Analogy

In this type a number is turned into its partner by some operation, and the same operation must turn the third number into the answer. It comes in three costumes. The plain stem: 11 : 121 :: 13 : ?. The "select the pair" form: one stem pair and four option pairs, and you pick the option pair with the same relation. And the multi-pair stem, where two complete pairs and one half pair are given, with the question mark in the middle or at the end (134 : 258 :: 310 : ? :: 125 : 240).

Why is this solvable at all? The paper-setter starts with one clean operation and picks numbers on which only that operation looks natural. So the relation is a power (n², n³, n² ± 1, n³ ± 1, n² + n), a multiplication or division (× k, × k ± c, n ÷ 2 ± c), or a plain addition or subtraction (+ c). Recent papers also state that numbers must be treated as whole numbers: work on 13 as thirteen, never as 1 and 3.

[[fig:0]]

A good order of testing saves time. This is a teaching order, not an exam law, so treat it as a habit, and let the options decide when two ideas fit.

• Power first. If the second number is close to a perfect square or cube, test n², n³ and the version with ±1. A big jump (4 → 65, 5 → 126) points to a power.
• Multiply or divide next. Test × k, then × k ± c, and halves (n ÷ 2 ± c). Do not assume that a falling number means division: 42 → 20 is n ÷ 2 − 1.
• Add or subtract. A small gap points to + c or − c.
• Digits last, and only if the stem does not forbid them. Under the "whole number" note, digit methods are out.

Sometimes the stem gives several pairs, such as 46 : 189 :: 33 : ? :: 21 : 89. Find the relation from the complete pairs (46 : 189 and 21 : 89): both follow × 4 + 5, so 33 × 4 + 5 = 137. The relation must hold on every complete pair.

A relation that fits the first pair but gives a value not among the options is the wrong relation. Stop at the first relation that the options support, and never at one that only the first pair supports.

A set of three numbers hides one relation, and the paper asks you to find the same relation in another set. This type comes in three forms. Option sets: two bracketed stem sets such as (15, 28, 64) and (63, 76, 112), and four option sets; a note says the numbers must be treated as whole numbers, so 13 is thirteen, not 1 and 3. Matrix: a 3 × 3 grid with a question mark, where one relation runs along every row (or every column). Chain: two strings of four numbers such as 5 – 20 – 60 – 120 and 8 – 32 – 96 – 192, where each step has its own operation and the right option repeats the same operations in the same order.

The logic behind all three is repetition. The paper-setter builds every set from one formula, so a relation is believed only when it works on all the given sets with the same constants. That is why the method has three moves.

Move 1: Target hunt. Ask which number is built from the others. Check the gaps first (b − a and c − b), then whether one number is a multiple or a part of another (b = k × a, b = a ÷ k). Next ask whether the largest number is made from the other two (c = a + b, c = a × b ± k, c = (a + b) × k), or the middle number is made from the ends (b = k (a − c), b = a + c + k).

Move 2: Two-set proof. Write the formula and substitute the second set. If it fails, throw the idea away. For (15, 28, 64) and (63, 76, 112) the gaps are 13 and 36 in both sets, so the relation is confirmed. For (8, 5, 38) and (9, 4, 34) the gaps differ, but a × b − 2 works on both.

[[fig:0]]

Move 3: Cut the options. Use the cheapest test first: one gap, one parity, the last digit, the size. Then do the exact check on the survivors. If none survives, or two survive, go back to Move 1; your formula is not yet the intended one.

[[fig:1]]

Relations that have appeared in real papers: c = a + b; fixed gaps (+8, +13); c = a × b − 2 for (27, 12, 322) and (24, 13, 310); b = a² × c for (8, 128, 2) and (9, 243, 3); c = b − a for (18, 20, 2) and (47, 48, 1); and b = a ÷ 3 with c = b − 0.2 for (42, 14, 13.8) and (54, 18, 17.8). Decimals are allowed when both sets show them.

Matrix form. Find a row whose three numbers are complete and see how the third comes from the first two. Check the same relation on the other complete row, and only then use it on the row with the question mark. In 28, 54, 82 / 19, 25, 44 / 31, 52, ? the sums 28 + 54 = 82 and 19 + 25 = 44 show c = a + b, so ? = 31 + 52 = 83. If the rows give nothing, try the columns.

Chain form. Write the operation between each pair of neighbours in the first string: 5 → 20 is × 4, 20 → 60 is × 3, 60 → 120 is × 2. The second string must give the same list (8 → 32 → 96 → 192). Then pick the option whose list is the same, in the same order.

[[fig:2]]

The options carry decoys: sets with equal gaps of a different size, sets that satisfy only half of a two-part relation, and sets that use + 2 where − 2 is needed. Prove the relation on every stem set, then test every option with the exact numbers, not with the look of the set.

In this type a pair of numbers produces one result. It comes as a bracket, (3, 5) : 34, or as a symbol joining two numbers: "If 15 Φ 3 = 10 and 24 Φ 4 = 12, find 35 Φ 5". In the symbol form the symbol is an unknown operation, and the given lines are your clues.

[[fig:0]]

The logic is simple: the setter has one formula of a and b, and the given lines are samples of it. You find the formula by asking what the first pair can produce. Start by computing the standard quantities of the first pair: the sum, the difference, the product and the quotient (a ÷ b), and also the squares and their sum. Then see which of them is connected to the result: equal to it, half of it, double it, one more, one less, or a square of it.

For (3, 5) : 34 the sum is 8, the difference is 2 and the product is 15. None of these gives 34. The squares are 9 and 25, and 9 + 25 = 34. So the relation is a² + b², and (6, 2) gives 36 + 4 = 40.

Division deserves a special look. If a ÷ b comes out whole in every given line (15 ÷ 3 = 5 and 24 ÷ 4 = 6), the setter has chosen the numbers on purpose and the formula is built on the quotient. Both results, 10 and 12, are double the quotient, so 35 Φ 5 = 2 × 7 = 14.

The common relations are a + b, a − b, a × b, a ÷ b, a² + b², (a − b)², (a + b) × k, (larger − smaller) × k, a × b ± c and a × b + a + b. In 3 ★ 5 = 23 and 4 ★ 6 = 34 the products 15 and 24 leave 8 and 10, which are exactly the sums 3 + 5 and 4 + 6. So a ★ b = a × b + a + b, and 5 ★ 7 = 35 + 12 = 47. A symbol like & changes nothing: treat it as an unknown operation and test the same list. Here 2 × a × b gives 32 and 50 for 4 & 4 and 5 & 5, so 6 & 6 = 2 × 36 = 72.

A relation that works on one line may fail on another. So confirm the relation on every given line before applying it. Verified examples: (38, 27) → 22 and (47, 23) → 48 follow (larger − smaller) × 2, and (4, 7) → 65 and (5, 9) → 106 follow a² + b². Compute the sum, difference, product and quotient first, and confirm the relation on every given line.

Letter analogy asks one thing: how was the first letter group turned into the second? There are two families of answers. In a shift, every letter moves a few steps along the alphabet. In a mirror, every letter is replaced by its opposite (A ↔ Z). The stem can show one complete pair (CAT : ECV :: DOG : ?), or three pairs of which two are complete and the question mark sits anywhere, even in the first place (? : EK :: PQ : KJ :: UG : PZ). First turn the letters into positions (A = 1 ... Z = 26).

In a shift analogy every letter of the first word is moved a few steps along the alphabet to make the second word, and the same steps are used on the third word. First turn the letters into positions (A = 1 ... Z = 26). Then write the shift row under the pair, one step for each position.

[[fig:0]]

There are two kinds of rows. In a fixed shift all steps are equal (+2, +2, +2). In a position-wise shift each place has its own step (+2, +3, +4), and the steps may have mixed signs (+2, −3, +3). Recent Mains-level sets prefer the second kind, with steps up to about 10 either way.

Take CAT → EDX. C (3) → E (5) is +2. A (1) → D (4) is +3. T (20) → X (24) is +4. The row is +2, +3, +4. To find DOG, add the row: D + 2 = F, O + 3 = R, G + 4 = K, so DOG → FRK.

When the move crosses Z or A, use the wrap-around. Forward: if the result is above 26, subtract 26. Backward: if it is 0 or below, add 26. For A (1) → X (24) the step is −3, because 1 − 3 = −2 and −2 + 26 = 24.

There is one special shift that is worth remembering: ±13. Because the alphabet has 26 letters, moving 13 forward or 13 backward lands on the same letter, so the two codes are the same: H ↔ U, E ↔ R, L ↔ Y, P ↔ C. Such a code turns HELP into URYC, and applying it again gives HELP back.

If the stem gives two complete pairs (as in KB : HX :: RM : ? :: DL : AH), the row from the first pair must also work on the third. That is a free check. Write one step for every position; never assume that all positions move together.

When the question mark is in the first place, work backwards. In ? : EK :: PQ : KJ :: UG : PZ the complete pairs give the row. P (16) → K (11) is −5 and Q (17) → J (10) is −7. Confirm on the next pair: U (21) → P (16) is −5, and G (7) → Z (26) is −7 (7 − 7 = 0, and 0 + 26 = 26). The row is −5, −7. The unknown letters are the ones that land on E and K after this row, so they sit 5 and 7 places higher: E (5) + 5 = J and K (11) + 7 = R. The cluster is JR.

Mirror (opposite letters). Write the alphabet forwards, and under it write it backwards. A sits above Z, B above Y, C above X, and so on. Each letter is now paired with its opposite letter. In every such pair the two positions add up to 27: A (1) + Z (26), B (2) + Y (25), M (13) + N (14).

[[fig:1]]

So the opposite of the letter at position p is the letter at position 27 − p. For R (18), the opposite is 27 − 18 = 9, which is I. A question like RAD : IZW is just this mirror: R → I, A → Z, D → W. When you see the first letter of the stem going to a letter near the other end of the alphabet (A → Z, B → Y), think "mirror" at once.

There is also a fast mental way. The letter in the p-th place from the start has its opposite in the p-th place from the end. G is 7th from the start, so its opposite is the 7th from the end: Z, Y, X, W, V, U, T. You never need to subtract if you can count from the end.

Sometimes the mirror carries one more step. In EG : WU, the opposite of E is V, but the code has W; the opposite of G is T, but the code has U. So the relation is "opposite letter, then +1". Always find the mirror first, then check whether the first pair leaves a constant gap. The same gap is then applied to the third word.

A useful check: the two letters of a mirror pair always add to 27, so there is no letter that is its own opposite. Use 27, not 26. Using 26 pushes every answer one letter off.

Not every letter pair is a simple shift. The first pair can show a procedure: reverse the word, replace each letter by its two neighbours, add the positions, or treat the letters of an alternate place differently. Your job is to spot which procedure was applied and then repeat it.

[[fig:0]]

The shape of the pair is the clue. If a whole word turns into a single letter (BED : K), think of the sum of positions: B (2) + E (5) + D (4) = 11, which is K. If the sum is above 26, subtract 26, and keep subtracting until it is 26 or less: H (8) + E (5) + N (14) = 27, and 27 − 26 = 1, which is A. A repeated letter is counted every time it appears.

If two letters turn into four, think of previous and next: C D → B D C E, because B and D surround C, and C and E surround D. The order is always the previous letter first, then the next letter, for each original letter in turn.

[[fig:1]]

If the letters of the second term are the same as the first but in a different order, it is a reversal or a rearrangement. If the letters at odd places and even places behave differently (the first and third move forward, the second and fourth move back), it is an alternating shift: find a row of steps with alternating signs.

Whole word to a number. The same sum idea can give a number instead of a letter. In "If WATER = 67 and EARTH = 52, then ELEMENT = ?" the sums are 23 + 1 + 20 + 5 + 18 = 67 and 5 + 1 + 18 + 20 + 8 = 52. So ELEMENT = 5 + 12 + 5 + 13 + 5 + 14 + 20 = 74. When the answer is a number, the sum is not wrapped. A related form writes each position as two digits side by side: QUIC becomes 17 21 09 03, so a single-digit position carries a leading zero.

Triads. Sometimes three words are linked in a chain, such as JOBS – JSBO – SBOJ, and the option must follow the same logic. Track what happens at each step: JOBS → JSBO swaps the 2nd and 4th letters, and JSBO → SBOJ moves the first letter to the end. The right option repeats both moves in the same order.

A third kind uses symbols such as `# : KPM :: ROT : %`. The symbols hide unknown letter groups, and the options supply them. Do not try to decode the symbols. Take each option, insert its groups, and test whether a single pattern works for both pairs. First identify the shape of the pair, then test the idea that fits that shape.

This type has three costumes. Letters : number, such as CF : 9 or TI : 58. Letters with a number on both sides, such as KG 13 : IC −2, where the letters shift and the number changes by its own relation. Clusters, such as BDF8 : CGK81 or ADK10 : ZCJ110, with a letter part and a number part. The key idea is the same in all three: the letters and the number are independent halves. Solve each half separately, and let the letter half cut the options first.

In a letter-number analogy the letters stand for their positions, and the number is built from them. The first job is the same every time: convert the letters into positions, and then solve the number part exactly as you would a number analogy.

[[fig:0]]

The usual ways to build the number from positions are the sum, the difference (larger minus smaller), the product, the square and the cube. Take CF : 9. C = 3 and F = 6. The sum is 9 and the difference is 3. The number is 9, so the relation is the sum, and ER gives 5 + 18 = 23. Take C : 27. C = 3, and 3³ = 27, so F gives 6³ = 216.

Test both the sum and the difference on the first pair, because both are common and one of them will be an option. If the first pair gives the same number for both ideas, the second pair or the options decide.

In the new cluster form, such as PQT14 : QTY225 or BDF8 : CGK81, the answer has a letter part and a number part. Treat them separately. The letters follow a shift row (for example +1, +3, +5), and the number follows its own relation (for example (n + 1)²). Check the letter part first, because it often removes three options at once. Then check the number part. The numbers are whole numbers; 14 is not split into 1 and 4.

Here is a full example. Take KG 13 : IC −2 :: NH 20 : LD 5, and ask for NW 24. For the letters, K (11) → I (9) is −2 and G (7) → C (3) is −4. The second pair confirms the row: N (14) → L (12) is −2 and H (8) → D (4) is −4. For the number, 13 → −2 is −15, and 20 → 5 is −15. Now NW 24: N − 2 = L, W (23) − 4 = S (19), and 24 − 15 = 9. The answer is LS 9. The letters alone already choose the option, because among LS 9, MS 11, MR 11 and LR 9 only LS 9 has L first and S second.

The number can also use the sum or product of the positions with a constant. TI : 58 is (20 + 9) × 2, so RM gives (18 + 13) × 2 = 62.

If the stem gives two complete pairs, the relations must fit both of them. Convert letters to positions first. Never start the alphabet from 0.

The examiner has exactly one relation in mind, and your first job is to name it. Take the first pair and speak it as a sentence in the same word order: "A barometer measures air pressure", "A judge works in a court", "Ohm is the unit of resistance". Now put the second pair into the same sentence. The word that completes it is the answer.

In the exam this comes in three forms. Single-word options: Judge : Court :: Doctor : ?, and you pick one word. Pair options: Guitar : Musician :: ?, where each option is a complete pair and you pick the pair with the same relation. Does NOT follow: a pair is given, and you pick the option pair that does not have the same relation (Gold : Ornament).

[[fig:0]]

Most word pairs come from a small set of families: quantity and unit, instrument and what it measures, worker and tool, person and workplace, country and capital or currency, disease and organ, animal and home or sound, synonym and antonym, producer and product. These are the relation banks from the start of the chapter. Knowing the family tells you what to look for before you even read the options.

The options are built with decoys. A decoy is a word on the right topic but with the wrong relation. For the pair Resistance : Ohm, the option "Ampere" is a real unit, but not the unit of power. For Barometer : Air pressure, the option "Rainfall" is weather, but not what an odometer measures. So decide the relation first and the topic second.

[[fig:1]]

The direction of the pair also matters. Resistance : Ohm is quantity : unit, and Ohm : Resistance is unit : quantity. If the stem gives quantity first, your answer must be a unit. Options often include both directions, so a student who ignores the order walks into the trap.

When the options are pairs, say the sentence of the stem pair and test every option pair with it. Name each relation in two words, such as tool → user, and keep only the option with the same two words in the same direction. In Guitar : Musician the relation is tool → user. Needle : Tailor has the same tag. Hospital : Doctor is place → person, Student : Teacher is learner → guide, and Destination : Pilot is goal → person.

Does NOT follow. Here three pairs share one relation and one pair breaks it. Find the relation of the majority first, then locate the pair that breaks it. In Gold : Ornament the relation is material → product: Cotton : Cloth, Wood : Paper and Clay : Pot fit, but Water : Ice does not, because ice is the same substance in another state and nothing new is made. If the first two pairs disagree, the odd pair may be one of them, so use the third and fourth to decide. The same question also comes with number pairs (2 : 8, 3 : 26, 4 : 63, 5 : 124: three follow n³ − 1 and 2 : 8 breaks it) and with letter groups (BDFH, RTVX and ACEG move in steps of +2; JKNP does not). A pair that only looks different is not necessarily the odd one: test the relation on every pair.

[[fig:2]]

When you meet an unfamiliar English word, translate it into Hindi (or the other way round) and decide the relation there. Say the relation as a sentence, carry the same sentence to the second pair, and eliminate the options that fit the topic but not the relation.

In this type the right-hand word of the first pair is the class (or the whole) of the left-hand word. Mango : Fruit says a mango is a kind of fruit. Page : Book says a page is a part of a book. The answer must therefore be the class, or the whole, of the new left-hand word. It must not be its property, its place or its product.

[[fig:0]]

The question you ask is short: "X is a kind of ___" or "X is a part of ___". Decide which of the two sentences the first pair fits, and then use only that sentence for the second pair.

Look at the four wrong answers that usually appear. For Tulip : ? the options might be Flower (class), Garden (place), Colour (property) and Petal (part). Only one of them answers "a tulip is a kind of ___". The others answer different questions: "where does it grow", "what is it like", "what is it made of". The examiner places the "kind of" and "part of" answers side by side on purpose, so the first pair must decide.

Direction. The order of the pair matters. In Disease : Malaria the class comes first and the member second, so the answer must also be class : member: Vehicle : Truck, not Truck : Vehicle. Truck : Vehicle is the same link read backwards, and it is almost always among the options.

The part that does the same job. In Bird : Wing :: Fish : ? the options may be Water, Fin, Gill and Tail. A wing is the organ with which a bird moves, so the matching part is the fin, the organ with which a fish moves. A gill is used for breathing, not for movement, and water is not a part of the fish at all. Literary forms follow the kind-of link: a novella is a kind of fiction, so a sonnet is a kind of poetry.

A recent form is the genus pair, such as Summer : Season. Summer is one kind of season, so Cotton, Hockey or Honesty needs its own class word: Fibre or Plant, Sport, Virtue. In abstract pairs (Honesty : Virtue, Greed : Vice) the words are ideas, not objects, but the link is the same: the right word is the group to which the left word belongs.

Some words have two classes. Cotton is a plant and also a fabric. The level of the first pair tells you which class is wanted. Pick the answer that completes the same sentence, not the one that is merely connected to the word.