Maths Tricks and Shortcuts

Mental Math with Tricks and Shortcuts

Addition

Technique: Add left to right

326 + 678 + 245 + 567 = 900, 1100, 1600, 1620, 1690, 1730, 1790, 1804, & 1816

Note: Look for opportunities to combine numbers to reduce the number of steps to the solution. This was done

with 6+8 = 14 and 5+7 = 12 above. Look for opportunities to form 10, 100, 1000, and etc. between numbers

that are not necessarily next to each other. Practice!

Multiplication & Squaring

Some useful formulae

Examples

(a+b)² = a² + 2ab + b², 49² = (40 + 9) ² = 1600 + 720 + 81 = 2401

(a-b) ² = a² –2ab + b² , 56² = (60 – 4) ² = 3600 – 480 + 16 = 3136

(a+b) (a-b) = a² – b², 64 x 56 = (60 – 4) (60 + 4) = 3600 – 16 = 3584

(a+b) (c+d) = (ac + ad) + (bc + bd) , 23 x 34 = (20 + 3) (30 + 4) = 600 + 80 + 90 + 12 = 782

(a+b) (c-d) = (ac – ad) + (bc – bd), 34 x 78 = (30 + 4) (80 – 2) = 2400 –60 + 320 – 8 = 2652

(a-b) (c-d) = (ac – ad) – (bc – bd), 67 x 86 = (70 – 3) (90 – 4) = 6300 – 280 – 270 + 12 = 5762

X = 1 to 9 & Y = Any Number

(X5) ² = 100X(X+1) + 25 , 65² = 600(7) + 25 = 4200 + 25 = 4225

25 x Y = (Y x 100)/4, 25 x 76 = 7600/4 = 1900

50 x Y = (Y x 100)/2 , 50 x 67 = 6700/2 = 3350

75 x Y = 3(Y x 100)/4, 75 x 58 = (5800 x 3)/4 = 17400/4 = 4350

Square any Two Digit Number (a = 10’s digit & b = 1’s digit)

(ab)² = 100a² + 20(a x b) + b², 67² = 100(36) + 20(42) + 49 = 4489

Multiply any Two 2 Digit Numbers (a & c = 10’s digit, b & d = 1’a digit)

ab x cd = 100(a x c) + 10[(b x c) + (a x d)] + (b x d), 53 x 68 = 3000 + 580 + 24 = 3604

Tricks using (X5) ²

(X5 – a) ² = (X5) ² – X5(2a) + a2), 63² = (65 – 2) ² = 4225 – 260 + 4) = 3969

(X5 + a) ² = (X5) ² + X5(2a) + a2), 67² = (65 + 2) ² = 4225 + 260 + 4) = 4489

Squaring Numbers 52 to 99

a² = [a - (100 – a)]100 + (100 – a) ², 93² = (93 – 7)100 + 7² = 8649

Squaring Numbers 101 to 148

a² = [a + (a – 100)]100 + (a – 100) ², 108² = (108 + 8)100 + 82 = 11664