Q8 · Advanced Data Sufficiency

Here is a kind of sufficiency that catches strong candidates, because it does not look like the sufficiency they are trained to recognise.

Usually we establish sufficiency by solving: we manipulate the information until a value falls out. But sometimes the information does not let you solve for anything — and is sufficient anyway, because it eliminates every possibility but one.

An example. Suppose m and n are positive integers with mn = 35, and the question asks for m + n.

There is no equation to solve here. But 35 factors as 35 × 1, 7 × 5, 5 × 7 and 1 × 35. So m + n is either 36 or 12. Two possibilities — insufficient so far.

Now add: both m and n are prime. Nothing has been solved. But 35 and 1 are not both prime, and 1 is not prime at all, so only 7 × 5 survives. m + n = 12. Sufficient — by elimination, not by algebra.

The signal that you are in a uniqueness item: the problem involves integers, factors, primes, or a small constrained set, and the statements read like conditions rather than equations. When you see that, stop looking for something to solve and start listing what is possible. Then check how many survive.

If one survives, it is sufficient. If two survive, it is not — and the item is usually built so that exactly two survive from one statement, which is what makes the answer C so often in this family.

Exam writers know that candidates judge statements partly by how much information they appear to contain. So they write statements that contain a great deal of information and still fail to answer the question.

The classic shape: the question asks for the median of a set, and the statement gives you the mean and the total. That is two facts and a number. It feels rich. It tells you nothing about the median, because the mean does not constrain the middle value.

Or: the question asks whether one quantity exceeds another, and the statement tells you a great deal about one of them and nothing about the other.

Or: the question asks for a specific value, and the statement gives a relationship — a ratio, a proportion, a comparison — without any absolute anchor. Ratios are seductive. "Adult tickets were three times child tickets" feels like solid information, and it fixes nothing on its own.

The habit that defends against this: after reading a statement, do not ask "does this tell me a lot?" Ask "does this tell me the specific thing the question asked for?" Then go back and read the question stem again to make sure you have not quietly substituted an easier question.

That re-read costs five seconds and is the highest-return habit in this format.

After enough practice, a pattern settles in: two statements arrive, neither seems to do the job alone, they combine neatly, the answer is C. It happens often enough to become a reflex.

Exam writers know this too, and they build items specifically to punish it. The two most common shapes:

The redundant pair. Both statements say the same thing in different clothing, and each is sufficient alone. The answer is D. Consider:

Statement (1): x³ > x² means x³ − x² > 0, so x²(x − 1) > 0. Since x² is never negative and cannot be zero here, this requires x > 1. So yes, x > 0. Sufficient. Statement (2): a reciprocal is positive only when the number is. Sufficient. Two very different-looking facts, same conclusion. D.

The insufficient pair. They combine, and the combination still does not answer the question. The answer is E. Consider:

Together, n is divisible by 12. Take n = 12: not divisible by 8. Take n = 24: divisible by 8. Both satisfy both statements, opposite answers. E.

The defence is the protocol from Q8.2, applied honestly. If you truly evaluate each statement alone before combining, neither trap can catch you — because you will have already discovered that (1) works alone, or you will test the combination properly instead of assuming it closes.

This is the section that saves the most time, and the hardest to actually do, because it asks you to stop when every instinct says finish.

You are never asked for the answer to the underlying question. You are asked whether the answer is determined. Those are different, and the gap between them is where your time goes.

Read this:

You could set up a + c = 340 and 12a + 5c = 2,916, solve, and get a number. That takes ninety seconds and a chance of arithmetic error.

Or you notice: two independent linear equations, two unknowns. That system has exactly one solution. Sufficient. Move on. Fifteen seconds, no arithmetic, no chance of error.

The general principle: you need to know that a unique answer exists, not what it is. Recognising a solvable structure is enough.

Structures worth recognising on sight:

• Two independent linear equations in two unknowns → one solution.
• A total plus all-but-one of the parts → the missing part is determined.
• A rate and a distance → the time is determined.
• A percentage and the base → the amount is determined.

Two warnings, because this technique has a sharp edge.

Check that the equations are actually independent. "x + y = 10" and "2x + 2y = 20" are the same equation twice. Two equations, two unknowns, no unique solution.

Non-linear systems can have multiple solutions. If a quadratic is involved, two equations in two unknowns may give two valid answer pairs — which is insufficient for a value question. Recognise the structure, but check the shape before you trust it.

Every error this chapter has described, in one place. When you get a Data Sufficiency item wrong, it is almost always one of these seven.

The practice for this chapter is on its own page. Six items where stopping at the mechanical answer is exactly the error: uniqueness, the C reflex, and the ones you should not solve.