Analogy
🔒 Log in to trackNumber analogy
🔒 Log in to trackA : B gives a numeric operation; repeat it on C. The operation is always simple and always uniform — the same expression turns a into b. Rank the usual suspects:
- Linear: b = k·a + r (e.g. ×3 + 2). Most common by far.
- Square/cube based: b = a² ± r, b = a³, (a+1)², a(a+1), etc.
- Digit based: b = (sum of digits) × k ± r, or reversal of digits.
- Powers: b = 2^a ± r or 3^a ± r.
Uniqueness habit: one pair (a, b) admits many rules, so the exam keeps distractors away from every natural rule. When your computed answer is missing from the options, your rule is wrong — drop to the next suspect, don't force-fit.
Detailed notes
What is a number analogy?
You get a pair like 4 : 20 and a third number, say 7. The first pair hides an arithmetic rule. You apply the same rule to 7. Here 20 = 4 × 5, so 7 × 5 = 35.
Step 1: the size test
Look at how big the second number is compared with the first. It tells you which family to try first.
| What you see | Family to try first |
|---|---|
| Second number a little bigger | add or subtract, or ×2 ± small number |
| Second number about k times bigger | ×k ± r |
| Second number near the square of the first | a² ± r, a(a + 1), (a + 1)² |
| Second number huge | a³ ± r |
| Two-digit numbers, answer small | digit sum, digit product |
| Digits look swapped | reverse the digits |
Step 2: know your squares and cubes
Squares 1 to 20: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. Cubes 1 to 12: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728. When you see 35, think 36 − 1. When you see 126, think 125 + 1. When you see 72, think 8 × 9 = a(a + 1) for a = 8.
The rule bank
- Linear: b = k·a ± r. Example 9 : 40 → 9 × 4 + 4 = 40; so 13 → 13 × 4 + 4 = 56.
- Square based: b = a² ± r, a(a + 1), a(a − 1). Example 6 : 35 → 6² − 1; so 9 → 80.
- Cube based: b = a³ ± r. Example 4 : 65 → 4³ + 1; so 5 → 126.
- Digit based: product of digits (47 → 4 × 7 = 28), sum of digits, square of the digit sum (36 → (3 + 6)² = 81), reversed digits (34 → 43).
- Chain of pairs: 4 : 20 :: 6 : 42 :: 8 : ? — two model pairs fix the rule. 4 × 5, 6 × 7, so 8 × 9 = 72.
Why you must test the rule, and use the options
One pair can fit many rules. 7 : 23 fits 7 × 3 + 2 and also 7 × 2 + 9. For 11 these give 35 and 31. The examiner keeps only one of these answers in the options. So:
- Find the simplest natural rule and compute the answer.
- If your answer is not in the options, try the next family. Do not force a rule.
- If two model pairs are given, your rule must fit both.
The whole-number note
Many papers add: "Operations should be performed on the whole numbers, without breaking them into digits." When this note is present, do not use digit sums or digit products. Only use operations on the full number (×, +, squares, cubes).
Pair-selection form
"Select the pair in which the numbers are related in the same way as 7 : 49." Find the link (a²), then check all four pairs. Only 9 : 81 keeps the link. Check each pair fully — the traps are near misses like 8 : 60.
Worked example
8 : 72 :: 12 : ? 72 is 8 × 9 = a(a + 1). So 12 × 13 = 156. Traps: 144 (12², forgot the +a), 132 (12 × 11, used a − 1).
Quick revision
- Size test first: small change → linear; near a square → a² ± r; huge → cube.
- Know squares to 20 and cubes to 12 by heart.
- Check the rule on the model pair before using it; answer missing → change family.
- Whole-number note present → no digit tricks.
- Pair-selection: one rule, test every pair completely.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Linear rule: b = k × a ± rvery common2 practice Q
The second number is a few times the first, plus or minus a small number (7 : 23, 9 : 40).
- Divide b by a to guess k (23 ÷ 7 ≈ 3).
- Find r = b − k·a (23 − 21 = 2).
- Apply k·c + r to the third number, then check the answer is in the options.
Why it works: most setters build number pairs with one multiplication and one small addition.
Example: 9 : 40 :: 13 : ?
9 × 4 + 4 = 40. So 13 × 4 + 4 = 56. (52 forgets the +4.)
Type 2: Square or cube based rulevery common2 practice Q
The second number is close to a square or cube of the first (6 : 35, 4 : 65, 8 : 72).
- Compare b with a² and a³.
- Note the difference (35 = 36 − 1; 65 = 64 + 1) or the product form (72 = 8 × 9).
- Apply the same form to the third number.
Why it works: squares and cubes grow fast, so a large jump almost always means a power.
Example: 6 : 35 :: 9 : ?
35 = 6² − 1. So 9² − 1 = 80. (81 forgets the −1.)
Type 3: Digit-based rule (sum, product, reversal)common2 practice Q
Two-digit numbers where the second number is small or looks like the digits rearranged (47 : 28, 36 : 81, 34 : 43).
- Try digit product, digit sum and the square of the digit sum.
- Try reversing the digits.
- Use this family only when the whole-number note is NOT given.
Why it works: digit rules explain pairs that no simple ×k ± r rule can.
Example: 47 : 28 :: 59 : ?
4 × 7 = 28 (product of digits). So 5 × 9 = 45. (14 is the digit sum trap.)
Type 4: Select the number pair with the same relationcommon2 practice Q
'Select the option in which the numbers share the same relationship as 7 : 49' — the options are complete pairs.
- Find the rule of the model pair.
- Test all four pairs with that rule.
- If two pairs fit, look for a sharper rule that fits the model and only one option.
Why it works: the traps are near misses (off by 1 or 2), which a full calculation catches.
Example: Select the pair related like 13 : 41. (17 : 53 / 15 : 44 / 19 : 55 / 21 : 62)
41 = 13 × 3 + 2. Test: 17 × 3 + 2 = 53 ✓; 15 → 47, 19 → 59, 21 → 65 ✗. Answer: 17 : 53.
Formulas
Solve k and r mentally from a and b; verify on the pair before applying to c.
Test squares first when b is close to a².
S(a) = digit sum; common when b is much smaller than a.
Shortcut tricks
⚡ Size test picks the family instantly
Compare b with a: if b is a few times a → linear; if b ≈ a² → square family; if b is 10-40 → digit rule when a is 2-digit; if b is small with big a → digit sum.
Example: Q. '7 : 50' :: '9' : ?
Sol. 50 ≈ 7² = 49, so square +1. 9² + 1 = 82.
Size test → family → one-line arithmetic.
⚡ Verify the rule on the model pair before the probe
Whatever rule you guess, check it reproduces b from a exactly. Guessing ×4+1 from 12 : 49 fails instantly (12×4+1 = 49 passes!) — then apply to c. The 2-second check prevents half the errors.
Example: Q. '6 : 42' :: '8' : ?
Sol. Guess a(a+1): 6×7 = 42 ✓. Then 8×9 = 72.
Model-pair verification is the cheapest correctness check in the paper.
Where students lose marks
Forcing a rule that ignores the model pair's exact value (fitting 12:49 with ×4+1 is right, but with ×4+0 it is wrong — check!).
Mixing two different rules for the two pairs (using squares for the model and cubes for the probe).
Digit-reversal missed: 27 : 72 looks like ×2+18 but the intended relation is reversal; if reversal reproduces b, prefer it.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 9 min · wrong answers go to your mistake notebook automatically.