Property of fractionsGiven two fractions h k and h ′ k ′ both in reduced form. I am unable to find a case when h + h ′ k + k ′ does not lie in the interval [ h k , h ′ k ′ ] . Is there such a case ?PS: I was able two prove no such case exists for consecutive terms of Farey series. But can't prove in general.

Question

Property of fractionsGiven two fractions       h    k   and             h                                  ′                            k                                  ′                     both in reduced form. I am unable to find a case when             h      +              h                                          ′                                      k      +              k                                          ′                               does not lie in the interval             [            h    k    ,            h                                  ′                            k                                  ′                                ]      . Is there such a case ?PS: I was able two prove no such case exists for consecutive terms of Farey series. But can&#039;t prove in general.

Pettanicej4lyy · Accepted Answer

Let             h      k        =  a and                     h        ′                    k        ′              =  bThen,                     h        +                  h          ′                            k        +                  k          ′                      =                    a        k        +        b                  k          ′                            k        +                  k          ′                    WLOG, let   a  &amp;lt;  b                    a        k        +        b                  k          ′                            k        +                  k          ′                      &amp;gt;                    a        k        +        a                  k          ′                            k        +                  k          ′                      =  a                    a        k        +        b                  k          ′                            k        +                  k          ′                      &amp;lt;                    b        k        +        b                  k          ′                            k        +                  k          ′                      =  b  a  &amp;lt;                    a        k        +        b                  k          ′                            k        +                  k          ′                      &amp;lt;  b  a  &amp;lt;                    a        k        +        b                  k          ′                            k        +                  k          ′                      &amp;lt;  b            h      k        &amp;lt;                    h        +                  h          ′                            k        +                  k          ′                      &amp;lt;                    h        ′                    k        ′            The case where   a  =  b and where   a  &amp;gt;  b is left to the reader as an exercise.

Marlie Cole · Accepted Answer

Let&#039;s prove that, for positive   a  ,  b  ,  c  ,  d, with       a    b    ≤      c    d  , it holds      a    b    ≤            a      +      c              b      +      d        ≤      c    d  The inequality on the left is equivalent to  a  b  +  a  d  ≤  a  b  +  b  cthat is,   a  d  ≤  b  c, which is true.The inequality on the right is equivalent to  a  d  +  c  d  ≤  b  c  +  c  dthat is,   a  d  ≤  b  c, which is true.Note that, if   a      /    b  &amp;lt;  c      /    d, the two inferred inequalities are strict too. The hypothesis about the fractions being in reduced form is redundant.

Property of fractions Given two fractions h k </mfrac> and h

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