Analogy

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Number analogy

⏱ 5 min read🧩 6 question types🎯 12 practice Q

The idea in one minute

In a number analogy the first pair hides one arithmetic rule. You find that rule and apply it to the third number. Most rules are simple: multiply and add, square or cube, or work on the digits. Always check the rule on the model pair before you use it.

What a number analogy asks

You get a pair like 4 : 20 and a third number, say 7. Find the rule that turns 4 into 20. Use it on 7.

Here 20 = 4 × 5. So 7 × 5 = 35.

Step one: the size test

Compare the second number with the first. The size tells you which rule to try first.

What you see	Try first
A little bigger	add, or × 2 ± a small number
About k times bigger	× k ± r
Close to the square of the first	a² ± r, a(a + 1), (a + 1)²
Very large	a³ ± r
Smaller than the first	square root, cube root, ÷ k
Two-digit numbers, small answer	digit sum or digit product

Tip: Here a is the first number of the pair, r a small number and k a multiplier.

Know squares and cubes by heart

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.

The rule bank

Multiply and add: 9 : 40 is 9 × 4 + 4. So 13 gives 13 × 4 + 4 = 56.
Square based: 6 : 35 is 6² − 1. So 9 gives 80.
Cube based: 4 : 65 is 4³ + 1. So 5 gives 126.
Product form: 8 : 72 is 8 × 9, a number times the next number.
Digit based: product of digits (47 gives 28), square of the digit sum (36 gives 81), digits reversed (34 gives 43).
Two model pairs: 4 : 20 :: 6 : 42 :: 8 : ? Here 4 × 5 and 6 × 7, so 8 × 9 = 72.

Why you must check the options

One pair can fit two rules. 7 : 23 fits 7 × 3 + 2. It also fits 7 × 2 + 9. For 11 these give 35 and 31.

The setter keeps only one of them in the options. So:

Find the simplest rule and work out the answer.
If the answer is not in the options, try the next family. Do not force it.
If two model pairs are given, the rule must fit both.

Rule: Before you use a rule, check it gives the second number of the model pair exactly.

The whole-number note

Many papers add a note: "operations should be performed on the whole numbers, without breaking them into digits". When you see it, do not use digit sums or digit products. Use only +, −, ×, ÷, squares and cubes of the full number.

Example: 8 : 72 :: 12 : ? Here 72 = 8 × 9, a number times the next number. So 12 × 13 = 156. The traps are 144 (just 12²) and 132 (12 × 11).

Pair-selection questions

"Select the pair related in the same way as 7 : 49." Here 49 = 7², so the rule is square.

Test all four option pairs with the square rule. 9 : 81 fits, because 81 = 9². A near miss like 8 : 60 fails, because 8² = 64. Compute each pair fully before you mark.

Watch: The traps pass a quick glance. Only a full check catches them.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Multiply and add: b = k × a ± r

How to spot it:

The second number is a few times the first, plus or minus a small number (7 : 23, 9 : 40).

b = k\,a + r

Method

Divide b by a to guess k.
Find r = b − k × a.
Apply k × c + r to the third number.
Check the answer is in the options.

Why it works:

Most setters build number pairs with one multiplication and one small addition.

Try this

Select the number that can replace the question mark. 9 : 40 :: 13 : ? (a) 52 (b) 54 (c) 56 (d) 60

Show solution

$9 \times 4 = 36$ and $36 + 4 = 40$ . Rule: × 4, then + 4.
$13 \times 4 + 4 = 52 + 4 = 56$ .

Answer

Type 2very common2 practice Q

Square or cube based rule

How to spot it:

The second number is close to a square or a cube of the first (6 : 35, 4 : 65, 8 : 72).

b = a^2 \pm r,\ a^3 \pm r,\ a(a+1)

Method

Compare b with a² and a³.
Note the gap (35 = 36 − 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 big jump almost always means a power.

Try this

6 : 35 :: 9 : ? (a) 81 (b) 80 (c) 82 (d) 72

Show solution

$6^2 = 36$ and $36 - 1 = 35$ . Rule: square, then − 1.
$9^2 - 1 = 80$ .

Answer

(b) 80

Type 3common2 practice Q

Digit rule (sum, product, reversal)

How to spot it:

Two-digit numbers where the second number is small, or looks like the digits rearranged (47 : 28, 36 : 81, 34 : 43). No whole-number note is given.

Method

Try the digit product and the digit sum.
Try the square of the digit sum.
Try reversing the digits.
Use this family only when the whole-number note is absent.

Why it works:

Digit rules explain pairs that no multiply-and-add rule can.

Try this

47 : 28 :: 59 : ? (a) 14 (b) 45 (c) 95 (d) 54

Show solution

$4 \times 7 = 28$ : the product of the digits.
$5 \times 9 = 45$ . (14 is the digit-sum trap.)

Answer

(b) 45

Type 4common2 practice Q

Select the number pair with the same relation

How to spot it:

"Select the option in which the numbers share the same relationship as 13 : 41." The options are complete pairs.

b = f(a)

Method

Find the rule of the model pair.
Test all four option pairs with it.
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 to 3), which a full calculation catches.

Try this

Select the pair related in the same way as 13 : 41. (a) 17 : 53 (b) 15 : 44 (c) 19 : 55 (d) 21 : 62

Show solution

$13 \times 3 + 2 = 41$ . Rule: × 3, then + 2.
$17 \times 3 + 2 = 53$ fits. 15, 19 and 21 would need 47, 59 and 65.

Answer

(a) 17 : 53

Type 5common

Second number smaller: root or division

How to spot it:

The second number is smaller than the first and the first is a perfect square or cube (196 : 16, 343 : 7).

b = \sqrt{a} \pm r,\ \sqrt[3]{a},\ a \div k

Method

Check if the first number is a square or a cube.
Take the root and compare it with b.
Note any ± r.
Apply the same steps to the third number.

Why it works:

A root is the reverse of a square, so perfect squares and cubes in the first place point straight to it.

Try this

196 : 16 :: 324 : ? (a) 18 (b) 20 (c) 22 (d) 16

Show solution

$\sqrt{196} = 14$ and $14 + 2 = 16$ . Rule: square root, then + 2.
$\sqrt{324} = 18$ and $18 + 2 = 20$ .

Answer

(b) 20

Type 6occasional

Next prime number

How to spot it:

Both numbers of the model pair are prime, and no other prime lies between them (19 : 23, 29 : 31).

Method

Check that both model numbers are prime.
Check that no prime lies between them.
Find the next prime after the third number.

Why it works:

A fixed gap like + 4 breaks as soon as it lands on a non-prime, so the prime rule is the only one that holds.

Try this

19 : 23 :: 23 : ? (a) 25 (b) 27 (c) 29 (d) 31

Show solution

19 and 23 are prime, and 20, 21, 22 are not. Rule: the next prime.
After 23: 24, 25, 26, 27, 28 are not prime. 29 is prime.
The + 4 trap gives 27, which is 3 × 9.

Answer

Formula sheet

Multiply-and-add rule

b = k a + r

a is the first number, b the second. Find k and r from the model pair, then check.

Square family

b = a^2 \pm r,\ a(a+1),\ (a+1)^2

Try these when b is close to the square of a.

Digit rules

b = \text{digit sum},\ \text{digit product},\ (\text{digit sum})^2

Not allowed when the whole-number note is given.

Shortcuts that save time

⚡ The size test picks the family

Compare b with a before anything else. A few times a: multiply and add. Near a squared: square family. Very large: cubes. Smaller: roots or digit rules.

Example

7 : 50 :: 9 : ?

Show solution

50 is close to $7^2 = 49$ , so the rule is square + 1.
$9^2 + 1 = 82$ .

Answer

⚡ Check the rule on the model pair

Whatever rule you guess, make sure it gives b from a exactly. The check takes two seconds and prevents most wrong answers.

Example

6 : 42 :: 8 : ?

Show solution

Guess a number times the next: $6 \times 7 = 42$ . It checks.
$8 \times 9 = 72$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using a rule that nearly fits (12 : 49 read as 12 × 4).

The rule must give the exact number: 12 × 4 + 1 = 49.

Mistake 02

Using squares for the model pair and cubes for the new number.

Use exactly the same rule on both pairs.

Mistake 03

Using digit sums when the whole-number note is printed.

With that note, use only operations on the full number.

Mistake 04

Forcing a rule when the answer is not in the options.

If your answer is missing, try the next family in the size table.

Mistake 05

Missing a simple reversal: 27 : 72 read as × 2 + 18.

If the digits are just swapped, the rule is reversal. Prefer the simpler rule.

Quick revision

Read this the night before the exam.

Size test first: a little bigger, k times, near a square, very large, or smaller.
Know squares to 20 and cubes to 12.
Common forms: $ka + r$ , $a^2 \pm r$ , $a^3 \pm r$ , $a(a+1)$ .
Check the rule on the model pair before using it.
Answer not in the options: change the family.
Whole-number note: no digit tricks.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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