Chapter C, Problem 1P

To determine

To find:

How to obtain 6 liters of a 40% solutions from a mixture of 20% dye solution and 50% dye solution.

Answer to Problem 1P

Solution:

2 liters of the 20% dye solution and 4 liters of the 50% dye solution should be mixed to obtain 6 liters of a 40% solution.

Explanation of Solution

General strategy for problem solving:

1. Understand the problem. During this step, become comfortable with the problem. Some ways of doing this are as follows:

Read and reread the problem.

Choose a variable to represent the unknown.

Construct a drawing whenever possible.

Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem.

2. Translate the problem into an equation.

3. Solve the equation.

4. Interpret the results: Check the proposed solution in the stated problem and state your conclusion.

Calculation:

We have to find how much 20% dye solution and 50% dye solution should be mixed to obtain 6 liters of a 40% solution.

Let x = the amount of the 50% of total solution; then

$(6-x)$ = the amount of the 20% of total solution.

Now we have to solve the problem in to an equation.

The amount of acid in each solution can be found by multiplying the acid strength of each solution by the number of liters.

The following table summarizes the given information.

	No. of Liters $·$ Acid Strength=Amount of acid
50% solution	x$·$ 50%=0.50x
70% solution	(6-x)$·$ 70%=0.70(6-x)
40% Solution needed	6 $·$ 40%=0.40(6)

The amount of acid in the final solution is the sum of the amounts of acid in the two beginning solutions.

In words:	Acid in 50% solution	+Acid in 20% solution	= acid in 40% mixture
Translate:	0.50x	+0.20(6-x)	=0.40(6)

Now we have to solve the above equation.

$0.50x+0.20(6-x)=0.40(6)$

We have to apply the distributive property, $a\left(b+c\right)=ab+ac$

$0.50x+1.2-0.2x=2.4$

Now combine like terms

$\begin{array}{l}0.50x-0.2x=2.4-1.2\\ 0.3x=1.2\end{array}$

Divide both sides by 0.3, we get

$\begin{array}{l}\frac{0.3x}{0.3}=\frac{1.2}{0.3}\\ x=4\end{array}$

Now we have to interpret the results.

The amount of the 50% of total solution (x) =4liters.

The amount of the 20% of total solution (6-x)$=6-4=2$ liters.

Conclusion:

If 4 liters of the 50% solution are mixed with 2 liters of the 20% solution, the result is 6 liters of a 40% solution.