Quick Tricks & Mnemonics

Memory aids, shortcuts, and exam-day hacks to help you remember formulas and solve problems faster

Subject

Difficulty

Algebra

(a-b)² Formula

basic

Formula

(a-b)² = a² - 2ab + b²

Mnemonic & Quick Trick

MNEMONIC: 'Minus in middle' - Same as (a+b)² but the 2ab term is negative.

Example

(5-3)² = 25 - 30 + 9 = 4

Explanation

The square of difference of two numbers equals the sum of their squares minus twice their product.

Algebra

(a+b)² Formula

basic

Formula

(a+b)² = a² + 2ab + b²

Mnemonic & Quick Trick

MNEMONIC: 'AaBbC' - First A squared, then 2aB, then B squared. Remember: middle term is always 2ab.

Example

(3+4)² = 9 + 24 + 16 = 49

Explanation

The square of sum of two numbers equals the sum of their squares plus twice their product.

Algebra

(a+b+c)² Formula

intermediate

Formula

(a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca

Mnemonic & Quick Trick

MNEMONIC: 'Three squares + three products' - All three individual squares, then pairs of products.

Example

(1+2+3)² = 1 + 4 + 9 + 4 + 12 + 6 = 36

Explanation

Expansion of square of three terms.

Algebra

a² - b² Formula

basic

Formula

a² - b² = (a+b)(a-b)

Mnemonic & Quick Trick

QUICK TRICK: For any two consecutive even numbers: 10² - 8² = (10+8)(10-8) = 18×2 = 36

Example

100 - 49 = (10+7)(10-7) = 17 × 3 = 51

Explanation

Difference of squares can be factored as product of sum and difference.

Algebra

a³ - b³ Formula

intermediate

Formula

a³ - b³ = (a-b)(a² + ab + b²)

Mnemonic & Quick Trick

MNEMONIC: For a³ + b³: plus in first term. For a³ - b³: minus in first term, plus in second bracket.

Example

64 - 8 = (4-2)(16 + 8 + 4) = 2 × 28 = 56

Explanation

Difference of cubes can be factored.

Algebra

a³ + b³ Formula

intermediate

Formula

a³ + b³ = (a+b)(a² - ab + b²)

Mnemonic & Quick Trick

QUICK TRICK: Always check if numbers are perfect cubes first. Use the factorization to solve quickly.

Example

8 + 27 = (2+3)(4 - 6 + 9) = 5 × 7 = 35

Explanation

Sum of cubes can be factored.

Arithmetic

Average Formula

basic

Formula

Average = Sum of all values / Number of values

Mnemonic & Quick Trick

QUICK TRICK: If average of n numbers is x, and 1 new number is added: New Average = (nx + new)/n+1

Example

Numbers: 10, 20, 30, 40. Average = 100/4 = 25

Explanation

Mean or average of a set of numbers.

Trigonometry

Basic Trigonometric Ratios

basic

Formula

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent

Mnemonic & Quick Trick

MNEMONIC: 'SOH CAH TOA' - Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent

Example

In 30-60-90 triangle: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3

Explanation

SOH CAH TOA mnemonic for sine, cosine, tangent.

Geometry

Circle Area Formula

basic

Formula

Area = πr²

Mnemonic & Quick Trick

MNEMONIC: 'Pie are squared' - Use π ≈ 22/7 for easy calculation. Circumference = 2πr

Example

r = 7, Area = 22/7 × 49 = 154 sq units

Explanation

Area of a circle with radius r.

Time & Work

Combined Work Formula

intermediate

Formula

1/Total Time = 1/Time_A + 1/Time_B

Mnemonic & Quick Trick

MNEMONIC: 'Add rates, flip for time' - Add individual rates, reciprocal gives total time

Example

A does work in 6 days (1/6), B in 8 days (1/8). Together = 1/6 + 1/8 = 7/24. Total = 24/7 days

Explanation

When two people work together, rates add up.

Arithmetic

Compound Interest Formula

intermediate

Formula

A = P(1 + r/100)^n

Mnemonic & Quick Trick

QUICK TRICK: For 2 years at r%, CI - SI = P × (r/100)². Always use this shortcut.

Example

P=1000, r=10%, n=2. A = 1000(1.1)² = 1000 × 1.21 = 1210. CI = 210

Explanation

Amount after compound interest for principal P at rate r% for n periods.

Arithmetic

Percentage Increase/Decrease

basic

Formula

Percentage = (Difference / Original) × 100

Mnemonic & Quick Trick

QUICK TRICK: If increase is x%, new value = original × (100+x)/100. If decrease, use (100-x)/100

Example

Price increased from 100 to 120. Increase % = (20/100) × 100 = 20%

Explanation

To find what percent one number is of another.

Geometry

Pythagorean Theorem

basic

Formula

a² + b² = c²

Mnemonic & Quick Trick

COMMON PYTHAGOREAN TRIPLES: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29)

Example

Sides: 3, 4, then hypotenuse = √(9+16) = 5

Explanation

In a right triangle, sum of squares of two sides equals square of hypotenuse.

Arithmetic

Ratio and Proportion

basic

Formula

If a:b = c:d, then ad = bc

Mnemonic & Quick Trick

MNEMONIC: 'Cross multiply' - In a:b = c:d, multiply diagonally: a×d = b×c

Example

2:3 = 4:6, so 2×6 = 3×4 = 12

Explanation

In proportion, product of extremes equals product of means.

Speed & Distance

Relative Speed Formula

intermediate

Formula

Relative Speed (same direction) = |S1 - S2|. Opposite direction = S1 + S2

Mnemonic & Quick Trick

QUICK TRICK: Same direction = subtract. Opposite = add. Time to meet = Distance / Relative Speed

Example

A at 60 km/hr, B at 40 km/hr. Same direction: 20 km/hr. Opposite: 100 km/hr

Explanation

Speed of one object relative to another depends on direction.

Arithmetic

Simple Interest Formula

basic

Formula

SI = (P × R × T) / 100

Mnemonic & Quick Trick

MNEMONIC: 'PRT over 100' - Always divide by 100. Amount = P + SI

Example

P=1000, R=10%, T=2 years. SI = (1000×10×2)/100 = 200

Explanation

Simple Interest = Principal × Rate × Time / 100

Speed & Distance

Speed-Distance-Time Formula

basic

Formula

Speed = Distance / Time

Mnemonic & Quick Trick

MNEMONIC: 'D over T for Speed' - Distance ÷ Time = Speed. Triple relationship: D=S×T, S=D/T, T=D/S

Example

Distance = 100 km, Time = 2 hours. Speed = 50 km/hr

Explanation

Speed is distance covered per unit time.

Geometry

Triangle Area Formula

basic

Formula

Area = 1/2 × base × height

Mnemonic & Quick Trick

QUICK TRICK: If height not given, use Heron's formula: s = (a+b+c)/2, Area = √[s(s-a)(s-b)(s-c)]

Example

Base = 10, Height = 6, Area = 1/2 × 10 × 6 = 30 sq units

Explanation

Area of any triangle is half the product of base and perpendicular height.

Trigonometry

Trigonometric Values (0°-90°)

basic

Formula

sin²θ + cos²θ = 1

Mnemonic & Quick Trick

MNEMONIC TABLE: sin(0°)=0, sin(30°)=1/2, sin(45°)=1/√2, sin(60°)=√3/2, sin(90°)=1. Reverse for cos.

Example

sin² 45° + cos² 45° = (1/√2)² + (1/√2)² = 1/2 + 1/2 = 1

Explanation

Fundamental trigonometric identity. Standard angles: 0°, 30°, 45°, 60°, 90°

Time & Work

Work Rate Formula

basic

Formula

Work = Rate × Time, or Rate = 1/Time

Mnemonic & Quick Trick

QUICK TRICK: If A completes in x days and B in y days, together = xy/(x+y) days

Example

A completes work in 10 days. Rate = 1/10 per day. In 5 days, work done = 5/10 = 1/2

Explanation

If a person completes work in T days, rate = 1/T of work per day.