Thank you!
Hi phrontister;
Okay, thanks for coming in. See you later.
Hi chen.aavazi;
Calling x = apples and y = bananas, we can reduce the system down to:
A)
100 - x > 3 y
3 y > 4 x
4 x > 100 - y
We can now graphically solve for the answer, I used Geogebra.
http://i.imgur.com/4A8ZlQC.png
The answer lies in the small darker triangle and must be an integer for (x,y). That reduces down the possibilities to just a few, as a matter of fact there is just one integer coordinate in there.
We can go a bit further with a trick that is used in numerical work which I invented?!
The three inequalities in A) can be changed to equations with the addition of what I call a slack variable... This is a term used in linear optimization but I give it an added meaning.
If 100 - x > 3 y we only have to add something to the RHS to pick up the slack. I add e ( slack variable) which balances the inequality and gives us the equality of 100 - x = 3 y + e. We do that with all 3 inequalities and hope for the best.
100 - x = 3 y + e
3 y = 4 x + e
4 x = 100 - y + e
this can be solved by ordinary means
x = 19.0476, y = 26.1905, e = 2.38095
Remembering that x = apples and y equals bananas we can guess and try the closest integers and get apples = 19 and bananas = 26. We test to see that we are right and we are done!
Here's an algebraic method with no trial and error.
Step 1 eliminate a variable so that it is a 2-D linear programming question. Using A, B and P and working in cents
P = 100 - A - B so
P > 2B becomes 100 > A + 3B
3A > P becomes 4A + B > 100
We also have
3B > 4A and 3A > 2B
If you were to plot (1) 3B = 4A (2) 3A = 2B (3) A + 3B = 100 and (4) 4A + B = 100 these lines enclose a quadrilateral and the solution must lie therein.
In fact a sketch is sufficient.
The top left corner is found by solving (1) together with (3) from that we find that A < 20 Continuing in this way you get to a single solution for A and B and hence for P.
Bob
]]>bobbym, silly question, where did the first and third equation derive from?
Remember the first constraint given?
banana + apple + peach = 100
You solve for the peach first and get peach = 100 - banana - apple, then you substitute that into the inequalities. Okay?
Great solution!
Uh oh! Great solutions are almost always wrong. I checked it again and it has a chance of being right because there is nothing great about it. It is a back engineering job since I already had the answer just a bit later than phrontister.
]]>Hi phrontister;
Okay, thanks for coming in. See you later.
Hi chen.aavazi;
Calling x = apples and y = bananas, we can reduce the system down to:
A)
100 - x > 3 y
3 y > 4 x
4 x > 100 - y
We can now graphically solve for the answer, I used Geogebra.
http://i.imgur.com/4A8ZlQC.png
The answer lies in the small darker triangle and must be an integer for (x,y). That reduces down the possibilities to just a few, as a matter of fact there is just one integer coordinate in there.
We can go a bit further with a trick that is used in numerical work which I invented?!
The three inequalities in A) can be changed to equations with the addition of what I call a slack variable... This is a term used in linear optimization but I give it an added meaning.
If 100 - x > 3 y we only have to add something to the RHS to pick up the slack. I add e ( slack variable) which balances the inequality and gives us the equality of 100 - x = 3 y + e. We do that with all 3 inequalities and hope for the best.
100 - x = 3 y + e
3 y = 4 x + e
4 x = 100 - y + e
this can be solved by ordinary means
x = 19.0476, y = 26.1905, e = 2.38095
Remembering that x = apples and y equals bananas we can guess and try the closest integers and get apples = 19 and bananas = 26. We test to see that we are right and we are done!
Okay, thanks for coming in. See you later.
Hi chen.aavazi;
Calling x = apples and y = bananas, we can reduce the system down to:
A)
100 - x > 3 y
3 y > 4 x
4 x > 100 - y
We can now graphically solve for the answer, I used Geogebra.
The answer lies in the small darker triangle and must be an integer for (x,y). That reduces down the possibilities to just a few, as a matter of fact there is just one integer coordinate in there.
We can go a bit further with a trick that is used in numerical work which I invented?!
The three inequalities in A) can be changed to equations with the addition of what I call a slack variable... This is a term used in linear optimization but I give it an added meaning.
If 100 - x > 3 y we only have to add something to the RHS to pick up the slack. I add e ( slack variable) which balances the inequality and gives us the equality of 100 - x = 3 y + e. We do that with all 3 inequalities and hope for the best.
100 - x = 3 y + e
3 y = 4 x + e
4 x = 100 - y + e
this can be solved by ordinary means
x = 19.0476, y = 26.1905, e = 2.38095
Remembering that x = apples and y equals bananas we can guess and try the closest integers and get apples = 19 and bananas = 26. We test to see that we are right and we are done!
]]>Catch you later.
]]>Witness the story of Andrew Wiles...
Wow! What persistence!! I just read the Wikipedia article.
]]>Even at my age I don't have centuries to wait for your 100-page proof to appear - and besides, I have to go to bed soon - but I'll hang around here for a bit longer in case you can get your idea to work.
All the best with it!
]]>I am looking at an idea now.
]]>Do you have any idea about how to solve this mathematically?
]]>