Aptitude Important Formulas

Aptitude Important Formulas

Problems on Trains	Time and Distance	Height and Distance
Time and Work	Simple Interest	Compound Interest
Profit and Loss	Partnership	Percentage
Problems on Ages	Calendar	Clock
Average	Area	Volume and Surface Area
Permutation and Combination	Numbers	Banker's Discount
H.C.F and L.C.M	Decimal Fraction	Simplification
Square Root and Cube Root	Surds and Indices	Ratio and Proportion
Chain Rule	Pipes and Cistern	Boats and Streams
Alligation or Mixture	Logarithm	Races and Games
Stocks and Shares	Probability	True Discount

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Important Formulas of Stocks and Shares -

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

1. Stock Capital:
   The total amount of money needed to run the company is called the stock capital.
   
   
2. Shares or Stock:
   The whole capital is divided into small units, called shares or stock.
   For each investment, the company issues a 'share-certificate', showing the value of each share and the number of shares held by a person.
   The person who subscribes in shares or stock is called a share holder or stock holder.
   
   
3. Dividend:
   The annual profit distributed among share holders is called dividend.
   Dividend is paid annually as per share or as a percentage.
   
   
4. Face Value:
   The value of a share or stock printed on the share-certificate is called its Face Value or Nominal Value or Par Value.
   
   
5. Market Value:
   The stock of different companies are sold and bought in the open market through brokers at stock-exchanges. A share or stock is said to be:
   1. At premium or Above par, if its market value is more than its face value.
      
      
   2. At par, if its market value is the same as its face value.
      
      
   3. At discount or Below par, if its market value is less than its face value.
      
      
   Thus, if a Rs. 100 stock is quoted at premium of 16, then market value of the stock = Rs.(100 + 16) = Rs. 116.
   
   
   Likewise, if a Rs. 100 stock is quoted at a discount of 7, then market value of the stock = Rs. (100 -7) = 93.
   
   
6. Brokerage:
   The broker's charge is called brokerage.
   
   
   (i)  When stock is purchased, brokerage is added to the cost price.
   
   
   (ii) When stock is sold, brokerage is subtracted from the selling price.
   
   
   Remember:
   1. The face value of a share always remains the same.
      
      
   2. The market value of a share changes from time to time.
      
      
   3. Dividend is always paid on the face value of a share.
      
      
   4. Number of shares held by a person
      
      

      =	Total Investment	=	Total Income	=	Total Face Value	.
      	Investment in 1 share		Income from 1 share		Face of 1 share

7. Thus, by a Rs. 100, 9% stock at 120, we mean that:
   1. Face Value of stock = Rs. 100.
      
      
   2. Market Value (M.V) of stock = Rs. 120.
      
      
   3. Annual dividend on 1 share = 9% of face value = 9% of Rs. 100 = Rs. 9.
      
      
   4. An investment of Rs. 120 gives an annual income of Rs. 9.
      
      
   5. Rate of interest p.a   =   Annual income from an investment of Rs. 100

      =		9	x 100		%	= 7	1	%.
      		120					2

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Important Formulas Of Alligation or Mixture

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

1. Alligation:
   It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.
   
   
2. Mean Price:
   The cost of a unit quantity of the mixture is called the mean price.
   
   
3. Rule of Alligation:
   
   
   If two ingredients are mixed, then

   		Quantity of cheaper		=		C.P. of dearer - Mean Price
   		Quantity of dearer				Mean price - C.P. of cheaper

   
   
   
   We present as under:

   C.P. of a unit quantity of cheaperC.P. of a unit quantity of dearer
   (c)	Mean Price (m)	(d)
   (d - m)		(m - c)

   
   
   
    (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c).
   
   
4. Suppose a container contains x of liquid from which y units are taken out and replaced by water.

   After n operations, the quantity of pure liquid =		x		1 -	y		n		units.
   					x

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Important Formulas Of Chain Rule -

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

1. 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)
   
   
2. Indirect Proportion:
   Two quantities are said to be indirectly proportional, if on the increase of the one, the orther 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.

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Important Formulas Of Square Root and Cube Root

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

1. 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.
   
   
2. 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

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Important Formulas Of H.C.F and L.C.M -

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

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 divided 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.

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Important Formulas of Permutation and Combination

STUDENT DUNIYA PRESENTS ALL APTITUDE IMPORTANT FORMULAS

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)}
      
      
   Examples:

   i.   11C4 =	(11 x 10 x 9 x 8)	= 330.
   	(4 x 3 x 2 x 1)

   
   
   

   ii.   16C13 = 16C(16 - 13) = 16C3 =	16 x 15 x 14	=	16 x 15 x 14	= 560.
   	3!		3 x 2 x 1

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Average - Important Formulas

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

STUDENT DUNIYA     

STUDENT DUNIYA      

STUDENT DUNIYA

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