Two solutions of different concentrations of acid are mixed creating 40 mL of a solution that is 32% acid. One-quarter of the solution is made up of a 20% acid solution. The remaining three-quarters is made up of a solution of unknown concentration, c.

Question

Which equation can be used to determine c, the unknown concentration?

30c + 10(0.2) = 40(0.32)
StartFraction 3 Over 4 EndFraction left-parenthesis c right parenthesis plus left-parenthesis StartFraction 1 Over 4 EndFraction left-parenthesis 0.2 right-parenthesis right-parenthesis equals 40 times 0.32.(c) + StartFraction 1 Over 4 EndFraction cup.(0.2) = 40(0.32)
(c)( (0.2)) = 40(0.32)
30(c)(10(0.2)) = 40(0.32)

Answer 1

To set up the equation to determine the unknown concentration \( c \), consider the following:

1. You have a total of 40 mL of a solution that is 32% acid, which means that the total amount of pure acid in the solution is \( 40 \times 0.32 = 12 \) mL of pure acid.

2. One-quarter of this solution (i.e., \( 40 \times \frac{1}{4} = 10 \) mL) is from a 20% acid solution. The amount of pure acid in this part is \( 10 \times 0.2 = 2 \) mL.

3. The remaining three-quarters of the solution is \( 30 \) mL (since \( 40 - 10 = 30 \) mL). This portion has an unknown concentration \( c \).

4. The amount of pure acid from this \( 30 \) mL part can be expressed as \( 30c \) mL.

From the above analysis, we can set up the equation based on the total amount of pure acid:

\[ 30c + 2 = 12 \]

Now you can solve for \( c \). Rearranging the equation gives:

\[ 30c = 12 - 2 \] \[ 30c = 10 \] \[ c = \frac{10}{30} = \frac{1}{3} \approx 0.3333 \text{ or } 33.33% \]

Thus, the correct equation to use is:

\( 30c + 2 = 12 \) (This translates into \( 30c + \frac{1}{4}(0.2) = 40(0.32) \)).

However, since \( 30c + 2 = 12 \) is the simplest form indicating the relationship between the concentrations, it is recommended to use this as your base equation.

None of the options listed directly represent this, but the correct process leads us to this conclusion.