How many quarts of a solution half of which is acid must be added to 10 quarts of a solution one-fifth of which is acid to from a solution which is three-tenths acid?

You would have to add 5 qt of the 50 % acid.

One-fifth acid means #"1 qt acid"/"5 qt solution"#

So the volume of acid in 10 qt of solution is

#"Volume of acid" = 10 color(red)(cancel(color(black)("qt soln"))) × "1 qt acid"/(5 color(red)(cancel(color(black)("qt soln")))) = "2 L acid"#

We can do a similar calculation for the other solutions.

We add #x " qt"# of one-half acid to #"10 qt"# of one fifth acid to get #10+x " qt"# of three-tenths acid.

#x color(red)(cancel(color(black)("qt soln"))) × (1 color(red)(cancel(color(black)("qt acid"))))/(2 color(red)(cancel(color(black)("qt soln")))) + 10 color(red)(cancel(color(black)("qt soln"))) × (1 color(red)(cancel(color(black)("qt acid"))))/(5 color(red)(cancel(color(black)("qt soln")))) = (10+x) color(red)(cancel(color(black)("qt soln"))) × (3 color(red)(cancel(color(black)("qt acid"))))/(10 color(red)(cancel(color(black)("qt soln"))))#

#x/2+10/5=(10+x)× 3/10#

#5x+20=3(10+x)#

#5x+20=30+3x#

#2x = 10#

#x=5#

You would add 5 qt of the 50 % acid.

Check:

#5/2+10/5=(10+5)× 3/10#

#25/10+20/10 = 15×3/10#

#45/10=45/10#

It checks!