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If S and T are non-zero numbers and Updated on: 04 Nov 2012, 15:06
Originally posted by SirGMAT on 04 Nov 2012, 08:20.
Last edited by Bunuel on 04 Nov 2012, 15:06, edited 1 time in total.
Renamed the topic and edited the question.
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33% (01:53) correct 67% (01:25) wrong based on 478 sessions

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If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

M18-19

Show Spoiler
Explanation provided:

1/S + 1/T = S+T;

T+S/ST = S+T→;

Cross-multiply: S+T=(S+T)∗ST;

(S+T)(ST−1)=0. Either S+T=0 or ST=1. Now, notice that if S+T=0 is true then none of the options must be true.

The correct answer is E


Question:
I understand the way of the equation, however, what I would have done is interfere at the following step:
S+T = (S+T)*ST
-> (S+T)/(S+T)=ST
-> ST = 1

Is there some rule which forbids me to take this step? Or is the only option to realize so, that you perform the above given equation as well and realise that "S+T = 0" negates all other options than E.... ??


Thanks in advance,
best regards


P.S. Sry if the format is terrible, this is the first question I am copying out of somewhere.

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Bunuel
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If S and T are non-zero numbers and 04 Nov 2012, 15:08
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If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.
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If S and T are non-zero numbers and 04 Nov 2012, 08:46
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SirGMAT
If S and T are non-zero numbers and 1S+1T=S+T, which of the following must be true?
A. ST=1
B. S+T=1
C. 1/S=T
D. S/T=1
E. none of the above
Explanation provided:
1/S + 1/T = S+T;
T+S/ST = S+T→;
Cross-multiply: S+T=(S+T)∗ST;
(S+T)(ST−1)=0. Either S+T=0 or ST=1. Now, notice that if S+T=0 is true then none of the options must be true.
The correct answer is E
Question:
I understand the way of the equation, however, what I would have done is interfere at the following step:
S+T = (S+T)*ST
-> (S+T)/(S+T)=ST
-> ST = 1
Is there some rule which forbids me to take this step? Or is the only option to realize so, that you perform the above given equation as well and realise that "S+T = 0" negates all other options than E.... ??
Thanks in advance,
best regards
P.S. Sry if the format is terrible, this is the first question I am copying out of somewhere.
Show more

Yes, this is not correct way of cancelling. I'll show you one example.
5* 0 = 3*0
if we cancel 0 on both side, we get 5=3. is it correct? No.

The crux (and the rule) is, we can cancel out a term only when we know its not 0. So the way it is done in explanation is absolutely correct and the right method.

Hope it helps :)
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If S and T are non-zero numbers and 04 Nov 2012, 09:43
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SirGMAT
If S and T are non-zero numbers and 1S+1T=S+T, which of the following must be true?

A. ST=1
B. S+T=1
C. 1/S=T
D. S/T=1
E. none of the above



Explanation provided:

1/S + 1/T = S+T;

T+S/ST = S+T→;

Cross-multiply: S+T=(S+T)∗ST;

(S+T)(ST−1)=0. Either S+T=0 or ST=1. Now, notice that if S+T=0 is true then none of the options must be true.

The correct answer is E


Question:
I understand the way of the equation, however, what I would have done is interfere at the following step:
S+T = (S+T)*ST
-> (S+T)/(S+T)=ST
-> ST = 1

Is there some rule which forbids me to take this step? Or is the only option to realize so, that you perform the above given equation as well and realise that "S+T = 0" negates all other options than E.... ??


Thanks in advance,
best regards


P.S. Sry if the format is terrible, this is the first question I am copying out of somewhere.
Show more

There was this fun derivation that my math teacher showed us in school. Just to demonstrate how cancelling of 0 could yield wrong results. He claimed that he could prove that 1=2 and hence all numbers are equal.

It goes as below:

Let,
\(a=b\)

Multiplying both sides by a, We get

\(a^2 = ab\)

Subtracting \(b^2\) from both sides

\(a^2 - b^2 = ab - b^2\)

\((a+b)(a-b) = b(a-b)\)

Cancelling \((a-b)\) on both sides,

\(a+b = b\)

Since \(a=b\)

\(a+a = a\)

\(2a = a\)

\(2=1\)
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If S and T are non-zero numbers and 04 Nov 2012, 15:06
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Yes, this is not correct way of cancelling. I'll show you one example.
5* 0 = 3*0
if we cancel 0 on both side, we get 5=3. is it correct? No.

The crux (and the rule) is, we can cancel out a term only when we know its not 0. So the way it is done in explanation is absolutely correct and the right method.

Hope it helps :)
Show more
Thanks - lol I am an idiot - ...! I even thought of the zero number things, but mistakenly memorized the prompt as telling me that the respective equation could not be "0", though it only said that each number alone is non-zero....

Thanks guys! ;)
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If S and T are non-zero numbers and 07 Nov 2012, 01:12
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Hi bunuel,

In this step:
s+t = (s+t)st can't we just cancel (s+t) and get ---> st =1?


thanks,
-K
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If S and T are non-zero numbers and 07 Nov 2012, 01:19
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.
Show more

Bunuel,
in your solution I need to ask one thing.
Inequalities, there is a rule that if you dont know the sign of denominator, then dont cross multiply. In your solution, how can you be so sure of the sign of ST. Please let me know if i am missing something
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If S and T are non-zero numbers and 07 Nov 2012, 02:30
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Hi bunuel,

In this step:
s+t = (s+t)st can't we just cancel (s+t) and get ---> st =1?


thanks,
-K
Show more

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) s+t = (s+t)st by (s+t), you assume, with no ground for it, that (s+t) does not equal to zero thus exclude a possible solution (notice that both st=1 AND (s+t)=0 satisfy the equation).

Hope it's clear.
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If S and T are non-zero numbers and 07 Nov 2012, 02:43
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

Bunuel,
in your solution I need to ask one thing.
Inequalities, there is a rule that if you dont know the sign of denominator, then dont cross multiply. In your solution, how can you be so sure of the sign of ST. Please let me know if i am missing something
Show more

We are concerned with the sign when we cross-multiply an inequality because this operation might affect (flip) its sign (> to <, for example) but it's always safe to cross-multiply an equation.
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If S and T are non-zero numbers and 10 Feb 2013, 06:03
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.
Show more

Bunuel,
Since S+T=0 OR ST=1 and the question asks what must be true, the answer is E ? Another way to answer the question. . Is my reasoning right?
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If S and T are non-zero numbers and 10 Feb 2013, 06:09
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.

Bunuel,
Since S+T=0 OR ST=1 and the question asks what must be true, the answer is E ? Another way to answer the question. . Is my reasoning right?
Show more

Yes, if for example s=1 and t=-1 (s+t=1-1=0), then none of the options is true (none of the options MUST be true).
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If S and T are non-zero numbers and 10 Feb 2013, 07:04
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I understand what you are trying to say bunuel..
my question is that since the equation results in 2 soln and we have a OR .. . that is soln 1 OR soln 2 and the question asks for MUST be true..

So based on this reasoning, can we say the answer is E..?

For something must be true , we cannot have soln 1 OR soln 2.. we need to have 1 soln / soln1 AND soln2..

Hope you are getting what I am trying to ask..
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If S and T are non-zero numbers and 10 Feb 2013, 14:55
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Sachin9
I understand what you are trying to say bunuel..
my question is that since the equation results in 2 soln and we have a OR .. . that is soln 1 OR soln 2 and the question asks for MUST be true..

So based on this reasoning, can we say the answer is E..?

For something must be true , we cannot have soln 1 OR soln 2.. we need to have 1 soln / soln1 AND soln2..

Hope you are getting what I am trying to ask..
Show more
I would say that don't generalize this point. You know that because there are two solutions, any of the given options need not be a MUST. But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.
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If S and T are non-zero numbers and 10 Feb 2013, 19:17
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Sachin9
I understand what you are trying to say bunuel..
my question is that since the equation results in 2 soln and we have a OR .. . that is soln 1 OR soln 2 and the question asks for MUST be true..

So based on this reasoning, can we say the answer is E..?

For something must be true , we cannot have soln 1 OR soln 2.. we need to have 1 soln / soln1 AND soln2..

Hope you are getting what I am trying to ask..
I would say that don't generalize this point. You know that because there are two solutions, any of the given options need not be a MUST. But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.
Show more


But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.

Ididn;t understand this.. if (s+t)(st-1)=0 then either s+t=0 or st-1=0.. we still have a OR here
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If S and T are non-zero numbers and 11 Feb 2013, 01:59
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Sachin9
I understand what you are trying to say bunuel..
my question is that since the equation results in 2 soln and we have a OR .. . that is soln 1 OR soln 2 and the question asks for MUST be true..

So based on this reasoning, can we say the answer is E..?

For something must be true , we cannot have soln 1 OR soln 2.. we need to have 1 soln / soln1 AND soln2..

Hope you are getting what I am trying to ask..
I would say that don't generalize this point. You know that because there are two solutions, any of the given options need not be a MUST. But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.


But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.

Ididn;t understand this.. if (s+t)(st-1)=0 then either s+t=0 or st-1=0.. we still have a OR here
Show more
If the question were to be this:
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above
F. (s+t)(st-1)=0

Then your generalization will go wrong as you have an ans choice F that must hold true.
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If S and T are non-zero numbers and 18 Mar 2013, 11:06
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.
Show more


Hello Bunuel,

How did u get S+T=(S+T)*ST --> (S+T)(ST-1)=0 ??

What did i miss in he below equation?? how come you got (S+T)(ST-1)=0 ?? Ca you please explain
1/S+1/T = S+T
(S+T) = (S+T) (ST)
divide both side by (S+T) we get
1=1(ST)
therefore ST=1

Thank you.
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If S and T are non-zero numbers and 18 Mar 2013, 11:12
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.


Hello Bunuel,

How did u get S+T=(S+T)*ST --> (S+T)(ST-1)=0 ??

What did i miss in he below equation?? how come you got (S+T)(ST-1)=0 ?? Ca you please explain
1/S+1/T = S+T
(S+T) = (S+T) (ST)
divide both side by (S+T) we get
1=1(ST)
therefore ST=1

Thank you.
Show more

Check here: if-s-and-t-are-non-zero-numbers-and-141887.html#p1140729

Hope it helps.
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If S and T are non-zero numbers and 18 Mar 2013, 11:14
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kuttingchai
Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.


Hello Bunuel,

How did u get S+T=(S+T)*ST --> (S+T)(ST-1)=0 ??

What did i miss in he below equation?? how come you got (S+T)(ST-1)=0 ?? Ca you please explain
1/S+1/T = S+T
(S+T) = (S+T) (ST)
divide both side by (S+T) we get
1=1(ST)
therefore ST=1

Thank you.
Show more

I think I got the answer

from your previous post I got "
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) s+t = (s+t)st by (s+t), you assume, with no ground for it, that (s+t) does not equal to zero thus exclude a possible solution (notice that both st=1 AND (s+t)=0 satisfy the equation).
"

therefore
1/S+1/T = S+T
(S+T) = (S+T) (ST)
0= (S+T) (ST) - (S+T)
0 = (S+T) (ST-1)

(S+T)=0 or ST = 1

Thank you
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If S and T are non-zero numbers and 27 Dec 2015, 08:46
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.
Show more


Bunuel,

Not able to understand the highlighted part. Kindly explain in detail...

Thx,
Arun
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Bunuel
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If S and T are non-zero numbers and 27 Dec 2015, 08:51
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Bunuel
If S and T are non-zero numbers and \(\frac{1}{S} + \frac{1}{T} = S + T\), which of the following must be true?

A. \(ST = 1\)
B. \(S + T = 1\)
C. \(\frac{1}{S} = T\)
D. \(\frac{S}{T} = 1\)
E. None of the above

OE:

\(\frac{1}{S} + \frac{1}{T} = S + T\) --> \(\frac{T+S}{ST}=S+T\) --> cross-multiply: \(S+T=(S+T)*ST\) --> \((S+T)(ST-1)=0\) --> either \(S+T=0\) or \(ST=1\). So, if \(S+T=0\) is true then none of the options must be true.

Answer: E.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.


Bunuel,

Not able to understand the highlighted part. Kindly explain in detail...

Thx,
Arun
Show more

\(S+T=(S+T)*ST\)

\((S+T)*ST-(S+T)=0\)

Factor s+t: \((S+T)(ST-1)=0\)

Hope it's clear.
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