Math calculations

I. Perfect Squares (1–30, Must-Know)

1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25

6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100

11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225

16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400

21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625

26² = 676, 27² = 729, 28² = 784, 29² = 841, 30² = 900

Quick Trick (Numbers ending in 5):

(10a + 5)² = 100a(a + 1) + 25

Example: 15² = 100 × 1 × 2 + 25 = 225; 35² = 1225

II. Cubes (1–12)

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125

6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000

11³ = 1331, 12³ = 1728

III. Pythagorean Triples (Geometry)

(3, 4, 5), (6, 8, 10), (9, 12, 15)

(5, 12, 13), (7, 24, 25), (8, 15, 17)

(9, 40, 41), (10, 24, 26)

IV. Fractions, Decimals & Percentages

Conversions (1/2 to 1/20):

1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75

1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8

1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875

1/10 = 0.1, 1/20 = 0.05

1/3 ≈ 0.333, 2/3 ≈ 0.667

1/6 ≈ 0.167, 5/6 ≈ 0.833

1/7 ≈ 0.142857 (repeating)

1/9 ≈ 0.111, 1/11 ≈ 0.0909

Common Percentages:

10% = 0.1, 20% = 0.2, 25% = 1/4, 50% = 1/2

75% = 3/4, 12.5% = 1/8, 37.5% = 3/8

V. Divisibility Rules (Quick Factoring)

  • 2: Last digit is even.

  • 3: Sum of digits is divisible by 3.

  • 5: Last digit is 0 or 5.

  • 9: Sum of digits is divisible by 9.

  • 11: (Sum of odd digits) minus (Sum of even digits) is divisible by 11.

VI. Multiplication Shortcuts

  1. Same Tens Digit (Tails sum to 10):
    (10a + b)(10a + 10 - b) = a(a + 1) × 100 + b(10 - b)
    Example: 43 × 47 = 4 × 5 × 100 + 3 × 7 = 2021

  2. Same Tens Digit (Any Tails):
    Example: 15 × 16 = 15 × 15 + 15 = 240

  3. Two-Digit × 11:
    "Spread the sides, add in the middle."
    Example: 23 × 11 = 253; 57 × 11 = 627

VII. Algebraic Identities & Roots

Formulas:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

  • Quick Calc: 98² = (100 - 2)² = 9604; 103 × 97 = 9991

Approximate Square Roots:

√2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236

√6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162

VIII. Factorials

1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720