Algbera question

[q4m_l]

Exactly seventy five percent of the members of a club fill exactly 5/6 of the chairs in the room. What is the smallest possible number of people in the club?

 

[Deleted member 4993]

Exactly seventy five percent of the members of a club fill exactly 5/6 of the chairs in the room. What is the smallest possible number of people in the club?

This is a Diophantine problem. Since we need to use EXACTLY 75%, the # of club members could be 4 or 8 or 12, or 16 etc.

Similarly the number of chairs could be 6 or 12 or 18 etc.

Please share your work/thoughts about this problem.

 

[JeffM]

As SK said, this is a Diophantine problem. Because you posted this in Intermediate Algebra, I am assuming that you have not been taught much, if anything, about Diophantine problems.

But you presumably know algebra.

What are your variables?

What equation or equations have you set up?

So do you have enough equations?

 

[q4m_l]

As SK said, this is a Diophantine problem. Because you posted this in Intermediate Algebra, I am assuming that you have not been taught much, if anything, about Diophantine problems.

But you presumably know algebra.

What are your variables?

What equation or equations have you set up?

So do you have enough equations?

That is true Sir I have not been taught much about Diophantine problems so I couldn't solve this then i referred to my math teacher to try and help me but he was struggling a bit and couldn't solve it so if possible could you perhaps tell me the way on how to solve this question if possible?
And Thanks for your cooperation

 

[Deleted member 4993]

This is a Diophantine problem. Since we need to use EXACTLY 75%, the # of club members could be 4 or 8 or 12, or 16 etc.

Similarly the number of chairs could be 6 or 12 or 18 etc.

Please share your work/thoughts about this problem.

The words EXACTLY and SMALLEST are very important in this problem.

Following guidelines set-up by @JeffM,

Let use start solving this problem by defining variables.

Let the number of members of the club = M​

​

Let the number of chairs in the club = C​

​

Now write an equation involving M and C from the information given to you in the problem statement.

 

[JeffM]

We do not initially provide answers at this site for student problems. We ask you to respond to questions first in order to develop your understanding. You ignored my questions. Let’s start over.

You understand from algebra about assigning letters to stand for unknowns, right?

What are the unknowns in this equation and what letters have you used as labels for them? Edit: Use SK’s labels.

You understand from algebra about setting up one or more equations to express the known relations among your unknowns, right?

What equation or equations do you get?

 

[yousef334453434d]

hello. SO i'm having trouble with this question that's coming up on my test tomorrow and I have no means of contacting my teacher, so I need help please. Its this: Exactly seventy five percent of the members of a club fill exactly 5/6 of the chairs in the room. The smallest possible number of people in the club is? I have no means of solving and don't understand anything except that i know that Diophantine equations are written in the form ax+by=c. my info might be incorrect. pls help, much appreciated

[moved from another thread]...................SK

 

[Dr.Peterson]

hello. SO i'm having trouble with this question that's coming up on my test tomorrow and I have no means of contacting my teacher, so I need help please. Its this: Exactly seventy five percent of the members of a club fill exactly 5/6 of the chairs in the room. The smallest possible number of people in the club is? I have no means of solving and don't understand anything except that i know that Diophantine equations are written in the form ax+by=c. my info might be incorrect. pls help, much appreciated

Who are you? You aren't the OP; are you someone else who was given the same problem? Where does it come from? (Were you actually told that this very problem would be on your test?)

It may be worth knowing that you don't need to know anything about Diophantine equations to solve this.

You're wrong in what you say about them, which applies only to linear equations; the defining property of Diophantine equations is merely that they require integer solutions. So my guess is that both of you haven't really learned anything about that, and the problem probably does not assume you have.

On the other hand, it would be very helpful to us if we knew something about your context, such as what level you are at in school, and what you are learning. Is this for an algebra class, or something else? It's possible to solve this even if you haven't learned any but the most basic algebra.

Exactly seventy five percent of the members of a club fill exactly 5/6 of the chairs in the room. What is the smallest possible number of people in the club?

If either of you had answered the direct questions that were asked, then you would have written an equation (or something else) we could talk about! Things would go so much quicker if people would cooperate ...

I'll write the equation: 3/4 M = 5/6 C. Do you see why?

Do you see that since both sides represent the number of chairs sat in, they must be a whole number? What must M be divisible by? What must C be divisible by? You could simply try numbers, starting at the smallest, to find a pair that work.

Or, using a little more thought, you could try solving for either M or C in terms of the other, and see what more that tells you about divisibility, reducing the number of things to try.

Basically, you just have to be willing to start trying things. Show us any sort of thought, and we can probably give you a little nudge in the direction of a solution. Our goal is to help you use whatever you know, and so far, we know nothing about that, which is why we can't do much for you yet.

 

[yousef334453434d]

Who are you? You aren't the OP; are you someone else who was given the same problem? Where does it come from? (Were you actually told that this very problem would be on your test?)

It may be worth knowing that you don't need to know anything about Diophantine equations to solve this.

You're wrong in what you say about them, which applies only to linear equations; the defining property of Diophantine equations is merely that they require integer solutions. So my guess is that both of you haven't really learned anything about that, and the problem probably does not assume you have.

On the other hand, it would be very helpful to us if we knew something about your context, such as what level you are at in school, and what you are learning. Is this for an algebra class, or something else? It's possible to solve this even if you haven't learned any but the most basic algebra.


If either of you had answered the direct questions that were asked, then you would have written an equation (or something else) we could talk about! Things would go so much quicker if people would cooperate ...

I'll write the equation: 3/4 M = 5/6 C. Do you see why?

Do you see that since both sides represent the number of chairs sat in, they must be a whole number? What must M be divisible by? What must C be divisible by? You could simply try numbers, starting at the smallest, to find a pair that work.

Or, using a little more thought, you could try solving for either M or C in terms of the other, and see what more that tells you about divisibility, reducing the number of things to try.

Basically, you just have to be willing to start trying things. Show us any sort of thought, and we can probably give you a little nudge in the direction of a solution. Our goal is to help you use whatever you know, and so far, we know nothing about that, which is why we can't do much for you yet.

Yes, i was told its coming on my test. 8th grade student here. Yes I know algebra, intermediate maybe as I would say. Unfortunately, I don't understand the question, or even what equation to write. It's the first time I've had a question like this.

 

[Dr.Peterson]

Yes, i was told its coming on my test. 8th grade student here. Yes I know algebra, intermediate maybe as I would say. Unfortunately, I don't understand the question, or even what equation to write. It's the first time I've had a question like this.

Again, my answer is to TRY SOMETHING. Anything! You probably do know what you need to know; you just don't yet see how to use it.

I often tell students, solving an unfamiliar problem is like being dropped in the middle of a forest and having to find your way out. You don't know the way out; but sitting there doing nothing is the worst possible thing to do! Take a few steps, in any direction you can, and maybe you'll see something that will help.

And since you're at a level where you wouldn't have been taught anything very specific about this sort of problem (just general ideas like divisibility and multiples, for example), you're free to start with the most basic ideas you have, rather than look for something impressive. Please try the very most basic suggestion I've given: Just try some numbers of students and numbers of chairs, and see what happens. You'll learn by doing.

For example, could there be 1 member? Of course not; 3/4 of 1 wouldn't be a living person. There must be at least 4. (There -- we've learned something about the problem already!)

Well then, could there be 4 members? Then 3 of them would be sitting. How many chairs would there have to be in all, if 3 chairs are used?

Keep thinking like that. And if your teachers haven't given you any opportunities to think for yourself like that, and yet they're giving you a problem like this that you've never seen before, then they're not being good teachers ... but you can show that you are better than that! You don't need to wait to be taught.

 

[Steven G]

Members: 3/4 = 6/8 = 9/12 = 12/16 = ... Note, for example, that 9/12 means 9 members out of 12 members.
Chairs: 5/6 = 10/12 = 15/18 = 20/24 = ... Note, for example, that 15/18 means 15 members out of 18 chairs.