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S.K. GUPTAPrincipal (Retd.)
Birla Vidya Mandir, NainitalFormer Chairman
Indian Public Schools’ Conference
O.P. MALHOTRAM.A. (Gold Medalist)Former Head of the
Mathematics DepartmentThe Doon School, Dehradun
ANUBHUTI GANGALM.A. (Gold Medalist), M.Ed.
Formerly, Senior Faculty MemberThe Daly College, Indore
Birla Vidya Mandir, Nainital
Mathematics Today
Book
7
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First Published in 1979First ICSE Edition 2006Thoroughly Revised Edition 2012 Revised Edition 2014This CD Edition 2016
ISBN : 978-93-5253-063-2 Code : 1014F514
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Mathematics Today: 7
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PREFACE TO THE REVISED EDITION
The authors wish to express their satisfaction and gratitude for the warmwelcome that has been accorded to this series all these years.
1. The text matter and answers have been thoroughly re-checked.
2. The special feature of this edition is the inclusion of Multiple ChoiceQuestions, Challengers [High Order Thinking Skills (HOTS)], Worksheets andChapter Tests.
3. The authors are grateful to all those teachers and students who have providedvaluable feedback for the improvement of the series.
AUTHORS
Disclaimer : While the authors of this book have made every effort to avoid any mistakes or omissions and have used their skill,expertise and knowledge to the best of their capacity to provide accurate and updated information, the authors and S. Chanddo not give any representation or warranty with respect to the accuracy or completeness of the contents of this publication andare selling this publication on the condition and understanding that they shall not be made liable in any manner whatsoever. S.Chand and the authors expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader of thispublication or not, in respect of anything and everything forming part of the contents of this publication. S. Chand shall not beresponsible for any errors, omissions or damages arising out of the use of the information contained in this publication.Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purelycoincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual.
The Most Popular Series which has been in use for more than 30 years.
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PREFACE
It gives us great satisfaction to be able to bring out this new version of our old MathematicsToday Series for Classes VI to VIII. The old series has been rehashed and redesigned incorporatingthe current global trends and International practices and the latest philosophy and policy ofproviding stress free education.
The salient features of this series are :
1. It follows strictly the new syllabus of the ICSE Council.2. All the mathematical concepts have been presented in a very simple and lucid form and
loading the course content with unnecessary and irrelevant details has been avoided. Theapproach and orientation is to lay a strong foundation for the students through adequateemphasis on the fundamentals.
3. It aims at complete involvement of the pupils in the learning process. The emphasis throughoutthe text is on a student-centered performance and the activity approach is freely used relatingthe mathematical concepts to real life situations.
4. Every unit is introduced by a motivating paragraph or story.5. To facilitate easy and better understanding each unit is divided into a number of subunits
with short and separate practice exercises on each subunit.6. An attempt has been made to expose the children more fully to the ‘Why’ of various
operations and made abundant use of diagrams, illustrations, cartoons, tables and chartsto stimulate the student’s interest in the subject and to clarify more difficult concepts.
7. Colour panels are used throughout as a teaching aid to emphasize important terms andrelationships and present useful tips.
8. The problems given in the books avoid tedious calculations and help in strengthening theunderstanding of basic principles honing the faculties of thinking and reasoning.
9. Each chapter contains a unit summary of key points at the end. It reviews the main pointscovered and helps the students in remembering them.
10. Mental maths exercises have been given to help the students acquire speed and sharpentheir intellect.
11. A special feature is the inclusion of ‘Mixed Review Exercises’ which would keep the studentsin constant touch with all the topics studied earlier.
12. Historical Notes, Quizzes, Just For Fun, Puzzles and Enrichment Material offer furtherintellectrual challenge to sharp students and help them not only to maintain their interest inthe subject and widen their horizon of knowledge but would also be of immense help inpreparing for such competitions such as Mathematics Olympiad at various levels.
It is hoped that this series of books will meet more than adequately, the needs of the studentsthey are meant for. Any suggestions for the improvement of the books would be most welcomeand gratefully acknowledged.
AUTHORS
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A NOTE FOR THE TEACHERSDear friends,
We feel happy to be able to present for your perusal and consideration the new andthoroughly revised and updated editions of our Mathematics Today Series for Classes VI,VII and VIII. No doubt it has been possible as a result of the motivation and feedbackreceived in the form of valuable comments, suggestions and criticism from the learned teachers.We strongly feel that a textbook howsoever good it may be is only a tool to help teachers toteach effectively. It is the teacher and only the teacher who is competent to decide his/herteaching strategies in the classroom and is the best judge of how to use the textbook to meetthe special needs of his/her class. It is earnestly hoped that this series will be able tosupplement your efforts effectively to create interest of your pupils in the subject and makethe study of mathematics interesting and enjoyable and gain mastery over the subject.
Howsoever best one performs or creates there is always scope for improvement. We wouldbe very happy rather grateful to receive your comments, appreciation/criticism andsuggestions for further improvement of the books.
With regardsYours sincerely
AUTHORS
A NOTE FOR THE STUDENTSDear students,
Wishing you the best in life. You are the best judge to evaluate whether the book you arestudying fulfills your needs and satisfies your thirst for knowledge or not. Please do nothesitate to write individually through the publishers or directly to the authors at the followingaddress if you come across any discrepancies or if you have some suggestions to make.
AUTHORS
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SYLLABUS
Teaching Points Teaching Notes
1. Sets Concepts Review of work done in Class VI - idea of notation, equal sets, equivalent sets, theempty set, the universal set, cardinal property of a set, finite and infinite sets. Unionand intersection of sets, disjoint sets, overlapping sets, complement set; Venn diagrams.
Examples should be drawn for the number systems with which the pupil is familiarand from real life situations; Operations on sets should be confined to the universal setand one or two of its subsets or two disjoint or overlapping sets.
2. Numbers Review of work done in class VI.
Natural numbers, whole numbers, the four fundamental operations, factors, repeatedfactors, exponents, prime factorisation, properties of exponents (confined to integralexponents); H.C.F. or G.C.D.; multiples, even and odd numbers, L.C.M.; perfect squarenatural numbers and their square roots.
Integers : the four fundamental operations.
Fractions : classification and comparison of fractions; the four fundamental operationswith fractions; simplification, percentages; ratio.
Decimals : the four fundamental operations; recurring decimals; approximation(rounding off). Powers and roots - elementary treatment, based on the multiplicationtables and drilling in the most frequently used powers and roots e.g. powers and roots- e.g. powers of 2 up to 28, powers of 3 up to 34; squares of all numbers up to 20 and cubesof all numbers up to 10; square roots of the perfect square natural numbers and perfectcube natural numbers corresponding to the previous item.
3. Arithmetic Problems Unitary method
Speed, time and distance, simple problems
Ratio, sharing in a ratio
Profit and loss
Simple interest
Averages (direct problems to be emphasized).
Note : pupils may be introduced to the idea of a multiplying ratio to prepare them fordirect and inverse proportion in higher classes.
4. Algebra
Fundamental Review of class VI work; concept of degrees and coefficients;
concepts like and unlike terms; polynomials with rational coefficients.
Fundamental Addition, subtraction, multiplication of a polynomial by a
operations monomial, binomial and trinomial (up to degree 2 only); division of a polynomial, inone variable only, by a monomial and binomial in one variable only. Using the rule;Dividend = Divisor times Quotient plus Remainder to check the result of division.
Formula Translation from words to symbols (construction of a formula) and from symbols towords. Use of formulae in real life situations as in simple interest, mensuration, geometry,physics etc. Changing the subject of a formula (simple other than the common factor).Substitution in formula. Substitution in an expression in which the variables are onlyup to power 2.
Exponents the following laws of indices - integral exponents only. Proofs are not required. Xm . Xn
= Xm + n ; Xm / Xn = Xm – n; (Xm)n; = Xmn; X0 = 1
Here x may be a rational number.
Products Special products as identities :(i) (x + a) (x + b) = x2 + (a + b) x + ab
(ii) (x + a)2 = x2 + 2xa + a2
(iii) (x – a)2= x2 – 2xa + a2
(iv) (x + a) (x – a) = x2 – a2
Factors Factors of -(a) polynomials with a common monomial(b) difference of two squares
Simplification Simplification, addition and subtraction of algebraic expressions with integraldenominators.
Relations and To be done through arrow diagrams, leading to listing of themapping matching pairs.
Classification of functions not included.Note : In the teaching scheme, the section on formulae and / or graphs or ordered pairsshould be linked with this section.
Equations and A mathematical sentence; an open mathematical sentence in one variable. Simpleinequations equations (in one variable) and the graphical representation of the solution. Problems
leading to simple equations. Simple inequations in one variable and the graphicalrepresentation of the solution.
Graphs Terms : rectangular coordinates, ordered pairs, abscissa ordinate; plotting.Representing a linear (first degree) equation in two variables, graphically.
5. Geometry
Lines, rays, line segments Review of work done in Class VI; lines, rays, segments, planes, angles.planes, angles Linear pair, vertically opposite angles; parallel lines, corresponding angles; alternate
angles, allied (interior opposite) angles. Problems on the above mentioned topics.Basic constructions Constructions ; bisection of a line segment and of an angle; construction of an angle
congruent to given angle. Construction of perpendiculars and parallels.Trianlges Classification; construction of triangles; appreciation of the fact that the sum of the three
angles of a triangle is two right angles or 180°; related problems.Symmetry, Simple, direct cases onlyReflection, RotationCongruency of triangles; Conditions for the congruency of two triangles namely, SAS, ASA, SSS, RHS - no formal
proofs (practical approach, use of movements e.g. sliding, turning to make congruenttriangles coincide); using congruency - single cases only ; some idea of similarity.
Polygons Quadrilaterals - different kinds of quadrilaterals and their properties.Polygons; sum of the interior angles of a polygon in the case of a polygon of a givennumber of sides.
Construction of Construction of a rectangle, given (i) a pair of adjacent sides (ii) one side and a diagonal.rectangles and squares Construction of a square given (i) one side (ii) one diagonal.Circles Revising terms associated with circles e.g. radius, diameter, secant, chord and arc.
Questions based on multiple radii and simple questions associated with angles of atriangle and angle at a point.(Concept : Ability to identify radii of the circle.)
6. Mensuration Perimeter and area of a rectangle and a square.Perimeter and area Volume and surface area of cubes and cuboids. Units of area and volume : cm2, cm3;Volume and surface area m2, m3
7. Statistics Handling of raw data, Making a frequency distribution table and calculating mean.Drawing a pie graph, column graph and line graph based on the data.
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UNIT I : SET THEORY
1. SETS CONCEPTS 1 – 11
2. OPERATIONS ON SETS AND VENN DIAGRAMS 12 – 22
UNIT II : PURE ARITHMETIC
3. NUMBER SYSTEM 23 – 34
4. INTEGERS 35 – 45
5. HCF AND LCM OF NUMBERS 46 – 53
6. FRACTIONS 54 – 74
7. DECIMALS 75 – 89
8. POWERS AND ROOTS 90 – 101
UNIT III : COMMERCIAL MATHEMATICS
9. RATIO AND PROPORTION 102 – 113
10. UNITARY METHOD 114 – 122
11. PERCENTAGE AND ITS APPLICATIONS 123 – 143(Percentage, Profit and Loss, Discount, Simple Interest)
12. DISTANCE, TIME AND SPEED 144 – 150
13. AVERAGE 151 – 156
UNIT IV : ALGEBRA
14. ALGEBRAIC EXPRESSIONS 157 – 168
15. FORMULA 169 – 172
16. EXPONENTS 173 – 184
17. SPECIAL PRODUCTS (Some Special Identities) 185 – 191
18. FACTORISATION OF ALGEBRAIC EXPRESSIONS 192 – 196
19. SIMPLIFICATION OF ALGEBRAIC FRACTIONS 197 – 199
20. LINEAR EQUATIONS IN ONE VARIABLE 200 – 210
21. LINEAR INEQUATIONS 211 – 215
22. GRAPHS 216 – 223
23. RELATIONS AND MAPPINGS 224 – 233
UNIT V : GEOMETRY
24. BASIC GEOMETRICAL CONCEPTS–LINES AND ANGLES 234 – 243
25. PARALLEL LINES 244 – 249
CONTENTS
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26. BASIC CONSTRUCTIONS 250 – 252
27. TRIANGLES 253 – 262
28. CONGRUENCE OF TRIANGES 263 – 274
29. POLYGONS 275 – 290
30. CIRCLES 291 – 296
31. SYMMETRY 297 – 302
32. REFLECTION 303 – 307
33. ROTATION 308 – 313
UNIT VI : MENSURATION
34. PERIMETER AND AREA 314 – 321
35. VOLUME AND SURFACE AREA OF SOLIDS 322 – 329
UNIT VII : STATISTICS
36. FREQUENCY DISTRIBUTION AND MEAN 330 – 335
37. GRAPHICAL REPRESENTATION OF DATA 336 – 341
PRACTICE TEST PAPERS 342 – 347
ANSWERS 348 – 381
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RevisionWe help you to revise here the basic set concepts that you have studied in Class VI.1. The word set is used for a collection of objects of any kind whatsoever, whereby given any object, we
can say with certainity whether the object belongs to that collection or not. In other words,
A set is a collection of well defined objects.
The following are some examples of sets :(i) a set of animals;
(ii) the set of teachers teaching you;(iii) the set of letters in the word “CRICKET”.The following are not well defined collections
and are, therefore, not sets :(i) the books that are a fun to read;
(ii) the movies that are good.2. Sets can be specified by any of the following methods :(i) Description method
(ii) Tabular form or the Roster method(iii) Rule method or Set Builder form
Notations1. The sets are usually denoted by capital letters A, B, C, etc., and the members of the set are denoted by
small letters a, b, c, ..., x, y, z, etc.2. The symbol x S means x is an element of set S or x belongs to S whereas the symbol x S means x
is not an element of S or x does not belong to S. For example, let C be the set of all countries in Asia.Then we have : India C, Ceylon C, Japan Cwhereas, England C, Canada C, France C.
Description MethodIn this method, we write a description of the members of the set stating clearly their common property. The
description is usually enclosed in braces ‘{ }’.Ex. 1. Write the following sets in description form :
(i) 0, 2, 4, 6, 8, 10 (ii) a, e, i, o, uSol. (i) {even numbers less than 12}
(ii) {vowels of the English alphabet}Tabulation form or the Roster Method
In this method, we list all the members of the set and separate them by commas. The list is enclosed withinthe braces ‘{ }’.
The order in which elements are listed is unimportant. Also, repetition is not done while listing theelements. Thus, if S is the set of letters in the word ‘PAPER’, it will be specified as S = { p, a, e, r }.
Belongs to Does not belong to
1
Set ConceptsUNIT I : SET THEORY1
2 Mathematics Today for Class VII [ICSE]
Ex. 2. Write each of the following sets in Roster form :(i) The set of prime numbers less than 15.
(ii) The set of colours of the rainbow.(iii) The set of letters in the word ‘REMEMBER ’.
Sol. The given sets can be written as(i) A = {2, 3, 5, 7, 11, 13}
(ii) B = {violet, indigo, blue, green, yellow, orange, red}(iii) C = {R, E, M, B}If the number of elements is large, we can write a few elements, then indicate the pattern by puttingthree dots and then in the end, write the last element.
Ex. 3. Write the set A of the set of non-zero multiples of 2 less than 100.Sol. The set of non-zero multiples of 2 less than 100 is A = {2, 4, 6, ..., 98}.Ex. 4. Write the set P of the integers greater than – 5.Sol. P = {–4, –3, –2, –1, 0, 1, 2, ...}.
Rule Method or Set Builder FormIn this method, we write a variable (say x) representing any member of the set, put the vertical
‘|’ or the colon ‘:’ which stands for the words ‘such that’.Then we state the property which is characteristic of all the objects of the set.Thus, if the members of a set S possess a property P and x is any element of S, then we write
the set S asS = { x | x has property P } or S = {x : x has property P }
Ex. 5. Write the following sets in the set builder form :(i) The set E of days of the week.
(ii) The set A of natural numbers less than 20.(iii) {–9, –6, –3, 0, 3, 6, 9, 12, 15, 18}.
Sol. (i) E = {d | d is a day of the week}(ii) A = {x | x N, x < 20}
(iii) Given set = {x | x = 3n, n Z and – 3 < n < 6}Ex. 6. Write the following sets in roster form and also in set builder form :
(a) The set of integers which lie between –5 and 4 (both inclusive).(b) The set of natural numbers which are divisible by 8 and less than 60.
Sol. (a) {–5, –4, –3, –2, –1, 0, 1, 2, 3, 4} or {x | x Z, –5 < x < 4}(b) {8, 16, 24, 32, 40, 48, 56} or {x : x = 8n, n N and x < 60}.
Ex. 7. Write the following sets both in roster form and description form : (i) {x : x = n2, n N and n < 7}. (ii) {x : x is a Ocean}(iii) {x : x is a letter in the word ‘EXAMINATION’}.(iv) {y | y = 2x –7, x W and x < 7}.
Sol. (i) Natural numbers < 7, i.e., less than or equal to 7 are 1, 2, 3, 4, 5, 6, 7.The squares of these numbers are 1, 4, 9, 16, 25, 36, 49
The given set can be written as {1, 4, 9, 16, 25, 36, 49} [Roster form]or {squares of first seven natural numbers} [Description form]
(ii) {Pacific, Atlantic, Indian, Arctic} [Roster form]{Oceans of the world} [Description form]
or :
Symbol forsuch that
Set Concepts 3
(ii) {E, X, A, M, I, N, T, O} [Roster form] or {letters of the word ‘Examination’} [Description form]
(iii) Whole numbers less than 7 are 0, 1, 2, 3, 4, 5, 6. Putting x = 0, 1, 2, 3, 4, 5, 6 respectively in 2x – 7 we get y = –7, –5, –3, –1, 1, 3, 5. So the given set can be written as {–7 –5, –3, –1, 1, 3, 5} [Roster form] or {odd integers lying between –8 and 6} [Description form]
Ex. 8. Write the following sets in builder form :
1 1 1 1, , , .......,
3 5 7 31
Sol. Since 3, 5, 7, ........ are odd numbers and general odd number can be written as 2n + 1 where n N,
we have Given set = 1: , and 15 .
2 1x x n N n
n
EXERCISE 1 (A)
I. List the members of the sets in questions 1 to 8 :1. {the first three months of the year}. 2. {first 4 prime numbers}.3. {odd numbers smaller than 8}. 4. {even numbers smaller than 10}.5. {the letters of the word ‘SCHOOL’}. 6. {even positive integers that are divisible by 5}.7. {the angles of ABC}. 8. {the numbers that differ from 5 by 1}.
II. Write down the following sets in the descriptive form in questions 1 to 6 :1. {1, 3, 5, 7} 2. {0, 1, 2, 3, 4, 5}3. {5, 10, 15, 20, 25} 4. {Tuesday, Thursday}5. {2, 3, 5, 7, 11} 6. {1, 4, 9, 16...}
III. List the following sets in Roster form :1. A = {x | x is a non-zero multiple of 2 less than 11}.2. B = {x | x is the shortest month of the year}.3. C = {x | x is a square number less than 20}. 4. D = {x | x is a prime number which is even}.5. E = {x | x is a vowel}.
IV. List the following sets in set-builder form :1. {2, 3, 5, 7, 11, 13, 17, 19}. 2. {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}.3. {1, 4, 9, 16, 25, 36, 49, 64}.4. {January, March, May, July, August, October, December}.5. {1, 2, 3, 4, 6, 9, 12, 18, 36}. 6. {1, 8, 27, 64, 125, 216, 243, 512}.
7. 1 1 1 1, , , .... .,3 4 5 10
V. Fill in the blanks with the symbol ‘’ or to make the statements true :1. 2 ________ {set of even numbers} 2. Carrot ________ {fruits}3. e ________ {set of vowels} 4. 7 ________ {x | x is a multiple of 7}5. m ________ {m, a, t} 6. 1 ________ {x | x is a prime number}7. 0 ________ {5 – 5, 5 + 5, 5 × 5, 5 ÷ 5} 8. b ________ {c, o, p, y}.
4 Mathematics Today for Class VII [ICSE]
Types of sets1. Finite set. A set whose members can be counted, i.e., in which the process of counting its
members comes to an end, is called a finite set.
For example :The following are all finite sets :
(i) A = {a, e, i, o, u} (ii) B = {x | x = 2n, n N, n < 30}(iii) The set of all persons on earth.
2. Infinite set. A set which contains unlimited number of elements, i.e., in which the counting ofelements does not come to an end, is called an infinite set.
For example :The following are all infinite sets :
(i) The set of prime numbers. (ii) The set of integers less than 0.(iii) The set of points on a line segment.
3. Singleton set. A set containing only one element is called a singleton set.
For example :{5} is a singleton set, containing only one element, namely 5.
4. Empty set or Null set. The set containing no elements at all is called the empty set. It isdenoted by the symbol ‘{ }’ or, by the symbol . The latter symbol is more commonly used.
For example :(i) = {pupils in your class over 70 years old} (ii) = {integers greater than 3 and less than 4}
(iii) = {squares with five sides}Cardinal number of a set. The number of distinct elements in a finite set A is called its Cardinalnumber and is denoted by n (A).
For example :(i) For the set A, shown in the picture n (A) = 4.
(ii) Let P = set of letters of the word ‘NANNY’. ThenP = {N, A, Y } has 3 distinct elements, and so n (P) = 3.
(iii) Let B = {x | x W, – 3 < x < –1}. Then B = , since there is no wholenumber between – 3 and –1.Since, contains no elements at all therefore n(B) = n () = 0
Remark : The cardinal set of an infinite set is not defined.
5. Equal sets. Two sets A and B are said to be identical or equal, written : A = B when they haveexactly the same elements.
To be more precise, A and B are equal if every element of A is an element of B and conversely every elementof B is also an element of A.
A =
A = B
,B =
Mathematics Today For Class 7
Publisher : SChand Publications ISBN : 9789352530632Author : S K Gupta AndAnubhuti Gangal
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