Mixture Problems

What we see: a Data Sufficiency mixture percent question. Solution \(S\) is \(2\) parts \(P\) and \(3\) parts \(R\). What percent of \(S\) is ammonia?

Step 1. Write what the percent depends on.
What we see: the mix is fixed at \(2\) parts to \(3\) parts, so \(P\) is \(\dfrac{2}{5}\) of \(S\) and \(R\) is \(\dfrac{3}{5}\).
What it means: percent ammonia in \(S = \dfrac{2 \times (\text{percent in } P) + 3 \times (\text{percent in } R)}{5}\).
What we do and why: the only unknowns are the strength of \(P\) and the strength of \(R\). A statement that gives both answers the question.

Step 2. Statement 1 alone.
What we see: \(P\) is \(10\%\) ammonia and \(R\) is \(3\%\).
What it means: percent in \(S = \dfrac{2(10) + 3(3)}{5} = \dfrac{29}{5} = 5.8\%\).
What we do and why: one answer. Sufficient.

Step 3. Statement 2 alone.
What we see: there were \(10\) liters of \(S\).
What it means: that tells us the amount, \(4\) liters of \(P\) and \(6\) liters of \(R\), but not how strong either one is.
What we do and why: the ammonia percent could be anything. Not sufficient.

The trap: thinking you need the total volume to find a percent, and picking both statements together. A percent compares two amounts, so the total cancels out. \(10\) liters or \(1{,}000\) liters of this mix are both \(5.8\%\) ammonia.
Faster way: once you see the mix ratio is given, ask only "do I know each part's strength?" Statement 1 says yes, Statement 2 says no.
Takeaway: A mixture's percent depends only on the mix ratio and each part's percent. The total amount never changes it.

Answer: A