SSC CGL maths formulas, topic-wise

the overlap holds multiples of the LCM

terms times average of first and last

split into co-prime parts only

repeat until small

same for sums

one-way rule

1 if n even, d - 1 if n odd

when 2 has exponent a

q in lowest terms

common primes, lowest powers

all primes, highest powers

Turn the percent into a fraction first; cancel where you can.

x is the percent that the part stands for.

B is the quantity that follows the word 'of'.

Only when both percentages are of the same whole.

Old value below the line.

Plus for a rise, minus for a fall.

a and b carry their own signs.

x% rise followed by x% cut.

x% more than becomes this much less than.

x% less than becomes this much more than.

Each 'of' multiplies.

r% added every year, compounding.

r% of value lost every year.

Divide by the chip power to go back.

Percents on valid votes only.

a = shortfall below pass, b = marks above pass, x and y are the two percents.

a is the shortfall.

Budget unchanged, price up x%.

Budget unchanged, price down c%.

Ratio after (or before) n years.

k falls over time for elder-younger pairs.

Two multipliers: x for incomes, y for expenditures.

Price one set, then scale.

The sum form is the one you use.

Adding k shifts the average by k; multiplying by k scales it.

n = how many terms.

n = old count; A' = new average.

A' = average of the remaining n members.

The count does not change.

N = total count, overall average known.

Reverse ratio of the distances.

A' = new average, d = rise.

Three-day windows; multiply by the overlap length.

One equation per chunk.

Any one of the four recovers from the other three.

'Amounts to' includes the principal.

Interest is only (n−1)P.

Same rupees every year.

One chip per year.

For rates that change yearly.

Rate halves, periods double.

Rate quarters, periods quadruple.

Linear, not powers.

Two-year case.

One term per instalment.

Interest the early payments save.

Must end at zero.

CI: divide by the chip power.

Divide at CI.

Subtract at SI.

Always on CP.

Plus for profit, minus for loss.

Divide by the chip.

When profit at S1 equals loss at S2.

Use negative b for a fall.

For two discounts.

Any number of discounts; customer pays this share.

On the marked price.

g = gain%, d = discount%.

m = markup%.

One chip per trader.

W = true weight, w = weight used; divide by w.

L = claimed loss%, c = short weight%.

b = extra taken while buying, s = shortfall while selling.

Per cent of total CP.

S = common selling price.

A = amount of the other ingredient, unchanged.

A uniform draw keeps the ratio.

Multiply one factor per round when the draws differ.

For a gain of g% when sold at cost price.

Only when every vessel holds the same volume.

Speeds in units per hour, with tank = LCM of the times.

Both pipes fill.

a = filling time, b = emptying time, b > a.

t₁ = normal time, t₂ = time with the leak.

lines cross once

parallel lines

same line

for px - q, substitute q/p

decides root nature

equality when x = y

at x = sqrt(b/a)

max for a < 0, min for a > 0

strict inequality, strict discriminant

A straight line totals 180 degrees; one full turn totals 360.

The two inside angles on one side of a transversal, between parallel lines.

Complement = 90 − x, supplement = 180 − x.

Each of these pairs is equal.

Exterior angle = sum of the two remote (far) interior angles.

G is the centroid on median AD; the vertex piece is twice the base piece.

I = incentre, where the angle bisectors meet.

O = circumcentre; the angle at O stands on the same arc BC as angle A.

H = orthocentre, where the altitudes meet.

Median to side a of a triangle with sides a, b, c.

a = equal side, b = base.

s = semi-perimeter (half the perimeter).

a = side, P = perimeter, K = area; k = scale factor.

Square a length ratio to get the area ratio; take a root to go back.

A line parallel to one side cuts the other two sides in the same ratio.

The join of two midpoints is half the third side and parallel to it.

The bisector of angle A splits BC in the ratio of the sides AB and AC.

h = hypotenuse (longest side, opposite the right angle).

Subtract when a leg is missing.

The median drawn from the right-angle corner.

The corner angles of a rectangle are right angles.

a = longest side; equal means right, greater means obtuse, smaller means acute.

Any four-sided figure.

Adjacent angles supplement; opposite angles equal.

Opposite corners on one circle.

d1, d2 = diagonals; they cross at right angles.

a and b are the two parallel sides; h is the gap between them.

Height is measured perpendicular to the base.

n = number of sides.

d = distance from centre to chord.

Equal chords sit equally far from the centre.

Both angles stand on chord BC.

P is d from the centre of a circle of radius r.

Tangent squared = outside part times whole secant.

Two chords crossing inside the circle.

The angle between a tangent and a chord equals the angle the chord makes on the far side.

d = distance between centres.

b = any side, h = perpendicular height on it.

a = side; height is root-three over two of the side.

s = half the perimeter.

a, b legs, c hypotenuse; from equating two areas.

K = area, s = half-perimeter.

Each median halves the area of a triangle.

Three linked facts; any two fix the third.

Diagonal is root-two times the side.

theta = angle between the two given sides.

Diagonals cross at right angles and halve each other.

a, b = the two parallel sides.

d = a diagonal; h1, h2 = perpendiculars onto it.

Take pi as 22/7 unless stated.

Same fraction of circumference and area.

R = outer radius, r = inner radius.

Count turns by dividing distance by one circumference.

Quarter and half of the circle area.

Six equilateral triangles of side a.

P = perimeter, a = apothem (centre to a side).

Equal turns around the boundary.

One angle plus the total for all n.

n sides give this many diagonals.

Lengths take k once; areas take k squared.

Works for two changes in a row, like both dimensions.

Area factor is the side factor squared.

Square the linear scale before converting units.

l, b, h are the three dimensions.

a = edge.

Corner to opposite corner.

Convert once, at the end.

3, 2, 1, 0 painted faces for n by n by n.

r = radius, h = height.

Reverse of the volume formula.

Same conversion as tanks.

Scale each symbol by its own factor.

Radius, height, slant: a right triangle.

One-third of the same-base cylinder.

Skirt alone, or skirt plus base.

Equal base and height.

One radius drives both.

Total adds the flat circle.

Divide the cubed length, then cube-root.

k = radius scale factor.

Base can be any polygon.

h = perpendicular height.

R, r = the two end radii.

P = base perimeter; L = length; slant for pyramid.

o = opposite, a = adjacent, h = hypotenuse.

Flip the fraction.

For 0, 30, 45, 60, 90 degrees; cos runs the row backwards.

Draw the 3-4-5 triangle and read every ratio off it.

Three engines from one identity.

Sum and difference are reciprocals.

Cross terms cancel in pairs.

The constant that kills cross terms.

Drop the co- or add it.

The two acute angles are partners.

Complementary tangents multiply to 1.

The bridge from a sum to a product.

After dividing every term by cosine.

Two sides given, Pythagoras gives the third.

Conjugates of cosec plus cot.

General angles; on 0 to 90 check the endpoints too.

Peak at 45 degrees.

For positive x; equality when x = 1.

Rewrite as one constant plus one square.

The angle sits at the observer. Height is opposite, distance is next to the angle.

L is the slanting line of sight, the hypotenuse of the triangle.

The two horizontal lines are parallel, so the angles are equal.

cot 30 = sqrt3, cot 45 = 1, cot 60 = 1/sqrt3.

Theta is the ladder's angle with the ground.

d is the distance walked; beta is the nearer, bigger angle.

d is the full distance between the two observers.

b = building height; beta from the foot, alpha from the roof.

alpha = first (farther) angle, beta = second (nearer) angle.

Use the angle at the car's current position.

d = fixed horizontal distance; angles of depression shrink as the balloon rises.

d comes from the lower triangle: d = pedestal height x cot alpha.

x = distance from the foot to where the top touches.

Use the same pattern for a leaving member with n-1.

Position in the sorted list.

Rough rule for mildly skewed data; rearrange for any missing one.

x is the midpoint of each class.

L: lower limit of median class; c: cf before it; f: its frequency; h: width.

f_m: modal class frequency; f_1, f_2: neighbouring frequencies.

Average of squared distances from the mean.

Adding c changes nothing; multiplying scales the SD.

The square of the sum of the first n naturals.

Base = the old / original value.

theta is the slice angle in degrees.

n_i is the count in each band or row.

$m$ options for step 1 and $n$ for step 2, joined by 'and'.

Two cases that cannot happen together, joined by 'or'.

$r$ slots from $n$ items, repeats allowed.

$r$ ordered slots, no item reused.

Order matters; $0! = 1$.

All $n$ letters distinct.

$p$, $q$, $r$ are the repeat counts of the repeated letters.

Fix one person; rotations are the same seating.

Flips look identical, so divide by 2.

Divide arrangements by $r!$ for selections.

Flip a large lower index before computing.

The number of subsets of an $n$-element set.

Handshakes, league matches, diagonals.

$n$ different people; divide again by $k!$ for $k$ equal groups.

$n$ items, $r$ people, no one empty.

$r$ different items, each choosing a person.

$n$ different items, at least one taken.

$m$ favourable of $n$ equally likely outcomes.

'Not the event'.

Subtract the overlap once.

For independent events.

The complement of 'none'.

$n$ fair coins tossed together.

Ordered pairs, not unordered.

Total for any two-card question.

$a$ is the first term, $d$ the common difference.

Works for any AP.

$l$ is the last term.

Only when the count $n$ is odd.

$k$ means create $k + 1$ gaps.

$r$ is the common ratio; power is $n-1$.

For $r \ne 1$; if $r = 1$, the sum is $na$.

Only when $|r| < 1$.

Their product is $a^3$.

$h$ is the drop height, $r$ the rebound fraction.

$a$, $d$ come from the AP of reciprocals.

The reciprocal of the AM of the reciprocals.

For positive numbers, AM $\ge$ GM $\ge$ HM.

For $|r| < 1$, series $a + (a+d)r + (a+2d)r^2 + \cdots$

The square of the natural-number sum.

The run must start at 1.

Compare each term with its position $n$.

A frequent hidden rule; test it when differences fail.

A set with $n$ elements; the power set has $2^n$ elements.

All subsets except the set itself.

All subsets except the empty set.

Drops both the empty set and the full set.

Order does not matter inside a subset.

Subtract the overlap counted twice.

A complement swaps the join sign.

At least one of the two groups.

The wing region outside B.

Only A plus only B.

Outside both circles.

$p$ are the pair intersections and $t$ the triple.

Petals only; the triple is removed from each pair.

Exactly two plus the triple.

The three wings together.

One set wholly inside the other.

$N$ is the size of the universal set.

Shrink the union to its largest member set.

Union fixed at $N$; everyone covered.

All three sets nested.

Same formula, percent units.

$ab$ is the both percent here.

Inclusion–exclusion in percent units.

Two groups; percent units.

Reject every input that makes the bottom zero.

The inside of a root stays on the non-negative side.

Complete the square; $k$ is the floor or the ceiling.

Substitute back to get the minimum or maximum value.

Replace $x$ by $-x$ and simplify fully before deciding.

$g$ acts first; work inside out.

Composition both ways must return $x$.

From swap-and-solve; memorise for speed.

Each unit step adds $k$.

Each unit step multiplies by $k$.

$k > 0$ lifts the picture, $k < 0$ lowers it.

Moves right by $a$; $f(x+a)$ moves left.

First flips over the $x$-axis, second over the $y$-axis.

The part below the axis is reflected up.

Minimum if $a > 0$, maximum if $a < 0$.

Base $a > 0$, $a \ne 1$; argument $b > 0$.

Multiplication becomes addition; division becomes subtraction.

The exponent comes out front.

Any common base works; base 10 is standard.

Log of 1 is 0; log of the base is 1; base and log cancel.

Two points at distance $d$ from $a$.

One interval; needs $a > 0$.

Two arms; the sign '>' splits.

Every $x$ between $a$ and $b$ achieves it.

Write both sides with the same base first.