In this problem, concentrations are the ratios of the pure vinegar volume to the total volume.



Let x be the amount of the 9%  vinegar needed (in milliliters), and

let y be the amount of the 14%  vinegar needed.



The amount of the "pure" vinegar in the 9% mixture  is 0.09x mL.

The amount of the "pure" vinegar in the 14% mixture is 0.14y mL.

The resulting mixture contains  0.09x + 0.14y mL of the pure vinegar and has the volume of 310 mL.


Thus you have these two equations


    x + y = 310    milliliters           (1)    (the total volume)

     = 0.13.                 (2)    (the resulting mixture concentration)



From equation  (1), express  x = 310 - y.  Substitute it into equation (2) and multiply both sides of this equation by 310. 
You will get


    0.09*(310-y) + 0.14y = 0.13*310.


From the last equation express y and calculate


    y =  = 248 mL of the 14% vinegar are needed.


Then from equation (1),  x = 310 - 248 = 62 mL of the 9% vinegar are needed.


Answer. 248 mL of the 14% vinegar  and  62 mL of the 9% vinegar are  needed.


Check.   = 0.13 = 13%.   ! Correct concentration !