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