SOLUTION: the reduced fraction m/n is less than 1/5 and m+n=100 find the greatest possible such fraction

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Question 1140817: the reduced fraction m/n is less than 1/5 and m+n=100 find the greatest possible such fraction
Found 3 solutions by ankor@dixie-net.com, ikleyn, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
the reduced fraction m/n is less than 1/5 and m+n=100
find the greatest possible such fraction.
m + n = 100
m = -n + 100
therefore:
%28-n%2B100%29%2Fn < 1%2F5
cross multiply, reverse the inequality
n > 5%28-n%2B100%29
n > -5n + 500
n + 5n > 500
6n > 500
n > 500/6
n > 83.333
n = 84 the nearest integer that satisfies this
therefore
16%2F84 is the greatest possible fraction

Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.

            The answer   m%2Fn = 16%2F84   which tutor @ankor found,   is not the solution to the problem,

            since the fraction  16%2F84  is not reduced.  I came to bring the correct answer and solution. 


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


So the inequality  m%2Fn < 1%2F5   takes the form


    m%2F%28100-m%29 < 1%2F5

    5m < 100 - m

    6m < 100

    m  < 100%2F6 = 16.66...

    m <= 16.


Now we should consider all the fractions  m%2F%28100-m%29  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%2F%28100-m%29-reduced or not    Value of m%2Fn


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


ANSWER.  The greatest possible such fraction is  13%2F87 = 0.149425287.

Solved, answered, explained and completed.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


m/n < 1/5 and m+n=100 means

m%2F%28100-m%29%3C1%2F5
5m+%3C+100-m
6m+%3C+100
m+%3C+100%2F6+=+50%2F3

Since m is an integer, the maximum possible value for m is 16.

To maximize the value of m/n, with the requirement that m+n=100, m must be as large as possible. So the possible values we need to look at for m are 16, 15, 14, ....

If m is even, then n=100-m will also be even, and m/n will not be a reduced fraction.

If m is a multiple of 5, then n=100-m will also be a multiple of 5, and m/n will not be a reduced fraction.

So m can't be 16, 15, or 14. So

m=13 --> n = 87

The largest fraction that satisfies the conditions of the problem is

m/n = 13/87