Question 449·Hard·Algebra

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A chemist has 20 liters of a solution that is 30% acid by volume. She plans to add some of a 50% acid solution and some of a 10% acid solution so that the final mixture will be 40 liters of a 35% acid solution.

How many liters of the 50% acid solution should the chemist add?

In a mixture problem, the cue is that solutions with different concentrations are combined to reach a specified final volume and concentration. First use the volumes to express both added amounts with one variable. Then balance liters of acid, not the percentages themselves, and solve the equation in Desmos. The tempting trap is taking the final percent of the starting volume instead of the final volume.

Hints

Hint 1
A volume balance counts all the liquid, regardless of acid strength. Subtract the starting volume from the final volume to find how much is added. If $x$ liters are $50\%$ solution, how much must be $10\%$ solution?
Hint 2
A concentration tells you what share of a solution is acid. Multiply each solution’s volume by its acid percentage, then add those acid amounts. The final percentage applies to which volume?

Step-by-step

Balance the acid in the mixture
Step 1
Find the total volume added
The mixture grows from $20$ liters to $40$ liters, so the added volume is $40-20=20$ liters.
Step 2
Write both added volumes with one variable
Let $x$ be the liters of $50\%$ solution added. Since the two additions total $20$ liters, the $10\%$ solution accounts for $20-x$ liters.
Step 3
Balance liters of acid
Each solution contributes its concentration times its volume in liters of acid. The original solution contributes $0.30(20)$ , and the additions contribute $0.50x$ and $0.10(20-x)$ . Together, they must equal $35\%$ of the final 40 liters, not $35\%$ of the starting $20$ :
$0.30(20)+0.50x+0.10(20-x)=0.35(40).$
Step 4
Solve for the requested volume
Type the acid equation in Desmos with $x_1$ in place of $x$ and $\sim$ in place of $=$ . That tells Desmos to find the value that makes the two sides equal. Under PARAMETERS, it reports $x_1=15$ . Since $x$ counts the $50\%$ solution, the chemist should add $15$ liters of it. Choice C.

Question 449·Hard·Algebra

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