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Data Sufficiency

Reasoning · Logical & Analytical Reasoning

5 Part5 worked examples12 practice questions

Imagine a detective with two clues who is asked: “Who is the thief?” He does not first look for the thief; he first checks whether the clues are enough to fix the thief’s name. Is the first clue enough alone? The second alone? Both together? Or not even both? This is the question of data sufficiency.

The question gives one question and two statements (I and II). You need not solve the question; you must say, from five options, whether the statements can give a definite answer and in what way.

In making every question of this chapter, each answer was checked by a computer trying all possible situations — so in every solution you will see which answers remain possible after a statement.

How to use this chapter

  1. Read — the five cases, the method and six kinds of question.
  2. Study the solved example — after every statement the list of possible answers is open.
  3. Try it yourself — first I alone, then II alone, then both.
  4. Know the level — Basic, Standard, Advanced.
NoteThe questions you solve are remembered in this browser; see your progress on the right (the “Contents” button on a phone).
1

Foundation — what sufficient means and how to check

Five cases, three steps

The definition of sufficiency, the method of checking and a solved example of each of the five cases.

What “sufficient” means and the five options

One definite answer = sufficient

The definition of sufficient (for a value and for yes/no) and the five standard options.

A statement is sufficient when it gives one definite answer to the question. If the question is about a value (“What is x?”) the answer must be a single number. If it is a yes/no question (“Is x even?”) the answer must be “yes” in every situation or “no” in every situation. If it is sometimes “yes” and sometimes “no”, it is not sufficient.

OptionMeaning
(a) Only I alone is sufficientI fixes the answer; II does not
(b) Only II alone is sufficientII fixes the answer; I does not
(c) Either aloneI fixes the answer and so does II
(d) Both together neededNeither alone, together they fix it
(e) Not even togetherEven together more than one answer remains
Remember“No” is also a definite answer. In the question “Is x > y?” if the statement forces x < y in every case, the answer “no” is firm and the statement is sufficient — even if you do not know the values of x and y.
BasicTry it yourself first
Question: Is x greater than y? Statement I: x − y = 3. Statement II: y is an odd number.
BasicTry it yourself first
Question: Is x greater than y? Statement I: x × y = 39. Statement II: x is a prime number.
BasicTry it yourself first
Question: Take a week as Monday to Sunday. On which day was the programme held? Statement I: It was after Thursday (in the same week). Statement II: It was between Tuesday and Saturday.

The three-step method

Understand the question → I alone → II alone → together

One order for every question, a truth table and the common cautions.

  1. Understand the question — what is asked: a value or yes/no? Of what?
  2. Take statement I alone (forget II) — how many answers does it leave possible? One → sufficient.
  3. Take statement II alone (forget I) — the same check.
  4. If both are insufficient alone, join them — does one answer remain now?
I aloneII aloneTogetherAnswer
sufficientnot—(a)
notsufficient—(b)
sufficientsufficient—(c)
notnotsufficient(d)
notnotnot(e)

Four cautions:

  • Test the statements separately. While testing I do not use the information of II — otherwise (a)/(b)/(c) go wrong.
  • No need to find the answer. Just see whether one definite answer will come. It saves time.
  • No outside information. Only common knowledge is allowed (a week has seven days, ages do not fall).
  • Prove “insufficient” by an example — show two different possible situations in which the statement is true but the answer differs.
TrickPick “both together” only when no single statement is sufficient. If one alone is enough, joining both is not needed — even though both together would also work.
BasicTry it yourself first
Question: Take a week as Monday to Sunday. Was the programme on a weekend? Statement I: It was between Wednesday and Saturday. Statement II: It was after Monday (in the same week).
BasicTry it yourself first
Question: Take a week as Monday to Sunday. Was the programme on a weekend? Statement I: It was not on Monday. Statement II: The day before it was Friday.
BasicTry it yourself first
Question: Take a week as Monday to Sunday. On which day was the programme held? Statement I: It was between Wednesday and Sunday. Statement II: It was after Friday (in the same week).
BasicTry it yourself first
Question: Take a week as Monday to Sunday. Was the programme on a weekend? Statement I: It was before Sunday (in the same week). Statement II: It was not on Sunday.

The five cases — one example each

What (a) (b) (c) (d) (e) look like

A solved example of each case followed by practice.

Below is an example of each of the five cases. In every solution see which answers remain possible after the statement. Where only one remains, the statement is sufficient.

Example 1 · Basic
Question: Is x an even number? Statement I: x is an even number. Statement II: y is an odd number.
Solution, step by step
  1. 1Statement I alone: it leaves one answer only: Yes — sufficient.
  2. 2Statement II alone: more than one answer is possible: No (x = 1, y = 1), Yes (x = 2, y = 1) — not sufficient.
Answer Statement I alone is sufficient, but statement II alone is not
(a): I alone gives one answer, II does not.
Example 2 · Basic
Question: Is x an even number? Statement I: x is a multiple of 3. Statement II: x is an even number.
Solution, step by step
  1. 1Statement I alone: more than one answer is possible: No (x = 3, y = 1), Yes (x = 6, y = 1) — not sufficient.
  2. 2Statement II alone: it leaves one answer only: Yes — sufficient.
Answer Statement II alone is sufficient, but statement I alone is not
(b): II alone gives one answer, I does not.
Example 3 · Standard
Question: Is x an even number? Statement I: x × y = 32. Statement II: x is an even number.
Solution, step by step
  1. 1Statement I alone: it leaves one answer only: Yes — sufficient.
  2. 2Statement II alone: it leaves one answer only: Yes — sufficient.
Answer Either statement alone is sufficient
(c): both reach the answer by different roads.
Example 4 · Standard
Question: What is the value of y? Statement I: y is an even number. Statement II: y is a multiple of 5.
Solution, step by step
  1. 1Statement I alone: more than one answer is possible: 2 (x = 1, y = 2), 4 (x = 1, y = 4), 6 (x = 1, y = 6) … — not sufficient.
  2. 2Statement II alone: more than one answer is possible: 5 (x = 1, y = 5), 10 (x = 1, y = 10), 15 (x = 1, y = 15) — not sufficient.
  3. 3Both together: it leaves one answer only: 10 — sufficient.
Answer Both statements together are needed — neither alone is sufficient
(d): neither alone, together one answer.
Example 5 · Standard
Question: What is the value of x? Statement I: y − x = 6. Statement II: y is a multiple of 4.
Solution, step by step
  1. 1Statement I alone: more than one answer is possible: 1 (x = 1, y = 7), 2 (x = 2, y = 8), 3 (x = 3, y = 9) … — not sufficient.
  2. 2Statement II alone: more than one answer is possible: 1 (x = 1, y = 4), 2 (x = 2, y = 4), 3 (x = 3, y = 4) … — not sufficient.
  3. 3Both together: more than one answer is possible: 2 (x = 2, y = 8), 6 (x = 6, y = 12) — not sufficient.
Answer Even both statements together are not sufficient
(e): even together they leave several answers.
Trick(c) comes least often but is missed most easily: if you find I sufficient, still test II alone — it may be sufficient too.
StandardTry it yourself first
Question: Take a week as Monday to Sunday. Was the programme on a weekend? Statement I: The day before it was Tuesday. Statement II: It was not on a weekend (Saturday or Sunday).
StandardTry it yourself first
Question: Is A older than B? Statement I: 7 years ago A was 4 times as old as B. Statement II: A is 21 years older than B.
StandardTry it yourself first
Question: Take a week as Monday to Sunday. On which day was the programme held? Statement I: The day before it was Thursday. Statement II: It was after Monday (in the same week).
StandardTry it yourself first
Question: What is A’s present age? Statement I: The sum of their ages is 61 years. Statement II: A is 3 years older than B.
StandardTry it yourself first
Question: What is A’s present age? Statement I: A is younger than 23 years. Statement II: B is older than 7 years.
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Six kinds of question

Numbers, ages, order, direction, codes, days

The special trick and practice of each kind — every answer checked by trying all possible situations.

Questions on numbers

x and y — equations and conditions

Questions on two unknown numbers: how many independent equations are needed, what conditions do, and what is simpler in yes/no questions.

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Questions on ages

Difference, sum, ratio, “after / before k years”

The nature of age statements: difference, sum, multiple relation and shifts in time.

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Questions on order and comparison

Who is the tallest? Who is second?

Questions on the order of four persons: positive statements, negative statements and joining links.

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Questions on direction

A to B, B to C — both legs are needed

Making coordinates in direction questions and seeing which link the statement carries.

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Questions on code languages

Catch the common word, strike off the rest

Finding the meaning or code of one word from code sentences and when it is not possible.

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Questions on days of the week

Days between, days excluded, the weekend

The method of cutting down the list of weekdays and the “all in the same class” test in yes/no.

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Strategy guide

One order for every question and the traps to avoid

Choosing the trick from the type of question and the common mistakes.

The trick by type

Choose the first step from the question

In each type, what is sufficient alone and what is not.

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Common traps and time-savers

The mistakes that keep happening

Reasons for losing marks in data sufficiency.

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4

Mixed practice

All kinds mixed

Questions of all six kinds in three levels.

Basic practice

One statement plainly sufficient or not

Simple questions on numbers, ages, order, codes and days.

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Standard practice

Both together, or either alone

Questions with (c) and (d) and with negative information.

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Advanced practice

Fine lines: (d) versus (e), tricks of yes/no

Questions that turn on whether even both together are sufficient.

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5

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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