If a number is added to the numerator of 4/9 and the same number is subtracted from the denominator the result is -9. find the number

What a delightful little problem !

You said that       (4 + x) / (9 - x) = -9

Multiply each side by  (9 - x) :      (4 + x) = -9 (9 - x)

Eliminate parentheses:                4 + x = -81 + 9x

Add  81  to each side:                85 + x  =  9x

Subtract  'x'  from each side:         85  =  8x

Divide each side by  8  :              85/8 = x    (or 10.625)

(4+n)/(9-n)=-9

4+n=-9(9-n)

4+n=-81+9n

4+81=9n-n

85=8n

n=85/8=80/8+5/8=10+5/8