# 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.

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 thats 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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Homework Helper
Gold Member

## 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 thats 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?

R + 11/S = R x 11/S

which got me nowhere.

Homework Helper
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?

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?

Homework Helper
Dearly Missed
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?

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.

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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 realise 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.

Homework Helper
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!

Homework Helper
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

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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

Homework Helper
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?