let the fraction be

p/q, where p and q are integers and q<>0

that is p is the numerator and q is the denominator

when 3 is added to both its numerator and denominator its value is decreased by 9/10

(p + 3)/ (q + 3) = p/q - 9/10

(p + 3)/ (q + 3) = (10p - 9q)/(10q)

(p + 3)10q = (10p - 9q)(q + 3)

10pq + 30q = 10pq + 30p - 9q^2 - 27q

- 9q^2 - 57q + 30p = 0

But when subtracted its value is increased by 9/10

(p - 3)/ (q - 3) = p/q + 9/10

(p - 3)/ (q - 3) = (10p + 9q)/(10q)

(p - 3)10q = (10p + 9q)(q - 3)

10pq - 30q = 10pq - 10p + 9q^2 - 27q

9q^2 + 3q - 10p = 0

by solving the system

- 9q^2 - 57q + 30p = 0

9q^2 + 3q - 10p = 0

we find

p = 7.2

q = 2.6667

click here to see the solution

https&amp;#x3a;&amp;#x2f;&amp;#x2f;www.quickmath.com/webMathematica3/quickmath/equations/solve/advanced.jsp#c=solve_advancedsolveequations&amp;amp;v1=-+9q%5E2+-+57q+%2B+30p+%3D+0+%0A9q%5E2+%2B+3q+-+10p+%3D+0&amp;amp;v2=p%0Aq

since the solutions are not integers follows the problem has no solutions - there is no such fraction.