SUBJECTABLE MATHMATICS

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2 SUBJECTABLE MATHMATICS

3 PREFACE (A NOTE TO THE READER) This book, A COMPLETE BOOK ON SUBJECTIVE (CONVEN- TIONAL) ARITHMETIC has been specially prepared for candidates appearing for competitive entrance examination held by Staff Selection Commission (S.S.C.). Syllabus of ARITHMETIC Questions are designed to test the ability of arithmetical computations of whole numbers, decimals and fractions. The questions are based on arithmetical concepts of various topics and the candidates have to give a detailed explanation in order to arrive at the final result of the problem. Every topic of arithmetic in this book starts with the introduction which is followed by plenty of solved examples with complete explanation that can help the candidates to secure full marks. At the end of every chapter, an excercise containing different types of questions is given for your practice. All questions are fully solved with proper explanation. I am confident that this book will be very useful to the candidates. Best of Luck. S.L. Gulati

4 CONTENTS Preface v 1. Numbers and Simplifications 1 2. H.C.F and L.C.M Ratio and Proportion Percentage Average Profit and Loss Interest Time and Work Time and Distance Mensuration Data Interpretation Partnership Alligation 302

5 1 NUMBERS AND SIMPLIFICATIONS 1. ARITHMETIC is a science that deals with numbers, and of the methods of computing by means of numbers. 2. Integers: Numbers like 3, 2, 1, 0, 1, 2, 3 etc. are called integers. 3. Rational numbers: All numbers of the type p/q where q ¹ 0 and p is an integer are called rational numbers; e.g. 2, 7 3, 4 9, 0, 2, 7 13 etc. are all rational numbers. 4. Irrational numbers: Numbers like 2, 3, 5 etc. are called irrational numbers. 5. Real numbers: All rational numbers, irrational numbers and a combination of rational and irrational numbers are called real numbers; e.g. 5, 7/3, 2, + 3 etc. real numbers. 6. Natural numbers: The numbers 1, 2, 3, 4, are called natural numbers. (i) Sum of the first n natural numbers, i.e n = n ( n + 1) 2 (ii) Sum of the first n square numbers, i.e n 2 = n ( n + 1) ( 2n + 1) 6

6 2 Subjective Arithmetic (iii) Sum of the first n cube numbers i.e n 3 = L NM n( n+ 1) Þ n 3 = ( n) Multiplication Table. 8. Division is the method of finding how often one given number, called the Divisor is contained in another given number, called the Dividend. The number expressing the times the divisor is contained in the dividend is called the Quotient. 9. Long Division: When the divisor is greater than 20, the process is called Long Division. The least number consisting of figures from the left of the dividend in which the divisor 536 is contained in 870 is called the first partial dividend. The next figure in the dividend 3344 is called the second partial dividend and 1282 is the third partial dividend. The last figure, which should be less than the divisor, is called the Remainder. This is clear that = Dividend = Divisor Quotient + Remainder. 11. Division by factors (successive division); complete remainder. Let us divide by 210, using factors and explain the rule to find the remainder. 2 O QP Divisor Dividend Quotient , Remainder 2 Now 210 = If we take 5, 6, 7 as d 1, d 2, d 3 and 3, 2, 2, as r 1, r 2, r 3. Remainder = = Quotient = 274 \ = , 7, 6, 1, 2, 6 1, 1, 5, 2, 2, 3 7 1, 9, 2, 0, 2 2, 7, 4, 2

7 Numbers and Simplifications 3 The remainder obtained by the Division by factors method is called the complete remainder or true remainder. Þ Complete remainder = r 1 + d 1 r 2 + d 1 d 2 r Metric System: The Government of India has introduced the metric system of weights and measures throughout the country. This system derives its name from the word "Metre" which is the standard unit of length in this system. The advantage of the metric system is the great simplification of calculation in different spheres of work. In this system, the various units of length, area (surface) volume, capacity and weight (mass) always bear a strictly decimal relation to each other. The international names of the five main units in the metric system of weights and measures are: Length measure unit is a Metre. Area measure unit is a square metre. Volume measure unit is a cubic metre. weight measure unit is a Gram. Capacity measure unit is a Litre. An are contains 100 sq. metres and A Hectare contains 100 areas or 10,000 sq. metres. A cubic metre of volume contains 1000 litres or 1 kilo litre. 13. Vulgar Fractions: A fraction is represented by two numbers written one above the other and sperated by a horizontal line. Thus the fraction two-fifths is written as 2/5. The upper number is called the numerator where as the lower number is called the denominator. The numerator and the denominator of a fraction are its terms. A fraction is zero when its numerator is zero alone. The denominator of a fraction is always non-zero. Fractions such as 3/5, 8 11, 7 25 etc. are called common or vulgar fractions. The value of the vulgar fraction is not altered by multiplying or dividing by the numerator and the denominator by the same number. 14. If the numerator and the denominator are large numbers, or if their common factors cannot easily be guessed, we may find their H.C.F.

8 4 Subjective Arithmetic 15. A fraction is said to be a proper fraction if its numerator is less than its denominator. Thus 5/7, 9 13, 19 are all proper fractions. An 23 improper fraction is the one whose numerator is equal to N greater than its denominator. Thus 7 7, 7 3, 15 are improper fractions. A 11 mixed fraction is one which consists of a whole number and a fraction. Thus 2 1 7, 15 2 are mixed fractions. 3 Complex Fraction is the one in which the numerator or denominator or both are fractions Thus 4 7/ 9, / 3, 4 7 are complex fractions Continued Fractions: or are called continued fractions. To simplify such fractions, begin at the bottom and work upwards. 17. Decimal Fractions: A decimal fraction or a decimal is a fraction which has 10 or any power of 10 for its denominator and is expressed in the decimal system of notation. A decimal fraction is not altered by annexing ciphers to the right of the last figure. Thus and are equal. 18. Addition of decimals: Write down the number under one another, placing units under units etc. then add as in the case of integers, and place the decimal point under the points in the given numbers. e.g. Add together 5.406, 0.8, and Ans.

9 Numbers and Simplifications SIMPLIFICATION: of vulgar fractions. (BODMAS). In questions on fractions, signs,, +,, of (means multiplication) and brackets are often involved. In simplifying these questions the following order B O D M A S i.e. (i) Remove the brackets (if any). (ii) Then numbers or fractions which are connected by of should be simplified. (iii) Then the division and multiplication in order must be carried out and (iv) Lastly the operations of addition (plus) or subtraction (minus) should be performed. 20. SIMPLIFICATION: by Algebric formulae. Remember the following. 1. x - y x - y = x + y and x - y x + y = (x y). 2. (x + y) 2 (x y) 2 = 4xy 3. ( x + y) + ( x - y) = 2. ( x + y ) x y x - xy + y x y x - xy + y = (x + y) = (x y) 6. (x + y) 3 = x 3 + y 3 + 3xy (x + y) 7. (x y) 3 = x 3 y 3 3xy (x y). 8. x 2 + y 2 + z 2 xy yz zx = 1 2 [(x y)2 + (y z) 2 + (z x) 2 ] x + y + z - 3xyz 2 ( x + y + z zx yx - yz) = (x + y + z) 10. If x + y + z = 0, then (i) x 3 + y 3 + z 3 = 3xyz (ii) x + y + z 3xyz = 1.

10 6 Subjective Arithmetic SOLVED EXAMPLES Example 1. In a division sum, the quotient is 195, the divisor is equal to the sum of the quotient and the remainder. Find the dividend. Solution: We know that Dividend = Divisior Quotient + Remainder Remainder = 195 Quotient = 105 Divisor = = 300 \ Dividend = = = Ans. Example 2. A number when divided by 899 gives a remainder 63. What will be the remainder, when the same number be divided by 29? Solution: A number when divided by 899 gives a quotient say Q. The remainder is given to be 63 Þ Such a number = 899 Q + 63 (i) Expressing this as multiple of 29, we have The number = 29 (31Q) + (2 29) + 5 \ The remainder obtained by dividing this number by 29 is 5. Ans. 5. Example 3. A certain number x is divided by 385 by dividing by three prime factors. The quotient is 102, the first remainder is 4, the second remainder is 6 and the third is 10. Solution: 385 = Let x be the required number. Y and Z be the respective quotients. X = 5Y + 4 Y = 7Z + 6 Z = = = Y = 7z + 6 = 7 (1132) + 6 = = 7930 (i) (ii) (iii) 5 X 7 Y 4 11 Z

11 Subjective Arithmetic : For All Competitive Exams 30% OFF Publisher : Cosmos Bookhive ISBN : Author : S L Gulati Type the URL : Get this ebook

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