What we see: a Data Sufficiency rate question. We need one number: how many papers Bill and Tim proofread together in 60 minutes.
Step 1. Set up what we need.
What we see: working together, their papers add up.
What it means: if Bill does \(b\) papers in 60 minutes and Tim does \(t\), together they do \(b + t\).
What we do and why: we need the value of \(b + t\). Knowing how the two compare is not enough; we need real counts.
Step 2. Statement 1 alone.
What we see: Bill does twice as many as Tim, so \(b = 2t\).
What it means: together they do \(3t\), but \(t\) could be 10 (total 30) or 50 (total 150).
What we do and why: a ratio gives the shape, not the size. Not sufficient.
Step 3. Statement 2 alone.
What we see: Tim does 30 fewer than Bill in 60 minutes, so \(b - t = 30\).
What it means: Bill 40 and Tim 10 gives 50; Bill 100 and Tim 70 gives 170.
What we do and why: a gap gives the difference, not the total. Not sufficient.
Step 4. Both statements together.
What we see: \(b = 2t\) and \(b - t = 30\).
What it means: put the first into the second: \(2t - t = 30\), so \(t = 30\) and \(b = 60\).
What we do and why: together they do \(60 + 30 = 90\) papers in 60 minutes. One value. Sufficient.
The trap: (A). Statement 1 feels like a full rate fact, but "twice as many" is only a comparison. Without one real number, the total can be anything.
Faster way: count the unknowns. Two unknowns (\(b\) and \(t\)) need two different equations. Each statement is one equation, and they are not the same equation, so together they solve it.
Takeaway: a ratio plus a difference between the same two amounts always pins both amounts down.
Answer: C
