Having a comprehensive formula sheet across all important topics in Quantitative Aptitude (QA) is crucial. Below is a more detailed breakdown of the key formulas covering Arithmetic, Algebra, Geometry, and Modern Mathematics, frequently tested in competitive examinations.

1. Arithmetic Formulas

Ratio and Proportion

  • Ratio:
    [
    \text{Ratio of two quantities} = \frac{\text{First Quantity}}{\text{Second Quantity}}
    ]
  • Proportion:
    [
    \frac{a}{b} = \frac{c}{d}
    ]
    [
    a \times d = b \times c
    ]

Percentage

[
\text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100
]

  • Profit and Loss:
    [
    \text{Profit \%} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100
    ]
    [
    \text{Loss \%} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100
    ]
  • Discount:
    [
    \text{Discount \%} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100
    ]

Simple Interest (SI)

[
SI = \frac{P \times R \times T}{100}
]
Where:

  • (P) = Principal
  • (R) = Rate of Interest (per annum)
  • (T) = Time (in years)

Compound Interest (CI)

  • Annually:
    [
    CI = P \left( 1 + \frac{R}{100} \right)^T – P
    ]
  • For compound interest compounded more than once a year:
    [
    CI = P \left( 1 + \frac{R}{n \times 100} \right)^{n \times T} – P
    ]
    Where (n) is the number of times interest is compounded per year.

Time, Speed, and Distance

  • Basic Formula:
    [
    \text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Distance} = \text{Speed} \times \text{Time}
    ]
  • Relative Speed (for objects moving in opposite/same directions):
  • Opposite direction: ( \text{Relative Speed} = S_1 + S_2 )
  • Same direction: ( \text{Relative Speed} = |S_1 – S_2| )
  • Average Speed:
    [
    \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
    ]

Time and Work

  • Work:
    [
    \text{Work} = \text{Rate} \times \text{Time}
    ]
  • Efficiency:
    [
    \text{Efficiency} \propto \frac{1}{\text{Time taken to complete the work}}
    ]
  • For (n) workers with work efficiencies (a, b, c, \dots), total work:
    [
    \text{Total Work} = \sum \frac{\text{Work Done}}{\text{Time Taken by each}}
    ]

2. Algebra Formulas

Quadratic Equations

[
ax^2 + bx + c = 0
]
The solutions (roots) of the quadratic equation are:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]

  • Sum of roots: ( \alpha + \beta = -\frac{b}{a} )
  • Product of roots: ( \alpha \beta = \frac{c}{a} )

Logarithms

  • Basic Rules:
    [
    \log_b(xy) = \log_b x + \log_b y
    ]
    [
    \log_b \left( \frac{x}{y} \right) = \log_b x – \log_b y
    ]
    [
    \log_b (x^n) = n \log_b x
    ]
    [
    \log_b 1 = 0, \quad \log_b b = 1
    ]

3. Geometry Formulas

Area and Perimeter of 2D Shapes

  • Rectangle:
  • Area = ( l \times b )
  • Perimeter = ( 2(l + b) )
  • Square:
  • Area = ( s^2 )
  • Perimeter = ( 4s )
  • Triangle:
  • Area = ( \frac{1}{2} \times \text{base} \times \text{height} )
  • Equilateral Triangle:
    [
    \text{Area} = \frac{\sqrt{3}}{4} a^2
    ]
  • Circle:
  • Area = ( \pi r^2 )
  • Circumference = ( 2\pi r )

Volume and Surface Area of 3D Shapes

  • Cube:
  • Volume = ( a^3 )
  • Surface Area = ( 6a^2 )
  • Cylinder:
  • Volume = ( \pi r^2 h )
  • Curved Surface Area = ( 2\pi rh )
  • Total Surface Area = ( 2\pi r (r + h) )

Pythagoras Theorem

For a right-angled triangle:
[
a^2 + b^2 = c^2
]
Where (a) and (b) are the legs of the triangle, and (c) is the hypotenuse.

4. Trigonometry Formulas

Basic Trigonometric Ratios

  • (\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}})
  • (\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}})
  • (\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin \theta}{\cos \theta})

Pythagorean Identities

  • (\sin^2 \theta + \cos^2 \theta = 1)
  • (1 + \tan^2 \theta = \sec^2 \theta)

Angle Sum and Difference Formulas

  • (\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B)
  • (\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B)
  • (\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B})

5. Modern Mathematics

Probability

  • Probability of an event occurring:
    [
    P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}
    ]
  • Addition Rule: ( P(A \cup B) = P(A) + P(B) – P(A \cap B) )

Permutation and Combination

  • Permutation (arrangement):
    [
    P(n, r) = \frac{n!}{(n – r)!}
    ]
  • Combination (selection):
    [
    C(n, r) = \frac{n!}{r!(n – r)!}
    ]

Binomial Theorem

For any positive integer (n):
[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
]
Where (\binom{n}{k}) is the binomial coefficient.

6. Mensuration

  • Area of Sector:
    [
    \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2
    ]
  • Surface Area of a Cone:
    [
    \text{Surface Area} = \pi r (r + l)
    ]
    Where (l) is the slant height.

These formulas cover the quantitative aptitude section of most of the competitive exams.

Share it with your friends.

 

Related Posts

Subscribe
Subscribe