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About Numbersmore_vert

Numbersclose

Natural Numbers : All Countable numbers N = {1,2,3,...,N}


Whole Numbers : Zero added to the set of natural numbers becomes the set of whole numbers. W = {0,1,2,3,...,N}

Integer : The integers are natural numbers (1,2,3...),their negatives (-1,-2,-3,...) and 0

Rational Numbers : Any number that can be expressed in the form pf p/q, where p and q are intergers and q!=0 is called a rational number.
Irrational Number : Any number that cannot be expressed in the form of p/q ,where p and q are integers and q!=0, is called an irrational number. They are generally non-recurring decimal fractions and non-terminating
Co-primes : Two numbers are said to be co-primes if their HCF is 1.
About Numbers (contd..)more_vert
Numbersclose

Real Numbers : Set of numbers that include both rational and irrational numbers is called real numbers. R=Q


Imaginary number : Square root of a negative real number denoted b the symbol 'i'.

Complex Number : A complex number consists of two parts : a real number and an imaginary number, and is expressed in the form a+ib. (ex: 3+i4)

Know me Bettermore_vert
Factors and Multiplesclose

If a number 'a' divides another number 'b' exactly, we say that 'a' is a factor of 'b'. In this case, 'b' is called a multiple of 'a'.


LCM : The least number which is exactly divisible by each one of the given numbers is called their LCM.

HCF : The HCF of two or more numbers is the greatest number that divides each of them exactly.

Product of two numbers = Product of their HCF and LCM.
Co-primes : Two numbers are said to be co-primes if their HCF is 1.
The product of least powers of common prime factors gives HCF.
LCM is the product of highest powers of all the factors.
More about Memore_vert
LCM and HCF of Fractionsclose

HCF = (HCF of Numerators / LCM of Denominators).


LCM = (LCM of Numerators / HCF of Denominators)

The HCF of (HCF of two numbers) and (third number) gives HCF of three numbers

Basic Rules
Commutativity : A ∪ B = B ∪ A
Associativity : A ∪ (B ∪ C)=(A ∪ B) ∪ C
Idempotency :A ∪ A = A
Distributivity : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
De Morgan's laws : (A ∪ B)′ = A′ ∩ B ′
a2 − b2 = (a − b) (a + b)
a3 − b3 = (a − b) (a2 + ab + b2)
a3 + b3 = (a + b) (a2 −ab + b2)
(a + b)3 = a3 + 3a2b + 3ab2+b3
(a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
Division Rules
Divisibility Test Condition Example
2 The Last Digit is even (0,2,4,6,8) 256 is divisible by 2 ,257 is not divisible by 2
3 The Sum of the digits is divisible by 3 471(4+7+1=12 ,12/3=4) is divisible by 3.
322 (3+2+2=7, 7 is not divisible by 3) is not divisible by 3
4 The Last Two Digits are divisible by 4 2516 (16/4 = 4) is divisible by 4 whereas 2414 is not divisible by 4
5 The Last Digit is 0 or 5 465 is divisible by 5 whereas 543 is not divisible by 5
6 The Number is divisible by both 2 and 3 78(is even and 7+8=15,15/3=5) is divisible by 6.
80 (is even but 8+0=8, 8 is not divisible by 3) is therefore not divisible by 6
7 Double the last digit and subtract it from the rest of the number
and the answer should either be 0 or a number divisible by 7		 343 (twice 3 is 6, 34-6=28 and 28/7=4) is divisible by 7
147 (twice 7 is 14, 14-14=0) is
192 (twice 2 is 4, 19-4=15!=0 nor a multiple of 7) is not divisible by 7
8 The Last three digits are divisible by 8 102192 (192/8=24) is divisible by 8
153190 (190 is not divisible by 8) is not divisible by 9
9 The sum of the digits should be divisible by 9 6561 (6+5+6+1=18 and 18/9=2) is divisible by 9
611 (6+1+1=8 which is not divisible by 9) is not divisible by 9
10 The number ends with 0 150 is divisible by 10
141 is not divisible by 10
11 The difference between the sum of digits in odd phases and the sum
of digits in even places should be either 0 or a number divisible by 11		 12364{(1+3+4)-(2+6)=0} is divisible by 11
4234 {(4+3)-(2+4)=1,!=0 or 11} is not divisible by 11
12 The number is divisible by both 3 and 4 144 (1+4+4=9, 9/3=3 also last digits 44 is divisible by 4) is divisible by 12
164 (1+6+4=11, 11 is not divisible by 3 but last two digits 64 is divisible by 4) is not divisible by 12.
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Ratios and Proportion:

  1. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a : b

  2. PROPORTION: The equality of two ratios is called proportion.

  3. if a : b = c : d, we write ,a : b :: c : d and we say that a,b,c,d are in proportion.
    Here a and d are called extremes, while b and c are called mean terms.
    Product of means = Product of extremes.
    Thus, a : b :: c : d == (b x c) = (a x d)

    • Fourth Proportional : if a : b = c : d, then d is called the fourth proportional to a,b,c
    • Third Proportional : if a : b = b : c, then c is called the third proportional to a and b.
    • Mean Proportional : Mean proportional between a and b is sqrt(a,b)

  5. COMPOUND RATIO : The compound ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf).

  6. if a/b = c/d, then ( a + b )/( a - b ) = ( c + d )/( c - d )

Time & Work :

  1. If A can do a piece of work in n days, then A's 1 day's work = 1/n.

  2. If A's 1 day's work = 1/n , then A can finish the work in n days.

    • If A is thrice as good as workman as B, then :
    • Ratio of work done by A and B = 3 : 1
    • Ratio of times taken by A and B to finish a work = 1 : 3

Partnership :

  1. Partnership : When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.

  2. Ratios of Division of Gains :
    1. When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.
      Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year :
      ( A's share of profit ) : ( B's share of profit ) = x : y
    2. When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking ( capital x number of units of time ). Now, gain or loss is divided in the ratio of these capitals.
      Suppose A invests Rs. x for p months and B invests Rs. y for q months, then
      ( A's share of profit ) : ( B's share of profit ) = xp : yq


  3. Working and Sleeping Partners : A partner who manages thee business is known as a working partner and the one who simply invests the money is a sleeping partner.

Percentage :

  1. Concept of Percentage : By a certain percent, we mean that many hundredths. Thus, x percent means x hundredths, written as x%.
    To express a/b as a percent : we have , a / b = { (a / b) x 100 }%

    • If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is
      {( R / [100 + R]) x 100 } %
    • If the price of a commodity decreases by R% , then the increases in consumption so as not to decrease the expenditure is
      [{ R / (100 - R)} x 100 ] %

    • If A is R% more than B, then B is less than A by
      [ { R / ( 100 + R )} x 100 ] %
    • If A is R% less than B, then B is more than A by
      [ { R / ( 100 - R )} x 100 ] %

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Permutations and Combinations :

  1. Factorial Notation : Let n be a positive integer. Then, factorial n, denoted by |_ or n! is defined as :
    n! = n x ( n - 1 )x( n - 2 ).... 3.2.1

  2. Permutations : The Different arrangements of a given number of things by taking some or all at a time, are called permutations.
    • Ex 1: All permutations made with the letters a,b,c taking 2 at a time are ( ab, ba, ac, ca, bc, cb ).

    • Ex 2: All permutations made with the letters a,b,c taking all at a time are (abc, acb, bac, bca, cab, cba)

    • NOTE: Number of all permutations of n things , taken all at a time = n!.

    • NOTE: Number of all permutations of n things, taken r at a time is given by
      nPr = (n! / {n - r}! )


  3. Combinations : Each of the different groups or selections which can be formed by taking some or all of a number of objects, is called a combination.
    • Ex 1: Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB,BC,CA.

    • Ex 2: All the cobinations formed by a,b,c taking two at a time are ab, bc, ca.

    • Ex 3: The only combination formed by a,b,c taking all at a time is abc.

    • NOTE : ab and ba are different permutations but represent the same combinations

    • NOTE : The number of all combinations of n things, taken r at a time is:
      nCr = ( n! / { r! } x {n -r}! )

Probability :

  1. Exxperiment : An operation which can produce some well-defined outcomes is called an experiment.

  2. Random Experiment : An experiment in which all possible outcomes are known and the exact output cannot be predicted in advance, is called a random experiment.

  3. Sample Space : When we perform an experiment, then the set S of all possible outcomes is called the Sample Space.

  4. Event : Any subset of a sample space is called an event.

  5. Probability of an Event : Let S be the sample space and let E be an event.
    P(E) = ( n(E) / n(S) )

Simple Interest :

  1. Principal : The money borrowed or lent out for a certain period is called the principal or the sum.

  2. Interest : Extra money paid for using other's money is called interest.

  3. Simple Interest : If the interest on a sum borrowed for a certain period is reckoned uniformly, then it is called simple interest.
    Let Principal = P, Rate = R % per annum (p.a.) and Time = T years. Then,
    • SI = ( {P x R x T} / 100 )

Compound Interest :

Compound Interest : Sometimes it so happens that the borrower and the lender agree to fix up a certain unit of time, say yearly or half-yearly or quarterly to settle the previous account.
In such cases, the amount after first unit of time becomes the principal for the second unit, the amount after second unit becomes the principal for the third unit and so on.
After a specified period, the difference between the amount and the moey borrowed is called the Compound Interest (C.I)

Let Principal = P , Rate = R% per annum , Time = n years.

  1. When the interest is compound annually :
    Amount = P ( 1 + {R/100})n

  2. when interest is compounded Half-Yearly :
    Amount = P ( 1 + { (R/2)/100 } )n/2

  3. When interest is compounded annually but time is in fractions i.e. 3(2/5)yrs:
    Amount = P ( 1 + (R/100)3 x ( 1 + {(2/5) x R } / 100 )

Time and Distance :

  1. Speed = (Distance / Time) , Time = (Distance / Speed) , Speed = (Distance / Time)

  2. X km / hr = ( X x 5/18 ) m /sec

  3. x m /sec = ( x X 18/5 ) km /hr

  4. If the ratio of speeds of A and B is a : b, then the ratio of the times taken by them to cover the same distance is 1/a : 1/b or b : a

  5. Suppose a man covers a certain distance at x km /hr and an equal distance at y km /hr. Then, the average speed during the whole journey is (2xy / x+y ) km / hr

Problems on Trains :

  1. Time Taken by a train of length 'l' metres to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover 'l' metres

  2. Time taken by a train of length 'l' metres to pass a stationary object of length 'b' metres is the time taken by the train to cover ( l + b ) metres.

  3. Suppose two trains or two bodies are moving in the same direction at u m /s and v m /s, where u > v, then their relative speed = ( u - v ) m /s.

  4. Suppose two trains or two bodies are moving in the opposite direction at u m /s and v m /s ,then their relative speed = ( u + v ) m /s.

  5. If two trains of length 'a' metres and 'b' metres are moving in opposite directions at v m/s and u m/s, then time taken by trains to cross each other = ( {a + b} / {u + v} )

  6. If two trains of length 'a' metres and 'b' metres are moving in opposite directions at v m/s and u m/s, then time taken by faster train to cross other train = ( {a + b} / {u - v} )

  7. If two trains (or bodies ) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then
    (A's speed ) : ( B's speed ) = ( sqrt(b) : sqrt(a) ).

Card Title

  1. Odd days : In a given period, the number of days more than the complete weeks are called odd days

  2. Leap year : A leap year has 366 days.
    • Every year divisible by 4 is a leap year, if it is not a century
    • Every 4th century is a leap year and no other century is a leap year
  3. Ordinary Year : The year which is not a leap year.It has 365 days

  4. Day of the week related to Odd days :
    No. of days Day
    0 Sun
    1 Mon
    2 Tue
    3 Wed
    4 Thurs
    5 Fri
    6 Sat
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