2410.34 – Wendy the Chemist

Wendy is a chemist. She has a test tube containing a dilute mixture of acid and water. She adds an ounce of water and the mixture is now 20 percent acid. Then adds a ounce of acid and the result is 33.3 percent acid.

What was the percent acid originally?

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Solution

Let the mixture contain ww ounces of water and aa ounces of acid. Let xx be the initial percentage of acid. Then

(1) x100=aa+w. \text{(1) }\dfrac{x}{100} = \dfrac{a}{a+w}.

After adding an ounce of water:

(2) aa+w+1=15, \text{(2) }\dfrac{a}{a+w+1} = \dfrac{1}{5},

and after the second addition:

(3) (a+1)a+w+2=13. \text{(3) }\dfrac{(a+1)}{a+w+2} = \dfrac{1}{3}.

Shaking these out gives

(1)100a=x(a+w),(2)5a=a+w+1,(3)3(a+1)=a+w+2=(a+w+1)+1.\begin{aligned}\text{(1)} & & 100 a &= x (a+w), \\\text{(2)} & & 5a &= a+w+1, \\\text{(3)} & & 3(a+1) &= a + w + 2 = (a+w+1) + 1.\end{aligned}

Taking 100a100a and plugging in (2), and then (3):

100a=20⋅5a=20(a+w+1)=20(3(a+1)−1)=60a+40.\begin{aligned}100 a &= 20 \cdot 5a \\&= 20 (a+w+1) \\&= 20 (3 (a + 1) - 1) \\&= 60 a +40.\end{aligned}

It follows that a=1a = 1 and w=3w = 3. So, to answer the question:

x100=aa+w=11+3=14=25100.\dfrac{x}{100} = \dfrac{a}{a+w} = \dfrac{1}{1+3} = \dfrac{1}{4} = \dfrac{25}{100}.

Wendy started with a 25 percent solution.