Cracking Number Analogy
In RPF Sub-Inspector reasoning, number analogy questions look harmless — two pairs of numbers, find the rule. But the same examiner who built the question is also counting on you to waste a minute trying the wrong operation. A trained candidate solves these in 12 seconds; an untrained one stares for a full minute. The difference is a system.
Definition: A number analogy is a reasoning question where two numbers in the first pair are linked by some mathematical operation (square, cube, addition, multiplication, sum of digits, etc.), and you must identify the same hidden link to complete the second pair. The standard form is "A : B :: C : ?".
Definition: A relation is any rule that maps the first number of a pair to the second. The skill is matching the relation found in the given pair to the missing number in the asked pair.
The five operations that cover 90% of questions
Almost every number analogy in RPF SI / SSC / RRB falls into one of five categories. Learn this list and you will recognise patterns instantly.
- Square — second number is the square of the first. Example: 4 : 16, 9 : 81.
- Cube — second number is the cube of the first. Example: 5 : 125, 6 : 216.
- Add a constant — second is first plus a fixed number. Example: 7 : 11 (+4), 12 : 16 (+4).
- Difference / Subtract — second is first minus a constant or vice versa. Example: 20 : 13 (−7).
- Multiply / Divide — second is first times (or divided by) a constant. Example: 6 : 36 (×6), 8 : 48 (×6).
A sixth family is the mixed operation, where you do something like n × (n+1) — for instance 5 : 30 means 5 × 6, and 7 : 56 means 7 × 8.
The SCADM order — your testing checklist
When you see a number analogy, run through the operations in this exact order: Square, Cube, Add, Difference, Multiply. Why this order? Because squares and cubes are the most common in objective tests, and they fail visibly in under two seconds — if 4 : 17 isn't 4², stop and move on. Addition and subtraction are next-easiest, multiplication tests last because it requires actual computation.
The cardinal rule is this: whatever rule fits the first pair must be tested on the second pair using the same operation. If 6 : 42 works as 6 × 7, then check whether 8 : ? could be 8 × 9 = 72. If the second pair refuses the rule, the rule is wrong — switch operations, not numbers.
The trickier patterns you must recognise
Beyond SCADM, three patterns trip up candidates regularly:
Sum / product of digits. The second number may equal the sum or product of the digits of the first. Example: 23 : 5 (because 2 + 3 = 5), or 14 : 4 (because 1 × 4 = 4).
Reverse digits. The second number is the digit reversal of the first. Example: 12 : 21, 34 : 43, 56 : 65.
Grouped numbers. When the question gives a triple like (3, 9, 27), look for geometric progression (each term × 3) or arithmetic progression. Triples are common in IBPS but appear in RPF / RRB level too.
Worked example
Question: 8 : 64 :: 12 : ?
Options: (a) 96 (b) 144 (c) 132 (d) 156
Solution:
Step 1: Apply SCADM. Try square: 8² = 64. Match! The rule is square.
Step 2: Apply the same rule to the second pair: 12² = 144.
Step 3: Match against options.
Conclusion: Answer is (b) 144.
Notice how fast that was — under five seconds once you trusted the rule.
A trickier one:
Question: 15 : 6 :: 24 : ?
Options: (a) 6 (b) 8 (c) 10 (d) 12
Solution:
Step 1: Try square (225 ≠ 6), cube (no), add a constant (15 + ? = 6 — no, but 15 − 9 = 6 — try the same subtraction). 24 − 9 = 15, not in options. Switch hypothesis.
Step 2: Try sum-of-digits: 1 + 5 = 6. Match!
Step 3: Apply same rule to 24: 2 + 4 = 6.
Conclusion: Answer is (a) 6.
Here SCADM failed on the first sweep — the secondary patterns (sum of digits) saved you. Always have those two backup checks ready.
Why it matters
In an RPF SI prelims paper of about 35 reasoning questions, 4–6 are pure analogy types (number + alphabet + meaning-based). At 12 seconds each, you bank a minute and a half of bonus time that you can spend on a harder data interpretation question. Examiners reward speed accumulation: the candidate who clears RPF SI is rarely the smartest in the room — they are the one who never gets stuck on a 30-second problem.
Real-world example
When a railway booking clerk verifies a PNR or a coach number against a manifest, she is doing exactly this — matching one pattern (the PNR digit sequence) to another (the seat allocation block). The mental skill of "spot the relation, apply it twice, verify" transfers directly. The RPF reasoning paper is testing on-the-job pattern recognition that an officer needs every day.
Common misconception
Many candidates assume that if the first relation they spot works for the given pair, it must be correct. Wrong. A pair like 4 : 16 could mean square (4²) OR multiply by 4 (4 × 4) OR add 12 — three different relations, all producing 16. You only know which one is intended by checking the same rule against the second pair (C : ?). If C = 5 and an option is 25, the rule is square. If C = 5 and an option is 20, the rule is "multiply by 4". Always test on both pairs.
A second mistake is to look at the options first, work backwards, and "engineer" a rule. This wastes time and fails when distractors are well-designed. Stick to: spot the rule on the given pair, then apply it forward.
| Pattern type | Example pair | How to test (≤ 2 sec) |
|---|---|---|
| Square | 7 : 49 | Is second = first × first? |
| Cube | 4 : 64 | Is second = first × first × first? |
| Add constant | 9 : 16 | Compute difference, check both pairs |
| Subtract / diff | 30 : 23 | Is first − second the same in both pairs? |
| Multiply / divide | 6 : 30 | Is second / first the same integer in both pairs? |
| n × (n+1) | 5 : 30 | Does second equal first × (first + 1)? |
| Sum of digits | 23 : 5 | Add the digits of the first |
| Reverse digits | 12 : 21 | Reverse the digit order |
- ✓- Number analogies test relations, not arithmetic — speed comes from recognising the family.
- ✓- Run the SCADM checklist: Square, Cube, Add, Difference, Multiply.
- ✓- If SCADM fails, switch to sum-of-digits, product-of-digits, or reverse-digit checks.
- ✓- Apply the SAME operation to both pairs to confirm — never assume.
- ✓- Eliminate options by quick mental arithmetic; do not guess.
- ✓- Watch for grouped triples — they often hide a GP or AP.
- ✓- An "odd one out" framing is asking the same question in disguise; same techniques apply.
- ✓- A 12-second target per analogy is realistic with practice.
SCADM — Square, Cube, Add, Difference, Multiply. Say "SCADM" out loud as you scan the pair and you cover the high-frequency patterns in under three seconds.
- ✓- Number analogies follow a small set of hidden mathematical rules.
- ✓- SCADM is your first-pass test; backup with digit-based patterns.
- ✓- Always verify the same rule on both pairs before locking an answer.
- ✓- Speed on analogies frees up time for the harder reasoning questions.
Position-Value Trick for Letter Analogy
Convert letters to position numbers (A=1...Z=26). MEMORY AID 'EJOTY': E=5, J=10, O=15, T=20, Y=25 — count forward/back from these milestones to fix any letter fast. Reverse positions: A=26, Z=1 (use 27-position). Common patterns: +1 skip (AB:CD), opposite letters (A-Z, B-Y sum=27 rule), gap series (A_C, skip one). For pairs like AZ:BY, note A+Z and B+Y both pair to position-sum 27. For DH:EI type, each letter +1. Always write the position numbers below letters to expose the gap. Backward alphabet (Z,Y,X...) is favourite in SI papers — practise reciting reverse alphabet.
Worked Example: Mixed Analogy
Q: 7 : 56 :: 9 : ? Solve: 7x8=56, so relation is nx(n+1). Apply: 9x10=90. Answer 90. Q: BD : FH :: JL : ? Positions B=2,D=4,F=6,H=8 — each pair is consecutive even letters increasing by 4 (B->F is +4, D->H is +4). Next: J=10,L=12, add 4 -> N=14, P=16, so NP. Q: 25:36 :: 49:? These are squares 5^2,6^2 then 7^2, so next 8^2=64. SHORTCUT: when both numbers are perfect squares of consecutive integers, the answer is the next perfect square. Verify by checking the difference pattern (11,13,... odd-number gaps confirm squares).
Number & Alphabet Analogy — Flashcards
Cover the answer, recall the rule, then check. 12 cards on number/alphabet analogy patterns tested in RPF SI.
Q1. First step in any A : B :: C : ? number analogy?
A1. Find the exact relation between A and B (difference, ratio, square, cube, or digit operation), then apply the same relation to C. Never guess from C alone.
Q2. 7 : 56 :: 9 : ? Find the rule and answer.
A2. 7×8 = 56 (n × n+1). So 9×10 = 90.
Q3. In alphabet analogy, what must you memorise cold?
A3. Letter positions A=1 … Z=26, and the reverse EJOTY / VUTSR aids. Most letter analogies are position arithmetic in disguise.
Q4. BD : FH :: JL : ? Solve.
A4. B(2)D(4) → F(6)H(8): each letter +4. J(10)L(12) → N(14)P(16) = NP.
Q5. 6 : 216 :: 8 : ? Rule and answer.
A5. 6³ = 216 (cube). So 8³ = 512.
Q6. Common RPF SI trap in "125 : 5 :: 343 : ?" style pairs?
A6. The relation is a root, not a power: 125 = 5³ so answer is cube-root of 343 = 7. Read the direction of the operation.
Q7. AZ : BY :: CX : ? What is the pattern?
A7. Opposite-letter pairs (A↔Z, B↔Y, C↔X). Next pair keeping the mirror: DW (D↔W).
Q8. 3 : 27 :: 5 : ? — is it cube or n×(n+something)?
A8. Both 3³=27 fit and 3×9=27. Check the answer options: 5³=125 vs 5×9=45. Pick whichever matches an option — RPF SI relies on you spotting the unique fit.
Q9. How do you handle a two-number-to-two-number analogy like 4:16 :: 5:25?
A9. Test squares first: 4²=16, 5²=25. Squares/cubes are the highest-frequency numeric relation.
Q10. DEF : 456 :: GHI : ? (position coding analogy)
A10. D=4,E=5,F=6; so G=7,H=8,I=9 → 789.
Q11. Fastest check when two rules seem to fit a number pair?
A11. Apply both rules to the third term and compare with the given options — only one produces a listed answer.
Q12. Biggest careless error in alphabet analogy?
A12. Miscounting positions past M (13). Use the EJOTY anchor (E5,J10,O15,T20,Y25) to jump, then adjust by ±1/±2.