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Thursday 7 December 2017

Fast Math Tricks & Shortcuts | Champions'Square


Simplification Techniques and Tricks - PDF

Simplification is one of the most important parts of Quantitative Aptitude section of any competitive exam. Today I am sharing all the techniques to solve Simplification questions quickly.

Rules of Simplification
V → Vinculum
B → Remove Brackets - in the order ( ) , { }, [ ] 
O → Of
D → Division
M → Multiplication
A → Addition
S → Subtraction


Important Parts of Simplification

  • Number System
  • HCF & LCM
  • Square & Cube
  • Fractions & Decimals
  • Surds & Indices

    Number System

    • Classification
    • Divisibility Test
    • Division& Remainder Rules
    • Sum Rules

     Classification 

    TypesDescription
    Natural Numbers:
    all counting numbers ( 1,2,3,4,5....∞)
    Whole Numbers:
    natural number + zero( 0,1,2,3,4,5...∞)
    Integers:
    All whole numbers including Negative number + Positive number(∞......-4,-3,-2,-1,0,1,2,3,4,5....∞)
    Even & Odd Numbers :
    All whole number divisible by 2 is Even (0,2,4,6,8,10,12.....∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19....∞)
    Prime Numbers:
    It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61....∞)
    Composite Numbers:
    Natural numbers which are not prime
    Co-Prime:
    Two natural number a and b are said to be co-prime if their HCF is 1.

    Divisibility

    NumbersIF A NumberExamples
    Divisible by 2End with 0,2,4,6,8 are divisible by 2254,326,3546,4718 all are divisible by 2
    Divisible by 3Sum of its digits  is divisible by 3375,4251,78123 all are divisible by 3.  [549=5+4+9][5+4+9=18]18 is divisible by 3  hence 549 is divisible by 3.
    Divisible by 4Last two digit divisible by 45648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4.
    Divisible by 5Ends with 0 or 5225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5.
    Divisible by 6Divides by Both 2 & 34536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6.
    Divisible by 8Last 3 digit divide by 8746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8.
    Divisible by 10End with 0220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10.
    Divisible by 11[Sum of its digit in
     odd places-Sum of its digits
    in even places]= 0 or multiple of 11
    Consider the number 39798847
     (Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)
    (23-12)
    23-12=11, which is divisible by 11. So 39798847 is divisible by 11.

    Division & Remainder Rules

    Suppose we divide 45 by 6
    Division Terms

    hence ,represent it as:
    dividend = ( divisorquotient ) + remainder
    or
    divisior= [(dividend)-(remainder] / quotient
    could be write it as 
    x = kq + r where (x = dividend,k = divisor,q = quotient,r = remainder)

    Example:
    On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder ?
    Number = 342k + 47
    ( 18 19k ) + ( 18 2 ) + 11
    18 ( 19k + 2 ) +11.
    Remainder = 11

    Sum Rules

    (1+2+3+.........+n) = 1/n(n+1)
    (12+22+32+.........+n2) = 1/n (n+1) (2n+1)
    (13+23+33+.........+n3) = 1/4 n2 (n+1)2
    Arithmetic Progression (A.P.)
    a, a + d, a + 2d, a + 3d, ....are said to be in A.P. in which first term = a and common difference = d.
    Let the nth term be tn and last term = l, then
    a) nth term = a + ( n - 1 ) d
    b) Sum of n terms = n/2 [2a + (n-1)d]
    c) Sum of n terms = n/2 (a+l) where l is the last term


      H.C.F. & L.C.M.

      • Factorization & Division Method
      • HCF & LCM of Fractions & Decimal Fractions

      Methods

      On Basis
      H.C.F. or G.C.M
      L.C.M.
      Factorization Method
      Write each number as the product of the prime factors. The product of least powers of common prime factors gives H.C.F.
      Example:
      Find the H.C.F. of 108, 288 and 360.
      108 = 22✘33, 288 = 25✘32 and 360 = 23✘5✘32
      H.C.F. = 22✘32=36
      Write each number into a product
      of prime factors. Then, L.C.M is
      the product of highest powers of all
      the factors.
      Examples:
      Find the L.C.M. of 72, 108 and 2100.
      72=23✘32,108=33✘22,
      2100=22✘52✘3✘7.
      L.C.M.=23✘33✘52✘7=37800
      Division Method
      Let we have two numbers.Pick the smaller one and divide it by the larger one. After that divide the divisor with the remainder. This process of dividing the preceding number by the remainder will be repeated until we got the zero as remainder.The last divisor is the required H.C.F.
      Example:
      How to find HCF Example

      H.C.F. of given numbers = 69
      Let we have set of numbers.
      First of all, find the number
       which divide at least two of
      the number in a given set of
       number.remainder and
      not divisible numbers
      will carry forward as it is.
      Repeat the process until
      at least  two number is
      not divisible by any number
      except 1.The product of
      the divisor and the
      undivided numbers is the required
      L.C.M.
      Example:
      Find the L.C.M. of 12,36,48,72
      How to find LCM Example
      H.C.F. & L.C.M. of Fractions
      H.C.F. =  H.C.F. of Numerator / L.C.M. of DenominatorsL.C.M. = L.C.M. of Numerator / H.C.F. of Denominators
      Product of H.C.F. & L.C.M.
      H.C.F * L.C.M. = product of two numbers
      Decimal numbersH.C.F. of Decimal numbers
      Step 1. Find the HCF of the given
      numbers without decimal.
      Step 2.Put the decimal point ( in the
      HCF of Step 1) from right to left
      according to the MAXIMUM
       deciaml places among the given numbers.
      L.C.M. of Decimal numbers
      Step 1. Find the LCM of the given
      numbers without decimal.
      Step 2.Put the decimal point ( in the
      LCM of Step 1) from right to left
      according to the MINIMUM
       deciaml places among the given numbers.

      Square & Cube 

      • Square & Cube
      • Square Root & Cube Root
      • Factorization Method
      Perfect Square
      Non-Perfect Square
      last digit is 1, 4, 9, 6, 5last digit is 2, 3, 7, 8
      Chart of perfect square image

      Square from 1 to 100 image

      Square Root & Cube Root
      Square root from 1 to 10 image
      last digit of square image
      Trick to solve Square root

      Last or Unit digit of Cube Image
      Tricks to solve Cube Root image
      • Fractions & Decimals
      On BasisExplanation
      Decimal Fractions
      A number with a denominator of power of 10 is a decimal fractions.
      1/10= 1 tenth; 1/100= 0.1;38/100=0.38
      Vulgar Fractions
      Conversion of 0.64(decimal number) into a Vulgar Fraction.First of all write the numeric digit 1 in the denominator of a number (like here 0.64) and add as many numeric zeros as the digit in the number after decimal point.After that removes the decimal point from the given number.At last step just reduce the fraction to its lowest terms. So, 0.64 = 64/100=16/25;25.025 = 25025/1000 = 1001/4
      OperationsAddition & Subtraction
      To perform the addition and subtraction of a decimal fraction could be done through placing them right under each other that the decimal points lie in one column.
      3.424+3.28+.4036+6.2+.8+4
        3. 424
        3. 28
          . 4036
        6. 2
          . 8
      +4______
      18. 1076


      Multiplication of a Decimal Fraction
      To find the multiplication of decimal fraction , first of all you need to remove the decimal point from the given numbers and then perform the multiplication after that assign the decimal point as many places after the number as the sum of the number of the decimal places in the given number.
      Step 1. 0.06*0.3*0.40
      Step 2. 6*3*40=720
      Step 3. 0.00720
      Multiplication of a decimal fraction by power of 10
      A multiplication of a decimal fraction by a power of 10 can be perform through shifting the decimal point towards right as many places as is the power of 10.
      like 45.6288*100=45628.8, 0.00452*100=0.452
      Division

      Comparison of FractionsTo compare the set of fractions numbers,first of all you need to convert each fraction number or value into a equal decimal value and then it will be became easy for you to assign them ( the numbers or value) in a particular way( ascending or descending order).
      3/5,4/7,8/9 and 9/11 Arranging in Ascending Order
      3/5= 0.6, 4/7 = 0.571, 8/9 = 0.88, 9/11 = 0.818.
      Now, 0.88 > 0.818 > 0.6 > 0.571
      8/9>9/11>3/5>4/7
      Recurring DecimalRecurring Decimal
      A decimal number in which after a decimal point a number or set of number are repeated again and again are called recurring decimal numbers.It can be written in shorten form by placing a bar or line above the numbers which has repeated.
      Recurring decimal example image
      Pure Recurring Decimal
      A decimal number in which all digits are repeated after a decimal point.
      Pure Recurring Decimal example image

      Mixed Recurring Decimal
      A decimal number in which certain digits are repeated only.
      Mixed Recurring decimal example image


      Surds & Indices 

      • Some Rules of Indices
      • Some Rules of Surds
      Rules fro law of indices and surds image
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