If a certain number is added to both the numerator and denominator of the fraction 5 6 56, the result is 3 4 34 . find the number.

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5. let x=the number to be added, then we have: 
(7+x)/(9+x)=5/7
7+x=5
x=-2
or
9+x=7
x=-2
another way:
Multiply each side by 7(9+x)---the LCM
7(7+x)=5(9+x) get rid of parens 
49+7x=45+5x subtract 7x and also 45 from both sides
49-45+7x-7x=5x-7x collect like terms 
4=-2x divide both sides by -2
x=-2 
CK
(7-2)/(9-2)=5/7
5/7=5/7 
7. 
Let x=one even integer; reciprocal=1/x
and (x+2)=the other; reciprocal=1/(x+2)