Finding the Value of S in the Sum Equals Product Problem

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Homework Help Overview

The problem involves finding the value of S in a scenario where the product of two numbers, R and 11/S, equals their sum. The context suggests a focus on algebraic relationships and integer solutions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss potential integer values for R and S, question the completeness of the problem statement, and explore the implications of the equation R + 11/S = R x 11/S. Some suggest considering factors and integer solutions, while others express uncertainty about the problem's requirements.

Discussion Status

The discussion is ongoing, with various interpretations being explored. Some participants have provided algebraic insights and suggested looking for integer solutions, while others have raised questions about the problem's clarity and completeness. There is no explicit consensus on a single approach or solution.

Contextual Notes

Participants note that the problem may not be fully stated and mention the need for additional equations to solve for two variables. There is also a reference to a graph that indicates multiple solutions, and some participants express confusion over the nature of the questions in the associated Olympiad document.

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Homework Statement



If the product of the numbers R and 11/S is the same as their sum, find the value of S.

Homework Equations



N/A

The Attempt at a Solution



I am suspecting that the only set of 2 numbers that have the same sum and product is 2 and 2.

So I guess R is 2, 11/S is also 2.

That gives S = 5.5

Please will anyone verify if this is correct?
 
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Is this the complete question?
 
It probably isn't. Graph by wolfram alpha attached.
Also integer solutions if that's what the question is asking: (-11,12);(-1,22);(11,10).
Source:http://www.wolframalpha.com/input/?i=x+11/y=x*(11/y)
EDIT: P.S. to solve a equation in two variables you always need two equations.
 

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tony24810 said:

Homework Statement



If the product of the numbers R and 11/S is the same as their sum, find the value of S.

Assuming, as others have suggested, that R and S are integers, what equation in which all terms are integers expresses the given condition? Can you deduce anything by thinking about factors?
 
dirk_mec1 said:
Is this the complete question?

That was the complete question, extract from Olympiad.
 
Enigman said:
It probably isn't. Graph by wolfram alpha attached.
Also integer solutions if that's what the question is asking: (-11,12);(-1,22);(11,10).
Source:http://www.wolframalpha.com/input/?i=x+11/y=x*(11/y)
EDIT: P.S. to solve a equation in two variables you always need two equations.

That's cool, haven't thought of the solutions you create.

I didn't understand your graph though.

Is there an algebra way of solving this though? I don't think graphical calculator is allow in exam.
 
haruspex said:
Assuming, as others have suggested, that R and S are integers, what equation in which all terms are integers expresses the given condition? Can you deduce anything by thinking about factors?

I am not sure what you mean by 'thinking about factor', I don't know the actual value of R and S so I cannot determine their factor. Is there any specific theorem that I should try?

the equation I made was:

R + 11/S = R x 11/S

which got me nowhere.
 
tony24810 said:
I am not sure what you mean by 'thinking about factor', I don't know the actual value of R and S so I cannot determine their factor. Is there any specific theorem that I should try?

the equation I made was:

R + 11/S = R x 11/S

which got me nowhere.

That's fine, but that equation has many solutions. R=2, S=11/2 works, and so does R=3, S=22/3. Etc, etc. Can you post a reference to the exact Olympiad question?
 
tony24810 said:
I am not sure what you mean by 'thinking about factor', I don't know the actual value of R and S so I cannot determine their factor. Is there any specific theorem that I should try?

the equation I made was:

R + 11/S = R x 11/S

which got me nowhere.

That gives you RS + 11 = R*11, or 11*(R-1) = R*S. Thus, R*S must be divisible by 11, and that suggests looking at some simple values like R = 11 or S = 11, etc, assuming you want positive integer values of R and S. If all you want are real values, you can just put S = 11*(R-1)/R and let R be anything you want (but not zero).
 
  • #10
tony24810 said:
That was the complete question, extract from Olympiad.
A reference would be nice...Were there options in the question.
tony24810 said:
That's cool, haven't thought of the solutions you create.

I didn't understand your graph though.
The graph just shows that there are infinite solutions possible. ie. Each point is a solution
Ignore the zig-zag lines at x axis, its just Mathematica having a fit.
 
  • #11
Dick said:
That's fine, but that equation has many solutions. R=2, S=11/2 works, and so does R=3, S=22/3. Etc, etc. Can you post a reference to the exact Olympiad question?

Here's the paper.

Well I also failed to do question 2, please help if possible^^. thanks!
 

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  • #12
Ray Vickson said:
That gives you RS + 11 = R*11, or 11*(R-1) = R*S. Thus, R*S must be divisible by 11, and that suggests looking at some simple values like R = 11 or S = 11, etc, assuming you want positive integer values of R and S. If all you want are real values, you can just put S = 11*(R-1)/R and let R be anything you want (but not zero).

O yes this is cool!

I have actually got to the equation you wrote, but didn't realize that it means infinite sets of solution!

Thanks!
 
  • #13
Enigman said:
A reference would be nice...Were there options in the question.

The graph just shows that there are infinite solutions possible. ie. Each point is a solution
Ignore the zig-zag lines at x axis, its just Mathematica having a fit.

There weren't options in the question.

I have attached the original document in previous post.

Some of the questions are somewhat confusing I think.

Like question 3, I think the answer can only be an expression, not a value, but it doesn't say. I think the answer is 11/6 log2 (Q) ], but there's no solution come with the document, quite annoying.
 
  • #14
Well, as you have to find four values: P Q R S
You will have to solve third question for R first before attempting the fourth one which requires the value of R to get S. To get R you will have to get Q and for Q you will need P.
So start from the start and solve question 1 for P first plug it into q.2 get Q plug into q.3 get R and use it finally for S.
 
  • #15
Very clever! So it is not four problems but a single one:biggrin:. And everything fits. R and S are really positive integers. ehild
 
  • #16
ehild said:
Very clever! So it is not four problems but a single one:biggrin:. And everything fits. R and S are really positive integers. ehild

And cruel, imagine the plight of the guy who does everything correctly except the first one...no marks for steps just answers.
Also it has a time multiplication factor, never seen that before.

EDIT: HEY! ehild, you just told OP the answer!
 
  • #17
Enigman said:
EDIT: HEY! ehild, you just told OP the answer!

I do not think so... You gave the principle of solution. :approve: It was very clever of you!

ehild
 
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  • #18
Enigman said:
Well, as you have to find four values: P Q R S
You will have to solve third question for R first before attempting the fourth one which requires the value of R to get S. To get R you will have to get Q and for Q you will need P.
So start from the start and solve question 1 for P first plug it into q.2 get Q plug into q.3 get R and use it finally for S.

OMG i feel so unintelligent, i thought these questions are all separate! No wonder I don't understand half of the questions in other events...

well thanks for pointing it out anyway
 
  • #19
Rather nasty of them to chain 4 questions into 1, but thankfully, all of them are fairly elementary. What level is this for, just out of curiosity?
 

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