M18-19

Question

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

Bunuel · Accepted Answer

Official Solution:

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

\frac{1}{s} + \frac{1}{t} = s + t ;
 
 \frac{t+s}{st}=s+t .
  
Notice that we cannot reduce the above by  s+t , because  s+t  can equal 0, ad we cannot divide by 0 But we can cross-multiply: 
  
 s+t=(s+t)*st ;
 
 s+t - (s+t)*st=0 .
  
Factor out  s+t : 
  
 (s+t)(1 - st)=0 .
 
 s+t=0  or  st=1 . 
  
Now, notice that if  s+t=0  is true, for example, if  s=10  and  t=-10 , then none of the options must be true. Therefore, E is the answer.

Answer: E

reddyMBA · Answer

When we solve we get 'Either S+T=0 or ST=1'. So, any of these is correct. We have 'ST=1' as option (A), why it is not correct? Please let me know!

Bunuel · Answer

Because it's not necessary that st=1. If s+t=0, then st could be some other number but 1.

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation.

Mo2men · Answer

Another approach for this great problem

Non-zero integers means negative or positive integers

Let see apply some cases with numbers to DISAPPROVE answer choices

S = T = 1............Equation is 2 = 2

S = T =-1............Equation is -2 = -2

S = 1 & T =-1....Equation is 0 = 0 or S = -1 & T =1....Equation is 0 = 0

A.  ST = 1  ............Case 3 does not apply.................... Eliminate A

B.  S + T = 1 ........No cases applies............................. Eliminate B

C.  \frac{1}{S} = T  .......Case 3 does not apply........... Eliminate C 
 
D.  \frac{S}{T} = 1  .......Case 3 does not apply..... Eliminate D 
 
E. none of the above........... Correct

Answer: E

SchruteDwight · Answer

Bunuel  I don't fully understand how  T+S=0  is possible:

\frac{ST}{(T+S)}=\frac{1}{(T+S)}  is what I got manipulating the original equation.

Then I thought, Okay, I can simply multiply by  (T+S)  as  T+S=0  is not possible (otherwise we would divide by 0).

Where's the mistake in that approach?

Bunuel · Answer

S + T = 0 IS possible, for example, if S = 1 and T = -1. In this case 1/S + 1/T = S + T = 0. The mistake is in your manipulation: when you get  \frac{T+S}{ST}=S+T , you cannot write this as  \frac{ST}{(T+S)}=\frac{1}{(T+S)}  specifically because S + T COULD be 0. You should proceed the way shown in the solution.

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Rebaz · Answer

Bunuel

Now, notice that if s+t=0 is true, for example, if s=10 and t=−10, then none of the options must be true.

Therefore, E is the answer .

Iam not sure what you really mean with the above reasoing at the end?

It all boils down that S+T=0 or ST=1.

Since we are dealing with Must be true, we can conclude that even though that ST=1 matches with one of the answer choices, but it doesnot have to be true, coz it could be that S+T=0.

But when i apply the same logic above(Must be true), S+T=0(Meaning S=-T), which doesnot match with any of the first 4 answer choices except E, This is too doesnot have to be true, coz it could be that the answer is that ST=1

My question is: Where do i go wrong in my reasoning?

Thanks in Advance!

Bunuel · Answer

Not fully following what confuses you, but the answer is E (none of the above) because none of the options A through D must be correct. Is it possible that A is not true? Yes. Is it possible that B is not true? Yes. Is it possible that C is not true? Yes. Is it possible that D is not true? Yes. Therefore, none of the options must be true.

M18-19

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards