Question 173693
lets call the numbers a and b  
:
{{{ab/ba=4/7}}}
:
but remember ab can also be written as 10a+b and ba can be written as 10b+a
:
so{{{(10a+b)/(10b+a)=4/7}}}
:
now cross multiply
:
4(10b+a)=7(10a+b)
:
40b+4a=70a+7b
:
66a-33b=0:.......eq 1---->or revised eq 1 {{{highlight(a=b/2)}}}
:
we also know that 
:
{{{(10a+b+16)/(10b+a-5)=7/4}}}(the reciprocal of 4/7)
:
so lets cross multiply again
:
4(10a+b+16)=7(10b+a-5)
:
40a+4b+64=70b+7a-35
:
-33a+66b=99.......eq 2
66a-33b=0........eq 1
:
multiply eq 1 by 2 and add the equations together
:
a's are eliminated and we are left with 99b=198
:
{{{highlight(b=2)}}}
:since we know that a=b/2 from revised eq 1
:
{{{highlight(a=2/2=1)}}}
:
so the original fraction is  {{{highlight(12/21)}}}...which reduces to 4/7
:
incidently other number that would have worked had it not been for the 2nd equation would be
24/42
36/63
48/84
:
which all reduce to 4/7