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SSC CGL Reasoning · Formula sheet

Classification formulas

9 formulas and 35 common traps for SSC CGL Reasoning Classification, grouped by subtopic.

Full notes with worked examples

Number Classification

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Prime and composite numbers

Prime test

n is prime if no prime p≤n divides nn \text{ is prime if no prime } p \le \sqrt{n} \text{ divides } n

Squares, cubes and divisibility tests

Last digits and divisibility

n2 ends in 0,1,4,5,6,93∣n when 3∣digit sum11∣n when 11∣alternating digit sumn^2 \text{ ends in } 0, 1, 4, 5, 6, 9 \qquad 3 \mid n \text{ when } 3 \mid \text{digit sum} \qquad 11 \mid n \text{ when } 11 \mid \text{alternating digit sum}

Common traps

Stopping at a rule that fits all four

If every option is even, being even is not the rule. The rule you want splits the options three against one; keep testing until one does.

Answering a different question from the one printed

Some stems ask for the number similar to a given list, or for the group whose members all belong together. There the answer is the option that follows the rule, not the one that breaks it.

Counting 1 as a prime

A prime has exactly two different factors. The number 1 has only one, so it is neither prime nor composite.

Trusting an odd last digit

An odd number can still be composite: 91=7×1391 = 7 \times 13 and 57=3×1957 = 3 \times 19. Run the digit-sum test for 3 and divide by 7 before you call a number prime.

Calling a number a square because of its last digit

A square must end in 0, 1, 4, 5, 6 or 9, but the reverse is not true: 26 ends in 6 and is not a square. The last digit can only rule a number out.

Missing that a number can be both a square and a cube

64 and 729 are squares and cubes at once. If the other options are cubes, such a number belongs with them; check what the other three share before you mark it.

Missing a run that goes the other way

5678 and 8765 both have digits in a run, one going up and one going down. When three options run one way, the fourth can be odd only because of its direction.

Number Pairs

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Pair rules made of multiplying, adding and fixed ratios

Common pair rules

y=kx+cy−x=dx:y=p:qy = kx + c \qquad y - x = d \qquad x : y = p : q
  • xxthe first number of a pair
  • yythe second number of a pair
  • k,c,dk, c, dfixed numbers shared by the three pairs that agree

Pair rules with squares, cubes and roots

Power rules for pairs

y=x2±cy=(x±c)2y=x3±xy=kx2y=x±cy = 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)=n2+n(n+1)2−n2=2n+1(n+1)3−n3=3n2+3n+1n(n+1) = n^2 + n \qquad (n+1)^2 - n^2 = 2n + 1 \qquad (n+1)^3 - n^3 = 3n^2 + 3n + 1

Common traps

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.

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(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.

Mixing up n(n + 1) and n(n − 1)

32×33=105632 \times 33 = 1056 and 32×31=99232 \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

14×14=19614 \times 14 = 196 but 14×15=21014 \times 15 = 210. A square in a list of neighbour products is the odd one, even though it looks the right size.

Number Triads and Sets

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Equal steps, equal ratios and growing gaps in a set

Steps and ratios in a set a, b, c

b−a=c−bba=cbb - a = c - b \qquad \frac{b}{a} = \frac{c}{b}

One number of a set made from the other two

Rules linking a set a, b, c

c=k(a+b)c=k(a−b)c=ab±ka+c=kbc = k(a + b) \qquad c = k(a - b) \qquad c = ab \pm k \qquad a + c = k b

Common traps

Mixing up equal steps and equal ratios

5, 20, 80 has steps 15 and 60 but ratios 4 and 4. 11, 17, 23 has steps 6 and 6 but no fixed ratio. Test both before you decide which rule the three options share.

Checking only one link of a triad

A triad can keep one link and break the other. If the first link holds in every option, the second link decides; skipping it means guessing.

Taking a rule that only two options obey

In 4, 5, 9 and 6, 3, 9 the sum works, but if the other two options fail it, the sum is not the rule. A rule must hold in exactly three options.

Throwing out an option because a step gives a fraction

When a rule divides by 3 and only one option is not a multiple of 3, that option is usually the answer, not a sign the rule is wrong. Check that the other three come out whole.

Letter Classification

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Letter positions and clusters with equal steps

Reverse position and wrapping round

reverse position=27−positionposition after Z=position−26\text{reverse position} = 27 - \text{position} \qquad \text{position after Z} = \text{position} - 26

Opposite letters, whose positions add to 27

Opposite letters

position(L)+position(opposite of L)=27\text{position}(L) + \text{position}(\text{opposite of } L) = 27

Common traps

Counting the starting letter as a step

Three places after D is G (E, F, G), not F. Count the letters you move onto, not the one you start from.

Stopping at Z

The alphabet wraps round. In a +4 cluster such as PTXB, X is followed by B, four places on. Subtract 26 from any position above 26.

Using 26 as the total for opposite letters

Opposite positions add to 27, because A (1) pairs with Z (26). With 26 as the total you get the letter one place off, such as U for E instead of V.

Letter Gap Patterns

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Common traps

Checking only the first gap

Two clusters can share their first and second gaps and differ only at the third. A cluster is odd if any one of its gaps differs, so write the full list before you decide.

Losing count when a step passes Z

From X, a step of +5 is Y, Z, A, B, C, so it lands on C. Counting stops at Z for many students and gives a wrong gap; add 26 to the later letter's position, or subtract 26 from a total above 26.

Dropping the direction of a step

Written without signs, KIM and KMI both read as gaps of 2 and 4, yet one goes back then forward and the other forward then back. Keep the plus or minus on every gap.

Wrapping the wrong way at the start of the alphabet

Going back from A or B, carry on from Z: two places back from A is Y. Adding 26 to the position before you subtract avoids a negative number.

Letter, Number and Symbol Groups

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Common traps

Checking the letters but not the numbers

A group can step its letters correctly and still carry one wrong number. For each term, test the number against the letter's position before you move on to the next term.

Counting from the wrong end of the alphabet

If the numbers are reverse positions (Z = 1), reading them as forward positions makes three groups look wrong at once. When no group fits A = 1, try Z = 1.

Stopping once the letters and numbers fit

A group whose letters and numbers are all correct can still be the odd one, because one symbol repeats. Read the symbols last, every time.

Missing a direction change

Letters going back and numbers going up is a pattern of its own. A group where both go back looks orderly, but it breaks the pattern the other three share.

Word Classification

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Common traps

Stopping at a group that holds all four

Saturn, Venus, Jupiter and the Sun are all bodies in the solar system, so that group splits nothing. Narrow it until exactly three fit: three planets and one star.

Taking the category word for a member

When one option is the name of the group the others belong to, it is not one more member. SSC's keys treat the category word as the odd one.

Answering from the broad group

Four rivers, four cities or four leaders cannot be split by their broad group. The answer is always one level down, so keep narrowing.

Confusing a big city with the capital

Guwahati is the largest city of Assam, but the capital is Dispur. When a question is about capitals, check the capital itself, not the best-known city.

Calling two near words opposites

Worry and fear are not opposites: both are kinds of unease. Decide for each pair whether the two words point the same way before you compare it with the others.

Reading the pair in the other order

"Shoes : feet" reads as something worn and where it is worn, not as a whole and its part. Say each pair's relation in the order printed before you match it with the others.

Ignoring a wrong fact inside a correct relation

Every pair can share one relation (a shape and its number of sides) while one gets the number wrong, such as a hexagon with 8 sides. Check each fact as well as the relation.

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