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Numerators sum also (Posted on 2014-01-13) Difficulty: 3 of 5
In the equation 4/5 + 5/275 = 9/11 note that 4+5=9 but one of the fractions actually reduces.

Find a similar equation of the form

a/d + b/e = c/f

Where:
a+b=c,
all six numbers are different positive integers,
none of the fractions reduces,
and the largest of the six is minimum.

No Solution Yet Submitted by Jer    
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Solution computer solution Comment 1 of 1

  5    Highallow=25
 10    dim Used(Highallow)
 20    for A=1 to Highallow
 30      Used(A)=1
 40      for D=1 to Highallow
 50        if Used(D)=0 then
 60         :Used(D)=1
 70         :if gcd(A,D)=1 then
 80            :F1=A//D
 90            :for B=1 to Highallow
100               :if Used(B)=0 then
110                 :Used(B)=1
120                 :for E=1 to Highallow
130                  :if Used(E)=0 then
140                   :Used(E)=1
150                     :if gcd(B,E)=1 then
160                     :F3=F1+B//E
170                     :C=num(F3):F=den(F3)
175                     :if C<=Highallow and F<=Highallow then
180                     :if Used(C)=0 and Used(F)=0 and A+B=C and B>A then
181                       :print F1;B//E;F3;tab(23);max(A,B,C,D,E,F)
184                     :endif
185                     :endif
310                   :endif
320                   :Used(E)=0
330                  :endif
340                 :next E
350                 :Used(B)=0
360               :endif
370            :next B
380         :endif
390         :Used(D)=0
400      next D
405     Used(A)=0
410    next A
 
  finds all such cases where no integer larger than 25 is used:
 
 1//2  3//10  4//5      10
 1//2  5//14  6//7      14
 1//2  7//18  8//9      18
 1//2  9//22  10//11    22
 1//6  2//15  3//10     15
 1//6  4//21  5//14     21
 2//3  4//21  6//7      21
 2//3  5//24  7//8      24
 3//2  9//10  12//5     12
 3//2  15//14  18//7    18
 3//20  9//4  12//5     20
 4//3  8//21  12//7     21
 5//3  10//21  15//7    21
 5//6  20//21  25//14   25
 5//12  10//3  15//4    15
 7//6  14//15  21//10   21
 7//12  14//3  21//4    21
 8//3  16//21  24//7    24

   
 reformatted and sorted:
                     largest
                     integer
 1/2 + 3/10 = 4/5      10
 3/2 + 9/10 = 12/5     12
 1/2 + 5/14 = 6/7      14
 1/6 + 2/15 = 3/10     15
 5/12 + 10/3 = 15/4    15
 1/2 + 7/18 = 8/9      18
 3/2 + 15/14 = 18/7    18
 3/20 + 9/4 = 12/5     20
 1/6 + 4/21 = 5/14     21
 2/3 + 4/21 = 6/7      21
 4/3 + 8/21 = 12/7     21
 5/3 + 10/21 = 15/7    21
 7/12 + 14/3 = 21/4    21
 7/6 + 14/15 = 21/10   21
 1/2 + 9/22 = 10/11    22
 2/3 + 5/24 = 7/8      24
 8/3 + 16/21 = 24/7    24
 5/6 + 20/21 = 25/14   25


 
The first, of course, is the answer, involving no integer larger than 10: 1/2 + 3/10 = 4/5.


  Posted by Charlie on 2014-01-13 17:40:03
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