased by p percent and then the resulting value 19 Dec 2016, 13:06
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MathRevolution
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?

(1) p is not a multiple of 10.

(2) q is not a multiple of 10.


If we modify the question by multiplying both sides by 100 and dividing by n, we get (100-p)(100+q)=10,000
There are 2 variables (p,q) and one equation (100-p)(100+q)=10,000. There are 2 more equations given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, p=98, q=4,900/ p=84, q=525. This is insufficient, as there is no unique answer.
For condition 2, p=20, q=25/ p=84, q=525. This is also insufficient for the same reason.
Looking at the condition together, however, we get p=84, q=525, which is a unique answer.
The answer is therefore (C).
The original question stem says:
N*(100-p / 100)*(100+q / 100) = N
can we really divide the equation by N?
There is no indication in the question that ''N is not zero''. So, why do we divide the equation by N?
Thanks...

Hi iMyself,

Because there is a condition that n is not 0.

Happy Studying!
Math Revolution
Show more
The question stem did not directly say that N is not 0, but it indirectly indicates that N is not zero by stating the line ''a number N is decreased by p percent.......''. If a number is decreased by a certain percent, then it indicates that N must be greater than zero, right MathRevolution? Thank you for your kind response.
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how did you come up two values for p&q that is 84 and 525 please explain
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Even if pretty loose the way I solved it is the following:

Once we arrive to (100-p)*(100+q) = 100*100 = 2^4∗5^4 we know the factorisation of (100-p)*(100+q)

Stm 1 tells us p is not a multiple of 10, therefore 100-p is not a multiple of 10 either. Thus 100-p contains only 2s or 5s in its factorisation, but we don't know in which quantity. i.e. 100-p can be 2^3 (p is 92) or 5^2 (p is 75). The rest of the factorisation of (100-p)*(100+q) will be covered by 100+q as long as this factorisation yields a number >100 --> No suff.

Stm 2 same as above but with different constraints. This time 100+q contains only 2s or only 5s. The limit here is that 100-p cannot be >100. --> No suff.

Stm 1 and 2 clearly here we know that either (100-p) contains only 2s or it contains only 5s (vice versa for (100+q))
Since (100-p) cannot be >100 (100-p) is 2^4 (p=84) and (100+q) is 5^4 (q=525) ---> C is the answer
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mattyahn
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santorasantu


Now, N*[1-(p/100)]*[1+(q/100)] = N

i.e. (100-p)*(100+q) = 100*100 = \(2^4*5^4\)



I cannot find an algebraic way to arrive at this equation. How is this determined?
I can get to { [1-(p/100)]*[1+(q/100)] = 1 }.
However, when multiplying both sides by 100, I get { (100-p)*(100+q) = 100 }..
Show more

I believe this should not be multiplied by 100 in both sides because multiplication is not distributive. I believe you would get (100-p)*(1+q/100)=100. and hence you would need another 100 to multiply.

I was having the same issue...hope this helps.
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If a number N is decreased by p percent and then the resulting value 16 Nov 2019, 06:42
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Even I also stuck at same equation.

Don't understand upto what level simplification should be done.

rahulkashyap
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Mo2men
Hi GMATinsight,

When combined 1 & 2, I found difficult to get the answer and in the real test it will be waste of time. Do you have any short cut to know there is only one unique answer such that the answer will be C?

Thanks

i.e. (100-p)*(100+q) = 100*100 = \(2^4∗5^4\)

Combining the two statements
Since and p and q are both NON-MULTIPLE of 10 so (100-p) and (100+q) also will be NON-MULTIPLE of 10

i.e. the values of (100-p) and (100+q) will include either powers of only 2 (i.e. 2 or 2^2 or 2^3 or 2^4) or powers of only 5 (i.e. 5 or 5^2 or 5^3 or 5^4)

Also, Note that p can NOT be greater than 100 because (100-p) must be POSITIVE

Only possible values of p are
q = 525 and p = 84
(100-p)*(100+q) = 16*625 = 100*100
SUFFICIENT

I Hope this helps!!!


on simplifying using another method, i arrive at
100(q-p) = pq

any help from here?
Show more
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If a number N is decreased by p percent and then the resulting value 15 Oct 2021, 06:44
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santorasantu
If a number N is decreased by p percent and then the resulting value is increased by q percent, the final result is equal to N. If both p and q are positive integers, what is the value of p ?.
Show more

N * ( 1 - p/100 ) ( 1 + q / 100 ) = N
( 1 - p / 100 ) ( 1 + q / 100 ) = 1
( 100 - p ) ( 100 + q ) = 10,000 = 10^4

(1) p is not a multiple of 10.
First case possibility

100 - p = 2^4, 100 + q = 5^4
100 - p = 16, 100 + q = 625
p = 84, q = 525

Second possibility
100 - p = 5^2, 100 + q = 2^4 * 5^2
100 - p = 25, 100 + q = 400
p = 75, q = 300

Clearly insufficient

(2) q is not a multiple of 10

100 - p = 2^4, 100 + q = 5^4
p = 84, q = 525

100 - p = 2^4 * 5, 100 + q = 5^3
100 - p = 80, 100 + q = 125
p = 20, q = 25


However when 1 and 2 is comb/ined we get p=84 and q=525
Therefore IMO C
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If a number N is decreased by p percent and then the resulting value 26 Mar 2025, 22:24
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