Question 1210328
.
A student requires at least three school uniforms and three social clothes. 
The uniform costs N600 and the social wear costs N360 each. 
If the student has N3 600 to spend and he decides to spend as much as possible of the N3 600, 
a) in how many ways can he spend the money? 
b) Which of these ways uses the much
(MOST, not "much") of the N3600
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In this problem,  the number of points under the interest is  4.


Imagine for a minute,  that there are  40  or  70  solutions in integer numbers
in such or similar problem.  How will you find them ?


There is very mechanical way to do it,  practically without thinking.


Plot the restriction lines in a graph paper.


Then you will see the integer grid points inside the feasible area
(may be,  including those on the boundaries).


Count all of them,  and you will get the answer. 


Such plot can be produced electronically,  using appropriate plotting tool, for example,  in this site

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


You only need to take care about units on the axes,  in order for
the points of interest were all integer points of the grid.


For it,  you should simplify your restriction equations to make them
as simple as possible - - - or to select the units in the axes by an appropriate way.


I do not state that this method will work everywhere or always,  but in many practical cases 
(in  95%  typical school  Math problems of this kind)  it will work effectively.


So,  keep it in your mind.