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?

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) }\frac{x}{100} = \frac{a}{a+w}.

After adding an ounce of water:

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

and after the second addition:

(3) (a+1)a+w+2=13. \text{(3) }\frac{(a+1)}{a+w+2} = \frac{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.\frac{x}{100} = \frac{a}{a+w} = \frac{1}{1+3} = \frac{1}{4} = \frac{25}{100}.

Wendy started with a 25 percent solution.