[Please Help] Math Problem

Hello again Melody :)

Thank you very much. I truly appreciate your help in resolving the problem.

Hi Arshan,

It is nice to meet you :)

I've been puzzling over this questions.

It is late - I am making excuses for the fact that my logic may be totally wrong... BUT

I divided all the number by 25 just to make them smaller and easier to work with:

Since the sugar in the 3 cakes is in the ratio  3:2:1

and the sale price of each cake is in the ratio 6:4:2  

(The ratios are the same but dollars is twice as big as grams.)

The maximum amount of sugar is 280*25g

So the maximum sale price must be no more than $560*25 = $14000

I drew this graph to help me.  I used x y and z for the number of A,B and C cakes produced.

https://www.desmos.com/calculator/bmfpvxwkoc

The maximum sale price is on the green line but it must also be in the shaded region so it seems to me that the maximum sale price is where between A=80, B=0 and C=40 and A=40, B=80 and C=0 as shown on the graph

BUT

to answer this question you do not need to understand all that because it only wants the maximum profit.

OK Maximum sale price is $14000 so

Max profit = 30% of  14000 = $4200

That is what I think anyway. :)

It would be good if another mathematician weighed in on this question.

I am pretty sure my answer is correct but there may be a much easier way to do it    

Hi again arshan,

You don't need to be sorry.

It is good so say so when you do not understand.  

A part of the reason that I didn't explain very well is becasue I am not entirely comfortable with my own logic.

If I do it again maybe I can become more comfortable with it. :)

Here is the problem:

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I can do the equations pretty much like you did.

\(4x+3y+2z\le400 \qquad (1)\\ 3x+2y+z\le280 \qquad (2)\\ \text{Multiply (2) by 2}\\ 6x+4y+2z\le560 \qquad (2b)\\ (2b)-(1)\\ 2x+y\le 160 \)

I haven't used this directly although I guess I could.  (And probably should)

I have gone directly  to using a graph.  But the graph re-confirms this equation.

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In the graph below the orange line represents where all the flour is used and

the blue line is there all the sugar is used

Where the blue and orange lines cross is where all the flour and sugar is used.

Because the ratio of the prices is also the ratio of the sugar I do not need another line.

The cross over points of the graphs changes as the quantity of  number of units of product C represeted by z changes.

BUT

The cross over always lies on the line  y=-2x+160

that is    160=y+2x      Which is the equation that was found algebraically.

This is the interactive graph that you do need to look at and play with.  (it is a bit different from the earlier one)

https://www.desmos.com/calculator/4u7lcelrbk

Here is the graph without the interactivity.  BUT you do need to click into the interactive one above!

So any combination of x,y, and z along that green line will give the maximum sales and therefore the maximum profits.

check.

These are all the combinations that will give the  maximum profit of $4200

NOW, if you are still confused I don't blame you :)

If you think I can help you understand then by all means ask.   :)