Formulae for HCF & LCM

1. Factors and Multiples:
   If number a divided another number b exactly, we say that a is a factor of b.
   In this case, b is called a multiple of a.
2. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):
   The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
   There are two methods of finding the H.C.F. of a given set of numbers:
   1. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.
   2. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.
      Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.
      Similarly, the H.C.F. of more than three numbers may be obtained.
3. Least Common Multiple (L.C.M.):
   The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
   There are two methods of finding the L.C.M. of a given set of numbers:
   1. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.
   2. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.
4. Product of two numbers = Product of their H.C.F. and L.C.M.
5. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.
6. H.C.F. and L.C.M. of Fractions:

   1. H.C.F. =	H.C.F. of Numerators
   	L.C.M. of Denominators

   2. L.C.M. =	L.C.M. of Numerators
   	H.C.F. of Denominators

7. H.C.F. and L.C.M. of Decimal Fractions:
   In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.
8. Comparison of Fractions:
   Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

Formulae for Combinatorics

1. Factorial Notation:
   Let n be a positive integer. Then, factorial n, denoted n! is defined as:
   n! = n(n - 1)(n - 2) ... 3.2.1.
   Examples:
   1. We define 0! = 1.
   2. 4! = (4 x 3 x 2 x 1) = 24.
   3. 5! = (5 x 4 x 3 x 2 x 1) = 120.
2. Permutations:
   The different arrangements of a given number of things by taking some or all at a time, are called permutations.
   Examples:
   1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
   2. All permutations made with the letters a, b, c taking all at a time are:
      ( abc, acb, bac, bca, cab, cba)
3. Number of Permutations:
   Number of all permutations of n things, taken r at a time, is given by:

   nPr = n(n - 1)(n - 2) ... (n - r + 1) =	n!
   	(n - r)!

   Examples:
   1. ⁶P₂ = (6 x 5) = 30.
   2. ⁷P₃ = (7 x 6 x 5) = 210.
   3. Cor. number of all permutations of n things, taken all at a time = n!.
4. An Important Result:
   If there are n subjects of which p₁ are alike of one kind; p₂ are alike of another kind; p₃ are alike of third kind and so on and p_r are alike of r^th kind,
   such that (p₁ + p₂ + ... p_r) = n.

   Then, number of permutations of these n objects is =	n!
   	(p1!).(p2)!.....(pr!)

5. 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.
   Examples:
   1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
      Note: AB and BA represent the same selection.
   2. All the combinations formed by a, b, c taking ab, bc, ca.
   3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
   4. Various groups of 2 out of four persons A, B, C, D are:
      AB, AC, AD, BC, BD, CD.
   5. Note that ab ba are two different permutations but they represent the same combination.
6. Number of Combinations:
   The number of all combinations of n things, taken r at a time is:

   nCr =	n!	=	n(n - 1)(n - 2) ... to r factors	.
   	(r!)(n - r)!		r!

   Note:
   1. ⁿC_n = 1 and ⁿC₀ = 1.
   2. ⁿC_r = ⁿC_{(n - r)}

Formulae for Average Problems

1. Average:

   Average =		Sum of observations
   		Number of observations

2. Average Speed:
   Suppose a man covers a certain distance at x kmph and an equal distance at ykmph.

   Then, the average speed druing the whole journey is		2xy		kmph.
   		x + y

Formulae for Age Problems

Important Formulas on "Problems on Ages" :

1. If the current age is x, then n times the age is nx.

2. If the current age is x, then age n years later/hence = x + n.

3. If the current age is x, then age n years ago = x - n.

4. The ages in a ratio a : b will be ax and bx.

5. If the current age is x, then	1	of the age is	x	.
	n		n

Formulae for Profit and Loss

IMPORTANT FACTS

• Cost Price: The price, at which an article is purchased, is called its cost price, abbreviated as C.P.
• Selling Price: The price, at which an article is sold, is called its selling prices, abbreviated as S.P.
• Profit or Gain: If S.P. is greater than C.P., the seller is said to have a profit or gain.
• Loss: If S.P. is less than C.P., the seller is said to have incurred a loss.

IMPORTANT FORMULAE

• Gain = (S.P.) - (C.P.)
• Loss = (C.P.) - (S.P.)
• Loss or gain is always reckoned on C.P.
• Gain Percentage: (Gain %)

  Gain % =		Gain x 100
  		C.P.

• Loss Percentage: (Loss %)

  Loss % =		Loss x 100
  		C.P.

• Selling Price: (S.P.)

  SP =		(100 + Gain %)	x C.P
  		100

• Selling Price: (S.P.)

  SP =		(100 - Loss %)	x C.P.
  		100

• Cost Price: (C.P.)

  C.P. =		100	x S.P.
  		(100 + Gain %)

• Cost Price: (C.P.)

  C.P. =		100	x S.P.
  		(100 - Loss %)

• If an article is sold at a gain of say 35%, then S.P. = 135% of C.P.
• If an article is sold at a loss of say, 35% then S.P. = 65% of C.P.
• When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by:

  Loss % =		Common Loss and Gain %		2	=		x		2	.
  		10					10
• If a trader professes to sell his goods at cost price, but uses false weights, then

  Gain % =		Error	x 100	%.
  		(True Value) - (Error)

Formulae for Time and Work

1. Work from Days:

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

2. Days from Work:

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

3. Ratio:
   If A is thrice as good a 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.

Formulae for Trains Problem

• • km/hr to m/s conversion:

    a km/hr =		a x	5		m/s.
    			18

  • m/s to km/hr conversion:

    a m/s =		a x	18		km/hr.
    			5

  • Formulas for finding Speed, Time and Distance
  • Time taken by a train of length l metres to pass a pole or standing man or a signal post is equal to the time taken by the train to cover l metres.
  • Time taken by a train of length l metres to pass a stationery object of length b metres is the time taken by the train to cover (l + b) metres.
  • Suppose two trains or two objects bodies are moving in the same direction at u m/s and v m/s, where u > v, then their relative speed is = (u - v) m/s.
  • Suppose two trains or two objects bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s.
  • If two trains of length a metres and b metres are moving in opposite directions at um/s and v m/s, then:

    The time taken by the trains to cross each other =	(a + b)	sec.
    	(u + v)

  • If two trains of length a metres and b metres are moving in the same direction at um/s and v m/s, then:

    The time taken by the faster train to cross the slower train =	(a + b)	sec.
    	(u - v)

  • 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) = (b : a)

Formalue for Quantitative Aptitude

Some Basic Formulae:

• (a + b)(a - b) = (a² - b²)
• (a + b)² = (a² + b² + 2ab)
• (a - b)² = (a² + b² - 2ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ + b³) = (a + b)(a² - ab + b²)
• (a³ - b³) = (a - b)(a² + ab + b²)
• (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc.

• Square Root:
  If x² = y, we say that the square root of y is x and we write y = x.
  Thus, 4 = 2, 9 = 3, 196 = 14.
• Cube Root:
  The cube root of a given number x is the number whose cube is x.
  We, denote the cube root of x by x.
  Thus, 8 = 2 x 2 x 2 = 2, 343 = 7 x 7 x 7 = 7 etc.
  Note:
  1. xy = x x y

  2.	x y	=	x	=	x	x	y	=	xy	.
  			y		y		y		y

  
  
• Direct Proportion:
  Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.
  Eg. Cost is directly proportional to the number of articles.
        (More Articles, More Cost)
• Indirect Proportion:
  Two quantities are said to be indirectly proportional, if on the increase of the one, the other decreases to the same extent and vice-versa.
  Eg. The time taken by a car is covering a certain distance is inversely proportional to the speed of the car. (More speed, Less is the time taken to cover a distance.)
  Note: In solving problems by chain rule, we compare every item with the term to be found out.

• 'BODMAS' Rule:
  This rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of given expression.
  Here B - Bracket,
  O - of,
  D - Division,
  M - Multiplication,
  A - Addition and
  S - Subtraction
  Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order (), {} and ||.
  After removing the brackets, we must use the following operations strictly in the order:
  (i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.
• Modulus of a Real Number:
  Modulus of a real number a is defined as

  |a| =		a, if a > 0
  		-a, if a < 0

  Thus, |5| = 5 and |-5| = -(-5) = 5.
• Virnaculum (or Bar):
  When an expression contains Virnaculum, before applying the 'BODMAS' rule, we simplify the expression under the Virnaculum.
• 1. Decimal Fractions:
     Fractions in which denominators are powers of 10 are known as decimal fractions.

     Thus,	1	= 1 tenth = .1;	1	= 1 hundredth = .01;
     	10		100

     99	= 99 hundredths = .99;	7	= 7 thousandths = .007, etc.;
     100		1000

  2. Conversion of a Decimal into Vulgar Fraction:
     Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.

     Thus, 0.25 =	25	=	1	;       2.008 =	2008	=	251	.
     	100		4		1000		125

  3. Annexing Zeros and Removing Decimal Signs:
     Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
     If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.

     Thus,	1.84	=	184	=	8	.
     	2.99		299		13

  4. Operations on Decimal Fractions:
     1. Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
     2. Multiplication of a Decimal Fraction By a Power of 10: Shift the decimal point to the right by as many places as is the power of 10.
        Thus, 5.9632 x 100 = 596.32;   0.073 x 10000 = 730.
     3. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.
        Suppose we have to find the product (.2 x 0.02 x .002).
        Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.
         .2 x .02 x .002 = .000008
     4. Dividing a Decimal Fraction By a Counting Number: Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.
        Suppose we have to find the quotient (0.0204 ÷ 17). Now, 204 ÷ 17 = 12.
        Dividend contains 4 places of decimal. So, 0.0204 ÷ 17 = 0.0012
     5. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
        Now, proceed as above.

        Thus,	0.00066	=	0.00066 x 100	=	0.066	= .006
        	0.11		0.11 x 100		11

  5. Comparison of Fractions:
     Suppose some fractions are to be arranged in ascending or descending order of magnitude, then convert each one of the given fractions in the decimal form, and arrange them accordingly.

     Let us to arrange the fractions	3	,	6	and	7	in descending order.
     	5		7		9

     Now,	3	= 0.6,	6	= 0.857,	7	= 0.777...
     	5		7		9

     Since, 0.857 > 0.777... > 0.6. So,	6	>	7	>	3	.
     	7		9		5

  6. Recurring Decimal:
     If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.
     n a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.

     Thus,	1	= 0.333... = 0.3;	22	= 3.142857142857.... = 3.142857.
     	3		7

     Pure Recurring Decimal: A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal.
     Converting a Pure Recurring Decimal into Vulgar Fraction: Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

     Thus, 0.5 =	5	; 0.53 =	53	; 0.067 =	67	, etc.
     	9		99		999

     Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
     Eg. 0.1733333.. = 0.173.
     Converting a Mixed Recurring Decimal Into Vulgar Fraction: In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated. In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

     Thus, 0.16 =	16 - 1	=	15	=	1	;   0.2273 =	2273 - 22	=	2251	.
     	90		90		6		9900		9900