Just a quick math question...

I wanted to confirm an equasion and was hoping to receive some feedback from the math guru's in the crowd.

I am selling 5 type of dogs.

Dog 4 sales cannot exceed l more than 10% of the total amount of dogs sales 1-3

Is the equation

X4 </= .1X1 + .1X2 +.1X3

which in standard form should be

.1X1 + .1X2 + .1X4 - X4 </= 0

Is this the way I would write this mathmatically?

Thanks!

 

You are off by one dog that you can sell

Dog 4 sales cannot exceed l more than 10% of the total amount of dogs sales 1-3

Dog4 = X4
Dog1 = X1
Dog2 = X2
Dog3 = X3

Thus: X4 <= 0.1*(X1 + X2 + X3) +1
or

X4 - 0.1*(X1 + X2 + X3) <=1
X4 - 0.1X1 - 0.1X2 - 0.1X3 <=1

 

I am not a guru math, but I am pretty sure you got the sign of that inequality wrong.

X4 </= .1X1 + .1X2 +.1X3

That one was correct (in my view), and would solve the problem (also in my view). However whenyou rearrange it, I think you do it wrong

.1X1 + .1X2 + .1X4 - X4 </= 0

This is how todo it.

X4 </= .1X1 + .1X2 +.1X3

Then you sustract X4 to BOTH sides of the inequality. Thus you get:

X4 - X4 </= .1X1 + .1X2 +.1X3 - X4

Reorganizaing again,


0 </= .1X1 + .1X2 +.1X3 - X4


Or,


.1X1 + .1X2 +.1X3 - X4 >/= 0


That´s your inequality (if I am correct). I don´t understand what Mike did so I may be totally wrong.



Regards

 

Unless you have to sell one and only one dog of type five.

 

Back to the problem.

Dog 4 sales cannot exceed l more than 10% of the total amount of dogs sales 1-3

Dog4 = X4

For example, if,

Dog1 = X1 = 10
Dog2 = X2 = 10
Dog3 = X3 = 10

Then by your formula: X4 </= .1(10) + .1(10) +.1(10) = 3

or 1 + 1 + 1 - 3 >/= 0, while it is = 0, it is not >= 0

if you use X4 <= 0.1*(X1 + X2 + X3) +1 then

X4 <= 0.1*(10+10+10) +1 = 4
which is 1 more than 10% of the total amount of dogs sales 1-3

and

X4 - 0.1*(10 + 10 + 10) <=1

4 - 3 <= 1

(BTW: no mention was made in the data of Dog 5, whihc would indicate no constraint on their sales).

 

I think this is the part I find confussing, Mike. X4 =0 is a solution (the trivial solution) even for that particular case you mention X1 = X2 = X3 = 10. I mean, 0 is clearly greater than or equal to zero.Thus 0 >=0; so the inequality, even in the specific case you mention, still holds (when X1 = X2 = X3 = 10). Just for the sake of arguing about an interesting subject, I don´t see the necessity to include that ONE in the right part of the inequality. Why? (I have read your explanation 5 times but I still don´t understand WHY?)

(I just hope we are not misleading Mr. Engineer who already have enough problems of his own )

 

Since the 5th type of dog has no bearing on this problem, your first equation looks fine.

The standard form, however, would look like this:

X4 - .1X1 - .1X2 - .1X3 <= 0

 

Yes, I agree with that too, PhD2B.

 

One minor point still, the question did not read

Dog 4 sales cannot exceed 10% of the total amount of dogs sales 1-3


But rather

Dog 4 sales cannot exceed l more than 10% of the total amount of dogs sales 1-3

.1X1 + .1X2 +.1X3 - X4 >/= 0

Has X4 being 1 less than the correct answer.

.1X1 + .1X2 +.1X3 - X4 </= 0

Would be correct as X4 can be equal to or greater than .1X1 + .1X2 +.1X3

 

Ok, I read l and not 1. I thought it was a typo after I read the solution Mr. Engineer proposed for this problem. This is then the origin of the discrepancies,and that explains everything.


Thanks

 

I also read l as a typo and not as 1.

Only Mr. Engineer knows for sure...

 

I actually went over this problem for 4 additional hours until I got it right. I appreciate all of the great responses. Like I said before, I think I am finally catching on to what is required for this class!