Word Problems

What we see: a Data Sufficiency ratio question. Two kinds of tickets, each at one fixed price. We need the ratio of Front Row money to Upper Level money.

Step 1. Write what the ratio is made of.
What we see: money from a kind of ticket is (number sold) times (price).
What it means: the ratio is \(\dfrac{F \times f}{U \times u}\), where \(F\) and \(U\) are the tickets sold and \(f\) and \(u\) are the prices.
What we do and why: we need the ratio of the counts, \(\frac{F}{U}\), and the ratio of the prices, \(\frac{f}{u}\). Knowing both ratios is enough; we never need the real numbers.

Step 2. Statement 1 alone.
What we see: 3 times as many Front Row tickets as Upper Level.
What it means: \(\frac{F}{U} = 3\), so the ratio is \(\dfrac{3f}{u}\).
What we do and why: nothing about prices. If the prices are equal the ratio is 3 to 1; if Upper Level costs 3 times more it is 1 to 1. Not enough.

Step 3. Statement 2 alone.
What we see: each Upper Level ticket costs 20 euros more.
What it means: \(u = f + 20\). That is a gap, not a ratio. And we know nothing about how many of each sold.
What we do and why: not enough.

Step 4. Both together.
What we see: ratio \(= \dfrac{3f}{f + 20}\).
What it means: try numbers. If \(f = 20\): \(\frac{60}{40} = \frac{3}{2}\). If \(f = 40\): \(\frac{120}{60} = 2\). Different prices give different ratios.
What we do and why: a 20 euro gap does not fix how the two prices compare. So both together are still not enough.

The trap: C. Statement 2 looks like it finishes the job, since it ties the two prices together. But a fixed difference (20 euros more) is not a fixed ratio (twice as much). A ratio needs a "times", not a "plus".
Faster way: after statement 1, ask only "do I know \(\frac{f}{u}\)?" Statement 2 gives \(u - f\), which never fixes \(\frac{f}{u}\) unless one price is known.
Takeaway: a ratio question needs ratios. "More than" or "less than" by a fixed amount does not give a ratio unless one actual value is also known.

Answer: E