Data sufficiency: the pitfalls

Why sufficiency goes wrong

The method of the previous lesson is simple; the errors come from what you assume while applying it. Each pitfall below either makes a statement look enough when it is not, or makes it look not enough when it is. As before, the exact wording of Bocconi's current answer choices is not published, so the checks here use our own four options, built so that each statement being enough on its own never arises.

Pitfall 1: counting equations

"Two equations, two unknowns, so both together" is a habit, not a rule. It fails in three ways:

• One equation can be enough when the question asks for a combination: $2x + 2y = 18$ fixes $x + y$.
• Two equations can be one: $3a + 2b = 13$ and $6a + 4b = 26$ are the same line.
• Non-linear equations can have two solutions: $x^2 = 25$ gives $5$ and $-5$.

Ask instead: does the information leave exactly one possible answer?

Pitfall 2: hidden whole numbers

Context often restricts the unknowns, and the restriction can make a statement enough. People, tickets and items come in whole, non-negative numbers. If pens cost €3, notebooks €5 and someone spent exactly €30, the only possibilities are 10 pens and no notebooks, 5 pens and 3 notebooks, or 6 notebooks and no pens. A statement that rules out two of these three fixes the answer, even though "one equation, two unknowns" says it cannot.

List the whole-number cases when the numbers are small. It takes less time than it seems, and it is the only reliable test.

Pitfall 3: assuming limits that are not there

The opposite error is just as common. Unless the question says so, a variable in pure algebra may be negative, zero or a fraction, and two different letters may be equal. Before declaring an inequality settled, try:

• a negative number;
• zero;
• a fraction between 0 and 1;
• two equal values.

Squares are the classic case: $x^2 > 9$ means $x > 3$ or $x < -3$, not just $x > 3$.

Pitfall 4: ratios, percents and counts

A ratio or percent is enough for a question about a ratio or percent, even with no counts at all. A count is enough for a question about a count. Mixing them is where marks go:

• "What percent of respondents were under 30?" is settled by "the number aged 30 or over was 1.5 times the number under 30": that is $\frac{u}{u + 1.5u} = \frac{1}{2.5} = 40\%$, whatever $u$ is.
• "How many respondents were under 30?" is not settled by that ratio unless a total is known.

Ask whether the question wants a relative or an absolute answer, then check whether each statement gives one of the same kind.

Pitfall 5: data sufficiency on a display

Sometimes the problem comes with a small table or a description of a chart. The display counts as given information, like the question's own facts. Read it with the care of the earlier lessons in this unit: the units, the period, and whether its figures are counts or rates. A statement that gives a rate for a group whose total is already in the table is enough to find a count.

A last check before choosing

When you think a statement is enough, try once to break it: find a second case that fits and gives a different answer. When you think it is not enough, make sure your two cases really satisfy the statement and every fact in the question, including the hidden ones.