Inequality question

[Audentes]

I don’t understand how to start working on this problem. How is the $750 divided amongst the people definitively?

thanks
Flo

 

[lev888]

I don’t understand how to start working on this problem. How is the $750 divided amongst the people definitively?

thanks
Flo

10 friends spent $750 at a restaurant. How much does each one pay?

 

[Deleted member 4993]

I don’t understand how to start working on this problem. How is the $750 divided amongst the people definitively?

thanks
Flo

Let

the # of people invited = P

cost of entry = T

Then total revenue collected R = T * P

T * P \(\displaystyle \ \ge \ \ \) 750 + P * 2.25

For how many people must attend (at least) we need to charge the upper bound of T (=3.25)

Continue....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[Audentes]

10 friends spent $750 at a restaurant. How much does each one pay?

$75

 

[Audentes]

View attachment 24150
Let

the # of people invited = P

cost of entry = T

Then total revenue collected R = T * P

T * P \(\displaystyle \ \ge \ \ \) 750 + P * 2.25

For how many people must attend (at least) we need to charge the upper bound of T (=3.25)

Continue....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

is this correct?

 

[lev888]

$75

Can you answer your question now? (How is the $750 divided amongst the people)

 

[Audentes]

Can you answer your question now? (How is the $750 divided amongst the people)

So, the equation would be?
3.25\(\displaystyle \ \le \ \ \)(750/P)+(P*2.25)\(\displaystyle \ \ge \ \ \)2.75

 

[JeffM]

You are asking the wrong question. The way to solve inequalities is to solve equations.

If the number of people who attend is p and the cost per snack is 2.25, how many people must attend so that the average cost per person is EXACTLY 2.75.

[MATH]\dfrac{750 + 2.25p}{p} = 2.75. [/MATH]
Can you solve that?

How would the equation change if the average is 3.25?

Now how do the answers to those equations help you answer the problem?

 

[Audentes]

You are asking the wrong question. The way to solve inequalities is to solve equations.

If the number of people who attend is p and the cost per snack is 2.25 and the admission price is 2.75, how many people must attend so that the money received EQUALS the cost.

[MATH]750 + 2.25p = 2.75p.[/MATH]
Can you solve that?

How would the equation change if the admission price is 3.25?

Now how do the answers to those equations help you answer the problem?

If the price is $2.75 then p=1,500
If the price is $3.25 then p=750 because the 2.75p becomes 3.25p?

So the answer would be
1500\(\displaystyle \ \ge \ \ \)p\(\displaystyle \ \ge \ \ \)750

am I correct?

 

[JeffM]

If the price is $2.75 then p=1,500
If the price is $3.25 then p=750 because the 2.75p becomes 3.25p?

So the answer would be
1500\(\displaystyle \ \ge \ \ \)p\(\displaystyle \ \ge \ \ \)750

am I correct?

You are.

Notice that I edited my response to make it conform more closely to the wording of the problem.

Do you see what I mean that you solve inequalities by solving equations first?

 

[Audentes]

You are.

Notice that I edited my response to make it conform more closely to the wording of the problem.

Do you see what I mean that you solve inequalities by solving equations first?

Yes, I see now, thank you for your help!

 

[lev888]

So, the equation would be?
3.25\(\displaystyle \ \le \ \ \)(750/P)+(P*2.25)\(\displaystyle \ \ge \ \ \)2.75

Not quite.
First, double check the signs - you want the cost per person to be greater than or equal one number and less than or equal the other number. What you have is greater than or equal twice.
The expression you are going for is the cost per person. It has 2 components: share of $750 (750/p) and the caterer fee, which is what?

 

[Audentes]

Not quite.
First, double check the signs - you want the cost per person to be greater than or equal one number and less than or equal the other number. What you have is greater than or equal twice.
The expression you are going for is the cost per person. It has 2 components: share of $750 (750/p) and the caterer fee, which is what?

My mistake, 3.25 is supposed to be greater than or equal to the middle.

the caterer fee is $750

 

[lev888]

My mistake, 3.25 is supposed to be greater than or equal to the middle.

the caterer fee is $750

No, the caterer fee is $2.25. You have p*2.25 instead. Seems like you do not yet understand the problem and the solution.

 

[Audentes]

OH I didn’t see that is the band

 

[Audentes]

So, with the 2.25 caterer fee it is 3.25 gtoet 750/p+2.25p ltoet 2.75

So, the equation would be?
3.25\(\displaystyle \ \le \ \ \)(750/P)+(P*2.25)\(\displaystyle \ \ge \ \ \)2.75

which is the above except for the wrong sign, right? Or am I missing something else

 

[Steven G]

Here is how I would do this problem. The fixed cost per person is $2.25. Hence we want the non fixed cost to be between $0.50 and $1 which is the fee for the band. Although the fee for the band is a fixed $750, it is not a fixed cost per person.

Let p = the number of people who attend the event.

So you want 0.5 < 750/p < 1 OR 1 < p/750 < 2 (I took the reciprocal of all three positive numbers and reversed the order of them)

Can you solve 1 < p/750 < 2 for p?

 

[Audentes]

Here is how I would do this problem. The fixed cost per person is $2.25. Hence we want the non fixed cost to be between $0.50 and $1 which is the fee for the band. Although the fee for the band is a fixed $750, it is not a fixed cost per person.

Let p = the number of people who attend the event.

So you want 0.5 < 750/p < 1 OR 1 < p/750 < 2 (I took the reciprocal of all three positive numbers and reversed the order of them)

Can you solve 1 < p/750 < 2 for p?

yes, that gives me the following:

 

[Audentes]

@lev888 @Subhotosh Khan @Jomo thank you for the help ??
and special thanks to @JeffM for being the first to get me to the answer

you are all better than my math teacher lol

 

[lev888]

So, with the 2.25 caterer fee it is 3.25 gtoet 750/p+2.25p ltoet 2.75

which is the above except for the wrong sign, right? Or am I missing something else

No, same mistake. The expression you are comparing to 2.75 and 3.25 is the cost PER PERSON. The cost of the band per person is 750/p - correct. The cost of the caterer per person is ?
Which makes the total cost per person 750/p+ ?