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!