UC-NRLF IN MEMORIAM FLOR1AN CAJORI " ^n OUTLINES -OF- NUMBER SCIENCE -BY- NATHAN NEWBY, Professor of Mathematics in the INDIANS STATE NORMAL SCHOOL, TERRE HAUTE, INDIANA. SECOND EDITION. TERRE HAUTE : GEORGE H. HEBB PRESS. 1884. [Entered according to Act of Congress in the year 1883. by NATHAN NEWBY, in the office of the Librarian of Congress.] ^Jd EXPlLANATORY. This little book is prepared to meet the author's convenience in giving Arithmetical instruction in the Normal School, and at the solicitation of a large num- ber of pupil-teachers, who have received in the class- room and have afterwards used in their own schools, much of the matter herein contained. The work is but an elaboration of a series of articles written by the same hand in 1873, and published in an educational paper in this State. An examination of the order of procedure will show that the book is not made for children, but for the mature mind already conversant with most of the facts of number as presented in the text books on Arithmetic. " The aim of the Normal School is not so much to teach the facts of the common school branches as to make a thorough study of the re- lations of those facts to one another. The study of these relations opens up new lines of thought that mate the common-school branches intensely interest- ing studies to most students." The province of the school as thus stated, being kept in view, the attempt has been made so to present the topics of number science that the lines which unify parts that are kindred, may be readily seen. The discussion begins with a general definition of M athematics, together with a brief consideration of the realms of Space and Time, These are defined as condi- tions: Space as the condition of extension, and hence preliminary to Geometry in its various phases; and Time as the condition of succession, and hence prelim- inary to the idea of Number. Starting with Time as the conditioning factor, it is sought to show the genesis of number in general. From this idea the mind readily passes to that of num- ber in particular. The term Integral unit or Unit one (from Davies), is fixed upon to designate the primary idea of number in particular. It must be borne in mind that the elements which enter into a science are men- tal and not material objects. A pencil, a horse or a box, is not an element of number science is not a unit; it is the idea one which the mind forms upon viewing the object as an entirety that constitutes the unit, or fun- damental element in the science of numbers. The student is next asked to re-think the nine general classes of numbers, to find the basis of each classification, and to give, by definition, the mark of each class. The classification of numbers is discussed thus early not because of its logical relation to that which precedes or succeeds it, but because of the basal char- acter which Number classification sustains in all com- putation. Number Representation is next discussed. Under the Arabic Notation it is observed that the characters are used to represent numerical values thought in three systems of numbers. Each of these systems, to- gether with its notation, is discussed. Number Reduction is next considered. The kinds of reduction are determined, and applied to numbers thought in the decimal, the fractional and the com- pound systems. Attention is called to the fact that reduction descending is effected by multiplication, and that reduction ascending is effected by division, whether the number to be reduced be an abstract integer, a fraction or a denominate number. Under Number Processes, the phases of synthesis and analysis are treated. The terms used, the mental acts involved, and the principles which guide the mind in computing are discussed. In formulating definitions and principles it is sought in the main, to "bring before the mind the act or process by which the concept to be defined is sup- posed to be constructed." It has been a special aim to make every definition sufficiently inclusive to embrace all that the term covers wherever found in the work: e. g. The definitions of multiplication and division as given in most text books on Arithmetic, are but par- tial since they do not include multiplication and divi- sion by a fraction. The definitions in this book are be- lieved to be ample. Both common and decimal fractions are treated together, the principles of the one being the principles of the other. The suggestions given for the treatment of Com- pound Numbers, will, it is hoped, enable the teacher to proceed more systematically and satisfactorily than by the methods usually presented. Most of the definitions and discussions under the applications of Percentage are omitted, not because they are unimportant, but because they are so well given in text books on Arithmetic that, their repeti- tion here is deemed unnecessary. Methods of solving representative problems have been freely inserted under these applications. Involution and Evolution are treated Arithmetic- ally instead of Geometrically and in a manner at once simple and exhaustive. "Results in teaching depend upon the clearness with which distinctions are made;" and the pupil can be brought to make clear distinctions, only by being held to rigidly logical modes of thinking. As an aid to this end, forms of solution are given for nearly all classes of arithmetical exercises. These forms or others equally logical should be strictly adhered to in order to secure to the pupil the maximum culture which the subject can give. A number of errors, typographical and otherwise, were observed after the work was in print. Some of these have been corrected with the pen. Others still remain, but they are of such a character as not to mis- lead the attentive reader. TEREE HAUTE, APRIL, 1884. CONTENTS. SECTION I. Number Genesis. Page 9 SECTION II. Number Classification Page 12. Integer 12 Fraction 12 Abstract 12 Concrete 12 Prime 12 Simple . 13 Denominate 13 Compound 13 Composite 12 SECTION III. Number Representation. Page 14. Notation 14 Koman 14 Arabic 16 Systems of Numbers ... 16 The Decimal System . The Fractional System The Compound System SECTION IV. Number Reduction. Page 24. Reduction Descending . . 24 | Reduction Ascending 16 21 23 25 SECTION V. Number Processes. Page 26. Rules for Squaring ... 35 Subtraction 38 Division 40 Disposition 44 Divisibility of Numbers . 45 Evolution . . 48 Computation 26 Addition 27 Multiplication 29 Composition 32 Involution 34 Tables of Squares .... 35 SECTION VI. Measures and Multiples. Page 5O. Greatest Common Divisor 50 | Least Common Multiple . 53 SECTION VII. Fractions. Page 55. The Primary Idea .... 55 Classes of Fractions ... 56 General Principles . . .58 Reduction . . . ... 59 Addition and Subtraction. 65 Multiplication 67 Division 71 Aliquots 80 SECTION VIII. Compound Numbers. Page 82. Diagram 82 I Application Time Mea- Order of Study 88 | sure 84 SECTION IX. Time and Longitude. Page 95. SECTION X. Areas and Volumes. Page 1OO. SECTION XI. The Decimal System of Measures. Page 1O4. SECTION XII. Percentage. Page 112. The Terms Used .... 112 Relations 113 General Cases 114 Applications 11 3 Profit and Loss 118 Commission 123 Stocks 127 Insurance 130 Taxes , 133 Customs 134 Interest 135 Discount 142 Exchange 147 Equation of Payments . . 148 SECTION XIII. Ratio and Proportion. Page ISO. Ratio. 150 I Proportional Parts. 158 Proportion ... .... 152 j SECTION XIV. Involution and Evolution. Page 159. Involution of Numbers to Second Power ... 160 Evolution of the second Root 163 Involution of Numbers to Third Power .... 166 Evolution of the Third Root 169 SECTION XV. Test Problems. Page 17O. SECTION XVI. Review Questions and Topics. Page 184. OUTLINES OF- NUMBER SCIENCE. SECTION I. NUMBER GENESIS. 1. Mathematics may be defined in a general way as the department of knowledge which exhibits the prop- erties and relations of extension and number. 2. A consideration of extension leads into the realm of space. Geometry and kindred branches are evolved from the properties and relations of extension. 3. A consideration of number leads into the realm of time for the primal elements of the science. The Basis of Number. 4. In General, a. Every conscious state of the mind is known to begin, to endure for a period and to end, giving way to another mental phenomenon. 6. Mental states are distinct from the mind and from one another. They are known to be distinct be- cause experienced in different periods of time or be- cause of a diversity of the states compared. 10 c. To give genesis to the idea of number it is nec- essary for a state to succeed a state in consciousness. The idea of number thus originates in the cognition of succession. Succession is possible only in time; hence time is conditional for the numerical idea. d. Arithmetic is a branch of the mathematics of number; it is both a science and an art. e. As a science, Arithmetic exhibits the facts of number, presents the relations sustained among the facts, formulates these relations into principles which bring the facts together until they appear in conscious- ness systematically arranged as an organic whole. As an art, Arithmetic is the application of the science in computation. 5. In Particular. If the attention be directed to an object, as a tree, a house, an apple, etc., among the at- tributes observed is that of oneness. The idea one may be abstracted from the conception of the object which occasions it, and as thus abstracted, it may be thought apart from its object. The Unit. The term integral unit, or unit one (from Davies) is fixed upon to designate the idea one as ab- stracted from the conception of an object thought as a whole, as not composed of identical parts, nor itself one of the identical parts which compose another whole. [In mathematics equality is identity. The reader is referred to Everett's Science of Thought, pp. 98-105, inclusive.] Remark. The primary, or integral one, arises in the mind as above stated; there are, however, two classes of secondary ones with which we shall have much to do in our investigations of number relations and processes. 11 1. The Fractional Unit. If an object be separated, into equal parts, and the attention be directed to a part thus found, the idea one arises. Definition. The idea one, which is applicable to a part that results from the separation of an object into equal parts is called a fractional unit. 2. The Multiple Unit. If the mind think sever- al objects as forming a group, the idea one arises. Definition. The idea one which is applicable to a group of wholes may be called a Multiple Unit. Remarks. 1. The integral unit, or unit one, is the primary idea in Arithmetic. All other units are definitely related to this primary unit through the objects from which the units are originally abstracted. 2. An object giving rise to the idea one may be called a unit-object. Most writers on Arithmetic call such an object a unit. 3. The object which gives rise to a fractional unit is one of the equal parts into which the unit-object of the integral unit is thought as separated. In view of this relation of part to whole, a fractional unit is often said to be derived from the in- tegral unit by the analysis of the latter into equal parts. The unit, which is the idea one, is a unity, and is incapable of division ; its object, only, can be divided, a new idea one arising when a part instead of a whole engages the attention. [See Unity in Fleming's Vocabulary of Philosophy.] A Number. A number may be defined as a unit or group of like units thought together. Remarks. 1. The units composing a number may be integral, fractional or multiple. 2. Aristotle did not include unity in the idea of number. He considered unity as the element of number. Locke included unity in the idea of number. This view is adopted by modern writers on Arithmetic. 12 SECTION II. NUMBER CLASSIFICATION. a. On the basis of integral or fractional units used in their formation, numbers are classified as integers and fractions. 6. An Integer. A number composed of one or more integral units is called an integer. 7. A Fraction. A number composed of one or more fractional units is called a fraction. Remark. An integer and a fraction thought as combined are together called a mixed number. 6. On the basis of application to objects numbers are classified as Abstract and Concrete. 8. Abstract. A number thought as independent of, or separate from an object, is called an abstract num- ber. 9. Concrete. A number thought as applied to an object is called a concrete number. Examples : 4 men, 12 horses, f of a dollar, etc. Remark. 4, 12 and f, in the examples are concrete num- bers because their objects are named. c. On the basis of divisibility, abstract integers are classified as Composite and Prime. 10. Composite. A number that can be divided into equal integers each greater than one, is a composite number. 11. Prime. A number that cannot be divided into equal integers each greater than one, is a prime num- ber. 13 d. On the basis of distinct or assumed unit-object, concrete numbers are classified a s Simple and Denom- inate. 12. ' Simple. A number whose unit-object is a dis- tinct whole, is a simple number. As 5 books, 3 pens, 7 horses; the unit-objects being the distinct wholes, book, pen and horse, respectively. Remark. An abstract number is often called a simple num- ber. 13. Denominate. A number whose unit-object is an assumed time, extent or degree of intensity, is a de- nominate number. As 3 days, 5 feet, 7 pounds ; the unit-objects being the day, the foot and the pound, respectively, each of which is an assumed unit-object. 14. Compound. Two or more denominate numbers having the same primary unit, are together called a compound number. Remarks. 1. A denominate number is often defined as a number whose unit (object) is named, but such definition is applicable to all concrete numbers and is, therefore, too inclu- sive. 2. A compound number is often defined as a number con- sisting of two or more denominations. Under this definition 5 ft. 3 Ib. 3 hr. is a compound number. The error in the defi- nition is apparent. 3. The classification of fractions is deferred until the sub- ject is treated in detail. 14 SECTION III. NUMBER REPRESENTATION. Notation. 15. Definition. Notation is a systematic method of representing numbers by symbols. Kinds of Notation. Remark Many kinds of notation have been in use in differ- ent times and places. But two of these, however, are usually presented in Arithmetic, viz., that used by the ancient Ro- mans and that introduced into Europe by the Arabs. The Roman Notation. 16. Characters. The alphabet of the Koman nota- tion consists of the seven letters, viz. : I. V. X. L. C. D. M. 17. Signification. I represents one, V five, X ten, L fifty, C one hundred, D five hundred, M one thousand. 18. Relation. \ T represents five times I. X two V. L " five " X. C " two " L. D " five " C. M two " D. 19. Limit. The Eoman notation is limited to the representation of integers, and is chiefly used in num- bering chapters, headings and divisions in books and papers. 15 20. Principles. Remark. As now used the Roman notation is effected in ac- cordance with the following principles : I. If a letter be written with its equal or with a combination of its equals, the letters combined repre- sent a value equal to the sum represented by the let- ters. II. If a letter be placed at the left of a letter rep- resenting a greater value, the letters combined repre- sent the diiference between the values represented by the letters taken separately. III. If a letter or combination of letters be placed at the right of a letter representing a greater value, the letters combined represent the sum of the two values, IV. If a d.ash be placed over a letter or combination of letters the value represented is multiplied by one thousand. 21. Exercises. Read each of the following : IX XIV, XXVII, LIII, XCV, CXXVIII, XII, XCVI, MDCCL, MCI, MDCCCL, LXXXI1I, DIV. Write in the Roman notation : Twenty-nine ; thirty- three ; fourteen ; one hundred six ; fifty-six ; one thou- sand two hundred sixty -four; seventy-eight ; one thou- sand seven hundred seventy-six ; ninety-seven ; forty- nine; five hundred thirteen; eleven hundred seventy. [Give additional oxercises.J 16 The Arabic Notation. 22. Characters. The alphabet of the Arabic nota- tion consists of the following ten characters called figures, viz. : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. 23. Signification. The first nine of these figures represent one, two. three, &c., units to nine. They are called significant figures, or digits because they signify or point out numbers. The tenth figure is called zero or nought. It expresses no numerical value. Remark. In Algebra the zero is classified among the symbols of quantity and is defined as the representative of an infinitely small quantity. 24. Systems. The Arabic figures are used in three distinct systems of numbers, viz.: (1.) The decimal. (2.) ThG fractional (3.) The compound. 1. The Decimal System. 25. A Scale. The series of units constituting the basis of a system of numbers is called a scale. 26. The Decimal Scale. The decimal system of numbers has for its basis a series of units each of which is ten times as great as the unit next below it in the series. This series of units is called the decimal scale. 27. Orders of Units. 1. The integral unit, or unit one, is called a unit of the first order. 2. Ten units of the first, or units' order are to- gether called a unit of the second order. 17 3. Ten units of the second, or tens' order are to. gether called a unit of the third order. 4. Ten units of the third, or hundreds' order are together called a unit of the fourth order. 5. Ten units of the fourth, or thousands' order are together called a unit of the fifth order. 6. Ten units of the fifth, or ten-thousands' order are together called a unit of the sixth order. Remark. Higher orders of units are formed by thinking together ten units, respectively, of the sixth, seventh, eighth, ninth, tenth, &c.. orders indefinitely. 28. Periods. 1. The first three orders of units in the decimal scale, viz.: units', tens' and hundreds', con- stitute a period called units' period. 2. The three orders of units next higher than units' period, constitute thousands' period. Units of thousands, tens of thousands and hundreds of thousands are thought as composing this period. 3. The three orders of units next higher than thousands' period constitute millions' period. Units of millions, tens of millions and hundreds of millions are thought as composing this period. 4. Other periods of three orders each are formed f-1 higher orders of units in the decimal scale. The periods above those named are those of billions, tril- lions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, etc., to in- finity. 5. Each period embraces three orders, viz.: units, tens and hundreds of that period. 18 29. Lower Orders. Remarks, a. The decimal scale may include units lower than the unit one. 6. The division of a unit into tenths, hundredths, thou- sandths, ten-thousandths, etc., is called a decimal division of the unit. 1. Units which result from dividing the unit one into ten equal parts are called tenths, or units of tenths' order. 2. Units which result from dividing one-tenth into ten equal parts are called hundredths, or units of hundredths' order. 3. Units which result from dividing one-hundredth into ten equal parts are called thousandths, or units of thousandths' order. 4. Orders of decimal units called respectively ten- thousandths, hundred-thousandths, millionths, ten-millionths, etc., are formed by dividing into ten equal parts the unit next higher in the scale. 30^ Principle of the Decimal Scale. Ten units of any order make a unit of the next higher order. Notation of Numbers Thought in the Decimal Scale. 31. Characters. The Arabic characters already given, together with a point or period, called the deci- mal point, are used in representing numbers thought in the decimal scale. 32. Signification. The digits and the zero have the signification already stated, while the decimal point is used to mark the place in a written number from which to start in determining its value. 19 33. Units' Place. The first place at the left of the decimal point is units' place. Remark. Any number of first order units from 1 to 9, may be represented by writing the proper figure in units' place. 34. Tens' Place. The place next at the left of units' place, is tens' place. Remark. Any number of second order units from 1 to 9, may be represented by writing the proper figure in tens' place. 35. Hundreds' Place. The place next at the left of tens' place is hundreds' place. Remark. Any number of third order units from 1 to 9, may be represented by writing the proper figure in hundreds' place. 36. Units of thousands, tens of thousands, and hundreds of thousands, respectively, may be repre- sented in the next three places at the left of those al- ready named. In the next three places may be repre- sented, respectively, units of millions, tens of millions and hundreds of millions, etc. 37. Tenths, hundredths and thousandths, etc., may be represented by figures written at the right of units' place in places corresponding to tens', hundreds', thou- sands', etc., at the left. 38. Scale. The term scale is used to name the series of successive units which form the basis of a sys- tem of numbers. In notation, however, the series of successive places in which the different orders of units may be represented is called a scale. This scale may be called the representative scale to distinguish it from the thought scale defined above. Remark. The simple value of a figure is determined by its form alone, while the local value of a figure is determined by both the form of the figure and the place it occupies in the representative scale. . 20 39. Limit. The decimal system of writing num- bers is limited to the representation of integers and such fractions as result from a decimal division of the the unit. Numeration and Beading Numbers. 40. Numeration. Naming the successive places in a written number is called numeration. Remark. In numerating a number it is both convenient and customary to begin with units' order. 41. Reading. Naming the numerical value repre- sented by a written number is called reading the num- ber. Remarks. 1. In reading a number it is both convenient and customary to begin at the highest order in which numerical value is represented. 2. In reading a number the word and should never be used except between the integral and the fractional parts of a mixed number, Thus 325 is read, three hundred twenty-five ; 4f is read, four and two thirds ; 420.005 is read, four hundred twenty and five thousandths. [Exercise in writing and reading numbers in the decimal scale.] Eead 46; 326; 460; 2346; 1785; 57689; 32567; 4567890; 45678834; 456784352; 46789716; 456.3; 34; 51.6; 317.04; 45607; 5678.17; 45.041; 32.117; 2.3456 ; 4.055, 56.78946 ; .346 ; .3 ; .5678 ; .56789047 ; .03456789 ; .45678; .5678789; Write each of the following in the decimal scale : Two hundred thirty-four ; seven thousand sixty-five; Five hundred forty-one ; seven thousand seventy-six ; two thousand ninety-eight ; Four hundred thousand 21 sixteen; forty-five thousand ten; sixty-six thousand ninety-four ; seventeen thousand five ; nineteen thou- sand nineteen ; five hundred thousand six ; six million four hundred thousand seventy-eight; three hundred million seven thousand six hundred nine ; three tenths ; nine tenths; fifteen hundredths ; seven hundredths; fourteen thousandths ; six thousandths ; two ten-thou- sandths; three hundred eleven thousandths; one hun- dred two ten -thousandths ; twenty-four hundred thou- sandths ; five hundred and seven hundred thousandths; sixteen millionths ; four hundred sixty-one millionths ; fifty-seven ten-millionths ; fifteen tenths; twenty-one tenths ; two hundred thirteen tenths ; five hundred tenths ; seventy tenths ; one thousand seventy-five tenths ; twenty tenths ; one hundred fifteen hundredths ; two hundred six hundredths ; four hundred forty-one hundredths; fifty hundredths ; six hundred hundredths ; four hundred fifty hundredths ; four hundred seventeen tenths; throe thousand hundredths ; four thousand two hundred eighty -four thousandths; two thousand seven thousandths ; one hundred tenths; forty-five thousand thousandths; forty tenthb, ten thousand ten-thou- sandths; two hundred and six tenths; forty and seven- teen hundredths ; eleven and five thousandths : seventy- five and six tenths ; one thousand and sixty-five ten- thousandths. 2. The Fractional System. 42. The primary idea of a fraction is composed of the following elements : a. The idea of a whole divided. b. The idea o equality of the parts. c. A number of the parts. 22 1. Two numbers are thus necessary to the con- ception of a fraction, a. The number of equal parts into which the unit is thought as separated, b. The number of those parts thac are thought as consti- tuting the fraction. 2. These two numbers are called the terms of the fraction. The number of equal parts into which the unit, or whole, is thought as separated is called the denominator of the fraction. The number of equal parts thought as constituting the fraction is called the numerator of the fraction. 3. Denomination. The fractional denomination of a fraction is the same as the ordinal of the denomina- tor. This is true of all fractions except those having the number two for a denominator. The denomina- tion of such fractions is half instead of second, the or- dinal of the denominator. The Notation of a Fraction. 43. Fractions which result from a decimal division of the unit may be written in the decimal (represent- ative) scale. [Art. 37.] Other fractions have a notation peculiarly their own. 44. Since two numbers are necessary to the thought fraction, two written numbers are necessary to notate a fraction. These written numbers have the same names, respectively, as the terms of the thought frac- tion which they represent. 45. a. The written denominator of a fraction is the figure or figures representing the number of fractional units into which a unit is. thought as separated. 23 b. The written numerator of a fraction is the fig- ure or figures representing the number of fractional units composing the fraction. 46. The written denominator is placed below the written numerator and separated from it by a short line. Remark. A fraction is thought as sustaining a definite rela- tion to the integral unit ; we, therefore, think of a written frac- tion as attached to units' place in the (representative) decimal scale. If the thought fraction require it, the written fraction may be attached to any other place in the representative scale. [Exercise in writing and reading fractions.] 3. The Compound. System. 47. A compound number is thought in two or more different orders of units that have the same primary, or standard unit. 48. In compound numbers the number of orders (denominations) in any "measure" is limited to the number of different unit-objects agreed upon for meas- uring the attribute under consideration. 49. The compound system of numbers is based, not so much on the fact that the several scales are varying, as that each "measure" has its own scale. Some of these scales are varying, and some of them are uni- form. Each of the common measures has a varying scale, while the "metric" measures, including the measure of U. $. money, has a uniform and decimal scale. In the old books will be found a duo-decimal "measure" which is, of course, uniform. 50. Each denominate number which forms part of a compound number, is thought in the decimal sys- tem, in the fractional system, or in both. 24 The Notation of a Compound Number. 51. The denominate numbers which compose a compound number, are written in a descending series, from left to right. Thus 4 bu. 3 pk. 5 qt. 1 pt.; 5 da. 16 hr.47 min.; I Ib. 3J oz. I pwt. 52. The names of the orders or denominations may be abbreviated, but the parts of a written compound number are not to be separated by anj r mark of punc- tuation. In reading a compound number the word and should not be used between any two adjacent denom- inate numbers in the series composing the compound number. Remark. The first compound number given under Art. 51 should be read 4 bushels 3 pecks 5 quarts 1 pint. The third should be read of a pound 3J ounces | of a pennyweight. [Exercise in writing and reading compound num- bers.] SECTION IV. NUMBER REDUCTION. 53. Reduction consists in the change by which a given numerical value is thought in another order or denomination. 54. Reduction Descending. Reduction descending consists in reducing a numerical value of any order or denomination to a lower order or denomination. It is effected by thinking the value of each unit of the given order or denomination in the number of units of the lower that are together equal to a unit of the higher. 25 55. Reduction Ascending. Reduction ascending con- sists in reducing a numerical value of any order or de- nomination to a higher order or denomination. It is effected by thinking as a unit of the higher the num- ber of units of the lower order or denomination that are together equal to a unit of the higher. Exercises. Reduce to lower orders each of the following: 4 thousands; 14 hundreds; 7 tens;. 3 units; 5 hun- dredths ; 4 tenths ; 2 thousandths, &c. Reduce 1, 2, 3, 4, 5, each to 3ds, 4ths, 5ths, 6ths, etc. Reduce to 4ths; f to 6ths; to 9ths; to 12ths; to 15ths. Reduce J, f and f each to 8ths ; to 12ths; to IHths. Reduce f to halves ; f- to 3ds; ^ to halves. Reduce $% to 4ths ; %% to halves ; to lOths; to 5ths. Reduce f to units ; f to units; - 1 / to units. Reduce 5 days to hr.; to min.; to sec. Reduce 1 gal. to qt.; to pt.; to gills. Reduce 3 bu. to pk. ; to qt. ; to pt. Reduce 128 pt. to qt. ; to pk. ; to bu. Reduce 96 gills to pt. ; to qt.; to gal. Reduce 500 units to tens; to hundreds. Reduce .2500 to thousandths; to hundredths; to tenths. Reduce .325 to hundredths ; to tenths. Reduce 220 units to tens ; to hundreds ; to tenths ; to hundredths. 26 SECTION V. NUMBER PROCESSES. Computation. 56. Definition. Computation consists in obtaining a number or numbers from other numbers in the light of definite relations existing between or among them. 57. The Mental Acts Involved. (1.) The first act of the mind concerned in computation is an act of com- parison. The numbers involved are compared in re- spect of one or more of the following points, viz.: a. Concrete denomination. b. Abstract denomination or order. c. Equality. d. Measurement. (2.) The mental act which effects a computation is an act of synthesis or analysis as the conditions of a given case may require. Eemarks. 1. The mind thus performs but three operations upon numbers, viz.: a. COMPARISON. b. SYNTHESIS. c. ANALYSIS. The first of these is preliminary in its nature while the other two are the processes by means of which all numerical compu- tation is effected. 2. Synthesis, as here used, means putting together, while analysis means separating, or taking apart. 58. The primary judgment in computation is a proposition of identity,^, e., something equals something. This judgment is called an equation. The values between which the relation of equality exists are called the members of the equation. 27 The Synthesis, or Combination of Numbers. i ADDITION. 59. Sum. The sum of two or more numbers is a number equal in value to them. 60. Addition. Finding the sum of numbers is called addition. 61. Addends. The numbers to be added are called addends. 62. The Mental Acts Involved. (1.) The mind com- pares two addends in respect of concrete denomination and order. (2.) The mind begins with one of the addends and from its number of units counts until the units of the other addend are used. The number reached is the sum of the two addends. Remarks. 1. An addition table is readily formed, the mas- tery of which enables the mind to give from memory the sum of any two addends within the limits of the table. 2. With the sum formed by the synthesis of two numbers, as above indicated, the mind may combine another number, and so may continue the act of combining one number with another so long as there are numbers to be added in a given case. 63. Principles. I. Only like numbers can be added together. Remarks. 1. Like numbers are abstract numbers having like units or concrete numbers having like unit-objects. And since a collection of objects can be named from a common at- tribute only, so only like numbers can be named together or added. 2. If unlike numbers are to be added together a common name or denomination must be found for them. They are then thought together under that name. 28 II. The sum is of the same order or denomination as the addends. III. The sum equals its addends. IV. The sum exceeds any of its addends. V. If the same numerical value be added to each of two equal values, the resulting sums are equal. 64. The Sign. A perpendicular cross (-f-) placed between two numbers indicates that they are to be added together. The sign is read plus. General Remarks. 1. In adding numbers of any order in the decimal scale, a sum exceeding 9 is often found. Since such a sum cannot be represented in the place in the written scale which corresponds to the order of units composing the sum, a part or all of the sum must be reduced to units of one or more higher orders in the scale before it can be uotated. Reduction ascending is in- volved. 2. The definitions, processes and principles stated above ap- ply equally well whether the addends be thought in the deci- mal, the fractional or the compound system of numbeis. Exercises in Addition. Remarks. For the exercises numbered in the left margin read across the page to the right. (1) (2) (3) (4) (5) (6) (7) (8) (9) 34 67 47 27 17 16 65 21 * (10) 76 98 58 39 19 27 47 18* (11) 26 46 74 66 28 94 86 31f (12) 38 78 23 72 30 89 75 481- (13) 46 96 17 87 48 67 49 59J (14) 71 84 29 93 85 86 87 29 (15) (16) (17) (18) (19) (20) (21) (22) (23) 3.5 26} 4.6 5 .1 3J 21* 4. 7 213 (24) 4.6 47i 3. 7 6 .7 5* 46| 8. 5 46.7 (25) 7.4 38 4. 1 8 .5 16 23i 9. 7 54.5 (26) 3.9 59| 5. 8 9 .4 14 48^- 6. 5 68.7 (27) 6.2 78|- 9 .9 6 .3 15J 56J 5.4 59.9 (28) 9.3 67i 9.8 5 .9 18| 87* 3.2 97.5 t (29) (30) (31) (32) (33) (34) (35) (36) 412 567 2678 4678 567.83 56789 54347 (37) '316 898 4513 8764 387.65 98765 45678 (38) 578 596 4678 9876 123.45 13759 94352 (39) 963 347 3542 6789 543.21 24680 24536 (40) 447 598 2345 8765 234.56 46'802 78579 (41) 567 678 5436 5678 654.32 20468 45678 (42) 324 345 7654 7654 345.67 57898 57890 (43) 568 234 4567 4567 765.43 85765 47875 (44) 718 432 8765 1234 456.78 45678 32567 (45) 678 561 5678 4321 876.54 57934 78579 II MULTIPLICATION. 65. Product. The product of two numbers is a number that sustains the same relation to one of them that the other does to 1. Remark. The term multiple is sometimes used for product. 66. Multiplication. Finding the product of two numbers is called multiplication. 67. Its Genesis. (1.) In addition it is not essential that the addends be compared in any respect except in that of denomination and order. They may, how- 30 ever, be compared in respect of equality. If the ad- dends are equal their addition may be called constant addition. (2.) If the sum of a constant addition be remem- bered so that it may be given when the number of equal addends and the value of each are known, the act is called multiplication. 68. The Mental Act. The act of synthesis involved is that of a constant addition previously performed, while the act called multiplication is but the recalling of the sum from memory. 69. Multiplicand. The multiplicand is usually de- fined as the number to be multiplied. 70. Multiplier. The multiplier is usually defined as the number by which the multiplicand is multiplied. Remarks. 1. If the multiplier be an integer the multiplicand is one of the addends of a constant addition while the multi- plier is the number of those addends. f 2. If the multiplier be a fraction the process of multiplica- tion consists in obtaining such part of the multiplicand as the multiplier is of 1. 71. Factors. The multiplicand and multiplier are called factors of the product. A factor of a number is one of its makers by mul- tiplication. 72. Principles. I. The multiplicand may be abstract or concrete. II. The product is of the same name and order as the multiplicand. III. The multiplier is an abstract number. IV. If both factors are abstract, they may be used interchangeably without affecting the value of the product. 31 V. If either or both factors be used in parts the sum of the partial products thus found equals the product of the factors as wholes. VI. The product sustains the same relation to the multiplicand that the multiplier does to 1. VII. If the multiplier is 1 the product equals the multiplicand. VIII. Multiplying either factor by a number mul- tiplies the product by that number. IX. If two equal values be respectively multiplied by the same number the resulting products are equal. X. A number is multiplied by multiplying one of its factors. Remark. The following principles are seen to be true in the light of definitions yet to be given. a. A number expressed in the decimal system of notation is multiplied by any power of 10 by removing the decimal point as many places to the right as there are units represented by the index of the given power of 10. b. Dividing any factor of a product by a number divides the product itself by that number. c. Dividing both factors of a product by the same number divides the product by the square of that number. d. If one of two factors be multiplied and the other be divided by the same number, their product is not changed. 73. The Sign of multiplication is an oblique cross (X). It is read times or multiplied by, and is placed be- tween two factors whose product is required. 32 Remarks. 1. If both factors are abstract it is immaterial which reading be given to the sign ; if, however, one of the fac- tors be concrete, it is thereby made the multiplicand, [Prin III,] and if it precede the sign, the sign is read multiplied by, while if the multiplier precede the sign,the sign is read times. 2. If a numerical value be expressed within a parenthesis or other sign of aggregation and another number be written with- out the sign but not separated from it by any sign, the sign of multiplication is understood between them. 3(4_|_2)=:3X (4+2)=l& 3. In effecting a multiplication a product exceeding 9 is often found. Since such a product cannot be repiesented in the place in the written scale which corresponds to the order of units composing the product, it becomes necessary to reduce to units of a higher order or orders so much of the product as is thus re- ducible without involving fractions. The reduction employed is reduction ascending. [Exercises in multiplying numbers thought in the different systems.] in COMPOSITION. 74. The continued product of several factors is found by multiplying the product of two of them by a third, the product of the three by a fourth, and so on until all the factors are used. 75. A Composite Number. The product of two in- tegral factors, each greater than 1, or the continued product of several such factors is a composite number. 76. Composition. The process of forming a compo- site number is called composition. 77. A Prime Number. A number that cannot be formed by composition is called a prime number. 78. Numbers are relatively prime if they have no common factor. 79. A Common Multiple of numbers is a product of which each of them is a factor. 80. The Least Common Multiple of numbers is the least product of which each of them is a factor. 81. A Common Measure of numbers is a factor that can be used in the formation of each of them. 82. The Greatest Common Measure of numbers is tLe greatest factor that can be used in the formation of each of them. 83. Principles. I. Both prime and composite factors may be used in composition. II. A multiple is the product of all its prime factors. III. A common multiple of numbers has in its composition the product of all the prime factors of each of the given numbers. IV. The least common multiple of numbers is the product of all the prime factors of each of the numbers, each factor being used in the composition the great- est number of times it occurs in the composition of any one of the numbers. V. A factor of a number is a factor of any multi- ple of that number. VI. A common factor of numbers is a factor of their sum. VII. The greatest common factor of numbers is the product of all their common prime factors. [Exercise in forming composite numbers, common multiples, etc.] 34 IV INVOLUTION. 84. Power. A power of a number is the number itself or the product of the number by itself one or more times. 85. Involution. The process of forming a power is called Involution. 86. Root. One of the equal factors used to form a power is called a root. 87. Second Power. The product of two equal fac- tors is called the second power, or square of either of them. 88. Second Root. Either of the two equal factors that compose a second power is called the second, or square root of the given power. 89. Higher powers are formed by the composition of a greater number of equal factors. Every such power is named by the ordinal of the number of equal factors used in the composition of the power. 90. A root is called the third, fourth, etc., if it be one of three, four, etc., equal factors that compose a power. 91. Any number is called both the first power and the first root of itself. A first power can enter into a synthesis with its equal and thus become a component of a higher power but, is itself not formed by involu- tion. 92' The index of a power is a small symbol of number written at the right and above a given number and indicates that the number is one of as many equal factors as there are integral units expressed by the index. Thus 2 3 =8, 6 2 ^36, etc. 35 93. Table of Squares from I 2 to 25 2 . 2 = 4 32^ 9 15 2 =225 41= 16 16 2 -=256 &= 25 17 2 =289 62 _ 36 18 2 =324 7 2 = 49 19 2 =361 32^ 64 20 2 =400 9 2 ^ 81 21 2 =--441 10^=100 22 2 25*=626 94. Table of Cubes from I 3 to 9 3 . 2 3 = 8 3 3 =27 ? 3; 4 3 =6 4 8 3 =512 9 3 =729. 95. Rules for Squaring Numbers. I. To Square a Number ending in . 1. Square the integer. 2. Add \ the integer. 3. Add -fa II. To Square a Number ending in \. C i. Square the integer. a I 2. Add the integer. ( 3. Add i. 6. Multiply the integer by the integer next greater and add J. III. To square a number ending in . 1. Square the integer. 2. Add f of the integer. 3. Add ^. IV. To square a number ending in 5. 1. Square the tens. 2. Add the tens. 3. Annex 25. 6. Multiply the simple value of the tens by the number next greater and annex 25. V. To square a number between 25 and 50. 1. Take 25 from the number. 2. Take the difference from 25. 3. Square the remainder. 4. Add the first difference as hundreds. VI. To square a number between 50 and 75. 1. Take 50 from the number. 2. Add the difference to 25. 3. Call the result hundreds. 4. Add the square of the difference. VII. To square a number between 75 and 100. 1. Take the number from 100. 2. Take the difference from the number. 3. Call the result hundreds. 4. Add the square of the first difference. VIII. To square a number ending in 25. 1. Square the hundreds. 2. Add J the hundreds. 3. Call the result ten-thousands. 4. Add 625. 37 i IX. To square a number ending in 75. 1. Square the hundreds. 2. A.dd f of the hundreds. 3. Call the results ten-thousands. 4. Add 5625. X. To multiply a number of two orders by 11. Think the sum of the terms of the multiplicand between them : As 34 X 11=374. 54x 11=594. [Exercises.] 96. Remarks on Synthesis. 1. Numerical synthesis classifies itself under four heads, viz , addition, multiplication, composition and involution. There is, however, but a single method of synthesis, and that is addition. In multiplication, including composition and in- volution, a mm is remembered, this sum having been previously found by the synthesis called addition. 2. In addition two numbers are combined by one impulse of the mind without regard to the equality of the numbers. 3. In multiplication a given number of equal numbers may be thought as combined at once. The number of equal num- bers is not limited to two, but may be any number whose sum, found by constant addition, can be given immediately from memory. This remark holds only with an integral multiplier. 4. In composition a definite number of factors are used in continued multiplication. 5. In involution a definite number of equal factors are used in continued multiplication. 38 The Analysis, or Separation of Numbers. Remark. Since a number may be obtained by adding to- gether any two parts which form it, it follows that the number may again be resolved into its parts. I SUBTRACTION. 97. Difference. The difference between two num- bers is a number which added to one of them will make a sum equal to the other. Remark. In Arithmetic the term difference may be denned as the numerical excess of one number over another. 98. Subtraction. Finding the difference between two numbers is called subtraction. Remark. Subtraction may be denned as taking a part of a number from the number. 99. The Minuend. The sww involved in subtraction is called the minuend. Remark. The minuend is often denned as the number to be diminished by the withdrawal of a part of it. 100. The Subtrahend. The known part of the minuend is called the subtrahend. Remarks. 1. The subtrahend is usually defined as the num- ber to be subtracted. 2. The term remainder is often used to designate the part of the minuend that is left after the withdrawal of the subtra- hend. The number so designated is, however, the difference between the minuend and subtrahend. 101. The Mental Acts. 1. If the sum of two num- bers and one of them be known, the other is found by the process called subtraction. The mental act con- sists in presenting the required part from memory, or it may consist in counting from the given part to the given sum. 39 2. If the difference between two separate numbers be required, the mind compares the two numbers in respect of concrete denomination and order. If the numbers are found to be similar, the mind proceeds to withdraw or think away from the greater, a part which equals the less. The number remaining is the excess of the greater over the less, and is, therefore, their difference. 102. The Sign. The sign of subtraction is a single dash, ( ) placed after the minuend and before the subtrahend in an indicated subtraction. 103. Principles. I. Only like numbers are used in subtraction. II. The sum of the subtrahend and difference equals the minuend. III. If the minuend and subtrahend be equally increased the difference between the yums thus ob- tained equals the difference between the minuend and subtrahend. IV. If the same value be taken from two equal values the remainders are equal. V. It either or both minuend and subtrahend be used in parts, the partial remainders combined equal the entire remainder. Remarks. 1. Since the minuend is a sum, it is greater (in Arithmetic), than the subtrahend. It sometimes occurs, how- ever, that there is a less number ol units of some order in the minuend than of the same order in the subtrahend. In such case the minuend must be prepared before the subtraction can be effected. This preparation consists in reducing a unit of the order next higher in the minuend to units of the required or- der and combining them with the units of that order. If there be no units of the order next higher in the minuend, the work of reduction must begin at the first order up the scale in which 40 numerical value is thought. A unit of that order is reduced to units of the next lower ; one of the units resulting from this reduction is then reduced to units of the order next lower, and so on until the number of units of each order in the minuend equals or exceeds that of each order in the subtrahend. The subtraction is then readily effected. The reduction involved is reduction descending. 2. The reduction mentioned in remark 1, may be avoided by effecting the subtraction in the light of Principle III, adding 10 units of the deficient order in the minuend to the units of that order, and then compensating this addition by adding 1 unit of the next higher order to the subtrahend. [Exercises in Subtraction.] II DIVISION. 104. Quotient. The quotient of one number by an- "other is a number that sustains the same relation to the first number that 1 does to the second. Remark. The quotient of one number by another is the fac- tor which, used with the second number, will produce the first. 105. Division. Finding a quotient is called division. Remark. In the light of remark under Quotient, division may be defined as finding one of two factors of a given product when the other factor is known. 106. Dividend. The number to be divided is called the dividend. Remark. The dividend is the given product of which the divisor is the known factor. 107. Divisor, The term divisor is usually defined as the number by which the dividend is divided. Remark. I. The divisor is the factor given with the dividend to determine the quotient. 2. A particular problem may beGiven the dividend and quotient to find the divisor. This probiem is solved by divid- ing the dividend by the quotient. The quotient of a preceding division thus becomes the divisor in the given problem. 108. Remainder. The term remainder is applied to a part of a given dividend that may remain undivided in any case. 41 The Genesis .of Division. 109. (1.) If a given subtrahend be taken from a given minuend, and again be taken from the remain- der, and again be taken from the second remainder, and so on until the given minuend is exhausted or gives a remainder less than the constant subtrahend, the several subtractions viewed together are called a con- stant subtraction. (2.) If, when the minuend and constant subtra- hend are known the mind gives from memory the number of subtractions necessary to exhaust the given minuend; or, if the minuend and the number of sub- tractions that can be made are known, and the mind gives from memory the constant subtrahend, the act is called (3.) If, when a product arid one of two factors that produce it are known, the mind gives from memory the other factor the act is called division. 110. The Mental Act. (1.) The mental act which effects a division is the presentation (from memory) of the quotient when the dividend and divisor are known. (2.) The act may be a memorized constant sub- traction; or, it may be the recalling from memory of one of two factors when the other and their product are known. Remark. In effecting the division of a decimal or a com- pound number, if the divisor be numerically greater than the number of unitp of the order or denomination to be divided, the latter number must be reduced to units of a lower order or de- nomination before the division can be effected without involv- ing a fractional quotient. Reduction descending is involved. 42 III. Principles. I. The product of the divisor and quotient equals the dividend. II. If the dividend be divided in parts the sum of the several partial quotients obtained is the entire quotient. III. If the dividend be divided by the factors of the divisor used in continued division, the final quo- tient is the quotient of the dividend by the entire divisor. Remark. In applying this principle a remainder may occur upon dividing by one or more of the factors of the divisor. These partial remainders do not constitute the ultimate or true remainder. The following is a method for determining the true remainder. Example. Divide 1377 by 294, using the prime fac- tors of the divisor, and determine the true reniainder. Solution. The prime factors of 294 are 2, 3, 7, 7. 2 Partial Remainders | 1377 True Remainders. 8 I 688 1 7 I 229 1 2 ~j 32 5 30 ~~4... 4 . 168 201 Explanation. a. 1. The entire dividend divided by 2 gives a quotient of 688 and a remainder of 1. 2. 688, which is approximately \ the entire divi- dend, divided by 3 gives a quotient of 229 and a re- mainder of 1. 43 3. 229, which is approximately J of the entire dividend, divided by 7 gives a quotient of 32 and a re- mainder of 5. 4. 32, which is approximately -^ of the entire dividend, divided by 7 gives a quotient of 4 and a re- mainder of 4. 5. 4, the final quotient is approximately the entire dividend, or the part required. b. 1. If 1 remain upon dividing the dividend, 2 times 1, or 2, would remain upon dividing the entire dividend. 2. If 5 remain upon dividing of the dividend, 6 times 5, or 30 would remain upon dividing the entire dividend. * 3. If 4 remain upon dividing^ of the dividend, 42 times 4, or 168, would remain upon dividing the en- tire dividend. 4. We thus find that upon dividing the entire dividend we should have remaining l+2-|-30-}-168j or 201, as the ultimate, or true remainder. The accuracy of this result may be tested by di- viding the entire dividend by the divisor as a whole. IV. The quotient sustains the same relation to the dividend that 1 does to the divisor. V. If the divisor is 1 the quotient equals the divi- dend. VI. If a division require the number of times that one number is contained in another, the divisor and dividend are like numbers and the quotient is abstract. 44 VII. If a division require one ot the equal parts of a number, the dividend and quotient are like numbers and the divisor is abstract. VIII. If two equal numerical values be divided by the same number, the resulting quotients are equal. 112. General Principles of Division. Remark. The following six principles are called gener; .034i 130. Common. A fraction whose fractional unit is other than a decimal part of the unit one, is called a common fraction. 131. This classification of fractions is often based upon the method of notation used in expressing the fractions; those expressed in the decimal notation being called decimal fractions, and those expressed in the fractional form being called common fractions. 132. With 1 as a basis, or standard of comparison, fractions are classified us proper and improper. 133. Proper. A fraction whose value is less than 1 is called a proper fraction. 134. Improper. A fraction who^e value equals or exceeds 1 is called an improper fraction. Remarks. 1. The primary idea of a fraction is a number of the equal parts of a unit. The classification of fractions as proper and iir. proper is thus seen to be inconsistent with the primary idea of a fraction. 2 The definition of a fraction as a' number of like fractional units is based upon a secondary idea, viz : A number of like parts of any number of like units. Under this definition any number of like fractional units is a fraction. 57 3. The two definitions of a fraction that we have given [Art. 125,] are fairly representative of the definitions found in tlie text books on Arithmetic. Under neither of these defini- tions is Jiere ground for classifying fractions as proper and im- proper. 135. On the basis of form, fractions are classified as simple, compound and complex. 136. Simple. A simple fraction is defined as a frac- tion each of whose terms is a single integer. 137. Compound. A compound fraction is defined as a fraction of a fraction. \ Remark. The simplification of a so-cailed compound frac- tion is effected in the light of principles which govern the mul- tiplication of one fraction by another, f of f does not express a class of fractions, but simply indicates that | is to be multi- plied by f. [See Kemark 2, under Art. 70.] 138. A complex fraction is defined as a fraction having a fraction in one or both of its terms. Remarks. 1. A complex fraction is read as an expression of division : e. g. |^ is read 2 -5- 3f and ^ is read \ -5-. In the light of the definition of denominator, the first of the above so-called complex fractions is derived from the division of a unit into 3| equal parts. The mind sees at once the im- possibility of such a division. An examination of the second so-called complex fraction given, renders still more manifest the absurdity of calling these numerical values fractions. 2. Since a simple fraction is expressed by a form in distinction from compound and complex, it disappears upon the disappear- ance of these as classes of fractions. 3. If it is found convenient to use the terms proper, im- proper, simple, compound and complex to distinguish certain phases of fractional notation or indicated processes, it may be well to retain those terms; but they are certainly not necessary to name any of the essential thought elements of a fraction. 58 139. General Principles. Remark. A fraction may be considered as a case of division. The numerator being the dividend, the denominator the divisor and tho fraction itself the quotient. The general principles of division become, therefore, the general principles of fractionsby a change of terminology. I. If the numerator be multiplied the fraction is multiplied by the same number. II. If the denominator be multiplied the fraction is divided by the same number. III. If both terms of a traction be multiplied by the same number greater than 1, the fraction is reduced to smaller fractional units but is not changed in value. IV. If the numerator be divided the fraction is divided by the same number. V. If the denominator be divided the fraction is multiplied by the same number. VI. If both terms of a fraction be divided by the same number greater than 1, the fraction is reduced to larger fractional units, but is not changed in value. VII. Fractions having a common denominator are to each other as their numerators. Remark. This principle may be thus stated : Like parts of numbers are to each other as the numbers themselves, e. g. the relation of f to J is the same as that of 2 to 4 ; i. e. $ of 2 bears the same relation to of 4 that the whole of 2 bears to the whole of 4. VIII. The numerator is as many times the value of the fraction as there are units in the denominator. Remark. This principle is seen in the relation of a fraction to division. The numerator is dividend and hence is the pro duct of the denominator (divisor) and the fraction (quotient.) A^product is as many times either of the two factors which compose it as there are units in the other. 59 140. Reduction of Fractions. Remarks. 1. For definitions and kinds of reduction, seepage 24. 2. In the light of general principles, I, II, IV, V, pupils will readily multiply or divide a fraction by an integer. The signs v and /. are convenient to use in some of the written forms. The former is read "since" or "because," and the latter is read "hence" or "therefore." 141. Reduction Descending. CASE I. An integer or mixed number to a fraction. Example. Reduce 3 to 8ths. Writtenform 3 =f- Thought form. 3x8 24 T\/8~~s' 3=f; and by multiplying both terms of f by 8 we have ^. Hence 3=V Remark. In reducing a mixed number to a fraction, the inte- ger is reduced to the denomination of the fractional part by the above method and the fractional part is then added. 142. Other Forms. Example. Reduce 3 to 8ths. =3'times = 4 . C *' l=f a] 3=3 t: ( /.3=V. 1=8 times , 38 " I 1=|. I 8=V- Remarks on c. 1. Make 1 = f > and then multiply both mem- bers of the equation by 3. We thus find that 3 = 60 2. In any similar example make 1 the first member of the first equation ; and for the second member take the equivalent of 1 in fractional units of the required denomination. Next multiply both members of the equation by the integer to be re- ed. __ HX4 12 Exercises. Perform each of the following reductions : 8 to 15ths; 19 to 4ths ; 8 to 7ths; 6 to 3rds; 5 to halves. [Give additional exercises.] Reduce each of the following to a fraction ; 10J ; 14fc 5i; 3; 4; 7J; 3; 2.5; 3.04; 5.2; 6.7. 143. CASE II. A fraction to smaller fractional units. Example. Reduce to 12th s. Written form. Thought form. . In the light of principle III, may be reduced to 12ths by multiplying both its terms by such a number as will produce 12 for the de- nominator of the resulting fraction. Both terms off multiplied by 4=^-. .-. %=-%. Example 2. Reduce 1 to a decimal fraction. Written form. 7 X 125^^75 = 875 Thought form. 8X125 1000 l n the light of Prin. Ill, * is re- duced to a decimal by multiplying both its terms by such a number as will produce a decimal denominator for the resulting fraction. Both terms of f multiplied by 125-,%%. /. 1=.875. Remarks. 1. A decimal fraction may be transferred from the fractional to the decimal notation by omitting the written de- nominator and supplying the decimal point. 2. A decimal fraction may be transferred from the decimal to the fractional notation bv writing the proper denominator under the written decimal and removing the decimal point from the written numerator. 61 3. The change of notation thus effected is not properly a re- duction, for the size and number of fractional units in the frac- tion remain unchanged. 4. If a decimal fraction is to be reduced to smaller decimal units, the application of the above form is best shown by ex- pressing the given decimal in the fractional notation. 144. Other Forms. Example. Reduce I to 12ths. 1=2 times A=A- .'. *=-* Written form Thought form. For the first equation we take l=|f . Equation (2) is found by dividing both members of equation (1) by 3. Equation (3) is obtained by multiplying both members of equa- tion (2) by 2. We thus find that f = Example 2. Reduce .2 to thousandths. p. -1=1.000, .1=.! of 1.000=.100; c ' } and .2=2 times .100=.200. t /. .2=.200. [Exercises.] Perform the following reductions, writing all the decimal fractions in the decimal notation: I to tenths ;Jg- to 24ths; f to27ths; f to 21st s;. 5 to lOOths ; .5 to lOOOths; f to 15ths ; 3i to 4ths; 3.2 to lOOths; f to20ths; f to lOOths; f to ISths; 3.2 to lOOOOths ; | to 54ths. [Give additional exercises.] 62 145. CASE III. Fractions to a common and to the least common denominator. Remark. This case is one of reduction descending in which the reduction is effected as in Case II. Definitvms. a. Fractions have a commoYi denom- inator if composed of like fractional units. b. Fractions have their least common denomina. tor if composed of the greatest possible like fractional units. Remarks. 1. Before reducing fractions to equivalent frac- tions having the 1. c. d. it is necessary that the terms of each fraction be relatively prime. 2. In case II, the denominator of the fraction resulting from the reduction is a multiple of the denominator of the given fraction. From the nature of the method employed in effect- ing the reduction, this must always be the case ; hence a com- mon denominator of fractions must be a common multiple of the denominators of the given fractions. Principle. The least common denominator of given fractions is the 1. c. m. of their denominators. Exaufple. Reduce f , f and -| to equivalent fractions having their least common denominator. Thought form. (1.) Prin. The 1. c. d. of given fractions is the 1. c. m. of their denominators. (2.) The 1. c. m. of 3, 4 and 6 is 12, hence each of these fractions must be expressed in 12ths. (3.) In the light of general principle III, each of these fractions may be reduced to 12ths by multiply- ing both its terms by such a number as will produce 12 for the denominator of the resulting fraction. (4.) Both terms off multiplied by 4=-^. f 3-. 63 (5.) /. f, I and reduced to equivalent fractions having their 1. c. d. equal, respectively, T 8 Y , T 9 ^ and i-|. Remark. Examples in this case may be solved by stating (1) and (2) as in the above thought form, and then reducing to the required denomination by any of the forms given under Case II. [Exercises.] Reduce to a common and 1. c. d. the following : if t; *,*,*;! f t; *,if; i, M; M .4, ^; f , .5, i ; .06, |, i, .05 ; f , M> t ; 2 J, f , f, 4i ; 3|, 2.5, 5J. 146, Reduction Ascending. CASE I. A fraction to an integer or a mixed number: or to the decimal scale. Example 1. l -- what integer or mixed number ? Written form. Thought form. 17-^-5 3f 02. I Since a fraction is reduced to larger 5 -=-5 1 F I units by dividing both its termfc by the same number, we may reduce y to integral ones by dividing both its terms by 5. The result, 3f-f-l,=3. Remark. Dividing both terms of a fraction by its denomina- tor reduces the fraction to the denomination of the unit 1. Example 2. Eeduce -fo to a decimal fraction. Written form. Thought form. ~*~^=== 08 400--4 100 is reduced to the denomina- tion hundredths by dividing both its terms by 4; the result, ^^=.08, a decimal fraction. Remark. The reduction of a common fraction to a decimal fraction may be effected by either reduction. The guiding thought is to multiply or divide both its terms by such a num- ber as will give a decimal denominator to the resulting fraction. 64 147. Other Forms. Example 1. y= what integer or mixed number? f U=17 5. a. I 175=31. !>. 1 y==as many 1's as 5 is contained times in 17, or 3f. Example 2. Reduce f to the decimal scale. Form, Remark. The denominator of a decimal fraction is a power of 10, the prime factors of which are 2 and 5. It follows, there- fore, that if any fraction (in its lowest terms) have in its de- nominator a factor other than those of 10, such fraction cannot be reduced to a decimal. A common fraction is reducible to a decimal if the denominator of the given fraction is divisible by 2 or 5, or by 2 and 5 and by no other prime number. If a fraction, in its lowest terms, have in its denominator a prime factor other than those ot a decimal denominator, a repetend or cir culate will result upon attempting to reduce the fraction to the decimal scale. ' Exercises. Reduce to integers or mixed numbers the following : JL6 JLJL 21 37 _2 3. i 3_9 4JL . 2.5 . JM 5 . 100 . 5 > 6 > 3' 5 > > 2 > 3 > 13? 7 ' 11 ^T ' 3 ' 2.5.3.. Reduce each of the following to the decimal scale or to a complex decimal. 11. 13. J.5. 3-7. 9.1. .2.1. 5.. 1. 2- 3. 1 . 5 . T (f > 16' 16' TTJ" > 25 ' ^TT >3' 3 7? 6'T> T' T ' 1 2" ' T2" ' 1 . 3 . 1 2 6 1'S ' T3 ' 15? 15? 16' 148. CASE II. A fraction to larger units, or lower terms. Example. Reduce -| to thirds. Written form. Thought form. 6~3_2 I In the light of Prin. VI, f may be re- 9-i-3 3 I duced to 3rds by dividing both its terms by such a number as will give 3 for the denominator of the resulting fraction. Both terms off divided by 3, Remark. A fraction is reduced to its lowest terms by divid- ing both its terms by their g. c. d. Exercises. Reduce each of the following to lower and lowest terms. **' 16.144- 10 f. 1006- 325. 117 . 7 8.25.48- 3TJTFT2' 3T ~5 ' 4TRT4 ' 6TO~'TT46> "2"! > "SIT > 4~0~ ) ^~0~ ? 1246 144 T6"8T> 1728' 149. General Remarks on Reduction of Fractions. 1. General principle Hi provides for the reduction of a frac- tion to smaller units; while general principle VI provides for the reduction of a fraction to larger units. 2. From the foregoing presentation of reduction of fractions we find that the several cases are readily classified under one or the other of the two kinds of reduction ; and that each ca&e in- volving reduction descending may be effected in the light of general principle III, while each case involving reduction as- cending may be effected in the light of general principle VI. 150. Addition and Subtraction of Fractions. Principle Only like units can be added. Principle Only like units can be used in subtrac- tion. Remark. In adding or subtracting fractions it is to be ob- served that the numerators are the numbers with which we deal while the denominators only give name to those numbers. Example. Add f, and f. Solution. Since only like units can be added, these fractions must be reduced to a common denom- inator before adding. Written form. 2 __ 8 3 _ 9 5 __ 10 T 12; 4 ~12" [ 6~W _8_+J__ | _10_ = 2^ ==2 i 12 12 ^12 12 Remarks. 1. A form for subtracting one fraction from an- other is similar to that for addition. 2. If mixed numbers are to be added or subtracted, the inte- gers and the fractions may be operated upon separately and .the results combined, or, the mixed numbers may be reduced to fractions and the result found by the form given for examples in addition, and by a similar form for examples in subtraction. 3. It is not essential that fractions expressed in the decimal notation be reduced to the same fractional denomination be- fore adding or subtracting. It is to be observed that only like orders of units can be added or subtracted. Exercises. (1.) Add |, f, f ; f, i, A; |, , A, i; |, *, i, *, f ; T \, |; .2, 4, i, |; 3%, ^ ^; f ^, 2, 5, 6*. (2.) Perform the operation indicated in each of the following : f-i;i-f; f-i; A -I; A- A; 3.4 -ij ; ^ - f; 2 T V - A; 3.15 - A; 6J - T V; 25.04 - 21.002; 364.006 162.1; 32.5 1.0001. 67 Multiplication of Fractions. 151. CASE I. To multiply by an integer. Example 1. Multiply T \ by 3. Thought form. Since a fraction is multiplied by multiplying its numerator, f\X3=^f. Example 2. Multiply fV by 2. Thought form. Srnoo a fraction is multiplied by dividing its denominator, Remark. . If the product should not be in its lowest terms or if it be reducible to an integer or mixed number it should be reduced. [Exercises.] Multiplicands:^] f; f ; f; T 2 ^; ^; T 4 5; fV; 84; jfo 4J; 51; 15fc 21f. Multipliers : 4, 5, 3, 2, 6, 7, 8, 10, 12, 15. 152. To multiply by a fraction. Example 1. Multiply 6 by * [6X1=6. 6= of 6=2. First farm -{ $=2 times 2=4. I The multiplier =j of 2, 6X2=12. 6Xiof 2=4 of 12=4. ' L/ . 6XI-4.' Principle. The product sustains the same relation to the multiplicand that the multi- plier does to 1. Third form In this exam P le tn ^ multiplier is of 1 ; hence the product is f of 6. if of 6=2. \ of 6=2 times 2=4. >. 6XI-4. r [Give principle and statement as in third form.] Fourth form of 6=J of 2 times 6, or 12, 4- of 12=4. V Written form. (1.) 6=6. (2.) X3=2. (3.) 6X1X3=6X2. (4.) 6X1=2X2=4. .-. 6XI-4. Thought form. 6=6. Also the multiplier multiplied by its de- \ nominator equals its numerator. Multiplying together the corresponding members of these equations gives equation (3). Dividing both members of equation (3) by 3, gives equation (4). The first member of equation (4) consists of the two factors whose product is required, while the second member is the result sough t. Fifth form( Example 2. Multiply f by f Remark. The forms of solution for this example do not dif fer from those given for example 1. Those forms are, however, applied to the solution of example 2. ix 1=1. f The multiplier is | of 5. T/iird form Principle. The product sustains the same relation to the multiplicand that the mul- plier does to 1. In this example the multiplier is f of 1 ; hence the product is f of f. I of !=A- f of f =5 times Y V=fl- [Give principle and statement as in third form.] Forth/arm ( f of f =4 of 5 times |, or - 1 /. of = 70 Written f win. (1.) 1X4=3. (2.) (3.) (4.) Fifth form /. etc. Thought form. The multiplicand multiplied by its denom- inator equals its numerator. <, The multiplier multiplied by its denomin- ator equals its numerator. Multiplying to- gether the corresponding members of these equations gives equation (3) ; and dividing both members of equation (3) by 28 gives equation (4). The first member of equation (4) consists of the two factors whose product is required, while the second member is the product \sought. 153. General Remarks on Multiplication of Fractions, 1 Mixed numbers may be reduced to fractions before multi- plying; or the multiplicand may be multiplied by the integral and the fractional parts of the multiplier (used separately) and the partial products combined. 2. If one or both factors be decimal fractions expressed in the decimal notation, any of the given forms may be used. The second form is, perhaps, the best to use in the multiplica- tion of decimal tractions. Example. Multiply 2.5 by .025. Form. The multiplier equals .001 of 25. 2.5X25=62.5. 2.5 X. 001 of 25=.001 of 62.5=.0625. -. 2.5X.025=.0625. 71 3. The product of one fraction by another may be obtained by multiplying together the numerators of the factors for the numerator of the product, and the denominators of the fac- tors for the denominator of the product. [See rule in any good text book ] Exercises. Multiplicands, f j 8 ; & ; 3| ; f ; j ^ ; f ; f ; 21 . 16; 12; .05;. 3; . 025; 3.2; 21.004; 8 J ; 4.07; t; .09 j .5 ; .008 ; 3 J ; 41 ; 6.7 ; 8.09 ; 6.3 ; 32.04 ; 2.004 ; 325.046 j 4.0064 ; 31.75. Multipliers, f ; f; t; ; f ; f; f ; j. 2.5 >5; .3;.7; .04; .005; .045; .008; ; 1.5;^; 2.3; 4.05; 6.25 ; 15.5 ; 7.5 ; .47 ; 31.056 ; .0045. Division of Fractions. 154. CASE I. To divide by an integer. Example 1. Divide |- by 3. Thought form. Since a fraction is divided by di- viding its numerator, -f-i-3=-f-. Example 2. Divide ^ by 5. Thought form. Since a fraction is divided by mul- tiplying its denominator, -r-5-f^. Exercises. Dividends. l;f;|; f; f; : -|j f; #;;; H; ft; H; - 2 ; - 25 ; 3 - 25 ; 2 - 004 ; 36 - 78 ; 47 - 3 ; 17 - 08 ' 6.25; .0625 ; 3$ ; 5| ; 24|; 14} ; 16f ; 26J; 32|. Divisors. 2, 3, 4, 5, 6 3 7, 8, 9, 10, 11, 12, 13, 14, 15. 72 155. CASK II. t To divide by a fraction. Example 1. Divide 5 by f . f 5--l=5. 5-1-4=7 times 5=35. Sec 1 nd form Third form. f The divisor equals | of 3. 1 5-^1 of 3=7 times f ^=llf. --rr=5 times i=4A =llf . Example 2. Divide f by f . Remark. For the first three forms see those given for the preceding example. Fourth form. \ f f-J-. Fifth form. One fraction may be divided by another by dividing the numerator of the dividend by the numerator of the divisor and the de- nominator of the dividend by the denom- inator of the divisor. If the terms of the dividend be not re- spectively divisible by the corresponding terms of the divisor, they may be made so by multiplying both terms of the dividend by the product of the terms of the divisor. Division of fractions is thus shown to be \the reverse of multiplication of fractions. 73 156. General Remarks on Division of Fractions. 1. If the divisor is a mixed number, it is, perhaps, better to reduce it to a fraction before dividing. 2< If the divisor is written wholly in the decimal notation, the above forms are as applicable as though it were written in the fractional notation. The second form is preferred. Example. Divide .0625 by 2.5. Form. The divisor is .1 of 25. .0625-f-25=:.0025. .0625-f-.l of 25=10 times .0025^.025. .-. ,0625--2.5=:.025. Exercises. Dividends. f ; f; I; $ ; f ; f ; f ; .5 ; .25 ; .025 ; 8.4 ; 3.2 ; 8.08 ; 10J ; 3fr ; 5* ; 4f ; 6* ; 34.567 ; 325.568 ; 354.56 ; 1.2342; 56.67895. .7 j .5 ; .3 ; .07 ; .002 j .626 ; 1,256 ; 3.4467 ; 2 ; 3i ; 4f ; 5| } 157. Exercises Involving Fractions. 1. 2 bu.=rwhat part of 3 bu.? 2. $4= what part of $10? 3. 15 apples = what part of 50 apples ? 4. ^ acre = what part of 4 acres ? 5. f of a gallon ='what part of 5 gallons ? 6- f of a bu.= what part of 10 bu ? 7. f of an orange = what part of f of an orange ? 8. f== what part of f? 9. f= what part of I? 74 10. $ = what part of | ? 11. ^= what part of ? 12. f=what part of ? 13. How much water will fill 4 tubs if each tul) holds 5J gallons? 14. What cost 9 apples at 1 \? each ? 15. If 81b. of sugar sell for $1, what is the price per pound? 16. At $6 per cord, required the cost of I ot a cord of wood. 17. Kequircd the cost of -f- Ib. of peaches at 35/ per pound. 18. If coal is $3 per ton, required the cost of 4-f tons. 19. If a train run 15 miles per hr., how far will it run in 3J hours? 20. At $| per bushel, required the cost of of a bushel of corn. 21. What cost f of a gallon of syrup, at %\\ per gallon ? 22. If a man cut f of a cord of wood in a day, how much can ho cut in f of a day ? 23. A boy divided ^ of a bushel of apples among 4 playmates; what part of a bushel did each receive? 24. A man had i of a barrel of pork and sold f of it; what part of a barrel remained ? 25. A man lost of his money and found \ us much :IH he had after his loss; what part of his original sum had ho then ? 26.^If 1 yard 4 of ribbon cost $^, how muny yards can bo bought tor 75 t 27. How many yards of cloth will $10 buy at $2-i por yard ? 28. If 2 men can do a piece of work in 4|- days, how long will it take 8 men to do it ? 29. What cost 3J boxes of oranges, if 2 boxes cost $9? 30. What cost 30 bushels of corn, if 3i bushels cost $1.20? 31. If a bushel of wheat cost $, what cost f of a bushel ? 4.2. If a bushel of wheat cost $ , how many bushels can be bought for $5.20 ? 33. A girl divided 10 apples among her companions, giving to each f of an apple ; how many companions had she? 34. 4=f of what number? 35. ^ of 14=f O f w h at num bcr? 36. f of 36= of what number ? 47. I of $40=^ of the cost of a horse ; required its cost? 37. After spending^ of his money, John had $42 re- maining ; how much had he at first ? 39. Iff of an acre of land be worth $15, what are 12 acres worth ? 40. A boy sold lemons at the rate of 6 for 8 cents ; how much did he receive for 3 lemons? For 8 lemons ? For 12 lemons? 41. If f of a yard of silk cost $3i ; what cost 4J yards ? 42. If of a yard of cloth cost $, what cost f of a yard? 76 43. A man gained $15 by selling a watch for 1^ times its cost ; required its cost. 44. Mary, after losing i of her flowers, had but 3 re- maining; how many had she at first? 45. If to $ the cost of John's coat $10 be added, the sum will be $21 ; required its cost. 46. If to | of William's age 8 years be added, the sum will be 1^ times his age ; how old is he ? 47. Two men hire a wagon for $9 ; A uses it 7 days, and B uses it 2 days; what should each pay? 48. John and James bought 22 apples for 11 cents ; John paid 7 cents while James paid 4 cents ; how many apples should each receive ? 49. Anna has 5 pinks more than Ruth, and together they have 19 ; how many has each ? 50. 4 boy said that 4 is 3 less than \ his number of marbles ; how many has he? 51. 10 years are 6 years more than f of John's age ; how old is he ? 52. If A and B do ^ of a piece work in a day, how long will it take them to do the entire work ? 53. George can plow a field in 8 days, and Henry can plow it in 12 days ; how long would it take them to do the work, working together ? 54. If % of a barrel of flour cost $4f , what cost of a barrel? 55. If A can do f of a piece of work in a day, how much can he do i 2 days ? 56. What cost 6 bushels of clover seed, if 2 bu. cost $12f? 77 57. If .3 of a pound of coffee cost 9/, what cost f of a pound ? 58.* A has $13 which is f of twice as much as B has . how much has B ? 59. A horse is sold for $60 which is f of of its val- ue; required its value. 60. Henry and George bought 30 nuts; Henry paid 18 cents and George paid 12 cents; how should the nuts be divided? 61. A man failing in business can pay 40 cents on the dollar ; what part of his debts can he pay? 62. i=f of what number ? 63. A has $3 more than B ; and both together they have $71; how many dollars has each? 64. f of twice what number? 65. If f of a box of berries cost $f-, what cost I of a box? 66. What is the number if its i increased by 10 equals 21 ? 67. Required the number if its added to its equals f. 68. f of a number -(-5=26 ; what is the number? 69. A farmer sold f of his grain, and' had 120 bushels remaining ; how much had he at first ? 70. Required the cost of 15 horses at the rate of 3} horses for $169. 71. How much will 3J acres of land cost at $64 for H acres? 72. The difference between f of a number and of it is 6; what is the number? 78 73. If 2J times a number exceed 2 times the number by 3f , what is the number ? 74. If a man walk |- of a mile in lOf minutes, how long will it take him to walk 5 miles ? 75. Eequired the cost of 4 dozen eggs at 25 cents for 10 eggs. 76. Eequired the cost of 45 apples at 10 cents per dozen. 77. $6=| of i of a sum of money ; required the sum. 78. How many yards of goods will make 3 dresses if 15 yards make f of a dress ? 79. | of the length of a pole is in the water and 15J feet are out ; what is the length of the pole ? 80. If 19 boxes of berries are worth 57 cents, requir- ed the value of f of a box. 81. If f of a yd. of cloth cost $lf, what cost $>f yd.? 82. f of John's money equals ^ of Harry's, and to- gether they have $55 ; how much has each ? 83 If on 1 orange I lose ^ of a cent, h^w many oranges must I sell to lose 6 cents ? 84. William has twice as many cents as Herbert, and together they have 24 ; how many has each ? 85. Two boys have 49 marbles ; one has 7 more than the other; how many has each ? 86. Henry received for his horse -J of its cost ; what part of the cost was the gain ? 87. A coat which cost $12 was sold for $16; the gain was how many hundredths of the cost ? 88. Goods bought at $12 were sold for $10 ; how many hundredths of the cost was the loss? 79 89. A boy bought some apples for 72 cents and sold them for 84 cents ; the gain was what part of the cost? How many hundredths of the cost? 90. A horse was bought for $60 and sold for $48 ; what part of the cost was the loss ? How many hun- dredths of the cost was the loss ? 91. f of $6^how many hundredths of $20. 92. For what must goods costing $50 be sold to gain ten hundredths of the cost ? 93. A paid $80 for a horse and sold it so as to lose O f its cost for what did he sell it ? 94. If a merchant sold goods at $2 per yard and thereby gained ^ 5 ^ of the cost, required the cost. 95. A cow was bought for $25 and sold for $30; what part of the cost was the gain ? 96. Henry can make f of a pair of boots in a day, and James can make % of a pair in a day ; how long will it take both to make 2 pairs of boots ? 97. In how many days can 3 men cut 15 cords of wood, if 1 man in 1 day cut f of a cord ? 98. A boy bought a certain number of apples at 2 cents each, and the same number at 4 cents each, and then sold out at the rate of 3 for 5 cents ; did he gain or lose and how much ? 99. Iff of an orange cost as much as I of a pineap- ple, required the price of two oranges in pineapples. 100. A man gained $10 by selling hia horse for If times its cost; what was the cost ? 101. A can do a piece of work in 5 days and B can do it in 3 days ; in what time can they together do it ? 80 102. M. bought f of 15J yd. of cloth for f of $241; required the price of the cloth per yd. 103. If 95 bushels of apples cost $110; what is the value of 3i bushels. ? 104. A lumber dealer bought siding at $18.75 per M and sold it at $2.875 per C ; how much did he gain per M? Aliquot Parts. 158. Table of Aliquots. 2| = J of 10. 18! = A of 100. 3 = J of 10. 20 = i of 100. 6J =3^ of 100. . 25 = J of 100. gJ^J^of 100. 33 = of 100. 12^ = | of 100. 62J | of 100. 16 = J of 100. 66f == | of 100. 125 = i of 1000. 159. Forms of Solution. Example 1. At 181 cents per Ib. required the cost of 32 Ib. of butter. Solution. At $1 per Ib. 32 Ib. of butter cost $32, but at 18! cents, or $^, 32 Ib. cost T \ of $32, or $6. .-. etc, Example 2. At 12^ cents per Ib. how many Ib. of rice will $24 buy? Solution. At 12 cents, or $ per Ib., $1 will buy 8 Ib., and $24 will buy 24 times 8 Ib., or 192 Ib. .-. etc. Example 3. At $87? per acre, how many acres of land can be bought for $4900? Solution, At $100 per acre, $4900 will buy 49 acres, but $87 J, or J of $100, $4900 will buy 8 times of 49 acres, or 56 acres. 81 160. Exercises. 1. At 6i/ each what cost 16 oranges ? 2. At $6.25 per barrel what cost 7 barrels of ftour? 3. At 2JX apiece what cost 20 pencils? 4. At $2.50 a box what cost 60 boxes of potatoes ? 5. At 8JX per yd. what cost 56 yd. of muslin ? 6. At $8.33 each what cost 30 calves? 7. At 12X a lb. what cost 64 Ib. of sugar? 8. $12.50 per acre what cost 17 acres of corn ? 9. At 16fX per lb. what cost 40 lb. of butter? 10. At 16fX each what cost 4 slates ? 11. At 18t/ per lb. what cost 32 lb. of steak? 12. At $18.75 each what cost 80 ponies ? 13. If 1 doz. eggs cost 20/ what cost 15 doz. ? 14. At 20/ each what cost 7 books. ? 15. If 1 cow cost $25, required the cost of 16 cows. 16. At 25/ each what cost 28 collars ? 17. At 3^/ apiece what cost 9 apples ? 18. At $3.50 each what cost 15 hats ? 19. At $33 per acre what cost 30 acres of land? 20. At 50/ each what cost 7 books ? 21. At 62|/ each what cost 17 pitchers ? 22. At $661 per head what cost 12 horses? 23., At $1.25 per rod what cost 10 rods of fencing? 24. At $125 peivhead what cost 14 horses. ? 25. At $20 per acre what cost 15 acres of land ? 26. At 37/ per yd. what cost 7 yd. silk cord ? -27. At $75 per head required the cost of 12 horses. 28. At 6J/ per spool required the cost of 11 spools of thread. 82 SECTION VIII. COMPOUND NUMBERS. 161. Compound numbers are classified on the basis of the kind of attribute measured, as 1. MEASURE OF DURATION. 2. MEASURES OF EXTENSION. 3. MEASURES OF FORCE. 162. Diagram exhibiting in classified form the "meas- ures" usually treated under compound numbers. Remarh. 1. The decimal measures are embraced in the dia- gram , but are treated separately. 2. The measure of value in most civilized countries is derived from the force of gravity, or weight. The primary units of value were weight units. 1. Of Duration. Time measure. f Long measure. I Surveyors' long measure, Length -{ Mariners' " | Decimal t Circumference 2. Of Extension Measures. 3. Of Force ( Square measure. Surfaced Surveyor's square " ( Decimal f Cubic measure. I Decimal cubic measure. Volume \ Dry measure. | Liquid measure. I Decimal capacity {Troy Weight. Apothecaries' weight. Avoirdupois Decimal {United States money. English Etc. 163. Order of Study. Remark. The following order should be followed in the study of each measure. (1.) The primary, or standard unit. How deter- mined. (2.) Other units and their relative value. (3.) Scale and table. (4 ) Reduction. a. Descending, ( 1. Integers to* integers of lower denomination. 1 2. Fractions " " ' " (3. fractions'- " b. Ascending. C 1. Integers to integers of higher denomination. X 2. " " fractions " " < <; ( 3. Fractions" " a u u (5.) Synthesis. a. Addition. b. Multiplication. (6.) Analysis. a. Subtraction. b. Division. 164. Tables. For the tables of compound numbers and many interesting and useful facts the pupil is referred to textbooks on Arithmetic and to Encyclopedias. 84 Applications of Compound Numbers. Remark. The "order of study" will be applied, in part, to "time measure." Each of the other measures should be treat- ed in a similiar manner. * 165. Time Measure. Time. That which renders succession possible is called time. The Primary Unit. The average solar day is taken as the primary unit of time. (1.) A Solar day is the interval of time between two successive transits of the vertical rays of the sun across a given meridian. This interval varies at dif- ferent times. (2.) The sidereal day is the interval between two successive transits of a fixed star across a given merid- ian. This interval is the same at one time that it is at another. (3.) The solar and the sidereal days would be of equal length if the earth did not revolve around the sun. While the earth is rotating upon its axis it is also moving forward in its orbit ; so that when it has made a complete rotation, it must make part of an- other before the sun's rays are vertical a second time upon any given meridian. The solar day is thus a little longer than the sidereal day. [About four min- utes.] Other Denominations. The other time units are either multiples or divisors of the day. The multiples of the day are the week, the month, the year and the century. The divisors of the day are the hour, the minute and the second. 85 Relations. Multiples. 1. The week equals 7 days. 2. The month equals 4|- weeks. Remark. The average calendar month is a little more than4^ weeks, or 30 days, while the lunar month is a small fraction more than 4 weeks. 3. The year equals 12 calendar months. 4. The century equals 100 years. Divisors, 1. The hour equals % of the day. 2. The minute equals -^ of the hour. 3. The second equals ^ of the minute. Scale. The units used in time measure may be written in a scale as follows: 100 12 4f 7 24 60 60 cen. yr. mo. wk. da. hr. min. sec. 11111111 Remarks. 1. The values expressed by the respective units of this scale increase from right to left in the written scale. In this respect the time scale is like the decimal scale. 2. Since the rate of increase varies, the time scale is called a varying scale. In this respect it is unlike the decimal scale, whose rate of increase is uniformly 10. 3. The time scale consists of but eight units. In this re- spect it is unlike the decimal scale, whose orders of units may be repeated in periods indefinitely. 4. In the time scale the units extend both above and below the primary unit. In this respect it is like the decimal scale. Table. For convenience the relations of the time units may be tabulated thus : 60 sec. = 1 min. 60 min. = 1 hr. 24 hr. = 1 da. 7 da. =1 wk. 4f wk. = 1 mo. 12 mo. = 1 yr. 100 yr. = 1 cen. Reduction. 166. Reduction Descending. Example 1. Eeduce 2 wk. 3 da. 12 hr. to min. First form, v 1 wk. = 7 da., 2 " == 2 times 7 da. = 14 da. 14 da. +3 da. = 17 da. v 1 da. = 24 hr., 17 " = 17 times 24 hr. = 408 hr. 408 hr.-f 12 hr. : = 420 hr. V 1 " = 60 min., 420 " = 420 times 60 min. = 25200 min. .-.2 wk. 3 da. 12 == 25200 min. Second form. v 1 wk. = 7 times 1 da., 2 *< = 7 " 2 " =14 da. 14 da.-f3 da. = 17 da. V 1 " =24 times 1 hr., 17 " = 24 " 17 " == 408 hr. 408 hr.-f 12 hr. = 420 hr. . 1 " = 60 times 1 min., 420 " = 60 420 " = 25200 min. .-. 2 wk. 3 da. 12 = 25200 min. Example 2. Koduce f wk. to smaller denominate units. First form, v 1 wk, . 7 da., | a =|of7da. = 3| da. v 1 da. == 24 hr., I " = f of 24 hr. = 21J hr. /I hr. = 60 min.,,, J " = J of 60 min 20 min. /. | wk. 3 da. 21 hr. 20 min. 87 Second form, v 1 wk. = 7 times 1 da., f " =7 times f da. = 3f da. v 1 da. = 24 " 1 hr., f =24 I " = 21J hr. V 1 hr. ^= 60 " 1 min., J " = 60 " J = 20 min. .-. | wk. = 3 da. 21 hr. 20 min. Example 3. .Reduce ^^ of a wk. to the fraction of an hour. First form, v 1 wk. = 7 da., vfa" = Tfa of 7 da. = .03 da. V 1 da. = 24 hr., .03 " = .03 of 24 hr. = f hr. wk. = hr. Second form, v 1 wk. = 7 times 1 da., Tfr " =7 " T*TF" =-03 da. v 1 da. = 24 " 1 hr., .03 " = 24 .03 " = -Jf hr. wk. == X hr. (1.) Two forms of solution have been given for a problem in each case under reduction descending. The first form in each case is that usually given for such problems and needs no comment other than the ob- servation that the multiplier is not taken from the table but is the number (taken abstractly) to be re- duced. (2.) The second form rests upon the following : PRINCIPLE. The numerical relation that exists be- tween given units exists between like multiples and also be- tween like parts of those units. [See remark under Prin. VII, page 58.] The first step consists in the statement of the relation existing between a unit of the denomination to be reduced and a unit of the denomination to which the reduction is to be made. The second step is made in the light of the above principle, e. g. Since 1 hr. =60 times 1 min., 420 hr. (a multiple of 1 hr.) equal 60 times 420 min. (a like multiple of 1 min.) It is ob- served that the multiplier is, in every instance, taken from the table. This form of solution is uniform and general in its application to the solution of all problems in the several cases of reduction descending in all the measures. (3.) A careful study of the second form given for the solution of problems in reduction ascending, will show the form to be uniform and of general application in all the measures. (4.) Any reduction, either descending or ascend- ing, may be made by a direct use of the equation. The first statement under the second form in each of the given examples is taken as the first equation. The second equation is obtained from the first by multiply- ing both its members by such a number as will give the number to be reduced for the first member of the second equation. In transforming the first equation it is to be observed that its second member consists of two factors, and~that the member is to be multiplied by multiplying its second facfor. [Prin. X, page 31.] 167. Reduction Ascending. Example 1. Reduce 25200 min. to integers of high- er denomination. First form, v 60 min. = 1 hr., 25200 ll = as many hr. as 25200 min. are times 60 min. which = 420. V 24 hr. = 1 da., 420 hr. = as many da. as 420 hr. are times 24 hr. which = 17. with a re- mainder of 12 hr. v 7 da. = 1 wk., . 17 da. = as many wk. as 17 da. are times 7 da. which = 2, with a re- mainder of 3 da. .-. 25200 min. = 2 wk. 3 da. 12 hr. Second form, v 1 min. = -fa of 1 hr., 25200 " == gV of 25200 hr. = 420 hr. v 1 hr. = fa of 1 da., 420 " = fa of 420 da. = 17 da. 12 hr. V 1 da. = | of 1 wk., 17 = \ of 17 wk. == 2 wk. 3 da. .-. 25200 min. = 2 wk. 3 da. 12 hr. Example 2. Eeduce 3 da. 21 hr. 20 min. to the frac- tion of a wk. First form. v60 min. = 1 hr., 20 u as many hr. as 20 min. are times 60 min. which = . 21 hr. -f J hr. = 21J hr. -.24 hr. = 1 da., 24 hr. = as many da. as 21J hr. are times 24 hr. which = %. 3 da. + f da. == 3| da. v 7 da. = 1 wk., 3| da. as many wk. as 3f da. are times 7 da. which = ^. .-. 3 da. 21 hr. 20 min. = f wk. 90 Remark. The foregoing form is the same as the first form used in solving Ex 1. The following form is more in accord- ance with the facts involved. v 60 min. = 1 hr., 20 " = such part of 1 hr. as 20 is part of 60 which is J. 21 hr. + J hr. = 21 J hr. v 24 hr. 1 da.,. 21J hr. = such part of 1 da. as 21^ is part of 24 which is f . 3 da. + f da. = 3f da. v 7 da. = 1 wk., 3f da. = such part of 1 wk. as 3f is part of 7 which is -|. /. etc. Second form, v 1 min. = ^ of 1 hr., 20 " = fa of 20 hr. == $ hr. 21 hr. -f i hr. = 21J hr. = *. hr. v 1 hr. = Jj of 1 da., .ayt br. = J of % 4 - da. = f da. 3 da. -f | da. = 3| da. = - 3 9 5 - da. v 1 da. = -J- of 1 wk., y da. = i of- 3 / wk, = | wk. . .-. 3 da. 21 hr. 20 min. = | wk. Remark. The several denominate numbers constituting the given compound number, may be reduced to the lowest de- nomination (min.) ; and this number of minutes may then be reduced to the fraction of a week. Mixed numbers are thus avoided. Example 3. Eeduce if hr. to the fraction of a wk. First form. [Use either the first form given under example 2 or that given in the remark under exam- pie 2.] 91 Second form, v 1 hr. = -^ of 1 da., if hr.= jfcofjf da. = .03 da. v 1 da. _ \ of 1 wk., .03 da. | of .03 wk. = ^ T wk. /. if hr. = T ^ wk . 168. General Remarks. *' 1. Such examples as the following are often given. 2 da. 5 hr. 15 min. equal what part of 2 wk. 4. da. 3 hr. ? In such cases each compound number should be reduced to the lowest denomination given in either. Solution. 2 da. 5 hr. 15 min. == 3195 min. 2 wk. 4 da. 3 hr. = 26100 mjn. v 1 min. = 26 ; 00 of 26100 min., 3195 min, = 3195 times -^-^ of 26100 min. = $& of 26100 min. .-. etc. 2. The following two solutions are given for the purpose of exhibiting a method that shall obviate the use of a complex fraction in stating the relation of the less unit to ihe greater in each example. Example 1. Reduce 33 yd. to rd. The first step (using the second form given in ex- amples 1, 2 and 3 in Reduction Ascending,) is to state the relation between 1 yd. and 1 rd., thus: V 1 yd. = -gij of 1 rd., etc. Since the relation as stated cannot be read as a fraction, it is well to express the relation by -fa which is the equivalent of - 5 \j. The solution should be as follows : fv ljd. J 33 " = (.-.33 u == Form. 4 33 " = T 2 T of 33 rd. = 6 rd. 92 Example 2. Eeduce 132 ft. to rd. ( V i. \ L-. 1 ft. = ^ Of 1 rd., Form. I 132 " = & of 132 rd. = 8 rd. 132 " = 8 rd. 3. Addition, multiplication, subtraction and division of com- pound numbers may be effected in the same manner and in obedience to the same principles that govern the synthesis and the analysis of numbers expressed in the decimal scale. No new principle is introduced and but one new fact, viz.: a va- rying scale is employed instead of a uniform scale. 4. The subject of time is placed first in the classification of the measures because the primary units of extension and weight are derived from a time unit. The primary unit of the common measures of extension is the yard, which is a definite portion of the length of a pendu- lum that beats seconds under certain conditions. The primary unit of weight is the pound which is the force of gravity that acts upon a certain volume of water under cer- tain conditions. The primary unit of extension is derived direct- ly from a time unit (the second), and the primary unit of weight is derived directly from a measure of extension (a volume) arid through that from the same time unit, (the second.) The primary unit of value is a certain weight of silver or gold. [NOTE Circumference measure and the decimal (metric) measures are exceptions to the above remark.] 169. Exercises. 1. Reduce 5 wk. 3 da. 4 hr. to min. 2. Reduce f da. T 2 5 hr. T 7 T min. to sec. 3. Reduce 2 bu. 3 pk. 5 qt. to pt. 4. Reduce I bu. f pk. to qt. 5. Reduce 5 yd. 2 ft. 7 in. to in. 6. Reduce 2 mi. to fathoms. 7. Reduce 60 acres to sq. ft. 8. Reduce 3 hhd. 24 gal. 3 qt. to gills. 9. Reduce 4 Ib. 15 pwt, to grains. 10. Reduce oz. f pwt. to grains. 11. Reduce 7$ 2 to grains. 93 12. Eeduce f | 3 9 to grains. 13. Eeduce 5 cwt. 14 Ib. 7 oz. to drams. 14. Reduce j- ton f cwt. -- Ib. -^ oz. to oz. 15. Reduce 7 wk. 3 da. 1 hr. 15 min. to seconds. 16. Reduce 3 gal. 2 qt. 1 gi. to gills. 17. Reduce 3 yd. 2 ft. 5 in. to inches. 18. Reduce 5 bu. 2 pk. 7 qt. to pints. 19. Reduce f Ib. Troy, to integers of lower denom- inations. 20. Reduce .3 da. to integers of lower denomina- tions. 21. Reduce f yd. to integers of lower denomina- tions. 22. Reduce .875 gal. to integers of lower denomin- ations. 23. Reduce y^Vs" da. to the denomination minutes. 24. Reduce .007 gal. to the denomination pints. 25. Reduce -^ yd. to the denomination inches. 26. Reduce -^ bu. to the denomination pints. 27. Reduce 54960 min. to integers of higher de- nominations. 28. Reduce 186 pt. to a compound number of higher denominations. 29. Reduce 57648 sec. to a compound number of higher denominations. 30. Reduce 35J qt. to a compound number com- posed of pk. and bu. 31. Reduce 211 inches to a compound number composed of in. ft. and yd. 32. Reduce 23400 grains to Troy integers of higher denominations. 94 33. Reduce 600 gr. to a compound number com- posed of pwt. and ounces. 34. Reduce 8211 oz. to a compound number com- posed of Ib. and cwt. 35. Add together 3 bu. 3 pk. 6 qt. 1 pt.; 5 bu. 2 pk. 5 qt. 1 pt.; 6 bu. 1 pk. 7 qt. 1 pt.; 1 bu. 3 pk. 2 qt. 36. Add 34 cwt. 17 Ib. 11 oz. 13 dr.; 19 cwt. 46 Ib. 7 oz. 4 dr.; 71 cwt. 10 Ib. 15 oz. 12 dr. 37. Add 5 mi. 6 fur. 21 rd. 4 yd. 2 ft. 9 in.; 9 mi. 7 fur. 17 rd. 5 yd. 1 ft. 11 in,; 14 mi. 21 fur. 14 rd. 2 yd. 2 ft. 4 in. 38. From 24 sq. yd. 7 sq. ft. 110 sq. in. take 16 sq. yd. 6 sq. ft. 136 sq. in. . 39. From 63 bu. 2 pk. 5 qt. take 54 bu. 3 pk. 6 qt. 40. From a hhd. of molasses f leaked out. How much remained? 41. What is the time from March 16, 1884 until July 4, 1884? 42. How long has it been since February 4, 1884 ? 43. A note was given Dec. 18, 1883 and was paid March 28, 1884; how long did it run ? 44. Multiply 5 Ib. 6 oz. 3 pwt. 11 gr. by 6. 45. Multiply 4 bu. 3 pk. 7 qt. 1 pt. by 5. 46. Multiply 4 rd. 3 yd. 2ft. 7 in. by 4. 47. Divide 5 da. 13 hr. 44 min. 18 sec. by 6. 48. Divide 6 cwt. 44 Ib. 12 oz. 10 dr. by 4. 49. Divide 3 bu. 2 pk. 5 qt. by 2 bu. 6 qt. 1 pt. 50. Divide 6 bl. 21 gal. 5 qt. by 5 bl. 12 gal. 7 qt. 51. Reduce f of aft. to the fraction of a rd. 52. Reduce .05 of a mi. to lower denominations. 53. -J of a dr. to the fraction of a Ib. 95 SECTION IX. TIME AND LONGITUDE. Distance measured on a parallel of latitude is called longitude. Remarks. 1. Since longitude is measured on the arc of a circle it is estimated in units of the circumference, viz.: degrees, minutes and seconds. 2. Unless otherwise designated, the meridian of ^Greenwich is taken as the prime from which longitude is reckoned. 170. Because of the daily rotation of the earth on its axis, any place on the surface of the earth except the poles, moves In 24 hours through 360 of space. " 1 hour " 15 " 1 min. 15' " " 1 sec. 15" " Hence if 15 units of* longitude lie between two places, the time registered at the places, respectively, differs by 1 time unit ; hours corresponding to degree?, minutes to minutes, and seconds to seconds. On the other hand, if the time of two places is found to differ by 1 time unit, the places are distant from each other (east and west) 15 corresponding longitude units. 171. All places on the earth's surface have the same absolute time, but not the same relative time. When the vertical rays of the sun reach the meridi- an of a place it is noon at all places on that meridian ;. while it is afternoon, or later in the day, at all places east, and before noon, or earlier in the day, at all places west of the given meridian. Thus a place east of a given meridian has later relative time and a place west has earlier relative time than that on the given meridian. 96 172. Standard Time. For the purpose of estimating time for the railway service, the country of the United States is divided into belts or strips marked by meridians of longitude 15 apart. The 75th meridian from Greenwich marks the middle of the eastern belt. The 90th meridian marks the middle of the central belt. The 105th me- ridian marks the middle of the mountain belt. The 120th meridian marks the middle of the Pacific belt. The local time on each of these meridians is taken as the "Standard" time for railway purposes at all places within the given belt; and since these meridi- ans are 15 apart, the time in each belt is registered one hour earlier than in the belt next at the east, e g. When it is noon on the 75th meridian, it is noon at nil places within the eastern belt, 11 a. m. on the 90th meridian and at all places within the central belt, 10 a. ra. on the 105th meridian and at all places within the mountain belt and 9 a. m. on the ]20th meridian and at all places within the Pacific belt. Remark. In each belt near its prime meridian workshops, manufactories, public schools and many other branches of business are now generally carried on by "Standard" instead of local time. 173. Forms of Solution. Example 1. The time between two places is 6 hr. 12 min. 10 sec.; required their difference in longitude. fkrm. -.' 1 hr. 1 min. 1 sec. correspond to 15 times 1 1' 1", 6 hr. 12 min. 10 sec- correspond to 15 times 6 12' 10"= 93 2' 30". .-. their difference in longitude is 93 2' 30". 97 Example 2. The difference in longitude between New York and Greenwich is 74 3"; required their dif- ference in time. Form. '.- 1 V 1" correspond to T V of 1 hr. 1 min. 1 sec., 74 3" " " T V " 74 " 3sec. =4hr.56 min. .2 sec. .-, etc. Example 3. When it is noon at New York, what is Greenwich time? Remark. Their time difference is given ab^ve Form. Since their time difference is 4 hr. 56 min. .2 sec., and since Greenwich is east of New York, its time is later than that of New York by 4 hr. 56 min .2 sec., or 56 min. .2 nee past 4 p. m. Example 4. When it is noon at Greenwich what is New York time? JSemark. Their time difference is given above. Form. Since their time difference is 4 hr. 56 min. .2 sec., and since New York is west, its time is earlier by 4 hr. 56 min. .2 sec., or 3 min. 59.8 sec. past 7 a. m. Exercises. 1. Two places differ in time 6 hr. 7 min. 10 sec. ; required their difference in longitude. 2. Two places are distant from each other 1 quad- rant ; what is their time difference ? When it is 10 a. m. at the place the farther west what time is it at the other? When it is 1 o'clock p. m. at the place the farther east what time is it at the other? 3. A man travels westward 517'11" ; is his watch too fast or too slow and how much ? 98 4. A man travels until his watch is 19 min. 55 sec. too fast ; has he traveled east or west, and how far ? 5. A man travels until his watch is too slow by 11 hr. 5 min. 16 sec. ; has he traveled east or we^t, and how far? 6. The longitude of St. Louis is 9010', while that of Cincinnati is 8426' ; required their difference in time. When it is noon at either place what is the time at the other ? 7. The longitude oi Bangor Me., is 6845', and that of San Francisco is 12225' ; required their time difference. When it is 7 a. m. at either place what time is it at the other ? 8. A vessel sailed due north at the rate of 14 knots per hour, while the sun apparently moved through 1 sextant 518' ; how long did she sail, and how many statute miles ? 9. f Washington is in longitude 772'48" west and Cincinnati is in longitude 8426' west ; when it is 6 a. m. at either place what time is it at the other? 10. If my watch keeps Terre Haute time and indi- cates 17 min. 10 sec. past 1 o'clock p. m. when it is noon by local time how far and in which direction am I from Terre Haute ? 11. A man travels from Pittsburg in longitude 80 2' west, until his watch is 1 hour and 45 minutes too fast ; how far and in which direction has he trav- eled ? 12. The longitude of Galveston is 9450' west while Constantinople is in longitude 2849' east ; when it is noon at either place what time is it at the other ? 13. A ship's chronometer, set at Greenwich, indi- cates 5 hr. 45 min. 24 sec. p. m. when the sun is on the meridian; required the longitude of the vessel. 14. A degree of longitude in the latitude of Boston is 44sr geographic miles-. How many more statute miles in 7 of longitude at the equator than in the latitude of Boston ? 15. At 3 o'clock and 35 min. a. m. in London, what is the time at Boston ? 16. At 5 o'clock p. m. in Eome, what time is it in Terre Haute ? 17. At noon in Pekin what time is it in San Fran- cisco ? 18. How many degrees of east or west longitude may a place have ? Why ? 19. What is the difference between the local and ''standard" time of Terre Haute ? 20. When it is 10 o'clock at Indianapolis, what is "standard" time at the same place ? 21. When it is 4 p. m., local time, at Omaha, what is ' standard" time at New York City ? 22. When it is 5 hr. 15 min. 10 sec. a. m., "stand- ard" time at San Francisco, what is local time at Pitts- burg ? 23. At 6 hr, 30 min., a. m. local time, Jacksonville, Fla., what is the "standard" time at San Antonio, Texas ? 24. When it is noon at Cincinnati, local time, is it before or afternoon and how much by "standard" time at Cleveland, Toledo, Detroit, Ft. Wayne, Mobile, Eichmond ? 100 SECTION X. ABBAS AND yOLUMES. 174. Areas. Example 1. A room is 12 ft. long and 8 ft. wide ; how many sq. ft. are in the floor ? Solution. A surface 1 ft. 1. and 1 ft. w. = 1 sq. ft. A surface 12 ft. 1. and 1 ft. w. = 12 times 1 sq. ft. = 12 sq. ft. A* surface 12 ft. 1. and 8 ft. w. = 8 times 12 sq. ft. = 96 sq. ft. .-. etc. Remark. Length, width and area are always expressed in concrete units, hence length cannot be multiplied by width ; even if such^multiplication were possible, the product would not be area, which is different in name from the multiplicand. If, however, the numbers representing the dimensions of a parallelogram be multiplied together, the product is the number representing the square units in the given surface. Example 2. A lot contains 192 sq. rd. and is 16 rd. long ; how wide is it ? Solution. A surface 1 rd. 1. and 1 rd. wide = 1 sq. rd. A surface 16 rd. 1. and 1 rd. wide = 16 times 1 sq. rd. = 16 sq. rd. A surface 16 rd. 1. must be as many rd. wide to contain 192 sq. rd. as 192 sq. rd. are times 16 sq. rd. which = 12. .-. The lot is 12 rd. wide. Remark. If the number of square units in a parallelogram be divided by the number of corresponding linear units in either dimension, the quotient is the number of linear units in the other dimension. Example 3. The base of a plane triangle is six inches and its altitude is 4 inches ; what is its area ? 101 (1.) The area of a plane triangle is one-half the area of a parallelogram having the same base and alti- tude as the triangle; hence find the area of such paral- lelogram as in Ex. 1, and divide it by 2. (2.) If the number of linear units in the base or altitude of a plane triangle be multiplied by one-half the number of like linear units in the other dimension, the product is the number of corresponding square units in the triangle. Example 4. The diameter of a circle is 14 inches; what is its area? (1.) The circumference of a circle is nearly 3-iJ- times the diameter. (2.) Any circle is practically equal to a rectangle whose length is the semi-circumference and whose width is the radius of the given circle. Solution. 3| times 14 inches = 44 in. the ap- proximate circumference of the given circle. Its equivalent rectangle is, therefore 22 in. by 7 in. 7 times 22 = 154. .'. the approximate area of the given circle is 154 sq. in. (3.) The area of any circle is .7854 of the area of its circumscribed square. Each side ot the square that circumscribes the given circle is 14 inches, the diameter of the circle. The area of the square is found by multiplying 14 by 14 and giving the denomination square inches to the product. .7854 of this product equals 154-f- sq. in., the area of the given circle. The rule is usually formulated thus: To find the area of a circle: Multiply the square of the diameter by .7854. 102 175. Volumes. Example 1. A common brick is 8 in. long, 4 in. wide and 2 in. thick ; how many cu. in. does it contain? Solution. A solid 1 in. 1. 1 in. w. 1 in. th. = 1 cu. in. A solid 8 in. 1. 1 in. w. 1 in. th. = 8 times 1 cu. in. =8 cu. in. A solid 8 in. 1. 4 in. w. 1 in. th. = 4 times 8 cu. in. = 32 cu. in. A solid 8 in. 1. 4 in. w. 2 in. th. = 2 times 32 cu. in. 64 cu. in. /. a brick contains 64 cu. in. Remark. If the numbers* representing the three dimensions of a rectangular solid be multiplied together, the product is the number representing the corresponding volume units in the given solid, Example 2. A bin containing 315 cu. ft. is 9 ft. long and 5 ft, wide, how deep is it ? Solution. - A solid 1 ft. 1. 1 ft, w. 1 ft. d. = 1 cu. ft. A solid 9 ft. 1. 1 ft. w. 1 ft. d. = 9 times 1 cu. ft. = 9 cu. ft. A solid 9 ft. 1. 5 ft. w. 1 ft. d. = 5 times 9 cu ft. = 45 cu. ft. A solid 9 ft. 1. 5 ft. w. must be as many ft. deep to con- tain 315 cu. ft. as 315 cu. ft. are times 45 cu. ft. which equal 7. .'. the bin is 7 ft. deep* Remark. If the number representing the volume of a rectan- gular solid be divided by the product of the numbers represent- ing two of its dimensions, the quotient is the number repre- senting its third dimension. Exercises. 1. How many square feet in a surface 12 ft. by 5ft.? 2. How many sq. yd. in a lot 15 yd. by 11 yd.? 103 3. A field is 40 rods long and 18 rods wide; how many acres in it ? 4. A room is 15 ft. 6 in. long by 24 ft. 4 in, wide ; how many square yards in the ceiling ? 5. A floor contains 192 sq. yd, and is 12 yards wide ; how long is it ? 6. A field contains 18 acres and is 60 rods long how wide is it? 7. A brick walk is f yd. wide ; how long must it be to contain 25 square yards ? 8. A triangular wall is 6 ft. long and 5 ft. high at its widest end ; what is the area of one side ? 9. A block is 5 in. by 3 in. by 2 in. ; what is its volume? 10. A piece of stone is 7 ft. long, 5-J-. ft. wide by 3 ft. thick ; how many cu. ft. in it? 11. A bin is 8 ft. by 6 ft. by 4J ft. ; how many bushels of wheat will it hold? 12. A cistern is 9 ft. by 5 ft. by 5 ft. ; how many gallons does it hold ? 13. How many perch of masonry in a wall 350 ft by 18 ft. by 2 ft. ? 14. A block contains 64 cu. ft., and is 4 ft. wide and -2 ft. thick ; how long is it ? 15. A box is 5 ft. long and 3 ft. wide ; how deep must it be to contain 60 cu. ft. ? 16. A tank is 10 ft. -Long and 4 ft. wide'; how deep must it be to hold 3 tons of water ? 17. A haystack is 36 ft. long and 9 ft. wide ; what is its hight if it contain 11 tons of hay ? 18". A horse is tied to a stake by a rope 100ft. long over how much ground can he walk ? 104 SECTION XI. THE DECIMAL SYSTEM OF MEASURES. Remark. The history of the decimal system of measures and numerous facts and arguments showing its simplicity, are found in the publications of the American Metric Bureau of Boston. The purpose of this section is to exhibit the system and to present methods of computing by means of it. 176. Primary Units. Of length, the meter. marked m. '* Capacity, " liter (leeter.) " 1. " weight, " gram, " g. Remarks. I. The ar is the unit for measuring land. Other surfaces are measured by the sq. m. or by decimal parts or deci- mal multiples thereof. The ar is marked a. 2. The ster is the unit for measuring wood. Other volumes are measured by the cubic meter or by decimal parts or deci- mal multiples thereof. The ster is marked s. 3. In measuring great weights the quintal and the tonneau (Metric ton) are used. These are marked Q. and T., respect- ively. 177. Secondary Units. Each of the secondary units used is either a deci- mal part or a decimal multiple of a primary unit, and is designated by using one of the following prefixes with the name of a primary unit. Decimal parts. Decimal Multiples. Dgei, meaning .1. Deka, meaning 10. Cent*, .01. Hgkto, " 100. Milli, .001. Kflo, 1000. Myria, 10000. Remarks. 1. In combining any one of the foregoing prefixes with the name of a primary unit, each word retains its own pronunciation and accent. 2. In abbreviating a word formed by combining a prefix with the name of a primary unit, it is customary to use the initial letter of each word, using a small letter to designate the .prima- ry unit or any of its decimal parts, and a capital letter to desig- nate any of its decimal multiples. 105 Scales. 178. Of Length Measure. 10 10 10 10 10 JO 10 Mm. Km. Hm. Dm. m. dm. cm. mm. 11 111111 179. Of Surface Measure. 100 100 100 100 100 100 100 Sq. Mm. sq. Km. sq. Hm. sq. Dm. sq. m. sq. din. sq. cm. sq. mm. 11111111 180. Of Land Measure. 10 10 10 10 10 10 10 Ma. Ka. Ha. Da. a. da. ca. ma. 11111111 181. Of Volume Measure, 1000 1000 1000 1000 1000 1000 1000 Cu. Mm. cu. Km. cu. Hm. cu. Dm. cu. m. cu. dm. cu. cm. cu. mm. 1 1 1 11111 182. Of Wood Measure. Remark. Only three units are used. 10 10 Ds. s. ds. 1 1 1 183. Of Liquid and Dry Measure. 10 10 10 10 10 10 10 Ml. Kl. HI. Dl. 1. dl. cl. ml. 1 1111111 184. Of Weight. 10 10 10 10 10 10 10 10 10 T. Q. Mg. Kg. Hg. Dg. g. dg. eg. mg lilt 11111 106 Relation of Decimal and Common Measures. 185. Of Length. 186. Of Surface. 1 cm. .3937 in. 1 sq. dm. = 15 sq. in. 1 dm. = * 3.937 in. 1 sq. m. = 1550 sq. in. 1m. = 39.37 in. _ ( 119.6 sq. yd. 1 Dm. = 32.8 ft. - { 1 sq. Dm. H m=. 328 ft. 1 in: 1 Ha. = 2.47 acres. 1 Km. = 3280 ft. f mi., nearly. 1 Mm. 6.2137 miles. 187. Of Volume. 1 cu. cm = .06 cu. in. 1 eu. dm. = 61 cu. in. 1 cu. m. \ C 35.317 cu. ft., or j ' i ^^* 1 ster. ) ( 1.308 cu. yd. 188. Of Capacity. 1 ml. = 1 cu. cm. = .27 fl. dr. = .061 cu. in. 1 cl. = 10 cu. cm. .338 fl. oz, =.61 " " 1 dl. = .1 cu. dm. = .845 gi. =6.1 " " 11. .= 1 cu. dm. = 1.05 qt. liq. = .9 qt. dry. 1 Dl. = 10 cu. dm. =. 2.64 gal. liq. = 9'.08 qt. dry. 1 HI. = .1 cu. m. = 26.41 gal. liq. = 2.76 bu. 1 Kl. = 1 cu. m. = 264.17 gal. liq. = 1.308 cu. yd. 189. Of Weight. 1 mg.=rthe wt. of 1 cu . mm. of water= .015 gr.Av. 1 eg. = " " 10 " " " " = .154 " " 1 dg. == " '.1 ' cm. " = 1.54 ' " 1 g. = " 4< " 1 4/ " " " = 15.43 " " 1 Dg. " " " 10 " " = .35 oz. " 1 Hg.= " " " 1 dl , " = 3.53 " " 1 Kg.= " 1 1. c " 2.2 Ib. " 1 Mg.= " " " 10 " " " 22.04 u " 1 Q. = " " " 1 HI. " " = 220.4 " IT. = " " ll 1 ci n. m. " =2204.b 107 Remark. The fractions in the above tables of relative value are not exact, but they are sufficiently approximate to meet most applications. Indeed where no great accuracy is required it is sufficient to use the following table of 190. Approximate Values. 1 dm. = about 4 in. 1m. = " 39i in. 5m. = " 1 rod. 1 Km. " f mi. 1 sq. m. " lOf sq. ft. 1 Ha. = " 2| acres. 1 cu. m. or ster. = u 1J- cu. yd., or i cord. 11. = u 1 qt. 1 HI. = " 2|bu. 1 g. = 15J gr. 1 Kg. == 2* lb. IT. = a 2200 lb. 191. Table of Specific Gravities; Water Being I. Air, .001 Ice, .93 Alcohol, pure, .79 Iron, cast, 7.1 " commercial, .83 " wrought, 7.7 Brass, 7.6 Lead,. 11.35 Brine, 1.04 Lime, 1.8 Coal, soft, 1.25 Marble, 2.7 " hard, 1.5 Mercury, 13.6. Copper, 9. Milk, 1.03 Cork, i- Silver, 10.5 Gold, 19.2 Zinc, 7.1 192. Important Facts. 1. The ar is an area equivalent to 1 sq. Dm. 2. The ster (pronounced stair} is a volume equiv- alent to 1 cu. m. 3. The liter is a volume equivalent to 1 cu. dm. 108 4. The gram isthe weight of a cu. cm. of distilled water at its greatest density, i. e., at 4 centigrade, or 39.2 Fahrenheit, The water is balanced in a vacuum. 5. 1 liter, or 1 cu. dm. of distilled water weighs 1 kilogram, while 1 cu. m. of water weighs 1 metric ton. 6. The legal letter weight in the United States is i oz. Av. The Postmaster General is authorized to substitute a 15 gram weight for the oz. weight in all postoffices that exchange mails with foreign nations and in other postoffices at his discretion. 7. A freshly coined nickel 5 cent piece (not the new nickel), is 2 cm. in diameter and weighs 5 grams. The silver coinage of the United States is worth 4 cents per gram. Three 5 cent nickels or 60 cents in silver coin constitute one letter weight. 193. Applications of Decimal Measures. Length. Write and read 1 m. as dm. cm. mm. Dm. Hm. Km. ( t n it 254m t " " u u li : f ; 2i : 3J. Compound, Two or more simple ratios, viewed together as to the product of their corresponding terms constitute a compound ratio. 227. Principles. Remark. Since the terms antecedent, consequent and ratio, signify dividend, divisor and quotient, respectively, the gener- al principles of division become, by a change in terminology, the principles of ratio. I. Multiplying the antecedent multiplies the ratio, II. Multiplying the consequent divides the ratio. III. Multiplying both antecedent and consequent by the same number does not change the ratio. IV. Dividing the antecedent divides the ratio. V. Dividing the consequent multiplies the ratio. 152 VI. Dividing both antecedent and consequent by the same number does not change the ratio, Remark. Prin. Ill may be thus stated : The ratio between like multiples of two numbers equals the ratio between the two numbers. Prin. VI may be thus stated : The ratio be- tween like parts of two numbers equals the ratio between the two numbers. Proportion. 228. Two equal ratios constitute a proportion. Remark. The ratios which form a proportion may both be simple, both compound, or one simple and the other com- pound. A Simple Proportion. A proportion that consists of two simple ratios is called a simple proportion. A Compound Proportion. A proportion containing a compound ratio is called a compound proportion. 229. Notation. A proportion is notated by writing a double colon between the two equal ratios and interpreting it by the word as. The proportion, 2:4 :: 5: 10, is read 2 is to 4 as 5 is to 10. A proportion is often notated by writing the two equal ratios with the sign of equality between them. The proportion, 2:3 = 4:6, is read The ratio oi 2 to 3 equals the ratio of 4 to 6. Remark. The antecedent of the first ratio and the conse- quent of the second ratio of a proportion are called the extreme terms and the other terms the mean terms of the proportion. 230. Principle. The product of the mean terms of any proportion equals the product of its extreme terms. Demonstration. In a ratio the antecedent is the product of two factors, viz.: the consequent and the ra- 153 tio. In a proportion the product of the mean terms is composed of three factors, viz.: The ratio and the consequent of the second ratio (which compose the an- tecedent of the second ratio,) and the consequent of the first ratio. The product of the extreme terms is composed of ihree factors, viz.: the ratio and the consequent of the first ratio (which compose the antecedent of the first ratio) and the consequent of the second ratio. It ib thus observed that the factors composing the product of the means are identical with the factors composing the product of the extremes ; hence the two products are equal. 231. Forms of Solution. Remarks. 1. Any problem that is solvable by proportion is readily solved by analysis. 2. It is to be remembered that a ratio exists between like numbers only. 3. In solving a problem by proportion two distinct steps are taken. a. The arrangement of the ratios, called the statement of the proportion. b. The reduction of the proportion, or the finding of the un- known term. Example 1. If 11 bu. of wheat cost $9, what cost 17 bu. ? Written form. Thought form. 9:x:: 11:17 | Since a ratio exists between like numbers only, a ratio exists, in this example, between 11 bu. and 17 bu., and between $9, the cost of 11 bu., and the cost of 17 bu., and these ratios must be equal. 9, which represents the cost of 11 bu., may be made either term of either ratio, and the required number, which we will represent by x, will be the other term of the same ratio. We choose to make 9 the antece 154 dent of the first ratio ; then is x the consequent oi the first ratio. $9 is the cost of 11 bu., while $# is the cost of 17 bu., a greater number than 11, hence the conse- quent, x, of the first ratio is greater than its antece- dent, 9 ; and since the two ratios must be equal to form a proportion, the consequent of the second ratio must be greater than its antecedent. Hence we make 11 (bu.) the antecedent and 17 (bu.) the consequent of the second ratio. We now have the two extremes and one mean of a proportion. Since the product of the extremes equals the product of the means, the required mean is found by dividing the product of the extremes by the" given mean, (using the numbers abstractly in performing the operation.) The required mean is 13.91. .'. 17 bu. cost $13.91. Example 2. If 75 men can build a wall 50 ft. long, 8 ft. high and 3 ft. thick, in ten days, how long will it take 100 men to build a wall 150 ft. long, 10 ft. high and 4 ft. thick ? Written form. Thought form, x: 10:: 75: 100 I In this problem a ratio exists be- 150:50 10:8 4:3 tween 75 men and 100 men ; 50 ft. length and 150 ft. length ; 8 ft. hight and 10 ft. hight; 3 ft. thickness and 4 ft. thickness ; 10 days and the required number of days which we may represent by x. 10 (days) may be made either term of either ratio, and x (days) will be the other term of the same ratio. We will make 10 the consequent of the first ratio, then will x be the antecedent of the first ratio. 10 days are required for 75 men to do a work while x days are required for 100 men to do the work. 100 155 men require a less number of days than 75 men, hence the antecedent, #, of the first ratio is less than its con- sequent, 10 ; and since the two ratios must be equal to form a proportion, the antecedent of the second ratio is less than its consequent ; we therefore, write 75 as the antecedent and 100 as the consequent of the sec- ond ratio. Again, 10 days are required to build a wall 50 feet long, while x days are required to build a wall 150 feet long, a greater length than 50 feet, hence x is greater than ten, and as the two ratios must be equal to form a proportion, the antecedent of the second ra- tio must be greater than its consequent ; we therefore write 150 as the antecedent and 50 as the consequent of the second ratio. Again, 10 days are required to build a wall 8 ft. high, while x days are required to build a similar wall 10 ft. high, a greater hight than 8 ft., hence x is more than ten, and since the two ratios must be equal to form a proportion, the antecedent of the second ratio is greater than its consequent ; we therefore write 10 as the antecedent and 8 as the consequent of the second ratio. Again, 10 days are required to build a wall 3 feet thick, while x days are required to build a similar wall 4 ft. thick, a greater thickness than 3 ft., hence x is more than 10, and, since the two ratios must be equal to form a proportion, the antecedent of the second ratio is greater than its consequent; we therefore, write 4 as the antecedent and 3 as the consequent of the second ratio. We now have given the factors of the means and the factors of one extreme of a compound proportion to find the other extreme. 156 Since the product of the extremes equals the pro- duct of the means, the required extreme is found by dividing the product of the factors of the means by the product of the factors of the given extreme. [The work of reducing the proportion may be shortened by con* sidering the factors of the means as factors of a divi- dend and the factors of the given extreme as factors of a divisor, and canceling common factors.] The re- quired extreme is found to be 37. .'. the required time is 37J days. Exercises. 1. If 70 horses cost $3500, what cost 160 horses? 2. If 5 Ib. of coifee cost $1.35, what cost 9 Ib. ? 3. If 12 tons of hay cost $87, what cost 17 tons ? 4. If a tarm cost $4800, what cost f of it ? 5. If i yd. of silk cost $1.65, what cost 8 yd.? 6. If 19 acres of land sell for $570, required the price of 40 acres. 7. How many men will do as much work in 84 days as 8 men do in 126 days? 8. If I borrow $3500 for 30 days, for what time may I return $900 to requite the favor? 9. If a 10 cent loaf weigh s 1 Ib. 2 oz. when flour is $7^ per barrel, what should it weigh when flour is $6 per barrel ? 10. What cost 147 bu. of corn if 35 bu. cost $28J? 11. If 15 sheep cost $15.90, how many sheep can be bought for $155.82? 13. At 36 d. per $, what is 1 worth in U. S. money ? 14, In what time can a man pump 54 barrels of water, if he pump 24 barrels in 1 hr. 14 min ? 157 15. If 10 bales of cotton can be carried 115 miles for $8, how far can 15 bales be carried for the same money ? 16. If $1200, in 2 yr. 3 mo. gain $162 interest at 6%, how much will $800 gain in 3 yr. 3 mo. at 8% ? 17. A mill makes 1265 barrels of flour by running 10 hr. per day for 13 days ; how many barrels would it make in 26 days, running 15 hours per day ? 18. If it cost $28 to carpet a room 12 ft. by 7 ft., what will it cost to carpet a room 30 ft. by 18 ft. ? 19. If 60 men dig a canal 80 rd. long, 12 ft. wide and 6 ft. deep in 18 days, working 16 hr. per day, how many men will be required to dig a ditch 30 rd. long, 9 ft. wide and 4 ft. deep in 24 days, working 12 hours per day ? 10. A min gained $3155 on the sale of 91 horses ; how much would he have gained on 245 horses sold at the same rate ? 21. If a tank 17^ ft. long, 11| ft. wide and 13 ft. deep, contain 546 barrels, how many barrels will a tank hold that is 16 ft. long, 15 ft. wide and 7 ft. deep ? 22. If the interest $346 for 1 yr. 8 mo. is $32, what is the interest of $215.25 for 3 yr. 4 mo. 24 da. at the same rate ? 23. A certain bin is 8 ft. by 4 ft. by 2* ft. and its capacity is 75 bu. ; how deep must a bin be to con- tain 450 bu., if it be 28 ft. long by 3J ft. wide? 24. If 366 men in 5 days of 10 hr, each, can dig a trench 70 yards long, 3 yards wide, and 2 yards deep ; what length of trench 5 yards wide and 3 yards deep, can 240 men dig in 9 days of 12 hr. each? 158 25. If it cost $64 to pave a walk 3 ft. wide and 32 ft. long, what will it cost to pave a walk 5 ft. wide and 64 ft. long? 26. If 7 men, working 10 hr. per da., make 6 wag- ons in 21 days, how many wagons can 12 men make in 16 days working 9 hours per day ? 27. If $8 yield $2 interest in 3 mo. 12 da, at 9%, how much int. will $140 yield in 9 mo. 22 da. at 6% ? 28. If it cost $320 to buy the provisions consumed in 8 mo. by a family of 7 persons, how much at the same rate will it cost to feed a family of twice the num- ber of persons for i as much time? 29. If a stone 2 ft. long 10 in. wide, and 8 in. thick, weigh 72 lb., required the weight of a similar stone 6 ft- long, 15 in. wide and 6 in. thick. 30. If 300 lb. of wool at 28/ per lb. are exchanged for 36 yd. of cloth 1J yd. wide, how many lb. of wool at 35/ per lb., should be given for 20 yd. of cloth f yd. wide? Proportional Parts. 232. A number is divided into proportional parts if separated into parts whose ratio equals the ratio of given numbers. Example. Divide 40 into two parts that are to each other as 3 to 5. 40 is to be divided into parts that are respectively, equi-multiples of 3 and 5; i.e., one of the required parts of 40 is as many times 3 as the other part is times 5. 159 40, the sum of the required parts is, therefore, the same number of times the sum of 3 and 5. 40 is 5 times the sum of 3 and 5. 5 times 3 = 15 and 5 times 5 = 25. Hence 15 and 25 are the required parts into which 40 is to be divided. Principles, a, The sum of the proportionals is to the sum of the required numbers as the less propor- tional is to the less of the required numbers. b. The sum of the proportionals is to the sum of the required numbers as the greater proportional is to the greater of the required numbers. SECTION XIV. INVOLUTION AND EVOLUTION. 233. Table for Determining the Number of Terms (Orders) in the Square Root of a Given Square Number. 1 2 =1 From the marginal table it is seen 9 3 =81 that a square consisting of one or two 10 2 100 terms contains one term in its square 99 2 =9801 root ; a square of three or four terms 100 I =10000 contains two terms in its square root; 999 2 998001 a square of five or six terms contains 1000 2 =1000000 three terms in its square root, etc. 9999 2 =99980001 .1 2 =.01 The principle may be formulated ,9^=.81 thus : .Ol 2 .0001 If a square number expressed in the .99 2 =:.9801 decimal scale be separated into periods .001 2 =.000001 of two terms each, beginning at the inter- .999 2 =.998001 val between units and tenths, there are as etc. many terms in its square root as there are periods in the square. 160 234. Table Indicating the Lowest Place which a Sig- nificant Figure can Occupy in the written Product of the Given Factors. Remark. The following abbreviations are used u for units, t for tens, h for hundreds, th for thousands, tth for ten thousands, .t for tenths, .h lor hundredth*, .th for thousandths, .tth for ten thousandths, etc. u a = u. t X u = t. u X .t = .t t f =h. h X u = h. u X .h = .h h 2 = tth. th X u = th. u X .th = ,th h 2 =m. h X t = th. .t X .h = .th .t 2 .h th X t = tth. .t X .th = .tth h 2 = .tth etc. etc. etc. 235. A Number Consisting of Tens and Units Involved to the Second Power. Example. M*= what ? Remark. In the light of the definition of a second power, 34 is squared by multiplying it by itself. In order that the par- tial products be readily seen, they should not be combined by addition until they are all found. The involution may be shown in the form that follows. = t.X u ) = t X u) 34 34 16"= 4 2 = u 2 . 120 = 30 X 4 =t.X gW=W = t 2 . 1156 = 34 2 = t 2 + 2t X u + u 2 . Remark. If any number consisting of tens and units be squared, it is readily seen that the same steps will be taken as in the above example, i. e., the square of a number consisting of tens and units is composed of the square of the tens plus twice the tens multiplied by the units plus the square of the units . .161 236. A Number Consisting of Hundreds, Tens and Units Involved to the Second Power. Example. 234! = what ? 234 234 16 = 4 2 = u 2 . 120 = 30 X 4 = t X u. 800 = 200 X 4 = h X u. 120 = 4_X 30 = t X u. 900 = 30* = t 2 , 6000 = 200 X 30 = h X t. 800 = 4 X 200 = h X u. 6000 = 30 X 200 = h X t. 40000 == 200 2 = h 2 . 54756 = 234* =_ h 2 + 2hXt -ft 2 -f 2[h+t]Xu+u 2 . Remark. If any number consisting of h, t and u be squared the same steps will be taken as in the above example ; i. e., the square of a number consisting of h, t and u is composed of the square of the h plus twice the h multiplied by the t plus the square of the t plus twice the sum of the h and t multi- plied by the u plus the square of the u. 237. A Number Consisting of Tenths and Hundredths Involved to the Second Power. Example. .24* = what ? ,24 .24 .0016 = .04 2 = .h 2 . .008 =.2 X .04=.t X ,h.. .008 == .04 X -2 = -t X -h. .04 = .2 2 = .t 2 . .0576 = .24 2 = .t 2 +2.tX.h+.h 2 . Remark. It is thus seen that the square of a number con- sisting of tenths and hundredths is composed of the square of the tenths plus twice the tenths multiplied by the hundredths plus the square of the hundredths. 132 * 238. A Number Consisting of Tenths, Hundredths and Thousandths Involved to the Second Power. Example. .246* = what ? .246 .246 .000036 = .006* = .th 2 . .00024 = .04 X .006= .h X -th. .0012 = .2 X -006 = .t X -th. .00024 = .006 X .04 = .h X .th. .0016 = .04 2 = .h 2 . .008 = .2 X .04 =-= .t X -h. .0012 =.006 X .2=.t X .th. .008 = .04 x .2 = .t X -h. .04 == .2 2 = .t 2 . .060516 = .246 2 =.t 2 +2.tX.h+.h 2 +2[.t+.h]X.th-hth 2 . Remark. It is thus seen that the square of a number con- sisting of tenths, hundredths and thousandths is composed of the square of the tenths plus twice the tenths multiplied by the hundredths, plus the square of the hundredths, plus twice the sum of the tenths and the hundredths multiplied by the thou- sandths, plus the square of the thousandths. 239. A Number Consisting of Tens, Units and Tenths Involved to the Second Power. Example, 32.6 2 = what ? 32.6 32.6 .36 = .6 2 = .t 2 . 1.2 = 2 X .6 := u X .t. 18. = 30 X .6 = t X .t. 1.2 = .6 X 2 = u X -t. 4. = 2 2 = u 2 . 60. = 30 X 2 = t X u. 18. = .6 X 30 = t X -t. 60. =2_X30 = tXu. 900. = 30 2 = t 2 . 1062.76 =3276 2 = 163 Remark. It is thus seen that the square of a number con- sisting of tens, units and tenths is composed of the square of the tens, plus twice the tens multiplied by the units, plus the square of the units, plus twice the sum of the tens and the units mul- tiplied by the tenths, plus the square of the tenths. 240. General Formula Embodying the Involution of any Number in the Decimal Scale to the Second Power. A general principle for exhibiting the elements which compose the second power of a number consist- ing of any number of terms (orders) in the decimal scale may be thus stated : The square of a number consisting of any number of terms is composed of the square of the first, or high- est term plus twice the first term multiplied by the sec- ond, plus the square of the second, plus twice the sum of the first two terms multiplied by the third, plus the square of the third, plus twice the sum of the first three terms multiplied by the fourth, plus the square of the fourth, plus twice the sum of the first four terms multiplied by the fifth, plus the square of the fifth, plus, etc. Remark. A fraction is involved to the second power by squaring both its terms. 241. Evolution of the Second Root. Example L i/TISS = what ? Written form. Thought form. 1156(34 900 6t)256 240 16 16 Since there are four terms in the square there are two terms in its square root,viz. tens and units. The square of a number consisting of t and u is composed of the t 2 -|-2tXu-j-u 2 . The sq. of t = h ; hence the h of the power con- tain the sq. of the t of the root. The greatest square 164 number of h in 11 h is 9 h, the sq. root of which is 3 t. 9 h taken from the power leave 256. This remainder contains a product of which twice the tens of the root, or 6 t, is one factor and the units of the root is the other factor. Since tXu = t, the 25 t of the remain- ing part of the power contain the required product. Since 25 t contain a product of which one factor is 6 t, tLe other factor is found by dividing 25 t b} T 6 t; the quotient, 4, is supposed to be the units of the root. The product of 6 t by 4 = 24 t, which taken from the remaining part of the power leave 16. This remain- der must contain the square of the u of the root. The square of 4 = 16, which taken from the remaining part of the power leave nothing. 1156 is thus found to be a square number ^of which 34 is the square root. Example 2. 1/54756 = what ? Written form. Thought form. 54756(234 4 Since there are five terms in the given square there are three terms in its square root, viz. h, t and u. The square of a number consisting of h, t and u, is composed of the square of the h, plus twice the h mul- tiplied by the t, plus the square of the t, plus twice the sum of the h and the t multiplied by the u, plus the square of the u. The sq. of h tth ; hence the tth of the power- contain the sq. of the h of the root. The greatest square number of tth in 5tth is 4tth ; the square root of which is 2h. 4tth taken from the power leave 14756. This remainder contains the product of two factors, one of which is twice the h of the root, or 4 h, 165 and the other the t of the root. Since hxt = th, the th ol the remaining part of the power contain the re- quired product .... Since 14 th contain a product of which one factor is 4 h, the other factor is found by di- viding 14 th by 4 h, the quotient, 3 t, is supposed to be the tens of the root. The product of 4h by 3t = 12 th, which, taken from the remaining part of the power leave 2756. This remainder contains the sq. of the t of the root. The square of 3t = 9h, which taken from the remaining part of the power leave 1856. This remainder contains the product of two factors, one of which is twi;e the sum of the h and the t of the root, or 46 tens, and the other the units of the root. Since t X u = t, the tens of the remaining part of the power contain the required product. Since 185 tens contain a product of which one fac- tor is 46t, the other factor may be found by dividing 185 tens by 46 tens, the quotient, 4, is supposed to be the u of the root. The product of 46t by 4 = 184t, which tat en from the remaining part of the power leave 16. This remainder must contain the square of the u of the root. The sq. of 4 = 16, which taken from the remaining part of the power leave nothing. 54756 is thus found to be a square number the square root of which is 234, Remark. It is observed that the tens and unite of the root as found are supposed to be the correct numbers for those or- ders, respectively. Sometimes the obtained quotient may be so great that its product by the divisor will, when subtracted from the remaining part of the power, leave a remainder too small to contain the partial products that are yet to be taken out. If a quotient is too great a less quotient must be used. Exercises. Evolve the second root of 4096; 9216; 7569; .0676; 54.76; 5476; 10201; 7744; 10509; 6889; 11025; 11236; 3844; .9409; 53.1441; 21025; 173056; 998001; 67305616. 166 242. Table for Determining the Number of Terms in the Cube Root of a Given Cube Number. 1 8 =1 From the table it is seen that a 9 3 =729 cube number consisting of three terms 10 3 1000 or less, contains one term in its cube 99 3 970299 root; that a cube number consisting of 100 3 1000000 four, five or six terms contains two 999 3 =997002999 terms in its cube root; that a cube number consisting of seven, eight, or nine terms contains three terms in its cube root ; etc. 243. Principle. There are as many terms in the cube root of a cube number as there are periods of three terms each in the number, counting from the interval between units' and tenths' orders. liemarks. 1. The highest period in a cube integer may con- tain but one or two terms. 2. The terms of the highest period of a cube decimal fraction may be wholly or partly represented by zeros. 3. The lowest period of a cube integer may be wholly or partly represented by zeros. Every period in a cube number may be partly represented by zeros. 244. Table Indicating the Lowest Place in which a Significant Figure can Occur in the Written Cube of Numbers in the Orders Named. u 3 - u. .t 3 = .th. [For other indicated pro- t 3 = th. .h 3 = .m. ducts see corresponding ta- li 8 == m. .th 3 = .b. ble, page 160.] th 3 = b. Etc., etc. 245. A Number Consisting of Tens and Units Involved to the Third Power. Example. 34 3 = what ? In the light of the definition of a 3d power, a number is cubed by multiplying its square by the number itself. 167 (30 -f 4) 2 = 30 2 -f 2 times 30 X 4 + 4 2 = 30 + 4 30 2 X 4 + 2 times 30 X 4" -f 4 3 30 s + 2 times 30 2 X 4 + 30 X 4 2 30 3 + 3 times 30* X 4 + 3 times 30 X 4 2 + 4 3 = 39304 Hence. The cube of a number consisting of tens and units is composed of the cube of the tens, plus 3 times the square of the tens multiplied by the units plus 3 times the tens multiplied by the square of the units, plm the cube of the units. 246. A Number Consisting of Hundreds, Tens and Units, Involved to the Third Power. Example. 234* what ? 234 = 200 + 30 + 4. (200 + 30 + 4) 2 =, 200 2 + 2 times 200 X 30 -f 30 2 + 2 times (200 + 30) X 4 + 4 2 [See Art. 236.] Re-writing this formula after removing the parenthesis we have 200 2 -f2 times 200x30-f-30 2 -|-2 times 200x4-}-2 times 30X4-J-4 2 Multiply this formula by . 200+30+4 and write the partial products below. Upon comparing these 2 times 30 X 4 2 partial products we find that 2 times 200 X 4 2 we have 200 s + 3 times 200 2 30 2 X 4 X 30 + 3 times 200 X 30 2 2 times 200 X ^0 X 4 + 30 3 + 3 times 200 2 X 4 200 2 X 4 + 6 times 200 X 30 X 4 -f 30 X 4 2 3 times 30 2 X4+3 times 200 2 times 30 2 X 4 X4 2 + 3 times 30 X 4 2 +4 3 . 2 times 200 X 30 X 4 Factoring the 5th, 6th 30 3 and 7th terms of this formu- 2 times 200 X 30 2 la reduces them to 3 times 200 2 X 30 (200 2 -f 2 times 200 X 30 200 X 4 1 + 30 2 ) X4. Observing that 2 times 200 X 30 X 4 the parenthetical quantity 2 times 200 2 X 4 equals (200 -f 30) 2 , the re- 200 X 30 2 duced expression becomes 3 2 times 200 2 X 30 times (200 + 30)* X 4. 200 3 168 Factoring the 8th and 9th terms of the formula above, reduces them to 3 times (200 -f 30) X 4 2 . Re-writing the formula, substituting 3 times (200 -f30) 3 X 4 for the 5th, 6th and 7th terms, and 3 times (200 -1- 30) X 4 y for the 8th and 9th terms, we have as the completed formula 200 3 -f 3 times 200' X 30 -f 3 times 200 X 30 2 -f 30 3 + 3 times (200 + 30) 2 X 4 + 3 times (200 + 30) X 4 2 + 4 3 . Any number consisting of h, t and u may be cubed in the same manner as the above ; hence, the cube of a number consisting of hundreds, tens and units is composed of the cube of the hundreds, plus 3 times the square of the hundreds multiplied by the tens, plus 3 times the hundreds multiplied by the square of the tens, plus the cube of the tens, plus 3 times the square of the sum of the hundreds and tens multiplied by the units, plus 3 times the sum of the hundreds and tens multiplied by the square of the units, plus the cube of the units. This formula is abbreviated thus : h 3 -j- 3h 2 X t -f 3 h X t ? -f t 3 + 3 [h + t] 2 X u + 3 [h -f t] X u 2 + u 3 . The cube of a number consisting of th, h, t and u is composed of th 3 -f 3 th 2 X h ' .+ 3 th X h* -f h 3 -f 3(th -f h) 2 X t -f 3(th + h) X t 2 -f t 3 + 3(th + h +t)' X u -f 3(th + h + t) X u 2 -f u 3 . Remarks. 1. A careful study of the above forms will discover the law by which a formula may be constructed for the cube of a number consisting of any number of orders in the decimal scale. 2. The cube of a fraction is found by cubing both its terms. 169 Evolution of. the Third Root. Example. ^103823 = what ? Written form. Thought form. 103823(47 Since there are six terms ia the 64 power, there are two terms in its cube 48h)39823 root ; viz. t and u. 336 The cube of a number consisting 6223 of tens and units is composed of \?-{- 588 3t 2 X u + 3t X u 2 -f- u 3 . The cube of tens is thousands, hence the th of the power contain the cube of the t of the root. The greatest cube num- ber of th in 103 th is 64 th, the cube root of which is 4t. The power diminished by 64th = 39823, which contains a product of which one factor is 3 times the square of the t of the root and the other is the u of the root. 3 times the square of 4 t = 48h. Since the product of h by u h, the hundreds of the remain- ing part of the power contain the required product. 398h -j- 48h = 7, which is supposed to be the u of the root. 7 times 48h = 336 h, which taken trom the remaining part of the power leave 6223. This re- mainder contains the product of 3 times the tens of the root by the square of the u of the root. 12 t X 49 = 588t, which taken from the remaining part of the power leave 343. This remainder must contain the cube of the u of the root. The cube of 7^=343, which taken from the remaining part of the power leaves nothing. We thus find that 103823 is a cube number and that 47 is its cube root. Remarks. 1. In the light of the foregoing form the obser- vant pupil will readily evolve the third root of any cube num- ber. 2. For the application of cube root in determining lines see text books on Arithmetic. 170 Exercises. Evolve the third root of 32768 ; 68921 ; 148877 ; 250047; 287-496; 328.509; .389017; .438976 ; 1061208; 1092727; 1124864; 1157.625; 1.191016; 870983875; 12977875 ; 5735339 ; 300763 ; 912673. Cube Boot by Endings. Example. 12167 is a cube number; what is its cube root ? Solution, a. Since 7 is the units of the cube, 3 must be the units of its cube root, b. The greatest cube in 12, (the left period) is 8, the cube root of which is 2 ; hence 2 is the tens of the cube root. The required root is, therefore, 23. SECTION XV. TEST PROBLEMS. 1. If a merchant mark his goods 25% above cost, and sell them at 25 % below the marked price, does he gain or lose and at what rate % ? 2. A load of hay weighs 15 cwt.21 lb. ; if 2 cwt. 11 lb. be sold, what part of the load remains ? 3. Multiply .025 by 2.5. 4. A owned f of a store and sold to B -J of his share and to C f of his share; what part of it did he still own ? 5. Eequired the cost of I of a yard of cloth, if | of a yard cost $7f . 6. The longitude of Washington is 76 56' west of London, what change would it be necessary to make in a time-piece in coming from London to Washington? 171 7. At what rate per cent, will $380 in 7 yr. 3 mo. yield $165.30 interest ? 8. = what part of 9 ? What per cent, of 9 ? 9. If 9 eggs weigh a pound, and a pound of eggs equal a pound of steak as food, at what price per dozen must eggs be bought in place of steak at 22 / per Ib. ? 10. What is the weight of the air in a room 25 ft. by 20 ft. by 12 ft., water weighing 770 times as much as air, and a cubic foot of ^vater weighing 1000 oz. Avoirdupois ? 11. If lead is 11.445 times as heavy as water, what is the weight of a piece of lead 1m. by 2 dm. by 5 cm. ? 12. A lot of goods was marked 40% above cost; if sold at 30% less than the marked price, was there a gain or loss and at w hat rate per cent. ? 13. Which would yield the better pay, a 1% bond at 115 or a 6% bond at 98, and at what rate better? 14. If a merchant sell flour at $9 per barrel, and wait 6 months for his pay, at what price could he af- ford to sell for cash if money is worth 2% a month? 15. For what sum must a 3 months' note be drawn so that when discounted by a bank at 7 per cent. I may get $400 ? 16. At what quotation must I buy a 6% stock to make as good an investment as from a 4% stock at 75 ? 17. A traveling salesman is allowed 12 % com- mission on his sales ; his employer's rate of profit is 20% on the goods sold; what is the first cost of goods which the salesman sells for $7.66 ? 18. A water tank is 3 ft. deep, 4 ft. long, and 4 ft. wide ; it is supplied from a flat roof 20 ft. by 30 ft. ; what depth of rain must fall to fill the tank ? 19. Eequired the present worth of $1320 due in 3 yr. 4 mo. without interest, if money is worth 6 % . 172 20. A man bought a horse for $72 and sold it for 25 per cent, more than it cost and 10% less than the asking price ; what did he ask for the horse? 21. 2 ft. 9 in. = what part of a rod ? 22. Cincinnati is 7 49' west of Baltimore; when it is noon at Baltimore, what time is it at Cincinnati ? 23. A man bought stock at 25% below par, and and sold it ai 25% above par; required his rate of gain. 24. How many yards of carpet f of a yard wide will cover a floor 18 ft. by 15 ft. ? 25. I -5- f = what? 26. Reduce to equivalent fractions with 1. c. d. , f and f . 27. How find the area of a circle if only the ra- dius be given? 28. If a wheel turn 17 30' in 35 minutes, in what time will it make a revolution ? 29. If rosin be melted with 20% of its weight of tallow, what % of the weight of the mixture is tallow? 30. Eequired the weight in kilograms of a bar of iron 3.6 m. long, 6 cm. wide, and 2 cm. thick, if iron is 7.8 times as heavy as water. , 31. With gold at 103 what rate of interest do I make on a $1000 5-20 bond bought at 106? 32. A and B are partners for 1 yr., A putting in $2000 and B $800 ; how much more must B put in at the end of 6 months to receive one-half the profits ? 33. How many grams does a Dl. of water weigh? 34. Demonstrate the principle: The product of the means of a proportion equals the product of the extremes. 35. The surface of the earth contains about 144000000 sq. mi. of water, and about 53000000 sq, mi. of land . What % of the earth's surface is water ? 173 36. f X f = what ? 37. If a watch sell for $60 at a loss of 22%, for what should it sell to gain 30 per cent. ? 38. A broker bought stock at 8% premium and sold it at 9% discount thereby losing $510; how many shares did he buy ? 39. What is the net tax in a town whose taxa- able property is assessed at $430000, at 12 mills per dollar, 5 per cent, being paid for collection ? 40. The difference of time between two places is 2 hr. 15 min. 10 sec. ; required their difference in longitude. 41. Eeduce .037 lb. Av. to drams. 42. Reduce 84.5 ars. to square meters. 43. An agent received $484.50 with which to buy sheep after deducting his commission at 2% ; how much money did he spend for sheep ? 44. A policy for $2675 cost $53.30 ; find the rate of insurance. 45. The difference of time between two places is 45 min. 30 sec., and the place having the earlier time is in longitude 85 40' west; required the longitude of the other place. 46. Reduce .096 of a bu. to the decimal of a pint. 47. A note of $125 dated May 3, 1883, and payable in sixty days, with interest at 5%, was discounted June 18, 1883, at 10% ; required the proceeds. 48. How many bu. of wheat will fill a bin 8 ft. by 5 ft. by 4 ft. ? 49. How many meters of carpet .7 m. wide will cover a floor 4 m. by 4.5 m. ? 50. Add | f and f 51. Multiply f by 2.3. 52. Divide 2.3 by f . 174 53. f = what part of 2.3? What % of 2.3? 54. Reduce 135 sq. rd. 54 sq. ft. to the decimal of an acre. 55. How much money must I remit to my agent to buy goods and pay himself $24 commission at 1%? 56. A sold a horse to B and gained \ of its cost ; B sold it for $80 and lost \ of what it cost him ; how much did A pay for the horse ? 57. Divide 3.45 by 1.5. 58. A garden contains 800 sq. rd. and is 33^ rd. long; how wide is it ? 59. What per cent, of f is f ? 60. What is J per cent, of .5 ? 61. 428 is 1% more than what number? 62. A man owes $300 due in 4 mo., $600 due in 5 mo., and $100 due in 6 mo. ; if he pay i of his indebt- edness in 2 mo., when in equity should he pay the balance ? 63. What is the difference between the true and the bank discount of $359.50 for 90 days without grace, money being worth 8% ? 64. A board is 20 ft. long and 9 in. wide ; what is it worth at $30 per M ? 65. At what time between 6 and 7 o'clock are the hour and minute hands of a watch together? 66. A broker bought 60 shares of stock at 106J, received a 5% dividend and then sold at 104; did he gain or lose and how much ? 67. How many ft,, board measure, each board be- ing 18 inches wide, can be cut from a squared log 16 ft. long, 18 in. wide and 10 in. thick, allowing in. for each cut of the saw ? 68. If 12 men can do a piece of work in 5 days, in how many days can 8 men and 5 boys do it, 1 man doing the work of 2? boys ? 175 69. If a note of $5000, for 4 mo., at 6% int. per annum, be discounted in bank, on day of making, at at 8% per annum, what will be the proceeds? 70. If it cost $312 to fence a field 216 rd. by 24 rods, what cost the fence of a sq. field of equal area? 71. A merchant bought goods at 20 cents per yd, and sold them at 40% profit, after allowing his cus- tomers 12% discount off; what was the marked price? 72. A vessel, at noon, sails due north ; after a certain time an observation shows the sun to have sunk toward the west 2 signs 15 degrees ; how long has the vessel been sailing ? 73. I sell a bill on London for 1675, at the rate of 24.3 cents per shilling ; how much do I receive? 74. If & 6000 of 6% stock be sold at 90, and the proceeds invested in 10% stock at 105, what will be the change in the income? 75. I buy $1500 worth of goods at 4 mo., $850 at 3 mo., $1750 at 5 mo. ; what is the equated time for the payment of the whole ? 76. The square root of .1369 plus the square root of 1296 equals what? 77. A owesB $1800; B offers to allow 5% off for cash ; A pays $1425, how much is still due ? 78. Evolve the second root of 15625. 79. What is the surface of a cube which contains 8 times the volume of a cube whose edge is -f of a foot ? 80. What is the capacity of a cylinder 20ft. long, whose radius is 2 ft. ? 81. I bought goods in Europe, paid 20% duties, a commission of 2% upon duties and cost, and sold them at $10 per yard, clearing 34% on invoice price; required the entire cost per yard. 82. A travels 5^ hours at the rate of 6 miles per hr., B then follows from the same point at the rate of 9 mi. per hr. ; how long will it take B to overtake A ? 176 83. At 24.2 cents per shilling, what cost 1050? 84. What principal in 3 yr. 4 mo. 24 da. at 5% will amount to $761. 44? 85. For what sum must a note dated April 5, 1883, for 90 da. be drawn, that when discounted at 7%, April 21, 1883, the proceeds may be $650? 86. A room is 26 ft. long, 16 ft. wide and 12 ft. high ; what is the distance from one of the lower cor- ners diagonally to the opposite upper corner ? 87. St. Petersburg is 30 19' east longitude, and Indianapolis is 86 5' west longitude; when it is 3 a. m. at St. Petersburg, what is Indianapolis time ? 88. Reduce 492 dekagrams to quintals. 89. How many bricks, each 8 in, long, 4 in. wide and 2? in thick, will be required for a wall 120 ft. long 8 ft. high and 1 ft. 4 in. thick, no allowance being made for mortar ? 90. If 6 men cai. build a wall 20 ft. long, 6 ft. high and 4 ft. thick in 16 days, in what time can 24 men build a wall 200 ft. long, 8 ft. high and 4 ft. thick ? 91. A man bought a square farm containing 140 acres and 100 sq. rd. ; required the length of one side. 92. Evolve the 3rd root of 592704. Of 2985984. 93. A farmer exchanged 100 bu. of wheat at $1.25 per bu. for corn at $.37-J per bu.; how many bu. of corn did he receive ? 94. The sum of two numbers is 785 ; their differ- ence is 27 ; what are the numbers ? 95. Divide Yr 8 b 7 iV 96. Reduce $ to a decimal fraction. 97. How many loads are contained in a pile of wood 40.16 ft. long, 7.04 ft. high and 4 ft. wide, if each load contains 1 J cords ? 98. If I exchange $12000 of 8% stock at 115 for 177 5% stock at 69, do I gain or lose on annual income, and how much ? 99. How large a sight draft can be bought for $259.52, exchange being lf% premium ? 100. What are the cubical contents of a cylinder that will just enclose a sphere 9 in. in diameter? 101. Evolve the second root of -^ to two decimal places. 102. Three farms contain respectively, 356, 898, and 1254 acres, which I desire to cut into building lots of the largest equal size possible ; how many acres will each lot contain ? 103. Divide f by f . 104. Multiply .803 by .03. 105. The sun at 12 o clock is over the meridian at Washington; over what meridian will it be after trav- eling through 5 signs and 5 degrees ? What time will it then be at Washington ? 106. How many grams does a liter of rain water weigh ? 107. At 7 cents per square foot, required the cost of a brick walk 6 feet wide around a lot 200 ft. by 300 ft. 108. What sum of money loaned ut 6% for 10 mo. will yield as much interest as $750 loaned at 4% for 11 months ? 109; Required the area of a circle whose radius is 10 feet. 110. A, B and C eat 8 loaves of bread of which A furnishes 3, and B 5. C pays A and B8 pieces of mon- ey of equal value ; how should they divicle the money? 111. For what sum must I make a bank note for 60 days, which discounted at 10% will pay a $1000 debt now due ? 112. A street 60 ft. wide is crossed at right angles by another 80 ft. wide, what is the distance between diagonal corners? 178 113. A wall is 83 meters long, 2 Dm. high, and 5 dm. thick ; what is its value at $3.30 per cu. meter?- 114. a. What is the value of the N. E. \ of the N. W. } of section 16, at $12 per acre ? 6, Make a plat of the congressional township and indicate the part described. 115. A miller takes for toll 4 quarts from every 5 bu. of grain ; what per cent, does he get ? 116. In what time will $375.40 yield $37 54 inter- est at 6% per annum? 117. A barri is 40 ft. wide; the comb is 15 ft. from the plate and the rafters are of equal length ; what is the length of each rafter ? 118. If money is worth 12%, what is the true dis- count of $235. ]0, due one year hence ? 119. In a cube whose edge is in., how many cubes each ^ of an inch wide ? 120. A rectangular field 15 rods wide contains 3 acres ; how long is it ? 121. How many yards of carpet 27 inches wide will cover a floor 18 ft. long by 14 ft. wide ? 122. If an article be sold for twice its cost, what is the rate per cent, of gain ? 123. If an article be sold for one-half its cost, what is the rate per cent, of loss ? 124. If an article is sold for one third its cost, what is the rate per cent, of loss? 125. Property worth $8760 is rented for $650 per annum; what rate of interest does the investment yield? 126. In what time will $2450 yield $725 int. at S%? 127. A man recei"es $280 interest, annually, on a 1% loan; what is the face of his loan? 128. What principal will amount to $5750 in 3 yr. 5 mo. 17 da. at Q% ? 179 129. For what ^sum must property worth $6000 be insured at 5% to* cover f of the property and the premium ? 130. What is the selling price of corn whoso first cost is40cts. per bu., freight 8% and rate of gain 16|-%? 131. A merchant sold two bills ot goods for $50 each; on one he gained 15%, and on the other he lost 15% ; required his gain or loss. 132. What is the rate of interest on an investment in U. S. 4 per cents, at 95 ? 133. A merchant is offered goods for $2500 cash, or for $2650 on 60 days time ; which is the better offer if money is worth 8% per annum? 134. For what must 5% stock be bought that it may yield 7% interest on the investment ? 135. U. S. 3f s bought at 98 pay what rate of in- terest on the investment ? 136. What sum of money will yield as much in- terest in 10 months at 6%, as $1500 will yield in 12 mo. at 4% ? 137. How much must be paid in U. S. currency for a draft of 210 10s. 6d., exchange being 102, bro- kerage %, and gold at 104? 138. Reduce -f, f, -^ to equivalent fractions having the 1. c. d. 139. Multiply by f ; -f T by .04; 2.06 by ;0013. 140. Divide -?- by f ; .004 by f ; 3.064 by .08. 141. Reduce 5 yd. 2 ft. 7 in. to inches. 142. Reduce I mile to lower integers. 143. A box is 4| m, long, 8 dm. wide and 3.5 dm. deep ; how many Kg. of distilled water will it hold ? 144. A pile of wood is 15.5 m. long, 12 dm. wide, and 1.8 meters high; how many sters does it contain? 145. f ==what % off? 180 146. Reduce -$ to the decimal^ scale. 147. Two men paid $150 for a horse ; one paid $90 and the other paid $60 ; they sold it so as to gain $75 ; what was the share of each ? 148. A man owes A $105, B $75. and C $120 ; he has only $125 ? how much should he pay to each? 149. Find g. c. d. of 348 and 1116. 150. What is the face of a note payable in 90 days, on which $3500 can be obtained at a bank, discounting at 6jg? 151. An irregular mass of metal immersed in a vessel full of water caused 2.25 1. of water to overflow the sides of the vessel ; what was the weight of the metal if its specific gravity was 7.2 ? 152. What is the price, at $11.65 per kilogram, of 1 liter of alcohol, it s. g. boing .79? 153. What is the capacity in liters, of a cylin- drical cup 16 cm. in diameter and 1.3 dm. deep? 154. A& B gain in business $4160, of which A is to have 8% more than B, how much will each receive? 155. A merchant insures a cargo ol goods for $3456 at 3%' the policy covering both property and premi- um ; what was the value of the property ? 156. If a stack of hay 82 ft. high, weigh 7 cwt., what weighs a similar stack 15 ft. high ? 157. For what sum must a note be drawn at 3 mo. that the proceeds, when discounted, without grace, at a bank, at 8% shall be $1274? 158. The contents of a cubical block of stone are 4913 cu % cm. ; required its superficial contents in sq. m. 159. I imported 12 casks of wine, each containing 48 gallons invoiced at $2.75 per gal. ; paid $108 for freight, and an Ad Valorem duty of 36% ; what is my rate of gain if I sell the whole for $3258 ? 181 160. A New York merchant gave $960 for a bill of 250 on London, required the rate of exchange. 161. What is the largest square stick that can be cut from a log 4 ft. in diameter ? 162 A sea captain set his watch at London, and found after traveling that it was 4 hours faster than the local time of the place he had reached ; how far had he traveled and in what direction ? 163. What is the area of a field whose parallel sides are 90 rods and 124 rods long, respectively, and the perpendicular distance between them is 50 rods ? 164. A has $550, and B has $330 ; what per cent, of the money of each is the money of the other ? 165. A farmer's wagon loaded with wheat weighed 4912 lb., and his wagon alone weighed 920 lb., what was his load of wheat worth at 95 cts. per bu. ? 166. If a bolt of paper is 8 yd. long and J yd. wide, how much will the paper cost at 30 cts. per bolt, to pa- per the walls and ceiling of a room 18 It. long, 15 ft. wide and 10 ft. high ? 167. A man paid $30.09 for the use $204 for 11 mo. ? what was the rate of interest ? 168. What is now due on a note given Sept. ], 1881, the principal being $430, the rate 7%, per annum, endorsed Aug, 4, 1882, $34, July 7, 1883, $118? 169. A railroai right of way 4 rods wide passes diagonally through a quarter section of land ; ho w many acres remain unoccupied by the road ? 170. Reduce f da., 3 min. to the decimal of a day. 171. A man owned a farm of 40 A. 3 R. 22 Sq. rd., and sold it at $45.50 per acre ; what sum did he realize? 172. The selling price of a house was 130% of the cost ; the gain was $240; required the cost. 173. The interest was $45, the rate 8% per annum, the time 1 yr. 3 mo. 18 da. ; required the principal. 182 174. .Reduce 6 oz., Av. to the decimal of a cwt. 175. How many boards 16 ft. long will be required to fence a lot 125 ft. by 80 ft., the fence being 5 boards high ? .Required its cost at $27 per M, each board be- ing 5 in. wide and 1 in. thick. 176. Divide sixteen ten-miilionths by twenty-five ten-thousandths. 177. How many bricks can be laid in a pavement 25 ft. 4 in. long by 15 ft. 3 in. wide, each brick being 8 in. by 4 in. ? 178. When it is 2 o'clock at Terre Haute, what is the time 47 13' 27" west of Terre Haute ? 179. The capacity of a cubical cistern is 74088 cu. in. ; how many sq. ft. in the bottom of it ? 180. What cost 2 Ib. 4 oz. of cloves at 87 cents per Ib. ? 181. At $11 per rod, what will be the cost of fenc- ing a lot in the form of a right angled triangle, whose sides forming the right angle are 264 rods and 23 rods, respectively ? 182. A grocer bought a hhd. of sugar weighing 3 cwt. 2 qr. 20 Ib. for $30, and sold it at 7J Ib. for $1 ; what per cent, does he gain ? 183. A 6 months' note for $5000, drawing interest at 6% per annum, was discounted not in bank 3 mo. after date at 10% per annum ; required the proceeds. 184. A physician bought 2 Ib. 9 oz. of quinine at $38 per Ib., Avoirdupois, and sold it in 10 grain doses at 20 cts per dose ; did he gain or lose and how much ? 185. Find the Amt. of $3350 for 3 yr. 9 mo. at 8%. 186. If you travel 360 miles in 12 days ot 8 hr. each, how many miles can you travel, at the same rate, in 60 days of 6 hr. each ? 187. What is the side of a cube which contains as many cu. it. as a box 8 ft. 3 in. long, 3 ft. wide, and 2 ft. 7 in. deep ? 183 188. What per cent, of 14 bu. is 5 bu, 3 pk. 5 qt. ? 189. Find the g. c. d. of 368 and 612. 190. I = what decimal fraction ? 191. If 50 men build 50 rods of wall in 75 days how many men will build 80 rods of similar wall f as thick and as high in 40 days ? 192. A farm is 125 rods square and a rectangular field containing the same area is 130 rods long; how wide is it ? 193. By what must I be multiplied that the pro- duct may be 12 ? 194. A flash of lightning is seen 47 sec. before its peal ot thunder is heard ; required the distance of the thunder cloud. 195. What is the difference betweeen the simple in- terest of $1000 at 6%, for 5 yr., and the true discount of the same sum for the same time at the same rate ? Which is the greater and how much ? 196. What is the value in English money, of $500 at $4.84 per ? 197. How many square yards of canvas would be required to make a balloon of globular form, 20 yards in diameter? 198. If linen is bought at 2 s. 9 d. per yd., for what must it be sold to gain 25% ? What is the selling price in U. S. money ? 199. If the fore wheels of a wagoii be 4 feet in di- ameter and the hind wheels 5 feet in diameter, how many more revolutions will the former make than the latter in going a mile ? 200. On a base of 120 rods, a surveyor wishes to lay off a rectangular field that shall contain 60 A. ; how far from his base line must he run out ? 201. A sells to B, tea worth 45/ for 48/; what should B charge A for sugar worth 9/ to balance the transaction ? 184 SECTION XVI. REVIEW QUESTIONS AND TOPICS. Define mathematics. What is conditional for extension? What branches of knowledge involve a study of extension ? What is known of every conscious mental state ? How are mental states 5 known to be distinct ? How does the idea of number arise? What is conditional for the num- erical idea? Define Arithmetic as a science; as an art. What attribute furnishes the basis of number in particular? By what process does 10 the mind obtain the idea one ? Define the integral unit, or unit one. How many classes of second^ ary ones are mentioned ? Define a fractional unit ; a multiple unit. What is the primary idea in Ar- ithmetic? How are other units related to the unit 15 one? What is a unit object? Define a number. Of what units may a number be composed ? On what basis are numbers classified as integers and fractions? Define an integer; a fraction. On what basis are numbers classified as abstract and 20 concrete ? Define an abstract number ; a con- crete number. On what basis are abstract inte- gers classified as prime and composite ? On what basis are numbers classified as simple and de- nominate? Define a simple number; a denomin- 25 ate number; a compound number. How is a de- nominate number sometimes defined ? What ob- jection ? How is a compound number sometimes defined? What objection ? Define notation. What kinds are usually presented in Arithmetic ? 30 What is the alphabet of the Eoman notation ? What does each letter signify ? What are the limits of the Boman notation? State the princi- 185 pies. What is the alphabet of the Arabic nota- tion? What is the signification of each charac- 35 ter? How many and what distinct systems of numbers make use of the Arabic characters? Define a scale; the decimal scale? What is a unit of the first order? How are units of dif- ferent orders formed? What are periods in the 40 decimal system of numbers? What orders are embraced by any period? What is meant by a decimal division of 1? How are lower orders ot units formed? State the principle of the decimal scale. What is the office of the decimal point? 45 Describe units' place. How many and what kind of units may be expressed therein? Describe tens' place. How many and what kind of units may be expressed therein? How may higher decimal units be expressed? How may lower decimal units 50 be expressed? What is the representative scale? What is the simple value of a figure? What is the local value of a figure? What are the limits of the decimal system of numbers? Define numeration ; reading numbers. In reading a number where 55 should the word and be used? What elements constitute the primary idea of a fraction? How many and what numbers are necessary to the idea of a fraction? What are these numbers together called? Define the denominator; the numerator. 60 What is the denomination of a fraction? What exception? How may decimal fractions be notat- ed? How are other fractions notated? What names are applied, respectively, to the terms of a written fraction? How is a compound number 65 thought? In compound numbers the number of denominations is how limited? What kind of a 186 scale has each of the common measures? Each de- nominate number forming a part of a compound num- ber is how thought? How is a compound number 70 notated? How are the parts of a compound num- ber written as to abbreviation and punctuation? How is a compound number read? Define reduc- tion; descending; ascending; and state how each is effected? Define computation? How are num- 75 bers compared previous to computing? What mental act effects a computation? How many and what operations does the mind perform upon num- bers? What is the primary judgment in computa- tion? What is an equation? What are the mem- 80 bers of an equation? Define sum, addition, addends. State the mental acts involved in addition. State and illustrate the principles of addition. Describe the sign of addition and state its use. Show by an example that reduction may be necessary in addi- 85 tion. Which reduction may be involved in addi- tion? Define product. What other term means the same? Define multiplication. State the gene- sis of multiplication. What act of synthesis is involved? What is the act of multiplication? 90 Define multiplicand ; multiplier. If the multiplier is an integer what relation do multiplicand and multiplier sustain? If the multiplier is a fraction in what consists the multiplication? Define a fac- tor. State and illustrate each of the ten principles 95 of multiplication. Describe and state the use of the sign of multiplication. Which reduction is often involved in multiplication? Show by exam- ple how reduction may be necessary in multiplica- tion. What is meant by "continued product?"- 100 Define a composite number ; define composition ; a 187 prime number. Under what conditions are num- bers relatively prime? Define a common multiple ; the 1. c. m.; a common measure; the greatest common measure. State and illustrate each of the 105 principles given under composition. Define a power; involution; a root; second power ; second root. How are higher powers formed and named? How are roots named? What is the first power and first root of a number? Describe and define 110 the index of a power. Repeat the table of squares given ; the table of cubes. State and apply rules for squaring numbers. Under how many and what heads is the synthesis of numbers discussed? How many and what methods of synthesis are there? 115 What relation do the other processes called syn- thetic sustain to the one method of synthesis? State in substance the last four remarks given under synthesis? On what ground is a number resolvable into parts? Define difference, giving 120 two definitions. Define subtraction; minuend; subtrahend; remainder. State the mental acts involved in subtraction. How is the difference between two separate numbers found? Describe the sign of subtraction and state its use. State 125 and illustrate each of the principles of subtrac- tion. Under what condition must the minuend be prepared before a subtraction is effected? In what consists the preparation? Which reduction is in- volved? How may reduction be avoided in sub- 130 traction? Define quotient; division; dividend; divisor. Define each of these terms in the light. of its relation to multiplication. What is meant by a .constant subtraction? State how division grows out of subtraction? How is division related to 188 135 multiplication? In what consists the mental act of division? Which reduction is sometimes in- volved in division? Illustrate. State and illus- trate each of the principles of division. State and illustrate each of the six general principles of divi- 140 sion. Define disposition. How is disposition re- lated to composition? How are the factors of a number found? State and illustrate each of the principles of disposition. Under what condition is a number known to be prime? Define evolu- 145 tion ; the index of a root. Describe and state the use of the radical sign ; exponent. What is signi- fied by a fractional exponent other than a frac- tional unit? Under how many and what heads is the analysis of numbers discussed? State the sub- 150 stance of the last four general remarks given under analysis. Define g. c. d. State and illustrate the principles under this head. How many methods are given for finding g. c. d.? Solve an example illustrative of each method. Define 1. c. m. State 155 the principle and solve an example, using the form given on page 53. If given numbers are not readily factored how may their 1. c. m. be found? Why? Define a fraction, giving two definitions. According to the primary idea of a fraction, what 160 is the maximum value of a fraction! ? How is an expression like 1 to be interpreted? How is a fraction notated? On what basis are fractions classified as decimal and common? Define a deci- mal fraction. How is a decimal fraction usually 165 notated? How may it be notated? What is a complex decimal? Define a common fraction. On what basis are fractions classified as proper and improper? Define a proper fraction ; an improper 189 X fraction. State wherein there is no ground for 170 this classification. On what basis are fractions classified as simple, compound and complex? De- fine a simple fraction ; a compound fraction ; a complex fraction. Show wherein this classification is not well founded. State the likeness of a frac- 175 tion to division. State and illustrate each of the general principles of fractions. State the principle under which reduction descending of fractions may be effected. Under what principle may reduction ascending of fractions be effected? How may a 180 fraction be reduced to its lowest terms? Under what condition is a common fraction not reducible to a decimal? Under what principle is addition of fractions effected? Under what principle is sub- traction of fractions effected? State the two cases 185 under which multiplication of fractions is present- ed. Solve an example under each case. State the two cases under which division of fractions is pre- sented Solve an example under each case, using a., common fractions; 6., decimal fractions. 190 Show that division of fractions is the reverse of multiplication of fractions. Divide -^ by ^ effect- ing the division by constant subtraction. Into what classes are compound numbers divided? On what basis? Write the diagram. State the order 195 to be pursued in the study of a " measure." Recite the tables of compound numbers. How many and what cases of reduction descending in compound numbers? Solve an example under each case. How many and what cases of reduction ascending 200 in compound numbers? Solve an example under each case. Define Longitude. What meridian is generally taken as prime? Make a table of corres- 190 ponding time and longitude units. What is meant by relative time? Absolute time? Show that all 205 places have not the same relative time. Explain what is meant by "Standard time," Solve exam- ples in time and longitude. What is area? What is a volume, or solid? Solve an example in which area is computed. Solve an example in which 210 volume is computed. Show that length and breadth are not multiplied together. If they could be would the product be surface? Why? Name the three primary units of the decimal system of meas- ures*. Which of these is the fundamental unit of 215 the system? How does the meter compare in length with the yard? Name the land unit. What is its area? By what units are most surfaces measured? What is the wood unit? What is its volume? By what units are most volumes measured? What 220 units are used in measuring great weights? How does each of these compare with the gram? Name each prefix designating a secondary unit. Con- struct tables for the measures of length, surface, volume, capacity, and weight, in the decimal sys- 225 tern. What relations exist by means of which capacity and weight are readily found from exten- sion? What is percentage? Name the essential terms of percentage. Define each. Define amount and difference, and state which of the essential 230 terms includes them. State the relations of per- centage. a. To multiplication. b. To factoring. c. To fractions. State each of the general canes of percentage, Give the solution of each, How many and what elements are involved in the first 235 class of applications of percentage? Name the principal applications of the first class. What ele- 191 ments are involved in the second class of applica- tions of percentage ? Name the principal applica- tions of the second class. Define profit and loss; 240 cost; selling price; gain; loss. State the relation of profit and loss to percentage by naming the cor- responding terms. What is an agent? A factor? A broker? A commission merchant? Define commission ; brokerage. State the relation of 245 commission and brokerage to percentage by nam- ing the corresponding terms. Define stock as used in percentage ; a corporation ; a firm ; a company ; a share ; a certificate of stock ; a dividend ; par value; market value; premium; discount. How 250 is stock quoted ? State the relation of stock to percentage by naming the corresponding terms. Define insurance; fire insurance ; marine insur- ance ; life insurance ; policy ; face of policy ; policy- holder ; underwriter ; premium. State the relation 255 of insurance to percentage by naming the corres- ponding terms. Define a tax ; a property tax ; a poll tax ; an income tax ; an excise tax ; an asses- sor; his duties; an invoice; tare; leakage; break- age; gross weight; net weight; specific duty; 260 advalorem duty. Define interest; principal ; amount; rate; legal rate. What is the time unit in interest? Define simple interest; a partial pay- ment. State the United States rule for computing interest in partial payments; the merchants' rule. 265 How many elements are involved in simple in- terest ? What relations are sustained among the elements? Define' compound interest; annual in- teiest; discount; true discount; present worth. State the relation of true discount to simple inter- 270 est by naming the corresponding terms. Define 192 bank discount; term of discount; proceeds; days of grace; face of a note; legal maturity; a protest. If an interest bearing note be discounted in bank, on what is the discount computed ? Stat e 275 the relation of bank discount to simple interest by naming the corresponding terms. Define commer- cial discount. State its relation to percentage. Define exchange; a draft ; a sight draft; a time draft; the face of a draft ; the course of exchange. 280 Who is the drawer or maker of a draft? The drawee ?- The payee ? State the relation of ex- change to simple interest by naming the corres- ponding terms. What is meant by the equation or average of payments ? Show the relation of 285 equation of payments to simple interest. Define Eatio. How many and what terms are used in thinking a ratio ? Define each, How is a ratio notated ? Define a simple ratio ; a compound ra- tio; an inverse ratio. State each of the six princi- 290 pies of ratio. Define proportion ? a simple pro- portion ; a compound proportion. How is a pro- portion notated? State and demonstrate the prin- ciple of proportion. Under what condition is a number divided into proportional parts? 295 State the principles. If two sides of a right-angled triangle be known, how find the third side ? The areas of similar surfaces are to each other as what ? How find the area of a trapezoid ? How find the two sides ot a rectangle, if the area and the ratio 300 of the sides are given ? How find the three di- mensions ot a rectangular solid, if the solid con- tents and the ratio of the sides are given ? How find the convex surface of a cylinder? How find the volume of a cylinder ? YB 17416 M306O22 PA 10 THE UNIVERSITY OF CALIFORNIA LIBRARY