# 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?

Aug 24, 2015

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

One-fifth acid means $\text{1 qt acid"/"5 qt solution}$

So the volume of acid in 10 qt of solution is

$\text{Volume of acid" = 10 color(red)(cancel(color(black)("qt soln"))) × "1 qt acid"/(5 color(red)(cancel(color(black)("qt soln")))) = "2 L acid}$

We can do a similar calculation for the other solutions.

We add $x \text{ qt}$ of one-half acid to $\text{10 qt}$ of one fifth acid to get $10 + x \text{ qt}$ 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

$5 x + 20 = 3 \left(10 + x\right)$

$5 x + 20 = 30 + 3 x$

$2 x = 10$

$x = 5$

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

$\frac{45}{10} = \frac{45}{10}$

It checks!