Finding the Value of S in the Sum Equals Product Problem

[tony24810]

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?

 

[dirk_mec1]

Is this the complete question?

 

[Enigman]

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.

 

[haruspex]

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?

 

[tony24810]

Is this the complete question?

That was the complete question, extract from Olympiad.

 

[tony24810]

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.

 

[tony24810]

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.

 

[Dick]

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?

 

[Ray Vickson]

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).

 

[Enigman]

That was the complete question, extract from Olympiad.

A reference would be nice...Were there options in the question.

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.

 

[tony24810]

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!

 

[tony24810]

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!

 

[tony24810]

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.

 

[Enigman]

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.

 

[ehild]

Very clever! So it is not four problems but a single one. And everything fits. R and S are really positive integers. ehild

 

[Enigman]

Very clever! So it is not four problems but a single one. 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!

 

[ehild]

EDIT: HEY! ehild, you just told OP the answer!

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

ehild

 

[tony24810]

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

 

[Curious3141]

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?