Question 1140817
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            The answer    {{{m/n}}} = {{{16/84}}}   which tutor @ankor found,   is not the solution to the problem, 


            since the fraction   {{{16/84}}}   is not reduced.   I came to bring the correct answer and solution.



<pre>
m + n = 100   ====>  n = 100-m.


So the inequality  {{{m/n}}} < {{{1/5}}}   takes the form


    {{{m/(100-m)}}} < {{{1/5}}}

    5m < 100 - m

    6m < 100

    m  < {{{100/6}}} = 16.66...

    m <= 16.


Now we should consider all the fractions  {{{m/(100-m)}}}  with  0 <= m <= 16 and select maximal of that which IS REDUCED.


I did it using EXCEL.  See the table below.



      T   A   B   L   E   


m     100-m      {{{m/(100-m)}}}-reduced or not    Value of {{{m/n}}}


16	84	    No	
15	85	    No	
14	86	    No	
13	87	    Yes                          0.149425287   <<<---===  The maximum value :  the solution and the answer
12	88	    No	 
11	89	    Yes	                         0.123595506
10	90	    No	 
9	91	    Yes	                         0.098901099
8	92	    No	 
7	93	    Yes	                         0.075268817
6	94	    No	
5	95	    No	
4	96	    No	
3	97	    Yes	                         0.030927835
2	98	    No	
1	99	    Yes	                         0.01010101


<U>ANSWER</U>.  The greatest possible such fraction is  {{{13/87}}} = 0.149425287.
</pre>

Solved, answered, explained and completed.