Data Sufficiency

An ordering question names one target, usually an end: the tallest, the lightest, the lowest marks. Sometimes it names a place, such as the second heaviest, or a pair ('who among P and Q scored more'). The statements give plain comparisons: A is heavier than B, C is taller than D. These are partial orders. One statement rarely lines everybody up, so the real question is always: does this statement fix the asked end?

The Kavach method is the Arrow Graph. For every comparison, draw an arrow from the larger to the smaller. For 'heavier', the arrow points to the lighter one.

• The lightest is a node with no arrow going out of it, and every other node has a path leading to it.
• The heaviest is a node with no arrow coming into it, and it has a path to every other node.

The word 'and' matters. In the figure, Q has no arrow going out, and P, S, T and R all lead to it, so Q is the lightest. But P and S both have nothing above them, so two nodes compete for the top and the heaviest is not decided. The same statements can fix the lightest and fail for the heaviest, so always check only the end that is asked.

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Now the second figure. When two arrows meet at one node (R → T ← S), T is the lowest of the three, but nothing tells you whether R or S is higher. This is the setters' favourite trap: 'P scored more than R' and 'Q scored more than R'. Both arrows enter R, so R is the lowest, yet P and Q are never compared. The top stays open even with both statements together.

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Watch the weak clue. 'M is not taller than K' is a weak arrow (K ≥ M); it says nothing about a third person H. A chain survives a weak link if one link is strict: A > B ≥ C still puts A above C. If everyone's height is different, 'not taller' simply means shorter.

A clue such as 'L is taller than only two trees' is a position clue: L has exactly two trees below it, so L is third from the bottom. A position clue can also remove a candidate. If L is taller than only one tree, L is second from the bottom and cannot be the shortest.

Finally, never assume a missing link. If two chains are never compared with each other, they stay apart, and the answer may be 'not sufficient even together'. That is a perfectly valid verdict.

In a seating question the statements say who sits where, in a row or around a table, and the question asks for one thing: who is at the left end, who is in the middle, who is opposite whom. You do not have to complete the seating. You have to decide whether the statement forces that one answer. A statement is sufficient only if every seating that fits it gives the same answer.

The danger is that one drawn picture can hide a second one. Take 'B sits between D and C, all three next to one another'. You draw D-B-C in seats 1 to 3 and feel sure. But the same block can sit in seats 2 to 4 or 3 to 5, or be written C-B-D. B may be in seat 2, 3 or 4. The statement allows many pictures, so it cannot fix B's seat.

The method is Branch-and-Check. Draw the statement. Wherever there is a choice (which end, which side, which order), split into cases and write each one out. Then read the asked target in every case. If all cases agree, the statement is sufficient, even though the picture is incomplete. If two cases disagree, it is not.

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In the figure, Statement I (A and E at the two ends) leaves B, C and D free in seats 2 to 4, so the middle seat can be B or D. Statement II alone leaves the block free to slide. Together, the ends are taken, the block must fill seats 2 to 4, and B, the centre of the block, sits in seat 3.

Three habits protect you. First, fix the facing direction before reading 'left' and 'right': in a row facing north, left is west; in a circle facing the centre, left is clockwise. Second, 'between' needs both neighbours, so 'B is between D and C' puts D and C on the two sides of B. Third, ask only whether the asked seat or person is fixed; a statement may be sufficient without placing everyone.

Now the most useful idea for a row: anchor and block. 'X sits immediately to the right of Y' only builds a block, and a block can slide along the row. In the second figure the block Q-R-S slides into three places, and the middle seat holds S, then R, then Q. So a block with no anchor cannot fix an absolute place. An anchor ties someone to a seat: an end, 'second from the left', 'only one person sits to the right of Q'. Look for the anchor first. A circle has no ends, so relative clues are enough there: put one person in seat 1 and count from that seat.

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In a circle of six seats, the person opposite you is three seats away, on either side.

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In these questions the answer is a path. Blood-relation items ask 'How is A related to B?'; direction items ask 'In which direction, or how far, is Y from X?'. In both, a statement is sufficient only when the path from the first person (or point) to the last is completely fixed, with no step left open.

Family chains. Draw each statement as a small tree: + for male, − for female, ? for unknown. First ask: do both named people appear in this tree? If not, the statement alone cannot join them. If no single statement does, join the trees at the joint, the person who appears in both. In the first figure, I makes Manoj (M) and Neha (N) brother and sister, and II makes Geeta (G) the mother of Neha. Now all three are in one tree, and Geeta is Manoj's mother too. If the two statements share no person at all, they cannot be joined.

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Gender decides whether the next step can be taken. Mother, father, son, daughter, brother and sister fix gender; parent, child, sibling and cousin do not (second figure). A name is a clue, not proof: Kiran or Jyoti may be a man or a woman, so take gender only from words such as son, Mr or she. A gender missing in the middle of a chain breaks the chain. At the very end, a gender-free answer such as 'sibling' is accepted when that is all the chain gives, as in the real paper below. And read the direction: 'How is P related to C?' asks what P is to C.

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Direction chains. Here the path is a walk. Treat the ground as a grid. Add the north–south legs on one line (north +, south −) and the east–west legs on another (east +, west −). The signs of the two totals give the direction, and √(x² + y²) gives the distance. This is Net Displacement. The order of the legs does not change the end point: in the third figure the blue and green walks both end at Q.

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Three situations come up again and again.
1. Two full walks to the same place. Each alone is sufficient, so the verdict is 'either'. A statement may walk back from the place to the start; reverse it (north ↔ south, east ↔ west) and compare.
2. A turn without a heading. 'He walked 10 m, turned right and walked 10 m' gives four different answers for the four starting headings (fourth figure). A left or right turn needs a known starting direction.
3. A distance without a direction. '12 km from the station' is a whole circle of places, not one point.

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Do not add distances that point in different directions. A place 9 km east and 12 km north is 15 km away, not 21 km.

A number question asks for a value: an age, a price, an average, a count of boys. It is decided by counting. The target needs some unknowns, and a statement is sufficient only if, by itself, it gives as many independent equations as there are unknowns. This is the safe default. The asked quantity can sometimes be fixed with fewer: x + y = 10 fixes the sum x + y itself. Conditions such as 'whole numbers' can also make an answer unique.

The method is the Unknown–Equation Counter:
1. List the unknowns the target needs.
2. Count the independent equations each statement gives by itself. 'Independent' means a new fact, not the same fact written twice. An inequality ('more than') or a loose link ('about twice') is not an equation.
3. If one statement alone gives as many equations as there are unknowns, it is sufficient. If not, merge the statements and count again.

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Count after merging too. With three unknowns (x, y, z) and only two equations, the system is not enough. The one exception is when the target happens to cancel the extra unknown. Say the target is the average age of three people. If the sum of two ages is given, and the third person is described in terms of one of those two, the total comes out as 'x + 60', which still depends on x. Three unknowns, two equations: not sufficient. Had the target been only 'the sum of the first two', one equation would have done.

A gap never gives an age. 'The father was 33 when Rahul was born' and 'Rahul is 29 years younger than his mother' are both gaps. A gap is the same in every year, so adding one number to every age keeps all the gaps true. Statements that are only gaps can never fix an age. You need a sum, a ratio, an 'after N years' clue or one real age.

Shortcuts for the usual themes:

• Averages and sums: total = count × average. The group total minus a pair total gives the third value.
• Ratio and fractions: 'two-fifths are girls' gives only a ratio. One absolute fact (a difference, a total) turns it into a count.
• Ages: a gap never gives an age; a sum, a ratio or a real age does.
• Boat and stream: still water = (downstream + upstream) ÷ 2.

Three statements. Some papers give I, II and III, and the options name pairs: 'I and II only', 'II and III only', 'any two of the three', 'all three'. Read the options first. Test each named pair without borrowing the third statement. With two unknowns, any two independent equations will do, so if every pair works the answer is 'any two'. Choose 'all three' only when no pair works, which happens with three unknowns.

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Never borrow a number from the other statement while testing one statement alone. That is Concept 1 of the chapter, and numbers tempt us to break it more than anything else.

A yes/no question asks 'Is N even?' or 'Is G taller than H?'. A statement is sufficient only if it makes the answer always yes or always no. 'Sometimes yes, sometimes no' is not sufficient. A definite no is as good as a definite yes, because it is still one single answer.

The method is Test-the-Extremes. For each statement, try two small values that obey it. If one gives yes and the other gives no, the statement is not sufficient, and you can stop. If every value you try gives the same answer, look for the reason why it must always happen. Only then call the statement sufficient.

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The figure shows the classic trap. 'N is a prime number' feels like 'N is odd'. Test N = 2: it is prime and even. Test N = 3: it is prime and odd. One yes and one no, so the statement does not fix the parity. The only even prime is 2, and that one exception is why prime numbers are favourite test values. Another favourite: 1 is neither prime nor composite.

Choose test values that satisfy the statement. A value that breaks the statement tells you nothing.

A few facts help you find the reason behind a 'always yes':

• odd + odd = even; even + odd = odd
• consecutive numbers alternate between odd and even
• N + 5 is odd exactly when N is even
• a number divisible by 12 is also divisible by 2, 3, 4 and 6

Comparison questions. For 'Is G taller than H?' do not test numbers; look for a chain between G and H. G > F > H gives always yes. A chain running the other way gives always no, which is just as sufficient. A clue such as 'H is not the shortest of G, H, F' leaves both G > H and H > G alive, so it is not sufficient.

Judgement questions. Some yes/no items have no numbers: 'Is Sana good at maths?'. Apply the Evidence-Direction Test: does the statement prove the quality, or only make it possible? A title, a rank or an expert rating proves it. Buying books, joining a class or owning equipment shows only exposure or interest. Two statements of possibility do not add up to proof, and you must not supply facts from your own guess. These items depend on the setter, so attempt them last.

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