Analytical Aptitude — Number Relations, Analogy, Deduction and Induction

The syllabus for this part is four items long — number relations and reasoning, analogy, logic-induction and deduction — and the last two are worth more attention than their length suggests, because they are the only place in General Aptitude where the question is about the shape of an argument rather than about its content. The distinction to carry in is this. A deduction is truth-preserving: if the statements are true, the conclusion cannot be false, and the test is whether you can construct even one situation where the statements hold and the conclusion fails. An induction generalises from cases: it can be strong or weak, but it is never guaranteed, and the honest answer to "what does this pattern prove?" is usually "nothing, but it suggests". GATE sets both under wordings that look the same, and the single most useful habit is to read the question stem for which one is being asked — must be true is deduction, most likely or best explains is induction — before reading any option. For number relations and analogies the method is different and simpler: name the relationship in words before you look at the options, because an unnamed relationship will match two options and a named one matches exactly one.

Deduction and Induction — Which One Is Being Asked

The stem tells you which logic applies
If the stem saysYou are being asked forAnd the test is
must be true, necessarily follows, can be concludedDeduction. Only what is guaranteed counts.Try to build one case where the statements hold and the option fails. If you can, the option is out.
most likely, best explains, the pattern suggestsInduction. The best-supported option wins, not the only possible one.Compare the options against each other. "Not guaranteed" is not a reason to reject one here.

Three deduction traps account for most of the losses, and all three are reversals or upgrades of something the statements did say:

  • Reversing an "all". "All the qualified candidates were interviewed" does not give "all those interviewed were qualified". The set of interviewees may be larger. Every "all A are B" question has this as an option.
  • Upgrading a "some". Some means at least one, and licenses nothing about most or all. It also does not imply "some are not" — "some of the bolts are steel" is perfectly consistent with all of them being steel.
  • Turning a necessary condition into a sufficient one. "Without calibration the reading is invalid" says calibration is needed; it does not say calibration makes the reading valid, because something else may also be needed.

Number Relations and Series — Test Differences First

A number series is a small search, and searching in a fixed order finds the rule far faster than staring at the numbers. Run the checks in this order and stop at the first one that fits: first differences (constant → arithmetic); second differences (constant → quadratic, so the terms involve n²); ratios (constant → geometric); then look for squares, cubes or primes near each term; then check whether the series alternates between two interleaved rules, which is what a series with an obvious break in it usually is.

🧠 Name the relationship before reading the options
For an analogy — verbal or numerical — write down what connects the first pair, in words, before looking at what is offered. "7 : 50" is not obviously anything; "the square plus one" is, and it picks 11 : 122 without ambiguity. The reason to do this first is that an unnamed relationship will find two options that both half-fit, and you will then choose between them by feel. A named one matches exactly one, and the question stops being a judgement call.

Key takeaways

  • Read the stem for the logic before reading the options: "must be true" is deduction, "most likely" or "best explains" is induction, and the two have different winning conditions.
  • To reject a deduction, build one case where the statements hold and the conclusion fails. One counterexample is enough, and finding it is faster than arguing.
  • "All A are B" never gives "all B are A". The reversal is offered as an option in almost every question of this type.
  • "Some" means at least one. It licenses nothing about most or all, and it does not imply that some are not.
  • A necessary condition is not a sufficient one: "not valid without X" does not mean X makes it valid.
  • For a series, test first differences, then second differences, then ratios, then squares and cubes, then an alternating pair of rules — in that order, and stop at the first fit.
  • Name the relationship in words before reading the options. An unnamed one half-fits two of them; a named one matches exactly one.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Statements: All the shortlisted applicants were interviewed. Some of those interviewed were offered a position.

    Which of the following must be true?

    1. Everyone who was interviewed had been shortlisted.
    2. Some shortlisted applicants were offered a position.
    3. At least one person was interviewed.
    4. Most of those interviewed were offered a position.
    Show answer

    Answer: C — At least one person was interviewed.

    The second statement says some of those interviewed were offered a position, and "some" means at least one — so at least one person was interviewed, which is option C. Option A reverses the first statement: all shortlisted were interviewed, but the interviewees could include people who were never shortlisted. Option B is the subtle one and does not follow: the people who got offers were interviewed, but nothing says they came from the shortlist rather than from outside it. Option D upgrades "some" to "most". Only C survives, and it survives because it claims the least.

  2. Select the number that will replace the question mark: 7 : 50 :: 11 : ?

    1. 110
    2. 121
    3. 122
    4. 132
    Show answer

    Answer: C — 122

    Name the relationship first: 50 = 7² + 1. Applying the same rule, 11² + 1 = 121 + 1 = 122. Option B is 11² alone, which is what you get by naming the relationship as "the square" and stopping — and note that it half-fits, since 49 is close to 50, which is exactly why an unnamed relationship is dangerous. Option A is 11 × 10 and option D is 11 × 12; both are products that land near the answer without following the rule the first pair sets.

  3. Find the next term: 3, 8, 18, 38, ?

    1. 58
    2. 68
    3. 78
    4. 88
    Show answer

    Answer: C — 78

    The first differences are 5, 10, 20 — they double each time, so the next difference is 40 and the next term is 38 + 40 = 78. Equivalently each term is twice the previous plus 2: 3 → 8 → 18 → 38 → 78. Option A comes from treating the differences as constant at 20, and option B from continuing them arithmetically as 5, 10, 20, 30. Testing the first differences before anything else is what finds this rule in seconds; staring at the terms themselves finds nothing, because none of them is a recognisable square or cube.

  4. A technician says: "Without calibration, a reading from this instrument is not valid." Which of the following follows from this statement?

    1. If the instrument is calibrated, its readings are valid.
    2. If a reading is valid, the instrument was calibrated.
    3. Calibration is the only thing that affects validity.
    4. An uncalibrated instrument gives no readings at all.
    Show answer

    Answer: B — If a reading is valid, the instrument was calibrated.

    The statement makes calibration necessary for validity: no calibration, no validity. The one thing that follows is its contrapositive — if a reading is valid then calibration must have happened, which is option B. Option A is the classic error of reading a necessary condition as a sufficient one: calibration may be required and still not be enough, because the instrument might also need to be in range or undamaged. Option C claims calibration is the only factor, which is more than the statement says. Option D confuses producing a reading with producing a valid one — an uncalibrated instrument reads perfectly happily, and that is the problem.

  5. In a laboratory, every sample that failed the hardness test also failed the fatigue test. Sample X passed the fatigue test. Which of the following must be true?

    1. Sample X failed the hardness test.
    2. Sample X passed the hardness test.
    3. Sample X was not tested for hardness.
    4. Nothing can be concluded about the hardness test.
    Show answer

    Answer: B — Sample X passed the hardness test.

    The rule is "hardness fail → fatigue fail". X did not fail fatigue, so by the contrapositive X did not fail hardness — that is, X passed the hardness test, option B. This is the same logical move as the previous question in the other direction, and it is the one deduction pattern GATE returns to most: from "if P then Q" and "not Q", conclude "not P". Option A applies the rule forwards from the wrong end. Option D is the safe-sounding answer that is wrong here precisely because the contrapositive is a guarantee — "nothing follows" is only correct when the given fact is the consequent being true, not the consequent being false.

  6. Pointing to a photograph, a man said, "She is the only daughter of my grandfather's only son." How is the woman in the photograph related to the man?

    1. His daughter
    2. His sister
    3. His cousin
    4. His aunt
    Show answer

    Answer: B — His sister

    Work from the inside out. "My grandfather's only son" — the man's grandfather has exactly one son, and the man is his grandfather's descendant, so that only son must be the man's father. The photograph is therefore "the only daughter of my father", which is the man's sister. Option A would require the man himself to be the grandfather's only son, which he is not — he is a generation below. Option C would need an uncle, and the word only rules one out; the phrase "only son" is in the puzzle precisely to close that door.

  7. A plant recorded its highest output in each of the last four months after installing new lighting. Which of the following is the strongest reason to doubt that the lighting caused the increase?

    1. The lighting was more expensive than the plant had budgeted for.
    2. A second production line began operating in the same week the lighting was installed.
    3. Some workers said the new lighting was too bright.
    4. The previous lighting had been in use for eleven years.
    Show answer

    Answer: B — A second production line began operating in the same week the lighting was installed.

    This is an induction question — the stem asks for the strongest reason to doubt, not for a disproof — and the strongest doubt is a rival cause that arrived at the same time. A second production line starting in the same week would raise output on its own, so the lighting is no longer needed to explain the figures. Option A is about cost, which has no bearing on whether output rose. Option C is about how the lighting was received, and a disliked change can still work. Option D explains why the plant replaced the lighting — a motive, not evidence about the effect.

  8. Statements: Some of the bolts in the batch are made of steel. All the steel bolts in the batch are galvanised.

    Which of the following must be true?

    1. Some bolts in the batch are not made of steel.
    2. Some bolts in the batch are galvanised.
    3. All the galvanised bolts in the batch are steel.
    4. No bolt in the batch is both steel and galvanised.
    Show answer

    Answer: B — Some bolts in the batch are galvanised.

    At least one bolt is steel, and every steel bolt is galvanised, so at least one bolt is galvanised — option B, and it is guaranteed. Option A is the most tempting wrong answer: everyday speech implies "some" means "not all", but logically some means at least one and is perfectly consistent with all of them being steel, so nothing follows about non-steel bolts. Option C reverses the second statement — galvanised bolts could include non-steel ones. Option D contradicts what the statements together establish. As in question 1, the answer is the option that claims the least.