SSC CGL Reasoning · Classification

Number Pairs

In three of four number pairs the second number is made from the first by one rule. Find the rule from the pairs that agree and test the last one exactly.

All Classification notes

Why this matters

Forty-seven questions in the bank, the biggest page of the chapter: 27 from the 2023 shifts and 20 from the 2024 shifts, 29 EASY and 18 MODERATE. The odd pair is usually off by only one or two, so the rule has to be checked exactly, not by feel.

Concept 1 of 3: Pair rules made of multiplying, adding and fixed ratios

When the second number is a few times the first, the rule is almost always 'multiply by k, then add or take away c'. Two pairs are enough to find k and c; the other two only need checking.

Definition

Order of tests: the difference (second minus first), then the ratio (second divided by first), then multiply-and-adjust, then squares and cubes (next concept).
Fixed difference: big numbers that differ by about the same amount, like 1984 and 1505, usually share one difference.
Fixed ratio: cancel each pair to lowest terms. $36 : 84$ and $66 : 154$ both become $3 : 7$ .
Multiply and adjust: guess k from the size of second ÷ first, then c = second − k × first. If c is the same in three pairs, that is the rule.
The rule can run backwards, from the second number to the first (first = 8 × second − 1). If nothing works forwards, try it the other way.
Fast route: with k and c found from two pairs, test the other two in your head. A pair that misses by 1 or 2 is the answer; there is no need to finish the arithmetic for the rest.

Common pair rules

y = kx + c \qquad y - x = d \qquad x : y = p : q

xthe first number of a pair
ythe second number of a pair
k, c, dfixed numbers shared by the three pairs that agree

Worked example

Find the odd pair: (a) 9 : 41 (b) 12 : 53 (c) 7 : 33 (d) 15 : 66

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 11 Sep 2024, Shift 1 · Q13Moderate

Example 1 · Classification · Number Pairs

The same operation(s) are followed in all the given number pairs except one. Find that odd number pair. (NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding /deleting /multiplying etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)

Reading k from the smallest pair

In a small pair the added number hides the multiplier: under the rule 4 × first + 5, the pair 1 : 9 looks like ×9. Guess k from the largest pair, where c matters least.

Only testing the rule from first to second

The setter may build the first number from the second, as in first = 8 × second − 1. When every forward rule fails, run the same tests from the second number back to the first.

Drill 19 more on pair rules made of multiplying, adding and fixed ratios

Concept 2 of 3: Pair rules with squares, cubes and roots

When the second number is far bigger than the first, the rule nearly always squares or cubes it. When the first number is a perfect square or cube, the rule often takes its root.

Definition

Compare the second number with the square of the first. If it is a little above or below, the rule is square plus or minus a fixed number.
If it is close to the cube, test cube plus or minus the number itself ( $n^3 + n$ , $n^3 - n$ ).
If the second number is a square, find its root and compare it with the first: $(x + 2)^2$ , $(x/2 + 1)^2$ .
If the first numbers are all squares or cubes, take the root and see what it becomes (half the root, root × 2 + 1).
A multiple of a square also appears: $3x^2$ , $2x^2$ , $8x^2$ . Divide the second number by the square of the first.
Fast route: the odd pair is usually the one whose second number is not near any square or cube. Check that one first.

Power rules for pairs

y = x^2 \pm c \qquad y = (x \pm c)^2 \qquad y = x^3 \pm x \qquad y = k x^2 \qquad y = \sqrt{x} \pm c

Worked example

Find the odd pair: (a) 5 : 29 (b) 8 : 68 (c) 11 : 125 (d) 7 : 51

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 12 Sep 2024, Shift 3 · Q16Easy

Example 2 · Classification · Number Pairs

Three of the following number-pairs are alike in some manner and hence form a group. Which number-pair does not belong to that group? (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)

Taking a square on the right as proof

In 4 : 36 and 7 : 64 both second numbers are squares, but only one fits

(n + 2)^2

. Work out what the rule gives for each first number, not just whether the result is a square.

Looking for a multiplier when the numbers grow too fast

If the second number is 10 or more times the first and the ratio keeps changing, no single k exists. Move straight to squares and cubes.

Drill 18 more on pair rules with squares, cubes and roots

Concept 3 of 3: Products and powers of neighbouring numbers

A number times its neighbour, or two squares next to each other, is a favourite SSC rule. The odd pair often uses the wrong neighbour, or the number times itself.

Definition

$n(n + 1)$ is a little more than $n^2$ : it adds n. $n(n - 1)$ is a little less: it takes n away.
If the second number is just above the square of the first, test $n(n + 1)$ ; just below, test $n(n - 1)$ .
For a pair of two squares, find both roots. Neighbouring roots (19 and 20) form the rule; a gap of 2 or 3 is the odd pair.
The gap between neighbouring squares is $2n + 1$ , and between neighbouring cubes $3n^2 + 3n + 1$ .
'The next perfect square after the first number' is another rule from this family.

Neighbouring numbers

n(n+1) = n^2 + n \qquad (n+1)^2 - n^2 = 2n + 1 \qquad (n+1)^3 - n^3 = 3n^2 + 3n + 1

Worked example

Find the odd pair: (a) 9 : 90 (b) 14 : 210 (c) 20 : 420 (d) 16 : 256

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 24 Sep 2024, Shift 2 · Q23Easy

Example 3 · Classification · Number Pairs

The second number in the given number-pairs is obtained by performing certain mathematical operation(s) on the first number. The same operation(s) is/are followed in all the number-pairs, except one. Find that odd number-pair. (NOTE: The relation should be found without breaking down the numbers into its constituent digits)

Mixing up n(n + 1) and n(n − 1)

32 \times 33 = 1056

and

32 \times 31 = 992

. Decide from the pairs that agree whether the rule adds n or takes it away, then test the fourth pair with that direction.

Treating n × n as n times its neighbour

14 \times 14 = 196

but

14 \times 15 = 210

. A square in a list of neighbour products is the odd one, even though it looks the right size.

Drill 7 more on products and powers of neighbouring numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Pair rules made of multiplying, adding and fixed ratios
Common pair rules
$y = kx + c \qquad y - x = d \qquad x : y = p : q$
Pair rules with squares, cubes and roots
Power rules for pairs
$y = x^2 \pm c \qquad y = (x \pm c)^2 \qquad y = x^3 \pm x \qquad y = k x^2 \qquad y = \sqrt{x} \pm c$
Products and powers of neighbouring numbers
Neighbouring numbers
$n(n+1) = n^2 + n \qquad (n+1)^2 - n^2 = 2n + 1 \qquad (n+1)^3 - n^3 = 3n^2 + 3n + 1$

Watch out for (6)

Reading k from the smallest pair
→ Pair rules made of multiplying, adding and fixed ratios
Only testing the rule from first to second
→ Pair rules made of multiplying, adding and fixed ratios
Taking a square on the right as proof
→ Pair rules with squares, cubes and roots
Looking for a multiplier when the numbers grow too fast
→ Pair rules with squares, cubes and roots
Mixing up n(n + 1) and n(n − 1)
→ Products and powers of neighbouring numbers
Treating n × n as n times its neighbour
→ Products and powers of neighbouring numbers

Test yourself on Classification

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Classification chapter test Drill all 47 questions