Two solution of `H_(2)SO_(4)` of molarities x and y are mixed in the ratio of `V_(1) mL : V_(2) mL` to form a solution of molarity `M_(1)`. If they are mixed in the ratio of `V_(2) mL : V_(1) mL`, they form a solution of molarity `M_(2)`. Given `V_(1)//V_(2) = (x)/(y) gt 1 and (M_(1))/(M_(2)) = (5)/(4)`, then `x : y` is

Explanation

`{:(,M_(1)'V_(1)'+M_(2)'V_(2)'=M_(3)'(V_(1)'V_(2)')),("In 1st case,",x xxV_(1)+y xx V_(2)=M_(1)(V_(1)+V_(2))" ...(i)"),("In 2nd case, ", x xx V_(2)+y xxV_(1)=M_(2)(V_(1)+V_(2))" ...(ii)"):}`
Dividing eqn. (i) eqn. (ii), we get
`(xV_(1)+yV_(2))/(xV_(2)+yV_(1))=(M_(1))/(M_(2))=(5)/(4)" Given)"`
`"or "4(xV_(1)+yV_(2))=5(xV_(2)+yV_(1))`
`"or "4xV_(1)+4yV_(2)=5xV_(2)+5yV_(1)`
`"or "(4x-5y)V_(1)=(5x-4y)V_(2)`
`"or "(V_(1))/(V_(2))=(5x-4y)/(4x-5y)=(x)/(y)" (Given)"`
`"or "((5x)/(y)-4)/((4x)/(y)-5)=(x)/(y)" (Dividing numerator and denominator by y)"`
`"or "(5x)/(y)-4=(4x^(2))/(y^(2))-4(x)/(y)" or "(4x^(2))/(y^(2))-10(x)/(y)+4=0`
`"Putting "(x)/(y)=z, 4z^(2)-10z+4=0`
`therefore" "z=(-b pmsqrt(b^(2)-4ac))/(2a)=(10pmsqrt(100-64))/(8)=(10pm6)/(8)=(16)/(8)" or "(4)/(8)," i.e., 2 or "(1)/(2)`
`"As "(x)/(y) gt 1," hence "(x)/(y)=2," i.e., " x:y = 2:1`