LIBRARY OF THE UNIVERSITY OF CALIFORNIA. ^Accession 9.0.2.5..7... Class MATHEMATICS FOR COMMON SCHOOLS A MANUAL FOE TEACHERS INCLUDING DEFINITIONS, PRINCIPLES, AND EULES AND SOLUTIONS OF THE MOEE DIFFICULT PROBLEMS BY JOHN H. WALSH ASSOCIATE SUPERINTENDENT OF PUBLIC INSTRUCTION BROOKLYN, N.Y. GRAMMAR SCHOOL ARITHMETIC BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1900 COPYRIGHT, 189!, BY JOHN H. WALSHc J. S. Gushing & Co. Berwick & Smith. Boston, Mass., U.S.A. CONTENTS (HIGHER ARITHMETIC MANUAL.) PAGE INTRODUCTORY i Plan and scope of the work Grammar school algebra Con- structive geometry. II GENERAL HINTS 5 Division of the work Additions and omissions Oral and written work Use of books Conduct of the recitation Drills and sight work Definitions, principles, and rules Language Analysis Objective illustrations Approximate answers Indicating operations Paper vs. slates. XIII NOTES ON CHAPTER TEN ......... 87 XIV NOTES ON CHAPTER ELEVEN 94 XV NOTES ON CHAPTER TWELVE ........ 117 XVI NOTES ON CHAPTER THIRTEEN .....,. 149 XVII NOTES ON CHAPTER FOURTEEN 185 iii 90257 iV CONTENTS XVIII NOTES ON CHAPTER FIFTEEN ..,,.,.. 225 XIX NOTES ON CHAPTER- SIXTEEN ........ 239 XX NOTES ON THE APPENDIX 282 SUPPLEMENT. DEFINITIONS, PRINCIPLES, AND RULES i General definitions Additions Subtraction Multiplication Division Factoring Cancellation Greatest common divi- sor Least common multiple Fractions Decimals Accounts and bills Denominate numbers Percentage Profit and loss Commercial discount Commission Insurance Taxes Duties Interest Partial payments Bank discount Ex- change Equation of payments Ratio Proportion Partner- ship Involution Evolution Stocks and bonds Notes, drafts, and checks. ANSWERS 1 MANUAL FOE TEACHERS INTRODUCTORY Plan and Scope of the Work, In addition to the subjects generally included in the ordinary text-books in arithmetic, Mathematics for Common Schools contains such simple work in algebraic equations and constructive geometry as can be studied to advantage by pupils of the elementary schools. The arithmetical portion is divided into thirteen chapters, each of which, except the first, contains the work of a term of five months. The following extracts from the table of contents will show the arrangement of topics : FIRST AND SECOND YEARS Chapter I, Numbers of Three Figures. Addition and Sub- traction. THIRD YEAR Chapters II, and III, Numbers of Five Figures. Multipli- ers and Divisors of One Figure. Addition and Subtraction of Halves, of Fourths, of Thirds. Multiplication by Mixed Num- bers. Pint, Quart, and Gallon ; Ounce and Pound. Roman Notation. 1 MANUAL FOR TEACHERS FOURTH YEAR Chapters IV, and V, Numbers of Six Figures. Multipliers and Divisors of Two or More Figures. Addition and Subtraction of Easy Fractions. Multiplication by Mixed Numbers. Simple Denominate Numbers. Koman Notation. FIFTH YEAR Chapters VI, and VII, Fractions. Decimals of Three Places. Bills. Denominate Numbers. Simple Measurements. SIXTH YEAR Chapters VIII, and IX, Decimals. Bills. Denominate Num- bers. Surfaces and Volumes. Percentage and Interest. SEVENTH YEAR Chapters XL and XII, Percentage and Interest. Commercial and Bank Discount. Cause and Effect. Partnership. Bonds and Stocks. Exchange. Longitude and Time. Surfaces and Volumes. EIGHTH YEAR Chapters XIII, and XIV, Partial Payments. Equation of Payments. Annual Interest. Metric System. Evolution and Involution. Surfaces and Volumes. INTRODUCTORY 3 While all of the above topics are generally included in an eight years' course, it may be considered advisable to omit some of them, and to take up, instead, during the seventh and eighth years, the constructive geometry work of Chapter XVI. Among the topics that may be dropped without injury to the pupil are Bonds and Stocks, Exchange, Partial Payments, and Equation of Payments. Grammar School Algebra, Chapter X., consisting of a dozen pages, is devoted to the subject of easy equations of one unknown quantity, as a preliminary to the employment of the equation in so much of the subsequent work in arithmetic as is rendered more simple by this mode of treatment. To teachers desirous of dispensing with rules, sample solutions of type examples, etc., the algebraic method of solving the so-called " problems " in per- centage, interest, discount, etc., is strongly recommended. In Chapter XV., intended chiefly for schools having a nine years' course, the algebraic work is extended to cover simple equations containing two or more unknown quantities, and pure and affected quadratic equations of one unknown quantity. No attempt has been made in these two chapters to treat algebra as a science ; the aim has been to make grammar-school pupils acquainted, to some slight extent, with the great instru- ment of mathematical investigation, the equation. Constructive Geometry, Progressive teachers will appreciate the importance of supplementing the concrete geometrical instruction now given in the drawing and mensuration work. Chapter XVI. contains a series of problems in construction so arranged as to enable pupils to obtain for themselves a working knowledge of all the most important facts of geometry. Applications of the facts thus ascertained, are made to the mensuration of surfaces and volumes, the calculation of heights and distances, etc. No attempt is made to anticipate the work of the high-school by teaching geometry as a science. 4 MANUAL FOE TEACHERS While the construction problems are brought together into a single chapter at the end of the book, it is not intended that instruction in geometry should be delayed until the preceding work is completed. Chapter XVI. should be commenced not later than the seventh year, and should be continued throughout the remainder of the grammar-school course. For the earlier years, suitable exercises in the mensuration of the surfaces of triangles and quadrilaterals, and of the volumes of right parallelopipedons have been incorporated with the arithmetic work. II GENERAL HINTS Division of the Work, The five chapters constituting Part I. of Mathematics for Common Schools should be completed by the end of the fourth school year. The remaining eight arithmetic chapters constitute half-yearly divisions for the second four years of school. Chapter L, with the additional oral work needed in the case of young pupils, will occupy. about two years; the re- maining four chapters should not take more than half a year each. When the Grube system is used, and the work of the first two years is exclusively oral, it will be possible, by omitting much of the easier portions of the first two chapters, to cover, during the third year, the ground contained in Chapters I., II., and III. Additions and Omissions, The teacher should freely supple- ment the work of the text-book when she finds it necessary to do so ; and she should not hesitate to leave a topic that her pupils fully understand, even though they may not have worked all the examples given in connection therewith. A very large number of exercises is necessary for such pupils as can devote a half-year to the study of the matter furnished in each chapter. In the case of pupils of greater maturity, it will be possible to make more rapid progress by passing to the next topic as soon as the previous work is fairly well understood. Oral and Written Work, The heading "Slate Problems" is merely a general direction, and it should be disregarded by the teacher when the pupils are able to do the work " mentally." The use of the pencil should be demanded only so far as it may 5 6 MANUAL FOR TEACHERS be required. It is a pedagogical mistake to insist that all of the pupils of a class should set down a number of figures that are not needed by the brighter ones. As an occasional exercise, it may be advisable to have scholars give all the work required to solve a problem, and to make a written explanation of each step in the solution ; but it should be the teacher's aim to have the majority of the examples done with as great rapidity as is con- sistent with absolute correctness. It will be found that, as a rule, the quickest workers are the most accurate. Many of the slate problems can be treated by some classes as " sight " examples, each pupil reading the question for himself from the book, and writing the answer at a given signal without putting down any of the work. Use of Books, It is generally recommended that books be placed in pupils' hands as early as the third school year. Since many children are unable at this stage to read with sufficient intelligence to understand the terms of a problem, this work shQuld be done under the teacher's direction, the latter reading the questions while the pupils follow from their books. In later years, the problems should be solved by the pupils from the books with practically no assistance whatever from the teacher. Conduct of the Kecitation, Many thoughtful educators consider it advisable to divide an arithmetic class into two sections, for some purposes, even where its members are nearly equal in attainments. The members of ono division of such a class may work examples from their books while the others write the answers to oral problems given by the teacher, etc. Where a class is thus taught in two divisions, the members of each should sit in alternate rows, extending from the front of the room to the rear. Seated in this way, a pupil is doing a different kind of work from those on the right and the left, arid he would not have the temptation of a neighbor's slate to lead him to compare answers. GENERAL HINTS 7 As an economy of time, explanations of new subjects might be given to the whole class ; but much of the arithmetic work should be done in "sections," one of which is under the im- mediate direction of the teacher, the other being employed in "seat" work. In the case of pupils of the more advanced classes, "seat" work should consist largely of "problems" solved without assistance. Especial pains have been taken to so grade the problems as to have none beyond the capacity of the average pupil that is willing to try to understand its terms. It is not necessary that all the members of a division should work the same problems at a given time, nor the same number of prob- lems, nor that a new topic should be postponed until all of the previous problems have been solved. Whenever it is possible, all of the members of the division working under the teacher's immediate direction should take part in all the work done. In mental arithmetic, for instance, while only a few may be called upon for explanations, all of the pupils should write the answers to each question. The same is true of much of the sight work r the approximations, some of the special drills, etc. Drills and Sight Work. To secure reasonable rapidity, it is necessary to have regular systematic drills. They should be employed daily, if possible, in the earlier years, but should never last longer than five or ten minutes. Various kinds are sug- gested, such as sight addition drills, in Arts. 3, 11, 24, 26, etc. ; subtraction, in Arts. 19, 50, 53, etc. ; multiplication, in Arts. 71, 109, etc. ; division, in Arts. 199, 202, etc. ; counting by 2's, 3's, etc., in Art. 61 ; carrying, in Art. 53, etc. For the young pupil, those are the most valuable in which the figures are in his sight, and in the position they occupy in an example ; see Arts. 3, 34, 164, etc. Many teachers prepare cards, each of which contains one of the combinations taught in their respective grades. Showing one of these cards, the teacher requires an immediate answer 8 MANUAL FOR TEACHERS from a pupil. If his reply is correct, a new card is shown to the next pupil, and so on. Other teachers write a number of combinations on the blackboard, and point to them at random, requiring prompt answers. When drills remain on the board for any considerable time, some children learn to know the results of a combination by its location on the board, so that frequent changes in the arrangement of the drills are, therefore, advisable. The drills in Arts. Ill, 112, and 115 furnish a great deal of work with the occasional change of a single figure. For the higher classes, each chapter contains appropriate drills, which are subsequently used in oral problems. It happens only too frequently that as children go forward in school they lose much of the readiness in oral and written work they possessed in the lower grades, owing to the neglect of their teachers to continue to require quick, accurate review work in the operations previously taught. These special drills follow the plan of the combinations of the earlier chapters, but gradu- ally grow more difficult. They should first be used as sight exercises, either from the books or from the blackboard. To secure valuable results from drill exercises, the utmost possible promptness in answers should be insisted upon. Definitions, Principles, and Kulee, Young children should not memorize rules or definitions. They should learn to add by adding, after being first shown by the teacher how to perform the operation. Those not previously taught by the Grube method should be given no reason for " carrying." In teaching such children to write numbers of two or three figures, there is nothing gained by discussing the local value of the digits. Dur- ing the earlier years, instruction in the art of arithmetic should be given with the least possible amount of science. While prin- ciples may be incidentally brought to the view of the children at times, there should be no cross-examination thereon. It may be shown, for instance, that subtraction is the reverse of addition, and that multiplication is a short method of combining equal GENERAL HINTS 9 numbers, etc. ; but care should be taken in the case of pupils below about the fifth school year not to dwell long on this side of the instruction. By that time, pupils should be able to add, subtract, multiply, and divide whole numbers ; to add and sab- tract simple mixed numbers, and to use a mixed number as a multiplier or a multiplicand ; to solve easy problems, with small numbers, involving the foregoing operations and others contain- ing the more commonly used denominate units. Whether or not they can explain the principles underlying the operations is of next to no importance, if they can do the work with reasonable accuracy and rapidity. When decimal fractions are taken up, the principles of Arabic notation should be developed ; and about the same time, or some- what later, the principles upon which are founded the operations in the fundamental processes, can be briefly discussed. Definitions should in all cases be made by the pupils, their mistakes being brought out by the teacher through appropriate questions, criticisms, etc. Systematic work under this head should be deferred until at least the seventh year. The use of unnecessary rules in the higher grades is to be deprecated. When, for instance, a pupil understands that per cent means hundredths, that seven per cent means seven hun- dredths, it should not be necessary to tell him that 7 per cent of 143 is obtained by multiplying 143 by .07. It should be a fair assumption that his previous work in the multiplication of common and of decimal fractions has enabled him to see that 7 per cent of 143 is -fa of 143 or 143 X .07, without information other than the meaning of the term " per cent." When a pupil is able to calculate that 15% of 120 is 18, he should be allowed to try to work out for himself, without a rule, the solution of this problem : 18 is what per cent of 120 ? or of this: 18 is 15% of what number? These questions should present no more difficulty in the seventh year than the following examples in the fifth : (a) Find the cost of -^ ton of hay at $12 per ton. (b) When hay is worth $12 per ton, what part of a 10 MANUAL FOR TEACHERS ton can be bought for $ 1.80 ? (c) If -^ ton of hay costs f 1.80, what is the value of a ton ? When, however, it becomes necessary to assist pupils in the solution of problems of this class, it is more profitable to furnish them with a general method by the use of the equation, than with any special plan suited only to the type under immediate discussion. In the supplement to the Manual will be found the usual defini- tions, principles, and rules, for the teacher to use in such a way as her experience shows to be best for her pupils. The rules given are based somewhat on the older methods, rather than on those recommended by the author. He would prefer to omit entirely those relating to percentage, interest, and the like as being unnecessary, but that they are called for by many success- ful teachers, who prefer to continue the use of methods which they have found to produce satisfactory results. Language, While the use of correct language should be insisted upon in all lessons, children should not be required in arithmetic to give all answers in " complete sentences." Espe- cially in the drills, it is important that the results be expressed in the fewest possible words. Analyses, Sparing use of analyses is recommended for begin- ners. If a pupil solves a problem correctly, the natural inference should be that his method is correct, even if he be unable to state it in words. When a pupil gives the analysis of a problem, he should be permitted to express himself in his own way. Set forms should not be used under any circumstances. Objective Illustrations, The chief reason for the use of objects in the study of arithmetic is to enable pupils to work without them. While counters, weights and measures, diagrams, or the like are necessary at the beginning of some topics, it is important to discontinue their use as soon as the scholar is able to proceed without their aid. I UNITERSITY , GENERAL HINT8V ^ J H Approximate Answers, An important drill is furnished in the " approximations." (See Arts. 521, 669, 719, etc.) Pupils should be required in much of their written work to estimate the result before beginning to solve a problem with the pencil. Besides preventing an absurd answer, this practice will also have the effect of causing a pupil to see what processes are necessary. In too many instances, w r ork is commenced upon a problem before the conditions are grasped by the youthful scholar ; which will be less likely to occur in the case of one who has carefully " estimated " the answer. The pupil will frequently find, also, that he can obtain the correct result without using his pencil at all. Indicating Operations, It is a good practice to require pupils to indicate by signs all of the processes necessary to the solution of a problem, before performing any of the operations. This fre- quently enables a scholar to shorten his work by cancellation, etc. In the case of problems whose solution requires tedious processes, some teachers do not require their pupils to do more than to indicate the operations. It is to be feared that much of the lack of facility in adding, multiplying, etc., found in the pupils of the higher classes is due to this desire to make work pleasant. Instead of becoming more expert in the fundamental operations, scholars in their eighth year frequently add, subtract, multiply, and divide more slowly and less accurately than in their fourth year of school. Paper vs. Slates. To the use of slates may be traced very much of the poor work now done in arithmetic. A child that finds the sum of two or more numbers by drawing on his slate the number of strokes represented by each, and then counting the total, will have to adopt some other method if his work is done on material that does not permit the easy obliteration of the tell-tale marks. When the teacher has an opportunity to see the number of attempts made by some of her pupils to obtain the correct quo- 12 MANUAL FOE TEACHERS tient figures in a long division example, she may realize the importance of such drills as will enable them to arrive more readily at the correct result. The unnecessary work now done by many pupils will be very much lessened if they find themselves compelled to dispense with the " rubbing out" they have an opportunity to indulge in when slates are employed. The additional expense caused by the introduction of paper will almost inevitably lead to better results in arithmetic. The arrangement of the work will be looked after ; pupils will not be required, nor will they be permitted, to waste material in writing out the operations that can be per- formed mentally ; the least common denominator will be deter- mined by inspection ; problems will be shortened by the greater use of cancellation, etc., etc. Better writing of figures and neater arrangement of problems will be likely to accompany the use of material that will be kept by the teacher for the inspection of the school authorities. The endless writing of tables and the long, tedious examples now given to keep troublesome pupils from bothering a teacher that wishes to write up her ^ecords, will, to some extent, be discontinued when slates are n-* longer used. IX NOTES ON CHAPTER SIX The previous work in mixed numbers should make the pupils reasonably familiar with the addition and subtraction of frac- tions having small denominators. In this chapter, the work is extended to cover the addition and subtraction of fractions whose common denominator is determinable by inspection. For the present, the teacher should be satisfied if her pupils acquire rea- sonable facility in performing the various operations, even if they are unable to formulate, in the language of experienced mathematicians, the reasons for the different steps. The children should be required to use correctly and intelligently such techni- cal terms as are required by the work of the chapter ; but they should not be compelled to memorize any definitions that convey to them no meaning. They should incidentally learn what is meant by numerator, denominator, common denominator, mul- tiple, etc., by hearing the teacher employ these words from time to time, rather than by commencing with what is to them an unintelligible jumble of words. 451. Whil'e systematic work in fractions belongs properly to the next chapter, the teacher should not hesitate to call ^ 2 -, -^ etc., " improper fractions," and to ask a pupil to state how they are changed to whole or to mixed numbers. 453. Do not, for the present, formulate the rule for changing a fraction to an equivalent one with higher terms. 458. The meaning of ''lowest terms" is given in No. 6. Leave the rule for the next chapter. After a pupil has rea- 45 46 MANUAL FOR TEACHERS soned out in his own way that 18 hours is f day, in No. 15, the teacher may explain that 18 hours can be written -|-f day, which is reducible to the answer given above. 463. Have pupils see that |- is larger than either -f- or -f, because 1 sixth is larger than a seventh or an eighth ; and this for the reason that the fewer the number of equal divisions made in a unit, the larger is each portion. Do not require scholars to change these fractions to equivalent ones having a common denominator. 467. For finding the difference between two mixed numbers when the fraction in the subtrahend is greater than that in the minuend, the method given in the text-book is the one generally employed. The teacher should always consider herself at liberty to use any other way of performing this and other operations, but she should not willingly adopt any method that is more tedious. Children should not, for instance, be required to change mixed numbers to improper fractions, and then to reduce these to a common denominator in order to subtract one from the other. 469. Pupils should now be required to pay more and more attention to the arrangement of the problem work, without, however, being permitted to use unnecessary figures or to waste time. In some good schools, the full written analysis of a prob- lem is occasionally used as an exercise in composition. When the pupils find difficulty in determining the operations necessary to the solutions of problems, the latter should be used as " sight " work. The alterations in the figures needed to sim- plify a problem should now be made by a pupil, instead of by the teacher, as recommended in previous chapters. The scholar that reads No. 1, for instance, might change 5J and 4-| yards, to 5 and 4, respectively. No. 2 can be solved as it stands. In No. 3, $150 might be substituted for $140.40, and $2 for $1.80. NOTES ON CHAPTER SIX 47 Work of this kind should gradually lead the pupil to form the habit of using some similar method of ascertaining for himself how to manage a problem. 471. Written in this form: 3)93f, No. 11 should give the children no trouble. If, however, they hesitate when the frac- tion is reached, the difficulty may be cleared up by making a concrete problem : Divide $ 93f equally among 3 persons. What is the share of each ? Under no circumstances should these dividends be changed to improper fractions. 484. In nearly all of the previous multiplication work in- volving mixed numbers, the latter have been used as -,,^7 multipliers. In No. 35, the mixed number appears as a multiplicand. The first six of these examples and the o\m~" last two should be used as sight work, the answers being ~i7ji written directly from the book. When the pupil reaches ^. g one that needs to be worked out in full, say No. 41, he -.^ should not be permitted to use 18- as a multiplier, as it is important that he should learn the proper method of working both classes of examples. 489. Many scholars will carelessly give 20 halves as the result obtained by dividing 500 halves by 25 halves. To pre- vent the possibility of a mistake of this kind, some teachers multiply the divisor and the dividend by the least common multiple of the denominators of the fractions. While this 18f)1387-^ method produces exactly the same figures as the X 4 X 4 one given in the text-book, it is probably less likely 75) 5550 to be followed by the error mentioned above. 497. Some teachers may prefer to write the example as is here given, although using 5 as the multiplier. Other $.05 teachers " analyze " as follows : At 1^ per lb., 157 X 157 pounds of sugar would cost $1.57; at 5^ per lb., the $7.85 48 MANUAL FOE TEACHERS $1.57 cost is 5 times $1.57. Business men pay no attention X 5 to these fine-spun distinctions ; they use as a multiplier $7.85 the most convenient number, and write the dollar sign and the period in the product alone. 500. In analyzing problems of this kind, it is better, perhaps, to emphasize the fact that multiplication is employed in obtain- ing the result. Thus, 32 base-balls @ 25^ = 32 times $ J = 32 quarters = $8 502. In No. 28, the price of 11 yards can be found by taking 11 times $f, or 33 quarters, etc. No. 24 is rendered easier by saying that 24 bushels at 1 quarter per bushel would cost 6 dollars ; and that at 3 quarters per bushel the cost would be 3 times 6 dollars, etc. Pupils should be encouraged to use the method best adapted to the particular example under considera- tion. 509. See Art. 306. 510. In finding, for example, the number of 50-cent knives that can be purchased for $20, it may be advisable to make the division idea prominent. The analysis can take some such form, as this : There can be bought as many knives as one half-dollar is contained times in 20 dollars or, as there are half-dollars in 20 dollars. Later problems involving division of fractions cause less trouble if the appropriate operation is always kept before the pupils, regardless of the method employed to shorten the solu- tion of questions of certain types. These short methods should, however, be used. To ascertain the number of 2-dollar knives obtainable for $24, the scholar turns naturally to division; and he should learn to see that he actually divides when he obtains 48 as the number of 50-cent knives that can be purchased for the same NOTES ON CHAPTER SIX 49 money. In the latter case, the numbers given are 24 and ^, from which 48 can result only when -| is used as a divisor. Many pupils that give the correct answer when 24 -^ -|- is placed upon the blackboard as a sight example will think that 12 is the quotient of -|-)24. For purposes of drill, this last form should occasionally be employed in sight work, as should be 24 the third form of division, * 2" 511. Example 6 : There can be bought as many bars of soap as there are quarters in $3^, or 13 bars. Example 8 : As many yards as there are quarters in $5f . Example 9 : As many bushels as there are quarters in $ 10f. While set forms of analysis should not be required in any grades, older pupils should be led to use such as are most likely to lead to an intelligent appreciation of mathematical principles. From the beginning of about the fifth year, the science of arith- metic should begin to receive some attention, but not so much as to lessen to too great an extent the time that should be devoted to arithmetic as an art. 516. These examples are introduced to lead up to division of Federal money. From their previous experience, the scholars will readily work the first example, for instance, by changing it to the form $24J- -*- $-J- = 4 *- = 49. No. 2 becomes f 12J^?J; No. 3, $26-3-$* = ^-^; etc. Without laying much stress upon the terms "abstract" and "concrete," the teacher should bring her pupils to understand that the quotient of the first example is 49, not 49 dollars. 517. While giving the answers to these exercises, the children should be able to state, after proper questioning, that the divi- dend must be of the same denomination as the divisor. In 2 ft. -^ 8 in., instead of changing the divisor to -| ft., they natu- rally reduce the dividend to 24 inches, even if in 2 ft. -^ 6 in. they may have used % ft. as the divisor. 50 MANUAL FOR TEACHERS 518. Changing the dividend to cents, No. 1 becomes 400 cents -*- 10 cents, or 400 *- 10. To No. 11, many will give 50 as the result, unless previously well taught. In No. 12, the denomi- nation of both terms being the same, the problem becomes 3 -s- -J-, or 12 quarters 4* 1 quarter, rather than 300 -*- 25. Nos. 13-20 are more readily worked by reducing each divisor to a fraction of a dollar. Pupils should understand that the answer is the same whether the dividend is changed to the same denomination as the divisor, or vice versa. 519. Some teachers write this example .36)27.00. It will be found safer to make the terms whole numbers by changing both to cents. 520. The method suggested for the first example, 11000 -5- 275, is the more general one, although longer, perhaps, than 110 H-2J. Do not permit long division in No. 2. In No. 3, after writing 14000 H- 560, the pupils should strike out a cipher from each term : this should be insisted upon whenever the divisor ends in a cipher. In No. 4, either 74-^- -r- -|- or 7450 -*- 50 should be accepted. If the work in No. 7 takes the form 75)$ 27.00, the answer should be $ .36 ; if the pupil writes 75)2700^, his answer should be 36 cents. The first form is the one employed in the work of preceding chapters, and no change should be suggested. 521. These drills in obtaining approximate results are intended to lead the pupil to such an examination of his answer as will prevent his being satisfied with one very much out of the way. It should not be expected that the same approximation will be obtained by all the members of a class. 2. 4200 +- 200. 6. 30 + 38. 3. x48. 7. 175^-25. 4. 12000 -*- 2000. 8. 19 x 10. 5. $2x99, or $1.95 x 100, 9. 87-50. or $2 X 100. 10. 5x5x5. NOTES ON CHAPTER SIX 51 526. Formal instruction in denominate numbers should be deterred for a year or more. The average scholar will be able to solve all these problems if left to himself. 528-532. See notes on previous drills of this kind, Arts. 286 and 350. Special exercises in multiplication are regularly given by some teachers in the following manner : 2, 12, 22, 32, 42, 52, 62, 72, 82, 92 X5 A horizontal or a vertical row of numbers ending in 2, for instance, is written on the board with, say, 5 as a multiplier. Attention is called to the fact that 2 X 5 is 10, so that all of these products must end in 0. The pupils are also reminded that when the multiplicand is a number of two figures, 1 must be carried to the product of 5 times the tens' figure. When the teacher points to 12, the pupil says 5, 6, 60 the first number (5) being the product of the multiplier and the tens' figure of the multiplicand ; the second, (6) being this product increased by the carrying figure 1 ; the third being the result, which has been completed by annexing the units' figure (0) of the first product (2 X 5). Pointing to 52, the pupil says 25, 26, 260 ; to 92, he says 45, 46, 460. After sufficient drill with 5 as a multiplier, it is replaced successively by 6, 7, 8, etc. The row of multiplicands is also changed to 3, 13, 23, etc. ; 4, 14, 24, etc. ; when the previous row has been employed with all of the multipliers, say from 5 to 12. With sufficient practice of this "kind, pupils become able to give the product of any number of two figures, by multipliers to 12, with great readiness. Some teachers, however, prefer in oral multiplication to use the method previously suggested of commencing to multiply at the tens' figure of the multiplicand. In finding 5 times 38, the pupil takes the latter number as it is given, thirty eight, and multiplies in the same order, obtaining 150 and 40, or 190. 52 MANUAL FOE, TEACHERS 534-535. Long-division drills. See Arts. 321 and 397-401. 538. See Art. 563, p. 55, and Arithmetic, Art. 385. 540. See Art. 384. 543. In problems of this kind, writing the given numbers in the places called for by tjie conditions of each example helps the pupil in his solution. After writing No. 5, as here indi- 1. 68 2. _?_ 5. ? 43 24)264 92 150 cated, he can see that addition is the required operation more readily than if he endeavors to determine it from the words of the problem. 546. Teachers should not weary pupils by giving too many items in the earlier bills. It is useful to employ occasionally such quantities and prices as will not require the use of a sepa- rate' piece of paper to perform the necessary multiplications. In No. 1, for instance, the pupil should be compelled to fill out the cost of each item without recourse to his slate. If he does not know the product of 16 X 5, he should multiply one figure at a time, writing the result in its proper place. Except, pos- sibly, No. 2, the other bills called for under this section should be made out in the way suggested for No. 1, the use of a slate or other paper not being permitted. The form given in the text-book is the one generally followed by business men. The first two vertical lines are kept to enclose the day of the month (see Arithmetic, Art. 642). The total cost of each item is placed in the first columns of dollars and cents, the amount of the bill being placed in the last columns, and on the line below the last item. When a single article is sold, its cost is placed directly in the first columns of dollars and NOTES ON CHAPTER SIX 53 cents, and is not written in the column of prices. Unnecessary words, at or @, for instance, per yd., lb., etc., are never used ; nor are commas employed after the names of the articles. It is now customary to omit the period after the date and after the name of the seller. The names of the articles are generally commenced with capitals, and the quantities are written with small letters. The heading given is the one most frequently used, though other forms are common ; such as ABRAHAM AND STRAUS Sold to MRS. H. T. SHORT Pupils should write the cost of 10J lb. of chicken @ 30^, in No. 4, as $3.08. Fractions of cents should not appear in the results ; those below ^ being rejected, and those of -^ and higher being considered 1^. 547-551. When pupils have become familiar with the nota- tion and numeration of decimals, the remaining decimal work of this chapter should not require much discussion. 554. After working Nos. 1-4, pupils should be left to them- selves to arrange No. 5. Nearly all of them will place the numbers in their proper places. Neither require nor permit unnecessary ciphers to be employed to fill out all the numbers to three decimal places. 555. The above suggestions apply here. 559. In the product of .36 by 3, pupils will naturally place the decimal point where it belongs, as they will in example No. 42. Before working No. 43, they should be required to deduce the rule for "pointing off." All unnecessary ciphers should be canceled in the product, the answer to No. 52 being read by the pupil as 960, not 960 and no tenths 960.0. 54 MANUAL FOR TEACHERS 560. These exercises should lead scholars to see that the number of decimal places in the multiplicand cancels a cor- responding number of ciphers in the multiplier. 561. In giving the quotient of No. 11, a pupil may write 9_32^_ if called upon to read this answer, he will see that 9.32 expresses the same result. He will readily understand that _| e_ i s a } so wr itten .086. After a few examples, he can state the rule. From this point, the teacher may change the method of dividing by a number ending in one or more ciphers. Instead of marking off, by a line, a corresponding number of figures from the right of the dividend, the pupil can locate the decimal point in the proper place. In No. 17, the decimal point will be moved one place to the left, to divide by 10 ; moving it two places to the left in the dividend of No. 18 will give its quotient ; etc. 563. The rule for " pointing off" should be deduced from the sight exercises of Art. 562. In dividing 8 and 64 hundredths by 2, the pupil will, without prompting, obtain 4 and 32 hun- dredths. When he comes to No. 5, 8.4 -*- 5, he can be led to see that this example is the equivalent of No. 4, 8.40 -s- 5, in which the quotient is 1.68. Nos. 9 and 10 also require the annexation of a cipher at the right of the dividend. When the scholars understand that the quotient must contain the same number of decimal places as the dividend, including any ciphers that may have been annexed to the latter, they should be taught the method a business man would employ. The latter, in dividing 120 by 64, does not find it necessary to write ciphers in the dividend, and then to count the number thus annexed, in order to determine the position of the decimal point in the quotient. He sees at once that the result is 1 and a decimal, and he places the point after the 1, before he writes the next quotient figure. NOTES ON CHAPTER SIX 55 In a short-division example, the pupils should write the decimal point in the quotient when they reach it in the divi- dend, placing it under the latter. In long division, the decimal point in the quotient is placed over the point in the dividend. While the scholars have been warned in their early work in short division against writing 02 as the quotient of 8)16, they will see the need of the prefixed cipher in the answer, .02, to 8). 16. From the inspection of a few examples of this kind, they will understand that each figure of the dividend after the decimal point requires a quotient figure (or cipher). In the long-division examples worked out in 1.875 Art. 563 of the Arithmetic, the partial products 64)120. are omitted, to show how some European countries 560 shorten work of this kind. The horizontal lines 480 given in the text-book are not used. 320 After the pupil writes the quotient figure 1, he subtracts by the " building-up " method. The second remainder, 48, is obtained by saying 8 fours are 32, and 8 (writing it) are 40; 8 sixes are 48, and 4 (carried) are 52, and 4 (writing it) are 56. See Arithmetic, Art. 385. 564. Teachers should not forget that systematic instruction in decimal fractions belongs to the sixth school year. They should be content if their pupils learn to place correctly the decimal point in the quotient. Nos. 21-30 should be considered rather as examples in divi- sion, than as examples in the reduction of common fractions to decimal ones. By ^ is meant 1 -2- 4 ; and to solve it, the pupil may write 4)1.00, as in No. 11. He should learn by degrees, however, that it is not necessary to write all the ciphers in the dividend in order to obtain the result. The answer to No. 22 can be derived by a bright scholar from 8)1.0 just as well as from 8)1.000. He may desire the first cipher as a starting- point, but he finds the others superfluous. 56 MANUAL FOR TEACHERS 565. These problems contain a few simple applications of the decimal work learned thus far. In No. 1, the value of the franc may be given in the fractional form also, 19 T 3 ^ ; and the problem worked fractionally and decimally. Approximate sight results might be asked before written work is begun. Taking the franc as about 20^, or $-J-, the pupil should say that the answer to No. 1 is less than $250. Assum- ing 40 inches, or 1-J- yd. as the length of the meter, 1800 meters would be equal to 2000 yards. No. 33 can be solved without using the pencil. In No. 38, the first result may be in the form of a common fraction, peck, to be changed, as in No. 22, to .125 peck. - 569. The remarks made in Art. 564 as to the formal teaching of decimal fractions is equally applicable to the subject of measurements. At one time, all instruction in mensuration was deferred until the last year of the common-school course. At present, this subject is generally taken up in connection with the systematic work in denominate numbers ; but there is no good reason why pupils compelled to leave school by the end of the fifth year of the course should not receive so much practice in finding the areas of rectangles as they have time for and can readily understand. In most city schools, the children of this grade know from their lessons in form and drawing what is meant by a square and a rectangle. If pupils are not familiar with these terms, they should be explained. That scholars may obtain a good idea of a square inch, they should be required to cut out a number of square pieces of paper, each side measuring an inch. These squares should then be used in determining the number of square inches in the two rectangles next drawn ; 2 inches by 1 inch, and 3 inches by 2 inches. Children that determine the areas of these rectangles by covering them with their paper squares will have a better knowledge of 2 square inches and 6 square inches than if they NOTES ON Ca^ffER.SIXa^hX 57 merely divide the rectangles by lines as suggested in the text- book. The larger rectangles, 6x3 and 4 X 4, may be cut up into inch squares by drawing lines, and the rule for obtaining the area of each deduced from an examination of the figures. In the first, the pupil will see that he has 3 rows of squares, 6 to a row (or 6 rows, 3 to a row), making 6x3, or 18, squares. The rule should take in his mind some such form as this : the num- ber of square inches in a rectangle is equal to the product of the number of inches in one dimension multiplied by the number of inches in the other ; but he should not be required to give it expression. The teacher should take care that he does not think that " inches by inches give square inches." 570. As has already been said, the chief use, in arithmetical instruction, of objects, diagrams, etc., is to enable pupils to work without them. After the scholars understand how to obtain the area of a rectangle, they should cease to draw the figure and to subdivide it into squares. It will be noted that the answers to the first 20 examples are to be given in square inches. In Nos. 11-20 each dimension should be reduced to inches before the multiplication is per- formed. NOTES ON CHAPTER SEVEN At this point regular fraction work should begin. From time to time, as occasion offers, the meanings of the technical terms should be elicited from the pupils ; but the teacher should neither accept a memorized definition that is not thoroughly understood, nor should she require absolute correctness in the phraseology of a definition made by a scholar. 577. While the denominators of the fractions should generally be small ; and while common denominators should, as a rule, be determinable by inspection, it is necessary, nevertheless, that the children be taught how to handle such other fractions as they may occasionally meet. It is not necessary that they should grasp the exact meaning of -|-||- in the answer to No. 4, although proper teaching may enable them to see later on that this fraction approximates f , f . In these earlier examples, the inspection method of determining the common denominator is continued. 580. Besides being necessary as a preliminary to subsequent work in fractions, expertness in determining the factors of a num- ber is useful in enabling pupils to shorten their work by cancel- lation. The teacher should use these and similar exercises again and again, for a few minutes at a time, until her scholars can give the answers with great rapidity. 581. The pupils will need to learn the difference between the three factors of 12, and three divisors of 12. The factors will be 2, 2, and 3, because their product, 2x2x3 equals 12. The divisors of 12 are 2, 3, 4, and 6. 58 NOTES ON CHAPTER SEVEN 59 These exercises, as well as those in Arts. 582-585, are not so valuable as to demand the reviews suggested for those in Art. 580. In finding three (or more) factors of a number, the scholar should commence with the smallest. The first of the three factors of 8 is 2 ; dividing 8 by this factor, 4 is obtained, of which 2 and 2 are the factors. The three factors of 18 are 2 X 3 X 3 ; of 20, are 2x2x5; of 27, 3 X 3 X 3 ; etc. ; etc. 582. It is customary to define a prime number as one that has no factor except itself arid unity. The omission of the last four words will not mislead any person, as there could be no prime numbers if 1 were considered a factor. When the factors of a number, say 20, are asked for, no one gives Ixlxlxlx 1x2x2x5 as the answer, or says that 20 has eight (or more) factors. 588. In reducing these fractions to lowest terms, it is not nec- essary that the pupils should use the greatest common divisor. See Arithmetic, Art. 592. On the other hand, they should not waste time in dividing each term by 5, if 25 is a common divisor. 589. Pupils should not be permitted to forget these tests of the divisibility of numbers. To those given in the text-book, there may be added that when a number divisible by 3 is even, it is also divisible by 6. While a teacher should know that 1001, with, of course, its multiples, 2002, 6006, 15015, etc., is divisible by 7, 11, and 13, she should not burden her scholars with the information ; nor should she dwell upon the test of divisibility by 8. 591. Beginners should be taught only one method of finding the greatest common divisor, and the one here given is applicable to all kinds of numbers. Teachers should not bewilder young pupils by endeavoring to make them understand the principles upon which this method is based. 60 MANUAL FOE, TEACHERS 595. Many teachers prefer to permit their pupils to write down all of the denominators, and . then to strike out any one that is repeated or that is the factor of any e ^ c - other. They think the pupil is less likely to make a mistake by following this plan. In no case should scholars be permitted to begin work before rejecting or striking out the unnecessary numbers. 605-606. Pupils that have had regular drills in the com- binations given in the previous chapters will be able to take the extra step required by these examples. See Arts. 286-290 and 350-352. 607. A scholar that can find mentally the cost of 47 articles at 25 $ each should be able to give the product of 47 X 25 or 25 X 36 without using the pencil, and the teacher should give him a chance to determine for himself the method of doing it. 609. Such questions as 18-| rt- 2-| can be worked by the method given in the last chapter ; viz., 56 thirds -s- 8 thirds 56 -*- 8 .= 7. Those contained in the 4th column should not be used until the pupils have had formal instruction in division of fractions. When they are taken up, the method followed should be that given above, the fractions being reduced to a common denominator, etc | -f- f = 3 sixths -*- 4 sixths = 3 -s- 1 4 = ; -*- = 8 twelfths -*- 9 twelfths f ; etc. See Arithmetic, Art. 639, note. 610. No. 9 : For $1.25 I can buy 5 times as many pounds as for 25^, or 15 pounds. No. 16: For 18^ there can be bought | lb., or f lb., or 12 oz. 613. In giving answers to these exercises, pupils should be permitted to write the fraction first and then the whole number. The object of these exercises is to accustom the scholars to dis- pense with writing unnecessary reductions in adding and sub- tracting simple fractions. NOTES ON CHAPTER SEVEN 61 614. Another method of finding the difference between 11-| and 6| is to take 6-| from 7, obtaining -|-, and to add to this the difference between 7 and llf, or 4-f. In some classes of sight exercises, those given in Arts. 587, 594, and 605-609, for instance, the pupils should not take pens to write the result until told to do so by the teacher, after sufficient time has been given to obtain the answer. In the exercises of Arts. 613 and 614, the pupils should be permitted to take their pens at once, and to write each part of the result as soon as it has been obtained. Arts. 584, 650-654, 699, etc., also contain exercises of this kind. 616. See Art. 563. The first quotient figure, 495i 4, is written. The pupil then says 4 sixes are 24 36)17837 and 4 (writing it) are 28 ; 4 threes are 12, and 2 343 (to carry) are 14, and 3 (writing it) are 17. This ^g^ gives the first remainder, 34. The next figure, 3, -tn is then brought down, and 9 is written in the quotient. The product of 9 times 36 is subtracted from 343 as given above, to obtain the next remainder, 19. The pupil says 9 sixes are 54 and 9 (writing it) are 63 ; 9 threes are 27 and 6 (to carry) are 33, and 1 (writing it) are 34. 618. While pupils should be encouraged to shorten their work by cancellation, the slower children should not be censured when they overlook some cases in which it is possible to employ this expedient. In these examples, however, all the scholars should be required to indicate the operations, and then to cancel. 619. It is not supposed that pupils should write answers to these questions, as is generally done in the case of the oral problems. These exercises are intended to lead up to the rule for multiplication of fractions. Diagrams should be drawn on the blackboard by the pupils, to illustrate the answers, but the teacher should refrain a much as possible from " explaining." 62 MANUAL FOR TEACHERS The board work should be done chiefly by the more backward members of the class rather than by the brighter ones. In illustrating fractions by diagrams, the unit employed should generally be a circle, the part dealt with being distinguished by shading. The zealous teacher should not become discouraged at the inability of some members of the class to thoroughly grasp the mathematical principles involved in this and other operations. Even the ability to handle fractions mechanically will be of great use in after life, and all the pupils can be taught at least this much. 624. No. 31 reduces to 9f -5- 3, which can readily be worked by the pupils without assistance. No. 34, 40|- -s- 8, may be difficult for soma; but the teacher should not offer help too soon. The second term of No. 36 is easily obtained; and No. 40 will give no trouble. 625. These exercises are to be used in the same way as those in Art. 619. Many children find it difficult to understand that 4 fourths -* 3 fourths = 1-^. They think that the answer should be 1-^, reason- ing it out in some such way as this : 3 fourths into 4 fourths goes 1 time and 1 fourth over. They fail to recollect that the remainder in division is written over the divisor, which would give them IsSL or 1-J-. If they have been well taught previously, they may remember that 4 fourths -=- 3 fourths 4 -H 3 = 1^. Even a fairly bright pupil, when asked how much tea at $f per Ib. can be bought for $1, will sometimes reply, "A pound and a quarter." When he is told that his answer is correct if he means by it a pound of tea and a silver quarter, he sees the mis- take and changes the result to " a pound and a third." 626. If 1 -*- is made concrete, a pupil can more easily show by a diagram that the result is 1^-. A problem of this kind may NOTES ON CHAPTER SEVEN 63 be given : If it requires -f yd. of material for an apron, how many aprons can be made from 1 yard ? A rectangle is drawn to represent the yard of material, and it is divided into thirds. Underneath, a rectangle two-thirds as long is drawn to represent the quantity required for an apron. When the pupil compares the two rectangles, he sees that the portion remaining after one apron is made will supply sufficient material for one-half of another. 627. While it is generally better in oral work to divide one fraction by another by reducing both to a common denominator, it will be found simpler in written work to have pupils invert the divisor. 631. " Invert the divisor, and proceed as in multiplication," is the rule generally followed. 634. No. 33 can be shortened by writing it as follows, before beginning work : - 2 I Q - X f X f , the divisor being inverted. No. 34 should be treated in the same way : (20 -f- $) X f = - 2 T - X -f X |. The divisor of No. 35 consists of two fractions, both of which should be inverted : 20 -f- (| X f ) = ^ X f X f . This method should be followed with Nos. 36, 41, and 42. The first of these becomes T 1 X -f- X f ; the next, ^ X f X ^ 8 X ^ X f ; and the next, 635. Each teacher must determine for herself what method of analysis should be encouraged in such questions as Nos. 4, 8, 13, and 14. While set forms should be avoided, children need direction in the solution of problems of this kind. In solving No. 4, for instance, the greater number of teachers prefer to have pupils first find the cost of ^ yd. When this' method is followed, care must be taken that all the pupils under- stand why ^ yd. costs one-half of 20 cents. This may be made clearer to some by writing the fractions in this way : If 2 thirds 64 MANUAL FOR TEACHERS (or parts) cost 20 cents, what will 1 third (or part) cost? A diagram similar to that given in Arithmetic, Art. 636, may help others to understand the method. Other successful teachers think the written work is benefited by treating these examples as problems in division. They lead their children to determine in each case what operation is in- volved, by requiring them to consider what they would do if the fraction were a whole number. In No. 1, for example, the cost of 16 balls at $3 each would be $3 X 16. In No. 2, the pupil would say, "If I paid $ 12 for base-balls at $ 3 each, the number of balls would equal 12 -z- 3. I must, therefore, divide." He mentally inverts the divisor, -|, then cancels, etc. 636. The scholars should be allowed sufficient time to work these out in their own way, 639. No. 4: 24 -*- 3J- = 4 -*- f = 49 -*- 7. Some pupils will see that time is lost in No. 6 by finding the cost of a pound. No. 7 is an example in division: l-|^-2^=-|-f-|- = fX-f; or if. -5- - 1 / = 10 -*- 15, etc. In No. 16, 36 hats will cost 3 times $ 7. 642. See Art. 546. 649. In multiplying by 25, the pupil is generally told to annex two ciphers and to divide by 4. In mental work espe- cially, the annexation of the ciphers confuses some scholars by giving them a larger dividend than is really required. The product of 25 times 19 may be obtained more easily by taking one-fourth of 19, or 4-|, and changing this quotient to 475, than by finding one-fourth of 1900. In No. 9, the pupil should see that at $100 per bbl., the pork would cost $5600, and that at $ 12.50 per bbl. (i of $ 100), it would cost | of $ 5600. 650. In No. 1, divide 837 by 4, and for the 1 remainder af- fix 25 to the quotient. In No. 4, annex two ciphers to the quo- tient of 508 by 4. In No. 9, aflSx 250 to the quotient of 837 *- 4. NOTES ON CHAFSjjjMEVEN ^K/ 65 In multiplying 6281 by 1% No. 18, divide 6281 by 8, obtaining 785, and annex 12^- for the 1 remainder, making the result 78,51% 654. When the divisor is a whole number, time should not be wasted in changing a mixed number dividend to an improper fraction. Nos. 64, 65, and 66 resemble those already worked. In No. 69, after obtaining the quotient 14, there will be a re- mainder 2J, which is changed to -f and divided by 5, giving ^ as the result. In No. 69, the remainder, 5^, is changed to %-, which gives -|^ when it is divided by 8. 656. Some mistakes would be avoided if pupils would learn to ask themselves if the answer they have obtained is a reason- able one. Permit the scholars to work out all these examples without giving them a rule for " pointing off." 669. See Art. 521. 3. 6x6. 7. 800 -s- 100. 4. 300-4-12. % 8. 8x8. 5. 86 x 1. 9. 7 X 11. 6. 36-5-4. 10. 64 Xf. 670. 3. 25x 12. 7. 800X.1. 4. 36 -* 6. 8. 8x8. 5. 86-f-l. 9. 7x12. 6. 32x5. 10. 64-*-^. 671. 2. $2^x200. 6. 25^x800. 3. 25^x4. 7. $2^x20. 4. $ 12 x 400. 8. 60^ x 1000. 5. $2x8. 9. $5000x 7. 10. $1x6. 66 MANUAL FOR TRACKERS 677. Teachers should carefully avoid giving unnecessary " rules." There is no good reason why an average pupil should not be able to determine for himself how to ascertain what part of $ 15 a man has spent when he has spent $ 5. While the in- troduction of fractions into such an example makes it more diffi- cult for the scholar to give the answer off-hand, his instruction up to this time should have taught him that the same process is to be employed. A pupil should be required to depend upon himself to at least a reasonable extent. 678. As a preliminary to the work in denominate numbers in the next three pages, the teacher should place on the board a few such examples as the following, to which the scholars should give answers at sight : 1 qt. 1 qt. 1 pt. 3 qt. 1 qt. 1 pt. + Ifr qt. + 1 qt. 1 pt. - 1 qt. 1 pt. X2 2)3 qt. 1 qt. 1 pt.)3 qt. Nearly every member of the class will be able to obtain the results in a moment, without any suggestions from the teacher. If the examples are left on the board, the pupils can refer to them for aid in working some of those found in the text-book. The teacher that wishes to develop power in her scholars should be careful not to give a particle more assistance than is necessary. She should permit the children to deduce from the above examples the rules necessary to solve the others, being patient if the pupils are somewhat slow in doing this work. When, however, a circuitous method has been employed, she should lead the class to see how the work can be improved by the use of a shorter way. 680. It may be necessary to take up again, for purposes of review, the preliminary exercises of the previous chapter. See Art. 569, pp. 56 and 57. NOTES ON CHAPTER SEVEN 67 681c As the table of square measure is not introduced until the next chapter, it will be necessary to reduce to yards the dimensions that are given in feet or inches. 2. 18 yd. by 21 yd. 8. 18 yd. by 2 yd. 3. 2 yd. by 3 yd. 9. 16 yd. by 15 yd. 7. 9 yd. by 32 yd. 10. 1 yd. by 24 yd. 682. No. 14. 14 yd. by f yd. No. 15. 8 pieces, each 36 yd. long and ff yd., or -| yd. wide. No. 18. See Arithmetic, Art. 818, problem 20. A modifica- tion of this diagram, showing four squares instead of four rec- tangles will be the drawing required, except that the squares above and below need not necessarily occupy the positions there indicated. XI NOTES ON CHAPTER EIGHT With this chapter begins the regular work in decimal frac- tions, and the pupils should now be taught the principles under- lying the various operations. 685. While pupils may know that - 2 ^- means that 23 is to be divided by 8, it may be well to lead them again to see that f is the same as 3-*- 4, or ^ of 3. After they understand that every common fraction may be considered an " indicated division,'' they will understand that the decimal fraction obtained by per- forming this operation is the equivalent of the common fraction whose denominator is used as a divisor and whose numerator is used as a dividend. See Arts. 563 and 564. 686. As previous work in decimals has been confined chiefly to three places, some review and extension of the notation and numeration exercises of Arts. 547551 may be necessary. 687. After writing each of these decimals in the form of a common fraction, a scholar should be able to determine at a glance whether or not it can be reduced to lower terms. This reduction is possible when the decimal is an even number or terminates in a 5. While it is inadvisable to waste time in calculating the great- est common divisor, pupils should be encouraged to use large divisors ; 4 rather than 2, when possible, and 25 rather than 5. 688. The common fractions contained in these exercises are such as do not require much calculating to change them to deci- 68 NOTES ON CHAPTER EIGHT 69 mals. The scholars should be able to write the numbers in vertical columns directly from the text-book, making the neces- sary reductions mentally. In reducing -^ to a decimal, it may be easier for some to con- sider it \ of \, or ^ of .25. The reduction of -ff is simplified by multiplying each term by 2, making it -$-$, or .46, instead of dividing 23 by 50, etc., etc. 690. Nos. 62, 64, 66, and 68 may be worked by using the common fraction given, and also by reducing this to a decimal before performing the multiplication. 691. See Art. 563, p. 55, and Art. 616. 692. The teacher should not permit the employment of long division in these examples. In No. 92, the children can see that changing the dividend to .18756 divides it by 100, and that .18756 -H 3 is the same as 18.756 by 300. See Arithmetic, Art. 668. 694. Ciphers at the right of a decimal should be rejected, excepting, perhaps, the final in cents. See Nos. 3, 4, and 10. 2. $.95x7.6. 6. $22x108.745. 3. 2.80x48.6. 7. $.75x148.6. 4. 21.30x39.25. 8. $ .13J x (2376 -*- 12). 5. $.68x 18.75. 9. $35x4.5. 10. $ 13.50 x [(28 x 12) -4- 144]. While it is inadvisable to confuse children by too many short methods in the earlier stages, they should be encouraged in ex- amples like the foregoing to use as a multiplier the number that will make the work easier, and to employ a common fraction instead of a decimal whenever the use of the former would lighten their labor. In No. 1, for instance, the result is obtained with fewer figures by multiplying 24.4 by 6J, instead of 6.25 X 24.4. 70 MANUAL FOE TEACHERS 695. The operation should first be indicated. $.90x38648 _ $.36x48576 ~^0~ ~W~ $ 5 x 18964 $ 1.83 x 691Q4 2000 2 x 56 etc., etc. 703-704. See notes on previous special drills, Arts. 286 and 350. 705. See Arts. 528 and 649. In multiplying 46 by 33^, divide 46 by 3, which gives 15-J, and substitute 33^ for the fraction in the quotient, thus obtaining the result, 1533^. 706. 975 -- 25 = 9f -*- = 9 x 4. 433^ -*- 33 = 4 +- $ - 4* X 3. 708. Use first as " sight " problems, if the pupils find the numbers too large to be carried in the mind. By degrees, how- ever, they should acquire the power to solve problems of this kind without seeing the figures, especially when the operations are not numerous or involved. 709. In such examples as Nos. 1, 9, 10, 11, and the like, many children fail to comprehend the form of analysis generally given. While they get some facility in applying the method, they do not understand the underlying principle. In finding a number, f of which is 180, they learn to divide by 5 and to multiply the quotient by 6, and to repeat the customary formula, without knowing the reasons for the different operations. There are only four fundamental processes in arithmetic, and children should be taught to determine for themselves which to use in a given example that is within their experience, rather than to depend upon a rule which they do not fully understand, and which they are likely to forget or to misapply. See Art. 635. A few dia- grams are here introduced, to be used by the teacher that does TJNITERSTTY NOTES ON CHAPTfetOJ^ 71 not wish her pupils to obtain in No. 1, for instance, the length of the room by dividing 15 by -J. The five spaces in the width are each 3 ft., which will make the length 18 ft. When a scholar understands this from the diagram, he can understand that when |- of a number is 15, ^ is 15 -?- 5, or 3. While, for purposes of drill, many " abstract " examples of the same kind are brought together in one place, care has been taken in the problems to avoid having two consecutive ones alike in character. Problem work to be of value should not be permitted to become mechanical. Pupils should need to study each prob- lem to determine the method of solving it. 710. See Arithmetic, Art, 384. 716. By placing the multiplier at the right of the multi- plicand, the pupil can use the latter as the first partial product, instead of writing it again, as he would be compelled to do if the multiplier were placed in its usual position. 717. Some teachers might prefer to place the product by 8 above the multiplicand, as being the form to which the scholars are more O /IAQ ^ 10 576 x 14 x ^jTrOo x io 2304 accustomed ; but m such an 44424 Am. example as No. 26, the 16128 latter part of the work can be shortened 169344 Ans on ^ ^y placing the product by the units' figure under the other. 48600 No. 28 should be worked as is here shown. Annexing two ciphers to the multiplicand and 3)97200 multiplying by f gives the product by 66f . 2300400 719-720. See Art. 521. 1. 241b. @fi 3. 64yd. 2. 24 horses $125. 4. 485 bu. @$1. 72 MANUAL FOE TEACHEES 5. 96 lb. @$f 18. 60-f-f. 6. 840yd. @$f 19. 64 -s- f. 7. 360yd. @ $f. 20. 28 -*- If. 8. '48cwt. @ff. 21. 17x4. 9. 92 hats @f If 22. 256 x f 10. 128 lb. @ $f 23. 25 x 16. 11. 27 -j-J. 24. 6x6. 12. 300 -v- If 25. 86 x 1. 13. 24 -^f. 26. 33x5. 14. 15-5-J-. 27. 800 Xf. 15. 60-*- 2}. 28. 8x8. 16. 32-*-f. 29. 7 X 11. 17. 70-;-f 30. 64 xf. 721. The decimals should be reduced to common fractions whenever the work is rendered easier by the change. 1. 360 xf 7. 72 xf 13. 84 xf 2. 560 x f . 8. 84 x f. 14. 15 X 6. 3. 240 X f . 9. 96 X aV 15 - 4 x 4 - etc. etc. etc. 722. The pupil should employ such method as is best adapted to the particular example : 1. 240 -i-f 9. 48-5-yfo. 17. 65-f-f. 2. 360 -f-f. 10. 72 -H V 18. 840-^-8. 3. 45 -5- f. 11. 92000 -f- 2. 19. ll-s-^. etc. etc. etc. 723. Nos. 1 to 8 are intended to furnish practice in sight cancellation. In Nos. 13 to 16, the reduction of the multiplier to an improper fraction will simplify the work for some pupils. 726. Whenever possible, the least common denominator should be determined by inspection. 735. Do not give " rules." See Art. 678. NOTES ON CHAPTER EIGHT 73 736-737. These exercises are introduced to accustom the pupils to add and to subtract simple mixed numbers without rewriting the fractions reduced to a common denominator. 740, In these and other similar examples, the teacher should not anticipate the work of the higher grades by systematic instruc- tion in advanced topics. All that should be done with respect to these problems is to show the pupils that, when a solution involves multiplication and division, time may frequently be saved by means of cancellation. The pupils should be permitted to work out No. 1 at length, if they wish ; after which they should be required to indicate the work by signs, and then to cancel. Division should, of course, be indicated by writing the divisor as a denominator. Some excellent teachers require their scholars before beginning work on a problem to indicate by signs all the operations neces- sary to its solution, thereby compelling them to study the con- ditions thoroughly at the outset. Too many pupils commence to add, subtract, etc., without fully realizing what is required in a given example. 742-744. See Art. 678. 745. The pupils should write the dimensions on each diagram, changing them, when necessary, to the denomination required in the answer. 746. The formal study of percentage belongs to the next year of the course, and teachers should not dwell too much on this topic. After the pupils understand the meaning of the term per cent, they should be able to work the examples given. Other technical terms, definitions, etc., should be omitted for the present. 753. The pupils will readily see that the words " Bought of," used in Arts. 546 and 642, are inappropriate in bills for work 74 MANUAL FOE, TEACHERS done. No. 5 may be made out in the form here shown or similar to the one given in Art. 546. See Art. 642 for a bill for goods bought at different times; or use the heading given in this article. 754. What has been said about percentage in Art. 746, is ap- plicable to this topic. Such children as hear their parents talk of savings-banks, etc., know sufficient about interest for the purposes of this chapter. No rules should be given. 756. The pupils should deduce their own rule for calculating the area of a right-angled triangle. 758. In Art. 653 the pupils have been taught to multiply 18f by 6 in one line ; in Art. 654, they have learned how to divide 18f by 2, which is the same as finding |- of 18f , so that nothing new is here presented. 763-764. Although these examples are not strictly practical, they are useful in giving the pupils the facility necessary to per- form readily operations involving fractions or decimals. While it is not necessary to work them all, the scholars should by this time have acquired such expertness in the fundamental operations as to be able to obtain the results in a very short time. 765. See Arithmetic, Art. 591. XII NOTES ON CHAPTEK NINE The technical terms used in denominate number work should now be regularly employed by teacher and pupil, but set defi- nitions should not be memorized. The scholars should be re- quired to arrange their work properly, and to perform the various operations with as few figures as are consistent with accuracy. 767. In reducing 16 gal. 1 qt. to quarts, the pupil should write 65 qt. at once. He multiplies by 4, saying 4 sixes are 24, and 1 are 25 writing the 5, etc. In reducing 31-J- gal. to quarts, the work should occupy but a single line. See Arithmetic, Art. 653. 770. No special rule should be given in Nos. 33, 34, and 35 for the reduction of a fractional or a decimal denominate unit. 773. A pupil should be permitted to work such examples as No. 2 in his own way. They do not occur frequently enough in practice to make it advisable to give them special treatment ; but the teacher should suggest, as in other exercises, the advis- ability of shortening the work by indicating operations and can- celling. Thus, 121. 121 12 min. 30 sec. = 12^- min. = -^ hr. = ^~- ^j da., etc. 9 5 ' 60x24 da ' 6. 7501b.=^^ r T. = f T.; $5x5| = $26.87i or $26.88. Ans. 75 76 MANUAL FOR TEACHERS 7. No. oftons = $18.76H-$5 = 3.752; .752 T. = (.752 x 2000) lb. = 1504 Ib. Ans. 3 T. 1504 Ib. 8. 7 T. 296 Ib. = 14296 Ib. ; ($ 35.74 -f- 14296) x 18748 = Ans. 9. 9 T. 1568 Ib. = 19568 Ib. ; $48.92 -*- 19568 = cost per Ib. $73.11 -*- ($48.92 -*- 19568), or ($73.11 X 19568) -f- $48.92 = number of pounds. Reduce to tons, etc. 774. By this time, the pupils should know how to add com- pound numbers, so that the chief duty of the teacher should be to see that the operation is not spun out too much. A scholar of this grade should not find the total number of ounces in 1 by adding each column separately; he should say 27, 36, 39 oz., or 2 Ib. 7 oz., without writing anything but the 7 oz., which is put in its proper column and 2 Ib. carried. In 4, the addition of the units' column of minutes gives a sum of 15. Since minutes are changed to hours by dividing by 60, which ends in a cipher, the units' figure of the remainder will be 5, so that this figure may be written in the total. Carrying one, the sum of the tens' column is 11, which contains 6 once with a remainder of 5. This is written in its place, making 55 minutes, and 1 hour is carried. The two columns of hours are added in one operation 21, 38, 43, or 1 day 19 hours. 6 should be treated in the same way, no side work being permitted. In 7, the pounds are reduced to tons by dividing by 2000, so that the sum of the units', tens', and hundreds' columns of pounds may be written in the total, the sum of the thousands' column being divided by 2 to reduce to tons. 775. Nothing should be written but the results. In 27, the addition of 1 ton to 1552 Ib. will change only one figure of the latter, and this change can be carried in the head. In 29, 320 rods should be added to 15 rods mentally and 24 rods deducted from the sum, only the answer being written. 779. In dividing 5 bu. by 4, 79, the answer is not to be given as 1J bu. ; the division should be continued through pecks. The result in 88 should contain weeks, days, hours, and minutes. NOTES ON CHAPTER NINE 77 784. While these drills seem somewhat difficult for mental work, they should not be too severe for children that have been studying arithmetic for over five years, especially if the previous drills have been faithfully attended to. The ability of many children to handle numbers seems to decrease after the fourth school year, the greater portion of the subsequent instruction being given to new topics to the neglect of continued practice in the fundamental processes. The conscientious teacher should remember that the bulk of the mathematical work of most of her scholars after they leave school will not extend much beyond what has been learned in the first four years. The ability to handle at sight or mentally such numbers as are here given, will be of use to the scholars in various ways. The average pupil attends to only one figure at a time ; and he is frequently unable, after a simple addition or multiplication, to see that his answer is very far astray. Practice with such drills as these, and in the sight approximations, will enable him to test his work in such a way as to detect any very serious error. Scholars find it easier to add or subtract such numbers as 163, 8610, etc., when they are read "one, sixty-three ; " " eighty-six, ten ; " etc. Following the order in which the figures are read seems the most natural way in mental work. When a pupil is asked to find the sum of 163 and 137, he is less likely to make mistakes if he proceeds in this way : 263, 293, 300 ; adding to the first number 163 100, 30, and 7 in the order in which the figures are repeated to him. 786. In multiplying 21 by 15, 41 by 14, etc., the scholar generally finds it easier to commence with the tens : 15 twenties are 300, 15 ones are 15315 ; 14 forties are 560, and 14 are 574. 48xl6| becomes of 48 hundred; 32x37| = f of 32 hun- dred, etc. 787. These exercises present rather more difficulty, and are probably not so useful, on the whole, as the others. For this reason, they should be employed as sight work chiefly. 78 MANUAL FOR TEACHERS 788. In 13f X5, multiply 13 first by 5, and then |, obtain- ing 65 + 3f, or 68f. In dividing 24 by 2f, reduce both to thirds - 72 thirds *- 8 thirds = 72 -^ 8 9. 790. The teacher should not neglect such addition exercises as are scattered throughout the book. 791. It happens occasionally in multiplying by 4846 a mixed number, that the units' figure of the in- x 3f teger and the numerator of the fraction are the 5^14538 same. In such a case, a few figures will be saved by following the method given in the text-book, A^QQ instead of writing again the product by 3 as shown above. etc. 792. The product by 100 may be placed above the number, if desired. In multiplying by 1000, the multi- 9761 x 999 plicand is subtracted from 1000 times itself. r^^nnn To find the product of 9832 by 990, multiply by 99, and annex a cipher to the result. Taking 2758239 Ans. one-fourth of 268400 gives the answer to 21. 800. The pupils should find for themselves in 5 the number of square inches in a square foot, etc. A drawing is asked in the first part of 14, so that children will see that the dimensions are not 4x6. The short method of finding the area of the fence in 15, by multiplying 900 by 10, should not be given yet: the scholars should be permitted, for the present, to calculate the area of one part at a time. In 16, it is suggested that the area of the walk be ascertained by subtracting from the whole area (250 X 200) sq. ft., the area of the part left for the garden (230 X 180) sq. ft. ; but the scholars should be encouraged to calculate the sur- face of the walk in another way, such as by taking the two ends as measuring each 250 ft. by 10 ft., and the sides as 180 ft. each by 10 ft. The number of square feet in the sidewalk of 17 will be (270 X 220) - (250 x 200) ; or (270 x 10) + (270 X 10) + (200 NOTES ON CHAPTER NINE 79 X 10) + (200 X 10). For 20, a modification of the diagram in Problem 20, Art. 818, is desired. 801. To show pupils what is required in 21, a pasteboard box, without a cover, may be opened out as is represented in Problem 2, Art. 871, the upper rectangle (the bottom of the box) repre- senting the ceiling. 802. 3. The sixth dose will be taken at 7 o'clock, the second at 3 o'clock, the fourth at 5 o'clock. 4. He works 6 days. 6. A fence 6 ft. long will require 2 posts ; a 12-foot fence will require 3 posts ; a fence 120 ft. long will require 21 posts. 803. In finding the time between two dates, the first date is excluded except when the contrary is expressly specified. 804. 11. 30 days +19 days. 12. days in October + 30 days in November + 30 days in December. 805. 1. In February, there are (29 6) days, or 23 days. 3 and 4. Leap year. 11. Jan. 8, 15, 22, 29; Feb. 5, 12, 19, 26 are Sundays. The man works 30 days in January and 28 in February, less 8 Sundays and 1 holiday. 807-809. See Arts. 746 and 754. 808. 3-13 should be worked as "sight" exercises, -J- being vised for 25%, | for 121%, e tc. 810. First compute the interest for one year. 1. $3.60 for a year ; of $ 3.60 for 2 months. 2. $ 3.60 for a year ; 1 of $ 3.60 for 60 days. 3. $ 5.00 for a year ; $ 5 X 2-i- for 2 yr. 6 mo. 4. $ 6.00 for a year ; ^ of $ 6 for 30 days. 5. $ 9.00 for a year ; of $ 9 for 90 days. 812. To take advantage of any opportunities for cancellation that may be offered, this method is given. It will afterwards be found useful in calculating the principal, the rate, or the time. 80 MANUAL FOR TEACHERS Pupils should not at this stage be taught more than one method of finding interest, and that the most direct and the most obvious one. 813. 2. In changing 2 mo. 12 da. to the fraction of a year, it is not necessary to reduce to the lowest terms. Change the time to 72 days, and write 360 underneath, -^Q-; the necessary re- duction can be made later in the cancellation. 6. Write 21 months as -f-^- years, canceling afterwards. The 100 in the denominator of an interest example should seldom be canceled, except as a whole or by 10. 815. 1. 6 hr. 17 min. 5 sec. 22625 sec.; 3 hr. 15 min. 25 sec. - 11725 sec. Ans. ^f| - f| 2. 3mi.96rd.x3t = 9mi.288rd. + lmi.32rd. = llnii. Ans. 4. A furnished ^ of the money, and should receive ^ of $ 1500, or $750; B should receive | of $1500, or $500; should re- ceive i of $1500. 5. If 5 T. 1000 lb., or 11000 lb., cost $30.25, 1 Ib. will cost $ 30.25^- 11000; and 7 T. 320 Ib, or 14320 Ib, - will -cost ($ 30.25 + 11000) X 14320. Cancel. IIQQQ 6. 25^x8JHf.= 250x8. Ans. 7. 2 yr. 7 mo. 8 da. - 31 mo. 8 da. = 31^ mo. = 31-^- mo. 10. 360 yd. @ 30^ cost $108. The number of square yards = 360 X H- = 360 X | = 270, on which the duty at 8^ per sq. yd. will be 8^ X 270, or $ 21.60. The duty on the value will be 50% of $108, or \ of $108, or $54; the total duty being $21.60 + $54 -$75.60. Ans. 816. 3. 5 bbl., 300 lb. each, @ 5^ per lb. 4. Interest on $ 200 for 6 mo. @ 6%. 5. 12 men take 24 days ; how long will 24 men take? 6. What decimal of 640 acres is 320 acres ? 7. 20 thousand bricks @ $ 20 per M. 8. 5600 lb. @ 87^ per bu. of 56 lb. NOTES ON CHAPTER NINE 81 9. 10 Ib. cost $8 ; find cost of 21 Ib. 10. Freight on 20 hundred Ib. @ 70^ per cwt. 817. 2. The wall 8 yd. X 4 yd. contains 32 sq. yd. ; the door is f yd. by 1-1- yd., and contains 4 sq. yd. ; 32 sq. yd. -- 4 sq. yd. = 28 sq. yd. Ans. 3. Number of square inches in the surface of the widest face = 8x4; in the surface of one side 8x2; in the surface of end 4x2. 4. (288x96) +-(8x4). 5. [(24xl2)x(8x 12)] -+-[8x2]. 6. See No. 20. Make four rectangles adjoining each other, each 8 inches high the first and the third being 4 inches wide; and the second and the fourth, 2 inches wide. Above and below the second, and connected with it, draw rectangles 2 inches wide and 4 inches high. These two rectangles may be drawn above and below either of the other rectangles, the above dimension being used if drawn above and below the fourth ; if drawn above and below the first or the third, they will be 4 inches wide and 2 inches high. The pupils should be permitted to make the diagram in their own way, and they should be encouraged to make one that differs from one drawn by a desk-mate. 8. The number of rolls will be (45x36) +-(24 X 1|). Cancel. 818. The scholars should make this table without any assist- ance. To obtain the number of acres in a square mile, indicate the number of square rods in a square mile, 320x320, and divide by the number of square rods in an acre, 160. 14. The number of yards = (5 + 3 + 4 + 7 + 3 + 6 + 12 + 10) X5f 15. Original dimensions 12 rods X 13 rods, making area 156 sq. rd. Present area = 156 sq. rd. (15 + 21) sq. rd. 19. ftof (80x60) ]+- 4840- Ans. 819. 1. 43 yd. = (43 -*- 5) rd. = (43 -H -U-) rd. - (43 X T 2 T ) rd.^-rd.-T.- rd. 82 MANUAL FOR TEACHERS 2. 43yd. = 7 T 9 T rd. ^ rd. = ( T \X 5*) yd. - (^ x 4) yd. = 41 yd. -4ns. 7 rd. 4J yd. 3. 43 yd. = 7 rd. 4| yd. - 7 rd. 4 yd. 1 ft. 4. 43 yd. - 7 rd. 4 yd. 1 ft, - 7 rd. 4 yd. 1 ft. 6 in. 824. 34. Carrying 1 to the column of yards, the total becomes 8 yd. or 1 rd. 4 rd. 3 yd. 1 ft. 2|-yd. Changing l yd. to 1 ft. 6 in., and 9 rd. 4 yd. 2 ft. addmg this to 17 rd. 2 yd. 1 ft. 6 in, 3rd. 1 ft. 6 in. the accompanying answer is obtained. 17 rd. J yd. 1 ft. 6 in. 38. 8rd. Oyd. lft.-2rd. Oyd. 2ft. 17 rd. 2 yd. 1 ft. 6 in. = 5rd. 4^-yd. 2 ft. = 5 rd. 4 yd. 2ft. + ^ yd. = 1 ft. 6 in. + 1 ft. 6 in. - 5 rd. 5 yd. 6 in. Ana. 17 rd 3 yd> Am 40. 5 rd. 4 yd. 2 ft. X 4 = 23 rd. 1 yd. 2 ft. -23 rd. 1 yd. 2 ft. + 1 ft. 6 in. = 23 rd. 2 yd. 6 in. 825. 10. The other dimensions would be 8 ft. and 4 ft., or 16 it. and 2 ft. 14. Number of cubic yards = ^ X -^ X f . Cancel. 15. % (yd.) X 2 (yd.) X width (yd.) = 1 (cu. yd.) ; or X 2 X x=l] # = 1 ; 1 yd. Ans. 16. A gallon contains 231 cu. in. ; a cubic foot contains 1728 cu. in. 1 cu. ft. = (1728 * 231) gal. = 7^ gal. - about 7-J- gal. J.ns. 17. About 1J cu. ft. J.715. 18. Number of gallons (21 X 15 X 22) -*- 231. 19. The decimal in the denominator is re- 36 x 28 X 6400 moved one place to the right, and a cipher is 2150 & 4 annexed to 64 in the numerator. NOTE. 2150.4 cu. in. is used instead of 2150.42 cu. in., the more correct equivalent, because the former is divisible by 6, 7, 8, etc. 25 . [$ 6.40 X (40 X 16J) X 4 x 3] -*- 24|. Cancel. NOTES ON CHAPTii^^fCtE iPO^i^ 83 826. 3. At 7^- gal. to cu. ft., a tank of 150 gal. will contain (150 *- 7-i-) cu. ftf = 20 cu. ft. The dimensions will be 2 ft. X 2 ft. X 5 ft., or 4 ft. X 1 ft. X 5 ft., etc., etc. 4. At 1^ cu. ft. to a bushel, the bin will contain 1-^ cu. ft. X 100 = 125 cu. ft. The dimensions will be 5 ft. X 5 ft. X 5 ft., or 5 ft. by 10 ft. by 2-i- ft., etc., etc. 5. 1000 bricks will build (1000-^20) cu. ft. A wall 1 ft. thick can be 10 ft. long and 2 ft. high, or 5 ft. long and 4 ft. high, etc. 6. 10 yd. X 5 yd. X 2 yd. (30 ft. X 15 ft. X 6 ft.), 4 yd. X 5 yd. X 5 yd., etc. 7. A gallon weighs about 8 Ib. ; a pint about 1 Ib. 8. A cubic foot of iron weighs about 7 times 64 pounds. 9. About| of $800. 10. About 4 years' interest. 831. 5. See Arithmetic, Art, 642. 6. See Arts. 829-830. 7. Three inches square = (3 X 3) sq. in. 832. 1. See Art. 1022, No. 15. 4. Including Sept. 19, the time is (28 + 30 + 31 + 30 + 31 + 31 + 19) days. 833. 6. The written analysis of an arithmetic example should be required occasionally as an exercise in composition. 7. 7000 gr. X 2| = number of grains in 2 Ib. 14 oz. Dividing by 480 grains, the number of Troy ounces is obtained (7000 X 2J)-^480. Multiplying $1.80 by this number, the cost of the urn is ascertained ($ 1.80 X 7000 X 2) *- 480. 835. 8. Since the denominator of a fraction indicates the number of parts into which a thing is divided, a larger denomina- 84 MANUAL FOR TEACHERS tor indicates a greater number of parts, and, therefore, smaller ones. 839. 1. If three-fifths of a bbl. cost $2.13, six-fifths will cost twice $2.13. 8. Commission at 1% would amount to $3; at -|-%, it is $1.50. 18. -| = 18^; i = 6^; |, or of , = of 6 841. Add without re-writing the fractions reduced to a common denominator. 845. 10. 75% of ^, or f of T %, or ^, is sold for $1710. Factory is worth $ 1710 -*-$. 13. First piece contains (20 X f) sq. yd., or 15 sq. yd. Width of second piece in yards = 15 -4- 12. 16. 8 men and 5 boys 8 men + 2-^- men 10|- men. If 7 men do a piece of work in 10-|- days, 1 man will do it in 1QJ- days X 7, and 101 men will do it in (10| days X 7) -f- 10|. Cancel. 17. Each of the six square faces of a cube contains (6 X 6) sq. in., or 36 sq. in. ; the whole surface will be, therefore, 36 sq. in. X 6. Each face contains (-J- X -J) sq. ft. = J sq. ft. ; whole surface = i sq. ft. X 6. Contents in cu. in. = 6 X 6 X 6 ; in cu. ft. = -J- X -J- X -J-. 18. || in water and if in mud = f, leaving ^ above water, or 5 ft. Length of post = 5 ft. -4- ^ 19. [(10 X 9) + (12 x 10) + (8 x 11) + (6 x 12) + (2 x 13) 4- (1x14)] -4- [10 + 12 + 8 + 6 + 2+1]. 21. From Oct. 25 to Dec. 31, inclusive, there are 7 + 30 + 31, or 68 days ; Oct. 30 is Sunday ; also Nov. 6, 13, 20, 27 ; Dec. 4, 11, 18, and 25 9 Sundays, Election Day, and Thanksgiving to be deducted, or 11 days, leaving 57 days, at $31 per day. 23. 12 Ib. tea cost $2.80 + $2.00, or $4.80'; value per Ib. 40 j*. NOTES ON CHAPTER NINE 85 24. House and lot, or 3| lots + 1 lot, or 4^ lots =-$8100; 1 lot = $ 8100 - 4| - $ 1800 ; house = $ 1800 x 3|. $ 18 x (20 x 12) x (15 x 12) x (6 x 12) 1000 X 8x4x2 27. 36 yd. 8 in. 1304 in.; 13 yd. 1 ft. 9 in. 489 in.; quantity left = 1304 in. - 489 in. = 815 in. ; fraction left = y 8 ^ = f ; decimal left = .625 ; per cent left 62^-. Ans. 28. Assessed value -80% of $ 30000 = $ 24000. Taxes on 24 thousand dollars -$21. 60 X 24. 846. 2. See Arithmetic, Art. 1251 ; angles E, F t G, and H\ and M, N, 0, and P. 3. Art. 1251 ; angles A and JB, and D. 4. Angles I and J, K and L. The scholars should under- stand that two lines can be perpendicular without one being a horizontal line and the other a vertical line. 5. The size of an angle does not depend upon the length of the lines that form the angle. Two short lines may meet at a very obtuse angle, and two long lines may form a very acute angle. 13. If the pupils have in their drawing lessons constructed triangles by means of compasses, these may be used ; otherwise, let them manage as best they can, no great accuracy being required. 15. Children are accustomed to seeing an isosceles triangle in only one position : they should learn that if a triangle has two equal sides, it is isosceles, no matter whether the unequal side is vertical, horizontal, or oblique. 16-22. Accustom the scholars to the occasional employment of an oblique line as a base in constructing squares, rectangles, etc. See Arithmetic, Art. 1265. A card may be used to make a square corner. 86 MANUAL FOR TEACHERS 24. See Arithmetic, Art. 929, No. 8, for a rectangle, a rhombus, and a rhomboid, having equal bases and equal altitudes. No. 5 shows three rhomboids of equal bases and equal altitudes, but differing in shape. 25. See Art. 929, No. 8. 847. 1. (i of 15 X 20) sq. in. The length of the third side does not enter into the computation. 6. Let the scholars find the area of the rectangle, 66 ft. by 63 ft., and the two triangles, 31 ft. each by 63 ft., and find the sum of the areas. Then lead them to see that bringing the right- hand triangle to the left of the rhombus would make a rectangle 97 ft. by 63 ft., whose area is the sum above found. 7. Find the area in square meters, saying nothing more about the meter than that it is largely used on the continent of Europe, and is a little longer than a yard. 8-10. Give no rules yet for calculating the areas of trapezoids and trapeziums. Let the pupils ascertain the areas of the figures from the data supplied. XIII NOTES ON CHAPTER TEN The formal study of algebra belongs to the high-school ; but some so-called arithmetical problems are so much simplified by the use of the equation that it is a mistake for a teacher not to avail herself of this means of lightening her pupils' burdens. In beginning this part of her mathematical instruction, the teacher should not bewilder her scholars with definitions. The necessary terms should be employed as occasion requires, and without any explanation beyond that which is absolutely neces- sary. 849. Very young pupils can give answers to most of these questions ; so that there will be no need, for the present, at least, of introducing a number of axioms to enable the scholar to obtain a result that he can reach without them. 850. Pupils will learn how to work these problems by work- ing a number of them. They may need to be told that x stands for 1 x ; and that, as a rule, only abstract numbers are used in the equations, the denomination dollars, marbles, etc. being supplied afterwards. While the scholars should be required to furnish rather full solutions of the earlier problems, they should be permitted to shorten the work by degrees, writing only whatever may be necessary. 4. a? + 2ar = 54. 8. x+2x+6x= 27000. 5. x+5x = 78. 9. 6. 7# + 5.r 156. 10. 7. 9#-3o;=:66. 11. 88 MANUAL FOR TEACHERS 12. 2x + IOx = 96. 13. Let x = the fourth ; then 4 x = the third, 12 x = the second, and 24 x = the first. 14. # = the second, 2 # = first, 9x = third. 15. 5* + 4* = 81. 17. 4^ = 340. 16. 24 a? = 456. 19. 3 x + 4 x = 175. 20. Let # each boy's share ; 2 a; = each girl's share. 21. # = number of days son worked; 2 # number father worked. 3 x son's earnings ; 8 # father's earnings. 3# + 8#= 165. 22. x = number of dimes; 2 x = number of nickels; 6x = number of cents. (10 X#) + (5x2*) +(1x6*0 -78, or 23. 15# 24. x + 4:X + x + 4x-= 250. 25. Let a: = cost of speller ; then 3 x cost of reader. -26. Let x = smaller ; then 5# = larger. 27. Let x = Susan's number ; 2 x = Mary's; 3 x = Jane's. 851. 10 : %x is the same as 852. Pupils already know that f means 3-*- 4, so that they can understand that means 3 # -s- 4, or J of 3x. When ^ of something (3 x) is 24, the whole thing (3 x) must be 4 times 24, Q /v. or 96 ; that is, when = 24, 3 x =- 96. 4 When ^ - 24, 2 y = 24 X 3, or 72. o When i2 - 20, 4 2 - 20 X 5, or 100. 5 From these examples can be formulated the rule for disposing of a fraction in one term of an equation, which is, to multiply NOTES ON CHAPTER TEN 89 both terms by the denominator of the fraction. In changing the first term of the equation, = 24, to 3 x, it has been multi- plied by 4, so that the second term must also be multiplied by 4. 853. In solving these examples by the algebraic method of " clearing of fractions," attention may be called to its similarity to the arithmetical method. To find the value of y in 2, the pupil multiplies 8 by 5 and divides the product by 2 ; as an ex- ample in arithmetic, he would divide 8 by --, that is, he would multiply 8 by -| ; the only difference being that by the latter method he would cancel. While ^ 8 may be changed to ^ = 4 by dividing both terms 5 5 by 2, beginners are usually advised to begin by " clearing of fractions," short methods being deferred to a later stage. 854. 6 may be written ^ + ^ = 92. 5 7 8. 2%x should be reduced to an improper fraction, making the equation, = 115. Make similar changes in 12, 14, 18, and 20. 8 855. 2. x + ^ = lW. 4 9. Let 5 x numerator ; 7 x = denominator. 7x 5# = 24; 2x 24 ; x 12. The numerator, 5 x, will be 5 times 12, or 60 ; the denominator will be 84 ; and the fraction, -||. Ans. 10. Let x = greater ; ^ = less. 90 MANUAL FOR TEACHERS Clearing of fractions, 1 x + x = 3360, Sx = 3360, x = 420, the greater number, - = 60, the less. 7 Or, let x = less ; 7 x greater. x + lx = 480, 8* = 480, x = 60, the less, 7# = 420, the greater. The employment of the latter plan does away with fractions ih the original equation. 11. 30* - 2; = 522, or#-^- =522. 13. Let x = number of plums; 4 x number of peaches. Then 2,x will be cost of plums, and 12 x the cost of the peaches. 15. tf-y = 80. 17 r Bx X -94 "8"~4~ 18. x + l%x + (l%x X 3) = 15. 19. Let x = price per yard of the 48-yard piece ; 2x = price per yard of the 36-yard piece ; 48 x will be the total'cost of one, and 72 #, of the other. 48 x +72 x = 240. 20. 160#+ 120^ = 840. 856. The pupils should be permitted to give these answers without assistance. In Art, 857 is explained what is meant by " transposing." NOTES ON CHAPTER TEN 91 858. While these exercises are so simple that they can be worked without a pencil, they should be used to show the steps generally taken in more complicated equations. _ In 1, for instance, the work should take the _ form here indicated, only a single step being taken at a time. In 19, the first step is to clear the equation of fractions by multiplying by 6 ; the second step is to transpose the unknown 2# 6 16 + - - quantities to the left side of the QA i Q o equation, and the known quantities yt) + O X ZiX . ,-t -i , ,1 .1 . T 2* = 96 + S6 * the "ght ; the th,rd step 1S to com- __ kme the unknown quantities into _ one, and to make a similar combina- tion of the known quantities; the last step is to find the value of x. After a little more familiarity with exercises of this kind, the pupil can take short cuts with less danger of mistakes ; for the present, however, it will be safer to proceed in the slower way. 859. 5. # + (#+75) + # + (#+ 75) = 250. x + x + x + x = 250 - 75 75. NOTE. The parentheses used here are unnecessary. They are employed merely to show that x + 75 is one side of the field. 6. # + (# + 8) = 86. 9. 7. # + # + 318-2436. 10. a;----=45. 3 4 8. #+ |+7 = 100. 2 11. # = one part ; 2# 6 other part. # + 2# 6 = 45. 12. # = John's money ; # + 5 William's money. # = 103. 92 MANUAL FOE TEACHERS 13. Let x = price of a horse ; x 80 = price of a cow ; 4 x cost of four horses ; 3 x 240 = cost of three cows. 4# + 3# -240-635, 7x -635 + 240 -875, x 125, price, in dollars, of a horse ; x 80 45, price, in dollars, of a cow. Other pupils may solve the problems in this way : x = price of a cow ; x + 80 price of a horse. 3#+ 4^ + 320-635, 7 x -635 -320 -315, x 45, price, in dollars, of a cow ; x + 80 125, price, in dollars, of a horse. 14. x = number of dimes; x + 11 number of five-cent pieces; 10 # value of dimes (in cents); 5# + 55 value of five-cent pieces. 100. 15. x greater ; x 48 less. # + #-48-100. Of, x = less ; x + 48 greater. x + x + 48 100. 17. x share of the first ; x + 2400 share of the second ; x + 2400 + 2400 share of the third. x + x + 2400 + x + 2400 + 2400 - 18000. 18. Let x less ; x + 33 greater. o; + 33-33? = ll. Bringing known quantities to the left side of the equation, and the unknown quantities to the right, 33 -11 = 3* -a;, 22 = 2x, NOTES ON CHAPTER TEN 93 Or, x-3x = Il 33, -2* = -22. Changing signs of both terms, 2* =22, x = 11. This problem may also be worked in this way : x = less ; 3 x + 11 greater. 19. x = number of 5-cent stamps ; x + 15 = number of 2-cent stamps ; x + 30 = number of postal cards. 5rr + 2a; + 30 + tf + 30 = 100. 20 . x = number of horses ; x + 1 7 = number of cows ; 2 x + 39 = number of sheep. R XIV NOTES ON CHAPTER ELEVEN With this chapter begins the regular work in percentage, and it is important that the pupils obtain, as soon as possible, a correct idea of what is meant by the term per cent. Many of the various subdivisions of this topic found in some books, are taken up only incidentally, while others are omitted altogether, the aim being to give the pupils a foundation upon which they can subsequently build, rather than to scatter their energies over too great a diversity of subjects. 860. The reduction of a common fraction to a per cent, con- sists in changing the former to a decimal of two places. In re- ducing J to a decimal, the result is .5, or 5 tenths ; in changing ~to an equivalent per cent, the result is 50 per cent, or 50 hundredths. In reducing i to a decimal, the answer is given in three places, .125, or 125 thousandths ; in changing it to a per cent, the division is stopped at the second place, and the remain- der written as a fraction, 12 per cent, or 12J hundredths. The denominator of a per cent being always the same, 100, the comparative value of several per cents is known at sight. To compare -| and -| as common fractions, they must be changed to -f-| and -f-|; if a further comparison is to be made between these and -f-%, a new common denominator must be employed, and the fractions reduced to /$, -j 2 ^-, and -j 3 ^-. Changing the fractions to decimals, 625 thousandths, 6 tenths, and 58-|- hun- dredths, simplifies the comparison ; but it is still easier to de- termine their relative value when they are expressed as 62-|- per cent, 60 per cent, and 58^ per cent. NOTES ON CHAPTER ELEVEN 95 The teacher must not be discouraged if the pupil fails to grasp at once the full meaning of percentage, Definitions will not help materially ; much practice in working examples is necessary to give the knowledge desired. 863. Many children find it difficult to distinguish between \/o and 50%. If the former is read in the business way, \ of one per cent, it may make the distinction plainer. 864. Per cents being generally given in two figures, scholars hesitate to give the correct answers: 300%, 250%, 125%, 1633%,420%,910%. 865. While pupils will find 33^% of 81 cows, by dividing 81 by 3, they should understand that they are really multiplying 81 by 331 hundredths, or 81 by . In 4, 6% of 150, or -^ of 150, may be obtained by multiplying 150 by 6 and cutting off two ciphers ; or by dividing 150 by 100, obtaining 1-|-, and multi- plying this quotient by 6 ; or by reducing 6% to -fa, and finding & of 150. In 9, the pupil should find 1% of $ 640 and take one- half of the result. The scholars should be permitted to use their own method of solving these problems, the different analyses given by the pupils furnishing their class-mates an opportunity to select a simpler method. 866. Although every pupil may not be able to determine at once the shortest way of calculating a given example, no one should be allowed to work 3, by multiplying by 33^. When the multiplication by ^ has been performed, the answer has been ob- tained, except as to the location of the decimal point, and the waste of time in multiplying by 3, repeating this product, and adding three columns should not be tolerated. No fault should be found with the average pupil for failing to recognize in 1, that 6|% is J^; or that in 12, 3^% is -^-. The general method should be to multiply by the figures given to represent the per 96 MANUAL FOR TEACHERS cent, except in such cases as 12%, 16f%, 25%, 33|%, 50%, and possibly a few others. Where the given numbers are used, they should be J made the multipliers and expressed as hun- ^ J^ _ * dredths. Nothing is gained in 5, by reducing ' ""' $1.55 fa Q ^ t a decimal; although in 13, writing - ! I , .055, might make it easier for some. 868. The rule generally given of finding the percentage, by multiplying by the rate expressed as hundredths, is here modified to the extent of using the common fraction to express hundredths, instead of the decimal, as being more in conformity with early algebraic methods. The teacher that prefers to ascertain the base or the rate by the older arithmetical method, will omit 30-41. 30. _?_x65 = Ans. 35. * + f=132; etc. 100 20 5 31. * = 26; etc. 37. s-f =78; etc. 20 o 32. | of* = 1^. 38. ^off 33. = 42; etc. 39. __ = 34. x + ^Ans. 40. _L f* = . Ans. 41. = 23; etc. 42. Let a; = rate. Then of 65 26, or -^- X 65 26. In an equation containing quantities to be multiplied, the mul- tiplication should be performed before the equation is cleared of fractions. This equation becomes 26, or 26, 100 ^0 it being immaterial whether canceling be done or not. NOTES ON CHAPTER ELEVEN 97 43. After a little experience with this class of examples, the equation may be written at once, in the order in which the terms are given : 44. ||of a=180,or^=180.v 45. * + - = 85. 4 46 5 = JL f ^ x _3 5~100 5' r i25~5' While the algebraic method is of no advantage to the bright scholar, it makes the employment of rules unnecessary in the case of the ordinary pupil. 50. fxf. 51. fff = 33. 52. of 800 = 100 ; \% of 800 = 1. 53. $175 + i of $175. 54. 2%% of x = 12.50 ; that is, = 12.50. 55. 6i X .16. r 9 ^ 1O 56. 34- = -^- of - ^^- ^ - 1U 57. -- of 389.50 = 124.64 ; 389 ' 50a; = 124.64 ; 389.5 x -12464; 3895 x = 124640. 58. f= 174.04; 95* =17404. -LUu 59. - + ^=1276. 98 MANUAL FOR TEACHERS '60. 984 - of x ; that is, j- = 984. 100 3 62. -.= 386.75. 4 2# 65. x = cost of oats ; x + = 1071. Divide the cost of the oats by 30^ to find the number of bushels. 68. Assessed value = f of $ 48000 = $ 32000. Taxes on 32 thousand dollars - $18.50 X 32. 869. In giving answers to these and to all other exercises, no " guessing " should be allowed. The pupil should be permitted to obtain the correct result in his own way that is, no inflexible rule should be given him to follow but he should be able to get the answer, using the algebraic method if that seems to him the easiest, as it may be in some instances. The examples are not arranged by " cases," so that each will have to be understood before it can be worked. ThQ careless pupil will probably give the wrong answer to 13 ; saying 6, instead of 600 ; he will be likely too, in 14, to use the larger number as a divisor, and to obtain 44f % instead of 225%. These mistakes are less likely to occur if he uses equations 3 = -^- and - = 20J. Even those scholars that have solved 200 100 in their arithmetic work of the lower grades, examples similar to 15, will have new light thrown on their method by using the equation, x + j = 20. In mental work, however, the first term 5# should be made - , to reduce it in size, so that it can be more 4 easily remembered. 24 is simplified by changing the fractions to whole numbers -f% is what per cent of ^f, 9 is what per cent of 10 before beginning to calculate the rate. In 25, 1-J- and 6f become f and ty, f and ^, 9 and 40. 870. 1-5 can be worked by the pupils without any explana- tion ; 6-20 present more difficulty. The beginner in algebra NOTES ON CHAPTER ELEVEN 99 desires to start at once with his X, without any preliminary cal- culations ; and the usual method of treating these examples requires him first to ascertain the gain or the loss before com- mencing his equation. The formula employed in the first five examples is : Cost X ~ = gain or loss. 100 When the pupil knows any two of these three terms, he can calculate the third ; and 6-15 furnish data from which the necessary two items can be obtained. The pupil must, however, be careful in 11, for instance, not to subtract the loss from the selling price to obtain the cost. In the following equations, cost X - is made equal to the gain or the loss. No canceling has been done. 6. =18;6*=18. 7 . ,^=1,93,0^=1293. 9. 908 - 40 *= 181.68. 10. i?=5.25; 84 x = 525. 13. = 43.75; 875 x = 4375. ., 934.56* 11fiR9 1012.6Qg_ 1fi( > 7> . 165" ~W~ In 16-20, the cost is represented by x. 16. x + - = 468.75. 18. x + |= 1646.08. 17. x --=73.84. 19. x -= O 100 MANUAL FOB. TEACHERS 20. *_l = 6630 27 " 3H* + t 100 28 x 16 *-3360 21. Gain = | of $275. ~100~ 22. x% of 60 = 15. 29. Gain = ty% of $8760. 23. , + =960. 30. ,-i-_S=70. 24. s% of 32 = 16. 31. 6000 - x% of 16000. 25. x% of 175 -25. 32. 6000 - x% of 24000. 26. x% of 200 25. 33. 1600 + 2^% of 1600. 34. 4200 - 31% of 4200. 871. In 1, the 30 cu. yd. are reduced to cubic feet by multi- plying by 27. Instead of performing the different multiplications, they are merely indicated, so that work may be saved by can- celing. Although 2 should be a simple problem for a bright pupil, it is apt to prove puzzling unless an x is introduced. A paste- board box may be used to represent the walls and the ceiling of a room, the sides and the top being then opened out to permit of its representation on the blackboard. 3. The area in square feet -|- of 132 X 110. This is reduced to acres by dividing by 9 X 30J X 160. 132X110X4 _ A 2 X 9 X 121 x 160 4. Number of strips 6 yd. -5- 27 in. 6 yd. -5- f yd. 6 X f . 7. The " development " of the fence will be represented by four adjoining rectangles, each marked 6 ft. high, the lengths being 25 ft., 100 ft., 25 ft., and 100 ft., respectively, the whole forming a rectangle 6 ft. X 250 ft. 8. A board's area in square feet 12 X -J- 6. Dividing number of square feet in the fence by 6, gives the number of boards, NOTES ON CHAPTER ELEVEN 101 9. The cost of a square foot is obtained by dividing $181.50 by (160 X 301 X 9) ; this, multiplied by (300 X 200), gives the cost of the plot. $181.50x4x300x200 160 x 121 x 9 The amount received for the lots will be $160 X 6. 10. Number of cakes = (320 X 160) -5- (4 X 2). 11. Number of cubic feet 320 X 160 X 1-J-. 12. (320 x 160 X 1) -s- (15 X 32). 13. Number of square feet originally 640 X 440. For build- ing purposes, there will be four pieces, each measuring 300 ft. by 200 ft. 14. The difference between the above areas will represent the number of square feet in the streets. 872. Many of these exercises can be used for mental and sight work. For methods of solution, see Art. 870. 873. As a preliminary to the formal study of interest, the teacher will need to see that her pupils understand what is meant by the term. She can explain that a person borrowing money should pay for its use, just as a person who rents a house, etc. 874. In changing 4 mo. 10 da. to the fraction of a year, many teachers prefer to reduce the time to days and to write the result over 360, -|~f$, leaving the reduction to lowest terms for the subsequent cancellation. In the same way, 1 yr. 5 mo. 15 da. is changed to (360 + 150 + 15) da., or 525 da. = ff- yr. The reduction to days is done very rapidly. 875. 1. ITSOXyfoXf. 5. $360x^X3^. 12. $ 84.75 Xyfox if 6. $ 94.43 3. $ 308.25 XyfoX^V ? $400x 4. $464.75x^x111. etc, etc. 102 MANUAL FOR TEACHERS 877. The teacher should explain that a person that owes money, frequently gives a note as an acknowledgment of the debt, etc. 878. There is no general method applicable to these problems. 1. Interest for a year is $12, or $1 per month, which gives $19 for 1 yr. 7 mo. 2. $3.60 per year is 1^ per day, 33^ for 33 da. 3. $ 6 per year, or $ 9 for 1-|- yr. 4. $ 6 per year, or $ 15 in 2 yr. 6 mo. 5. If $ 50 produces $6 in 2 yr., it will produce $3 in 1 yr. ; rate, therefore, is 6%. 6. $ 18 per year is $1 for 20 da., or ^ yr. 8. 4% per year == 1% for 90 da. ; 1% of $ 150 - $ 1.50. 9. 5% per year - \% for 36 da. ; |% of $240 = \ of $2.40. 11. $1 is 100% of $1; at 5% per year it will take 20 yr. to make 100%. 12. At 6% it will takelGf yr., or 16 yr. 8 mo., to make 100%. 13. Disregarding $14.90, it will take 25 yr. at 4% to make 100%. 14. -i-% per month = 8% for 16 mo. ; 8% of $ 90 - $ 7.20. 15. 5% for 360 da. = 1% for 72 da. 16. 360 da. -*- 4 = 720 da. -+ 9 = 80 da. 17. 5% for 1 yr. = 1% for 72 da. ; 1% of $75 = 75 cents. 18. l%of$63. 20. 1% of $840. 22. 1% of $275. 19. 1% of $570. 21. 1% of $150. 23. 2% of $360. 879. 1. 30 rd. 5 yd. 1 ft. = 511 ft. ; 8 rd. 4 yd. 2 ft. = 146 = Ana. 2. Number of feet deep = (36 X 5) -5- (6 X 4). NOTES ON CHAPTER ELEVEN 103 3. 3 mi. 96 rd. = 1056 rd. ; 3 hr. 16 min. - 3 T \ hr. ; 1056 rd. x 3^ = 1056 rd. X f f = ^-^^ rd. - 3449f rd. - 10 mi. 249f rd. Ans. 4- (f + i) X (f X ft) + (f X ^) = V X f X ^ X t X / = 2TT. .4ns. 8. The first two figures express 1800 ; the second two, 5.4. 22. $48.37 -^-8f 880. 2. Provisions that will supply 450 men for 5 months will supply 5 times 450 men for 1 month, and will supply (5 times 450 men) -f- 9 for 9 months, or 250 men. The number that must be discharged = 450 men 250 men 200 men. Ans. 15 16. D bought X f X | X f of the ship. 18. 100 a; + 50 a;- 340 X 75. 19. x = number distributed by each new man ; 2x = number distributed by each experienced man. 16# + 32# = 36000. 20. *--= 1972.65. 4 881. 10. 100 cents -^1.13. 883. 3. $1.10 + 15% of f 1.10. 4. 9876 = 87.T + 45. 5. 640 is what per cent of (640 + 560), etc. 6. 43 gal. 3 qt. 1 pt. = 43| gal. ; $70.20 -4- 43f = Ans. 7. (48 x 32) -*- (16 x |). 8. 20 is what per cent of 160? 20 is what per cent of 180? . 3450 23 Ala/ 25 1 . . x 23 Let x cost per bbl. x -- = 104 MANUAL FOR TEACHERS 11. 100 884. 425+99 is 1 less than 425 + 100; 425 + 999 is 1 less than 425 + 1000. 885. 565 - 99 is 1 more than 565 - 100 ; 1424 - 999 is 1 more than 1424 - 1000. 886. 24 x 21 = (24 x 20) + (24 x 1). See Art. 786. 887. 16 -*- .25 = 16 -*- i = 16 x 4 ; 36 -*- .75 - 36 -*- f - 36 X|-12x4. 888. 7i-^J=V--i = - 4 -i = 30 ^- 3 - When the dividend is a whole number, it is frequently better to perform the division in the ordinary way : 63 -*- 3 = 63 -*- f = 63 X \ = 9 X 2. 889. 1. $|X48. 2. (48xf)sq. yd. 3. (48 -4- f) yd. 7. 9 into 83, 9 times and 2, or |, remaining ; 9 into 9 quarters, . Ans. 9J. 10. $lx!9. 11. $lx!20. 12. (120 -s- If) yd. = (120 X ^) yd. = (8 X 8) yd. 14. Dimensions of field = 80 rods X 80 rods. 16. 95^- 4f = 95 -s-J^- = 95 X -^- = 5 X 4. 17. 4000^2. 19. 3 T. will cost $15; 480 lb.@i^ per Ib. will cost $1.20 ; total, $ 16.20. 20. For $10, I can buy 2 tons ; for 80^, I can buy (80 X 4) Ib, or 320 Ib. 23. 347 + 495 = (347 + 500) 5. 25. One man can do -^ in 1 day; the other can do ^g- in 1 day ; both can do -^ + ^-g- in 1 day, or -fa + -fa ^ 3 g- = -^Q in one day, thus requiring 16 days to do the whole work. 890. 1. $150, at 4%, for 3 years. 2. 12 cu. ft., at 60 Ib. to cu. ft. 3. $12 is 3% of what? 4. 250 is what per cent of 500? 5. 12 is what per cent of 4? 6. 20M@$30perM. NOTES ON CHAPTER ELEVEN 105 7. 4 bbl, 300 Ib. each, @ 5^ 9. 84 -f- 4. 8. 18-J-4J. 10. 75 @$ 79 (or $80). 892. 4. Find -^ and annex cipher. 13. See Art. 791. 893. See Arts. 792, 716, 717, 714. 26. Multiply by 36, by subtracting 4 times the number from 40 times the number; multiply by 45, by subtracting 5 times the multiplicand from 50 times the multiplicand. 895. See Art. 563, p. 55. 897. 2. 18*+ 15^ + lSx + 15* + (18x15) -930. 6. Floor space = (30 X 24) sq. ft. Air space (30 X 24 X 15) cu. ft. 7. (30x24)^- ff 8. Reducing to yards : [(10 X 5) + (8 X 5) + (10 X 5) + 8 x 5) + (10 x 8)] -*- 3. 10. Commencing at lower right-hand corner: (15 + 3+12 f 9 + 8 + 18 + 10 + 15 + 9 + 15) rd. 12. [(22 x 12) x (14 x 12) x (9 x 12)] + 2150.4. 15. Dimensions : 1000 yd., 2 yd., 3 yd. 16. Dimensions of pipe space : 1000 yd., 1-J- yd., 1% yd. 900. It may be necessary for the teacher to supplement the n formation given the pupils in connection with the demand lotes in Art. 877. The present note is payable at a fixed time, md the place of payment is specified ; but it does not bear nterest. If, however, it is not paid at maturity it bears interest Tom that time at the legal rate. While savings banks loan money only on good security, gener- ally real estate, banks of deposit will advance money on a note, if the officers feel certain that it will be paid at maturity. When William Brown & Sons present the note for discount, they endorse 106 MANUAL FOR TEACHERS it by writing their name on its back. This transfers the owner- ship of the note to any subsequent holder, and also makes the endorsers liable for the amount in case the maker fails to pay it at maturity. The discounting bank thus has two parties upon whom to depend for the money. The sum charged by the bank for this service is the interest on the face of the note for the time it has to run. This sum is called the discount, as it is deducted from the sum named in the note; and the difference called the avails or proceeds is given to the owner of the note. When the above note is due, it is sent to the Park National Bank for collection. If Thomas Tierney, or some other person, does not pay the money before the close of banking hours, the note is protested ; that is, a notary public certifies that payment has not been made, and notifies the endorsers, William Brown & Sons, of their liability. 901. In states in which days of grace are no longer allowed, the pupils should not employ them even in calculating discount on notes made in places that still have days of grace. Two answers are given to each problem in discount, one including days of grace ; the other, enclosed in parentheses, in which days of grace are not employed. 902. These exercises are nothing more than examples in interest, except that in some states, three days are to be added to the time mentioned. Deducting the discount from the face of the note gives the proceeds. 903. As will be seen in Art. 906, the exact number of days is taken for periods less than a year. 904. Pupils should be led to see that banks are entitled to interest only for the number of days they have to wait for repay- j ment. A failure to understand this, leads to frequent mistakes. Many careless scholars find the difference in time between the NOTES ON CHAPTER ELEVEN 107 two dates named in the example, from Feb. 27 to March 9, in 25, disregarding entirely the time for which it is drawn. Instead of explaining how to calculate the discount on the notes given in Art. 905, the teacher should permit the class to attempt to ascertain the result by themselves. In case of a failure to obtain the correct answer, a discussion of the matter will lead to a proper understanding of the principles involved. 906. 1900 is not a leap year. See Arithmetic, Art. 1303, Time Measure. 907. Find the total number of meters in the first twelve pieces, and ascertain their value at the price named. Do the same for the remaining four pieces. Reduce the total number of meters in the sixteen pieces to yards, by multiplying by 39,37 and dividing the product by 36. 908. See Arithmetic, Art. 758. 910. In 64i X 11-?-, the product by is found by multiplying the product of 7, already ascertained, by 4. 912. 8. A yard is 36 in. If ribbon is 36 in. long, its width must be (144 -*- 36) in. to contain 144 sq. in. 14. Some pupils will say without reflecting, 200% not see- ing that the profit is equal to the cost, 100%. 15. 1% of $1500. 20. 10 - what per cent of (40 + 10) ? 27 . The remainder = 20 % , or = $ 2000. 29. The thoughtful teacher must determine for herself just how much time she can afford to waste in giving the pupils a number of useless facts about taxes, brokerage, commissions, bonds, etc., etc. The time allotted to arithmetic should be spent chiefly in developing " power " in her scholars. If the latter can correctly apply mathematical principles in ordinary problems suited to their present experience, they will not find it difficult 108 MANUAL FOR TEACHERS in later life, after they understand the conditions, to solve such new problems as come up. In this example, it will be sufficient to say that the " premium for insuring " means " cost of insuring." 913. 1. See Art. 685. 3. The pupils should attempt to frame the definitions asked. 5. (a) Jof 832 = Ans. (J) 832-1- of a? = ; etc. 8 6.^=3750. 8. The first boy gains J^ on a 1^ apple, or 25% ; the second gains | or 20%. 9. Sold (20 X 20) sq. rd. + 16 sq. rd. 15. The pupil should be able to state the rule. 19. x + ^x = 1050. 21. ([(35 + 23 + 35 + 23) x 13] + [35 X 23]) -H 9. 914. Use first as sight problems. 3. a? + 4. If -| of A's money ^ of B's, A's money = -f- of B's X f = fB's. Let x = B's, = A's. x += 165 ; - 165; 5 5 5 dividing by 11, -=15; x = 75. B's -$75, A's -$90. 5 8. After 2 days, there will be enough to feed 4 horses 4 days, or 1 horse 16 days, or 5 horses 3-J- days. NOTES ON CHAPTER 915. 7. An ounce avoirdupois contains 7000 gr. -4- 16 ; an ounce troy contains 480 gr. 8. 4 Ib. 8 oz. avoirdupois 7000 troy grains X 4. 9. The pure silver amounts to 192.9 gr. X.9; divide by 480 to reduce to ounces, and multiply 75^ by the quotient. 751 X 192.9 X .9 480 11. Number of square feet [(16 + 14 + 16 + 14) X 8] + 16 x 14. NOTE. When the bottom of a tank is covered with sheet lead, the side strips will be -fa in. less than 8 ft. high, etc., but this small difference may be neglected in these four problems. 12. Multiply the number of square feet by -^5-, and divide the product by 12. Cancel. 21. Assessed value, f of $24000, or $18000. Taxes = If % of $18000. 917. 8. Dividend - 2|% of ($50 X 65). 918. 1. Area of surface to be papered : [(16 + 14 + 16 + 14) < 10]- 174 ; divided by area of roll, 24 ft. by 1% ft. 4. When he sells 31 gills, the grocer charges for 32 gills, or 1-^Y times the correct quantity, thereby charging for -^ more than he gives. In 2 hhd. of 58 gal. 2 qt. 1 pt. each, there are 117 gal., the dishonest gain on which is ^ T of 117J gal. worth $.80 X ^ of 117i 6. 30% of cost (x) = 21. 919. 1. (a) $ 48.50 + ($ 48.50 X ^ X ^ ) - $ 48.50 + .51 -$49.01. Am. Omitting days of grace, $48.50 + ($48.50 X y-jfo X ^) = $48.50+ .48^ - $48.98^ say $48.99. Ans. These examples should be worked with days of grace or without days of grace, but not in both ways. See Art. 901. Days of grace were not abol- ished in New York until January 1895. 110 MANUAL FOR TEACHERS (b) With days of grace, this note is due Dec. 17. Term Dec. 1 to Dec. 17 = 16 days. Discount on $49.01 for 16 days at 6% -$49.01 X yfo. X ^ = 13 Proceeds -$49.01 - .13 -$48.88 Am. Omitting days of grace, the note is due Dec. 14. Term 13 days. Discount on $48.99 for 13 days at 6% $48.99 X yf X 3W = 11 Proceeds - $48.99 - .11 - $48.88. Ans. 2. With days of grace, the amount due at maturity will be $175 + ($175x T f X-^ 3 1T )-$175 + 2.71 + -$177.71+. The term of discount 93 days 33 days = 60 days. Interest of $177.71 for 60 days at 6% will be $1.78 nearly. Proceeds - $177.71 - 1.78 = $175.93. More accurately, $177.7125 - 1.7771 - $175.9354 or $175.94 -. Without days of grace, the amount due at maturity will be $175 + ($ 175 X TJhr X inrW = $ 177.63 - . Interest of this amount for 57 days $1.69 . Proceeds $177.63 - 1.69 -$175.94. 920. See Arithmetic, Arts. 821, 822. 921. 9. 4) 183 14g. 8d. 13. Total number of days' work 32+ 53 + 41 126. Value of 1 day's work $283.50^126. Share of first man ($283.50 -* 126) X 32. Cancel. 14. Amount furnished, $ 12000. The one furnishing $3000, or i, is entitled to \ of $ 1800 ; the second to ^ of $ 1800, etc. 15. After 10 days, there are rations for 1200 men for 30 days ; which will last 1 man 30 days X 1200 ; and will last 1200 men + 300 men, (30 days X 1200) -*- 1500 Ans. 16. Train leaving B goes 1-|- times as fast as the other, so that meeting place will be 1^ times as far from B as it is from A. If x represents distance from A, \\x will represent distance from B, and x + \\x 120, or x 48. Trains meet 48 mi. from A, or 72 mi. from B. The first train takes |~| hr. to travel the distance, or 2 hr. 24 min. ; second train requires the same time, NOTES ON CHAPTER ELEVEN 111 -|| hr., or 2 hr. 24 min. Time of meeting = 9 A.M. + 2 hr. 24 min. = 24 min. past 11. Or, let x = time required to reach meeting point ; then 20 x = distance travelled by one train, and 30 x = distance trav- elled by the other, and 20 x + 30 x = whole distance = 120, or a = 2f. Time is 2| hr., etc. 17. x + 3 x + 6 x + 10 # = 900. 18. Let x = share of third ; x + 75 = share of second ; and x + 75 + 48 = share of first. x + x + 75 + x + 75 + 48 = 540. 19. If 4 men need 105 hr., one man would need 420 hr., and 6 men would need 70 hr., or (70 *- 10) da. 20. Let x number of dozen bought, 15 x cost in cents ; x 1^ = number of dozen sold, 16 times (x 1^) = 16 x 20 = selling price ; 16 x - 20 = 15 x, or x = 20. He bought 20 dozen, or 240 eggs. 21.. The interest on $250 for 8 mo. is the same as that on $ 1 for 8 mo. X 250, and on $400 for (8 mo. X 250) -*- 400, or 5 mo. 22. Provisions for 3000 men would last 1^- times as long as for 4000 men, or 18 wk. X 1-J- = 24 wk. Ans. Or, (18 wk. X 4000) -*- 3000. 24. Area of first plank in square feet, 20 X 1 ; of second, 24 x x. 24 x = 20. x = f Ans. in feet, or (f X 12) in. 25. He can pay -f$ff of his debts. Mr. Smith should receive $ 576 x $$$. Cancel. 922. 6. See table, Arithmetic, Art. 795. 923. 10. 23 x 11 X x = 2749. 13. 48 X 72 x x = 2150.4 X 40. _ 2150.4 x 40 _ 48x72 22. 55 cts. X 6 X 54 X . 112 MANUAL FOR TEACHERS 924. 1. The pupils should gradually become accustomed to .business methods of obtaining results. In calculating the amount to be paid, a clerk writes the discount at T^o^l OQ "I fj once under the gross price. He takes -^ by dividing ' by 2 and placing the first quotient figure one place to $554.23 the left. 2. In the first example, a discount of 5% is made for prompt payment ; the discount here allowed is a reduction from what is called the " list " price. Catalogues are issued by some merchants on which the prices named are not the ones regularly charged, but are much larger so as to mislead persons that do not know the rate of discount allowed. Information as to this rate is com- municated to customers, and varies from time to time owing to fluctuations in the market, the " list " price seldom being changed. The list price is sometimes called the "gross" price, the "net" price being the one actually paid. A bill for the Roman candles would be made out as follows : 16 gross Roman Candles, $26.75, $428. - Less 60%, 256.80 Net, $171.20 The product by .60 is written under the "gross" total, the first figure being written two places to the left. The net cost can be directly obtained by multiplying $428 by .40. 7. Two, three, and even more discounts are very frequent in business. An article catalogued at $100 is sold, for instance, at $70, and customers informed that the discount is 30%. A later reduction in price is accompanied by a notice that a fur- ther discount of 10% will be allowed. This does not signify 30% + 10%, or 40%, from the " list " price ; it means that the regular price of $ 70 is to be reduced $ 7, making the new price $63. A third discount of 5% means 5% of the last price, $63; etc., etc. In writing these discounts, the per cent mark is written only after the last. NOTES ON CHAPTER ELEVEN 113 11. On a bill of $100, 40 and 10% gives a " net " amount of $60-$6, or $54; 30 and 20% gives $70 -$14, or $56; the former being better for the buyer by $2. 12. $100 less 331 and 10% -$60. The discount is $40 on $100, or 40%. The net is 60%. 13. 100-40 = 60; 20% of 60 = 12; discount = 40% + 12% = 52%. Ans. 14. Let # = "list" price. After first discount of -|- is de- 2>x ducted, there will remain - Deducting 10% of this, or -fa of ^5 it, there will remain ^5- of it. & of =' = 60. 3 15 15. The first reduction is 100% -20%, -or 80% of the list; the second is 100% - 10%, or 90% of the former. 90% of 80% = T 9 o of 80% = 72%. Ans. 16. 80% of 90% = 72%. Ans. The net price is the same for the same discounts in whatever order they are taken. 925. 3. ~- of 5000 (cents) = 5 (cents) ; 50 x= 5; etc. 11. Value at par, $50 X 96 = $4800. Discount = $4800 -$4476 = $324 = *% of $4800, i.e., '324 = 48 a: ; etc. 15. 926. 3. 50% + [10% of (100% - 50%)] = 50% + 5% = 55%. Ans. 4. 30% + [30% of (100% - 30%)] = 30% +(30% of 70%) = 30% + 21% = 51%. Ans. 5. -& of gross price (x) = 729 ; -^x = 81 ; x = 810. acres \ '. 21. Cost per acre = $ 40293 -*- 396, at which price 112 res were sold. 25. 40^XffXff. Cancel. 26. Number of hours = 365 -*- 114 MANUAL FOR TEACHERS 928. 2. *X T |hFX=|j=180; 9^ = 14400; etc. 4. 4250 X yjhr X x = 765. 6. 2020 X yjfo X a; = 606. 5. a;X T |oX3 = 240. 7. 6000x^x-900. 1UO 2 929. 1. The pupils should not be shown how to calculate these areas. 2. If any difficulty is experienced in finding the areas of these triangles, the pupils should be referred to 1 ; after which they should be led to deduce the rule. Thus the area of the second triangle may be calculated from the figure in 1 by adding J- of (60 X 50) to of (60 X 50) ; that of the third by adding of (60 X 60) to i of (60 X 40) ; and that of the fourth by adding i of (60 X 70) to of (60 X 30). Each of these will be found equal to -j- of (60 X 100). 3. The second rectangle is divided into three triangles, two of them right-angled. By deducting from the area of the rec- tangle the sum of the areas of the two right-angled triangles, they will obtain the area of the remaining triangle. 4. The area of each of these triangles can be ascertained by referring to the corresponding triangle of 3. Let the scholars do this for themselves. 5. The areas of the oblique-angled triangles constituting the first and second quadrilaterals, can be calculated by the pupils that have benefited by the work in 4. If they see that the area of each triangle of a parallelogram is equal to % (base X altitude), the area of the latter is equal to base X altitude. For definitions of quadrilaterals see Art. 1265. 6. The area of the first is equal to the area of the rectangle, (50 X 60), plus the area of the triangle, of (50 X 60) ; or 4500 sq. ft. The second is made up of a rectangle and of two triangles ; its area is also 4500 sq. ft. NOTES ON CHAPTER ELEVEN 115 The pupils should be led to see that if, in the fourth, the upper left triangle were cut off and placed below, and if the lower right triangle were cut off and placed above, as indicated by the dotted lines, the resultant figure would be a rectangle 60 X 75. Cutting off both triangles in the third, and placing them above, will make a 60 X 75 rectangle. The area of each trapezoid is equal to [-J- of (50 + 100)] X 60. 7 . The area of each of these quadrilaterals equals \ of (30 X 100) + i of (40 X 100), or [i of (30 + 40)] X 100. The first three quadrilaterals are trapeziums. The last is a trapezoid. Which are the parallel sides ? 8. A strip of paper of any uniform width may be used. Carefully cut a rectangle by making square corners with a card. Using the base of the rectangle as a measure, place two dots on the lower edge of the strip to mark the extremities of the base of a parallelogram equal in length to the base of the rectangle, and above these, at any convenient distance to the right or to the left, two others to mark the extremities of the opposite side of the parallelogram. Draw lines forming the right and left sides, and cut along these lines. That the parallelogram is equal in area to the rectangle, may be shown by carefully drawing a perpen- dicular at one corner; cutting off the triangle thus made, and placing it, in a reversed position, on the opposite side of the parallelogram. 9. See 6, third and fourth trapezoid. 930. 2. Four faces will measure 6 ft. by 4^- ft. each, and two will measure 4^ ft. by 4|- ft. each. 4. Dimensions of floor, 57 ft. by 18 ft., or 19 yd. by 6 yd. 5. Volume in cubic feet, 4 X 4 X 12. Multiply by 1000 to get the weight in ounces of an equal volume of water. Multiply by 2.8 to get weight of marble in ounces. Divide by 16 X 2000 to reduce to tons. 4x4 x~12 X 1000 X 2.8 16 X 2000 116 MANUAL FOR TEACHERS 9. Outer dimensions, 14 X 14 X 14, or 2744 cu. in. See if the same number of cubic inches of wood is obtained by calculating the volume of the wood in 6 2 pieces, 12 X 12, 1 in. thick; 2 pieces, 12 X 14, 1 in. thick ; 2 pieces, 14 X 14, 1 in. thick. 10. A cube of water 2 ft. long contains (2x2x2) cu. ft., or 8 cu. ft, At 1000 oz. to a cubic foot, it weighs if^ Ib. X 8, or 500 Ib. The cube of' iron weighs 8 times as much as an equal volume of water. A cube of iron 1 ft. long also weighs 8 times as much as a corresponding cube of water, or 8 times J-f- Ib. = 500 Ib., or \ ton. A 3 ft. cube of iron contains 27 cu. ft., weighing 8 times as much as a corresponding cube of water, or 216 times 1000 oz. = 6f tons. XV NOTES ON CHAPTER TWELVE 931. While problems requiring the pupil to find the principal, the rate, or the time have very little " practical " value, they can be so readily taught by the algebraic method that the time spent upon them need not be very great. A pupil that is able to calculate one of a series of related items is benefited by being required to calculate the others, if he is not compelled to resort to a series of ill-understood rules in order to obtain the results. Although there is no real difference between the algebraic method and the arithmetical one, a great number of scholars fail to obtain a thorough understanding of the latter. They can work a number of examples, following a model solution at the head of the lesson ; but they fail to grasp the underlying prin- ciples. By the algebraic method, x is used to represent the number of years or dollars, or the rate, instead of the 1 year, $ 1, or 1 / , of the other ; but this method seems to require the formu- lation of a number of rules, as against practically none in the case of the other. After pupils have learned to work examples by the algebraic method, they can be encouraged to discontinue the use of the x ; but they should not be taught both methods at one time. 933. Represent the required item by x. Simplify the first member before proceeding to solve the equation. 1. 2000 x X 3 = 300. 2. 1800x^X^ 144. 117 118 MANUAL FOR TEACHERS 3. 4. 38 + 38 x x x 2 = 40.28. J.UU Or, 38 X X 2 = 40.28 - 38 = 2.28. 5. 140 X 100x2 7 . 15. Principal - f 97.57 - $ 7.57 = $ 90. 90 X TW x * = 7 - 57 - 21. Let a; = principal. i # X 4 x 846 _ lg6 92 100 x 360 The interest is then found by subtracting from $196.92, the value obtained for x. 22. First find the principal (x). 25. See 15. 934. The recommendation so frequently made, that all writ- ten work be preceded by oral exercises of the same character, should not be followed without some modifications. Oral work is necessarily accompanied by a number of devices that tend to simplify the task of handling numbers that are not seen ; written work should follow general rules in order to be learned by a majority of the pupils, although later they may adopt some of the short-cuts of their oral exercises. Even the oral addition of two numbers of two figures each, is done in a manner different from the ordinary slate method, the operation in the former case being commenced generally with the tens' figures, and in the NOTES ON CHAPTER TWELVE 119 latter case with the units' figures. The reduction to a common denominator recommended in oral division of fractions, is seldom employed in slate work. The average scholar is able to handle " mental " problems con- taining small numbers in a way that he cannot always explain, although he may endeavor to stultify himself by repeating a prescribed form of analysis. It is next to impossible, with the average teaching, to get the same pupil to work some varieties of " written " problems containing the same conditions. In order to furnish a general method of treating some classes of examples, it has been thought best to commence with written work, leaving the mental exercises with their various devices until the former task is accomplished. The accompar ving exercises are so simple as not to need ex- planation by the teacher ; but sufficient time should be given the pupil to work them out in his own way. They differ in this respect from the oral examples of a single operation containing larger figures, but which do not require any effort on the part of the scholar to determine which process is required. 1. Yearly interest is $6; a year and a half will be needed to make the interest $ 9. 2. The yearly interest is $ 8, making the rate 4%. 3. Yearly interest is $4, requiring a principal of $100, at the given rate. 5. The pupils may remember (Art. 878, No. 15) that 5% for a year is 1% for 72 da. 6. 4% per year is 1% for 90 da. 11. 2 mo. 12 da. = 72 da. See 5. 12. 1% for 80 da. is (360 H- 80) % for a year. 17. 2% for 6 mo. 18. $ 3.60 per year is 1 cent per day. 20. See 18. 120 MANUAL FOR TEACHERS x 2 x 935. First payment = -, leaving remainder ; second pay- 3 o ment % of = ^, leaving - remainder ; third payment f of 33 3 = - ; last payment, = $ 2000. The total cost of the house, x = 3 5 the sum of the payments, - + + - + 2000. 335 936. The books contain many methods of calculating interest, but it is questionable whether it is not time wasted in giving so much attention to this topic. The average person is required to do comparatively little work in this line ; while those called upon to compute interest often, learn short methods of their own or use interest tables. If a second method is to be taught at all, the one by aliquot- parts is the most useful, as modifications of this method may be applied to other operations. 6. See Arithmetic, Art. 384. 937. A modification of the so-called " 60-day method.'* 16. See Art. 901 as to days of grace. 938. 21. 10% gives 2 years' interest; then 1 yr. (-J- of the foregoing) ; 6 mo. ; 1 mo.; 18 da. (y 1 ^ of 6 mo.). 942. 46. Term, 57 da. (54 da.). 47. Term, 92 da. (89 da.). 49. Term, 34 da. (31 da.). 48. Term, 16 da. (13 da.). 50. Term, 187 da. (184 da.). 943. 9. See Table, Arithmetic, Art. 1303. 944. 11. The net price of goods catalogued at x dollars, and "\ sold at a discount of 20 and 10%, will be (x-, or - 100 100 100 100 ' NOTES ON CHAPTER TWELVE 121 13. If the selling price of the above is $360, =360; 100 = 36000; x = 500. Catalogue price = $ 500. Aw. 14. 750 -(i of 750) -500; 500 - (^- of 500 ) = 500 - 5 a; = net price. 500 -5 a? = 450. Transposing, 5x = 50. Changing signs of both terms, 5# 50, x -10. 945. 7. Let x selling price of muslin. (84 x 40) + 105^ = (84 x 55) + (105 x 20). Another way : He loses 15^ per yard on 84 yd., which is a loss of 15^ X 84. This he must make up on 105 yd., which is (15 ^ X 84) -* 105 on each yard, or 12^. Selling price of muslin, 20^+12^, or 32^. Ans. 8. J of them brought $120; -J- of remainder, or J of them, brought $96; -J- of remainder, or of them, brought $40; re- mainder, or i of them, brought $30. Total amount received, $286. 9. Proceeds of gas stock, $25 X 165 $4125. Cost of lots, $4125 - $27 - $4098. Number of square feet in lots, (32 X 115) + (30 X 105) 3680 + 3150 = 6830. Value per square foot, $ 4098 H- 6830 = $ 0.60. Ans. 10. Two walls, each 16 X 14, and two others, each 12 X 14, contain (32 + 24) X 14, or (56 X 14) sq. ft. = 784 sq. ft. The ceiling contains (16 X 12) sq. ft. = 192 sq. ft. Adding this to the walls, makes a total of 976 sq. ft. The deductions are (8 X 4) sq. ft. X 2, and (7 X 3) sq. ft. X 3, or 64 sq. ft. + 63 sq. ft. 127 sq. ft. Number of square feet to be plastered = 976 127 = 849. Cost at -^ per square foot = 2^x849 -$16.98. Ans. 122 MANUAL FOR TEACHERS 946. 1. A can do ^ of the work in 1 hr., and B can do $ f it in 1 hr. ; together they can do in 1 hr. (-|- -f- y) of the work, or -|~| of it ; and to do the whole work it will take as many hours as !" is contained times in 1. l-^lx-- 2 ' Ans ' 2. Commission of 2|-% -fa of amount collected $1.60. Amount collected = $ 1.60 X 40 = $64. Amount remitted $64 -$1.60 = $62.40. Ans. 3. i% of (f of $ 12000) = i% of $9000 -i of $90 = $22.50. Ans. NOTE. It may be advisable to explain to the pupils that property is seldom insured for its full value, because it is not likely that a fire will completely destroy a building, and insurance companies reimburse the per- son insured, only to the extent of his loss. 4. 32x# = 6x4; 32^ = 24; a? = |f = f. Ans. f yd. or 27 in. 5. 5% for 360 days = 1% for 72 days = 2% for 144 days. 2% of $&T = Ans. , 6. 2% of $176. 7. Let x commission ; 40 x = amount invested ; x + 40 x = 41 x = 8200 ; x = 200. Ans. $ 200. 8. $ 500 is -J of cost, $4000. 9. Let x = loss, or 20% of cost; 5# cost; 5# x = 4x = selling price. x, the loss, is J of selling price, 4 x. 10. Let x = gain, which is 20%, or -|-, of the cost of the goods ; 5 x cost ; 5 x + x, or 6#, selling price. x, the gain, is ^ of selling price, 6#. NOTE. The amount of money given in these two examples, $1200, does not affect either result. It may be used or not, as the pupil prefers. 11. 3 men earn $72^-8 in one day, or $3 per day each. 5 men earn $15 a day, or $165 in 11 days. 12. 3 qua NOTES ON CHAPTER TWELVE 123 .2. 3 quarters of the cost, or 5 - 225. Cost - $300. By selling for $325, there is a gain of $25, or ^ of the cost. ^ Ans. 13. 2| yd., or f yd. cost 40^; 1 yd. costs 40^-s-f, or | of 4 yd. 1 ft., or 4 yd., cost 15^ X 4J. 947. The following is the solution without days of grace : Let x = face of the note. Then, , X JL X -L=J|L = discount; x - = proceeds = 1000. 200 #- a; = 200000, 199 x -200000, x = Face of note $ 1005.03. Ans. Proof. Face of note, $ 1005.03 - Deduct 30 days' discount, \%, 5.03 Proceeds, $1000.00 949. 1. When days of grace are omitted, the term of dis- count is 90 da. 1.0. Find the term, and add the number of days to March 15. 950. 2. (a) 1 trillion, 500 billions, etc. 5. The first quarter of 1888 contained (31 + 29 + 31) da., or 91 da. The man was employed 60 da., and unemployed 31 da. His $3 additional paid the expenses of the working days. Deducting $2 X 31, or $62, for the expenses of the other days, his nst income = $ 350 - $ 62 = $ 288. Ans. 124 MANUAL FOE TEACHERS 6. Apr. 16, 79 to Mch. 19, '86, 83 T V mo., @ $8, $664.80 Mch. 19, '86 to Mch. 4, '87, 11^ 12, 138.00 Apr. 16, 79 to Sept. 1, '80, 16 " " 2, 33.00 Apr. 16,- 79 to Nov. 22, '82, 43 " " 2, 86.40 Am. $922.20 10. Last quarter's salary = $287 82% of previous quarter's salary - - 287 ; x = 350. Salary of three quarters @ $350 $1050 ; add last quarter's, $287. Total for year, $1337. Ans. 951. See Art. 784. 953. To find 19 times 91, subtract 91 from 20 times 91, or 1820 - 91. 82 X 19 - (82 X 20) - (82 X 1). 51 X 29 - (51 x 30) - 51. 27 x 99 - 2700 - 27. 954. See Art. 706. 675 -*- 374 = 64 -s- 4 = 54 -*- 3. A 99 355. 136 x J - 136 - (-J. of 136) = 136 - 17. 290 X ^..= 290-29. 64^ 5 = : 22 x!9* - (22 X 20) - (22 x |). 45 x 9|f = (45 X 10) - to of 45). 160 -*- 1| -320^3. ISf-H- 1| = 94-*-9 = 10f. 956. 3. At 50^ each, the cost would be $8; at 1^ apiece less, the cost is $8.00 $.16 = Ans. 4. (| of 100 Ib.) x 27 - 100 Ib. x (f of 27) = 100 Ib. X - = 100 Ib. x 20J. NOTE. The 100 should not be used until the end; even then, 20J i changed to 2025 without thinking of multiplication, J being considered 25 and annexed to 20. See Art. 649. NOTES ON CHAPTER TWELVE 125 6. 900 -s- 75 = (900 4- 100) -*- (75 -+ 100) = 9 + f . 7. See Art. 955. 9. (10! x 4)+! of 101 = 42 12. of (33 x 42) = 33 x 21. 15. $16i-r$li = 65-^5. 20. 161 x 2J = (161 X 2) + (i of 957. 1. T V(rof(27^x56x37ixf). 2. [($4.875 x 17350) -*- 196] + [$4.9375 x 122.75] + [$.0825 X 2240 x 2J]. 3 - # *fo # = 49739.55f . 12. Duty - [^V of (55^ x 45 x 38)] + [20^ x (45 X 38 X ff)]. 958. In multiplying by 427, the first figure of trie product by 42 (7 X 6) is placed under the 2 ; in multiplying by 832, the first figure of the product by 8 is placed under the 8, and the first figure of the product by 32 (8 X 4) is placed under the 2. 959. These exercises contain some examples worked by short methods explained in previous chapters. See Arts. 650, 714, 791, 792, and 891. 960. See Arithmetic, Art. 384. 964. 2. It won 17 games out of 30. 3. 1600%. 4. f is what per cent of /-? % is what per cent of ? T \ is what per cent of T % ? 5 is what per cent of 6 ? = 83%%. 6. The deduction of the first discount leaves 80% of the list price ; the deduction from this of 10% of itself leaves 90% of 80%, or 72%. 7. One fills % of tank in 1 hr., the other fills % in 1 hr. ; both together fill % + % in 1 hr., or ^ + -&, or ^ ; to fill f of tank, it will take 24 hr. -*- 7 = 3^- hr. 126 MANUAL FOR TEACHERS 8. 6% for 60 da. = 800; for 12 da., | of 800 or 160; for 72 da., 800+160 = 960. Or, $4.80 for year, and of $4.80 for 72 da. 9. 16%= A; 420 = ; etc - 965. 5. Selling price = | of $1.50- $1.12-1-; gain = 2210 = J of 900. 6. Selling price $9.60, a reduction of $2.40 from marked price, or of $12, or 20%. 7. The rug is sold for $24. If this is |- of marked price, the latter is $30. 8. See 7. NOTE. It is not to be expected that all the pupils' work will be short- ened to this extent, but the majority of the class should be able to give answers at sight to these four examples. 9. Find y 1 ^ of 83 2s. 6c?. by compound division ; do not reduce to pence. 21 x 966. 4. Let x profits first year ; then - = profits second 91 $. 18. Let x '= cost of the horse. x + i) = asking price = ~2cf He sold it at ffi of asking price ; i.e., of -^, or ~ 400 20. Length of field in rods === (1600 + 146) -s- 18. 21. Let 10 x represent cost, then loss #; and selling price 9 x = $117; a? = 13; cost $130. A gain of 10% would be $13, making the price at which he should have sold it to gain 10% -$130 + $13 -$143. Ans. 22. Wife receives ^; son, | of f, or ; daughter, $5000. 3 339 3 9 977. The ten problems of this section will call for no special treatment. An occasional problem of this kind has already been given, although, perhaps, with smaller numbers. After the pupils understand in 1, that the joint capital, $700, is the basis upon which the profits are distributed, they will have no diffi- culty in understanding that B is entitled to f-^- of $182, and that C is entitled to f$$ of $182. Cancellation should be em- ployed. See Art. 1121, No. 5. There is no need in this connection of discussing the subject of business partnerships that are continued for a year or longer. The division of profits in these cases is the subject of a special NOTES ON CHAPTER TWELVE 133 agreement, and is rarely made solely on the basis of the amounts invested. A yearly gain of $4000 made in regular business by two partners, one of whom invested $1000, and the other $3000, might be divided in various equitable ways. The partner invest- ing the larger sum, might first take out $ 120 as interest on his excess of capital ; and the remaining $ 3880 might be equally divided, giving one of them $1940, and the other $2060. Another arrangement might permit each partner to withdraw a fixed sum for services, say $1000, leaving $2000 to be divided on the basis of 1 to 3. This would make the shares ($1000 + $500) and ($1000 + $1500), or- $1500 and $2500. 10. Cases of this kind are found only in the books. 979. The cost of the first item is given. The sebond item contains 451 sq. ft. ; the third contains (4 X 42 X ?--) sq. ft., or 1204 sq. ft. ; the fourth contains (3 X 43 X 7f) sq. ft., or 989 sq. ft. ; the total of the three items being 2644 sq. ft. The cost @ IQd. is 110- 3-4 First item, 24- 7-8 134-11 Less 2|% (^), 3- 7-3J + ' Ans. 131- 3-8f The fraction of a penny in the discount is ^5-, which is nearly 1 farthing, written \d. 980. German currency being a decimal one, the bill is com- puted in the ordinary way. The duty is found by reducing the marks to dollars by multiplying by $.238, and taking 35% of the result. 981. Too much stress cannot be laid upon the importance of requiring children to estimate the probable answer to every " written " problem before placing a figure on paper. The mere drill on the " approximations " found in the text-book is of com- 134 MANUAL FOR TEACHERS paratively little value, unless it leads pupils to the employment of this device throughout all of their work. Besides preventing a pupil from making a very serious mistake in an ordinary com- putation, the habit of careful reading that is necessarily formed by the scholar that is not satisfied with a simple guess, will tend to make his methods simpler and more accurate. He will learn to apply to the apparently more difficult numbers of the written problem the processes employed in solving the compara- tively simple " mental " questions. While all of the pupils should not be expected to give the same answer to each of these * examples, they should gradually approach more and more closely to the correct result. 1. 480 is what per cent of 960 ? 2. 52 bu. is about 3 times (17 bu. 37 lb.). 3. 500 cu. ft. *. 128 cu. ft. Less than 4 cords. 4. 120 cu. ft. *- about 1 J- cu. ft. 5. 1500 sq. rd. -*- 160 sq. rd. Less than 10 acres. 6. A little less than 70 X 70. 7. (6 X 4 X 5) X about 7-J-. 8. 64 ^- T V, or 64.3 -^ T V More than 643. 9. About 200 @ $ 4.80 to . 10. About 4 marks to $1. Over 400 marks. 982. 'The teacher that is allowed any discretion should omit all problems relating to Bonds and Stocks. The average gram- mar-school pupil cannot be made to understand the subject without the expenditure of more time and energy than should be given to a topic that he will learn by himself -when he grows older if he gets the proper foundation. If, however, the course of study requires that this topic be taken up, the teacher should aim to interest the pupils by making some local corporation the basis of the work. A certificate of stock should be obtained, or at least a copy made, which might be placed upon the blackboard. NOTES ON CHAPTER TWELVE 135 The scholars should be led to see that large undertakings, such as the construction of a railroad, the building of water-works, and the like, require more money than any individual might have, or would care to risk. It then becomes necessary to interest a number of persons that will be willing to invest more or less money in the new enterprise. It is found, for instance, that $ 50000 will be needed to build and equip the street rail- road mentioned in 1. The projectors divide this amount into 500 shares, each of which represents a one-five-hundredth interest in the profits. It may happen sometimes that it is considered advisable to interest people of small means, and who are unwill- ing to take a $100 share. In these 'cases the original (par) value of the shares may be fixed as low as $10 each. When the par value is not given, it is understood to be $100. In the distribution of profits, the owner of 10 shares is entitled of $2000, or $40. 2. These profits are generally distributed annually, semi- annually, or quarterly. Before the time comes for " declaring " the dividend, the directors of the company meet and determine how much money shall be thus distributed. It may be con- sidered advantageous to reserve a portion of the profits for the purchase of new cars, or for the extension of the road, etc. ; so that the amount distributed at any time does not necessarily include all that has been gained. The dividend is generally announced as a per cent of the capital, which in this case is $50000; so that the semi-annual dividend is 4%, equal to 8% per year. 3. The owner of a $100 share, on which he receives $8 per year, is not likely to be willing to sell it for $ 100 if he can obtain only $4 per year interest on that sum of money deposited in a savings bank. The person desirous of obtaining stock after it is reasonably certain that the railroad is going to prove suc- cessful, will have to pay more than $100 per share. Mr. H. pays $150 per share, or 150% of the par value. 136 MANUAL FOE TEACHERS 4. Mr. H. receives 4% dividend on $3000, or $120. From the savings bank he would obtain 2% on $4500, or $90. 5. The $120 semi-annual dividend is 2|% on the $4500 invested, equivalent to 5\% per year. 6. The words " per cent " are not used in stating the price of stock. A $100 share at 164% is worth $164^; 30 shares are worth $1641x30. 7. Assuming the par value of each to be $ 100, the first pays $6 per year on $150, or 4%; the second pays $7 per year on $175, or 4%. The par value of $100 is assumed for convenience; a par value of $50 would make the cost of a share of gas stock 150% of $50, or $ 75, and its annual dividend would be 6% of $50, or $3, the rate on the amount invested being 4%. 8. The buyer of stock at 125 wishes to receive 4% of $125, or $5. As the dividend is based on the par value ($100), the rate must be 5% per year, or 2%% semi-annually. 9. [93|% of ($50 x 17)] + [102f% of ($10 X 143)]. 10. A person that desires to buy stocks is not always likely to know where he can find any for sale; so he goes to a stock- broker, who makes a business of buying and selling stocks on commission. This commission is a small per cent of the par value, the charge of \% for buying or selling a share of the par value of $100 being 12-^, whether the actual value of the stock be $150 or $50. This broker receives, therefore, \/ of [($50x17) + ($10x143)]. 11. A bond is a note issued by a corporation, and is generally secured by a mortgage on its property. It is a much larger document than the note of an individual, and frequently contains at the bottom a number of " coupons," one for each half-year's interest, upon which is engraved the date when due and the sum NOTES ON CHAPTER TWELVE 137 payable. A 10 years' U. S. 4% coupon bond for $1000 would have 20 coupons, each worth $20 when due. At the expiration of each 6 months, the holder of the bond cuts off the proper coupon and presents it for payment. A " registered " bond con- tains no coupons, a check for the interest being mailed to the owner. Although the railroad company in this example receives only $95 for each $100 bond, it promises to pay $4 interest per year, and $ 100 at the end of 20 years. In considering the rate of interest received by the owner of such a bond, it is not customary to complicate the example too much by requiring the pupils to take into account the additional $ 5 above the cost received when the bond is redeemed at the end of 20 years, although buyers of bonds include it in their calcula- tions. For our purpose, at present, it will be sufficient to assume that the purchaser of one of these $ 100 bonds receives $ 4 on each $95 invested, the rate being 400^-95, or 4^-%. 12. Omitting the question of redemption, at which time the purchaser for $116.50 would receive only $100, the rate is 400^116^, or 3#H&. The holder of a U. S. bond knows that the face value of the bond will be paid in full at maturity, and that the interest pay- ments will be made on the dates when due ; in the case of the bond of a railroad, or the like, there is always the possibility that something may occur to prevent the company from meeting its obligations. 13. The rate of income from stocks may vary at each dividend period, depending upon the amount of business done, etc. ; the rate of income from bonds is fixed as stated on their face. Bonds are redeemed at the time specified ; there is no reason why a successful company should sell out and divide the proceeds among its stockholders. When, however, the property of a cor- poration is sold, the claims of bondholders and all other obliga- tions must be satisfied before the stockholders receive anything. 138 MANUAL FOR TEACHERS 14. The stock-broker's fee is called brokerage, or commission. 15. A cotton-mill obtains material through a cotton " factor " ; property is purchased through a real estate agent ; a grocer may buy butter, eggs, etc., from a commission merchant, the seller in each case remitting the amount received to the owner after deducting his fee, or commission. 16. The base in insurance is the sum for which the property is insured ; in taxes, it is the assessed value of property ; in brokerage, it is the par value of stocks or bonds ; in commission, it is the sum for which goods are bought or sold ; the principal is the base upon which interest is calculated ; the face of the note is the base in bank discount ; in commercial discount, the gross price is the base in the first instance, the base for each subsequent discount being the successive remainders left after the deduction of the previous discounts ; in stocks and bonds, the base is the par value. 17. The assessed value of property is the value for purposes of taxation, and is fixed annually by officers chosen for this duty, generally called assessors. - 18. 2^% of assessed value (x) = $540; assessed value = $24000. This is f of the actual value, or 66|%. For various reasons, the assessed value is placed below the sum that would be realized by the sale of the property under favor- able conditions ; but care is taken that all property is assessed upon the same basis. 19. If all property were assessed at its actual value, the same amount of taxes would be produced by a lower rate. To obtain $540 taxes on property assessed at $36000, the rate would be 21. 1674 ft. -558 yd.; 558 yd. -s- 5-J- yd. = 1116 half-yd. -s- 11 half-yd., which gives 101 rd. and 5 half-yd. = 101 rd. 2 yd. - 101 rd. 2 yd. 1 ft. 6 in. 22. $ 8575 -*- $ 245 = 35 = number of shares. Quarterly divi- dend = 2%% of ($100 x 35). NOTES ON CHAPTER TWELVE 139 983. 27. Each figure of the product is written two places to the right of the corresponding figure of the multiplicand. Principal, $375. 3% 11.25 ' _ $386.25 Amount yr. 3% 11.5875 $397.8375 Amount 1 yr. 3% 11.9351 $409.7726 Amount 1 yr. 3% 12.2931 $422.0657 Amount 2 yr. 1% ) 4.2206 \% I 2.1103 $428.3966 Amount 2 yr. 3 mo. Principal, 375. $53.3966 Interest 2 yr. 3 mo. Ans. $428.40, amount; and $53.40, interest. It is not necessary throughout the work to carry the multi- plication beyond four places of decimals. 985. 6. After deducting the first 25%, or J, f of list price remains ; a second discount of of this remainder leaves f of this remainder, or f of f of list price, or T 9 g- of list price = 90^. List price = 90 ^-f-^ = 90 **X ^- = $1.60. 7. All together can do (% + % + -J-) of the work in 1 hr. 8. Selling price, $60 = f of value of cow ; J of value = $60 -*-3 = $20, the loss. 9. Selling price, $60 = f of value of cow ; J of value $ 60 -5- 5 = $12, the gain. 986. 6. The wrong weights are ^^, or f \ of correct weights, so that the customer receives for $352, fj of this amount, the 140 MANUAL FOR TEACHERS gain to the grocer being ? 3 of $352, or $16.50. By selling 16|- oz. to the pound, the grocer gives |-| of the proper amount, his loss being -fa of $ 320, or $ 10. The net gain is $ 16.50 - $ 10 = $6.50. Am. 7. Cost of alcohol, $2.50 X 42 = $105 ; 3 yr. interest on $105 = $18.90. Amount to be realized, $105 + $18.90 = $123.90. Number of gallons to be sold, 42 7 = 35. Selling price per gallon, $123.90 -4- 35 = $3.54. 8. 4500 = $4.85x4500 = $21825. Income on consols at 3%= $654.75. Selling price of consols, $21825 X. 96; value of U. S. bonds, $21825 X .96 -f- 108. Canceling, we obtain $19400; 6% of which gives the income on bonds = $1164. Difference = $ 1164 - $654.75 = $509.25. Ans. 9. Number of cubic feet = 9| X 9^ X 6f = 609 ; weighing 609000 oz. ; etc. 988. 11. Agent collected 80% of $4500 = $3600; on this, his commission at 7^-% is $270; making the amount to be given me = $3600 - $270. " is. tf X X-, or ^=15.12. 100 2 40 Find the discount for 17 da., the time from June 20 to July 7. 15. Ignoring the price if he could buy 80 Ib. with 81|-% of his money, he could buy with 100%, (80 Ib. -f- 81) X 100. 17. Let # = cost of each cow; 6# = cost of 6 cows; com- mission = - of 6 x = - ; cost and commission = 6 x -\ JLUvj _LUv/ = 525.30. 18. The note is discounted 35 da. after it is made, so that it has (93 35) da. to run, or 58 da. [Without grace, 55 da.] The interest for a year is $36, which is 10 cents a day, or $5.80 for 58 days ; etc. 19. If the selling price (regardless of its amount) is six-fifths of the cost, the gain is of cost,. or 20%. NOTES ON CHAPTER TWELVE 141 989. 6. The walls contain [(20 + 15 + 20 + 15) X 10] sq. ft. ; the ceiling contains (20 X 15) sq. ft. Dividing by ^-|, the width in feet of the paper, gives the number of feet of paper required, which is then reduced to yards. 991. 7. Dividend is 3% of ($9562.50 -f- 1.271). 993. Mr. Smith wishes to pay Mr. Thompson the exact amount of his bill. A check on a Memphis bank for $3475.86 would not be sufficient, as Mr. Thompson would have to pay a New York bank for collecting the check in Memphis. As the charge may not always be the same, Mr. Smith cannot know how much to add to the amount of his bill to cover this expense. From a banker that has an account in a New York bank, he can obtain a draft, payable "in that city, for the exact amount, by giving the Memphis banker $3475.86 + $5.21, or $3481.07. Exchange is at a premium when the cost of a sight draft is greater than its face ; it is at a discount when the cost of a sight draft is less than its face. BANK CHECK. Quogue, N.T., jt /o 2. Interest - x X X - = -. 100 360 1000 3. The rate is x dollars per $1000 premium; on $1800, the premium - X x The interest on $ 1800 for 60 days @ 1000 5 Q 6% 1% of $1800 = $18. The equation becomes +1782 -1778.85; -1778.85 -1782- -3.15; 9s = -15.75; x 5 1.75. The minus sign indicates that it is a discount. Ans. $1.75 discount per $ 1000. 6. Let x time in years. The equation becomes 602.25 -36 x -598.95; - 36 x - 598.95 - 602.25 -- 3.30; 36 a; = 7. .60# = 0; x years ; i.e. sight. 8. The interest - 1200 x-^X^ = ~ The equation becomes -21.70; 31^-217; a?=7. ^Iws. 7%. - 9. Rate a? per M. Premium -^- X ar ; x 0. A "T> J-UUU O Ans. Par. 1088. 4. 10 so^ch. = 1 A. = 160 sq. rd. 1 sq. ch. 16 sq. rd. 1 chain Vl6 rods 4 rods 16-J- ft. X 4. 5. Each face contains 1350 sq. in. -5- 6 225 sq. in. Length V225 in. 15 in. 6. Hypotenuse 5 in. 7. Hypotenuse = 3^ in. 1089. Another method of subdividing square O is here shown, Fig. 3. From each side of C, a right-angled triangle is cut, of the same dimensions as the original triangle m, leaving a small square X. In Fig. 4, is shown a rearrangement of these NOTES ON CHAPTER THIETEEN 173 FIG. 1. FIG. 2. FIG. 9. FIG. 4. FIG. 5. triangles, making a polygon equal in surface to the sum of the squares A and B, Fig. 5. 1091. 1. To find 4^, divide by 4, placing each quotient figure one place to the right. Nothing should be written beyond what is shown in the Arithmetic. To find the interest, see Art. 983, 27. 2. 1% each quarter. In dividing by 100, the dividend is repeated, with each figure two places to the right. 3. 3% each half year. Multiply by 3, placing each figure of the product two places to the right. $1500." 45. $1545. iyear 46.35 $1591.35 1 year. 1092. In some cities, the quotations of stocks give the price per share. A share whose par value is $50 and which sells for $48, is quoted in the New York papers as worth 96 ; i.e. 96% 174 MANUAL FOR TEACHERS of $50. In a few places, the price is quoted as 48, meaning $48 per share. In the examples given in the Arithmetic, the rate always means the per cent of the par value. 1- (87f + i)% of ($10x240). 2. Brokerage = \% of ($100 X 120) = $30. Value of stock = $11460 - $30 = $ 11430 ; value per share = $11430 -f- 120 -$95.25. 3. Let x = par value per share. Then (87-} + )% of (a: X 150) = 5265; 2L*lj^ = ^^^ = 5265. 5265 x 100 X 4 Using the cancellation method, x = lOU X oOl 4. \% of $27500. 5. 102|% of ($25 X 200) = selling price. Deduct there- from \% of ($25x 100). 6. His income is 18% of the par value ($100x60) = $1080. His investment is 450% of ($ 100 X 60) = $27000. $1080 is 4% (Ans.) of $27000. Or, irrespective of number of shares, 18% -+- 4^- 4%. Ans. 7. The stock = $50 X 4000 $200000. The rate of divi- dend - $20000 -*- $200000 = ^ - 10%. Ans. 10% -*- 1.75 = 5f%. Ans. 8. $10000 interest must be paid to the bondholders, leaving ($47500 - $ 10000) $37500 to be paid as dividends on $1000000 of stock. 37500 *- 1000000 - .0375 - 3|% . 9. Let a; = brokerage. 168 ^ + - of 25x360 = 15176.25; 9000 (168J- + a?) == 1517625 ; etc. 10. Let x = amount of brokerage. 107f % of (100 x 250) - x = 26875; 26937.50 - x - 26875 ; x = 62.50; brokerage -$62.50. Ans, Etc. NOTES ON CHAPTER THIRTEEN 175 11. $ 35050 -*-1.75i = $ 20000 = par value of stock; 71% of Ans. 12. Mr. Tower* receives $30 interest, and $100, the face value of the bond, in six years, $130 in all. His investment was $104, on which he has received $130 $104, or $26 interest in 6 yr., or $4.331 per year. 41 -*- 104 - .041 =4%. Ans. 1093. 1. A corporation is an association of a number of persons legally empowered to transact business as a single indi- vidual. The charter specifies the name of the corporation, the amount of capital, the business it is authorized to carry on, the powers and privileges conferred, etc. The stock is the money invested in the business of the corporation ; a share is one of the equal parts into which the stock is divided ; a shareholder is the owner of one or more shares ; a stockbroker is a person engaged in the business of buying and selling stocks on commission ; a dividend is &pro rota division of profits among the stockholders ; an assessment is a sum levied upon stockholders to meet some unexpected expenses, losses, etc. 2. Income from bonds = 6% of $125 X 109 -*- 1.09 ='6% of $12500 = $750, an increase of $ 68.75 over the previous income of $681.25. 3. Sum loaned = $59.57 -+-1 = $ 59.57 X 6. 4. In 3 yr. at 7%, $1 will amount to $ 1.21 ; i.e. a payment of $1.21 in 3 yr. is considered equal to a cash payment of $1. The "present worth" of the delayed payment = $4235 -+- 1.21 - $ 3500, which is a loss of $ 3675 - $ 3500 = $ 175. Ans. 5. The rate on one = 5% -*- (.981 + .OOJ) = 5 -H .981; O n the other -6% -5- 1.09. The cost of a $100 5% bond is $98.25 + 25^ brokerage = $98.50; the interest on the bond is 5% of $100 = $5; the rate is, therefore, 5% -^ .981 ; etc. 6. The time of the place being 1 hr. later than the time of the starting-point, the traveler is 15 east of the latter place. 176 MANUAL FOR TEACHERS 8. As the note is stated to be due in 3 mo., which may be assumed to include days of grace where allowed, the date of its maturity is Sept. 20. The amount of the note -for 92 da., at 9%, $2455.20. The discount on this sum at 6% from Aug. 8 to Sept. 20, 43 da. = $17.60. Proceeds = $2455.20 - $17.60 - $2437.60. Ans. 10. $4800 less 63 days' interest, $50.40, = $4749.60. ($4797.58 less 60 days' interest, $47.98, -$4749.60.) 1097. 4. Having found that VlO = .316 +, the pupils should see that VI = .316 +. In pointing off, begin at the units' place, pointing off two places to the right or the left. 7. 1.6 is made 1.60. See 10. 1098. 7. Find the amount of $467.50 at 6% from July 5, 1881, to Dec. 19, 1885. 8. Amount stolen = $650. The first sender is entitled to |5o. of $523.25 ; the second, to fff of $523.25 ; etc. 9. WASHINGTON, D.C., Dec. 1 ? 1896. THE UNITED STATES To JAMES RYAN, Dr. To Services, Nov. 19-30, 12 da. 45 65 " Traveling Expenses, 12 da., $4.00 48 00 $12.50 $17.40 $3.25 " Stage Fare St'mb't Fare Telegraph 33 15 $126 80 10. The account for the first quarter, exclusive of box rents, is $124.96. On $50, the commission allowed =$50. On the remaining $74.96, the commission is 60% $44.98. Total = $77 + $50 + $44.98 = $ 171.98. Salary for the year = $171.98 +$194.01 +$174.08 +$167.87 -$707.94. NOTES ON CHAPTEE THIRTEEN 177 1099. Base 2 + perpendicular 2 = hypotenuse 2 ; H^E 1 + P\ 1. 225 + 64 = tf 2 ; # 2 289; # 17, hypotenuse. 2. 1225 + x 2 = 1369 ; of - 1369 - 1225 = 144 ; x= 12, per- pendicular. 3. x 2 + 225 = 1521 ; x* = 1521 - 225 - 1296 ; x = 36, base. 1100. 11 . Let x perpendicular. Area = % (200 + 160) X a; = 180 x -32400; a; =180. 12. 1(20 + 00x15=225; 8QO + I5x = 225; 300 + 15*- 450; 15 #- 450 -300 =150; a? =10. 13. $( x + x + fy X 10=(a? + 3)x 10- 10 x + 30 -150; 10a;-150-30-120; a? - 12, x + 6 = 18. 12 rd. and 18 rd. Am. 14. The base 7 rd. ; the altitude, or perpendicular, 24 rd. Area (24 X 7) sq. rd. 15. Number of stones (84 X 36) -*- (6 X 3). Cost $1 X (28 X 12). 16. The angle of 90 indicates a right-angled triangle. Hypotenuse = 20 chains. Number of chains of fence 12+16 + 20 - 48. 1 chain - 66 ft. - 4 rd. Number of rods 48 X 4 - 192. 17. A perpendicular let fall from the upper right corner would form a right-angled triangle, whose perpendicular is 40 ft., base 30 ft., making the hypotenuse = Vl 600 +900 ft. 50 ft., the fourth side. The number of yards of fence (40 + 70 + 50 + 100) X 5J-. Area in acres [ of (70 + 100) X 40] 4- 160. 18. The diagonal is the hypotenuse of a right-angled triangle whose other sides measure, respectively, 90 yd. and 120 yd. 19. The diagonal Vl296 + 729 chains 45 chains. The distance by the road 27 chains + 36 chains 63 chains. The saving is 18 chains, of 22 yd. each. 178 MANUAL FOR TEACH EUS 20. A 40-acre field contains 6400 sq. rd. Each side measures V6400 rd. = 80 rd. The diagonal = V80 2 + 80 2 . 1101. 1. The loss is -j- of the cost, $300. 2. If 3 boys solve 3 problems in 3 min., 1 boy will solve 1 problem in 3 min., and 6 boys will solve 6 problems in 3 min. Or, 3 boys will solve 6 problems in 6 min., therefore 6 boys will solve 6 problems in 3 min. 3. The 3 boys eat 12 cakes, 4 each ; for which the third boy pays 12^, or 3^ apiece. The boy that brought 7 cakes supplies 3 ; for which he should receive 9^. The other boy furnishes 1, and is. entitled to 3^. 4. The cost of the article is 50^ ; a sale for $2.00 is a gain of $1.50, or 300%. 6. A 6-ft. fence needs 2 posts, one at each end ; a 12-ft. fence, 3 posts ; a 30-ft. fence, 6 posts. 7. One half is profit, the other half is the cost. The profit equals the cost, and is 100%. 8. ixx-&x = ^x* = e>0', 3a; 2 = 1200; a* = 400 ; a: = 20. 9. ^^90. 400 10. The difference between 28 and 75, inclusive, =(75-28) + 1. 1102. 2. The quantity of provisions for each man 2^ Ib. X 20 = 45 Ib. To last 24 da., the allowance should be 45 Ib. *- 24 - If Ib. = I Ib. 14 oz. 5. 80 min. X 112 -*- 46. Cancel. 6. See Arithmetic, Arts. 821, 822. 7. First beam contains (66 X 10 X 8) cu. in. The second contains (x X 12 X 12) cu. in. The contents of the latter %\ 4 - times the contents of the former. NOTES ON CHAPTER, THIRTEEN 179 r-12-]2- 3024x66Xl X8 ~^24~ __3024x66x 10x8- lop 924 x 12 x 12 Ans. 120 in. = 10 ft. 9. The weight of the provisions = 3 Ib. X 32 X 45. Dividing by 40 gives the daily allowance for the increased crew ; dividing this by their number, 32 + 16, gives the allowance of each. 3 Ib. X 32 X 45 40x48 12. The difference in deposits = $450, which sum produces $ 18 interest. The rate is 4% . 14. $7500 at x% for 3 years produces $ 1125 interest. x =8000. 16. [yfo of (x + 400)] - (ffa of *) = 30 ; 5 a: + 2000 x 100 100 5 x + 2000 -4 x -3000, x = 1000, x + 400 = 1400. Ans. $1000 at 4%, $1400 at 5%. 19. A receives --$ of selling price; he invested, therefore, of $600; etc. 20. F receives y 7 ^. E's share is -^ more than D's. If y 2 ^ -$90, -^ = $45; Fs share, ^, = $315. 21. The bank receives at the end of 63 days, the sum loaned, $593.70, + $6.30 interest. The problem is: At what rate will $593.70 produce, in 63 days, $6.30 interest? 180 MANUAL FOR TEACHERS 593 - 70x ilj x lr 6 - 30 ' _ 630x100x360 _ A 126 59370 x 63 T ^' 22. The two supply pipes fill ^-f-J-, or , of tank in 1 hour. If all the pipes are set to work when the tank is full, the ex- haust pipe takes off each hour ^ of the tank more than the others supply. To empty the tank would, therefore, require 6 hours. 1103. 7. The diagonal = V137i 2 + 9. Let 100 # = value. Buying price 90 # ; selling price = 110 a: ; gain = 20 z, which is f of the cost, 90 z, or 22f %. 10. 420 j* -+-($p-*p) = number. 1106. 1. $ 500 -*- f = cost of one portion ; 500 -*- = cost of other portion. 4. If f of farm is sold for f cost, the selling price of the whole farm at the same rate would be f- cost -5- -8- = cost, mak- < o o ing the gain |- cost = 12^-%. 7. 10^ per bu. of 60 Ib. 16|^ per 100 lb., or f ^ higher than 16 f per 100 lb., or -fa of 16 p higher, or 4|%. 8. See Art. 1084, 15. 10. If -| of the selling price is profit, the cost must be f of the selling price ; the latter is therefore -| of the cost. This is a profit of - of the cost, or 66-|%. Ans. Or, a profit of 2-fifths on a cost of 3-fifths is -- of the cost. 11. x + ^x = 60J-. 14. \\ ? -, 0) From 84 (ft 84 subtraction, it may be made \v j ^ J. . -i -I /7\l 1 JLctKc Tit/ ZjtJ _ 49 4. 95 one m addition (6) by chang- ing the signs of the subtrahend. It may then be written 84 49 + 25. 1211. The pupil will readily ascertain that a parenthesis pre- ceded by a plus sign may be removed without any alteration being required in the signs of the quantities enclosed within it. 57 + (33 _ 16) = 74, may be written 57 + 33 -16 = 74. In 2, the signs of the quantities within a paren- ^ i 09 thesis must be changed. The first number m -, . nc . , . ,, ,, . . .,, . . lake + od + 25 within the parenthesis, being without a sign, is go positive ; it therefore becomes negative when 63 25 ^ e P aren ^ nes i s i g taken away. The equation - -- then becomes 92 63 - 25 = 4. In 5, the multiplier 4 affects only the quantity within the parenthesis. The brackets heretofore used, have been omitted in 5 and 6, to make these arithmetical equations resemble more closely the algebraic equations of Art. 1213. 5 becomes 75 + 60 - 40 = 95 ; 6 becomes 75 60 + 40 = 55. 1213. It has not been considered necessary to give any pre- vious practice in multiplying simple algebraic polynomials by an ordinary number. The average pupil will readily understand that 6 times two x = twelve x. 1. 12.x 30 5rr+12. 12o;-5# = 12 + 30; 7^ = 42; * = 6. Proof. 6 (12 - 5) = 30 + 12 ; i.e., 6 times 7 = 42. 228 MANUAL FOR TEACHERS 2. 7 a: +14 3o; + 50; etc. 3. 15 + 5a?+16 = 61; etc. 4. 48-3a? = 52 4z; etc. 7. 2a? 2 4a? + 38 = 3# 9; etc. 8. 12 x -30 5a? = 12; etc. 9. 5a;-12a; + 30 = -12; etc. 10. 11 3ff+10a? = 38; etc. 1215. 12. Zx- 3-2^ + 4 = 13. 6x-6- 15. 14a-16 Transposing, 14 a? - 18 a; - I2x + 60; = 15 - 12 + 16 + 36 t Combining, 10 a; = 55. Changing the signs of both members, 10*= -55; Or, a? = -5f 18. 2^^3 + -5-+- 4 55 19. +9-2.; + - 4 52 22. a? -20 -^+60. 1216. l. Multiplying by 16, 3| x - 7 = 48 ; or ^ - 7 = 48. 3 Clearing of fractions, 11 x 21 = 144 ; etc. Q ~ 2. ^ = ^-60; etc. NOTES ON CHAPTER FIFTEEN 229 3. z + 6. x + 12 son's present age ; 2 x + 12 = father's present age. 3^ = 138-24 = 114; 5; = 38, the son's age 12 yr. ago ; 2x = 76, the father's age 12 yr. ago. The present age of the son is 50 yr. (x + 12) ; the present age of the father is 88 yr. (2x + 12). Ans. 7. (80 + 3? = 2.(60-a?); etc - 8. 2(ll + #) = 25 + a;. 9. Let x = the number of gallons originally in the cask. - = amount drawn off, leaving in the cask. 60 = num- 4 4 4 ber of gallons required to fill the cask. 24 = 60 - ; 96 = 240 3#; etc. 10. ?-40=104. x x Clearing of fractions, x + 430 = 4 x + 76. Transposing, x 4 x = 76 430. Combining, 3 x 354. Changing the signs of both members, 3 x = 354. Or, x 118, the smaller number; x + 430 = 548, the larger number. 230 MANUAL FOR TEACHERS 12. Let x the number of $2 bnls. Then 29 - x = the number of $ 5 bills. 2# + 5(29-#) = 103; efcc. 13. 4 + 4) = # + 34. 14. a? + 3)x -225. Multiplying, x Clearing of fractions, 180 x + 540 = 225 x ; etc. 15. 33 (#+1) -40 x +12. 16. The numbers are x and x + 17. Then # + #+17=47; etc. 17. Let x number of years. The mother's age will then be 41 + x, and the son's 5 + x. o ; etc. 1218. 1. Substituting the value of 8x, the equation becomes 16 + 7y = 44; etc. 2. 9 + 52; = 34; 5z = 25; 2 = 5. Ans. 1219. 11. 3#+14y=78 Subtracting, x = 12 Substituting this value of x in the first equation, 36 + 14y = 78; etc. 14. Multiplying the first equation by 2, we Have 2# + 2y = 30. Subtract the new equation from the second one, 2# + 3y = 38, thus finding the value of y. Substitute this value in either of the original equations. NOTES ON CHAPTER FIFTEEN 231 17. Multiplying the first equation by 3, and the second by 2, we have 6a + 9y = 120 and 6# + 4y 70. 18. Multiply the first by 2, and the second by 7. 19. Multiply the first by 9, and the second by 5. 20. Multiply the first by 8, and the second by 3. 1221. 21. If we add, we have 2x = 22. Or, subtracting, 2y 14. By adding, y is eliminated; by subtracting, x is eliminated. From this example, pupils should see that either addition or subtraction may be employed to eliminate one of the unknown quantities ; and that either of the two may be elimi- nated, as may be found convenient. 27. 5x-6y= 5, 28. 3x 5y = 4. 2#- y = 12. 29. 10 x + y = 1, (1) Subtract (1) from (2). . $ x + y = 9. (2) 30. 3^ + 87/^204, 31. 3^ + 2y 252, 34. 3* + 7=15y-20; 3a?-15y = -27. (1) 7*-6 = 10y+ 6; 7ar-10y= 12. (2) Multiply (1) by 7 and (2) by 3, to eliminate x\ or (1) by 2 and (2) by 3, to eliminate y. 35 . 51 x + 44 y = 804, (1) Multiply (1) by 8, and (2) 45#-32y = 72. (2) by 11. Add. 37 . 2 x - 22 2 y + 18 = 6, The pupils should be taught h e common de- 15 4- 135 32 T/ 96 nominator of y 3 and 15, as 15 (y 3), which contains y 3, 15 times ; and 15, (y 3) times. 39. 2a; + 5y + 3 = 18a;-24y 12; - 16#+ 29y = 15. 232 MANUAL FOR TEACHERS 1222. 1. # + y = 37; This problem can also be solved 2#-j- 3 + 900 - 2500 - lOOtf + x 2 ; 100* =1600; a; =16. AO= 16 ft., OB, the part broken off = 50 ft. - 16 ft. = 34 ft. Ans. Or, making BC=x, AC= 50 x. Then, (50-#) 2 + 30 2 = # 2 . 2500- 100* + * 2 + 900 = ^ -100^ = -3400; x = 34, the length in feet of the part broken off. 9. 60 2 + (58-o;) 2 = 56 2 + z 2 ; 3600 + 3364 - 116 x + x* = 3136 + of ; # 2 - x* - 116 x = 3136 - 3600 - 3364 - 116 x = - 3828, or 116 x = 3828 ; x = 33 = AE. The_length of the ladder in feet = V56 2 + 33 2 = V3136 + 1089 - V4225. 65 ft. 10. From ABD, the square of J5D = 13 2 - (15 - x}\ From BOD, the square of BD = 4 2 - x\ Therefore 13 2 -(15-o;) 2 = 4 2 -^; or, 169 - (225 - 30 x + * 2 ) = 16 - of. 238 MANUAL FOE, TEACHERS Kemoving the parenthesis, 169 225 + 30 x x 2 = 16 x 2 . Transposing and combining, 30 x = 16 - 169 + 225 = 72; _ _ = V4 2 - 2f - Vl6 -i== Altitude = 3J ft. Am. 11. AF= -JAB 1 - BF 2 = V1156 - 256 = V900 = 30 ; FC= V^C 2 - BF 2 = V400 - 256 - Vl44 = 12; ^(7 -^4^+ ^^-30 +12 = 42. 42ft. ^Ins. Let ^LJ?=a?; ^7(7- 42 -a:; ^7D 2 - ^D 2 - AE 2 = 26 2 - x 2 = 676 - ^ ; ED 2 = DC 2 - EQ 2 = 40 2 - (42 - x) 2 - 1600 Therefore 676 - x 2 = 1600 1764 + 84 x - x 2 - 84 x = 1600 - 1764 - 676 - - 840 ; x -10. ED = V26^10* - V576 - 24 ; or, = V40 2 -32 2 = V576"= 24. 24 ft. Am XIX NOTES ON CHAPTER SIXTEEN The geometry work contained in this chapter should be com- menced not later than the seventh year of school, and should be continued throughout the remainder of the grammar-school course. 1251. No formal definitions of lines, angles, etc., should be given at the beginning. After drawing angles of various sizes and with lines of different lengths, the pupils will be able to understand that " an angle is the difference in direction of two straight lines that meet in one point, or that would meet if produced." 1255. The semi-circular protractor is better than the com- mon rectangular one for beginners, as they see more clearly by using the former that an angle is measured by the arc of a circle. Two protractors are printed on a fly-leaf in the back of the text- book, for the use of such pupils as cannot procure others. Pro- tractors made of stout manilla paper can be obtained from the Milton Bradley Co., New York, at one cent each in quantities. A large protractor is needed for blackboard use. This can be made of pasteboard ; or wooden ones can be bought of the Keuffel & Esser Co., New York. Many scholars that are able to measure an angle one of whose sides is horizontal, Fig. 1, find it difficult at first to ascertain the number of degrees in an angle formed by two oblique lines, Figs. 2 and 3. They should be permitted to discover the method 239 240 MANUAL FOR TEACHERS for themselves. All that is necessary, is to place the center (A) of the base of the protractor on the vertex of the angle, Figs. 1-3, and the edge of the protractor on one of the sides, the other side cutting the circumference. In Figs. 2 and 3, the number of degrees in the angle XAZ is determined by the number of degrees FIG. 2. in the arc BM, and the upper row of figures is used, having tho zero mark at B. In Fig. 1, the number of degrees in the angle is measured by the arc CM, which requires the use of the lower row of figures. 1256. The average class will find the 100 exercises to Art. 1269, inclusive, sufficient for the first year's work. This will give three per week, and leave some time for review. Pupils should work the exercises at home without any preliminary discussion in class. After the exercises are brought in, they should be done on the blackboard, at which time the mistakes made can be pointed out. While first-class drawing cannot be expected from the instruments used by school-children, the teacher should exact the best work possible under the circum- stances. A hard pencil, kept sharp, is necessary to secure the requisite fineness of line. 1. In drawing an angle, commence at the vertex. This exercise is given to remove the impression sometimes formed, that the size of an angle depends upon the length of the lines, instead of their greater or less difference in direction. In this and all other exercises, the pupils should be encouraged NOTES ON CHAPTER SIXTEEN 241 commence occasionally with an oblique line. No two results should be exactly alike. If two pupils compare notes, it should be for the purpose of producing a different drawing. One pupil's angle may have its vertex at the right, another at the left ; one vertex may be above, another below ; etc. The better the teaching, the greater will be the variety of results in exer- cises that permit of variety. 3. It is expected that the pupils will see for themselves that each arc will contain \ of 360. 4. Using the ruler, draw the first line of any convenient length and in any direction. Placing A of the protractor at either end, mark off 45, being careful to use the proper row of figures. Kemove the protractor ; place the ruler so that its edge just touches the end of the line and the 45 point, and draw the second line. This latter should not be of the same length as the first, unless for some good reason ; so that pupils will not consider that the lines forming an angle should be equally long. Write the number of degrees in each angle. 5. The teacher should not inform the pupils in advance how many degrees they will find in the second angle. They should measure it for themselves, using the protractor. In drawing these angles, the figure in the book should not be followed. The second line should be drawn to the left in some cases ; the lower angle may be mad 60 ; etc. When two lines meet to form two angles, it is not at all necessary that the point of meeting should be at the center of one line. 1257. Pupils should be taught that horizontal lines are lines parallel to the surface of still water. Floating straws are hori- zontal, and may point in any direction. A spirit level is used by the carpenter to determine whether or not a beam, for instance, is horizontal. A vertical line is one that has the direc- tion of a plumb line, which is used by a mason to ascertain if a wall is perpendicular. 242 MANUAL FOE TEACHERS In drawings, however, lines that will be horizontal when the paper is placed upon the wall, are called horizontal lines ; and lines that will be vertical when the paper is placed upon the wall, are called vertical lines. 6. The perpendicular need not be drawn to the center of either of the others, nor need it always be drawn above. The teacher should encourage variety. 8. The pupils should draw these lines, and mark in each angle the number of degrees it contains. Encourage the greatest possible variety in the size of the angles and the direction of the lines. 9. While pupils may be able by this time to give the result without drawing the angles and measuring the second one, the teacher should not fail to give them the necessary practice in constructing angles of a given number of degrees, and in measuring the contents of others. Many scholars make as ridiculous mistakes in the measure- ment of angles as they do in their work in numbers, frequently reading the wrong figures, and figures from the wrong row marking an angle of 45, for instance, 135; etc. They should learn to " approximate" the size of an angle, as well as to " esti- mate " the probable answer to an arithmetical problem. An acute angle should not be marked as containing over 90 ; etc. 10. Having learned by observation that the sum of two adja- cent angles is 180, the pupils should now discover that the sum of any number of angles formed on one side of a straight line is 180. When they have learned this from drawing the first exercise, they may be permitted to calculate the result in the other two, especially as the protractors are not marked for fractions of a degree. The first exercise should show the same variety in the work of the different pupils as has been recommended for previous work. 12. In constructing a square, the protractor is used to erect a perpendicular at each corner. These perpendiculars are made NOTES ON CHAPTER SIXTEEN 243 equal to each other and to the original line. A fourth line is drawn. The accuracy of the work may be tested by measuring, with the protractor, the two upper angles. The base lines used by different pupils should be of different lengths. The pupils should be permitted, also, to construct the square in their own way. Some may erect a perpendicular at one end of the given line, and at the extremity of the second line erect another perpendicular. Some pupils may not measure the first line, drawing the second and third lines lightly of indefi- nite length, and using compasses to make them equal to the first. In this case, the light lines should not be erased ; but the square should be marked off by heavier lines. It is a good practice to have the pupils give a written descrip- tion of their method of working one of these exercises, which should be accepted as a regular composition. The language should be correct ; the proper technical terms should be em- ployed ; and there should be sufficient detail to enable any one not familiar with the work to understand just how it was done. 13. Pupils should be permitted to learn for themselves from this exercise and from 14, that vertical, or opposite, angles are equal. 17. After drawing the required lines, the scholar should mark in each angle its contents in degrees. 19. This exercise should enable the pupil to see that the sum of all of the angles formed about a point will be 360. 20. The teacher should not give unnecessary assistance. If the scholars have a few days in which to work out an exercise, they should find no difficulty in managing this. The word " adjacent " in geometry is applied to each of the two angles formed by one straight line meeting another. In 19, the two lower angles are adjacent ; but none of the upper three angles is adjacent to any other, because three straight lines are used to construct two angles in each case. No two angles in 20 are adjacent, and no two are vertical. 244 MANUAL FOE TEACHERS 21. There are no adjacent angles. They are all vertical, because the lines forming each are produced to form an opposite angle. 22. The pupil is not yet ready to do this in the geometrical way. The teacher should be satisfied if he adds 65 and 25, and uses the protractor to make an angle of 90 ; etc. 24. If the pupil examines a clock, he will see that the num- ber of degrees between 12 and 1 is -^ of 360, or 30. He has learned already that the length of the sides has nothing to do with the magnitude of the angle. 25. The minute hand goes 90 in a quarter of an hour. The hour hand goes 30 in an hour ; 15 in ^ hr. ; 7-|- in \ hr. To ascertain the angle at 12 : 15, the pupils should draw a clock face, locating the hands properly. Some will place the hour hand at 12, forgetting that it has gone 7-^ in -J hour; and will give the answer as 90 instead of the correct one of 90 7-1-, or 82 30'. At 6 : 30, the minute hand is at 6, and the hour hand is half way between 6 and 7, or \ of 30 15. At 8 : 20, the minute hand is at 4, and the hour hand \ of the way between 8 and 9 the number of hour spaces being 4^-, corresponding in degrees to 30 X 4^- = 130. Pupils should understand that while the angle at 4 o'clock is 30 X 4, or 120, and while the angle at 5 o'clock is 30 X 5, or 150, the angle at 7 o'clock is not 30 X 7, or 210. By making a drawing, they will see that in the last case the angle should be measured on the left, which will make it 30 X 5, or 150. NOTE. Angles of 180, 210, etc., may be left for more advanced work. 1260. From 26 should be learned that two lines perpendicular to a third line are parallel to each other ; and from 27, that two lines running in the same direction and making the same angle with a third line, are parallel to each other. 29. DE will be drawn parallel to EC by means of the pro- tractor, an angle of 58 being made at the intersection of AB and NOTES ON CHAPTER. SIXTEEN 245 FIG. 4. DE. Six of the twelve angles will contain 58 each ; the remaining six will each measure 180 58 = 122. The pupils should be permitted to ascertain this for themselves. The card suggested in the note is to be used in schools in which small wooden triangles are not obtainable. 32. To draw from P, a line parallel to the oblique line AB, place the perpendicular of the triangle on the line AB, Fig. 4, arid place the ruler XY against the base of the triangle. Hold- ing the ruler in position, slide the triangle along it until the perpendicular passes through the point P. A line PE drawn along this side of the triangle will be parallel to AB. Time should be given the pupils to discover this or a similar method of drawing by means of a ruler and a triangle a line parallel to another line. The method may be made the sub- ject of a composition. 33. QR and U Fare drawn parallel by means of the ruler and the triangle. They may lie in any position, care only being taken to cut them by a line making angles of 50 and 130 with one of them. Three of the remaining six angles will measure 50 each ; and the others, 130 each. 1261. 35. If the work is done as it should be, the angle at C will measure 80. The line AB does not need to be horizontal ; nor should all the pupils draw AB of the same length. 36. The third angle will measure 60. 37. There are 68 in a, and 57 in c. In b, there are 180 - (68 + 57), or 55. In d, which is vertical to b, there are 55. 38. The angle e should measure 28, and/ 120. There are 180 in e (28) + g +/ (120). 246 MANUAL FOR TEACHERS 39. PEQ = 180 - (70 + 60) = 50. PRS = 180 - 50 130. PRS is therefore equal to the sum of the angles P and Q. 40. 180. Ans. 41. 180 - (36 + 65). 45. In measuring the side of a triangle, use the smallest frac- tion marked on the ruler. When the ruler in use has the denominations of the metric system on one face, that face should be used, and the length of the line given in millimeters. This will not require any teaching of the metric system beyond show- ing pupils how to read their rulers, and it will do away with the need of using fractions. 46. The length of each side should be marked. If two of them are not found exactly equal, the construction is faulty. 47. The three sides should be equal. 50. Each of the oblique angles will contain 45. 52. Angles 2 and 3, 90 each ; 4, 50 ; 5 and 6, 40 each. 53. Angles 1 and 4, 67^ each ; 2 and 3, 90 each ; 5 and 6, 22i each. 54. The angle p contains 120; ra, 30; n, 30. 1266. 56. Construct this parallelogram by drawing two lines of the given lengths, meeting at an angle of 60. By means of the ruler and the triangle, construct the other two sides. If the work is properly done, these two sides will measure 2 in. and 3 in., respectively; and the remaining angles will measure 120, 60, and 120, respectively. From this exercise, the pupils should learn that the opposite sides and the opposite angles of a parallelogram are equal, and that the sum of the four angles is 360. The angles being oblique, the figure is a rhomboid. The work of the scholars should show the variety suggested in previous exercises. It is not essential that the longer of the two Ojf YS. NOTES ON CHAPTEj^ Sl|*Wl 247 given sides should be taken as the base, nor tnai'TTie base should be parallel to the lower edge of the paper. 57. Different pupils will construct this trapezoid in different ways. Some, seeing that one angle is a right angle, will use the triangle to draw the second side, 3 in. ; and at the extremity of this side, will draw, by the same means, an indefinite perpendic- ular line to form the third side, which is parallel to the base. The fourth side is drawn to make an angle of 60 with the base. The remaining angles will measure 90 and 120, respectively ; and the sides will measure 5 in., 3 in., nearly 3^ in., and nearly 31 in. 58. The triangle cut off will form a rhombus when opened out, unless the base and the perpendicular are equal. In this case, the paper, when opened, will form a square. Making one angle of the triangle 30 (or 40) will give a rhombus containing 60 (or 80). 60. When the three sides of a triangle are equal, its three angles are equal ; but the rhombus has four equal sides without having equal angles. 61. A triangle that contains three equal angles, has its sides equal ; but an oblong has four angles of 90 each, with unequal sides. 62. To construct the rhomboid, draw a base of 2^- in. Two inches above, draw a parallel line 2|- in. long, with the extremi- ties of the latter on the right or the left of the extremities of the base. If the two remaining sides are exactly equal to each other, the work is correctly done. It is not necessary to draw the altitude, though a broken line may be used. While different methods may be employed, the use of an incorrect one should not be permitted. The work done on the board by a pupil, should be criticised by the class if it be faulty ; or a better way may be suggested, if the one employed by the pupil at the board require too much time or unnecessary work. 248 MANUAL FOR TEACHERS The lengths of the two remaining sides should be measured, and marked on the papers. Those of different pupils should be different, there being no limit except the size of the paper: they must, however, be longer than 2 in. each ; and they should not be just 2J- in., which would make the figure a rhombus. 63. If the line that shows the altitude of the rhomboid can be drawn within the figure, a right-angled triangle can be cut from one side and transferred to the other, making the figure a rectangle. Arithmetic, Art. 929, 5, last parallelogram. In the case of a rhomboid whose altitude does not fall FIG. 5. within the figure, several cuts will be necessary to change it into a rectangle. See Fig. 5. 65. The areas will be equal, because each rhomboid is equal in area to a rectangle 3 in. by 2 in. 66. The three rhomboids in Fig. 6 have their respective sides equal each to each, but their angles are unequal ; hence their altitudes and their areas are unequal. To construct these rhomboids, draw base lines of three inches, and inclined to each, at any angle except one of 90, a two-inch line. Use the ruler and the triangle to complete the figures. NOTES ON CHAPTER SIXTEEN 249 68. As the altitudes are not given, the areas of the figures drawn by different pupils should vary. The protractor or the triangle should be used in drawing the altitudes, which should then be carefully measured. 69. See 62, making the upper side 2-|- in. The fourth trape- zoid in Arithmetic, Art. 929, 6, shows the manner of determining the dimensions of the required rectangle. 1267. 78. The diameters of these circles should be such as not to admit of the use of the protractor in drawing them. To ascertain the extremities of an arc of 120, two lines are drawn meeting in the center of the circle at an angle of 120. The por- tion of the circumference intercepted by these lines will constitute an arc of 120. The remainder of this circumference will form an arc of 240. 79. The pupil should be permitted to make the first attempts in his own way. He will doubtless soon discover that the distance between the points of his compasses in drawing the cir- cle, is the length of the chord required, and that by placing one point of the compasses on any portion of the circumference of the circle just drawn, the other point will indicate on the circumfer- ence the other extremity of the chord. To measure the length of the arc in degrees, draw radii to its extremities, and use the protractor to determine the angle made by these radii, which may be produced, if necessary. If the work is properly done, the angle should measure 60, which is the length of the arc. 80. Draw two light lines meeting at an angle of 72. Using the vertex of the angle as a center, and any radius, draw an arc be- tween the lines. This arc will measure 72. FIG 7 Darken the lines from the arc to the vertex to show the radii of the circle of which the arc is a part. 250 MANUAL FOR TEACHEKS 81, 82. Figs. 8 and 9 show sectors of 90 and 270, respec^ lively ; Figs. 10 and 11, segments of 120 and 240, respectively. FIG. 8. FIG. 9. FIG. 10. FIG. 11. 126S. 86. A diameter will divide the circle into two equal parts. A second diameter perpendicular to the first, drawn by means of the protractor or the triangle, will divide the circle into four equal parts. To divide the circumference into four equal parts, it will not be necessary to draw the diameters. When he has the ruler in the proper place to draw the first diameter, the pupil needs to mark only the two points where the ruler cuts the circumference. The third point can be indicated when the triangle is placed at the center of the circle and against the ruler ; etc. While the scholars may be permitted in the beginning of this work to draw a number of unnecessary lines, and while it may be an advantage to even require it, they should gradually learn to make as few lines as possible. The construction lines that are employed, should be drawn very lightly and should not be erased. Other lines should be made more conspicuous. Careful pupils may be allowed to use ink for this purpose. suggested 1269. It is not intended that the methods here should be communicated in advance to pupils. Each should be allowed to try the problem in his own way. The discussion oJ the method employed afterwards on the blackboard, will suggest other and possibly better modes of procedure. 88. To inscribe a regular pentagon in a circle, it will be neces- sary to divide the circumference into five arcs of 72 each. The NOTES ON CHAPTER SIXTEEN 251 protractor should be used to obtain the first arc ; the remain- ing ones can be set off by the compasses, the first being used as a measure. Careful work should be exacted by the teacher. 89. Many pupils will have learned in their drawing lessons the regular method of inscribing a hexagon in a circle. Those unfamiliar with this way, should not be shown it until after they have constructed the hexagon by means of the protractor. It is a pedagogical mistake to suggest " short-cuts " to pupils before they thoroughly understand a general method. For this reason, the teacher should permit the members of her class to use the protractor in the construction of the inscribed triangle, leav- ing it to themselves to discover a simpler way. She should encourage, also, the employment of a variety of methods even if some of them are not very direct. The experiments made by pupils to discover a new mode of constructing a polygon, will help them in their geometrical study. The chord of 60 being equal in length to the radius (79), the shortest method of inscribing a hexagon is to apply the radius as a chord six times. Two of these divisions of the circum- ference will make arcs of 120, the chords of three of these form- ing the sides of an inscribed equilateral triangle. 90. Each of the six angles at the center contains 60. Since the two sides AC and AB, enclosing any central angle, are radii of the circle, and therefore equal to each other, the angles oppo- FIG. 12. FIG. 13. site those sides are equal ; that is, angle A angle B. The angle at O being 60, A + B == 180 - 60 120, and A = B = 60. Angles 1 and 2 (see Arithmetic), therefore, measure 60 252 MANUAL FOR TEACHERS each, and the whole angle contained between two adjoining sides of the hexagon, measures 120. After determining by this method the number of degrees in each angle of a regular hexa- gon, the pupils should be required to construct one, and to mark in each angle its contents in degrees, as in Fig. 13, verifying the result by using the protractor. 91. A careless scholar, measuring the number of degrees in each angle of. a regular pentagon (Arithmetic, Art. 1268), will sometimes read from the wrong row of numbers on the protrac- tor, getting the result 72, instead of the correct one of 108. As there are 72 in each division of the circumscribing circle, he will have no doubt of the correctness of his answer, unless he has been trained to estimate the size of an angle. In this case, he will see that each angle of a regular pentagon is obtuse, and, therefore, greater than 90. One method of calculating the number of degrees, is to divide the pentagon into five equal triangles, one of which is shown in Fig. 14. The angle at C is 72. The sides CA and A"^ CJ3, being radii, are equal ; which makes equal angles at A and JB, each of which is % of (180 - 72), or 54. Each of these angles is the half of one of the angles of the pen- tagon, so that these latter angles measure 108 each. 92. A circumscribed square touches the circle at four points, each side constituting a tangent. A tangent being perpendicular to the radius drawn to the point of contact, the square may be constructed by drawing perpendiculars to two diameters inter- secting at right angles, using the triangle or the protractor for the purpose. The ingenious pupil will discover other ways ; drawing, for instance, at each extremity of the two diameters, a line parallel to the intersecting diameter, by means of the ruler and the triangle ; etc. No method should be permit- ted that merely approximates accuracy, such as determining that a line is parallel or perpendicular by the eye alone. The NOTES ON CHAPTER SIXTEEN 253 r FIG. 15. average class will contain many members intelligent enough to pass upon the correctness of a given method, and they should be called upon to give reasons for any criticisms they may have to offer. / In circumscribing some polygons, a hex- agon for instance, many pupils prefer locat- ing points X and Y, instead of using the triangle and the ruler to draw a tangent XF. After dividing the circumference (Fig. 15) into six equal parts at 1, 2, 3, 4, 5, and 6, they draw a diameter from 1 to 4. Through 2 and 6 they draw a secant XA } making MX and MA each equal to the radius. Through 3 and 5 a second secant is drawn, and NY and NB are also made equal to the radius. A line drawn through X and Y will form one side of the circumscribed hexagon. One extremity of this side can be determined by producing the diameter 3(76 to F, and the other by a line through 2C"5. NOTE. A secant is a line that cuts a circle at two points; a tangent is a line that touches a circle' at one point. 93. The smallest number of triangles into which a pentagon can be divided, is three (Fig. 16). The three angles of each triangle contain 180, making the sum of angles 1-9, 540. Since l = A, 2 + 4: = , 6 + 7 - C, 8 = D j 3 -(- 5 = E, the sum of the five equal angles (A, B, C, D, and E) of a regular pentagon = 540, and each equals 108. This is the result that was found in 91 by another method. 94. The hexagon is divisible, as above, into four triangles, containing 180 X 4, or 720; making each angle 720 -s- 6, or 120. 95. A quadrilateral is divisible into 2 triangles; a pentagon, 254 MANUAL FOB TEACHERS into 3; a hexagon, into 4; a heptagon, into 5; an octagon, into 6, being 2 triangles less in each case than the number of sides in the polygon. 96. The number of degrees in each angle of a regular octagon = [180 x(8-2)]-s-8. 97. At each end of the 2-inch line, draw a 2-inch line at an angle of 108. At the farther extremity of each of those lines draw a line at an angle of 108. These last lines meet at an angle of 108 if the work is correctly done, and are each two inches long to the point of their intersection. 98. A line drawn to each extremity of the base at an angle of 60 will form an equilateral triangle. Use angles of 90 for the square, 108 for the pentagon, 120 for the hexagon, nearly 129 for the heptagon, 135 for the octagon, 140 for the non- agon, etc. The first line should be placed near the cen- ter of the bottom of the paper to give room for the successive polygons. See Fig. 17. Bright pupils will calculate the necessary angles and continue to construct polygons as far as the space will permit. 99. By drawing the diameters AB and XY (Fig. 18), the inscribed square will be divided into four triangles, while the circumscribed square contains eight, being double the area of the inscribed square. 1270. These problems are given to en- able the pupils to learn how to bisect lines, erect perpendiculars, construct angles, etc., by means of the ruler and the compasses, and incidentally to learn a number of geometrical facts. The use of other instruments is unnecessary, and should, therefore, not be tolerated. FIG. 17. NOTES ON CHAPTER SIXTEEN 255 1. The distance between the centers = 1-J- in. + 1 in. 2. 1^- in. 1 in. 4. The line XY joining the two points of intersection of the equal circles (Fig. 19), bisects the line AS connecting the centers. The radii AX, EX, AY, BY are each 2 inches long. 5. The previous exercise should lead the pupils to see the steps neces- sary to the construction of the re- quired triangle. The 3-inch base AB is first drawn. The next requirement is to find a point X (Fig. 19) 2 inches from A and from B. A circle of 2-inches radius with B as a center, will contain every point that is 2 inches from B. A similar circle with A as a center, contains every point 2 inches from A. The intersections, X and Y, are each 2 inches from both points A and B. Using either one as a vertex, draw lines to A and B, forming the required triangle. Authorities differ somewhat as to the advisability of requiring pupils to employ circles rather than arcs in geometrical problems. While it may be better at first to use circles, the point to be finally reached is the employ- ment of the shortest lines possible. This should not be inconsistent with the acquirement of geometrical knowledge. A scholar should know after a very few exercises that each point of an arc is 2 inches from the center, as well as he does when he draws the entire circle. In his later construction of an isosceles triangle, the pupil should know that the vertex is above (or below) the center of the base. For this reason the first arc need not be a very long one. The second should.be still shorter. 6. Fig. 19 will suggest the necessary steps. Using each end of the 3-J-inch. base as a center, and with a radius of 4 inches, draw two circles. Draw lines corresponding to XA and XB for the required triangle. Placing the ruler on X and Y will give the perpendicular, which should not extend below the base. If arcs are used, the upper intersection determines the position of the vertex. A lower intersection is used to determine the 256 MANUAL FOR TEACHERS direction of the perpendicular. The pupil will gradually learn that while a definite radius, 4 inches, is required to locate the vertex of the triangle, intersecting arcs, each of 3 inches or 5 inches or any other radius, will serve to locate the second point, used with the vertex to determine the direction of the perpendic- ular. The point of their intersection may be above the base or below it, according to convenience. A point below secures greater accuracy, by being probably at a greater distance from the vertex than one above is likely to be. 7. This is a variation of 6, but without directions as to length of sides. If the perpendicular is correctly drawn, it will bisect the base. See Exercise 48, Art. 1263. The compasses should be employed to determine the equality or inequality of the seg- ments of the base. 8. The same procedure is required as in 7, except that the sides of the triangle are not drawn. The bisecting line should be extremely short. 9. With a 2-inch radius, draw intersecting arcs, using as centers the two extremities of a 2-inch line. -11. Either side may be used as the base ; and different pupils should use a different base, although the greatest number will probably take the longest side. Using the 2-inch side as a base, draw from one end, as a center, an arc with a radius of 1 inch ; and from the other, an arc with a radius of 1^- inches. The intersection of the arcs will be the vertex of the required triangle. With the 3-inch line as a base, the radius of the arcs will be 2 inches and 2-J- inches, respectively. Besides employing different bases, the pupils should use oblique lines and vertical lines as bases, and the vertex in some instances should be below the base ; etc. 12. A scholar should be permitted to discover for himself that the intersection of the 2-inch arcs will be at the center of the 4-inch base. After endeavoring, also, to make a triangle whose NOTES ON CHAPTER SIXTEEN 257 sides shall measure 1, 2, and 3 inches, respectively, he will understand that the third side of a triangle must be shorter than the combined lengths of the other two. 13. If the pupil, in drawing arcs to locate the bisecting line, employs the radius used in drawing the circle, he will discover that one intersection will take place at the center of the circle. This will lead him to see that only one set of intersecting arcs is necessary the one beyond the circle, the center of the circle serving as a second point. The teacher should be in no hurry to inform the pupil using two sets of arcs, that a single set will be sufficient ; knowledge that comes in some other way than merely by direct telling, is apt to last longer. . 14. Before dividing a sector (Figs. 8 and 9) into two equal parts, some scholars may consider it necessary to draw the chord. Permit them to do so at first. 15. The pupils have already learned that a line drawn from the vertex of an isosceles triangle to the center of the base, is per- pendicular to the base. The method employed in bisecting a line is practically to consider it the common base of two isosceles triangles having their vertices on opposite sides of the base, although the equal sides of the triangles are not drawn. This bisecting line is perpendicular to the assumed base. The pupils have probably discovered that the perpendicular line bisecting the chord in 13, would, if produced, pass through the center of the circle. From this previous knowledge, they will doubtless be able to answer the last question of 15. The scholars that have drawn the chord in dividing the sector in 14 into two equal parts, will be likely to see that the radius CX (Fig. FIG. 20. 20) is perpendicular to the bisected chord AS. 16 shows that a perpendicular that bisects a chord, AE (Fig. 20), also bisects the arc AXB. 258 MANUAL FOR TEACHERS It will readily be seen by the pupils that an arc AXB can be bisected without drawing the chord AB. 17. This problem is the same as 8. The drawing, however, should show a longer line, and one that does not cut the given line, the requirement being that the perpendicular be drawn to the latter. A perpendicular drawn to a horizontal or to an oblique line from below it, would be correct ; although it might not be accepted as satisfactory if the more common wording of the problem were employed : Erect a perpendicular at the mid- dle point of a line. 18 requires no explanation. 19. Bisecting one of the four divisions of a circumference gives the dividing point between two one-eighths. A ruler placed upon this point and on the center of the circle will indicate the location of another. The distance between two points can be ascertained by the compasses, which can then be used to locate the remaining two dividing points. The lines used to divide a circumference should not be too long. The thoughtless pupil sometimes fails to see, when he employs his compasses to measure the distance between two points on a circumference, that he is measuring the chord of an arc, and not the arc itself. It may be necessary for the teacher to explain this in connection with 25. 22. Unless the pupils have learned in their drawing work the method of erecting a perpendicular at the end of a line, some of them may experience a little difficulty in solving this problem. One way of drawing the 'circumscribed square is to construct on A Y (Fig. 18) an isosceles triangle A2Y equal to ACY. Produce 2A to 1, making A\ equal to 2 A. Produce %Y to 3 in the same manner. Lines from 1 through* X, and 3 through JB, will intersect at 4, which completes the square. 24. The object of 23 and 24 is to lead the pupils to see again that the side of an inscribed hexagon is equal in length to the radius of the circle. NOTES ON CHAPTER SIXTEEN 259 26. The four triangles will together constitute a circumscribed equilateral triangle. 27. Draw lightly an arc less than 90 in length. After cut- ting off 60, darken this portion. No radii ^ . should be drawn. To construct an angle of 60, draw an in- definite line AB (Fig. 21). With any con- venient radius, as AM, draw an arc MN. With M as a center, and using the same radius, cut the arc at X. This makes MX an arc of 60. From A draw a line through X. 28. An arc of 30 is constructed by the bisection of an arc of 60. To construct an angle of 30, draw AB (Fig. 21) and the arc MN\ and cut off MX, as in 27. With the same, or any other convenient radius, and using M and X as centers, draw two arcs intersecting on the right of MX. A line drawn from A through this intersecting point will make with AB an angle of 30. 29. To construct an angle equal to 60 + 30 (or 30 + 60), bisect the arc XM (Fig. 21), and from the bisecting point, which is 30 from M, lay off an additional 60. A line drawn from A through this point will make with AB an angle of 90. 30. One method of erecting a perpendicular at the end of a line is given in 29. A somewhat similar method consists in prolonging the arc, and marking off at Q and R two divisions of 60 each. Using these two points as centers, draw arcs intersecting at F. From D draw a line through F. The line DF bisects the arc QE, which makes the angle 260 MANUAL FOR TEACHERS 31. The bisection of VP gives an angle of 45. Do not draw DF. The recommendation so often made as to getting a variety of drawings from the various members of the class, applies to these problems. The erection of a perpendicular at the end of a vertical line should call forth at least four different results. The perpendiculars may be erected at either end, and may run to the right or to the left. In constructing an angle of 45, a greater variety is possible. The first line may be horizontal, verti- cal, or oblique ; the second line may start from either end ; and it may extend above or below, to the right or to the left; the lines forming the angles need not be of the same length on all papers, nor need both of them be of the same length on any one paper. 22J- = -J- of 45; 135 = 90 + 45 ; 15 = of 30; 75- 60 +15. 33. The preceding problem should give a hint as to the method of solving the present one. In ^j 32, a circle was drawn with A as a center and AX as a radius. This circle cut XY (Fig. 23) in M. To erect a perpen- ^ dicular at A, the center of the circle, the ^~~ A M extremities X and M of the diameter XM were used as centers to draw arcs intersecting at J. A perpendicular was then drawn from A through J. This problem does not furnish a circle, nor is one necessary. The line XY is given, and the point A at which the perpendic- ular is to be erected. With A as a center, and a radius equal to AX, cut off AM. This gives us the diameter of the circle employed in the preceding problem. It will be noticed that a second set of intersecting arcs on the opposite side of XM is not needed, the point A answering instead. This problem amounts to the bisection of an arc of 180, of which only the chord XM is drawn, A being the center of the circle. 34. The : NOTES ON CHAPTER SIXTEEN 261 FIG. 24. 34. The first two divisions of this problem are reviews of parts of 17 and 30. The erection of a perpendicular at a point between the end and the center, is a variation of the preceding problem. Let 8 (Fig. 24) be the required line and A the point. With A as a center, and any convenient radius, lay off the points X and M, which will be equidistant from A. Using X and M as centers, draw arcs intersecting at J. Draw the required perpendicular from A through J. In problem 33 the point M was located at a distance from A, equal to AR\ and while greater accuracy is obtained by having the points Jkf and X as far apart as possible, the present method is suggested to show that the only essential requirement as to their location is to have them equidistant from A. 35. The base need not be a horizontal line. 36. Proceed as in the construction of a right-angled triangle, base 2 inches, perpendicular 2 inches ; but do not 8 draw the hypotenuse. With E and H as cen- ters (Fig. 25), and a radius of 2 inches (BP or PIT), draw arcs intersecting at 0. BO and HO will form the remaining sides. To construct the rectangle, PH (Fig. 25) should be made 3 inches long, BP measuring 2 FlG - 25 - inches. With B as a center, and a radius of 3 inches, draw an arc. Intersect this by a second arc drawn with H as a center, and a radius of 2 inches. From the point of intersection, draw lines to B and H. Some scholars may prefer to erect a perpendicular at each end of the base, etc. 38. Draw a base line, WV (Fig. 26), 3 inches long. At any convenient point, M, erect a perpendicular, MP, 2-J inches long. -H 262 MANUAL FOE, TEACHERS At its extremity P, draw the perpendicular PO. With W as a center and a radius of 3 inches, locate the point X on the line PO ; TF^will be the second side of the rhombus. On PO draw XN 3 inches ; and connect NV. This last line should measure 3 inches if the work is properly done. O FIG. 26. To- construct a 3-inch rhombus containing an angle of 60, draw WV\ at W. construct an angle of 60, and make WX 3 inches. At X or at V, draw a 3-inch line making an angle of 120 with XW or VW, etc. ; or, using JTand Fas centers, and with a radius of 3 inches, draw arcs intersecting at N] draw NXwdNV. 39. Proceed as in the first problems of 38 ; but make IFF and XN each 4 inches long. 40. See BPH (Fig. 25) ; draw EH. Some pupils, preferring to commence with a horizontal base, calculate the number of degrees in each angle at the base, A- (180 - 90), or 45. To each end of a base line of any length (Fig. 27), they draw a line making with the base an angle of 45. If one angle is 120, the angles at the base will be 30 each; if one is 135, the angles at the base will measure 22 each. FlG - 27 - The method shown in Fig. 27 requires the construction of two angles ; and is, therefore, not so direct as drawing two equal lines meeting at the given angle. 41. Erec NOTES ON CHAPTER SIXTEEN 263 B FIG. 28. 41. Erect a 3-inch perpendicular at the middle of a 3-inch line. A perpendicular, AB (Fig. 28), bisects the angle at A of the equilat- eral triangle. Erect a 3-inch perpen- dicular at .Z?, on an indefinite line CD\ at A, construct two angles of 30 each, as shown in the figure ; draw AX and A Y, forming the required equilateral triangle AXY. Test the work by meas- uring XY. 42. To construct a scalene triangle of the required dimen- sions, erect the 3-inch perpendicular at a point not in the center of the 3-inch base, nor at either extremity. To construct the obtuse-angled triangle, produce the 3-inch base by a dotted line, and erect the 3-inch altitude at some point outside of the base. 43. This problem may puzzle the pupils at first. If the tri- angle were isosceles, XYM, for instance, there would be no diffi- culty. The solution of the prob- lem consists practically in making an isosceles triangle, although the line 'XM\& not drawn. With X as a center (Fig. 29), and XY &$ a radius, the point M is located (withou 4 drawing the arc shown in the figure). Using M and Y as centers, and with any con- venient radius, arcs are drawn intersecting at It. XR, gives the direction of the altitude. 44. The given triangle in this case will be XMZ (Fig. 29). Produce ZM indefinitely towards Y, and with XM as a radius, cut the base at Y; etc. JTFis, of course, not drawn ; MY should be a light line or a broken line ; etc. X R M FIG. 29. 264 MANUAL FOR TEACHERS 45. This is the same problem, in another form. Let YZ (Fig. 29) be the given line, and X the given point, from which a perpendicular is to be let fall upon YZ. In 43-44, the arc described with X as a center, passes through one extremity of the line. The pupils should see that this is not essential, and that it would be very inconvenient in the case of a very long line. All that is necessary is to intersect the line "YZ at two points sufficiently far apart to secure accuracy. 47. Each angle will measure 60; the arc on which it stands measures 120 ; an angle at the circumference, therefore, is measured by one-half the arc intercepted by its sides. Apply this to the regular pentagon (Arithmetic, Art. 1268). The angle at the top of the figure stands on an arc containing 72 + 72 + 72, or 216 (made up of three arcs) ; it contains, therefore, % of 216, or 108. See Fig. 30. An angle of a square stands on an arc of 180 ; its contents are, therefore, \ of 180, or 90 ; etc. 48. The number of degrees in each angle of a regular hexagon is 120 (94, Art. 1269). From each end of the given line AE (Fig. 31) draw a line equal in length to AB, and making with it an angle of 120 ; etc. While this is not the best way to construct a regular hexagon, it gives the pupils an opportunity to try the general method on a polygon of six sides. 49. For the construction of an octagon, the angles at A, , C, etc., should measure 135 (Problem 31). NOTES ON CHAPTER SIXTEEN 265 By a later problem, the pupils will learn how to find the center of the circumscribing circle after two sides of the octagon are drawn ; for the present, however, they should be required to repeat the construction of the angle of 135 at the required number of corners. The work should be as accurate as possible, considering the necessary imperfections of the tools employed. 50. A right angle whose vertex is at the circumference, sub- tends an arc of 180, or a semi-circumference. 51. By means of a ruler, placed on the center of the circle, mark points A and B (Fig. 32) the extremi- ties of a diameter (not drawn). The arc A MB is one-half of the circumference, or 180 ; and the angles X, Y, and Z, whose sides XA and XB, YA and YB, ZA and ZB, subtend this arc, are each measured by one-half of it, and contain, therefore, -J- of 180, or 90. 52. The hypotenuse of an inscribed right- angled triangle is always the diameter of the circle. In Fig. 32 draw the diameter ACB (or any other). Bisect the arc A XYZB, which gives the vertex of the required trian- gle. Connect this point with the points A and B. NOTE. It may be necessary to have the scholars understand that all the vertices of an inscribed polygon must lie in the circumference ; an inscribed triangle, therefore, has three vertices in the circumference. 53. The diameter of a circle having a radius of 2 inches, will be the diagonal of the inscribed square. The location of the remaining two corners is easily determined. The square may also be constructed by drawing 4-inch diagonals bisecting each other at right angles ; etc. The rhombus will consist of two 3-inch equilateral triangles having a common base. . 54. Every angle being on the circumference, each will be measured by one-half the intercepted arc, and the sum of the 266 MANUAL FOR TEACHERS three angles will contain one-half the number of the degrees con- tained in the three arcs ; i.e., -J- of 360. 55. See the rhombus in 53. 56. The two triangles will be equal in all respects, as will those in 57. 58. From 56 and 57 the pupils should learn that two triangles are equal in every respect (a) when two sides and the included angle of one are equal to two sides and the included angle of the other, each to each ; and (&) when two angles and the included side of one are equal to two angles and the included side of the other, each to each. From 59, they should learn that when two triangles have the three sides of the one equal to three sides of the other, each to each, they are equal throughout. From the present problem, they should learn that any number of triangles can be constructed having their corresponding angles, equal each to each, but whose corresponding sides are unequal, each to each. 60. Impossible, because the sum of three angles of 75 each is greater than 180. 61. The radius of each circle must be one-half of the hypote- nuse, or 1-J- in. See Problem 52. Any inscribed triangle having one side the diameter of the circle will answer the conditions of the first case. The 2-inch base of the second triangle is obtained by taking A as a center (Fig. 32), and with a radius of 2 inches, drawing an arc at, say, Y; ^4 Y" will be the required base, YB the per- pendicular, and AB the hypotenuse. Using a radius of 1-J- inches, as above, will give a perpendic- ular, AX, of 1-J- inches, etc. NOTE. The term base does not necessarily imply a horizontal line; nor perpendicular, a vertical one. 63. To construct an angle at F(Fig. 33) equal to the angle at X, take JTas a center and any convenient radius, and draw the arc Aa\ then take ]Tas a center and the same radius, and draw the arc Oc. The angle X is measured by the arc A, the NOTES ON CHAPTER SIXTEEN 267 length of whose chord is determined by means of the compasses. Lay off the same on Cc, making the arc CD equal to the arc AB. An angle formed by drawing a line from Y through D Y \* FIG. 33. will be equal to the angle at X, as each angle is measured by an arc of the same number of degrees. See note to Problem 19, Art, 1270. 64. Draw the first line at any angle ; and by the preceding problem, draw the two intermediate lines, making, with the horizontal line, angles equal to that made by the first line. The pupil will have no difficulty with the oblique line at the other extremity of the horizontal line. 65. Proceed as in the second part of 64, using a horizontal 3-inch line for the diagonal, drawing from one end a 2-^-inch oblique line running upward at any angle, and from the other end a similar line running downward at the same angle. The lines divide the 3-inch diagonal into 5 equal parts, each part measuring -| inch. 66. The scholars should gradually learn that it is unnecessary to measure the oblique lines, Km and Ln (Fig. 34). In setting off the divisions JT1, 1 2, etc., La, ab, etc., it will be found unnecessary to use a definite measure ; all that is required is that they be equal. This equality is obtained by opening the compasses slightly and marking off the divisions with them. There is no need of completing the FIG. 34. 268 MANUAL FOR TEACHERS rhomboid, nor of drawing the lines 4a, 3&, 2c, etc. Place the ruler on 4 and on a, and use a short line on KL to mark off one division; etc. It is expected that the teacher will not give this method at the outset. To draw a line exactly f of an inch in length, a construction line, 3 inches long, KL (Fig. 34), is drawn lightly, also Km and Ln of indefinite length. Only the first division is required on Km. Placing the ruler on 1 and d, mark off on KL -f- inch, and darken this portion One-seventh of a 5-inch line = -7- inch. 67. When the base of a right-angled triangle measures 3 inches, and the perpendicular measures 2 inches, the hypotenuse will measure V3 2 + 2 2 = V9 + 4 = Vl3 inches. 68. A line Vl3 inches long is drawn by constructing a right- angled triangle having a 3-inch base and a 2-inch perpendicular. The hypotenuse is the required line. A base of 2 inches and a perpendicular of 1 inch give a hypot- enuse of V5 inches. A hypotenuse of 4 inches and another side of 3 inches give a remaining side of A/7 inches. For the construction of this tri- angle, see Problem 61. 69. The side opposite the angle of 30 is one-half as long as the side opposite the angle of 90. 71. The chord of an arc of 60 is 2 inches long ; the chord of an arc of 180 is the diameter, and is 4 inches long. The chord of an arc of 300 is 2 inches long. 72. The perpendicular "erected" at one end of the chord should be turned downwards if the chord is drawn above the cen- ter of the circle. The perpendicular and the diameter meet at the circumference. See Problem 61. 73. The triangle will retain its shape, because only one tri- angle can be drawn with sides of a given length. Problem 59. The rectangle will not retain its shape, because an indefinite vari- NOTES ON CHAPTER SIXTEEN 269 ety of parallelograms can be constructed having their corre- sponding sides equal, each to each. See Exercise 66, Art. 1266. 77. From Problem 15, the pupils have learned that a perpen- dicular bisecting a chord, passes through the center. If there are two lines passing through the center, the latter must be located at their intersection. 78. To inscribe a circle in any triangle, draw lines bisecting the three angles. See problem 28. The intersection of these three lines will be the center of the circle. In practice, only two lines are drawn ; but the third serves to test the accuracy of a pupil's work. 79. The sides of an inscribed triangle are chords of the cir- cumscribing circle. In Problem 77, we have learned that the intersection of two perpendiculars that bisect chords, locates the center of the circle. With this center, and with a radius equal to the distance from the center to the vertex of any angle of the triangle, draw the circle. 80. The larger the circle the better for beginners. The average scholar may consider it necessary to draw two adjacent chords, but every purpose will be served by cutting the circum- ference by three short lines to mark the boundaries of two adjoining arcs. The bisection of these arcs by perpendiculars will locate the center of the circle. 82. Use a cup to draw the arc. Divide it at random into two parts. Bisect each, as above. C 83. The altitude is 2 in. m *~ ^"*>^ 4? Problem 70. 84. Experienced draughts- men save time by changing the radius as infrequently as possi- ble. To construct a square on a 3-inch line AE (Fig. 35), take a radius of 3 inches, and with A as a center draw the arc Bm. With B as a center, mark off En = 60. Bisect n, 270 MANUAL FOR TEACHERS using the same radius to obtain the arcs intersecting at o, with B and n as centers. BX ' 30 ; place the compasses on JT, and cut the first arc at (7, which makes XC 60, and EG 90. O is the third corner of the square. Using B and C as centers, and with a radius of 3 inches, draw arcs intersecting at D, the fourth corner of the square. R m r 86. We have found in Problem 26, that constructing an equilateral tri- angle on each side of an inscribed equilateral triangle gives a circum- scribed equilateral triangle. Locate m, n, and o, 120 apart. With each as a center, and with a radius equal to mo, draw arcs intersecting at A, B, and C. Do not draw the triangle mno, shown in the figure. 87. See Problem 41, second part. 89. See Arithmetic, Art. 1089 (Fig. 2). 90. Construct a right-angled triangle having a base and a perpendicular measuring 2 inches and 3 inches, respectively. The hypotenuse will be the side of the required square. Do not construct squares on the other two sides. 91. The hypotenuse is 3 inches, etc. See Problem 68. 92. The square constructed on the hypotenuse of a right- angled triangle whose base and perpendicular measure 3 inche each, will be double the area of a 3-inch square. When the 3-inch square is constructed, measure the length the diagonal (without drawing it) ; and on a line of this length as a base, construct the required square. 93. Four. See Fig. 36. 94. Nine ; sixteen ; twenty-five. 95. 1, 1, 2 inches. 96. 3, 4J, 6 inches. 97. The radius of the second circle is 2 inches. NOTES ON CHAPTER SIXTEEN 271 FIG. 37. 98. A square described on the base of an isosceles right- angled triangle will be one-half the size of the square described on the hypotenuse. A a + a = 2 a ; a = ^A. The area of an equilateral triangle whose base is YZ, is one-half that of one whose base is XY. The area of a circle whose radius is YZ (or XZ\ is one-half that of a circle whose radius is XY. NOTE. If the square A in Fig. 37 is drawn above XY instead of below it, the ratio of the large square to that of each small one will be more apparent. Construct an isosceles right-angled triangle having a hypote- nuse of 2 inches (Problem 52). The base of this triangle will be the radius of the required circle. The area of a circle of 2 inches radius 2 2 X TT = 4 IT. The radius of the other circle = V2 ; its area is (V2) 2 X TT 27T, which is one-half of 4?r. 99. The side of the required triangle measures V2 inches. See 98. 100. Cab (Fig. 38) is one-sixth of an inscribed regular hex- agon, and CAB is one-sixth of a circum- A X B scribed regular hexagon. Calling the radius of the circle 2 inches, the altitude CX si the large triangle is 2 inches. Since ab is 2 inches, ax = 1 inch ; Ca = 2 inches. In the right-angled triangle Cax, ~Cx + ax = "&? ; ~C~x + 1 r= 4 ; ~Cx = 4 - 1 = 3 ; Ox - VS. The areas of the two triangles will be proportional to the squares of their respective altitudes. ~Gx ~ 3 and CX* 4. That is, the area of the larger triangle is 1-|- times the area of the smaller ; hence the area of the circum- scribed hexagon is 1-|- times the area of the inscribed hexagon. The area of the circumscribed square is double the area of the 272 MANUAL FOE, TEACHERS inscribed square ; and the area of the circumscribed equilateral triangle is four times the area of the inscribed equilateral triangle. 1271. 3. At the middle point JTof the perpendicular of the right-angled triangle (Fig. 39), cut Xm parallel to the base. He-arranging the two parts gives a rectangle whose dimensions are 4 inches and 1-J- inches. Figs. 40 and 41 show how the FIG. 39. FIG. 40. FIG. 41. other two triangles are divided to make rectangles of the same dimensions. 4-6. See 56-59, Art. 1270. 7. The two triangles have two sides of one, AC and EC, equal to two sides of the other, CE and CD; and the angle ACE equal to its opposite angle DCE] hence the third side, DE, of one triangle is equal to the third side, AE, of the other. 9. The area of the larger triangle is four times that of the smaller. 11. Another method of dividing a line into equal parts. See Problem 66, Art, 1270. 1273. Much interest is added to this work by employing the method here given, in calculating heights and distances in the neighborhood of the school. A comparison between the results obtained by calculations and those obtained by actual measure- ments, will be useful in teaching the pupils the necessity of great accuracy in their preliminary work. 2. The hypotenuse of each triangle represents a ray of light from the sun, etc. NOTES ON CHAPTER SIXTEEN 273 3. Fr=110ft. 4. CD:DE:: CB:BA> 50 : 90 : : 1200 : BA; 5^1 = 2160 ft. The width of the river = 2160 ft. - 100 ft. = 2060 ft. Avis. 5. -02)= 00-1)0 = 6 ft. -4 ft. = 2 ft.; JEH=m ft. + 3 ft. = 180 ft ; AH= 120 ft. ; AB = 120 ft. +4 ft. = 124 ft. 6. The two triangles are similar, because the angles of one are equal to the angles of the other. Angle D angle A = 90. The two angles at C are vertical angles, and, therefore, equal to each other. The remaining angle B must be equal to the remaining angle E, because the sum of the angles in each triangle is the same. O:DE:: AC'.AB* 3.25 : 5 : : (12 - 3.25) : AB. 7. 10J- : (10 + 195 + 15) : : (12 - 4|) : hf. hf = 157-j- ft. ; fi = 157J ft. + 4 ft. = 162 ft. Ans. 8. RP:RN::PQ:MN\:(12Q + G):::MN. MN= 84ft. Ans. 9. The tree is 3 ft. X 36, or 108 ft. high. Ans. AO=AB, because angle 0= angle B 45. 10. A0= AB. A line CB makes an angle of 45 with AC, angle A = 90 ; angle ABC= 45. NOTE. In the succeeding problems, the triangle and the protractor may be employed when necessary. 1274. 1. The circumference of the circle, or 360 = 4 in. X 3.1416. The arc of 60 is % of the circumference; the arc of 120 = of .it; etc. 2. Draw the circle and the various chords. The chord of 60 chord of 300 radius 2 inches ; the chord of 180 = diameter = 4 inches. If the pupils draw the chord of 120, ZX (Fig. 42), they will find that it bisects the ra- dius YO, making MO I inch. CX 2 inches ; therefore MX == Vf^l = V3 = 1.732 + ; ZX-= 1.732 + in. x 2 = 3.464 + in. chord of 120 = chord of 240. 274 MANUAL FOE TEACHERS Diagrams should be employed, unless the pupils can do satis- factory work without them. Since the arc of 180 is three times as long as the arc of 60, the more careless members of the class may jump to the conclusion that the chord of 180 is three times as long as the chord of 60. This mistake cannot be made if the circle is constructed, and the chords are drawn and measured. 5. See Exercise 98, Art. 1269. 6. Each side of the hexagon measures 1 inch ; the perimeter 1 inch X 6 = 6 inches. Ans. Circumference = 2 in. X 3.1416 = 6.2832 inches. Ans. 7. The pupil should use the ruler to ascertain the length of the apothem, which is about - in. It can be calculated as follows : Cb (Fig. 38) is the radius = 1 inch ; bx one-half the side of the inscribed hexagon = \ inch ; Cx apothem. Ox = Cb - bx=l-\ = .75 ; Cx = V775 - .866+, or nearly f 8. The base of each triangle measures one inch, so that AB (Fig. 43) = 3 in. AX (apothem) x = in. nearly. 9. The base of each triangle measures about f in. so that the base of the rectangle (the half FlG ' 48 * perimeter) will be nearly 3 inches and the apothem about -j-f in. Area about 3 X |-f sq. in. about 2if sq. in. 10. AB X AX= 3 X I = 2f . Ans. 2f sq. in. 11. See 9. 12. The perimeter of a 16-sided polygon will be greater than that of an octagon. The 16-sided polygon will have the greater apothem. 13. As the number of sides increases, the perimeter approaches more and more closely the circumference, 6.2832 inches; and the apothem approaches the radius, 1 inch. NOTES ON CHAPTER SIXTEEN 275 14. The base of the rectangle will be 3.1416 inches, one-half the perimeter (circumference) ; the apothem will be 1 inch (radius). Area = 3.1416 sq. in. Ans. 15. One-half the circumference = 2 X 3.1416 = 6.2832, multi- plied by the radius, 2, gives answer in square inches, 12.5664 sq. in. 16. 78.54 sq. in. Ans. 17. 3.1416 sq. in. Ans. 18. 314.16 sq. in. -* 6 = Ans. 19. Subtract from the area of the outer circle, 113.0976 sq. in., the area of the inner circle, 28.2744 sq. in. Ans. 84.8232 sq. in. 1282. Right prisms, cylinders, etc., are meant when the word oblique is not used. 1. See Arithmetic, Art. 818, Problem 20. The upper, squares need not be drawn, as only the convex surface is required. 2. Three rectangles and two triangles. See 1. 3. Three rectangles, each 3 inches high, bases 1, 1, and 2 inches, respectively. 5. A hollow paper cylinder, without bases, can be opened out into a rectangle whose base is the circumference of the base of the cylinder, and whose height is the altitude of the cylinder. 6. The slant height of a square pyramid is the distance from the apex to the center of one side of the base. This problem requires the pupil to draw, side by side, four isosceles triangles, the base of each being 2 inches and the altitude 3 inches. After constructing the first (Fig. 44), by erect- ing a 3-inch perpendicular at the center of a 2-inch base, he should use B 11 and A as centers, and radii equal to B"B* and AB" to locate B'". On AB' construct a third triangle, using A and B 1 as centers and the radii previously given. Upon one side of this triangle construct a fourth. FIG. 44. 276 MANUAL FOR TEACHERS The pupils can discover for themselves the method of drawing geometrically the required convex surface. The altitude should be carefully measured. It will be equal in length to the perpendicular of a right-angled triangle whose hypotenuse is the slant height of a pyramid, 3 inches ; and whose base is the distance from the center of one edge of the base of the pyramid (the foot of the slant height) to the foot of the alti- tude, 1 inch. Ans. V inches = 2.83 inches nearly about 2-J-f inches. 7. The area of each convex face of a regular pyramid is found by multiplying one side of the base by one-half the slant height; therefore, the area of all the faces forming the convex sur- face, is obtained by multiplying the sum of all the sides of the base, that is, the perimeter of the base, by one-half the slant height. 8. The pupil requires the slant height in order to proceed as in 6 ; and he should obtain it by drawing it rather than by cal- culating it. He should be able to see that the altitude is a line drawn from the vertex to the center of the base, and that its foot is 1 inch distant from the foot of the altitude. Constructing a right-angled triangle whose base is 1 inch, and whose perpen- dicular is 3 inches, will give a hypotenuse equal to the required slant height. Some scholars will bring in a prism whose slant height is 3 inches. The mistake should be pointed out, but not the mode of correcting it; and a pyramid of the required dimensions should be insisted upon. 1283. When the diameter of the base of a cone is 2 inches, the arc 13 DC, which forms the circumference when folded, measures 2 IT inches, or 6.2832 inches. NOTE. The Greek letter TT (pi) represents 3.1416. 9. The semi-circumference of paper = Sir inches, which is the circumference of base of cone. The diameter of base of cone 3 TT inches -5- TT 3 inches. The radius of base = 1 J inches. NOTES ON CHAPTER SIXTEEN 277 The slant height = radius of paper 3 inches diameter of base of cone = twice radius of base of cone. 10. The area of any sector of a circle = radius X % length of arc. The arc when folded becomes the circumference of base, and the radius becomes the slant height ; so that convex surface = -- circumference X slant height circumference X slant height. 11. Length of arc of 90 ^ circumference = \ of 6?r inches = 1^ TT inches. This is the circumference of the base of the cone, which makes the diameter = 1^ TT inches -*- IT 1^ inches. The slant height is 3 inches. Length of arc of 60 = % circumference ^ of 6 IT inches = ir inches. The diameter of the base of the cone TT inches -f- ?r = 1 inch ; slant height, 3 inches. 12. The circumference of the base of the cone 3?r. This equals the length of the arc of the required sector. If the slant height is 5 inches, the circumference of which the sector is a part = 10 TT. The arc of the sector is, therefore, y 3 ^ of the circumfer- ence ; and its length is T %- of 360 - 108. 13. XY represents the base of the pyramid; and AC, its altitude. The slant height of two faces, AM (Fig. 45), is the hypotenuse of a right-angled triangle, base 1| in., perpendicular 4 in. Using this as the perpendicu- lar of a new triangle, with a base M Y, gives as the hy- potenuse AY (Figs. 45 and 46), one of the edges. To draw the development, take AY as a radius, and draw an arc. On this lay offsucces- FIG. 45. 278 MANUAL FOR TEACHERS sive chords of 3 in., 2 in., 3 in., and 2 in., connecting the extremity of each with the center A. These four triangles ^ _- " ti c r Sin. PIG. 46. constitute the convex faces of the pyramid. On one of them, construct a rectangle 3 inches by 2 inches for the base. 14. In this pyramid, AC (Fig. 45) measures 12 inches, CM measures ^ of 18 inches, or 9 inches, making AM, one slant height, = Vl44 + 81 inches = 15 inches. The other slant height, drawn to the center of OY, = Vl44 + 25 inches = 13 inches. The pupils should be encouraged to construct stout paper pyramids and cones of required dimensions, making the necessary calculations themselves. The previous fourteen problems will present no difficulty whatever to pupils that are interested. As examples in dry calculation, they may prove somewhat tiresome. Many scholars will, of themselves, construct models of solids much more complicated than the foregoing. 1284. If there are no Solids at hand, the pupils should con- struct a supply of paper ones, or make them of modeling clay, turnips, etc. Drawings are not sufficient for effective instruc- tion. NOTES ON CHAPTER SIXTEEN 279 'he short method of ascertaining the convex surface should not be given until 18. Each of the convex faces is a trapezoid, whose parallel sides measure 4 inches and 8 inches, respectively, the altitude being 10 inches. 16. Two inches apart, draw two parallel lines, AB and CD (Fig. 47), measuring 1 inch and 2 inches, respectively, a 2-inch perpendicular, XY, connecting the middle point of each. Draw AC and BD, and produce them until they meet in M. With this as a center, and MC as a radius, draw an arc ; on which three other chords equal to CD are laid off, as in Fig. 46. With M as a center and a radius MA, lay off another arc, on which chords are laid off equal to AB\ etc. See Arithme- tic, Art. 1296. 17. The entire surface = convex surface + 4 sq. ft. + 9 sq. ft. 18. The pupils can readily understand this rule, using the frustums of problems 15 and 17 as illustrations. 19. The pupil should first locate the apex of the cone. This, he can do by following the method shown in 16 ; making AB (Fig. 47) 2 inches; CD, 1% inches; and AC, 2 inches. This makes MA 5 inches, the slant height of the whole cone. The circumference of the base of the cone = 2|-7r. The slant height, 5 inches, is the radius of the required sector, its circumference being 10 IT.. 2^-Tr, the length of the arc of the sector, being one- fourth of 10 TT, shows that the required sector is a quadrant. 20. The number of square inches in the convex surface = [cir- cumference (perimeter) of upper base (9 X 3.1416) + circumfer- ence of lower base (6 X 3.1416)] X \ slant height (2). Adding to this, the area of the bottom (3 2 ) 9 sq. in. X 3.1416 (Art. 1124, 2) gives the number of square inches of material required. 280 MANUAL FOR TEACHERS 22. Circumference of upper base = 6 X 3.1416 Circumference of lower base 10 X 3.1416 One-half sum = 8 X 3.1416 Multiplying by slant height gives 48 X 3.1416 Add to this the area of the upper base, 9 X 3.1416 And the area of the lower base, 25 X 3.1416 Total in square yards, 82 X 3.1416 or ( [(3 + 5) X 6] + 9 + 25) x 3.1416. 23. EA : EC : : 6 : 8 x:x + 9 :: 6:8 54 # = 27. Ans. 27 ft. The slant height of the whole cone 27 ft. +9 ft. =36 ft. 24. The convex surface of the whole cone = J (8 X 3.1416 X 36) sq. ft. ; of the part cut off = (6 X 3.1416 X 27) sq. ft. 1287. A sphere (a croquet ball, for instance) and a hemisphere should be used to illustrate these problems. On the plane face of the latter can be drawn the lines AD, FG, HI, CI, etc. ; while on the curved face can be drawn II YI, FXG, etc. 25. \ of 25,000 miles. 26. IH= chord of 60 of the great circle = radius of the great circle = 4000 miles. IB = \ of IH= 2000 miles. 27. The diameter, HI, of the small circle is -|- diameter FQ of the great circle; the circumference HYI=\ of 25,000 miles, or 12,500 miles. 28. The length of a degree of longitude on the 60th parallel is about one-half of the length of a degree on the equator. (See Art. 995, Problem 10.) NOTES ON CHAPTER SIXTEEN 281 29. On the plane face of the hemisphere suggested above (Art. 1287), draw diameters AD and FG at right angles ; and 45 from G, a chord NM parallel to FG. (This chord will not bisect AC.) As MCG = 45, WCM= 45 ; and the triangle ' WCM is a right-angled isosceles tri- angle, and WM* --= \CM' i J WM= .7071 OAT. (See Art. 995, Problem 12.) If CG measures 4000 miles, CM= 4000 miles, and WM= V8,000,000 = 2828.4 miles. 1289. The pupils have already learned that the volume of a rectangular prism is equal to the area of the base multiplied by its altitude ; these problems are intended to show that the same is true of all prisms and of the cylinder (6). 1292. 8. The volume of the frustum is obtained by deduct- ing the volume of the part cut off from the volume of the whole pyramid (Problem 7). The rule is given later. 10. Fig. 49 gives the method of calculating the slant height. The illustration shows a sec- tion of the frustum formed by passing a plane through the center of the frustum perpendicular to the base. The center of the upper base of the frustum of a right pyr- amid is directly above the center of the lower base, so that a perpendicular let fall from A will fall on CD at a point X, 5 in. from C. In the right-angled triangle AXC, AX= altitude 12 in. = 144 + 25-169; Sin. FIG. 49. SUPPLEMENT DEFINITIONS, PRINCIPLES, AND RULES A Unit is a single thing. A Number is a unit or a collection of units. The Unit of a Number is one of that number. Like Numbers are those that express units of the same kind. Unlike Numbers are those that express units of different kinds. A Concrete Number is one in which the unit is named. An Abstract Number is one in which the unit is not named. Notation is expressing numbers by characters. Arabic Notation is expressing numbers by figures. Eoman Notation is expressing numbers by letters. Numeration is reading numbers expressed by characters. The Place of a Figure is its position in a number. A figure standing alone, or in the first place at the right of other figures, expresses ones, or units of the first order. A figure in the second place expresses tens, or units of the second order. A figure in the third place expresses hundreds, or units of the third order ; and so on. A Period is a group of three orders of units, counting from right to left. RULE FOR NOTATION. Begin at the left, and write the hun- dreds, tens, and units of each period in succession, filling vacant places and periods with ciphers. 11 SUPPLEMENT RULE FOR NUMERATION. Beginning at the right, separate the number into periods. Beginning at the left, read the numbers in each period, giving the name of each period except the last. ADDITION Addition is finding a number equal to two or more given num- bers. Addends are the numbers added. The Sum, or Amount, is the number obtained by addition. PRINCIPLE. Only like numbers, and units of the same order can be added. EULE. Write the numbers so that units of the same order shall I e in the same column. Beginning at the right, add each column separately, and write the sum, if less than ten, under the column added. When the sum of any column exceeds nine, write the units only, and add the ten or tens to the next column. - Write the entire sum of the last column. SUBTRACTION Subtraction is finding the difference between two numbers. The Subtrahend is the number subtracted. The Minuend is the number from which the subtrahend is taken. The Kemainder, or Difference, is the number left after subtracting one number from another. PRINCIPLES. Only like numbers and units of the same order can be subtracted. The sum of the difference and the subtrahend must equal the minuend. RULES. I. Write the subtrahend under the minuend, placing units of the same order in the same column. DEFINITIONS, PRINCIPLES, AND RULES 111 Beginning at the right, find the number that must be added to the first figure of the subtrahend to produce the figure in the corre- sponding order of the minuend, and write it below. Proceed in this way until the difference is found. If any figure in the subtrahend is greater than the corresponding figure in the minuend, find the number that must be added to the former to produce the latter increased by ten ; then add one to the next order of the subtrahend and proceed as before. II. Beginning at the units column, subtract each figure of the subtrahend from the corresponding figure of the minuend and- write the remainder below. If any figure of the subtrahend is greater than the corresponding figure in the minuend, add ten to the latter and subtract ; then,- (a) add one to the next order of the subtrahend and proceed as before ; or, (b) subtract one from the next order of the minuend and proceed as before. MULTIPLICATION Multiplication is taking one number as many times as there are units in another number. The Multiplicand is the number taken or multiplied. The Multiplier is the number that shows how many times the multiplicand is taken. The Product is the result obtained by multiplication. PRINCIPLES. The multiplier must be an abstract number. The multiplicand and the product are like numbers. The product is the same in whatever order the numbers are multiplied. RULE. Write the multiplier under the multiplicand, placing units of the same order in the same column. Beginning at the right, multiply the multiplicand by the number of units in each order of the multiplier in succession. Write the IV SUPPLEMENT figure of the lowest order in each partial product under the figure of the multiplier that produces it. Add the partial products. To multiply by 10, 100, 1000, etc, RULE. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier. DIVISION Division is finding how many times one number is contained in another, or finding one of the equal parts of a number. The Dividend is the number divided. The Divisor is the number contained in the dividend. The Quotient is the result obtained by division. PRINCIPLES. When the divisor and the dividend are like num- bers, the quotient is an abstract number. When the divisor is an abstract number, the dividend and the quotient are like numbers. The product of the divisor and the quotient, plus the remainder, if any, is equal to the dividend. RULE. Write the divisor at the left of the dividend with a line between them. Find how many times the divisor is contained in the fewest fig- ures on the left of the dividend, and write the result over the last figure of the partial dividend. Multiply the divisor by this quotient figure, and write the product under the figures divided. Subtract the product from the partial dividend used, and to the remainder annex the next figure of the dividend for a new dividend. Divide as before until all the figures of the dividend have been used. If any partial dividend will not contain the divisor, write a cipher in the quotient, and annex the next figure of the dividend. If there is a remainder after the last division, write it after the quotient with the divisor underneath. DEFINITIONS, PRINCIPLES, AND RULES V FACTORING An Exact Divisor of a number is a number that will divide it without a remainder. An Odd Number is one that cannot be exactly divided by two. An Even Number is one that can be exactly divided by two. The Factors of a number are the numbers that multiplied to- gether produce that number. A Prime Number is a number that has no factors. A Composite Number is a number that has factors. A Prime Factor is a prime number used as a factor. A Composite Factor is a composite number used as a factor. Factoring is separating a number into its factors. To find the Prime Factors of a Number. RULE. Divide the number by any prime factor. Divide the quotient, if composite, in like manner ; and so continue until a prime quotient is found. The several divisors and the last quotient will be the prime factors. CANCELLATION Cancellation is rejecting equal factors from dividend and divisor. PRINCIPLE. Dividing dividend and divisor by the same num- ber does not affect the quotient. GREATEST COMMON DIVISOR A Common Factor (divisor or measure) is a number that is a factor of each of two or more numbers. A Common Prime Factor is a prime number that is a factor of each of two or more numbers. The Greatest Common Factor (divisor or measure) is the largest number that is a factor of each of two or more numbers. Numbers are prime to each other when they have no common factor. Vi SUPPLEMENT The greatest common divisor of two or more numbers is the product of their common prime factors. PRINCIPLES. A common divisor of two numbers is a divisor of their sum, and also of their difference. A divisor of a number is a divisor of every multiple of that number ; and a common divisor of two or more numbers is a divisor of any of their multiples. To find the Common I rime Factors of Two or More Numbers. RULE. Divide the numbers by any common prime factors, and the quotients in like manner, until they have no common factor ; the several divisors are the common prime factors. To find the Greatest Common Divisor of Numbers that are Easily Factored. RULE. Separate the numbers into their prime factors ; the product of those that are common is the greatest common divisor. To find the Greatest Common Divisor of Numbers that are not Easily Factored. RULE. Divide the greater number by the less; then divide the last divisor by the last remainder, continuing until there is no remainder. The last divisor is the greatest common divisor. If there are more than two numbers, find the greatest common divisor of two of them; then of that divisor and another of the numbers until all of the numbers have been used. The last divisor is the greatest common divisor. LEAST COMMON MULTIPLE A Multiple of a number is a number that exactly contains that number. A Common Multiple of two or more numbers is a number that is a multiple of each of them. The Least Common Multiple of two or more numbers is the smallest number that is a common multiple of them. DEFINITIONS, PRINCIPLES, AND RULES vii PRINCIPLES. A multiple of a number contains all the prime factors of that number. A common multiple of two or more numbers contains each of the prime factors of those numbers. The Least Common Multiple of two or more numbers contains only the prime factors of each of the numbers. To find the Least Common Multiple of Two or More Numbers, RULE. Divide by any prime number that is an exact divisor of two or more of the numbers, and write the quotients and undivided numbers below. Divide these numbers in like manner, continuing until no two of the remaining numbers have a common factor. The product of the divisors and remaining numbers is the least common multiple. FRACTIONS Fraction is one or more of the equal parts of anything. The Unit of a Fraction is the number or thing that is divided into equal parts. A Fractional Unit is one of the equal parts into which the num- ber or thing is divided. The Terms of a Fraction are its numerator and its denominator. The Denominator of a fraction shows into how many parts the unit is divided. The Numerator of a fraction shows how many of the parts are taken. . A fraction indicates division ; the numerator being the divi- dend and the denominator the divisor. The Value of a Fraction is the quotient of the numerator divided by the denominator. Fractions are divided into two classes Common and Decimal, A Common Fraction is one in which the unit is divided into any number of equal parts. A common fraction is expressed by writing the numerator above the denominator with a dividing line between. Vill SUPPLEMENT Common fractions consist of three principal classes Simple, Compound, and Complex, A Simple Fraction is one whose terms are whole numbers. A Proper Traction is a simple fraction whose numerator is less than its denominator. An Improper Fraction is a simple fraction whose numerator equals or exceeds its denominator. A Compound Fraction is a fraction of a fraction. A Complex Fraction is one having a fraction in its numerator, or in its denominator, or in both. A Mixed Number is a whole number and a fraction written together. The Reciprocal of a Number is one divided by that number. The Eeciprocal of a Fraction is one divided by the fraction, or the fraction inverted. PRINCIPLES. Multiplying the numerator or dividing the de- nominator multiplies the fraction. Dividing the numerator or multiplying the denominator divides the fraction. Multiplying or dividing both terms of a fraction by the same number does not alter the value of the fraction. Eeduction of fractions is changing their terms without altering their value. To reduce a Fraction to Higher Terms, EULE. Multiply both numerator and denominator by the same number. To reduce a Fraction to its Lowest Terms, EULE. Divide both terms of the fraction by their greatest common divisor. A fraction is in its lowest terms when the numerator and the denominator are prime to each other. DEFINITIONS, PRINCIPLES, AND RULES IX To reduce a Mixed Number to an Improper Fraction, RULE. Multiply the whole number by the denominator ; to the product add the numerator ; and place the sum over the denom- inator. To reduce an Improper Fraction to a Whole or to a Mixed Number, RULE. Divide the numerator by the denominator. A Oommon Denominator is a denominator common to two or more fractions. The Least Oommon Denominator is the smallest denominator common to two or more fractions. To reduce Fractions to their Least Oommon Denominator. RULE. Find the least common multiple of all the denomi- nators/or the least common denominator. Divide this multiple by the denominator of each fraction, and multiply the numerator by the quotient. ADDITION OF FRACTIONS PRINCIPLE. Only like fractions can be added. RULE. Reduce the fractions, if necessary, to a common denom- inator, and over it write the sum of the numerators. If there are mixed numbers, add the fractions and the whole numbers separately, and unite the results. SUBTRACTION OF FRACTIONS PRINCIPLE. Only like fractions can be subtracted. RULE. Reduce the fractions, if necessary, to a common denom- inator, and over it write the difference between the numerators. If there are mixed numbers subtract the fractions and the whole numbers separately, and unite the results. MULTIPLICATION OF FRACTIONS RULE. Reduce whole and mixed numbers to improper frac- tions ; cancel the factors common to numerators and denomina- tors, and write the product of the remaining factors in the numer- ators over the product of the remaining factors in the denominators. SUPPLEMENT DIVISION OF FRACTIONS RULES. I. Reduce whole and mixed numbers to improper fractions. Reduce the fractions to a common denominator. Divide the numerator of the dividend by the numerator of the divisor. II. Invert the divisor and proceed as in multiplication of frac- tions. To reduce a Complex Fraction to a Simple One. RULES. I. Multiply the numerator of the complex fraction by its denominator inverted. II. Multiply both terms by the least common multiple of the denominators. DECIMALS A Decimal Fraction is one in which the unit is divided into tenths, hundredths, thousandths, etc. A Decimal is a decimal fraction whose denomination is indi- cated by the number of places at the right of the decimal point. The Decimal Point is the mark used to locate units. A Mixed Decimal is a whole number and a decimal written together. A Complex Decimal is a decimal with a common fraction written at its right. To write Decimals. RULE. Write the numerator ; and from the right, point off as many decimal places as there are ciphers in the denominator, prefixing ciphers, if necessary, to make the required number. To read Decimals. RULE. Read the numerator, and give the name of the right- hand order. PRINCIPLES. Prefixing ciphers to a decimal diminishes its value. DEFINITIONS, PRINCIPLES, AND RULES xi Removing ciphers from the left of a decimal increases its value. Annexing ciphers to a decimal or removing ciphers from its right does not alter its value. To reduce a Decimal to a Common Fraction. KULE. Write the figures of the decimal for the numerator, and 1, with as many ciphers as there are places in the decimal, for the denominator, and reduce the fraction to its lowest terms. To reduce a Common Fraction to a Decimal, RULE. Annex decimal ciphers to the numerator, and divide it by the denominator. To reduce Decimals to a Common Denominator, RULE. Make their decimal places equal by annexing ciphers. ADDITION AND SUBTRACTION OF DECIMALS Decimals are added and subtracted the same as whole numbers. MULTIPLICATION OF DECIMALS RULE. Multiply as in whole numbers, and from the right oj the product, point off as many decimal places as there are decimal places in both factors. DIVISION OF DECIMALS RULE. Make the divisor a whole number by removing the decimal point, and make a corresponding change in the dividend. Divide as in whole numbers, and place the decimal point in the quotient under (or over) the new decimal point in the dividend. ACCOUNTS AND BILLS A Debtor is a person who owes another. A Creditor is a person to whom a debt is due. Xii SUPPLEMENT An Account is a record of debits and credits between persons doing business. The Balance of an account is the difference between the debit and credit sides. A Bill is a written statement of an account. An Invoice is a written statement of items, sent with merchan- dise. A Eeceipt is a written acknowledgment of the payment of part or all of a debt. A bill is receipted when the words, " Received Payment," are written at the bottom, signed by the creditor, or by some person duly authorized. DENOMINATE NUMBERS A Measure is a standard established by law or custom, by which distance, capacity, surface, time, or weight is determined. A Denominate Unit is a unit of measure. A Denominate Number is a denominate unit or a collection of denominate units. A Simple Denominate Number consists of denominate units of one kind. A Compound Denominate Number consists of denominate units of two or more kinds. A Denominate Fraction is a fraction of a denominate number. A denominate fraction may be either common or decimal, Eeduction of denominate numbers is changing them from one denomination to another without altering their value. Eeduction Descending is changing a denominate number to one of a lower denomination. RULE. Multiply the highest denomination by the number re- quired to reduce it to the next lower denomination, and to the prod- uct add the units of that lower denomination, if any. Proceed in this manner until the required denomination is reached. ^ DEFINITIONS, PRINCIPLES, WJ^JElULESQ^gJ^ Xlll Eeduction Ascending is changing a denominate number to one of a higher denomination. RULE. Divide the given denomination successively by the numbers that will reduce it to the required denomination. To this quotient annex the several remainders. To find the Time between Dates, RULE. When the time is less than one year, find the exact number of days; if greater than one year, find the time by com- pound subtraction, taking 30 days to the month. PERCENTAGE Per Gent means hundredths. Percentage is computing by hundredths. The elements involved in percentage are the Base, Bate, Per- centage, Amount, and Difference. The Base is the number of which a number of hundredths is taken. The Kate indicates the number of hundredths to be taken. The Percentage is one or more hundredths of the base. The Amount is the base increased by the percentage. The Difference is the base diminished by the percentage. To find the Percentage when the Base and Kate are Given, RULE. Multiply the base by the rate expressed as hundredths. To find the Kate when the Percentage and Base are Given, RULE. Divide the percentage by the base. To find the Base when the Percentage and Kate are Given, RULE. Divide the percentage by the rate expressed as hun- dredths. To find the Base when the Amount and Kate are Given. RULE. Divide the amount by 1 + the rate expressed as hun dredths. Xiv SUPPLEMENT To find the Base when the Difference and Kate are Given, RULE. Divide the difference by I the rate expressed as hun- dredths. PROFIT AND LOSS Profit or Loss is the difference between the buying and selling prices. In Profit and Loss, The buying price, or cost, is the base. The rate per cent profit or loss is the rate. The profit or loss is the percentage. The selling price is the amount or difference, according as it is more or less than the buying price. COMMERCIAL DISCOUNT Commercial Discount is a percentage deducted from the list price of goods, the face of a bill, etc. The Net Price of goods is the sum received for them. In Commercial Discount, The list price, or ^ . The face of the bill } 1S The rate per cent discount is the rate. The discount is the percentage. The list price diminished by the discount is the difference. In successive discounts, the first discount is made from the list price or the face of the bill ; the second discount, from the list price or face of the bill diminished by the first discount ; and so on. COMMISSION Commission is a percentage allowed an agent for his services. A Commission Agent is one who transacts business on com- mission. DEFINITIONS, PRINCIPLES, AND RULES XV A Consignment is the merchandise forwarded to a commission agent. The Consignor is v the person who sends the merchandise. The Consignee is the person to whom the merchandise is sent. The Net Proceeds is the sum remaining after all charges have been deducted. In buying, the commission is a percentage of the buying price; in selling, a percentage of the selling price; in collecting, a per- centage of the sum collected ; hence : The sum invested, or ) ,-, -, > is the base. The sum collected ) The rate per cent commission is the rate. The commission is the percentage. The sum invested increased by the commission is the amount. The sum collected diminished by the commission is the differ- ence. INSURANCE Insurance is a contract of indemnity. Insurance is of three kinds Fire, Marine, and Life, Fire Insurance is indemnity against loss of property by fire. Marine Insurance is indemnity against loss of property by the casualties of navigation. Life Insurance is indemnity against loss of life. The Insurance Policy is the contract setting forth the liability of the insurer. The Policy Face is the amount of insurance. The Premium is the price paid for insurance. The Insurer, or Underwriter, is the company issuing the policy. The Insured is the person for whose benefit the policy is issued, In Insurance, The policy face is the base. The rate per cent premium is the rate. The premium is the percentage XVl SUPPLEMENT TAXES A Tax is a sum of money levied on persons or property foi public purposes. A Personal, or Poll Tax, is a tax on the person. A Property Tax is a tax of a certain per cent on the assessed value of property. Property may be either personal or real. Personal Property consists of such things as are movable. Eeal Property is that which is fixed, or immovable. In Taxes, The assessed value is the base. The rate of taxation is the rate. The tax is the percentage. DUTIES Duties are taxes on imported goods. Duties are either Specific or Ad Valorem. A Specific Duty is a tax on goods without regard to cost. An Ad Valorem duty is a tax of a certain per cent on the cost of goods. In Ad Valorem Duties, The cost of the goods is the base. The rate per cent duty is the rate. The ad valorem duty is the percentage. INTEREST Interest is the sum paid for the use of money. The Principal is the sum loaned. The Amount is the sum of the principal and interest. The Kate of Interest is the rate per cent for one year. The Legal Eate is the rate fixed by law. Usury is interest at a higher rate than that fixed by law. Simple Interest is interest on the principal only. DEFINITIONS, PRINCIPLES, AND RULES XV11 To find the Interest when the Principal, Time, and Eate are Given. RULE. Multiply the principal by the rate expressed as hun- dredths, and this product by the time expressed in years. To find the Time when the Principal, Interest, and Eate are Given, RULE. Divide the given interest by the interest for one year. - To find the Eate when the Principal, Interest, and Time are Given, RULE. Divide the given interest by the interest at one per cent. To find the Principal when the Interest, Eate, and Time are Given, RULE. Divide the given interest by the interest on $ 1. To find the Principal when the Amount and Time and Eate are Given, RULE. Divide the given amount by the amount of $ 1. INTEREST BY ALIQUOT PARTS. To find the Interest for Years, Months, and Days. RULE. Find the interest for one year and take this as many times as there are years. Take the greatest number of the given months that equals an aliquot part of a year and find the interest for this time. Take aliquot parts of this for the remaining months. In the same manner find the interest for the days. The sum of these interests will be the interest required. To find the Interest when the Time is Less than a Year, RULE. Find the interest for the time in months or days that will gain one per cent of the principal. Find by aliquot parts, as in the first rule, the interest for the remaining time. The sum of these interests will be the interest required. XViii SUPPLEMENT INTEREST BY Six PER CENT METHOD. To find the Interest at 6%. EULE. Por Years.: Multiply the principal by the rate ex- pressed as hundredths, and that product by the number of years. For Months : Move the decimal point two places to the left, and multiply by one- half the number of months. For Days i Move the decimal point three places to the left, and multiply by one-sixth the number of days. To find the interest at any other rate per cent, divide the in- terest at 6% by 6, and multiply the quotient by the given rate. To find Exact Interest. RULE. Multiply the principal by the rate expressed as hun- dredths, and that product by the time expressed in years of 365 days. ANNUAL INTEREST Annual Interest is interest payable annually. If not paid when due, annual interest draws simple interest. To find the Amount Due on a Note with Annual Interest, when the Interest has not been Faid Annually, RULE. Find the interest on the principal for the entire time, and on each annual interest for the time it remained unpaid. The sum of the principal and all the interest is the amount due. COMPOUND INTEREST Compound Interest is interest on the principal and on the un- paid interest, which is added to the principal at regular inter- vals. The interest may be compounded annually, semi-annually, quarterly, etc., according to agreement. To find Compound Interest, RULE. find the amount of the given principal for the first period. Considering this as a new principal, find the amount of DEFINITIONS, PKINCIPLES, AND RULES xix it fw the next period, continuing in this manner for the given time. Find the difference between the last amount and the given principal, which will be the compound interest. PARTIAL PAYMENTS Partial Payments are part payments of a note or debt. Each payment is recorded on the back of the note or the written obligation. UNITED STATES EULE. Find the amount of the principal to the time when the payment or the sum of two or more payments equals or exceeds the interest. From this amount deduct the payment or sum of payments. Use the balance then due as a new principal, and proceed as before. MEKCHANTS' EULE. Find the amount of an interest-bearing note at the time of settlement. Find the amount of each credit from, its time of payment to the time of settlement ; subtract their sum from the amount of the principal. BANK DISCOUNT Bank Discount is a percentage retained by a bank for advanc- ing money on a note before it is due. The Sum Discounted is the face of the note, or if interest-bear- ing, the amount of the note at maturity. The Term of Discount is the number of days from the day of discount to the day of maturity. The Bank Discount is the interest on the sum discounted for the term of discount. The Proceeds of a note is the sum discounted less the bank dis- count. Problems in bank discount are calculated as problems in interest. XX SUPPLEMENT In Bank Discount, The sum discounted is the principal. The rate of discount is the rate of interest. The term of discount is the tirr^e. The bank discount is the proceeds. EXCHANGE Exchange is making payments at a distance by means of drafts or bills of exchange. Domestic Exchange is exchange between places in the same country. Foreign Exchange is exchange between different countries. Exchange is at par when a draft, or bill, sells for its face value ; at a premium when it sells for more than its face value ; at a discount when it sells for less. The cost of a sight draft is the face of the draft increased by the premium, or diminished by the discount. The cost of a time draft is the face of the draft increased by the premium, or diminished by the discount, and this result diminished by the bank discount. To find the Cost of a Draft. RULE. Find the cost of $ 1 of the draft; multiply this by the face of the draft. To find the Pace of a Draft. RULE. Divide the cost of the draft by the cost of $1 of the draft. EQUATION OF PAYMENTS Equation of Payments is a method of ascertaining at what time several debts due at different times may be settled by a single payment. The Equated Time of payment is the time when the several debts may be equitably settled by one payment. The Term of Credit is the time the debt has to run before it becomes due. DEFINITIONS, PRINCIPLES, AND RULES XXI The Average Term of Credit is the time the debts due at different times have to run, before they may be equitably settled by one payment. To find the Equated Time of Payment when the Terms of Credit begin at the Same Date, RULE. Multiply each debt by its term of credit, and divide the su7n of the products by the sum of the debts. The quotient will be the average term of credit. Add the average term of credit to the date of the debts, and the result will be the equated time of payment. To find the Equated Time when the Terms of Credit begin at Different Dates, RULE. Find the date at which each debt becomes due. Select the earliest date as a standard. Multiply each debt by the number of days between the standard date and the date when the debt becomes due, and divide the sum of the products by the sum of the debts. The quotient will be the average term of credit from the standard date. Add the average term of credit to the standard date, and the result will be the equated time of payment. RATIO Ratio is the relation one number bears to another of the same kind. The Terms of the ratio are the numbers compared. The Antecedent is the first term. The Consequent is the second term. The antecedent and consequent form a couplet. PRINCIPLES. See Fractions. PROPORTION A Proportion is formed by two equal ratios. The Extremes of a proportion are the first and last terms. The Means of a proportion are the second and third termSc SUPPLEMENT PRINCIPLES. The product of the means is equal to the prod- uct of the extremes. Either mean equals the product of the extremes divided by the other mean. Either extreme equals the product of the means divided by the other extreme. KULE FOR PROPORTION. Represent the required term by x. Arrange the terms so that the required term and the similar known term may form one couplet, the remaining terms the other. If the required term is in the extremes, divide the product of the means by the given extreme. If the required term is in the means, divide the product of the extremes by the given mean. PARTNERSHIP Partnership is an association of two or more persons for busi- ness purposes. The Partners are the persons associated. The Capital is that which is invested in the business. The Assets are the partnership property. The Liabilities are the partnership debts. To find the Profit, or Loss, of Each Partner when the Capital of Each is Employed for the Same Period of Time, RULE. Find the part of the entire profit, or loss, that each partner s capital is of the entire capital. To find the Profit, or Loss, of Each Partner when the Capital of Each is Employed for Different Periods of Time, RULE. Find each partner s capital for one month, by multi- plying the amount he invests by the number of months it is employed ; then find the part of the entire profit, or loss, that each partner s capital for one month is of the entire capital for one month. DEFINITIONS, PRINCIPLES, AND RULES INVOLUTION A Power of a number is the product obtained by using that number a certain number of times as a factor. The Pirst Power of a number is the number itself. The Second Power of a number, or the Square, is the product of a number taken twice as a factor. The Third Power of a number, or the Cube, is the product of a number taken three times as a factor. An Exponent is a small figure written a little to the right of the upper part of a number to indicate the power. Involution is finding any power of a number. To find the Power of a Number, RULE. Take the number as a factor as many times as there are units in the exponent. EVOLUTION A Eoot is one of the equal factors of a number. The Square Eoot of a number is one of its two equal factors. The Cube Eoot of a number is one of its three equal factors. Evolution is finding any root of a number. Evolution may be indicated in two ways : by the Radical Sign, ~v , or by & fractional exponent. The Index of a root is a small figure placed a little to the left of the upper part of the radical sign, to indicate what root is to be found. In expressing square root, the index is omitted. In the fractional exponent, the numerator indicates the power to which the number is to be raised ; the denominator indicates the root to be taken of the number thus raised. To find the Square Eoot of a Number, EULE. Point off in periods of two figures, commencing at units. Find the greatest square in the first period and place the root in the quotient. Subtract this square from the first period, and bring down the next period. XXIV SUPPLEMENT Multiply the quotient figure by two, and use it as a trial divisor. Place the second figure in the quotient, and annex it also to the trial divisor. Then 7nultiply the figures in the trial divisor by the second quotient figure, and subtract. Bring down the next period, and proceed as before until the square root is found. To find the Square Eoot of a Fraction. RULE. Reduce the fraction to its simplest form, and find the square root of each term separately. To find the Cube Koot of a Number. RULE. Point off in periods of three figures each, beginning at units. Find the greatest cube in the first period and place the root in the quotient. Subtract this cube from the first period, and bring down the next period. Multiply the square of the first quotient figure by three and annex two ciphers for a trial divisor. Place the second figure in the quotient. Then, to the trial divisor add three times the prod- uct vf the first and second figures, also the square of the second. Multiply this sum by the second figure and subtract. Bring down the next period, and proceed as before until the cube root is found. To find the Oube Koot of a Traction. RULE. Reduce the fraction to its simplest form, and find the cube root of each term separately. STOCKS AND BONDS. Capital Stock is the money or property employed by a corpora- tion in its business. A Share is one of the equal divisions of capital stock. The Stockholders are the owners of the capital stock. The Par Value of stock is the face value. The Market Value of stock is the sum for which it may be sold. DEFINITIONS, PRINCIPLES, AND RULES XXV Stock is at a premium when the market value is above the par value ; at a discount, when below par. Bonds are interest-bearing notes issued by a government or a corporation. A Dividend is a percentage apportioned among the stockholders. A Stock Broker is a person who deals in stocks. Brokerage is a percentage allowed a stock broker for his services. In Stocks and Bonds, The par value is the base. The rate per cent premium, or discount, is the rate. The premium, \ discount, or > is the percentage. dividend J , . . v ( amount, or The market value is the \ 7 . ( difference. NOTES, DRAFTS, AND CHECKS. A Promissory Note is a written promise to pay a specified sum on demand, or at a specified time. The Face of a note is the sum named in the note. The Maker is the person who signs it. The Payee is the person to whom the sum specified is to be paid. The Indorser is the person who signs his name on the back of the note, thus becoming liable for its payment in case of default of the maker. An Interest-bearing Note is one payable with interest. If the words "with interest" are omitted, interest cannot be collected until after maturity. A Demand Note is one payable when demand of payment is made. A Time Note is one payable at a specified time. A Joint Note is one signed by two or more persons who jointly promise to pay. SUPPLEMENT A Joint and Several Note is one signed by two or more persons who jointly and severally promise to pay. In a joint note, each person is liable for the whole amount, but they must all be sued together. In the joint and several note, each is liable for the whole amount, and may be sued separately. A Negotiable Note is one that may be transferred or sold. It contains the words " or bearer," or " or order." A Non-negotiable Note is one not payable to the bearer, nor to the payee's order. The Maturity of a note is the day on which it legally falls due. A Draft, or Bill of Exchange, is a written order directing the payment of a specified sum of money. The Face of a draft is the sum named in it. The Drawer is the person who signs the draft. The Drawee is the person ordered to pay the sum specified. The Payee is the person to whom the sum specified is to be paid. A Sight Draft is one payable when presented. A Time Draft is one payable at a specified time. An Acceptance of a time draft is an agreement by the drawee to pay the draft at maturity, which he signifies by writing across the face of the draft the word " accepted " with the date and his name. A Check is an order on a bank or banker to pay a specified sum of money. ANSWERS. GRAMMAR SCHOOL ARITHMETIC. Page 202. 19. 332/ ff . 35. 11}. Page 208. 16. 44/ . 20. 193. 36. 8J. 6. $ 1.40. 17. 137f 37. 25J. 7. $ 2.00. 18. 131A. 38. 17 T V 8. 13J pounds. 19. 72J. Page 206. 39. 25 iV 9. $9.64. 20. 150}. 11. 27^V- 40. 10J. 10. $20. 21. 259A- 12. 54f 41. 23 rV 11. 360. 22. 201^- 13. 10&. 42. 19f. 12. $36. 23. 363}. 14. 30^. 43. 69 T V 13. $3.40. 24. 261&. 15. 97?V 44. 3 it' 14. $48. 25. 121f. 16. 36^. 45. 8 fa. 15. $ 9.30. 17. I 3 46. 7JL 18. 6 A- 47. 5JJ. Page 209. Page 204. 19. 78^. 48. 106^. 16. 16 marbles. 1. 07 9 Vtfg. 20. 8H- 49. 12f}. 17. 20 stamps. 2. 137}. 21. 382}. 50. 7|}. 18. 64} yards. 3. 176f. 22. 291f. 51. 2}|. 19. 35 cents. 4. 40}. 23. 109f. 52. 45|. 20. 2 days. 5. 35jV 24. 115f. 53. 25^. 6. 51H- 25. H- 54. 38A. 1. 258. 7. 61|. 26. 24f 55. 21 A- 2. 2449. 8. 205f. 27. 599f. 56. 3f|. 3. 228. 9. 85. 28. 860J. 57. 18 H- 4. 568. 10. 234}. 29. 85^. 58. iff. 5. 124. 11. 144}. 30. 4|f. 59. 7A- 6. 9920. 12. 151^. 60. 15}}. 7. 339. 13. 223}. 8. 199}. 14. 563^V Page 207. 1. 7 T V yards. 9. 293J. 15. 1096f. 31. 2|. 2. 57f gallons. 10. 1231f 16. 7491. 32. 17J. 3. 26 yards. 11. 31J. 17. 1256}$. 33. 6f. 4. 31 cents. 12. 14J. 18. 99|. 34. 5. $ 2.60. 13. 14i ANSWERS. 14. 14J. 45. 309 T V Page 216. 17. $53.07. 15. 16J. 46. 155,090. 41. 24. 18. $103.60. 16. 21-j-V 47. 6,108,5381. 42. 40. 19. $1591.62. 17. 50 T V. 48. 3,761,048f 43. 17. 20. $4879.77. 18. 45. 49. 25,01 If 44. 61. 19. 75&. 50. 28,5081. 45. 9. 20. 144^. 51. 69,763f. 46. 90. Page 225. 21. 246 T V 52. 598,686. 47. 6. 1. 40 days. 22. 302 T V 53. 3,600,925f 2. 1050 yards. 54. 21.436.213J. 1. $1283.35. 3. 25 sheep. Page 211. 2. $845.95. 4. 149 pounds. 1. 225,506,736. Page 214. 3. $2121.75. 5. 300 bushels. 2. 804,580,398. 55. 7,045,5061. 4. $857.62. 6. 1402 pounds 3. 561,276,891. 56. 8,325,706f. 5. $1247.80. 7. 3i6 cents. 57. 2,795,877^. 6. $769.89. 8. 66f cents. Page 212. 58. 6,285,440. 7. $530.25. 1. $609,340.37. 59. 15,389,967|. 8. $853.58. 2. $680,494.41. 60. 12,313,351if 9. $1250.93. Page 226. 3. $840,200.33. 10. $712.50. 9. 522 rabbits. 10. $2837.50. Page 213. Page 215. 25. 12f. 1. 6. Page 217. 26.- 19J. 2. 12. 1. $1115.02. Page 227. 27. 421 3. 12. 2. $505.44. 1. 180 hours. 28. 79J. 4. 24. 3. $1592.64. 2. 180 hours. 29. 80f. 5. 4. 4. $9263.05. 3. 7200 seconds, 30. 128f 11. 5. 5. $1526.25. 4. $12.75. 31. 101J. 12. 15. 6. $967.20. 5. $2.48. 32. 169 T V- 13. 13. 7. $7133.80. 6. 40 pints. 33. 215 T 4 T . 14. 31. 8. $1072.56. 7. 160 hf. pt. 34. 177 T V. 15. 5. 9. $23.76. 8. 20 packages. 35. lllf. 16. 11. 10. $58.24. 9. 25 cents. 36. 96J. 17. 17. 37. 163 J. 18. 25. 38. 266J. 19. 5. Page 218. Page 228. 39. 245|. 20. 8. 11. $92.88. 10. 6 gallons. 40. 348f 31. 16. 12. $989.90. 11. 18,000 sec. 41. 244. 32. 8. 13. $102.52. 12. 10,080 min. 42. 378 T V 33. 20J. 14. $133.38. 13. 800 minutes 43. 468 T 4 T . 34. 30J. 15. $140.60. 14. 512 quarts. 44. 257. 35. 10. 16. $34.98. 15. 757 ounces. ANSWERS. 16. , 14 Ib. 13 oz. 19. 857,375. 3. $550. 18. 748.311. 17. 24 hr. 54 mm. 20. 274,170. 4. $1120. 19. 1062.556. 18. 1008 hours. 21. 72,243. 5. $105. 20. 799.511. 19. 744 hours. 22. 109,243,616. 6. 5 cents. 20. 98 inches. 23. 115,242f. 7. $1536.65. Page 243. 21. 128,000 oz. 24. 16,610,750. 8. 49 T. 820 Ib. 41. 1.08. 22. 9280 ounces. 9. 22. 42. 400.4. 23. 750 pounds. 10. $80. 43. 780.8. 24. 15 cwt. Page 235. 11. $1.25. 44. 780.8. 25. $1. 25. 93/ft. 12. 25 cents. 45. 2.68. 26. | ton. 26. 2175 V 13. 60 yards. 46. 1.536. 27. 4 da. 16 hr. 27. 1025i. 47. 1.536. 28. 50 yards. 28. 296 7 VAV 48. 55.272. 29. $8.25. 29. 53QHf*. Page 238. 49. .485. 30. 32 cents. 30. 6S43#fc. 14. $22. 50. 4344. 31. | cwt. 31. 5565^ ? V 15. 10 papers. 61. 330. 32. 348 pints. 32. 15,168t 7 ff. 52. 960. 33. 2708. 1. $12.04. 53. 18. 34. 920}J. 2. $ 19.63. 54. 3801. Page 233. 35. 870ff 7 ff. 3. $107.52. 55. 59.13. 1. $134,083.44. 36. 2isHttt- 4. $8.29. 56. 725.56. 2. $108,350.78. 57. 376.68. 3. $ 27,437.70. 1. 39. 58. 334.508. 4. $56,284.66. 2. 11. Page 264. 59. 62.7. 5. $11,672.66. 3. 12. 5. $122.75. 60. 1.8. 6. $96,229.43. 4. 88. 5. 181. Page 245. 6. 77. Page 241. 31. $241.25. Page 248. 7. 690. 5. 122.995. 32. 1968.5 yd. 7. 1,000,342. 8. 17. 6. 293.056. 33. 35.4 pounds. 8. 854,822. 9. 11. 7. 59.556. 34. 1212.5 Ib. 9. 3,649,094. 10 123. 8. 404.529. 35. 2.64 tons. 10. 11,810,804. 9. 390.732. 36. $ 3364.02. 11. 77,472,965. 10. 300.417. 37. 19 pints. 12. 33,845,968. Page 236. 11. 480.507. 38. .125 peck. 13. 27,749,898. 11. 13. 12. 939.186. 39. $58.50. 14. , 28,338,290f. 12. 20. 13. 1180106. 40. $ 1751.56J. 15. 32,136,750. 14. 104.231. 41. 22j ft. 16 . 6,248,365J. Page 237. 15. 23.495. 42. 44 cents. 17 , 29,654,230. 1. $3. 16. 18.168. 43. 25 bu. 3 pk 18 . 63,257,616f. 2. $3. 17. 2359.925. 4qt. ANSWERS. Page 246. 1. 182 sq. in. 5. $15. 6. $48.75. Page 251. 1. 2, 43. Page 253 3. f 2. 153 sq. in. 7. $2.40. 2. 3, 29, 4. if. 3. 126 sq. in. 8. 672 sq. rods. 3. 2, 2, 2, 11. 5. f 4. 345 sq. in. 9. 30 sq. yd. 4. 2, 3, 3, 5. 6. -rV 13. 180 sq. in. 10. 4 sq. yd. 5. 7, 13. 7. f. 14. 144 sq. in. 6. 2, 2, 23. 8. f. 15. 192 sq. in. Page 249. 7. 3,31. 9. T V 16. 360 sq. in. 1. 22*1 8. 2,47. 10. f. 17. 450 sq. in. 2. 12 T V_. 9. 5, 19. 11. T v 18. 1419 sq. in. 3. 18iff. 10. 2, 2,,2, 2, 2, 3. 12. f. 19. 1180 sq. in. 4. 48ttJ. 11. 2, 2, 5, 5. 13. T V 20. 2205 sq. in. 5. 47 T V 12. 2, 2, 2, 3, 5. 14. f 6. 90JJ. 13. 2, 3, 5, 7. 15. T V Page 247. 7. 3A- 14. 2, 2, 2, 2, 3, 5. 1. 168 sq. ft. 8. 48if. 15. 2, 2, 2, 3, 3, 5. Page 255. 2. 255 sq.ft. 9. 103f. 16. 2, 2, 2, 2, 2, 2, 1. 120. 3. 209 sq. ft. 10. 99i|. 3, 3. 2. 900. 4. 345 sq. ft. 17. 2, 2, 2, 3, 5, 7. 3. 840. 5. 288 sq.ft. 18. 2, 2, 2, 2, 2, 2, 4. 240. 6. 348 sq. ft. Page 250. 2, 3, 3. 5. 600. 7. 186 sq.ft. 11. 8ii 19. 2, 2, 2, 2, 2, 2, 6. 48. 8: 186 sq. ft. 12. ejf. 3, 3, 3. 7. 360. 9. 300 sq. ft. 13. 38|. 20. 2, 2, 2, 2, 2, 3, 8. 231. 10. 300 sq. ft. 14. 17if. 3,7. 9. 720. 11. 423 sq.ft. 15. 19f 10. 420. 12. 444 sq.ft. 16. 16. . , 13. 308 sq. ft. 17. 24tf. 1. 16ff 14. 426 sq. ft. 18. 22ff Page 252. 2- 84JJ. 15. 386 sq. ft. 19. 6262|J. 1. f 3. eerffr. 16. 843 sq.ft. 20. 199JJ. 2. 11. 4. 31|. 17. 321f sq.ft. 21. 74. 3. f . 5. 61 T V 18. 320 sq. ft. 22. 3&. 4. f. 6. 2dfr. 19. 400 sq.ft. 23. 16Jf. 5. H- 7. 82J. 20. 150 sq. ft. 24. 318. 6- J. 8. lOlVHfr 25. 4 T V 7. . 9. 41 T %. Page 248. 26. 22f. 8. if 10. 23^. 1. 10,937.5 sq.ft. 27. 46f 9. f 2. 2f sq. yd. 28. 2437. 10. f. Page 256. 3. 41 sq. yd. 29. 5071. 11- M- 11. 8&. 4. 117 sq. m. 30. A- 12. f. 12. 9 Z V ANSWERS. 13. 85^. 18. $119.46. 8. 2221f 7. A- 14. 74fJ- 19. 58J miles. 9. 3H5 T V 8. 2f. 15. 79ff. 20. $ 6.90. 10. 15. 9. W- 16. 223 T 9 ff \. 11. 9. 10. T^V 17. 471&V Page 263. 12. 1*. 11. !& 18. 1T 9 6 (TO-- 1. 135. 13. 115. 12. f- 19. 43Jf. 2. 105|f. 14. 3J. 13. 1J. 20. 9 TiHhr- 3. 117#. 15. 37|. 14. 2|. 21. TV 4. 121f|. 16. 3. 15. I- 22. ?v 5. O1 Q 7 8 ^^ 97" 17. 42. 16. T 4 T- 23. 17& 6. 344^ 18. 3fl- 17. 6. 24. 6fJ. 7. 213|f|. 19. t 18. fc 25. 13*. 8. 168fff. 20. J. 19. TV 26. 674J. 9. 420 T 5 169,344. 15. 49f|. 6. 17,675 Ib. sec. 16. 212,175. 16. 42H- 7. 180 quarts. 5. 1 ft. 4 in. 17. 710,046. 17. 37tf. 8. 600 pints. 6. 18 gal. 2 qt. 18. 301,392. 18. 6SJ. 9. 79 pecks. 7. 21 bu. 3 pk. 19. 16Jf 10. 632 quarts. 8. 3 quarts. Page 308. 20. 228ft. 11. 62 Ib. 8 oz. 9. 1 wk. 3 da. 2; 42 gal. 2 qt* 21. Hi 12. 62 Ib. 8 oz. 10. 4 T. 912 Ib. 3. 637 gal. 2 qt. 22. 3i 13. 2 ft. 4 in. 4. 25,000 times. 23. 210f Page 313. 5. 9} yards. 24. 52i 1. 60 Ib. 15 oz. 1. 42 days. 6. 1501 min. 25. 15f. 2. 11 yards. 2. 12 days. 7. 66 days. 26. 3A- 3. 21 da, 13 hr. 3. 28 days. 8. 41 hr. 15 27. 248. 4. 28 min. 14 4. 56 men. rain. 28. 84*. sec. 5. 33 horses. 9. 1440 steps. 29. 4f 5. 4 T, 1314 Ib. 6. 18 lines. 10 ANSWERS. 7. 900 steps. Page 315. 5. $21.87. 12. 7J sq. ft. 8. 3072 bricks. 28. 12. 6. $7.47. 13. 750 sq. ft. 9. 11 hours. 29. 15. 7. $3.99. 14. 61J sq. ft. 10. 77 cents. 30. 13. 8. $ 22.26. 15. 308J sq. ft 11. 12 days. 31. 16. 9. $5.08. 32. 11. 10. $ 7.46. 1. 121. 33. 18. 11. $87.99. 2. 210J. 12. $6.30. 3. 88J, Page 314. 1. 1125 sq. ft. 13. $ 13.42. 4. 331*. 12. 72 days. 2. 432 sq. ft. 14. $64.97. 5. 139*. 13. 1. 3. 48 sq. yd. 15. $2.55. 6. 118f. 4. 12 sq. yd. 16. $56.25. 7. S01*. 1. 37 Ib. 5 oz. 5. 13 sq. yd. 17. $ 49.02. 8. 382 T V 2. 22 hr. 10 6. 44 sq. yd. 18. $ 3.99. 9. 247ff inin. 7. 7975 sq. ft. 19. $95.02. 10. 263f. 3. 49 T. 835 Ib. 20. $96.58. 11. 5*. 4. 69 bu. 3 pk. Page 317. 21. $23.20. 12. 13*f*. 5. 14 wk. 2 da. 1. $548.80. 22. $189. 13. 21*. 6. 21 yd. 2 ft. 2. $37.45. 23. $568. 14. 81f. 7. 73 minutes. 3. $72. 24. $225. 15. 12H- 8. 19 gal. 2 qt. 4. $187.60. 25. $589.60. 16. 11*. 9. 22 feet. 5. $137.10. 26. $51.30. 17. 10J. 10. 9 yards. 6. 11 pupils. 27. $ 62.40. 18. 21*. 11. 4 Ib. 9 oz. 7. $480. 28. $320. 19. H. 12. 3 gal. 2 qt. 29. $133.25. 20. 12f 13. 6hr.27min. Page 318. 30. $52.92. 21. 72* 14. 5 bu. 1 pk. 8. $ 333. 31. $ 13.50. 22. 30*. 15. 5min. 13 sec. 9. 3 words. 32. $2.75. 23. 40H- 16. 2 yd. 2 ft. 10. 60 cents. 33. $55. 24. 30f- 17. 1 ft. 11 in. 25. 32B. 18. 8 T. 1234 Ib. Page 319. Page 322. 26. 21*. 19. 2 wk. 6 da. 1. $27.56. 4. 37J sq. yd. 20. 4 yd. 2 ft. 3 2. $31.40. 5. 900 sq. ft. Page 324 in. 3. $18.99. 6. 2 sq. ft. 1. 2ff- 21. 4. 4. $45.52. 7. 3f sq. ft. 2. ITF. 22. 6. 8. 8f sq. ft. 3. *. 23. 8. Page 321. 9. 13 J sq. ft. 4. 6*. 24. 9. 1. $11.46. 10. 8100 sq. ft. 5. 1*. 25. . 7. 2. $ 29.10. 6. *i 26. 9. 3. $18.77. Page 323. 7. 3*, 27. 8. 4. $11.37. 11. 5062J sq. ft. 8. H ANSWERS. 11 9. 1. 13. 51 gal. 44. 27 bu. 1 pk. Page 330. 10. 15. 14. 162 gal. 3 qt. 5 qt. 3. 34 yd. 2 ft. 6 11. 3.679+. 15. 7 gal. 3 qt. 45. 93 Ib. 7 oz. in. 12. .005. Ipt. 46. 10 yd. 1 ft. 4. 40 da. 19 hr. 13. .004375. 16. 34 gal. 2 qt. 1 in. 55 min. 14. 3.78. 1 pt. 47. 54 gallons. 5. 186 gal. 3 qt. 15. 102.390561. 17. 17 gal. 1 qt. 48. 2 mi. 236 6. 22 hr. 30 min. 16. 19,700. 1 pt. rd. 28 sec. 18. 42 gal. 3 qt. 49. 39 gal. 3 qt. 7. 18 T. 862 Ib. Page 325. 19. 15 gal. 3 qt. 1 pt. 8. 9 wk. 1 da. 17. .125; 8. 20. 88 gal. 3 qt. 50. , 2 miles. 9. 53 mi. 294 rd. 18. 90. 1 pt. 10. 76 yr. 8 mo. 19. 1.36. Page 328. 11. 866 T. 899 Ib. 20. .26285. 1. .0015 T. 12. 140 Ib. 3 oz. Page 327. 2. sh da y- 13. 38 hr. 40 min. 1. T 3 T- 21. 633 inches. 3. 45 minutes. 2 sec. 2. TV 22. 7594 yards. 4. 45 minutes. 14. 180 gal. 3 qt. 3. ili- 23. 2391 quarts. 1 pt. 4. f. 24. 2507 ounces. Page 329. 15. 137 yd. 7 in. 5. it 25. 2271 inches. 5. .00625 day. 16. 194 mi. 183rd. 6. H- 26. 611 quarts. 6. $76.87. 17. 128 yr. 4 mo. 7. * 27. 192 feet. 7. 3 T. 1504 Ib. 21 da. 8. Hf 28. 510 pints. 8. $46.87. 18. 36 wk. 5 hr. 9. 1 29. 102 quarts. 9. 14 T. 1244 19. 22 hr. 24 min. 10. f 30. 34,369 Ib. Ib. 22 sec. 11. A- 31. 54,960 sec. 10. H yard. 20. 17 bu. 4 qt. 12. tttt 32. 827 hours. 11. .890625 bu. 33. 120 hours. 12. 3 pk. 6 qt. Page 331. Page 326. 34. 165 yards. 13. 75 cents. 21. 17 Ib. 7 oz. 1. 131 pints. 35. 40 ounces. 14. $6.86. 22. 3 bu. 2 pk. 5 2. 220 pints. 36. 52 yd. 4 in. 15. 2 qt. 1J pt. qt. 3. 128 pints. 37. 29 Ib. 11 oz. 16. T 4 A- 23. 7 yd. 2 ft. 7 4. 129 pints. 38. 22 bu. 3 pk. 17. Hi in. 5. 279 pints. 1 qt. 18. 1 bu. 1 pk. 24. 10 da. 9 hr. 20 6. 252 pints. 39. 6 da. 35 min. 1 qt. min. 7. 77 pints. 40. 2 T. 972 Ib. 19. .885 day. 25. 3 gal. 2 qt. 1 8. 85 pints. 41. 3 mi. 12 rd. 20. 5280 feet. pt. 9. 217 pints. 42. 14 gal. 2 qt. 26. 13 hr. 44 min. 10. 39J pints. 1 pt. 1. 46 Ib. 7 oz. 30 sec. 11. 39 gal. 43. 2 hr. 38 min. 2. 43 bu. 2 pk. 27. 246 T.I 676 Ib, 12. 19 gal. 3 qt. 3 sec. 1 qt. 28. 11 wk. 16 hr. 12 ANSWEKS. 29. 15 mi. 311 53. 30 Ib. 8 oz. 80. 1 hr. 10 102. 11 gal. 1 qt rd. 54. 16 hr. 16 min. min. 1 pt. 30. 11 yr. 3 mo. 15 sec. 81. 5 Ib. 13 oz. 103. 22 bu. 2 pk. 31. 16 Ib. 12 oz. 55. 138bu.2pk. 82. 18bu.3pk. 2 qt. 32. 8 bu. 3 pk. 2qt. 7 qt. 104. 17 yd. 1 ft. 9 6 qt. 1 pt. 56. 224 gal. 3 qt. 83. 16 yd. 2 ft. in. 33. 8 yd. 1 ft. 9 1 pt. 9 in. 105. 31 mi. 108 rd. in. 57. 202 pounds. 84. llda.Shr. 4yd. 34. 12 da. 23 hr. 58. 5 hr. 1 min. 19 min. 106. 25 da. 23 hr. 45 min. 57 sec. 85. 93 gal. 3} qt. 48 min. 35. 56 gal. 1 pt. 59. 7 bu. 6 qt. 86. 5hr.35min. 36. 67 yr. 6 mo. 60. 9 gal. 2 qt. 5 sec. Page 334. 37. 42 mi. 245 rd. 1 pt. 87. 22T.8251b. 108. 13 gal. 1 pt. 38. 38T.5461b. 61. 54 years. 88. 2 wk. 4 da. 110. 17 Ib. 3 oz. 39. 16 Ib. 12 oz. 62. 94 wk. 6 da. 4 hr. 48 111. 4 T. 960 Ib. 63. 74T.5001b. min. 112. 1 mi. 110 rd. Page 332. 64. 77 yards. 89. 18 mi. 180 113. 3 yr. 6 mo. 40. 38 wk. 3 da. 65. 65 mi. 160 rd. 114. 12 bu. 3 pk. 17 hr. rd. 90. 5 yr. 9 mo. 2| qt. 41. 10 gal. 1 qt. 66. 1 da. 14 hr. 91. 13bu.3pk. 115. 17 yd. 4 in. 1 pt. 18 min. 6qt. 116. 5 hr. 20 min. 42. 17hr.24min. 92. 25 gal. 2 qt. 10 sec. " 35 sec. Page 333. 1 pt. 117. 18 gal. 1 qt. 43. 8 yd. 1 ft. 10 67. 1323 gal. 93. 33 min. 33 1 pt. in. 68. 4bu. 2pk. 4 sec. 118. 3 da. 5 hr. 20 44. 57 bu. 1 qt. qt. 94. 2 wk. 5 da. min. 45. 38 da. 18 hr. 69. 2 hr. 34 min. 12 hr. 119. 14 T. 110 Ib. 55 min. 5 sec. 95. 5 yd. 6 in. 120. 16 yd. 2 ft. 46. 13 bu. 1 pk. 70. 94 wk. 3 da. 96. 7 bu. 2 pk. 11 in. 6 qt. 20 hr. 1 qt. 121. 14 gal. 3 qt. 47. 16 gal. 2 qt, 71. 20 bu. 2 pk. 97. 1 da. 6 hr. IJpt 1 pt. 7 qt. 49 min. 48. 6 hr. 29 min. 72. 73 yd. 2 ft. 98. 3 qt. 1 pt. Page 336. 40 sec. 3 in. 99. 2 yd. 2 ft. 1. 1 hr. 18 min. 49. 3 Ib. 10 oz. 73. 21 days. 2 in. 17 sec. 50. 25 bu. 1 pk. 74. 54 yr. 10 mo. 100. 10 wk. 2 2. 25 bu. 3 pk. 7qt. 75. 41 gal. 1 qt. da.lShr. 4qt. 51. 27 bu. 3 pk. 76. 5 Ib. 3 oz. 15 min. 3. 3J inches. 4qt. 77. 7 ounces. 101. 50 hr. 50 4. 2 pk. 6 qt. 52. 76 gal. 3 qt. 78. 3 quarts. min. 50 5. 14 mi. 17 rd. Ipt. 79. 1 bu. 1 pk. sec. 6. 10 hr. 28 min, ANSWERS. 13 7. 1 ft. 10J in. Page 341. Page 345. 13. Tuesday. 8. 4hr. 43 min. 31. 4187HH- 21. 160 sq. yd. 14. 66 days. 30 sec. 32. 62,132 T \/ 7 . 22. 160 rods. 15. August 15. 9. 27 min. 10 33. 9555f|f. 23. 64 yards. 16. 170 days. sec. 34. 9593 itf?t- 24. 18 sq. yd. ; 10. $37.50. 35. 5O n^7^ 8054 ^j,v/ / "TTfoVz" 20} yd. 11. 9 If cents. 25. 59 T V sq. yd. Page 348. 12. 202} miles. 1. 57,537 Ib. 17. 44 yr. 4 mo. 12 2. 44 T. 13 cwt. 1. 18 hours. da. 3 qr. 2. 31 days. 18. 4 yr. 1 mo. 11 Page 340. 3. $821.96+. 3. 7 o'clock ; 3 da. 1. 1129f 4. 12 T. 18 cwt. o'clock ; 5 19. 8 yr. 4 mo. 14 2. 10,665. 22 Ib. o'clock. da. 3. 8077 T V 5. $ 356,400. 4. $18. 20. 128 yr. 2 mo. 4. 28,813|. 6. $980. 9 da. 5. 31.523J. 21. Mar. 4, 1841. 6. 61,903^. Page 342. Page 346. 22. 33 yr. 1 mo . 8 7. 206,783^. 2. 100 envel- 5. 25 cases. da. 8. 403,270. opes. 6. 21 posts; 2 23. 3 yr. 9 mo. 15 9. 834,085f 3. 24 rugs. posts ; 3 da. 10. 15,940,572. 4. 72 boards. posts. 24. 49 yr. 3 mo. 15 11. 775,665. 7. 31 days ; 29 da. 12. 933,273. Page 343. days. 25. July 21, '61 13. 601,227. 7. 1944 bricks. 8. 43 days. 26. 117 yr. 5 mo. 14. 542,817. 8. 72 tiles. 9. 23 chapters. 27 da. 15. 2,758,239. 9. 240 boards. 10. 27 problems. 16. 8,296,695. 10. 264,000 17. 1,232,766. stones. Page 349. 18. 3,855,141. 11. 4816 sq. yd. Page 347. 1. $34.56. 19. 9,733,680. 12. 4840 sq. yd. 1. 238 days. 2. $15.30. 20. 7,467,570. 13. 80 by 121, 2. 140 days. 3. 469 bushels. 21. 67,100. 40 X 242, 3. 109 days. 4. 108 cows. 22 310,700. etc. 4. 76 days. 5. 216 yards. 23. 108,662}. 14. 16 times. 5. 151 days. 6. 192 soldiers. 24. 324,133f 15. 9000 sq. ft. 6. 284 days. 7. 116 gallons. 25. 113,437}. 7. 179 days. 8. 31 cents. 26. 216,500. Page 344. 8. 139 days. 9. $39. 27. 426,300. 16. 41 ,400 sq.ft. 9. 91 days. 10. $1.56. 28. 2150. 8600 sq.ft. 10. 151 days. 11. 39 cents. 29. 105,300. 17. 9400 sq. ft. 11. $196. 12. 240 hours. 30. 690,300. 18, , $2800; $330. 12. 235 days. 13. 2 hr. 24 min. 14 ANSWERS. 14. $ 1.77J. 5. 1728 bricks. 14. 9 rd. 1 yd. 39. 12rd.4yd.6in. 15. $2.20. 6. 112 sq. in. 1 ft. 6 in. 40. 23rd. 2 yd. 6 in. 16. 27 days. 7. 36 sq. ft. 15. 9 rd. 1 yd. 41. Ill rd. 1 yd. 6 8. 45 rolls. 1 ft. 7 in. in. Page 350. 9. 24 lots. 16. 9 rd. 2 yd. 42. 3 rd. 4 yd. 2 1. $ 34.40. 11. 9000 sq. ft. 1 ft. 6 in. ft. 6 in. 2. 48 cents. 12. $1200. 17. 9 rods. 43. 3 rd. 4 yd. 1 ft. 3. $ 15.60. 18. 9 rods. 6 in. 4. $3.90 Page 355. 19. 9 rods. 5. 35 cents. 13. 16 fields. 20. 9 rods. Page 359. 14. 275 yards. 21. 7 rd. 2 yd. 5. 70 cu. in. Page 351. 15. 120 sq. rd. 2 ft. 1 in. 8. 46,656 cu. in. 6. $10.40. 16. 5 acres. 22. 4 rd. 5 yd, 1 9. i cu. yd. 7. $22.40. 17. 405 sq. yd. ft. 10. 8 ft. X 4 ft. ; 8. $1. 18. 18,755 sq. 23. 6 rd. 4 yd. 1 16 ft. x 2 ft, ; 9. $108. yd- ft. 1 in. etc. 10. $ 9.60. 24. 14 rd. 2 ft. 11. 5184 cu. ft. Page 356. 25. 5 rd. 3 yd. 1 12. 3x7x11; Page 352. 19. J acre. ft. 6 in. 6x7x5J; etc. 1. Mi 5' 20. 160 sq. in. 26. 7 rd. 2 yd. 1 13. leu. ft. smaller. 2. 11 miles. ft. 14. $132. 3. 3 bu. 7 qt. 1. 7 T 9 T rods. 27. 17 rd. 2 yd. 4. A, $750; 2. 7 rd. 4 yd. 1 ft, 3 in. Page 360. B,$500; 3. 7 rd. 4 yd. 28. 6rd. 2yd. 1 15. 3 feet. C, $250. lift. ft. 6 in. 16. About 7J gal. 5. $39.38. 4. 7 rd. 4 yd. 1 29. 7 rd. 5 yd. 17. About l}cu. ft. 6. $2.15. ft. 6 in. 10 in. 18. 30 gallons. 7. $ 1407. 5. 13 rd. 1 ft. 30. 990 inches. 19. 30 bushels. 8. $7.05. 6 in. 31. 5 rods. 20. 9 cords. 9. 48 tiles. 6. 12 rods. 32. 1422 inches. 21. 162,000 bricks. 10. $ 75.60. 33. 7 rd. 1 yd. 10,368,000 cu. 11. $216.66f. Page 357. in. 12. 12 bu. 3 pk. 7. 8 rd. 5 yd. Page 358. 22. 31,104 bricks. 4 qt. 8. 8 rd. 5 yd. 34. 17 rd. 3 yd. 23. 27 bricks. 9. 8 rd. 5 yd. 35. 15 rd. 2 yd. 24. 40,000 bricks. Page 353. 10. 8 rd. 5 yd. 2 ft. 6 in. 25. $2048. 1. 18,500 sq.it 11. 9 rd. 1ft. 6 in. 36. 13 rods. 2. 28 sq. yd. 12. 9 rd. 1 yd. 37. 22 rd. 3 yd. Page 362. 1 ft. 6 in. 2 ft, 6 in. 1. $874.80. Page 354. 13. 9 rd. 1 yd. 38. 5 rd. 5 yd. 6 2. $31.50. 4. 864 bricks. 1 ft. t) in. in. 3. 4 Ib. 8 oz. ANSWERS. 15 4. 1400 pills; 30. 431-3 3 ^. Page 368. 21. Father, 30 da.; 1152 pills. q-i i(=;2i 9. lOfg-. 9. 81 sq. ft. son, 15 da. 5. 12 spoons. 32. 13}|. 10. 735 sq. yd. 22. 3 dimes, 6 nick- 33. 16ft. els, 18 cents. 1. $693. 34. 12fJ. 23. 25 yards. 2. 45 cents. 35. 24ft. Page 370. 24. 25 rods; 100 3. fff day. 36. 31^. 2. 20 and 80. rods. 4. $2.39. tyr-f oq 11 O 1 . ^G-J^g-j. 3. $2000; 25. Speller, 15 ^ ; 5. 24 miles. 38. 22fJ. $ 4000 ; reader, 45^. 39. 30 Ar 1 . $12,000. 26. 60 and 12 1. 38ft. 40. 20 T V 4. 18 girls ; 36 27. 18 nuts; 9 2. 37A- boys. nuts ; 27 3. 58ft. Page 364. 5. 13 and 65. nuts. 4. 31. 1. 460.12. 6. 13. 5. 61ft. 21,355.74. Page 374. 6. 68ft. 2. .4551. 1. 24. 7. 66^. 3. Aft. Page 371. 2. 24. 8. 95^- 4- Ml- 7. 11. 3. 42. 9. 38Jf. 5. 11^^. 8. $3000; 4. 84. 10. 89.fr. 6. 725ftJ. $6000; 5. 24. 7. T Vg-. $ 18,000. 6. 70. Page 363. 8. .675. 9. 12 and 60. 7. 72. 11. 6|. 9. $227.60 A- 10. 9 marbles ; 8. 40. 12. 13ft. 10. $3800. 18 marbles ; 9. 360. 13. 54Jf. 11. 23 hr. 2min. 27 marbles. 10. 160. 14. 67ff. 8f- sec. 11. 36 years; 11. 18. 15. 17ft. 12. $11.08J. 6 years. 12. 18. 16. 61f 13. 1} yards. 12. 8. 13. 8. 17- 37JJ. 14. 1. 13. 1; 4; 12; 24. 14. 16. 18. 18||. 15. $187.50. 14. 30; 15; 135. 15. 12. 19. 39Jf 16. 7 days. 15. 9 pounds. 16. 20. 20. 25i}. 16. 19 rods. 17. 900. 21. 131}. Page 367. 17. 85 feet. 18. 60. 22. 211}. 1. 150 sq. in. 19. 60. 23. 663f 2. 1536 sq. yd. Page 372. 20. 32. 24. 185A- 3. 117sq. yd. 18. Son, $40; 25. 103ft. 4. 1225 sq. ft. daughter Page 375c 26. 95f. 5. 1554 sq. yd. $80. 21. 36. 27. 81}. 6. 6111 sq. ft. 19. 25 days. 22. 222. 28. 98|. 7. 924 sq. m. 20. Girl, $80; 23. 180. 29. 513Jf 8. 81 sq. ft. boy, $40. 24. 72. 16 ANSWERS. 25. 320. 10. 3. 19. 5 five-cent 32 - 26. 7. 11. 28. stamps ; 20 two- 4 12. 96. cent stamps; 35 33. 168. 1. 15 and 75. 13. 144. postal cards. 34. 2. 28 ; 71f 14. 18. 20. 8 horses; 5 3. $816. 15. 24. 25 cows; 55 sheep. 35. 110. 4. $180. 16. 6. 36. ^. 5. 89. 17. 32. Page 382. 3 6. 100. 18. 18. 1. $6.34. 37. 117. 7. 40; 15. 19. 12. 2. $4.50. 38. 8. fi 20. 20. 3. $9.60. 150 9. f|. 4. $1.55. 39. 500%. 2. 15. 5. $4.158. 40. --. Page 376. 3. 9. 6. $415.80. 800 10. 60; 420. 4. 15 marbles; 7. $10.32. 41. 18,400. 11. 540; 18. 33 marbles. 8. $1255.80. 42. 40%. 12. 9. 9. $9.16. 43. 1331 13. 20 peaches; Page 379. 10. $'3.68. 44. 72. 5 plums. 5. 25 ft. ; 100 ft. 11. $1.60. 45. 68. 14. $200; $600; 6. 39 acres ; 47 12. $1.58. 43. 75%. $700. acres. 13. $1.62}. 15. $60; $140. 7. 1059 votes ; 14. $326.80. 16. $300. 1377 votes. 15. 3 cents. Page 384. 17. 64 marbles. 8. 62 years. 16. $13.09. 47. 83f%. 18. $2; $3;-$10. 9, 84; 12. 17. $19.98. 48. II dol 19. $4; $2. 10. $108. 18. $1.17. 200 20. 3 horses; 12 11. 17; 28. 19. $3.16. 0. $800. cows. 12. $16; $11. 20. $7.50. 50. f . 13. Cows, $45; 21. $3.21f. 51. 37J%. Page 377. horses, $125. 22. $11.32. 52. 99. 1. 19. 14. 3 dimes ; 14 23. $63.04. 53. $218.75. 2. 22. half dimes. 24. $19.95. 54. $500. 3. 47. 15. 74 and 26. 25. $550. 55. 1. 4. 14. 16. 21 boys ; 33 26. $44.40. 56. 500%. 5. 9. girls. 27. $323.40. 57. 32%. 6. 10. 28. $4.65. 58. $183.20. 7. 6. Page 380. 59. 1100. 8. 33. 17. $3600; Page 383. 60. 738. $6000; 30. 1** 61. 10,800. Page 378. $ 8400. 20 62. $1547. 9. 27. 18. 44; 11. 31. 40%. 63. $764.80. ANSWERS. 17 Page 385 Page 388. 3. 11*%. 42. $309.14. 64. $4120. 24. 50%. 4. $26.40. 43. 8%. 65. $170. 25. 14f%. 5. $2.80. 44. 7A%. 66. 3500 bu. 26. 121%. 6. $1.22}. 45. $21. 87. $1440. 27. 37} cent*. 7. 45o/ . 46. $677.25. 68. $592. 28. $40. 8. 1350%. 47. $7692. 29. $219. 9. 4}%. 48. $2.36. 30. $200. 10. *%. 49. 7}%. Page 386. 31. 37}%. 11. 25%. 50. $21.55. 1. Selling price, 32. 25%. 12. 33*%. $ 2157.40. 33. $1646. 13. 16f%. Page 393. 2. Selling price, 34. $4053. 14. $75. 1. $112.50. $29.40. 35. $113.75 + . 15. 21%. 2. 90f cents. 3. Selling price, 16. $35. 3. 85f cents. $1181.25. Page 389. 17. $71.34}. 4. $19.52-. 4. Selling price, 1. 3 feet. 18. $19.20. 5. $1.65. $831.25. 2. 18 feet. 19. 33}%. 6. $1.16-. 5. Selling price, 3. * acre. 20. 12%. 7. $19.55. $1051.38. 4. 1430 yards. 21. $1.92. 8. $2.94. 5. 8 strips. 22. $24.24. 9. $1.114. 23. $39.20. 10. $67.72}. Page 387. Page 390. 24. $1.57. 11. $49.77+. 6. 3%. 6. 3600 sheets. 12. $235.75}. 7. 331 o /o . 7. 250 ft. 1500 Page 392. 8. 15o/ . sq. ft. 25. $18.33. Page 394. 9. 20%. 8. 250 boards. 26. 119%. 13. $18.88. 10. 6J%. 9. $710. 27. $200. 14. $7.80. 11. 6J%. 10. 6400 cakes. 28. $893.20. 15. $18.74-. 12. 20%. 11. 76,800 cu. ft. 29. $1. 16. $1050.80}. 13. 5%. 2208 tons. 30. 2%. 17. $1035.73}. 14. 12J%. 12. 160 feet. 31. $18. 18. $33.73-. 15. 16}%. 13. 281,600 sq. 32. $626.05. 19. $49.04-. 16. Cost, $375. ft.; 240,- 33. $1.57. 20. $154.87}. 17. Cost, $92.30. 000 sq. ft. 34. $68.18. 21. $884.53+. 18. Cost, 14. $2773}. 35. 9Jft%. 22. $6191.20. $ 1234.56. 15. 96 lots. 36. 14f%. 23. $2841.81-. 19. Cost, $ 240. 16. 164 sq. yd. 37. 18ff%. 24. $2344.50+. 20. Cost, $63.75. 38. $27.20. 25. $161.00+. 21. $55. Page 391. 39. 15 cents. 26. $835.31-. 22. 25%. 1. 11*%. 40. $6300. 27. $886.17-. 23. $800. 2. 10%. 41. $871.83. 28. $411.65+. 18 ANSWERS. Page 395. 5. 7ttHfttf. 7. $36.23+ 23. Proceeds, 29. $1550.21-. 6. 9809ff|ff. ($36.25+). $95.58+; 30. $118.14-. 7. 49,167if?ff. 8. $791.42+; ($95.63-). 8. 9631|-||-|f. ($791.89-). 24. Proceeds, Page 399. 9. 7288ifff?|, 9. $176.74^; $162.65+: 1. 165. 10. 6462ifff-f. ($ 176.85). ($162.76+). 2. 252. 10. $985.41f; 25. Proceeds, 3. 4048. 1. 630 boards ; ($986). $81.91+; 4. 910. 140 posts. 11. $3.10; ($81.95+). 5. 2594J. 2. 10 feet. ($2.80). 6. 5646f. 3. $1.50. 12. $2.60; 1. $24.28. 7. 1977f ($ 2.48). 2. $10.15+. 8. 6373J. Page 401. 13. $5.64; 3. $2.52+. 9. 5067 T V 4. 4340 sq. yd.; ($5.46). 4. $367.42-. 10. 8852J. about 70 14. $1.74-; 5. $10.99+. 11. 46,018. yards. ($1.69+). 6. $2.13. 12. 79,520. 5. 400 rods. 15. $5.38-; 7. $11.23+. 13. 25,554f. 6. 18 sq. ft.; ($5.29+). 8. $12.78+. 14. 106,908. 270 cu. ft. 16. $2.52; 15. 65,471. 7. 106f yards. ($2.40). 16. 65,635^. 8. 86f bundles. 17. $5.58; Page 405. 17. -86,855. 9. 810 sq. yd. ($5.40). 1. 371|. 18. 27.702J. 10. 114 rods. 2. 5301. 19. 37,411. 3. 129 r V 20. 26,969. Page 402. 4. 127ff. 21. 41,382. 11. 20,736 gal. Page 404. 5. 5171. 22. 58,975. 12. 72 cords. 18. Proceeds, 6. 289 T V 23. 899,100. $87.34 ; 7. 233ff. 24. 426,000. 1. $.41+; ($87.38+). 8. 260|f. 25. 16,800. ($.37i). 19. Proceeds, 9. 734if. 26. 1,172,880. 2. $.55+; $122.66+; 10. 1078}. 27. 4290. ($.46-). ($122.75-). 11. 85. 28. 71,400. 3. $2.89+; 20. Proceeds, 12. 89f. 29. 67,716. ($2.75+). $502.05-; 13. 264if. 30. 293,249|. 4. $.37-; ($.32). ($ 502.34+). 14. 361if 5. $6.89-; 21. Proceeds, 15. 337^. Page 400. ($6.72). $71.65-; 16. 6741J. I- 6975f?fff. ($ 71.68+). 17. S1 T V 2. 6126JHW- Page 403. 22. Proceeds, 18. 401^- 3. ll,202ff|if , 6. $173.59.V $ 232.99- ; 19. 421f. 4. ll,699|fjf| ($173.70). ($233.10+). 20. 434}. ANSWERS. 19 Page 406. 5. $724.85+; 21. 5 months. 9. Outside di- 1. $46.80. ($724.69-). 22. 24 weeks. mensions, 2. $18.40. 14x14x14; 3. 17 Ib. 11 oz. 1. 38 rd. 3 yd. Page 410. 2744 cu. in. 5 pwt. 19 11 in. 1. $554.23. wood and g r - 2. 113rd. 3 yd. 2. $171.20. marble ; 4. 23,220 gr. 1 ft. 6 in. 3. $1782.67J. 1728 cu. in. gold; 3. 175 rd. lyd. 4. $158.40. marble ; 2322 gr. 1 ft. 6 in. 5. $28.50. 1016 cu. in. silver ; 4. 16 13s. 6. $60.56J. wood. 258 gr. 4d. 7. $50.85. 10. 8 times ; copper. 5. 21,090d. 8. $ 13. iton; 5. 24 spoons. 6. 26 5s. 9. $7.45. 6f tons. 6. $2800. 7. $27.37+. 7. i; 109*%. 8. 108 4s. 8. $105. 6d. Page 411. Page 416. 9. 27+ cents. 9. 45 18 s.8 d. 10. $27.84-. 1. 5%. 10. 23 A gal. 10. 773^ oz. 11. 40 and 10%; 2. 2 years. 11. N 704 sq. ft. 11. $175. $2 differ- 3. $96. ence. 4. 3%. Page 407. 12. $ 60 ; 40 % 5. $1.43-. 12. 300 pounds. discount ; 6. $36. 13. A, 43i%; Page 409. 60% net. 7. 1 yr. 6 mo. B, 36%; 12. $123. 8. $42.17+. C, 20J% 13. $72; Page 413. 9. $5000. 14. 672 yards. $119.25; 1. 121J- cu. ft. 10. 6%. 15. $315. $92.25. 2. $162. 11. 1 yr. 9 mo. 16. 82$%. 14. $450; 2 da. $ 750 12. $ 144. 1. $49.01-; $600. Page 414. 13. $83.26J. ($ 48.99-) ; 15. 24 days. 3. $126. 14. 3%. $48.88-; 16. 2| hours; 4. $85.50. 15. 2 yr. 1 mo, ($48.87-). 48 miles 5. 16.8 tons. 7 da. 2. SI 75.94-; from A. 6. 2 pieces, 16. $80. ($175.94-). 17. $ 45 ; $ 135 ; 12x12; 17. $2181.99-. 3. $350.55+; $270; 2 pieces, $72. ($ 350.55+). $450. 12X14; 18. $237; $189; 2 pieces, Page 408. $114. 14 x 14. Page 418. 4. $846.26-; 19. 7 days. 7. 44ff Ib. 2. ? 41.99+. ($845.77+). 20. 240 eggs. 8. 5280 Ib. 3. $951.13+. 20 ANSWERS. 4. $112.74-, Page 421. 11. $853.27+; 32. 10,757|. 5. $119.43-. 33. $1473.52-. ($853.71-). 33. 21,873^. 6. $13.91-. 34. $6.55+. 12. $56.62}; 34. 24,292}. 35. $278.16-. ($57.50). 35. 485,072. 36. $1.23-. 13. $1500. 36. 167,276. Page 419. 37. $196.64+. 14. Dec. 30, 1895. 37. 24,418}. 7. $147.19-. 38. $1.10. 38. 24,130f. 8. $8.10-. 39. $389.60. Page 428. 39. 651,329 T 5 T . 9. $52.33-. 40. $6.11+. 1. 1,287,400. 40. 1,932,560. 10. $1005.50. 41. $4.09+. ' 2. 3,370,185. 41. $277,133.11 12. $967.78. 42. $56.32-. 3. 598,969. 42. $60,887.10. 13. $21.79-. 43. $16.72-. 4. 2,883,736. 43. $48,554.Q8. 14. $16.53-. 44. $95.43-. 5. 816,669. 15. $1274.21-. 6. 5,127,460. Page 430. Page 422. 7. 2,455,038. 44. 7 5 s. 9 d. 4. $196.17. 8. 42,327,198. 45. 11 yd. 1 ft. Page 420. 5. $600.01+. 9. 2,513,420. 11 in. 16. ($860.50-). 6. $1506.12. 10. 22,944,747. 46. 12 bu. 2pk. 17. $24.74-; 7. 4269.22+ 5qt. ($23.94). francs. Page 429. 47. 11.76+. 18. $761.06-; 8. $1563.55-. 11. 857,712. 48. 13.72+. -($761.45). 9. $1563.55-. 12. 6,482,112. 49. 14.41+. 19. $43.99-. 10. $1547.37. 13. 1,230,828. 50. 13.34+. 20. $786.39-. 11. 1^ ; $500. 14. 921,776. 51. 19.05-. 21. $65.40. 25 ' 15. 3,460,704. 52. 22.30-. 22. $625.03+. Page 425. 16. 5,888,304. 23. $98.49+. 1. $106.33+; 17. 1,460,025. 1. 79.98%. 24. $993.27+. ($ 106.38). 18. 10,563,960. 2. 5.02%. 25. $61.68-. 2. 7%; 19. 3,911,322. 3. 4.28%. 26. $ 252.37+ ; (7*%). 20. 2,982,840. 4. 3.83%. ($252.52-). 3. 24 days. 21. 714,186. 5. 3.810/0. 27. $13.09-; 4. $1200. 22. 3,277,719. 6. 2.44%. ($12.87}). 5. $4.68; 23. 456,375. 7. 0.54%. 28. $486.10-; ($ 3.90). 24. 174,600. 8. 0.10%. ($486.43-). 6. 5%. 25. 362,250. Total, $872,- 29. $2.33+; 7. 72 days. 26. 104,787}. 270,283. ($1.94+). 8. $1200; 27. 128,550. 30. $989.67-; ($ 1260). 28. 625,975. Page 431. ($990). 9. $304.26-; 29. 213,966|. 1. 237.49%. 31. $1938.43-. ($304.05+). 30. 413,866f. 2. 234.60%. 32. $8.06+. 10. Apr. 8,1894. 31. 86501 3. 563.36%. ANSWERS. 21 Page 432. Page 434. Page 438. 8. 864 bales; 4. 0.04%. 3. 8750 sq. yd. 8. 60 bushels. 540 bales ; 5. 25%. 4. 3000 + 30 x- 9. 720 pounds. 396 bales. 6. 20%. 80 yards. 10. 4600 sheets. 9. $12; $20; 7. f 24; $30. 5. 100 a;; 40 11. $312. 10. $60; $72. 8. $3. yards. 12. 8 da. 4 hr. 6. 60 z + 1200; 13. $28.80. 2. 326.9843. 1. 4|f. 80 yd. and 14. 96 rods. 3. 6408. 3. A- 120 yd. 15. 50 days. 5. 4.2633; 4. 16ft 7. 10,000 sq. 16. $45. 1.405712. 5- T 2 sV yd. 17. 120 men. 6. 8l|. 8. 1920 flag- Page 441. 7. 73; 240. stones. 6. 16.5393. 8. 4||. 9. $9.26-. Page 439. 9. 28.165; 9- T 9 T ; T V 10. 46f yards. 18. 360 men. 305.36721642. 10. Largest, f ; 11. 90Jflb. 19. 10 hours. 10. .3375. smallest, 12. 36& s q- ft - 20. $1.20. 12. Sum, foff 21. 32 days. 8.08690625. Page 435. 22. 20 days. 14. .019104141. 1 O 1 O f f ft i r> IK o^- O^Qfi 1 ^ J.O. 1U ID. O ID. 14. 950 bushels. 1. $78; $104. J.t. .vU , .vGC/O '2"3"3" /O' 25%. 4. 37 30'. 12. 1331%; Page 451. 5. 1 hr. 14min. Page 444. 75%. 12. 1104f|mi. 52 sec. 17. $459.371 13. 24 days. 18. $24,000"; 14. $24. 1. $828.45; 66f%. ($828.87). Page 454. 19. 11%. Page 448. 2. $397.30; 6. 33 30' east 20. $18,000. 1. $3481.07+. ($397.50). longitude. 21. 101 rd. 2 yd. 2. $1874.771. 3. $554.40; 7. 39^f^ miles 1 ft. 6 in. 3. $2461.76+. ($554.68). per hour. 22. 35 shares; 4. $1000.50+. 8. 1800 Ib. $87.50. 5. $1632. 9. $1000.80. 23. 37 rd. 4 yd. Page 452. 10. 8 yd. 6 in. 11 in. Page 449. 4. $625.33+. 11. 15 ft. 7 in. 24." $ 106.12. 6. $1845.90+. 5 3959 a. 12. 257bu.2pk. 7. $946.04-. '. 4000 ' 2qt. Page 445. 8. $632.65-. /3961z\ 13. 20 spoons. 25. 229 rd. 4 yd. 9. $985.22+. \ 4000 j 14. 16 40'. 2 ft. 6 in. 10. $326.34. a 11979 -2x 26. $427.67+. 10 Page 455. 27. $428.40-; 1. 9hr. 45min. / i 1. $880.86+. $53.40-. 2. 14 J minutes ( 10 } 2. $1229.01-. 28. $119.10+ past 5. r, 7956 + 8x. 29. $149.14+. 3. 4 ft. 6 in. 1 . , D Page 456. / 15 92+ ^ 3. $193.70+. Page 446. Page 450. V 5 / 4. $446.33-. 30. 51 cents. 4. 748 plants ; 8. $1200; 5. 4188.48+ 754 plants. ($1199.39+). marks. 1. 15%. 5. 20 da. 6 hr. 9. 30 days ; 6. 8490.44- 2. $400. 40 miri. (33 days). . francs. 3. $102.50. 6. 5544 revo- 10. $1.50 discount 7. 307 7s.6| 4. $2.70. lutions ; per $ 1000. d. nearly. 5. $40. 1320 revo- ($2 discount 8. $2050.72+. 6. $1.60. lutions. per $ 1000). 9. $4138.97+. ANSWERS. 23 Page 457. 2. 16. 4. m- 23. 90 da. 10. $908.87-. 3. 18. 5- m- 24. 18 da. 4. 24. 6. 4. 25. $475; Page 459. 5. 26. ($474.76-) 1. $224.46+. 6. 36, Page 469. 2. $261.19-. 7. 35. 7- JW- Page 471. 8. 42. 8. Equal. 1. 567. Page 460. 9. 44. 9- A- 2. 915. 4. $772.37-. 10. 51. 10. - 2 /. 3. 144,200. 11. 53. 11. 4 4. 25 758. 1. $137.61+. 12. 54. 12. A. 5. 2,114,000. 2. $12.39-. 13. 61. 13. ff. 6. 86,526. 14 3 *7 1 Oft OOO Page 461. JL*. OO. 15. 72. 1. $2167.09-. 8. 374,625. 1. 1J yards. 16. 75. 2. $543.16-. 9. 19,800. 2. 18.72 feet. 17. 83. 3. $1335.23+. 10. 7312. 3. $78.75. 18. 84. 4. $911.66-. 11. 466,000. 4. 8 ft. 4 in. 19. 91. 12. 50,133. 5. $14,910.75. 20. 95. Page 470. 13. 98,500. 6. 108JJ sq.ft. 5. $784.54. 14. 4180. 7. 44.17875 sq. 1. 7568. 6. $724.03+. 15. 423,000. ft. 2. 5107. 7. $1586.88-. 16. 40,096. 8. 2970 bu. 3. 6008. 8. $1008.10+. 17. 419,904. 9. 252 gallons. 4. 6285tff |f|. 9. $2607.48-. 18. 310,050. 10. 146 sq. yd. 5. 6098. 10. $718.00-. 19. 230JV. 11. 3hr. 12min. ; 6. 3007. 11. 4}%. 20. 2271 4hr. 6^-min. ; 7. 98,640. 12. $600. 1 hr. 48min. 8. 75,064. 13. 4 yr. 6 mo. 1. 12: 6 P.M. 9. 70,921. . 9 da. Page 462. 10. 78,905. 14. 2 yr. 8 mo. Page 472. 13. 40i- yards; 1 da. 2. 9 : 36 A.M. $ 44,02 J. 15. $600. 3. 142 55' 30" 14. 1J ft.; 108 Page 466. 16. 4%. West. sq. ft. 1. $183.27-. 17. 63 days. 4. 52 36' East 15. 984 sq. ft.; 2. 1508s.4d. 18. $400. 5. '5 : 16 A.M. 71 ft. 19. $295.35; 16. $11,200. ($ 295.50). Page 473. 17. $56. Page 468 20. 7%. 6. 8: 24 A.M. 1. j|f. 21. $4.10; 7. 12:1:20 P.M Page 465. 2. f ($4). 8. 143 3' West. 1. 14. 3- A 8 *- 22. $2750. 9. 82 40' West. -2A ANSWERS. 10. 11 : 33 : 12 5. 70. 4. 10%. 16. 23. A.M. 6. 15fJ. 5. $8.96. 17. $ 6000 ; 11. 3 : 37 : 20 P.M. 7. H $ 14,000. 12. 88 4' East. 8. f Page 481. 18. 50%. 13. 32 34' East. 9. if- 6. 60 cents. 19. 4%. 14. 2: 45: 42A.M. 10. A 7. 20%. 20. $372. 15. 23 West. 11. i. 8. $ 150. 21. Iff days 12. iff. 9. $120. 3. if 13. 6. 10. 20%. 1. 7|iJ. 6. if- 2. Mi 7. if. Page 478. 1. 254. 8. ft- 2. $ 625.92. 2. 27.1. Page 484. 9. H- 3. 3968 revolu- 3. 1.37, 3. 22 T J . 10. If. tions. 4. 26.8. 4. t; 4s. 6 d. 11. ft- 4. 27 kilo- 5. 3.76. . 5. 13 T \ 3 . 12. n meters. 6. .838. 6. %% Q acre ; 13. 3J. 7. 16.27. 2181b.l2oz. 14. 3J. Page 479. a .4876. 7. T?; tt- 15. If 5. 770 meters ; 9. .306. 8. i 8"' 16. 2A- 42 kilos. 10. .069. 9. 42ff. 17. 2H- 6. $504. 10. 1. 18. - 3 &- 7. 8J- days. Page 482. 11. 10. 19. 12J. 8. $ 82.50. 1. $6.24. 12. f; T 3 o- ' da. 20. 16i. 9. 14 yards. 2. 4hr. 24min. 13. 8. 21. 10}. 10. $2.45. 3. $ 22.50. 14. 13s. 3J|< 4. 150 barrels. 15. T 8 2 8 3T- Page 476. Page 480. 5. $ 153.60 ; 16. 3 / ^ /J*J . IZJ. 1. 4. Orr. X O, V O. 35. x = 8, y = 9. Page 533. Page 537. 2. 5. gg a; = 11 n = 15 9. 2 x + 4. 1. 15. 3. 12. 37. z=23, y = 18 10. Sx + l. 2. 96. 4. 7. 38. a = 13, y = 9. 11. - 4 x + 9. 3. 20 years. 5. 24. 39. a;=10, y = 5. ANSWERS. 29 1, , 15 and 22. Page 546. 17. a; 2 - 14 x + 45. Page 551. 2. , 47 and 19. 61 O 1 Q . a; = L T/ = IO. 18. z 2 + 10o; + 25. 6. 2500. 3. 5 and 4, 7. a; = 12, y = 6. 19. x z + 5 x - 24. 7. 12 rods; 20 4. , 23 and 42. 8. aj=6,y = 5. 20. s* + x - 42. rods. 9. a;=19J, 21. a; 2 -11 x + 28. 8. 12 yards. Page 543. y 17. 22. 6a; 2 -24a; + 24. 9. 8 feet. 5. 1 Q twn rlnllar 23. 6 a; 2 + 4 x 42 10. 24 and 25 bills ; 1. $30,000. 24. 13 five-dollar 2. A, 20 chest- 25. 6x 2 -13a;+6. Page 553. bills. nuts ; B, 2 26. 4 a; 2 - 9. 1. x + 3 = 7. 6. 10 pigs; 15 chestnuts. 27. 8z 2 + 24z-54. 2. a -6 =8. sheep. 3. $5; $3. 3. a;-4=6. 7. Oranges, 3 f ; 4. 17,38, and 45. 4. x - 8 = 5. peaches, 2 p. Page 550. 5. a: + 9 =10. 8. Tea, 60 j* ; Page 547. 1. 7. 6. x + 1 = 5. coffee, 25 p. 5. 21 and 32. 2. 5. 7. x7=8. 9. 4 horses, 16 6. $18; $32; 3. 5. 8. aj-ll=12. cows, 32 $16. 4. 5. 9. a? + 7 =10. sheep, 2 pigs. 7. $600. 5. 5. 10. aj-ll=13. 10. Raisins, 12^; 8. $150. 6. 24. cheese, 17 ^. 9. 66 acres ; 7. 24. 1. 7or-l. 11. 12 and 7. 38f acres ; 8. 5. 2. 18 or -6. 12. If 25-j- 1 ^ acres. 9. 13. 3. 6 or -8. 13. Ml- 10. 70 yards. 10. 5. 4. 5 or - 23. 14. 40 pounds. 11. $20. 11. 7. 5. 13 or 1. 12. 1. 6. 10. Page 544. Page 548. 13. 8. 7. 5 or -25. 15. 40 pounds 1. x 2 + 7 x + 10. 14. 4. 8. 2 or - 28. green tea ; 2. a; 2 + 17a; + 72. 15. 6. 9. 24. 60 pounds 3. 2a; 2 + 9a; + 10. 16. 7. 10. 24 or - 16. black tea. 4. 2 x 2 + 26 x 17. 13. 11. 3 or 1. + 72. 18. 6. 12. 5 or - 35. Page 545. 5. 3a; 2 + 22 19. 13. 13. 1 or - 29. 2. x = 2, y = 3, + 7. 20. 13. 14. 4 or - 26. z 5 6. 4- T^ 1 4 y 1 1 15. 8. 3. z = 7, y=13, 7. ^ */ T^ Tt Jj ~r J.. a? + 6 a? -27. 1. 60 rods; 30 16. 2 or -38. 2=1. 8. a; 2 + a;-42. rods. 4. & = 12,y = 31, 9. a; 2 -25. 2. 4 inches. Page 554. 2=19. 3. 10 and 8. 1. 3 or -4. 5. aj = 10 f y=13, Page 549. 4. 45. 2. 5 or -2. 2=16. 16. x- 2 -14z + 49. 5. 50. 3. - 1 or - 4. 30 ANSWERS. 4. 8 or - 1. Page 587. 8. Base 3 in. 21. 175 sq. in. 5. _4or-5. 11. fjin. Perpendicular 22. 257.6112 sq. 6. 7 or 4. about % in. (.866+ in. 7. _6or-7. 1. 60 feet. in.). 23. 27 feet. 8. 19 or -4. 2. 48 feet. 9. Half perim- 9. 18or-l. eter about 3 in. Page 598. 10. -18or-l. (3.06+ in.). 24. 452.3904 sq. Page 588. Apothem about ft.; 1. 3, -2. 3. 109^ feet. yfin. Area about 254.4696 sq. 2. 4, -5. 4. 2160 feet; 2if sq. in. ft. 3. 3, - 7. 2060 feet. 25. 4166f mi. 4. 8, 2. 5. 124 feet. Page 592. 26. 2000 miles. 5. 9, -7. 14. Dimensions, 27. 12,500 miles. 6. 3, -6. Page 589. 3.1416 in. 28. 1:2. 7. 5,4. 6. 13 T 6 3 chains. by 1 in. 29. 2828.4+ mi. 8. 1, -8. 7. 162 feet. 15. 12.5664 sq. 9. 2, -9. 8. 84 feet. in. Page 599. 10. 4, 1. 16. 78,54 sq. in. 30. 113.0976 sq. Page 590. 17. 3.1416 sq. in. Page 555. 9. 108 feet. in. 31. 56.5488 sq. 1. 4 and 8. 10. 119 yards. 18. 52.36 sq. in. in.; 2. 80_feet. 19. 94.8232 sq. 28.2744 sq. 3. 60 yards. 1. 2.0944 in.; in. in. ; 4. 25 yards. 4.1888 in. ; 84.8232 sq. 5. 5 feet. 6.2832 in. ; Page 595. in. 6. 4 feet. 8.3776 in. ; 10. 14.1372 sq. 7. 68 rods. 10.4720 in. in. 1. 72 cu. in. 2. 2 inches ; 11. ljin.,3in. ; 2. 36 cu. in. Page 556. 3.464+ in. ; 1 in., 3 in. 3. 36 cu. in. 8. 34 feet. 4 inches ; 9. 65 feet. 3.464+ in.; Page 596. Page 600. 10. 3feet. 2 inches. 12. 108. 4. 36 cu. in. ; 11. AC= 42 ft.; 3. 35 ; 40; 14. Two, 15 in. ; 124.71- cu. #D=24ft. 105; 105. two, 13 in. ; in. 384 sq. in. 5. 690 cu. in. Page 586. 15. 240 sq. in. 6. 62.882 cu. in 9. 1:4. Page 591. 17. 40 sq. ft. 10. 90, 5 in. ; 5. 140. Page 601. 1 in., 2 in., 6. 6 in. ; Page 597. 7. 144 cu. in. ; 2J in. ; 37, 6.2832 in. 20. 122.5224 sq. 18 cu. in. 53, 90. 7. About in. in. 8. 126 cu. in. ANSWERS. 31 9. 1350cu. in.; 50 cu. in. Page 602. Page 603. 11. 1300 cu. in. 14. 1927.3716 Page 605. 16. 3053.6352 cu. 10. 1300 cu. in. ; 12. 65i cu. ft. cu. in. in. 520 sq. in. ; 13. 2232 cu. in. 15. 929 ; 9136 cu. 17. 381.7044 cu. 770 sq. in. in. ; in.; 4.0256- gal. 190.8522 cu. in. ADVERTISEMENTS Elementary Mathematics AtWOOd's Complete Graded Arithmetic. Presents a carefully graded course, to begin with the fourth year and continue through the eighth year. Part I, 30 cts.; Part II, 65 cts. Badlam's Aids tO Number. Teacher's edition First series, Nos. i to 10, 40 cts.; Second series, Nos. 10 to 20, 40 cts. Pupil's edition First series, 25 cts.; Second series, 25 cts. Branson's Methods in Teaching Arithmetic. 15 cts. Hanus's Geometry in the Grammar Schools. 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HEATH & CO.,Publishers, Boston, New York, Chicago Supplementary Reading A Classified List for all Grades GRADE /. Bass's The Beginner's Reader ..... .25 Badlam's Primer ........ .25 Fuller's Illustrated Primer ...... .25 Oriel's Glimpses of Nature for Little Folks .... .30 Heart of Oak Readers, Book I . . . . . . .25 GRADE II. Warren's From September to June with Nature . . .35 Badlam's First Reader ....... .30 Bass's Stories of Plant Life ...... .25 Heart of Oak Readers, Book I . . . . . . .25 Wright's Nature Readers, No. i ..... .25 GRADE III. Heart of Oak Readers, Book II 35 Wright's Nature Readers, No. 2 ..... .35 Miller's My Saturday Bird Class ..... .25 Firth's Stories of Old Greece . . . . . . .30 Bass's Stories of Animal Life ...... .35 Spear's Leaves and Flowers ...... .25 GRADE IV. Grinnell's Our Feathered Friends .... .30 Heart of Oak Readers, Book III . . . . . .45 Kupfer's Stories of Long Ago . ..... .35 Wright's Nature Readers, No. 3 . . . . . .50 GRADE V. 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