Aptitude Mathematics Formulas

Mathematics Formula Collection- Algebra & Mensuration

  1. (a+b)2 = a2 + 2ab + b2
  2. (a – b)2 = a2 – 2ab + b2
  3. a2 + b2 = (a+b)2 – 2ab = (a – b)2 + 2ab
  4. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
  5. (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
  6. (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
  7. a2 – b2 = (a – b)(a + b)
  8. (a + b)3 = a3 + b3 + 3ab(a + b)
  9. (a – b)3 = a3 – b3 – 3ab(a – b)
  10. a3 – b3 = (a – b)(a2 + ab + b2)
  11. a3 + b3 = (a + b)(a2 – ab + b2)
  12. an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1) | n is a natural number.
  13. aman = am+n 
  14. (ab)m = amb
  15. (am)n = amn
  16. am/ an = am-n
  17. (a/b)n = an/ bn
  18. a0 = 1
  19. a-n = 1/ an
  20. logamn = logam + logan
  21. loga(m/n) = logam – logan
  22. logamn = n.logam
  23. logba = logka / logkb
  24. logba = 1/ logab
  25. (x+a)(x+b) = x2 + (a+b)x + ab

Progression(Series)

 1-Arithmetic Progression(AP)
   a, a+d, a+2d, a+3d, …, …, …,l(last term)
   a-First Term, d-Common Difference, l-Last Term

  • nth Term = Tn = a + (n-1)d
  • Sum of n Terms = Sn = n/2[2a + (n-1)d] = n/2[first term + last term]

  2-Geometric Progression(GP)
   a, ar, ar2, ar3, ar4, ….
   a-First Term, r-Common Ratio

  • nth Term = Tn = arn-1
  • Sum of n Terms = Sn = a(1-rn) / (1-r)
  • Sum of ∞ Terms = S∞ = a / (1-r)

  3-Harmonic Progression(HP)
   a1,
 a2, a3, a4, …, …. are said to be in Harmonic Progression(HP) if and only             if 1/a1, 1/a12, 1/a3, 1/a4, … are in Arithmetic Progession(AP).  

VBODMAS Rule

   V- Vinculumor or Bar
   B- Bracket
   O- Of
   D- Division
   M- Multiplication
   A- Addition
   S- Subtraction 

Average

  • Average of First n Natural Numbers = (n+1)/2
  • Average of First n Even Numbers = n+1
  • Average of First n odd Numbers = n
  • Average of Consecutive Numbers = (First No. + Last No.) / 2
  • Average of squares of First n Natural Numbers = (n+1)(2n+1) / 6
  • Average of Cubes of First n Natural Numbers = n(n+1)2 / 4

Percentage

y% of x = x * y/100

  • 5% of a number = number/20
  • 10% of a number = number/10
  • 20% of a number = number/5
  • 25% of a number = number/4
  • 50% of a number = number/2
  •  

Profit and Loss

  • Profit = Selling Price – Cost Price
  • Loss = Cost Price – Selling Price
  • Profit% = (Profit/Cost Price) * 100
  • Loss(Loss/Cost Price) * 100
  • Selleng Price = (100 + Profit%) * Cost Price / 100
  • Selleng Price = (100 – Loss%) * Cost Price / 100
  • Cost Price = 100 * Selling Price / (100 + Profit%)
  • Cost Price = 100 * Selling Price / (100 – Loss%)

Discount

  • Discount = Marked Price – Selling Price
  • Discount% = (Discount/Marked Price) * 100
  • Selling Price = Marked Price – Discount
  • Selling Price = Marked Price * (1 – r/100)

Simple Interest

  • Simple Interest = (Principal * Rate * Time)/100
  • Total Amount = Principal + Simple Interest = P + SI = P + PRT/100
    = P(1 + RT/100)
  • Principal = (Amount * 100) / (100 + RT)
  • Simple Interest = (Amount * Rate * Time) / (100 + RT)
    P = Principal Amount, A = Total Amount, R = Rate of Interest (Annual)
    T = Time

Compound Interest

  • In Compound Interest the amount at the end of each year becomes the principal for the next year.
  • Total Amount = P(1 + R/100)T
  • P = Principal Amount, R = Rate of Interest(Annual), T = Time

Factorial

  • n! = n(n-1)(n-2)(n-3)……*1

Combination

  • Number of Combination of n different things when r things are taken into consideration. (Order is not important)

nCr = n! / r!(n-r)!

Permutation

  • The number of Permutation of n different things taken r at a time

nPr = n! / (n-r)!

Probability

Probability of happening of an event = (Number of Favourable events) / (Total Number of possible outcomes)

Mensuration

Triangle

Equilateral Triangle:
  • All three sides of the Equilateral Triangle are equal.
  • Each angle of the Equilateral Triangle is 60o 
Isosceles Triangle:
  • 2 Sides and 2 Angles are equal.
  • Altitude drawn on non equal side bisect it. 
Scalene Triangle:
  • All the 3 sides are unequal.
Right Angle Triangle:
  • One of the angle is 90o 
Isosceles Right Angle Triangle:
  • One of the angle is 90o 
  • 2 Sides are equal. 
Properties of Triangle:
  • Sum of any 2 sides of any triangle is greater than the third side.
  • For a constant Perimeter, Equilateral Triangle will have the maximum area.
  • Median: The line joining the midpoint of any side of triangle to the opposite vertex.
  • The median of a triangle divides the triangle into triangle of two equal area.
Inner Circle and Circumcircle of Equilateral Triangle:
  • 2 Let the side of equilateral triangle is a
Quadrilateral:
  • Figure enclosed by 4 sides.
  • Sum of all the 4 angles = 360o 
Parallelogram:
  • Opposite sides are parallel.
  • Diagonals of a parallelogram bisects each other.
  • Diagonal of parallelogram divides it into 2 triangles of equal area.
  • Opposite angles of a parallelogram are equal.
Square:
  • It is a parallelogram which has all 4 sides equal and each angle is 90o 
  •  Diagonal of a square are equal & bisect each other at 90o 
Rectangle:
  • It is a parallelogram with equal opposite sides and each angle is 90o 
  • Diagonals of a rectangle are equal.
  • Diagonal of a rectangle bisects each other.
Trapezium:
  • It is a quadrilateral with one pair of opposite sides parallel.
Rhombus:
  • It is a parallelogram with all 4 sides equal.
  • Opposite angles of rhombus are equal but not 90o .
  • It has unequal diagonal but they bisect each other at 90o .
Circle:
  • Let r be the radius of circle.
  • l is the length of Arc.
Semi Circle:
  • A circle when separated into 2 parts along its diameter.
Circular Ring:
Volume Surface & Area
Cube
  • A solid body having 6 equal faces with equal length, breadth and height is called cube.
  • Each face of cube is a square.
  • Let a be the side of cube.
Cuboid:
Cylinder:
Hollow Cylinder:
Cone:
Sphere:
Hollow Sphere or Spherical Shell:
Hemisphere:

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