Wendy, the chemist, had a test tube containing a mixture of acid and water. She added an ounce of water, resulting in a mixture that was 20% acid. Then, she added an ounce of acid to that new mixture, and the result was 33⅓% acid. What was the percentage of acid in the original mixture?
At the start: a is the ounces of acid, while w is the ounces of H2O and
| acid | = | x | . |
| total mixture | 100 |
| So (I) |
|
|||||||||
| Then (II), |
|
|||||||||
| Then (III), |
|
Shaking these out, we have
(I) 100a = x(a + w),
(II) 100a = 20(a + w + 1), and
(III) 3(a + 1) = a + w + 2 → 3a + 3 = a + w + 2 → 2a + 1 = w.
(II) 100a = 20(a + w + 1), and
(III) 3(a + 1) = a + w + 2 → 3a + 3 = a + w + 2 → 2a + 1 = w.
Plug (III) into (II):
100a = 20(a + 2a + 1 + 1)
→ 100a = 20(3a + 2)
→ 100a = 60a + 40
→ 40a = 40
→ a = 1 and w = 2a + 1 = 3.
→ 100a = 20(3a + 2)
→ 100a = 60a + 40
→ 40a = 40
→ a = 1 and w = 2a + 1 = 3.
So, returning to the initial (I) above,
| a | = | x | → | 1 | = | 1 | = | x | , so she started with a 25% solution. |
| a + w | 100 | 1 + 3 | 4 | 100 |