Just write the strength of available solution (i.e. #80%# and #20%#) at two ends (as shown below). Then write the desired strength (this value will be between the two earlier value) which here is #65%#.
Draw a cross and find the differences as shown. Here they are #45%# and #15%#. Simplify to get the ratio #3:1#.
Now divide desired quantity (of desired strength), which is #100# gallons in this ratio.
In this case we need #(100xx3)/(1+3)=75# gallons of #80%# acid solution and
#(100xx3)/(1+3)=45# gallons of #30%# acid solution.
Well more formal way is as follows :
Let us add #x# gallons of #80%# acid solution and #100-x# gallons of #20%# acid solution.
While #x# gallons of #80%# acid solution will have #x xx80/100# gallons of acid; #60-x# gallons of #20%# acid solution will have #(20(100-x))/100# gallons of acid and total acid would be
#(80x)/100+(20(100-x))/100#
But this should be #65%# of #100# iter i.e. #65# gallons.
Hence #(80x)/100+(20(100-x))/100=65#
or #(4x)/5+20-x/5=65#
or #3x=(65-20)xx5=225#
i.e. #x=225/3=75#
Hence, we need #75# gallons of #80%# acid solution and #100-75=25# gallons of #20%# acid solution.
