After winning \(80\%\) of his first \(40\) matches, Igby won \(50\) percent of his remaining matches. How many total

What we see: a percent Data Sufficiency question. Igby won 80 percent of his first 40 matches and 50 percent of the rest. We need his total wins.

Step 1. Name the one unknown.
What we see: 80 percent of 40 is 32 wins. The rest of the matches are unknown.
What it means: call the remaining matches \(r\). Total matches \(= 40 + r\). Total wins \(= 32 + 0.5r\). Total losses \(= 8 + 0.5r\).
What we do and why: everything depends on \(r\). A statement is enough if it pins down \(r\).

Step 2. Statement (1): at 50 percent overall he would have lost 12 more.
What we see: a "what if" about losses.
What it means: at 50 percent overall, losses would be \(0.5(40 + r) = 20 + 0.5r\). Real losses are \(8 + 0.5r\).
What we do and why: the difference is \((20 + 0.5r) - (8 + 0.5r) = 12\). The \(r\) cancels out. This is true for every value of \(r\), so it tells us nothing. Statement (1) is not enough.

Step 3. Statement (2): at 80 percent overall he would have won 18 more.
What we see: a "what if" about wins.
What it means: at 80 percent overall, wins would be \(0.8(40 + r) = 32 + 0.8r\). Real wins are \(32 + 0.5r\).
What we do and why: the difference is \(0.3r = 18\), so \(r = 60\). Total wins \(= 32 + 30 = 62\). One value, so statement (2) is enough.

The trap: treating statement (1) as an equation and thinking it gives \(r\). It looks like one, but the \(r\) cancels, so it is always true and holds no new information.
Faster way: the first 40 matches are 30 percent above the 50 percent rate (32 wins against 20), which is exactly 12 matches. So statement (1) just restates the question's own facts. For (2), the later matches are 30 percent below 80 percent, so the 18 missing wins are 30 percent of \(r\).
Takeaway: When a variable cancels out of a statement, that statement adds no information, no matter how specific its number looks.

Answer: B