**Quant Short-tricks & Formulae**

**Multiplication**

**Multiplying 2-digit numbers**

**Type-I**

AB **×** CD = A **×** C **/** A **×** D **+** B **×** C **/** B **×** D

E.g. 1. 34 **×** 57 = 3 **×** 5 **/** 3 **×** 7 **+** 4 **×** 5 **/** 4 **×** 7

= 15 **/** 21 + 20 **/** 28

= 15 **/** 41 **/** 28

= 1(5 **+** 4)(1 **+** 2)8

= 1938

E.g. 2. 56 **×** 48 = 5 **×** 4 **/** 5 **×** 8 **+** 6 **×** 4 **/** 6 **×** 8

= 20 **/** 40 + 24 **/** 48

= 20 **/** 64 **/** 48

= 2(0 **+** 6)(4 **+** 4)8

= 2688

**Type-II**

AB **×** AC = A^{2} **/** A (B **+** C) **/** B **×** C

E.g. 1. 64 **×** 65 = 6^{2} **/** 6(4 **+** 5) **/** 4 **×** 5

= 36 **/** 54 **/** 20

= 3(6 **+** 5)(4 **+** 2)0

= 4180

E.g. 2. 54 **×** 56 = 5^{2} **/** 5(4 **+** 6) **/** 4 **×** 6

= 25 **/** 50 **/** 24

= 2(5 **+** 5)(0 **+** 2)4

= 3024

**Type-III**

AB **×** CC = AC **/** (A **+** B)C **/** BC

E.g. 1. 24 **×** 33 = 2 **×** 3 **/** (2 **+** 4) **×** 3 **/** 4 **×** 3

= 6 **/** 18 **/** 12

= (6 **+** 1)(8 **+** 1)2

= 792

E.g. 2. 43 **×** 55 = 4 **×** 5 **/** (4 **+** 3) **×** 5 **/** 3 **×** 5

= 20 **/** 35 **/** 15

= 2(0 **+** 3)(5 **+** 1)5

= 2365

**Multiplying 3-digit numbers**

ABC**×**DEF=A**×**D**/**A**×**E**+**B**×**D**/**A**×**F**+**B**×**E**+**C**×**D**/**B**×**F**+**C**×**E**/**C**×**F

E.g.321**×**546=3**×**5**/**3**×**4**+**2**×**5**/**3**×**6**+**2**×**4**+**1**×**5**/**2**×**6**+**1**×**4**/**1**×**6

= 15 **/** 12 **+** 10 **/** 18 **+** 8 **+** 5 **/** 12 **+** 4 **/**0 6

= 15 **/** 22 **/** 31 **/** 16 **/ **06

= 1(5 **+** 2)(2 **+** 3)(1 **+** 1)(6 **+** 0)6

= 175266

**Easy techniques for multiplication**

To multiply with 4, multiply with 2 twice

To multiply with 5, multiply with 10 and then divide by 2

To multiply with 6, multiply with 3 and then 2

To multiply with 9, multiply with 3 twice

**Divisibility test**

A number is divisible by 2, if the last digit is even i.e. 0,2,4,6 or 8

A number is divisible by 3, if the sum of all the digits is divisible by 3

A number is divisible by 4, if the last two digits are divisible by 4

A number is divisible by 5, if the last digit is 0 or 5

A number is divisible by 6, if the number is divisible by 2 and 3

A number is divisible by 7, if the new number obtained after removing the last digit and subtracting its twice from the truncated original number, is divisible by 7

A number is divisible by 8, if the last three digits are divisible by 8

A number is divisible by 9, if the sum of all the digits is divisible by 9

A number is divisible by 10, if the last digit is 0

A number is divisible by 11, if the difference of sum of digits at odd and even places is divisible by 11

**Square of a number**

- No square number ends in 2, 3, 7 or 8.
- The number of zeros at the end of a perfect square is always even.
- Square of even numbers are even and squares of odd numbers are odd.
- A perfect square leaves remainder 0 or 1 on division by 3.
- A perfect square leaves remainder 0 or 1 on division by 4.
- The unit’s digit of the square depends on unit’s digit of the number.(See Table)

**Unit’s digit of square of numbers**

** **

Unit’s digit ofthe number | Unit’s digit of the square of number |

0 |
0 |

1 |
1 |

2 |
4 |

3 |
9 |

4 |
6 |

5 |
5 |

6 |
6 |

7 |
9 |

8 |
4 |

9 |
1 |

**Squares and cubes of first 30 natural numbers**

Number |
Square |
Cube |

1 | 1 | 1 |

2 | 4 | 8 |

3 | 9 | 27 |

4 | 16 | 64 |

5 | 25 | 125 |

6 | 36 | 216 |

7 | 49 | 343 |

8 | 64 | 512 |

9 | 81 | 729 |

10 | 100 | 1000 |

11 | 121 | 1331 |

12 | 144 | 1728 |

13 | 169 | 2197 |

14 | 196 | 2744 |

15 | 225 | 3375 |

16 | 256 | 4096 |

17 | 289 | 4913 |

18 | 324 | 5832 |

19 | 361 | 6859 |

20 | 400 | 8000 |

21 | 441 | 9261 |

22 | 484 | 10648 |

23 | 529 | 12167 |

24 | 576 | 13824 |

25 | 625 | 15625 |

26 | 676 | 17576 |

27 | 729 | 19683 |

28 | 784 | 21952 |

29 | 841 | 24389 |

30 | 900 | 27000 |

**Square of a number if all the digits are same(1,3 or 9)**

11^{2 }= 121

111^{2} = 12321

1111^{2} = 1234321

11111^{2} = 123454321

111111^{2} = 12345654321

1111111^{2} = 1234567654321

11111111^{2} = 123456787654321

33^{2} = 1089

333^{2} = 110889

3333^{2} = 11108889

33333^{2} = 1111088889

333333^{2} = 111110888889

3333333^{2} = 11111108888889

3333333^{2} = 1111111088888889

99^{2} = 9801

999^{2} = 998001

9999^{2} = 99980001

99999^{2} = 9999800001

999999^{2} = 999998000001

9999999^{2} = 99999980000001

99999999^{2} = 9999999800000001

**Square root of a number**

Complete square root of a number is only possible if its last digit is 0, 1, 4, 5, 6 or 9. If the last digit is 2, 3, 7 or 8 then there is no complete square root.

Last digit ofthe number | Last digit of completesq root |

0 |
0 |

1 |
1 or 9 |

2 |
No Complete square root |

3 |
No Complete square root |

4 |
2 or 8 |

5 |
5 |

6 |
4 or 6 |

7 |
No Complete square root |

8 |
No Complete square root |

9 |
3 or 7 |

**Square roots and cube roots of first ten natural numbers**

Numberx |
Square Rootx ^{1/2} |
Cube Rootx ^{1/3} |

1 | 1.000 | 1.000 |

2 | 1.414 | 1.260 |

3 | 1.732 | 1.442 |

4 | 2.000 | 1.587 |

5 | 2.236 | 1.710 |

6 | 2.449 | 1.817 |

7 | 2.646 | 1.913 |

8 | 2.828 | 2.000 |

9 | 3.000 | 2.080 |

10 | 3.162 | 2.154 |

**Even and odd numbers**

Even + Even = Even

Odd + Odd = Even

Even + Odd = Odd

Even × Even = Even

Odd × Odd = Odd

Even × Odd = Even

**Prime numbers**

To check if a number is prime

- Find approximate square root of the number
- Check if the no. is divisible by any prime number less then its square root
- If not divisible then the number is a prime number

E.g. Number 461 is a prime number or not.

Approx sq. root = between 21 and 22

Prime numbers less then square root are 2, 3, 5, 7, 11, 13, 17 and 19

Since 461 is not divisible by any of these, it is a prime number

**NOTE:**

- If complete square root exist then no. is not prime
- There are fifteen prime numbers from 1 to 50, twenty five from 1 to 100.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,43, 47

53, 59, 61, 67, 71, 73, 79, 83, 89, 97

**Number of divisors**

If N is any no. and N = a^{n} **×** b^{m} **×** c^{p} **×** …. where a, b, c are prime no.

No. of divisors of N = (n **+** 1) (m **+** 1) (p **+** 1) ….

e.g. Find the no. of divisors of 720

N = 720 = 2^{4} **×** 3^{2} **×** 5^{1}

No. of divisors = (4 **+** 1) (2 **+** 1) (1 **+** 1) = 30

**Percentage**

Percentage means per every hundred.

Properties

- p % of q = q % of p= (p × q)/100
- What percentage of p is q: q/p × 100
- Percentage change = (change/initial value) × 100
- Increase p by q% = p( 1+ q/100 )
- Decrease p by q% = p(1 – q/100)
- If the value changes by x%, the percentage change required to bring back to original value is 100x/(100 + x)%.
- If p is x% more/less than q, then q is 100x/(100 + x)% less/more than p.
- If there are successive percentage changes of a% and b%, then the effective percentage change is:

(a + b + (ab/100))%

**Geometry**

**Triangles and quadrilaterals**

**Equilateral triangle**

A triangle is called equilateral if all the angles or sides are equal

Properties

- All sides are equal
- All angles are 60
^{o}

**Isosceles triangle**

A triangle is called isosceles if any two angles or sides are equal

Properties

- Two sides are equal
- Two angles are equal

**Parallelogram**

A quadrilateral is called a parallelogram if opposite sides are parallel, equal or opposite angles are equal.

Properties

- Opposite sides are parallel and equal
- Opposite angles are equal
- Consecutive angles are supplementary
- A diagonal of a parallelogram divides it into two triangles of equal area
- Parallelograms on the same base and between the same parallels are equal in area
- Diagonals bisect each other

**Rhombus**

A quadrilateral is called a rhombus if all the sides are equal.

Properties

- All sides are equal and opposite sides are parallel
- Diagonals bisect the angles at vertex.
- The diagonals are perpendicular bisectors of each other

**Rectangle**

A quadrilateral is called a rectangle if all the angles are right angle.

Properties

- Opposite sides are equal
- All angles are right angles
- Diagonals are equal and bisect each other

**Square**

A quadrilateral is called a square if all the sides are equal and all the angles are right angle.

Properties

- All sides are equal
- All angles are right angles
- Opposite sides are parallel
- Diagonals are perpendicular bisectors

**Note:**

Parallelogram –> If all angles are equal –> Rectangle

Parallelogram –> If all sides are equal –> Rhombus

Parallelogram –> If both angles and sides are equal –> Square

**Properties of triangle**

- There are six exterior angles in a triangle.
- Sum of interior angle & corresponding exterior angle is always 180°.
- Sum of interior opposite angles is equal to exterior angle
- Sum of the length of any two sides is always greater than the length of the third side.
- Difference of the length of any two sides is always less than the third side.
- Side opposite to the greatest angle is the longest side.
- A triangle must have at least two acute angles.
- Triangles with same or equal base and between the same parallels have equal areas.
- If a, b, c denotes the length the sides of a triangle then

If c^{2}< a^{2}**+**b^{2}, Triangle is acute angled.

If c^{2}= a^{2}**+**b^{2}, Triangle is right angled.

If c^{2}> a^{2}**+**b^{2}, Triangle is obtuse angled. - Any median divides the triangle in two parts with equal areas.
- The point of intersection of the medians is called the Centroid of the triangle.
- The centroid divides any median in the ratio 2 : 1.
- Midpoint Theorem: In any triangle, the line joining the mid points of any two sides is parallel to the third side and is half in length.

**Properties of polygon**

- Sum of all the exterior angles of a polygon is 360°
- Each exterior angle of a regular polygon is 360° / n
- Sum of all the interior angles of a polygon is (n – 2) x 180°
- Each interior angle of a regular polygon is (n – 2) / n x 180°
- Number of diagonals of a polygon is n(n – 3) / 2
- The ratio of sides a polygon to the diagonals of a polygon is 2 : (n – 3)
- Ratio of interior angle to exterior angle of a regular polygon is (n – 2) : 2

**Properties of circle**

- One and only one circle passes through three fixed points.
- From a point on the circle only one tangent can be drawn.
- From any exterior point of the circle only two tangents can be drawn.
- The lengths of two tangents drawn from an exterior point of the circle are equal.
- The tangent at any point of a circle and the radius through the point are always perpendicular to each other.
- If two circles touch each other internally or externally, then their centers & the points of contact are collinear.
- If two circles touch externally,then the distance between their centers is equal to the sum of the radii of the circles.
- If two circles touch internally, then the distance between their centers is equal to the difference of the radii of the circles.
- Circles having same centre are concentric circles.
- Points lying on the same circle are called concyclic points.

**Surface area and volume**

Lateral surface area and curved surface area are nearly same. It’s called lateral surface area for cube and cuboid and curved surface area for cylinder, cone and hemisphere.

**Cube**

- Lateral Surface Area = 4a
^{2} - Total Surface Area = 6a
^{2} - Volume = a
^{3}

**Cuboid**

- Lateral Surface Area = 2(lh
**+**bh) = 2 (l**+**b) h - Total Surface Area = 2(lb
**+**bh**+**lh) - Volume = l
**×**b**×**h

**Cylinder**

- Curved Surface Area = 2πrh
- Total Surface Area = 2πrh
**+**2πr^{2 }= 2πr(r**+**h) - Volume = πr
^{2 }h

**Cone**

- Curved Surface Area = πrl
- Total Surface Area = πrl
**+**πr^{2 }= πr(r**+**l) - Volume = 1
**/**3 πr^{2 }h

**Hemisphere**

- Curved Surface Area = 2πr
^{2} - Total Surface Area = 2πr
^{2 }**+**πr^{2 }= 3πr^{2} - Volume = 2
**/**3 πr^{3}

**Sphere**

- Surface Area = 4πr
^{2} - Volume = 4
**/**3 πr^{3}

**Frustum**

- Curved Surface Area = π(r
_{1 }**+**r_{2 })l - Total Surface Area =π(r
_{1 }**+**r_{2 })l**+**πr_{1 }^{2 }**+**πr_{2 }^{2} - Volume = 1
**/**3 πh (r_{1 }^{2 }**+**r_{2 }^{2 }**+**r_{1 }r_{2 })

**Algebra**

**Quadratic equation**

The general form of a quadratic equation in x is

ax2 + bx + c = 0 , where a , b , c ∈ R & a **≠** 0.

The solution of the quadratic equation , ax² + bx + c = 0 is

x = [– b ± √(b^{2} – 4ac)]**/**2a

Discriminant of a quadratic equation

ax² + bx + c = 0 where a, b, c ∈ R & a**≠** 0

D = b^{2} – 4ac

- If D > 0 ⇔ Roots are real & distinct
- If D = 0 ⇔ Roots are real & coincident
- If D < 0 ⇔ Roots are imaginary

**Important points**

- If all the coefficients of a quadratic equation are real (a, b, c ∈ R & a
**≠**0) and p + iq (p , q ∈ R & i = √-1) is one root of the equation, then the other root must be p – i q(conjugate of p + iq) & vice - If all the coefficients of a quadratic equation are real (a, b, c ∈ Q & a
**≠**0) and p + √q (p , q ∈ Q) is one root of the equation, then the other root must be p – √q & vice - A quadratic equation whose roots are α & β is

(x – α)(x – β) = 0

x² – (α + β) x + α β = 0

- A quadratic equation whose sum and product of roots is given can be written as

x² – (sum of roots) x + product of roots = 0

x² – (α + β) x + α β = 0

- The condition that a quadratic function f (x , y) = ax² + 2hxy + by² + 2gx + 2fy + c may be resolved into two linear factors is

abc + 2fgh – af² – bg² – ch² = 0

**Logarithmic inequalities**

- For a > 1, the inequalities 0 < x < y & log
_{a}x < log_{a}y are equivalent. - For 0 < a < 1, the inequalities 0 < x < y & log
_{a}x > log_{a}y are equivalent. - If a > 1, then log
_{a}x < p ⇒ 0 < x < a^{p} - If a > 1, then log
_{a}x > p ⇒ x > a^{p} - If 0 < a < 1, then log
_{a}x < p ⇒ x > a^{p} - If 0 < a < 1, then log
_{a}x > p ⇒ 0 < x < a^{p}

**Algebraic identities**

- (x
**+**y)^{ 2}= x^{2}**+**2xy**+**y^{2}= (x – y)^{ 2}**+**4xy - (x – y)
^{ 2}= x^{2}– 2xy**+**y^{2}= (x**+**y)^{ 2}– 4xy - x
^{2}**+**y^{2}= (x**+**y)^{ 2}– 2xy = (x – y)^{ 2}**+**2xy - x
^{2}– y^{2}= (x**+**y)(x – y) - (x
**+**y)^{3}= x^{3}**+**3x^{2}y**+**3xy^{2}**+**y^{3}= x^{3}**+**y^{3}**+**3xy(x**+**y) - (x – y)
^{3}= x^{3}– 3x^{2}y**+**3xy^{2}– y^{3}= x^{3}– y^{3}– 3xy(x**+**y) - x
^{3}**+**y^{3}= (x**+**y)(x^{2}– xy**+**y^{2}) = (x**+**y)^{ 3}– 3xy(x**+**y) - x
^{3}– y^{3}= (x – y)(x^{2}**+**xy**+**y^{2}) = (x – y)^{ 3}**+**3xy(x – y) - x
^{4}– y^{4}= (x – y)(x**+**y)(x^{2}**+**y^{2}) - x
^{6}– y^{6}= (x**+**y)(x^{2}– xy**+**y^{2}) = (x – y)(x^{2}**+**xy**+**y^{2}) - x
^{6}**+**y^{6}= (x^{2}**+**y^{2})(x^{4}– x^{2}y^{2}**+**y^{4}) - (x
**+**y**+**z)^{2}= x^{2}**+**y^{2}**+**z^{2}**+**2xy**+**2yz**+**2xz - (x
**+**y**+**z)^{3}= x^{3}**+**y^{3}**+**z^{3}**+**3(x**+**y)(y**+**z)(z**+**x) - (x – y – z)
^{ 2}= x^{2}**+**y^{2}**+**z^{2}– 2xy**+**2yz – 2xz - x
^{3 }**+**y^{3 }**+**z^{3 }– 3xyz = (x**+**y**+**z)(x^{2 }**+**y^{2 }**+**z^{2 }– xy – yz – xz) - x
^{4}**+**x^{2}y^{2}**+**y^{4}= (x^{2}– xy**+**y^{2})(x^{2}**+**xy**+**y^{2}) - (x
**+**a)(x**+**b) = x^{2}**+**x(a**+**b)**+**ab - (x – a)(x
**+**b) = x^{2}**+**x(b – a) – ab - (x – a)(x – b) = x
^{2}– x(a**+**b)**+**ab