DE No. RATIONAL GRAMMAR SCHOOL ARITHMETIC BY GEORGE W. MYERS, Ph.D. PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY, SCHOOL OF EDUCATION, THE UNIVERSITY OF CHICAGO SARAH O. BROOKS CITY NORMAL SCHOOL, BALTIMORE, MD. CHICAGO SCOTT, FORESMAN AND COMPANY 1903 COPYRIGHT, 1903, BY SCOTT, FORESMAN AND COMPANY EPUCATION tS, CHICAGO. TYPOGRAPHY BY MARSH, AITKKN Ji CURTIS COMPANY, CHICAGO PREFACE The present book is an outgrowth of the notion that arithmetic as a science of pure number, and arithmetic as a school science, must be treated from two essentially different stand- points. Viewed as a finished mental product, arithmetic is an abstract science, taking its bearings solely from the needs of the subject; but viewed as a school subject, arithmetic should bean abstracted science, taking its bearings mainly from the needs of the learner. The former calls only for logical treatment, while the latter calls for psychological treatment as well. In other words, to be of high educational value the school science of arithmetic must take into full account the particular stage of the pupil's development. The abstract stage must be approached by steps which begin with the learner, rise with his unfolding powers, and end leaving him in possession of the outlines of the science of arithmetic. To break vital contact with the learner at any stage of the unfolding process is fatal. A controlling principle in the development of the various topics of this book is that any phase of arithmetical work, to be of value, must make an appeal to the life of the pupil. But the social and industrial factors in American communities enter largely into the pupil's life. This renders material drawn from industrial sources and from everyday affairs of high pedagog- ical value for arithmetic. The recent infusion of new life into the curricula of elementary schools through the wide introduction into them of nature study, manual training, and geometrical drawing furnishes a basis for a closer unifying of the pupil's work in arith- metic with his work in the other school subjects. Wide use has been made of all these sources of arithmetical material. A rational presentation of the processes and principles of arith- metic can be secured as well through material representing real conditions as through material representing artificial conditions. Not only have most of the problems been drawn from real sources IV PREFACE but a yejry sa-rriest effort has also been made to have all data of 'problems absolutely correct and consistent, to the end that infer- 'frotai titepo. may be relied upon. With so rich a store as the corifams, however, it is perhaps too much to hope that no errors remain. The authors will deem it a favor to be notified of any errors that may be detected. It is well known that the majority of the pupils of the elementary school never reach the high school. Even these pupils, whose circumstances cut them off from advanced mathe- matical study, have a right to claim of the elementary school some useful knowledge of the more powerful instruments of algebra and geometry. For those who will continue their studies into the high school it is important that the roots of the later mathe- matical subjects be well covered in the soil of the earlier. The present book seeks to meet the needs of both classes of pupils through the organic correlation of the elements of geometry and algebra with the arithmetic proper. Treated thus, the geometry serves to illustrate the work of arithmetic and algebra, and the algebra emerges from the arithmetic as generalized number. In particular, this text aims to accomplish four main pur- poses, viz. : (1) To present a pedagogical development of elementary mathe- matics, both as a tool for use and as an elementary science ; (2) To base this development on subject-matter representing real conditions; (3) To open to the pupil a wide range and variety of uses for elementary mathematics in common affairs to aid him to get a working hold of his number sense ; and (4) To give the pupil some training in ways of attacking com- mon problems arithmetically and some power to analyze quanti- tative problems. It is not intended that teachers should have their classes solve all the problems; but rather that each teacher should select such topics and problems as have a particular interest for his school. We trust that the suggestiveness of this book as to sources of problems and ways of handling them in the arithmetic class will be appreciated by many teachers. Such teachers will doubtless PREFACE T prefer to work out some topics more fully than is done in the text. It is thought that what is given in the text will make the fuller working out of special topics practicable and not difficult. The work makes continual call for estimating magnitudes and for actual measurement by the pupils. Let the children be sup- plied with measures, foot-rules, yardsticks, meter-sticks, etc., and encourage their constant use. Make so regular a feature of this work that pupils form the habit of estimating distances, areas, volumes, weights, etc., always correcting their estimates by actual measurement. All models, scales, or standards of measure made by pupils should be carefully kept and used in the later work. Pupils should also be supplied with instruments with which the exercises in constructive geometry may be actually done. A cheap pair of compasses is the only necessary purchase. It is better that the pupil make the rest of the apparatus needed. If for any reason the pupil cannot actually do the constructive work it should be omitted altogether. The Introduction which precedes the work in the formal operations is a departure from current text-book procedure believed to be worthy of special remark. It consists of thirty pages of simple, practical problems, relating to matters with which the pupil's experiences, in school and out of school, have familiarized him in an indefinite way, for the right understanding of which the use of numbers and of the arithmetical processes is necessary. This chapter, by furnishing to the pupil a gradual transition from his vacation experiences to the rather severe study of the formal processes of arithmetic, by impressing the pupil with a sense of the real need and purpose of such study, thus preparing his inter- est for it, will be recognized as a deviation from common practice resting upon sound pedagogical grounds. Especial attention is invited to this introductory chapter; to the chapter on measurement, 76-86, preceding and laying the foundation for common fractions; to the treatment of propor- tion, 90-92, 125, 126 ; to the lists of data for original problems, 36, 42, 50, 140; to the work in geography, 30, 34, 60, 166; in farm account keeping, 17, 41; in commerce, 35; in nature VI PREFACE study, 51, 101, 130, 132, 164, 165, 190, 191; in physical measurements, 8-10, 133, 141; in paper-folding, 183; on the locomotive engine, 175; in constructive geometry, 75, 177-188, 197; and in applied algebra, 217. Most teachers will approve dropping many of the time-honored but anti- quated subjects and the curtailing of other topics of slight utility, and the introduction in their places of more timely and more real subjects. The many lists of data for original problems, 36, 42, 50, 140; the treatment of longitude and time, 192, 193, the practical work leading up to this topic; the extensive use of graphs to put meaning into arithmetical measures and to bring out the laws involved in numerical data will commend themselves to teachers as useful and instructive means of keep- ing the number faculty employed on practical material. Another special feature is the numerous lists of problems that bear on the development of some important idea or law, having an interest on its own account. In these lists each problem is a step in a connected line of thought culminating in an important truth. This plan furnishes numerous problems, miscellaneous as to process, thereby requiring original mathematical thought, and still organically related to a central idea, thereby calling for the constant exercise of judgment. Examples of this may be seen in any section of the introduction; in the problems on geography, commerce, nature study, and elementary science and also in the following sections: 72, 73, 76-89, 101, 118, 123-126, 141, 144, 145, 164, 165, 166, 168, 169, and in practically all the matter from p. 271 to the end of the book. For the maturity of pupils of the later grades this is believed to be an important feature. It avoids the danger, always present with lists of promiscuous problems when classified under the arithmetical processes to be exemplified, of reducing to the mechanical what should never be allowed to become mechanical, viz. : the analysis of relations. Throughout the book the instruction is addressed to the pupil's understanding rather than to his memory. But while accomplishing this, due regard has been had to the necessity of sufficient drill in pure number to enable the pupil to obtain both a conscious recognition of processes and considerable PREFACE '.*:: ^2 V v i l facility in their automatic use. This is done&ut that the fundamental arithmetical operations should be reduced to the automatic stage as early as possible, consistently with a clear understanding of them. The attention of teachers is called to the section on Short Methods and Checking at the close of the book. After pupils have clearly grasped the meaning of the arithmetical processes and have acquired some mastery of their uses, a relief from the tedium of long arithmetical calculations becomes a matter of great impor- tance. Xo one can become a rapid computer without short methods and, considering that the problems of daily life that call for arithmetical treatment do not have answers with them, no one can be certain of his results without means of checking calcula- tions. Every expert accountant uses them and the more expert he is the more does he use them. In fact, expertness consists very largely in the ability to shorten calculations and to apply rapid checks. Training in the use of short cuts and checks should con- stitute a much more important part of the pupil's work in arith- metic than is common. Constant use of these sections should be made through the seventh and eighth grades. Definitions, processes, rules, and even special subjects are worked out under the guidance of the principle: "First its informal, though rational, use, and afterwards conscious recogni- tion of its formal use." Definitions of merely technical terms are given when called for by the development and where the need for them arises. Unusual attention has been paid to the numerous illustrations. The publishers have spared no pains to make them an important aid to the teacher in the development of the subject. None have been inserted for mere adornment and none are mere pictures of things or relations that are perfectly obvious to the pupil. Their aim throughout is to secure clearness, precision, and certainty of thought. It is thought the book may lay rightful claim to an innovation in this particular. For convenience in reviews and for reference, a synopsis o'f definitions and a full index are given. The authors' acknowledgments for suggestions and ideas are PREP ACE ibxuumetotts arithmetical writers and teachers. They desire to express their obligations in particular to Miss Katheriiie M. Stilwell of the School of Education, University of Chicago, for valuable help throughout the book; to Mr. W. S. Jackman, Dean of the School of Education (to whom is due the credit for the skiameter work as given in 190, p. 299), for the privilege of using much valuable material; to Mr. J. B. Eussell, Super- intendent of Schools, Wheaton, 111., and to Miss Ada Van Stone Harris, Supervisor of Primary Schools, Rochester, N. Y., both of whom read much of the proof. They desire most heartily to thank Mr. Stephen Emery of Lewis Institute, whose unceasing diligence and pains with the proofs have wrought distinct improve- ments on nearly every page of the book. If the book shall in some measure aid in putting the teaching of arithmetic on a more rati nal basis, thereby bringing a-bout results more nearly c mmensurate with the time and energy put upon the subject in the elementary schools, the authors will deem their efforts repaid. THE AUTHORS. CHICAGO, July, 1903. RATIONAL GRAMMAR SCHOOL ARITHMETIC INTRODUCTION ORAL WORK FIGURE l Bookcase 1. Scale Drawing. 1. If one side of a triangle is 4 in. long, how long a line will represent this side in a drawing to a scale of 1 to 4, or ? 2. If the two other sides are 2 in. and 3 in. long, how long must the lines be to rep- resent them in the same drawing? in a drawing to a scale of -j^? 3. What is the scale of a drawing in which a line 1 in. long represents 1 ft.? in which 1 in. represents 40 ft.? 100 ft.? 4. In Fig. 2, which is a scale drawing of a schoolroom, one- sixteenth of an inch represents 1 ft. How long is the room? the teacher's desk? How long and how wide are the pupils' desks? 5. In what direction do the pupils face, when seated at their desks? 6. Find by measurement how far it is from the northwest corner of desk 1 to the southwest corner of desk 4; from the west edge of desk 2 to the east wall of the room. 7. Make and answer other questions on this plan. 8. Make a similar scale drawing from actual measurements of your schoolroom and the fixed objects within it. 9. Make a scale drawing from measurements of your school- house and grounds. .1 i G FIGURE 2 -.2: &AXIONAL GRAMMAR SCHOOL ARITHMETIC ; ;T(*wii Blofck Cartel Lots. Fig. 3 represents a scale map of a town block. Scale: 1 in. equals 100 ft. If possible, pupils should measure and draw to scale a block or field in the neighborhood of the schoolhouse, and in all problems use the numbers they obtain from their own measurements in preference to those given in the exercises. RACE ST. B K G H w- MARXET ST. .1 in. divided into tenths FIGURE 3 Measure the drawing in Fig. 3, and find how long the block is between the sidewalks; how wide. NOTE. Use the scale of tenths given in the figure. Lay a strip of paper having a straight edge beside the marked inch and mark short lines on the strip to indicate the tenths. INTRODUCTION 3 WRITTEN WORK 1. How many square feet in the area of the block mnop? 2. At $20 per ft. of frontage on Market street, what is lot L worth? 3. Make problems like 2 for other lots on Market street, using price per front foot of lots where you live. 4. At $12.50 per ft. of frontage on Mathews avenue, what is lot H worth? 5. Make similar problems for other lots on Mathews avenue. 6. At $18 per ft. of frontage 011 Race street, what is lot G worth? Similar problems should be made by the pupil. 7. Make problems for any, or all, of the lots, at the price per front foot where you live. Each property holder is taxed to provide funds for founda- tion material, brick, and labor to pave in front of his property to the middle line of the street. This is called an assessment. 8. The streets are to be paved with brick. The cost of exca- vating to the proper depth is 30^- per square yard of surface;* find the cost of excavating a strip 1 yd. wide extending from outside edge of sidewalk to middle of Race street. Ans. $3.00. 9. What would be the cost of excavating a similar strip 5 yd. wide? a strip as wide as frontage of lot E? Ans. $15.00 ; $50.00. 10. The cost of foundation material is 36^ per square yard ; find the cost of enough such material for the strip described in problem 8. Ans. $3.60. 11. Eind the cost of foundation material for each of the two strips mentioned in problem 9. 1st Ans. $18.00. 12. The cost of the brick to oe used is $10 per M (thousand) and 78 bricks are needed to cover 1 sq. yd. What is the cost of the brick needed for the strip of problem 8? for each of the two strips of problem 9? 1st Ans. $7.80. 13. What is the total cost of excavating, of foundation mate- rial, and of brick for the strip of problem 8? for each of the two strips of problem 9? 1st Ans. $14.40. * Use prices current in your community whenever they can be obtained, instead of prices given. 4 RATIONAL GRAMMAR SCHOOL ARITHMETIC 14. Find this total for other lots fronting on either Market or Eace street. 15. The labor of construction costs 75^ per sq. yd. What is the cost of labor on a strip of the size mentioned in problem 8? Am. $7.50. 16. What will be the assessment against lot F for paving for each foot of frontage? 17. Make similar problems for other lots, not including the corner lots. 18. Henry and Mathews avenues, which are 30 ft. wide between sidewalks, are to be paved in the same way as Market and Race streets. Rates being the same as above, what will be the total assessment against lot L? 19. Make and solve problems like 18 for other lots. 20. The owner builds a house on lot L. The length of the house is 36 ft. and the width is 30 ft. How many square feet of the lot are covered by the house? 21. The southeast corner of the house is located 30 ft. from the south and east lines of the lot. Locate the three other corners. 22. Find the cost, at 45^ per sq. ft., of the concrete walk 3 ft. wide in lot L, as shown in the drawing. 3. House and Furnishings. ORAL WORK 1. What is meant by a 1-brick wall?* a 2-brick wall? 2. If it takes 7 bricks to make 1 sq. ft. of wall surface when laid in a 1-brick wall, how many bricks per square foot are needed for a 2-brick wall? a 3-brick wall? a 5- brick wall? 3. From the dotted lines in Fig. 4, can you tell how measurements may be taken on a wall to avoid counting corners twice? OHtside-Measure pw riTT} ,, 4 4. A brick is 2 in. by 4 in. by 8 in. ; cTGlTH/E 4 J J how many cubic inches are there in it? * A 1-brick wall is a wall one brick thick, bricks lying on the largest surfaces. INTRODUCTION 5 5. How many square inches in one end? one edge? one of its largest surfaces? 6. In locating the foundation of the house on lot L (Fig. 3), in what direction would you wish the line of the front foundation wall to run with reference to the Market street line? 7. Point out some of the square corners on the plans in Fig. 5. 8. Can you point out any corners that are not square? WRITTEN WORK The owner of lot L, which fronts on Market street, builds the house whose foundation plans are given in Fig. 5 and whose first and second floor plans are given in Fig. 6. FOUNDATION FlGFRB 5 WALLS 1. What will it cost to excavate a rectangle 21' x 36' to a depth of 6 ft. , for the foundations and cellar, at 20^ per cubic yard? Think of the foundation as made up of straight walls such as are shown in the second part of Fig. 5. The mark (') means foot or feet, and (") means inch or inches. 2. The foundation is inclosed by 2 side walls, each 36 ft. long, and 2 end walls, each 18J ft. long. Point out these walls in both parts of Fig. 5. The foundation walls are all 8 ft. high. Find the area of the north surface of the north side wall. Ans. 288 sq. ft. 3. What other wall has an outer surface equal to the surface mentioned in problem 2? 6 RATIONAL GRAMMAR, SCHOOL ARITHMETIC 4. Find the area of the outer surface of the east end wall. Ans. 148 sq. ft. 5. What other outside surface equals this one in area? G. Masons reckon that it takes 14 bricks for each square foot of outer surface to lay a solid 2-brick wall. If all walls are 2-brick walls, and solid, how many bricks will be needed for the north foundation wall? for the east wall? the south? the west? 1st Ans. 4032; 2d Ans. 2072. 7. How many bricks will be needed for all four of the outside walls? 8. There are two inside foundation walls each 18 ft. long, also one 10^ ffc. long, and one 11 ft. long. All are 2-brick walls 8 ft. high. If these walls contain no openings, how many bricks will be needed for the inside foundation walls? Ans. 6552. 9. The outside walls contain 8 window openings each 1^- ft. by 3 ft. If masons allow for one-half the area of all openings in computing the number of bricks, how many bricks should be deducted for the outside walls? Ans. 252. 10. The inside walls contain 4 door openings each 2^- ft. by 8 ft. How many bricks should be deducted for the inside walls? Ans. 560. 11. Find the total number of bricks needed for the foundation walls and their cost at $9 per M.* 12. If hauling costs 75^ per load of 1^ T. and each brick weighs 6 lb., find the cost of hauling the brick for the foundations.! 13. If the mortar costs $1.25 per M. bricks, what will be the cost of the mortar? 14. Four brick piers 2 ft. by 2 ft. and 4 ft. high support the porch columns. How much will the bricks needed for these col- umns cost at $12 per M., counting 22-J bricks for each cubic foot? 15. While the house was building the owner decided to replace weather -boarding by stained shingles on a belt running around the house and extending to a distance of 9 ft. below the eaves. Each square yard, thus changed, cost $1.75 extra, no allowance * In computing the cost of brick use the nearest whole thousand, t When the last load is fractional, the price for a full load is charged. INTRODUCTION 7 being made for openings. How much does this change add to the cost of the house? 16. Hardwood floors were decided upon later to take the place of pine floors in the dining-room and front hall. This increased the price by !%

House ^ Oats and /^%^ Grounds '///!&' \iM 5. Fencing a Farm. A farm was divided into fields and seeded as shown in the drawing (Fig. 7). The scale of the drawing is 1 in. to 80 rd. This means that 1 in. in the drawing represents 80 rd. in the farm, and that all other lines longer or shorter than 1 in. represent distances in the farm proportionately longer or shorter than 80 rd. ORAL WORK 1. How long is the corn- field north and south? how wide? 2. How long and how wide is the wheatfield? the north oatfield? 3. How long and how wide is the meadow? the south oatfield? 4. The drawing of the road is T V in. wide; how wide is the road? 5. How long and how wide is the farm? 6. How long must a wire be to reach entirely round the farm? 7. How many rods of barbed wire will be needed to inclose the farm with a 3-wire fence? how many miles (320 rd. = 1 mile)? WRITTEN WORK 1. How many rods long is the partition fence running across the farm from east to west? what part of a mile is its length? How many rods of wire will make it a 4-wire fence? 1st Ans. 160. 2. How many rods of wire will run a 4-wire fence across the farm from the middle of the north side to the middle of the south side? Ans. 640. 3. How many rods of wire will run a 4-wire fence between the wheatfield and the north oatfield? between the meadow and the south oatfield? 4. How many bales of wire of 100 rd. each will be needed for these cross fences and division fences? Ans. 19.2. FIGURE 7 12 RATIONAL GRAMMAR SCHOOL ARITHMETIC 5. How much will the wire cost for the 3-wire inclosing fence @ $3.50 a bale of 100 rods? Ans. $67.20. 6. The posts for all the barbed wire fences are set 1 rd. apart. How many posts will be needed to fence around the farm? 7. How much will these posts cost @ 25^ apiece? 8. How much^ will the wire cost @ $2.60 a bale for the two 4-wire cross fences, and the two division fences between the wheat and the north oatfield and between the meadow and south oat- field? 9. How much will the posts for these fences cost @ 15^ apiece? 10. Find the total cost of wire and posts for these fences. 11. The house and barn lot is separated from the pasture by a 5-board fence. The boards used are 1 in. thick, 6 in. wide, and 12 ft. long. How many boards will reach once along the north side of the house and barn lot? Ans. 55. TWO FENCE PANELS ^ H()W m ^ oax ^ rf^ reaCQ r IGURE o * once along the east side of the lot? 13. How many boards will be needed for the 5-board fence along both sides? 14. The posts are set 6 ft. between centers, that is, a post is set every 6 ft. (Fig. 8) . How many posts will be needed for the north side? for the east side? for both? 1st Ans. 110; 2d Ans. 109. 15. How much will these posts cost @ 23^ apiece? Lumber is sold by the board foot. A board foot is a board 1 ft. long, 1 ft. wide, and 1 in., or less, thick. 16. How many board feet of lumber in a board 1 in. thick, 1 ft. wide, and 5 ft. long? the same thickness and width and 10 ft. long? 12 ft. long? 17. How many board feet in a board 1 in. thick, 10 ft. long, and 2 ft. wide? the same thickness and length and 6 in. wide? 9 in. wide? 18 in. wide? 1 in. wide? 8 in. wide? 18. How many board feet in a board 12 ft. long, 4 in. wide, and 1 in. thick? 2 in. tbick? INTRODUCTION" 13 19. How many board feet in a "two by four" scantling 10 ft. long (2" x 4" x 10')? 16 ft. long? 20. How many board feet in a "four by four" scantling 10 ft. long? 12 ft. long? 18 ft. long? 21. How many board feet in a fencing plank 1" x 6" x 12'? in 5 such planks? 22. How much will the fencing lumber of problem 12 cost @ $18 per M board feet? 23. What will be the total cost of the lumber and posts for the fence on the north and east sides of the lot? 24. It cost 8^ apiece to have the post holes bored, and it took 4 days' work by 3 men @ $1.50 per day to build the fence. Three patent gates costing $12 apiece were put in the fence. Find the total cost of the lot fence, including gates. 25. What was the total cost of all the fencing done on the farm? 6. Areas of Fields. ORAL WORK 1. The whole farm contains 160 acres; how many acres are there in the cornfield? in the wheatfield? in the meadow? in the south oatfield? in the pasture? in the lot around the house and barn? 2. What is the width of each of these fields in rods? 3. If the meadow were divided by a north and south central line, how many rods wide would each half be? How many acres would there be in each half? 4.. How many square rods in a strip 1 rd. wide and 80 rd. long? 5. How many square rods in an acre? 6. How wide must a strip of land 80 rd. long be to contain an acre? 40 rd. long? 7. Compare the sizes of the wheatfield and the north oatfield; of the cornfield and the wheatfield ; of the meadow and the wheat- field; of the meadow and the south oatfield; of the meadow and the house and barn lots; of the lots and the pasture; of the pasture and the south oatfield. 14 RATIONAL GRAMMAR SCHOOL ARITHMETIC WRITTEN WORK 1. What would be the gross income from the farm in 1 yr. if it were all seeded to corn and the average yield were 47 bu. per acre and were sold at 28^' per bu.? What would be the income per acre? 2. If the farm were rented @ $5 per acre, how much would remain for the tenant * from each acre? from the entire farm? 3. What would be the owner's income from the whole farm, not allowing for expenses? 4. Which is the more profitable way for the owner to rent his farm, @ $6 per acre cash, or for J of all the crop delivered to mar- ket, supposing that the whole farm is planted to corn and that a yield of 48 bu. per acre and a price of 30^ per bushel can be obtained every year? NOTE. Rent of the first sort is called "cash rent," of the second sort, "grain rent." 5. If this same farm will produce 20 bu. of wheat per acre and a price of 60^ per bushel can be obtained for it, which is the more profitable crop to the owner, wheat or corn? how much more profitable? 6. Compare the profits to the owner of a grain-rented farm from an oats crop of 40 bu. per acre and a price of 22^ per bushel with the profits of the wheat crop of problem 5 ; also with corn crop of problem 4. 7. In problem 5 which would bring the larger income to the owner, and how much, cash rent @ $5 per A., or grain rent @ $ delivered? 8. Answer the same question for the oats crop mentioned in problem 6. / 9. A cornfield of 68 acres produced an average yield of 48 bu. per acre. How many bushels did the farm yield? 10. It cost $5.85 per acre to raise the crop. At 46^ per bu. what was the net value of the crop? 11. If the tenant paid cash rent @ $5.50 per acre, what was his net profit from the crop? How much did the owner receive? * The tenant is the farmer who raises the crop on another man's farm. INTKODUCTKW 15 7. Habits of Animals. Certain pupils made observations of the different sorts of beasts, birds, and insects which inhabited the neighborhood of their school. The following table shows the classified results of 100 observa- tions of this kind: 1 Migrate as winter approaches 18 2 Store food for winter 15 3 Remain and feed abroad in winter 20 4 Hibernate without food 10 5 Die as winter comes on 27 6 Appear only in winter 10 Total 100 NOTE. Pupils are urged to make their own observations, to arrange them as in the table, and to use them as indicated in the problems. 1. What fractional parts of the entire number of different kinds of animals do you find in the 6 classes separately? 2. The number of class 2 equals what part of the number of class 1? of class 3? of class 5? 3. How many different kinds of animals per 100 (kinds) hiber- nate without food? 4. What per cent of all the different kinds of animals of the region migrate? NOTE. "What per cent" means "how many in a hundred" or "how many hundred ths." 5. What per cent of the different kinds store food for winter? 6. What part of the number of animals of the first 5 classes equals the number of animals of class 6? 7. The animals of classes 1, 4, and 5 disappear from the land- scape in winter. How many disappear? 8. What per cent of all the animals in this collection disap- pear from the winter landscape? 9. The first five classes appear in summer. Only classes 2, 3, and 6 appear in winter. The number of animals which enliven the winter landscape equals what per cent of the animals of the sum- mer landscape? 16 RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. Physical Measurements (a): Spirometer. The school should be supplied with one tin can or pail 18 in. high and 10 in. in diameter, and another about 9 in. in diameter and - * * " A a 9 to 12 in. in height. If the school does not possess this apparatus, have it made I L^^^J I by a tinner. Fill the large can nearly full (3 Xfl ^ wa ^ er an ^ insert the small can, mouth downward, within the water. With an arrangement of pulleys and weights, w w, which may be cans containing sand, the inner can may be counterbalanced until it moves easily either upward or down- ward. FIGUKB 9 A rubber tube passed under the mouth of the inverted can and stretched over the end of a piece of glass tubing, which is held in an upright position inside the inverted can by light cross-braces, may be used to con- vey air into the can. If the stopper at A is drawn, the can sinks readily in the water; after which the stopper is inserted air-tight. A foot rule, graduated to eighths or sixteenths of an inch, stuck with putty to the side of the inside can, permits the read- ings to be taken. Two pails of different sizes may serve as a very satisfactory sub- stitute for the cans. The method of fitting them up for use is plain from Fig. 9. 1. If the inside diameter of the inverted can is 8.8 in., for each inch the inner can sinks into the water, there will be 60.85 cu. in. of water inside the can, the stopper at A being removed. How many cubic inches of water will there be in the can when it has sunk 2 inches? 5 inches? 8 inches? NOTE. If a pail is used, the corresponding number of cubic inches is found by measuring the distance in inches across the mouth of the pail, multiplying one-half this distance by itself, and the product by 3f. 2. How many cubic inches of water will there be in the can when it is sunk in the water only 1J inches? If inches? 2-J- inches? 2 T V inches? 3. Let the can now be pushed down as far as it will go and the stopper at A be pushed in tightly. Read the scale. When a pupil blows through the tube until the inside can rises 1 inch, how INTRODUCTION 17 many cubic inches of air has he expelled from his lungs into the can? How many when the scale indicates that the can has risen I inch? 1 T V inches? 1J inches? l T 5 g- inches? If inches? 4. Fill your lungs full and blow into the tube as long as you can without danger. What is the capacity of your lungs in cubic inches? 5. The average lung capacity for a man 5 ft. 8 in. tall is 204 cu. in. This average capacity may be called the normal lung capacity. How much does your lung capacity fall short of the normal value for a man? 6. The normal lung capacity in cubic inches for a man is 3 times his height in inches. For a woman, it is 2.6 times the height. Divide your lung capacity by your height in inches, and compare your quotient with these numbers. 7. The chest measure of a man should be not less than half his height. How much greater or less is your chest measure than it should be according to this law? 9. Physical Measurements (b). 1. Fill out in your notebook a record like the one below, which is a copy from a student's notebook : Name, John Morrow; age, 10 yr.; height, 52 in.; weight, 61 Ib. ( Inspiration, 28 in. Chest measures, < Expiration, 2Jfo in. Lung capacity, 104 cu. in. ( Mean,* 26\ in. Height in inches by 3 (if a boy). Result 52 X 3 = 156 cu. in. Height in inches by 2.6 (if a girl). Average to each inch of height, 2 cu. in. , for 104 -*- 52 = 2. Chest average = S6\ in. Half height = 26 in. 2. How much do your measures exceed or fall short of the normal values? 3. Do your measures show that you need chest exercise? 4. How many in the class or room are above or below the normal? wan of ; : numbers is half their sum. 18 RATIONAL GRAMMAR SCHOOL ARITHMETIC 5. Compare class or room averages to see whether the average for class or room is about normal. 6. Make the comparison of problem 5 after a course of exercises in physical training, and record the changes due to the exercises. 7. Every month or two during the year make the computa- tions called for in problems 2 and 5 ; keep your results, and show what changes are taking place in your measures. 10. Physical Measurements (c). The following table gives the average height in inches and the average weight in pounds for boys and for girls year by year from 4 to 15 years of age. The num- bers are the averages of careful measures of heights and weights of hundreds of boys and girls in the schools of Chicago, Boston, Cincinnati, and St. Louis. These averages may be called normal values for boys and girls of school age : HEIGHT YEARLY GROWTH WEIGHT YEARLY GROWTH AGE Boys Girls Boys Girls Boys Girls Boys Girls 4yr. 39.2 in. 39.0 in. 37.31b. 35.31b. 5 41.6 41.4 40.6 39.7 6 43.7 43.3 44.7 43.0 7 45.8 45.5 48.7 47.0 8 47.9 47.6 53.8 52.0 9 49.7 49.5 58.7 57.1 10 51.6 51.3 64.8 62.2 11 53.5 53.4 70.1 68.1 12 55.2 55.8 76.7 77.4 13 57.3 58.5 84.9 88.4 14 60.0 60.2 95.0 98.3 15 62.4 61.3 106.5 105.0 1. Subtract each number in the column of heights from the number next below it and find the growth in height for boys and for girls from year to year. Do the same for the weights. 2. When is the yearly growth in height greatest for boys? for girls? What is the yearly growth in each case? 3. Make and solve similar problems for weights. 4. Compare your own height and weight with the normal values for your own age. How much do you exceed or fall short of the average? INTRODUCTION 19 5. How do the averages for your room compare with the values in the table for the same years? 6. How much do your height and weight exceed or fall short of the averages for pupils of your own age in your room? 11. Vital Statistics. The New York Board of Health gwes the annual death rate per thousand of population as follows : 1886 26.0 1887 26.3 1888 26.4 1889 25.3 1890 24.9 1891.. ..26.3 1892 25.9 1893 25.3 1894 22.8 1895 23.1 1896 (1st half year) 21. 5 1. What is the average death rate for 1886-89? NOTE. The average death rate for 4 years is \ of the sum of the rates for the single years. 2. What for the years 1890-93? 3. What is it for the years 1894-96, assuming that the death rate of the first half will be the rate for the entire year? 4. In 1894 special efforts were directed toward street cleaning and better city housekeeping generally. What reduction in death rate was made in the one year from 1893 to 1894 as a consequence? 5. If the population of New York City for 1893-94 be taken as 2,800,000, the lives of how many people were saved by the improved conditions during 1894? 6. Compare the death rate of your own town, city, or county with the values of this table. How much does it exceed or fall short of the largest 'value of the table? the smallest value? the average value? 12. Wind Pressure. The velocity of wind in miles per hour and the pressure per square foot are as given here : Light breeze 3| mi. .75 oz. Moderate breeze . . 6 " 3.33 " Fresh breeze 16^ " lib. 5 " Stiff breeze 32i " 5 " 3 " Strong gale 56| mi. 15 Ib. 9oz. Hurricane 79^ " 31 " 4 " Violent hurricane 97 " 46 " 12 " NOTE. Use measures from your own schoolhouse or from other build- ings in your neighborhood in preference to the numbers in the problems. 20 RATIONAL GRAMMAR SCHOOL ARITHMETIC FIGURE 10 1. What is the total pressure in pounds on the west side of a house 30 ft. by 25 ft., due to a light breeze from the west? to a stiff breeze? to a strong gale? 2. What is the pressure on the side of a tall building 265 ft. by 80 ft., due to a hurricane blow- ing squarely against it? 3. Find the wind pressure against the side of a load of hay 25 ft. long and 12 ft. high, due to a strong gale blowing squarely against it. 4. Find the number of pounds pressure against a signboard fence 100 ft. by 28 ft., due to a strong gale blowing squarely against it. 5. What is the wind pressure tending to over- turn a square chimney 30 ft. wide at the base, 15 ft. wide at the top, and 175 ft. high, due to a violent hurricane blowing souarely against one of the flat faces? NOTE. To obtain the area of the effective surface against which the wind is blowing, multiply the height of the chimney by its breadth halfway up. This breadth is the half sum of the widths at the top and bottom (Fig. 10). 6. Find the wind pressure on the side of to a violent hurricane * a tree, due blowing against the triangular top, ft. across the base and 43 ft. high (see Fig. 11). 7. Find the wind pressure on the side of a passenger coach 70 ft. long and 15 ft. high, due to a stiff breeze blowing squarely against it. (a) Suppose the wind to blow against the whole rectangle 15'x70'; then, (b) Suppose the wind is obstructed by only -J- 4V x 70', from the track to the bottom running the length of the coach. FIGURE 11 of the strip of the coach box and INTRODUCTION 13. Dairying. The following table is an extract from the milk record of the University of Illinois herd of milk-cows : WEIGHT OF MORNING (A.M.) AND EVENING (P.M.) MTLKINGS IN POUNDS 1901 QUEEN BEAUTY BEECH- WOOD MYRTLE SPOT ROSE IV rw TOTAL i" Of a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m. 1 6.5 6.5 11.0 10.4 12.5 10.2 7.3 5.7 12.0 11.5 11.0 9.6 5.6 6.0 11.0 8.4 11.0 11.0 7.2 5.6 11.5 10.5 11.3 8.8 6.4 6.0 10.8 9.5 11.7 11.5 7.3 5.7 12.0 10.4 11.2 9.8 6.0 6.5 12.5 10.2 12.6 13.0 7.2 6.0 11.5 11.5 11.0 9.8 6.0 7.5 11.4 10.2 12.0 11.4 6.7 5.8 11.6 11.3 11.0 10.2 e 6.6 6.8 12.0 10.0 8.0 11.6 8.0 5.0 12.5 11.8 11.7 10.0 7 7.4 8.5 10.8 12.4 12.0 11.7 7.0 7.2 12.0 12.5 11.6 12.0 8 7.8 7.3 11.2 11.6 13.2 12.2 7.2 5.7 11.5 11.3 11.5 10.1 9 7.5 6.4 11.6 10.4 12.0 11.6 6.6 5.4 12.0 12.2 10.7 100 10 7.8 7.2 11.0 10.4 12.0 12.0 6.7 4.7 11.5 8.5 11.8 10.6 11 6.8 7.3 12.2 11.0 13.6 11.5 7.7 5.3 9.0 10.0 12.3 10.0 12 7.8 7.7 12.5 12.0 13.0 11.6 8.0 5.5 12.0 12.6 12.2 10.6 13 7.6 8.0 13.4 11.5 13.5 13.0 7.2 5.8 11.1 13.0 11.8 112 14 7.0 8.0 12.0 11.8 14.0 12.7 7.7 5.7 12.5 12.7 11.5 10.3 15 8.3 7.5 13.0 10.7 15.0 12.0 7.6 5.0 14.0 12.4 12.3 9.0 16 9.0 7.0 12.5 11.0 15.2 13.7 7.0 5.5 12.7 12.3 14.1 8.5 17 8.0 7.0 12.5 10.5 14.0 13.0 6.1 6.4 10.0 9.0 12.0 10.5 18 8.1 7.7 13.4 10.9 14.0 11.5 6.6 5.0 12.5 11.5 11.0 9.1 19 7.0 7.0 11.5 11.4 13.6 11.0 6.2 5.3 12 2 11.7 11.1 10.0 20 7.5 6.5 12.6 10.5 12.7 12.0 6.5 4.9 1L8 11.4 11.5 9.5 21 7.0 7.6 12.0 10.2 13.7 11.3 6.1 5.2 13.0 11.0 10.6 9.6 22 8.0 7.5 12.2 11.9 14.7 11.2 6.7 6.0 12.4 10.7 11.3 9.5 23 7.5 9.0 12.2 11.0 14.4 12.0 7.0 5.5 12.8 10.8 11.8 10.4 24 7.0 7.1 13.5 10.6 13.8 11.5 7.2 5.1 13.0 10.7 12.0 10.4 25 7.4 8.0 12.5 11.2 14.0 10.5 7.4 5.7 12.7 11.7 12.0 11.0 26 6.5 8.4 12.0 11.8 12.0 12.0 6.1 5.8 13.2 11.5 11.5 10.5 27 7.5 8.5 13.7 11.8 14.0 13.0 6.1 5.8 13.0 12.7 11.4 10.5 28 7.2 9.2 12.0 12.0 13.6 13.8 6.6 4.9 13.1 12.5 11.7 10.5 29 6.5 9.0 13.5 12 6 15.0 13.5 6.7 5.0 10.2 12.6 12.5 12.6 30 7.2 8.0 11.7 12.7 14.2 12.2 6.0 4.3 12.4 11.5 12.7 10.3 1. How many pounds of milk did Queen give the first week of Nov., 1901, at the morning milkings? at the evening milkings? at both milkings? 2. At which milking did Queen give most milk and how much more than at the smallest milking? 3. Make and answer similar questions for the remaining weeks and for any or all of the 6 cows. 22 KATIOKAL GRAMMAR SCHOOL ARITHMETIC 4. What was the total weight of milk at the morning milkings for the whole herd on Nov. 1st? at the evening milkings on the same date? at both milkings? Put the answer to the last question in the column of totals at the right. 5. Make and answer such questions as 4, for other dates and for each week of November. 6. If a quart of milk weighs 2.2 lb., what is the morning milking of the herd worth at 6$ per quart for Nov. 1? the even- ing milking? 7. What is the milk of the herd worth each week if milk sells at 6^0 per quart? what for the month? 8. Make such problems as 7, for single cows and solve your own problems. 9. Has there been any gain, or loss, in the weekly milk yield of the herd, or of individual cows, during the month? If so, how much? 14. The Thermometer. The official record of hourly temper- atures from 6 p.m. February 27 to 6 p.m. March 1, 1902, as given in a daily paper, is as follows : FEB. 27; NIGHT FEB. 28; DAY FEB. 28 ; NIGHT MARCH 1; DAY 6pm 38 Gam 40 6pm 39 6am 34 C 7 p. m 38 7 a. m 41 7 p. m 38 7 a. m 34 8pm. 37 8am 41 8pm 37 8am 33 9pm. 37 9am 41 9pm 37 9am 33 10 p m. 39 10 a. m 43 10 p m 37 10 a m 32 11 p m .39 11 a m 44 11 p m 37 11 a m 32 12 midnight 38 12 m 38 12 midnight 36 12 m 32 Average. ..... Average 1 a. m 36 1 p. m 31 12 midnight 38 12 m 38 2am 35 2pm 31 1 a. m 38 1pm 35 3am 34 3 p. m 30 2am. .38 2pm 35 4 a. m . . 34 4 p. m 29 3 a. m 38 3 p. m 37 5 a. m . . . 34 5pm 28 4 a. m 37 4 p m. 39 6am . . 34 6 p. m. 28 5 a. m 39 5 p. m . . 40 Sum .... Sum . 6 a. m 40 6 p. m 39 Average Average. . Average. ..... Average NOTE. The average of temperatures, or numbers, is found by adding them and dividing the sum by the number of temperatures, or numbers, which have been added. To average means to find the average. INTRODUCTION 80-: 60-: 5CK 1CH |zo-i 30-: 1. Average the temperatures on Feb. 27 from 6 p.m. to midnight; from midnight to 6 a.m. 2. What was the average temperature on Feb. 28 from 6 a.m. to 12 m.? from 12 m. to 6 p.m.? 3. What is the difference between the two aver- ages of problem 1? of problem 2? What do these differences mean? 4. What is the difference between the lowest and highest temperatures from 6 p.m. Feb. 27 to 6 p.m. Feb. 28? This is called the range of temperature for the day. 5. The average temperature for Feb. for 30 yr. is 26. How much does the average temperature for the 24 hr. following 6 p.m. Feb. 27 exceed the thirty- year average? 6. Find the average temperature for the night of Feb. 28; for the day of March 1. 7. Find the difference between these averages. What does this difference show? 8. Make and solve such problems from outdoor thermometer readings at your schoolhouse. 9. What is the reading of the thermometer of Fig. 12? What would the thermometer read if the top of the mercury column stood at "freezing"? at "summer heat"? at "blood heat"? 10. Average the thermometer readings taken at your own school for each hour from 9 a.m. to 4 p.m. for a month, and keep in a notebook a record of your readings and averages. 11. Lay off on cross- lined paper the read- ings of problem 10 as the read- ings of the table from 9 a.m. to 4 p.m. of Feb. 28 are laid off in Fig. 13; draw a smooth curve, free-hand, through the points. Such a curve gives a picture of temperature changes. FlGUKE 12 50 40 30" 20 10 0, 9 10 11 12 1 2 3 4 A.M. A.M. A.M. M. P.M. P.M. P.M. P.M. February 28 FIGURE 13 9 10 11 12 1 234 P.M. P.M. P.M. M. A.M. A.M. A.M. A.M. February 28 March l FIGURE 14 24 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. From a daily paper obtain the hourly readings from 9 p.m. to 4 a.m. and lay them off to scale as the readings of the table from 9 p.m. Feb. 28 to 4 a.m. March 1 are laid off in Fig. 14. Draw the free-hand curve as before and find whether the day or night temperatures are the steadier. 13. Average your readings from 9 a.m. to 4 p.m. and draw a straight, horizontal line on the diagram of your temperature curve, at a distance above the horizontal zero line equal to the average, as the line av of Fig. 13 is drawn. This is the average line for these 8 hours. 45 40 10 C 5- 910H121 23456789 10 11 12 1 23456789 A.M. A.M,A.M.MI P.M. P.M. P.M. P.M. P.M. P.M. P.M. P.M. P.M. P.M. P.M. M. A.M. A.M. A.M. A.M. A.M. A.M. A.M. A.M. A.M. Changes from 9 a.m. February 28, to 9 a.m. March 1 FIGURE 15 14. From a daily paper, or by other means, obtain the hourly readings for 24 hr. in succession. Draw the curve for them as the curve of Fig. 15 is drawn for the readings from 9 a.m. Feb. 28 to 9 a.m. March 1. 15. Lay oft to scale the noon temperatures from day to day for a month on a piece of cross-ruled paper, and draw as smooth a free-hand line as possible, uniting the points. This curve show? the changes of noon temperatures for the whole month. 16. Lay off the averages, as in problem 13, for each day for a week and obtain the changes of temperature for this week. 17. Extend the curve of problem 16 for a month. 18. In the same way draw a curve through points obtained bj laying down to scale the average values for the 12 mo. of any INTRODUCTION" year since 1872. The vertical lines should correspond to months of the year. (See Chicago Weather Bureau Summary.) 19. Draw a curve for the monthly averages for the 29 years. 15. November Meteors. The earth in its yearly journey around the sun passed through a swarm of meteors Nov. 13-15, 1901. The meteors were seen in great numbers as shooting stars by observers at different places, and were counted and recorded as in the table below. 1. The swarm was so large that it took the earth, JEJ, which moves 18.6 mi. per second, 48 hr. . pat i, to move across it, from a to b. How long is aft? 2. The meteors were thicker at some places than at others and the greatest number of shooting stars was seen when the earth was where the meteors were thickest. From the table below, on which day did the earth go through the densest part of the swarm? COUNTS OP SHOOTING STARS, NOVEMBER, 1901 FIGURE 16 HOUR A.M. NUMBER SEEN Nov. 13 AT CLAREMONT, CAL. NUMBER SEEN NOVEMBER 14 NUMBER SEEN Nov. 15 AT MINNEAPOLIS AT CLAREMONT, CAT,. AT MINNE- APOLIS 12 to 1 19 51 5 5 1 to 2 27 85 19 28 2 to 3 20 95 54 35 3 to 4 22 238 62 , 87 4 to 5 30 682 106 19 5 to 5:30 19 306 132 11 TOTALS 3. Find the whole number of shooting stars counted at Minne- apolis on Nov. 14; at Claremont. 4. Answer the same question for Nov. 15 for Minneapolis, and for Nov. 13 for Claremont. RATIONAL GRAMMAR SCHOOL ARITHMETIC 16. The Barometer. Close the upper end of a glass tube 40 inches long and -f-$ of an inch, or more, in diameter, by heating it in a flame until the glass is soft, and drawing off a small piece near the top by a gradual stretch. Draw the end of a piece of rubber tube 8 or 10 in. long over the lower end of the glass tube. Insert into the other end of the rubber tube a piece of glass tube about G inches long and open at both ends. Straighten out the entire tube and pour into it some 44 in. of mer- cury. Bend the rubber tube and tie the whole to a board by a string or wire, as shown in Fig. 17, with the mouth of the short glass tube upward. Provide a home-made scale in inches and tenths (use the scale of tenths of Pig. 3) at the top and bottom of the board, extending a short distance above and below the tops of the mercury columns. The graduations should be numbered in inches above the same hori- zontal line, as ab, across the lower end of the board. The scale at the right gives the length of the long column and that at the left the length of the short column. The difference of these lengths is the length of the mercury column which is supported by the air through the mouth of the short tube. NOTE. If desired, these differences may be obtained by measuring with a yard-stick the lengths ad and be and sub- tracting. FIG. 17 Nov. 11. Long col. 33.15ft. Short col. 2.75 ft. Diff.... 30.40 ft. MARIAN JOHNSON. With an apparatus like this, a fourth grade class measured the two columns on 8 successive days, and found the difference between their lengths. This difference be- tween the lengths of the mercury columns is called the "reading" of the barometer. The readings taken were laid down to scale on the vertical lines of a drawing, as shown in Fig. 18, and a smooth curve was then drawn free-hand through the points. This curve shows to the eye how the noon heights of the mercury columns varied from Nov. 11 to Nov. 20. 1. Can you tell from the words "clear, cloudy" in Fig. 18, what kind of weather follows a high barometer, a low barometer, a falling barometer, a rising barometer 9 INTRODUCTION O.b ' 0.0" 9.5" 3 \ s~ N \ \ / V "^^^ xZ _-/ ON NOON NOON NOON NOON NOON NOON NOON 0.4 29.8 29.35 29.8 30.2 29.8 29.7 29.65 M.J. R.W. F.C. M.F. J.I. R.B. J.M. R.Q. FIGURE 18 barometer reading en- able one to predict what sort of weather we are likely to have? 29.5 NOTE. The pupils whose initials stand be- low the readings were the readers and record- ers of the measured dif- ferences. TABLE OF HIGHEST, LOWEST, AND MEAN BAROMETER READINGS IN INCHES 1900 HIGHEST LOWEST MEAN 1901 HIGHEST LOWEST MEAN Oct. Nov. Dec. 30.48 30.61 30.52 29.46 29.60 29.37 Oct. Nov. Dec. 30.40 30.58 30.52 29.58 29.63 29.60 Autumn Averaj Autumn Range y e . . Autumn Average Autumn Range 1901 HIGHEST LOWEST MEAN 1902 HIGHEST LOWEST MEAN Jan. Feb. Mar. 30.72 30.61 30.28 29.45 29.63 29.41 Jan. Feb. Mar. 30.94 30.51 30.58 29.71 28.73 29.11 Winte Winte r Average r Range 5 Winte Winte r Average r Range 1901 HIGHEST LOWEST MEAN 1902 HIGHEST LOWEST MEAN Apr. May June 30.51 30.30 30.18 29.36 29.55 29.71 Apr. May June 30.31 30.35 30.20 29.25 29.68 29.37 Spring Spring Average Range ... Spring Spring Average 1 Ran ire 1901 HIGHEST LOWEST MEAN 1902 HIGHEST LOWEST MEAN July Aug. Sept. 30.18 30.20 30.34 29.68 29.81 29.58 July Aug. Sept. 30.19 30.23 30.27 29.66 29.75 29.56 Summer Averag Summer Ranee 'Q Summer Averag Summer Ranee. e 28 RATIONAL GRAMMAR SCHOOL ARITHMETIC 3. Find the mean reading by taking the sum of the highest and lowest readings for October, 1900; for November; for Decem- ber. 4. Average these means and write the average in the means column opposite "Autumn Average." 5. Treat all 8 parts of the table in the same way. 6. What is the range (difference between the greatest and least readings) of the barometer during the autumn of 1900? When did the greatest reading occur? the least? , 7. Answer questions similar to 4 for each of the 8 seasons of the table. 8. For which season of the year is the average barometric height greatest? least? What is the range for each of the 2 years? 17. Farm Account Keeping. The forms below show how a cer- tain farmer keeps a systematic account of receipts and expendi- tures with each of his fields. The fields referred to are those shown in the drawing of Fig. 7, p. 11. WRITTEN" WORK The items for which money was paid out or received were put down with the dates of payment or of receipt in a day-book thus: Day-Book for Forty- Acre Cornfield 1. Paid for 4 da. work removing stalks @ 1.15, Mar. 18. 2. Paid for 9 da. plowing @ 1.15, Apr. 30. 3. Paid for 2 da. harrowing @ 1.00, May 1. 4. Paid for 3fc da. planting corn @ 1.20, May 18. 5. Paid for 6 bu. seed corn @ 90^, May 18. 6. Paid for 2 da. replanting corn @ 1.15, May 30. 7. Paid for 5 da. harrowing corn @ 1.15, May 30. 8. Paid for 7 da. cultivating @ 1.15, June 15. 9. Paid for 6 da. cultivating @ 1.15, June 30. 10. Paid for 6 da. cultivating @ 1.15, July 15. 11. Paid for cutting 240 shocks corn @ 8^, Sept. 15. 12. Paid for husking 1392 bu. corn @ 3^, Nov. 20. 13. Paid for shelling 1336 bu. corn @ f 0, Jan. 10. 14. Sold 1336 bu. corn @ 38^, Jan. 10. 15. Received pay for 3 mo. pasture @ 1.50, Feb. 28. 1. These items are here arranged in the form of a receipt and expenditure account. Fill out the vacant columns, and find the totals and the net profit for this forty-acre cornfield: INTRODUCTION In Account with Forty-Acre Cornfield EXPENDITURES RECEIPTS Mar. 18 Removing stalks, 4 da., @ Jan. 10 1 336 bu. corn, 380 $1.15 Feb. 28 3 mo. pasture @ $1.50 Apr. 30 Plowing, 9 da., @$1.15 May 1 Harrowing, 2 da., @ $1.00 TOTAL " 18 Planting corn, 3y z da., @ $1.20 Expenditures to be de- " 18 Seed corn, 6bu.,@ 900 ducted " 30 " 30 Replanting, 2 da., @ $1.15 Harrowing corn, 5 da., @ Net profit from 40 acres " " per acre $1.15 June 1 5 30 Cultivating, 7 da., @ $1.15 6 da., @$1.15 Account closed March 1, July 15 6 da., @ $1.15 1901 Sept. 15 Ciittiiig 240 shocks corn @, 8^0 Nov/20 Husking 1392 bu. corn @ 30 Jan. 10 Shelling 1336 bu. corn @. X* TOTAL 2. From the following list of day-book items for the twenty- acre wheatfield the account below is arranged. Fill out the vacant columns and find the totals, the net profit on the whole field, and the net profit per acre. Day-Bookfor Twenty- Acre Wheatfield 1. Paid for 4 da. plowing @ 12.50, Oct. 20. 2. Paid for 3 da. harrowing and rolling @ $2.25, Oct. 25. 3. Paid for 25 bu. seed wheat @ $1.25, Oct. 25. 4. Paid for 2 da. drilling wheat @ $2.50, Oct. 30. 5. Paid for cutting 20 acres wheat @ 75^, July 15. 6. Paid for 2 da. shocking wheat @ $1.75, July 15. 7. Paid for threshing 480 bu. wheat @ 6/-, Aug. 30. 8. Paid for 3 da. help in threshing @ $1.50, Aug. 30. 9. Sold 425 bu. wheat @ 68/, Dec. 22. 10. Received pay for 4 mo. pasture @ $1.50, Mar. 2. In Account with Twenty-Acre Wheatfield EXPENDITURES RECEIPTS Oct. 20 Plowing, 4 da., $2.50 Dec. 22 Wheat, 425 bu., @ 680 " 25 Harrowing and rolling, Mar. 2 Pasture, 4 mo., @ $1.50 3 da., @$2.25 " 25 Seed wheat, 25 bu., @ $1.25 Expenditures to be de- " 30 July 15 Drilling, 2y z da., @ $2.50 Cutting 20 acres wheat @ ducted Net profit from the field per acre " 15 Shocking wheat, 2 da., @ $1.75 Aug. 30 Threshing 480 bu. wheat @, 60 " 30 Help threshing, 3% da., @ $1.50 TOTAL 3. Draw up these facts into an account, and find the totals and the net profit, and treat in the same way as above : 30 RATIONAL GRAMMAR SCHOOL ARITHMETIC Day-Book for Thirty- Acre Pasture 1. Bought 120 bu. corn @ 37/, Apr. 23. 2. Sold 5 yearling steers @ 18.00, Apr. 25. 3. Sold 3 yearling heifers @ $15.00, July 6. 4. Sold 2 milk-cows @ $46, July 8. 5. Bought 4 two-yr. old heifers @ $25, July 10. 6. Sold 8 hogs weighing 2064 lb., @ $5 per 100 lb., July 20. 7. 8 head of hogs, worth $12 each, died July 25. 8. Bought 20 pigs @ $2.50, Aug. 10. 9. Sold 3 two-yr. old colts @ $72, Sept. 30. 10. Sold 2 draught horses @ $195, Sept. 30. 11. Sold 9 four-yr. old steers @ $68, Oct. 25. 12. Bought 10 yearling calves @ $12, Nov. 1. 13. Sold 3 Jersey milk-cows @ $45, Nov. 15. 14. Paid 6 mo. wages for one man @ $30, Dec. 23. 15. Sold 8 head of hogs @ $15, Jan. 10. 16. Bought 12 head of hogs @ $9.50, Feb. 14. NOTATION AND NUMERATION 18. Digits. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 1. How many characters are given above? By what names do you know the first character? We shall call it zero. These ten characters are called digits o? figures; and by means of them all numbers, no matter how large, are expressed. 19. Place and Name Values of Digits. 1. Read the following numbers: $30 3 30 30,000 300 mi. .3 300 300,000 3 yd. .03 3000 T \ "What is the same in all these numbers? What is different in the numbers? 2. Eead: 7 feet 7 days 7 hundreds 7 millions 7 tenths How many feet? 7 what? How many hundreds? 7 what? How many millions? 7 what? In all these cases, what is the same? What is different? 3. In the number 666, what does the first 6 on the right denote? the second 6? the third? 1st Ans. 6 ones or 6 units. 4. If the value which depends on the name of a digit is called its -name value, what shall we call the value which depends on its place? 5. How many values has a digit? What gives the first value? the second? Which is the permanent value? Which is the changeable value? t 20. ORAL WORK 1. Give the place value of 2 in each of the following numbers : 20 2 200 .2 20,000 2000 2222 2. In 286, for what number does the 2 stand? the 8? the 6? 31 32 RATIONAL GRAMMAR SCHOOL ARITHMETIC We may read the "number 2 hundreds, 8 tens, 6 units, or two hundred eighty-six, meaning two hundred eighty-six units. Writing the names of the three places, they appear: III 286 meaning 200 + 80+6. 3. Read the following numbers, first as hundreds, tens, and units, and then as units: 729 916 335 494 845 642 538 739 21. Periods. For convenience in reading, the digits of large numbers are grouped into periods of three digits each. These periods have names depending upon their position ; but the three places composing each period are always hundreds, tens, and units.* The first seven periods are units, thousands, millions, billions, trillions, quadrillions, and quintillions. The first three periods are sufficient for common purposes. QUINTIL- QUADHIL- TRIL- BlL- MIL- THOU- TTVTT LIONS LIONS LIONS LIONS LIONS SANDS M T x I 1 . *a 1 1 *a I 1 E I S E ! 1 *s i 13 1 . 1 *a H HP H H P s ^ P w P a P WHP w $ P 253 462 571 324 918 645 284 WRITTEN WORK Write the following in figures : 1. Forty -five thousand, two hundred eighty -four; six thousand, nine hundred thirty-four ; three hundred twenty-one thousand, one hundred thirteen ; five thousand, five. NOTE. Fill all vacant places with zeros. *In Great Britain a notation different from that explained in 21 is used. The method of 21 is called the French notation and is used altogether in the United States, France, and Germany. The English notation groups the digits into periods of six figures each : Billions Millions Units In the English notation a million millions make a billion, a million billions a trillion, etc. In the French notation a thousand millions make a billion, a thousand billions a trillion, etc. NOTATION AND NUMERATION 33 2. Nine tnousand, ten ; ten thousand, nine ; six hundred thou- sand, six hundred; five million, fifty thousand, three ; eighty thou- sand, eighty. 22. Decimal Notation. ORAL WORK 1. 10 equals how many units? 1 equals what part of 10? 2. 1 hundred equals how many tens? how many units? 1 ten equals what part of 1 hundred? 1 unit equals what part of 1 hun- dred? 3. 1 thousand equals how many hundreds? 1 hundred equals what part of 1 thousand? 4. The number denoted by each digit in 1111 is how many times as great as the number denoted by the digit to its right? The number denoted by each digit equals what part of the num- ber denoted by the digit to its left? 5. How many tenths in 1 unit? 1 unit equals how many times 1 tenth (.1)? 6. 1 hundredth (.01) equals what part of 1 tenth? 1 tenth equals how many times 1 hundredth? 7. How does the place value of the unit change from right to left? from left to right? 23. Reading Numbers. 1. Read the following numbers : 23,910 800,007 20,390 1,001,001 3,007 .6 365,834 .27 4,004,246 1.07 Mean radius of the earth 636,739,510 centimeters Mean distance to moon 76,429,120 rods Mean radius of sun 138,560,000 rods Mean distance to sun 92,897,500 miles Mean distance to sun = 149,500,000,000 meters Light travels in one minute - 11,199,000 miles 34 RATIONAL GRAMMAR SCHOOL ARITHMETIC Number of vibrations for red light = 482,000,000,000 per second Number of vibrations for green light = 584,000,000,000 per second Number of vibrations for violet light = 707,000,000,000 per second Mass of moon = 75,000,000,000,000,000,000 tons Mass of sun . 26,892,000 moon masses Distance from sun to nearest star = 18,600,000,000,000 miles 24. Writing Numbers. WRITTEN WORK Write in numerals the following : 1. Four billion, two hundred thirteen million, one hundred twenty-two thousand, six hundred seven; forty-five billion, four hundred fifty million, three hundred twenty-seven thousand, one. 2. Seven hundred eighteen billion, two hundred million ; one hundred ninety thousand, four hundred two ; three million, one hundred twenty-four; fifteen thousand, nine; ninety; one hun- dred eighty-six billion; one hundred forty-seven million, one hun- dred thousand, one hundred. NOTE. Persons who work with long numbers, containing many zeros use a shorter way of writing them. They write 25,000,000,000 thus, 25 X 10 9 , and 675,000,000,000,000 is written 675 X 10 12 . The small 9 or 12, written above and to the right of the 10, tells how many zeros are to be written after the first part of the number. This is called the index notation. Write in this index notation the numbers above which contain sev- eral zeros. The digits, excepting zero, were first used by the Hindus, but they were introduced into Europe by the Arabs, and we call the numbers written with these digits Arabic numerals. It would be more just to call them Hindu numerals. 25. Roman Notation. The Romans used letters instead of digits to represent numbers. This notation is still used upon monu- ments, and in numbering chapters and volumes of books. In the Roman notation, 1=1 L = 50 D = 500 V = 5 = 100 M = 1000 X = 10 ADDITION 35 When a letter of less value, as I, is placed before a letter of greater value, as V, the value of the less is to be taken from that of the greater; thus, IV means 4 and XC means 90. When a letter of less value follows one of greater value, its value is added to that of the greater; as, VI for 6, XV for 15, and CX for 110. Repeating a letter repeats its value; as, XX for 20, CO for 200. A horizontal line over a letter increases its value a thousand- fold; as, C meaning 100,000, D meaning 500,000. In interpreting numbers written in the Roman notation, always begin on the right. 26. Change of Notation. WRITTEN WORK 1. Change from Roman to Arabic: XCV CMXIX CDLXXXIV DXXV MLXXX MDCCLXXVI DCIV MCCLXX LXIX XCIX DLVI CCXI 2. Change from Arabic to Roman: 10 65 100 572 619 1902 30 77 500 78 584 1774 60 80 140 50 400 1565 49 95 46 246 2000 2590 }27. Definitions. ADDITION ORAL WORK 1. A bicyclist rides 19 mi. Monday and 10 mi. Tuesday; how far does he ride both days? 2. I paid $28 for a suit of clothes and $20 for a bicycle; how much did I pay for both? 3. A four-sided lot has sides of the lengths: 25 yd., 4 yd., 11 yd., and 28 yd. ; how far is it around the lot? 4. A man works for me 7 hr. one day, 6 hr. the next day, 7 hr. the next, and 8 hr. the next; how many hours does he work for me in all? 36 RATIONAL GRAMMAR SCHOOL ARITHMETIC 5. A newsboy sold 18 papers Wednesday, 12 papers Thursday, and 16 papers Friday; how many papers did he sell in all? 6. The number of hours the sun shone on each of the days of a certain week was: 8 hr., 6 hr., 3 hr., 7 hr., 5 hr., 9 hr., 2 hr. How many hours of sunshine were there during the week? 7. A farmer's fields are of the following sizes : 40 acres, 80 acres, 16 acres, and 34 acres. How many acres in all? 8. How many dollars are $25 and $35 and $20? 9. How many feet are 12 ft., 8 ft., 6 ft., 12 ft, and 4 feet? The answer to problem 1 is the same as the answer to the question: "What single distance is just as long as the distances 19 mi. and 10 mi. combined?" Answering problem 2 answers the question: "What single amount equals $28 and $20 combined?" Combining numbers into a single number is addition. The result of addition is called the sum or amount. The numbers to be combined or added are the addends. Thus in the problem Addend. 29 mi. Sum 19 mi. and 10 mi. are the addends and 29 mi. is the sum. The sign of addition is -f. It is read "plus." In some cases it may be read "and," though it is better to use the correct read- ing, "plus," at all times. The sign = when written between two numbers means that they are equal. It should never be read "is" or "are," but always "equals" or "is equal to." 10. Eead and answer these questions : (1)3 + 8 + 4 = ? (4) 12 + 7 + 8 = ? (7) 3 + 15 + 4 + 9 = ? (2)7 + 6 + 8 = ? (5) 6 + 13 + 7 = ? (8)6 + 16 + 8 + 9 = ? (3)9 + 8+5 = ? (6) 5 + 18 + 9 = ? (9)5 + 14 + 9 + 7 = ? 11. In these questions, what numbers should stand in place of the question mark? (1) 2 8's + 5 8's = ? 8's. (5) 9 #'s + 8 y's = ? y's. (2) 6 9's + 7 9's = ? 9's. (6) 15 z's + 10 z's = ? z's. (3) 12 13's + 7 13's = ? 13's. (7) 8 a's + 17 a's = ? fl's. (4)* 7 z's + 5 's = ? 's. (8) 16 i's + 20 J'g = ? J's. * Read 4 thus, "7 cc's plus 5 cc's equal how many x's?" ADDITION 37 Exercises. WBITTEX WORK 1. In five trips a street-car carried the following number of passengers: 30, 42, 45, 60, 65. Find the whole number of pas- sengers. When the sums can not be seen mentally it is convenient to arrange the addends in this form: CONVENIENT- FORM SOLUTION. Writing the addends so that units, tens, hundreds, etc., shall stand in the same vertical column, we add the units first; thus, 5, 10, 12 units = 1 ten and 2 units. Write the 2 units in units column under the line and the 1 ten in tens column. Then, 6, 12, 16, 20, S3 tens = 2 hundreds and 3 tens. Write the 3 in tens column and the 2 in hundreds column. Then add the partial 12 sums as shown. 23 242 2. On these trips the conductor collected $1.50, $2.10, $2.25, $3.00, $3.25. The fares for the five trips amounted to what? 3. In one week my street-car fare was 350, 250, 400, 250, 300, 300. Find the whole amount. 4. In one evening a telegraph operator sent messages contain- ing the following numbers of words: 2, 6, 3, 12, 6, 3, 3. What was the total number of words dispatched? 5. Fred bought a note book for 100, a lead pencil for 50, a foot rule for 100, a compass for 250, and a bottle of ink for 100. What was the amount of his bill? 6. A train moves the following distances in five successive hours: 40 mi., 45 mi., 50 mi., 50 mi., 48 mi. What distance is traveled in the given time? Find the total sales in each of the following problems : 7. Silk sales: 2| yd., 3 yd., f yd., 2J- yd., 3 yards. 8. Linen sales: 2J yd., 5 yd., 6 yd., 2 yd., 5 yards. 9. Ribbon sales: 1 yd., \ yd., 3 yd., 2 yd., 12 yards. 10. Thread sales: 25 spools white cotton, 6 spools black cot- ton, 3 spools black linen, 4 spools A silk, 6 spools twist. 11. Handkerchief sales: 6, 12, 3, 6, 12, 6, 3, 2. How many dozen were sold? 38 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. Notion sales: 6 papers pins, 2 papers safety pins, 6 papers needles, 2 cards darning wool, 2 combs, 12 bunches tape, 4 bunches braid. How many articles were sold? 13. Coffee sales: -J lb., 2 lb., 2- lb., 3 lb., f lb., l\ lb., 5 lb., 4 pounds. 14. Sugar sales: 18 lb., 12 lb., 12 lb., 20 lb., 5 lb., 6 pounds. 15. Flour sales: 50 lb., 150 lb., 2S lb., 15 lb., 25 pounds. 16. Potato sales: 2 bu., 1 bu., 4 bu., 6 bu., 2 bu., 1 bu., 3 bushels. 17. Apple sales: 2 bu., 1 bu., 3 pk., 2 pk., 1 pk., 2 bu., 1 bu. , 4 bushels. (Answer in bushels. ) 18. Egg sales: 7 doz., 3 doz., 6 doz., 4 doz., 8 doz., 5 doz., 2 dozen. 19. Butter sales: 5 lb., 3 lb., 3 lb., 2 lb.,l|lb., 3 lb., 5 pounds. 29. Problems. 1. A fifth grade staked out a -275' -- * rectangular plot of ground to be used for a playground and school ^ garden. The eighth grade pupils ? fenced the plot in and ran a cross fence separating the parts. The entire plot was 126 ft. wide by 275 ft. long (126' x 275'). How many feet of fence were needed? f | 1 C j Playground .7 *i Garden ; ! I 1 FIGURE 19 SOLUTION. Arrange the addends in convenient form. Beginning at the bottom of units column, think the sums thus, 12, 17, 23, 28 equals 2 tens and S units. Write the 8 in units column. Add the 2 tens to the tens digits, thinking (not speaking) 4, 6, 13, 15, 2'2 tens, equals 2 hundreds and 2 tens. Write the 2 tens in tens place and add the hundreds to the hundreds in the third column, thus 3, 4, 6, 1 9 hundreds. Write the 9 in hundreds place, and since we are adding feet the sum is 928 ft. Test the correctness of the sum by beginning at the top of each column and adding downward. If the sum is the same the work is probably correct. This is called check- ing the work. 2. Add 187, 892, and 478. CONVENIENT FORM 275 ft. 1 126 ft! I 275 ft. } Addends 126 ft. I 126 ft! J ^x f<- Q ADDITION 39 3. In a certain school there were enrolled one month : 96 children in the kindergarten 100 in the fifth grade 182 in the first grade 143 in the second grade 133 in the third grade 123 in the fourth grade 1)5 in the sixth grade 83 in the seventh grade 76 in the eighth grade How many pupils were in the whole school? 4. During the same month, the numbers of cases of tardiness were as follows : Kindergarten ... 10 First grade 12 Second grade... 4 Third grade 8 Fourth grade 5 Fifth grade 9 Sixth grade 12 Seventh grade.. 4 Eighth grade ... 2 What was the total number of cases of tardiness? 5. The numbers of cases of absence for the same month were as follows, beginning with the kindergarten: 87, 60, 46, 21, 15, 20, 15, 14, 9. Find the total number of cases. Teachers may take reports of their own or of other schools and make similar problems. 6. Fill out for each of these cities the total number of rains during May and June, 1902, and the total rainfall in inches. Do not rewrite the numbers. CITY NUMBER or RAINS RAINFALL IN INCHES MAY JUNE TOTAL MAY JUNE TOTAL Chicago 11 13 14 13 13 10 15 16 18 15 16 13 4.46 4.57 3.46 5.02 2.83 2.21 6.00 6.76 6.16 4.19 7.29 9.08 Des Moines Detroit Kansas City Omaha St. Louis The length in feet, the number of officers and men, the dis- placement in tons, and the indicated horse power of eight of the first-class battleships of the United States navy are as given in the table on the following page. 40 RATIONAL GRAMMAR SCHOOL ARITHMETIC NAME LENGTH MEN DISPLACE- MENT HORSE POWER Alabama 368 585 11 525 11 366 Wisconsin 368 585 11 525 10 000 Kearsarge 368 520 11 525 11 954 Kentucky 368 520 11 525 12*318 Iowa 360 444 11 340 12 105 Indiana Massachusetts 348 348 465 424 10,288 10 288 9,738 10 403 Oregon 348 424 10 288 11 111 Total 7. If these 8 vessels stood in a straight line with their ends touching, how far would they reach? 8. How many officers and men are needed to man the 8 ships? 9. The displacement is the number of tons of water the vessel pushes aside as it floats. What is the combined displace- ment? 10. What is the combined horse power of the engines of the & ships? 11. At the close of 1900 the numbers of teachers, schools, and pupils in the 6 provinces of Cuba were as follows : PROVINCE TEACHERS SCHOOLS PUPILS Havana 904 904 41 383 Puerta Principe 246 246 9,355 Santa Clara 879 879 44 872 Santiago 645 645 33 983 Pinar del Rio. . . .... 274 274 13 282 Matanzas. ... . . 619 619 29 398 Total Find the totals and tell what they mean. 12. The 6 European countries from which most immigrants came to the United States in 1901 were as follows : ADDITION 41 COUNTRY IMMIGRANTS IN 1901 IMMIGRANTS IN 1900 MALE FEMALE TOTAL MALE FEMALE TOTAL Italy 106,306 78,725 54,070 12,894 12,875 29,690 34,665 31,187 17,667 10,456 76,088 79,978 60,091 16,672 10,262 24,047 34,499 31,066 19,058 8,388 Austria-Hungary Russia Ireland Sweden Total . Fill out all the totals and tell what they mean. 13. The published daily circulation of a city newspaper from week to week is here given : FIRST WEEK SECOND WEEK THIRD WEEK FOURTH WEEK FIFTH WEEK DAY COPIES 1... 255, 572 "2... 303, 062 3... 323,513 4... 312,724 5... 306, 009 6... 297, 918 DAY COPIES 8... 305, 725 9... 303, 489 10... 304, 991 11.,. 304, 746 12... 305, 515 13... 299, 095 DAY COPIES 15... 304, 636 16... 302, 173 17... 302, 650 18... 304, 255 19... 302, 942 20... 297, 684 DAY COPIES 22... 304, 836 23... 304, 698 24... 310, 870 25... 299, 780 26... 301,455 27... 298, 446 DAY COPIES 29... 303, 383 30... 302, 005 Total Total Total Total Total Total for mo. Find the weekly totals and the total for the whole month. 14. In the year 1891 the United States imported 94,628,119 Ib. of coffee, which was 54,262,757 Ib. less than was imported during the two previous years. How many pounds were imported during the two previous years? 15. The United States imported 79,192,253 Ib. of tea in 1889; 83,494,956 Ib. in 1890; and 82,395,924 Ib. in 1891. How many pounds of tea were imported in the three years? 16. In 1889 $27,024,551 worth of molasses was imported into the United States; in 1890 $31,497,243 worth; and in 1891 $2,659,172 worth. How many dollars' worth was imported during the three years? RATIONAL GRAMMAR SCHOOL ARITHMETIC 17. Of the yearly internal trade of New York state, $1,050,- 000,000 worth of the freight passes over the railroads; $150,000,- 000 over the eauals; and $250, 000, 000 over Long Island Sound and the lakes. What is the total value of the internal trade? THE TEN LARGEST CITIES OF THE UNITED STATES POPULATION New York 3,437,202 Chicago 1,698,575 Philadelphia 1,293,697 St. Louis 575,236 Boston' 560,892 Baltimore 508,957 Cleveland 381,768 Buffalo 352,387 San Francisco 342,782 Cincinnati 325,902 THE TEN LARGEST FOREIGN CITIES POPULATION London 4,433,018 Paris 2.536,834 Canton 2,000,000 Berlin 1,677,304 Vienna 1,364,548 Tokyo 1,268,930 St. Petersburg 1,267,023 Pekin 1,000,000 Moscow 988,610 Constantinople 900,000 18. Find the total population of the ten largest cities of the United States. Test the correctness of your addition hy adding the columns from top downward. Teachers may omit the following method of checking if thought too difficult. A very useful method of checking long problems in addition is known as casting out the nines. To cast out the nines from a number add its digits and whenever the sum equals or exceeds 9, drop 9 and continue adding the next digits to what remains, dropping 9 whenever the sum equals or exceeds 9. The last remainder is called the excess. ILLUSTRATION. Cast the 9's out of 647,255. Beginning on the left, 6 + 4=10; drop 9, giving the remainder 1. 1 +- 7 + 2 = 10, drop 9. 1 + 5+5=11, drop 9. The last remainder is 2 and this is the excess of 647,255. To check addition by casting out nines, cast the nines out of the addends and the sum. Then cast out the nines from the excesses of the addends. If this last excess of the excesses equals the excess of the sum, the work is probably correct. ILLUSTRATION 8,465 3,282 4,497 2,957 7,642 EXCESS 5 6 6 5 1 26,843 5 5 5 = excess of the sum. 5 = excess of ^xcesses of addends. To obtain the excess of the excesses of the addends : 5 + 6 = 11, drop 9, giving 2. 2 + 6 + 5 = 13, drop 9, giving 4. 4 + 1 = 5, the excess of the ex- cesses of the addends. Since this equals the excess of the sum, the work is checked. ADDITION 43 10. Find the population of the ten largest cities of Europe and Asai. (See table, p. 42.) 20. Find the total population of these twenty cities. 21. Find the population of those cities in both lists having more than 900,000 inhabitants. 22. Make and solve similar problems based on the table. 30. Distribution of Population in the United States in 1900. FIGURE 20 On the map the main geographical divisions are surrounded by heavy full lines : . The divisions are separated into sections by heavy dotted lines : . Thus the New England states are the eastern section of the North Atlantic division; and New York, Pennsylvania, and New Jersey are the western. Teachers may select such problems from this list as have a special geographical interest to their school. 1. From the table of statistics given on the following page, find the total population of the New England states for 1900. 2. Of the North Central division. 3. Of the Western division. 4. Of the United States with outlying territory. KATIONAL GRAMMAR SCHOOL ARITHMETIC 5. Find the total area in square miles of the United States with outlying territory. AREA AND POPULATION OP STATES AND TERRITORIES STATE OB TERRITORY AREA IN SQ. MILES POPULA- TION 1900 PUPILS IN ELEM. & SEC. SCHOOLS STATE OR TERRITORY AREA IN SQ. MILES POPULA- TION 1900 PUPILS IN ELEM. & SEC. SCHOOLS Me. . N. H Vt 33,040 9,305 9,565 8,315 1,250 4,990 49,170 7,815 45,215 694,666 411,588 343,641 2,805,346 428,556 908,420 7,268,894 1,883,669 6,302,115 130,918 65,193 65,964 474,891 64,537 155,228 1,209,574 315,055 1,151,880 Ky Tenn. . . . Ala 40,400 42,050 52,250 46,810 48,720 265,780 39,030 53,850 31,400 2,147,174 2,020,616 1,828,607 1,551,270 1,381,625 3,048,710 398,331 1,311,564 392,060 501,893 485,354 376,423 360,177 196,169 578,418 314,662 99,602 Mass R I Miss La Tex Okl Ark Conn N. Y N J Penn Ind. T. . . . North At- lantic divi- sion South Cen- tral divi- sion Del .. 2,050 12,210 70 42,450 24,780 52,250 30,570 59,475 58,680. 184,735 1,188,044 278,718 1,854,184 958,800 1,893,810 1,340,316 2,216,331 528,542 33,174 229,332 46,519 358,825 232,343 400,452 281,891 482,673 108,874 Mont. . .. Wy Col N. M 146,080 97,890 103,925 122,580 113,020 84,970 110,700 84,800 69,180 96,030 158,360 243,329 92,531 539,700 195,310 122,931 276,749 42,335 161,772 518,103 413,536 1,485,053 39,430 14,512 117,555 36,735 16,504 73,042 6,676 36,669 97,916 89,405 269,736 Md D. C Va W.Va.... N. C so. . . Ariz Utah Nev Idaho Wash. ... Ore Ga. Fla South At- lantic divi- sion Cal Western division Ohio Ind Ill Mich Wis Minn . ... Iowa Mo 41,060 36,350 56,650 58,915 56,040 83,365 56,025 69,415 70,795 77,650 77,510 82,080 4,157,545 2,516,462 4,821,550 2,420,982 2,069,042 1,751,394 2,231,853 3,106,665 319,146 401,570 1,066,300 1,470,495 829,160 564,807 958,911 498,665 445,142 399,207 554,992 719,817 77,686 96,822 288,227 389,583 U. S. with- out outlying territory Alaska ... Hawaii... Phil. Is.. Tutuila . . Guam . . . Porto ) Rico f 590,884 6,449 114,410 77 150 3,531 63,592 154,001 6,961,339 6,100 9,000 953,243 N.D S. D Neb Kan North Cen- tral divi- sion Total U. S. with outly- ing territory ADDITION 45 6. Find the area in square miles of the North Central division. 7. Of the South Central division. Which has the greater territory? 8. What state in the Union contains the smallest number of square miles? the greatest? What state supports (contains) the greatest population? the smallest? How do the areas of these two states compare? 9. Find the number of pupils attending school in the New England states; in the North Central division. 10. Find the total number of children attending school in the United States. 11. Make such original problems as these: Find the total population in 1900 of the 13 original states; the total area of the 13 original states; of the states east of the Mississippi river, etc. 12. Check the correctness of your work in problems 2 and 7 by adding the columns first as a whole and then adding the foot- ings of the separate sections and comparing the two sums. 13. Find the population and the area of the eastern North Atlantic states ; of the western. 14. Find the number of pupils attending the elementary and secondary schools in both the sections mentioned in problem 13. 15. Check the work of problems 1 and 4 by casting out the nines. 31. Measurements. A square unit, or a unit square, is a square each of whose sides is 1 unit long. The area of a surface is the number of square units in it. 1. What is a square inch? a square foot? a square yard? a square mile? a square rod? 2. If a rectangle is made up of 5 rows of 10 unit squares each, what is its area? How many tens of unit FIGURE 22 squares are in the rectangle? 4C RATIONAL GRAMMAR SCHOOL ARITHMETIC 3. If a rectangle is 12 units long and 6 units wide, how many twelves of unit squares are there in the area? 4. If a rectangle is x units long and FIGURE 23 9 units wide, how many z's of unit squares are in its area? NOTE. Seven x's is written 1x and is read FIGURE 24 "seven a?." 5. How wide must a rectangle z inches long be to contain 6z sq. in.? 152 square inches? 6. How long must a rectangle 15 in. wide be to contain I5a square inches? 7. The area of one rectangle is Sz sq. in. and of another 7z sq. in. ; how many square inches in their sum? 8. How many 9's are 8 9's and 7 9's? How many 12's are 8 12's and 7 12's? How many z's are Sx and 7x? 8z + 7z = ? 9. Add these numbers : (1) 9a (2) Sa (3) 26z (4) 48* (5) 76y (6) 695 Sa . 60 47z 982 49z/ 795 (7) Qm (8) 68 (9) 84c QO) 73^ (11) 78G^ (12) 960s 8w 75^ 76c 39d 31 9a; 9m SUBTRACTION 32. Definitions. ORAL WORK 1. A man earns $180 a month and spends $140. What is the difference between his earnings and expenditures? 2. One farmer owns 320 A. of land, another 80 A. Find the difference in size of the two farms? 3. From a cheese weighing 43 lb., 23 Ib. were sold; how many pounds remained? 4. From a bin containing 72 bu. of oats, I use 32 bu. How many bushejs remain? SUBTRACTION 47 5. The sum of two numbers is 48, and one of the numbers is 28 ; what is the other? G. What is required in the first two problems of this section? in the second two? in the fifth problem? 7. What is subtraction? 1 What is the result in subtraction called? Subtraction means either of two things : (1) The way of finding the difference, or remainder, of two numbers. (2) The way of finding either one of two addends when their sum and the other addend are known. With the first meaning, the number from which we subtract is the minuend. The number to be subtracted is the subtrahend. The result is called the difference, or remainder. With the second meaning, the known sum is the minuend. The known addend is the subtrahend, and the unknown addend is the difference, or remainder. The sign of subtraction is read "minus," and when placed between two numbers means that their difference is to be found. The minuend is always written before the sign. 8. Tell what number the letter stands for in these problems : (!) 8-5 = 3 (:3) 16-7 = (5) 15-8 = * (3) 17 -9 = 3 (4) 25 - y = 8 (6) 16 - z = 8 9. What digit should take the place of the question mark in these problems? (1) 7 S's-5 8's = ?8's (4) 9 a- G x = 1x (2) 8 12's - 3 12's = ? 12's (5) 15 x - 7 x = ? x (3) 4 x - 2 x = ? x (6) |15 x - ? x = 8 x WRITTEX WORK 1. A traveler has a journey of 437 mi. to make and he has already traveled 199 mi. of it; how much farther must he travel? SOLUTION. 437 means 400 + 30 + 7, and 199 means 100 +90 + 9. We arrange the numbers conveniently, thus : Minuend 437 mi. Subtrahend 199 mi. Remainder 238 mi. Beginning on the right, 9 units can not be taken from 7 units. We take one of the 3 tens and add it to the 7 units, giving 17 units. Then 17 units 9 units = 8 units. Write 8 in units column. Passing to tans 48 RATIONAL GRAMMAR SCHOOL ARITHMETIC column, we can not take 9 tens from the 2 tens remaining, so we take one of the 4 hundreds and add it to 2 tens, making 12 tens. Then 12 tens 9 tens = 3 tens. Write 3 in tens column. Finally, 3 hundreds 1 hundred = 2 hundreds. Write the 2 in the hundreds column. The remainder or difference is 238 miles. 2. How can you prove 238 to be the correct remainder or difference in problem 1? 3. The minuend is 1284, and the subtrahend is 876. What is the difference, or remainder? Prove. 4. When the water was dried out of 38 oz. of natural soil, the dry soil weighed 29 oz. Make a problem from these facts, solve it, and prove your solution correct. 5. A subtrahend is 99, and the difference is 338. What is the minuend? How is it found? 6. From 72 hr. subtract 1 da. and 4 hours. NOTE. A correct answer to this problem would be 72 hr. 1 da. and 4 hr. But the answer is to be expressed in a single unit. To do this, first change 1 da. and 4 hr. to hours. 1 da. = 24 hr. ; 24 hr. -f- 4 hr. = 28 hr. 72 hr. 28 hr. = 44 hours. 7. From 36 ft. take 5 yd. What must first be done to obtain the result in a single unit? Make complete statements. 8. 20 qt. - 2 pk. = x qt. What is the first thing to be done? Make complete statements. Find the value of x. 9. From f yd. take f yd. What is the unit of f yd.? of f yd.? What then must be done before subtracting? Why? NOTE. Only numbers expressed in the same unit can be subtracted if the result is to be expressed in a single unit. 10. Restate the last problem, using the sign of subtraction. 33. Exercises. 1. From a barrel of vinegar containing 31 J gal., suppose the following quantities to be removed, and give successive remainders : 6 gal. ; 2i gal. ; 5 gal. ; 3 gal. ; 2* gal. ; 2J gallons. 2. From a wagon box containing 42 bu. of potatoes, a farmer sold the following quantities: 2 bu., 4 bu., 3 bu., 2 bu., 5 bu., 6 bu., 9 bu., 4 bu. How many bushels remained? 3. On March 31 the gas meter read 30,000 cu. ft., and on April 30, 45,000 cu. ft. How many cubic feet of gas had been used during the month? SUBTRACTION 49 4. A horse and his rider weigh 1375 Ib. The man weighs 160 Ib. What is the weight of the horse? 5. Find and read the differences in these exercises: (1) W (3) $9274.75 $4246.28 1,024,637 yd. 1846.19 1571.16 725,448 " 6. Find the differences : (1) (2) 6,420,014 rd. $3472.06 1,382,741 " 2189.24 7. During one week a merchant's transactions at his bank were as follows: deposits, $217.40, $343.27, $290.00, $365.18, $380.24, $415.17; withdrawals, $200, $320.25, $195.75, $286, $240.15, $395.25. Which item was the greater at the close of the week, and how much? 8. During one week of November, 1901, 63,188 cattle were received at the Chicago stock yards; during the corresponding week of 1899, 32,722 were received. What was the increase? 9. Great Salt Lake is 4200 ft. above sea level, and Lake Su- perior 602 ft. x is the difference in elevation. Find value of x. 10. Cuba contains 45,884 sq. mi., and the Philippines, 114,410 sq. mi. Find the difference in areas. For the numbers for problems 11-17, see the tables, p. 50. 11. Find the total receipts at Chicago for the week Nov. 25-30, 1901, of cattle; of calves; of hogs; of sheep. 12. Find the total shipments. 13. Find the difference between receipts and shipments. 14. Compare the receipts with those of the previous week; with those of the corresponding week of 1900; of 1899. 15. Compare shipments in the same manner. 16. Find total receipts for one week in the four markets, Chi- cago, Kansas City, Omaha, and St. Louis, of cattle; of hogs; of ep. 17. Compare Chicago receipts with those of the other cities. 50 RATIONAL GRAMMAR SCHOOL ARITHMETIC A WEEK'S RECEIPTS AND SHIPMENTS OF LIVE STOCK. CHICAGO, Nov. 30, 1901 RECEIPTS CATTLE CALVES HOGS SHEEP Monday Nov. 25 16,078 6,941 17,583 5,857 800 537 735 481 162 40 39,630 41,654 51,275 44,090 25,000 26,170 18,297 12,393 12,462 2,000 Tuesday ' 26 Wednesday, * 27 Thursday, ' 28 (Holiday) . . Friday, 29..... Saturday, 30 Total 63,188 53,965 32,722 3,814 1,822 1,335 273,426 191,104 159,724 104,528 55,046 61,718 Previous week Corresponding week 1900 . . Corresponding week 1899 SHIPMENTS CATTLE CALVES HOGS SHEEP Monday Nov 25 2,578 1,904 5,647 1,875 400 o 1 16 205 30 7,543 1,495 5,712 4,308 5,000 2,852 4,518 4,501 2,127 1,000 Tuesday 4 20 Wednesday, * 27 Thursday, ' 28 (Holidav) . . Friday k 29 Saturday ' 30 Total 18,699 18,438 9,556 324 352 210 29,486 23,575 16,637 24,429 14,332 3,182 Previous week Corresponding week 1900 . Corresponding week 1899 . LIVE STOCK RECEIPTS FOR ONE WEEK AT FOUR MARKETS CATTLE HOGS SHEEP Chicago 47,300 201,600 71,300 Kansas City 29,300 89,000 14,900 Omaha 16 800 65 200 12 800 St Louis 10,700 38,100 5 800 Total : Previous week 144,600 481,700 173,600 Corresponding week 1900* 111,500 339,800 85,000 Corresponding week 1899 91 700 206 100 81 300 Corresponding week 1898 ... 128 400 365,500 95 600 Corresponding week 1897 143,000 374,100 105,600 SUBTRACTION 51 18. Compare the total receipts of the week with those of the previous week; of the corresponding week of 1900; 1899; 1898; 1897. 34. Geography. 1. Lake Titicaca is 12,645 ft. above the level of the sea, and Lake Superior 602 ft. Find the difference in eleyation. 2. Mt. Everest is 29,002 ft. above sea level, and Mt. Blanc 15,744 ft. What is the difference in altitude? 3. The Mississippi basin has an area of 1,250,000 sq. mi.; the Amazon, of 2,500,000 sq. mi. How much greater is the basin of the Amazon than that of the Mississippi? 4. The area of the Philippines is 114,410 sq. mi. ; that of California, 158,360 sq. mi. The state is how much larger than the islands? 5. Compare the same islands with Texas; with Illinois; with Rhode Island. (See table, p. 44.) 6. Of a population of 6,302,115 in Pennsylvania, 1,151,880 are pupils in elementary and secondary schools. How many per- sons are not attending those schools? 7. In a certain year the Hawaiian Islands produced 221,694 T. of sugar, while Cuba produced 880,372 T. How much more did Cuba produce than the Hawaiian Islands? 8. In 1890, 25,403 manufacturing establishments in New York City paid for wages $230,102,167; for materials, $366,422,- 722; for miscellaneous expenses, $59,991,710. The value of the products was $777,222,721. ^What was the total profit for all the establishments? 9. The battle of Lexington was fought in 1775, and the battle of Santiago in 1898. How many years elapsed between the two battles? For the numbers for problems 10-19, see table, p. 44. 10. What is the difference of the areas of the largest and the smallest states? 11. How many more persons are in the state with the largest population than in the one with the smallest? 52 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. How many more children attend school in one of these two states than in the other? 13. Make and solve other problems, comparing the areas, total populations, and school populations of the different states. 14. Which is the larger, and by how much, the North Atlantic division or the South Atlantic division? the North Central or the South Central? the South Central or the Western? 15. Compare the total populations of these groups. 16. How many persons are not attending school in Maine? in Mississippi? How do these two results compare? 17. Find how many persons are not attending school in your own state; in bordering states. 18. How many more persons are not attending school in New York than in Illinois? How does this result compare with the difference in the total populations of these states? 19. Make and solve other problems of interest to your school. 20. Find, by comparing the populations for 1890 (p. 53) with those for 1900 (p. 44), the increase of population for the ten years in each state of the North Atlantic division ; in the whole division. 21. Find the increase of population in your own state from 1890 to 1900. What is the average yearly increase? 22. How many years since Illinois was admitted as a state? since your native state was admitted into the Union? 23. Make similar problems from the table opposite and the table on p. 44, solve your problems, and check your work. 24. Each principal division of states is separated into two or more sections by heavy dotted lines on the map, p. 43. Find how much greater, or less, the grow'th of population since 1890 has been in one section of each division than in another. 35. Commerce. 1. Find the total export trade of the United States with the twelve countries given in the table on p. 54, in 1891 and in 1901. 2. Find the increase, or decrease, for each country, and place the difference in the last column. Whenever the difference is an increase write + before it. When it is a decrease write before it. Make the subtractions without rewriting the numbers. SUBTRACTION 53 DATES OF ADMISSION AND POPULATIONS FOR 1890 OF STATES AND TERRITORIES GEOGRAPHICALLY GROUPED DATE OF ADMISSION POPULA- TION 1890 DATE OF ADMISSION POPULA- TION 1890 Me Mar 15, 1820 661 086 Ky June 1 1792 1 848 635 N H June 21, 1788 376,530 Tenn. . . . June 1 1796 1,767,518 Vt Mar 4, 1791 332,422 Ala Dec. 14, 1819 1,513,017 Mass .... Feb. 7, 1788 2,238,043 Miss Dec. 10, 1817 1,289,600 R I May 20 1790 345 506 La April 30 1812 1 118 587 Conn. . . . N Y Jan. 9, 1788 July 26 1788 746,258 5 997,853 Tex Okl Dec. 29, 1845 May 2 1890 2,235,523 61 834 N J Dec. 18 1787 1,444,933 Ark June 15, 1836 1,128 179 Penn. ... Dec. 12, 1787 5,258,014 Ind. T. . . June 30, 1834 180,182 North , sion Atlantic divi- South sion . Central divi- Del Dec. 7 1787 168 493 Mont Nov 8 1889 132,159 Md. April 28 1788 1,042,390 Wy July 10 1890 60,705 DC.. Mar. 30, 1791 230,392 Col. . . . Aug 1 1876 412,198 Va June 25, 1788 1,655,980 N. M Sept. 9, 1850 153,593 W. Va. . . N. C s c. . June 20, 1863 Nov. 21, 1789 Mar. 23 1788 762,794 1,617,947 1,151 149 Ariz Utah Nev Feb. 24, 1863 Jan. 4, 1896 Oct 31 1864 59,620 207,905 45,761 Ga Fla Jan. 2, 1788 Mar. 3, 1845 1,837,353 391,422 Idaho. . . . Wash. . . . Ore July 3, 1890 Nov. 11, 1889 Feb. 14, 1859 84,385 349,390 313,767 South j Atlantic divi- Cal Sept. 9, 1850 1,208,130 sion Westen i division Ohio Ind 111. Mich.. .. Feb. 19, 1803 Dec. 11, 1816 Dec. 3, 1818 Jan. 16, 1837 3,672,316 2,192,404 3,826,351 2,093,880 U. S. lying without out territory Wis Minn. . . . Iowa Mo .... May 29, 1848 May 11, 1858 Dec, 28, 1846 Aug 10 1821 1,686,880 1,301,826 1,911,896 2 679 184 Alaska . . Hawaii. . Phil Is July 27, 1868 April 30, 1900 32,052 89,990 N. D Nov 2 1889 182 719 Tutuila S. D Neb Kan Nov. 2, 1889 Mar. 1, 1867 Jan. 29, 1861 328,808 1,058,910 1,427,096 Guam . . . Porto I Rico J North sion . Central divi- Total U Iviner . S. with out- territory . . . RATIONAL GRAMMAR SCHOOL ARITHMETIC UNITED STATES EXPORT TRADE COUNTRY 1901 1891 TEN-YEAR DIFFERENCE United Kingdom $598 766 799 $482 295 796 Germany 184,678,723 90 326 332 Canada 107,496 522 41,686,882 Netherlands 85,643,804 31,261,766 Mexico 36 771 568 15 371 370 Italy . . 34 046 201 14 447 004 British Australasia . 30 569 814 13 564 931 British Africa 24,994 766 3 511 668 Japan 21 162 477 3 839 384 Brazil Argentina 11,136,101 11 117 521 15,064,346 1 909 788 Russia 6 504 867 5 400 357 Total VALUES OF MANUFACTURED AND INDUSTRIAL PRODUCTS MANUFACTURED ARTICLES 1902 1897 DIFFERENCE Agricultural implements 82,075,609 8 243 466 Books, maps, etc 988,195 470,358 Carriages and cars 913,513 80,065 Copper ingots . . 198,438 31 583 Cotton cloths . 385,086 1 499 769 Cotton manufactures, other Cycles and parts of . 1,634,642 98,476 983,661 339 563 Builders' hardware 735,165 377,549 Sewing machines . 182,710 69,756 Other machinery 894,330 1,222,708 Total OTHER ARTICLES 1902 1897 DIFFERENCE Corn $1 468 390 $1 770 531 Wheat 3 769 577 2 548 778 Wheat flour 638 361 2 415 519 Coal . . 5 473 177 6 987 856 Cotton 4 509 205 2 626 679 Fruits and nuts 1 345 60 566 584 Furs and fur skins 667 164 195,534 Cot ton -seed oil. 261 688 47 069 Beef salted or pickled 240 978 208 195 Bacon 557 827 365 419 Hams . 218 995 188 116 Total SUBTRACTION 55 3. The preceding columns show the values of the different kinds of goods exported by the United States to Canada during the nine months ending March, 1902, and March, 1897, respectively. Fill out the vacant column of differences, marking the difference + whenever it denotes an increase, and when it denotes a decrease. 36. Data for Individual Work. SCHOOL STATISTICS FOR THE THIRTY LARGEST CITIES OP UNITED STATES CITY POPULA- TION. CENSUS 1900 POPULA- TION. CENSUS 1890 SCHOOL ENROLL- MENT. 1900 NUMBER TEACH- ERS. 1900 SCHOOL EXPENDI- TURES. 1900 New York 3,437,202 2,492,591 559,218 12,212 21,040,810 Chicago 1,698,575 1,099,850 262,738 5,951 7,929,496 Philadelphia 1,293,697 1,046,964 151,455 3,591 4,677,860 St. Louis Boston 575,236 560,892 451,770 448,477 82,712 91,796 1,751 2,018 1,526,140 3,664,298 Baltimore Cleveland Buffalo 508,957 381,768 352,387 434,439 261,353 255,664 65,600 59,635 56,000 1,600 1,303 1,300 1,279,936 ,933,965 ,408,000 San Francisco 342 782 298,997 48 517 1 017 152 631 Cincinnati Pittsbur " . 325,902 321,616 296,908 238,617 44,285 50,000 993 1 000 ,064,047 ,757 381 New Orleans Detroit 287,104 285,704 242,039 205,876 31,547 40,303 782 966 455,073 1,251,825 Milwaukee 285,315 204,468 37,000 900 733,510 Newark 246,070 181,830 41,870 851 1,213,660 Washington Jersey City 218,196 206,433 188,932 163,003 40,069 32,174 1,043 586 634,153 Louisville Minneapolis Providence 204,731 202,718 175,597 161,129 164,738 132,146 27,626 38,591 23,485 650 892 682 555,811 841,000 682,000 Indianapolis . 169 164 105 436 27 334 650 729,106 Kansas City, Mo. . . St. Paul Rochester Denver . . 163,752 163,065 162,608 133 859 132,716 133,156 133,896 106 713 28,280 26,000 24,896 27 181 700 610 692 530 524,065 672,350 682,018 750 180 Toledo 131 822 81 434 21,467 455 471,314 Allegheny 129,896 105 287 20,104 377 835,634 Columbus Worcester Syracuse 125,560 118,421 108,374 88,150 84,655 88,143 18,855 19,600 21,090 502 574 485 771,132 529,937 409,073 Total Problems like the following may be made from the table, and assigned to different pupils. 56 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1. Find the totals of all columns of the table for the cities whose inhabitants numbered over 500,000 at the 1900 census. 2. Find the totals for the cities whose populations in 1900 were between 250,000 and 500,000; for cities with populations between 160,000 and 250,000. NOTE. These intervals may be shortened or lengthened at will. Indi- vidual pupils may do different parts of the work, thus obtaining a large number of individual problems. In each case the pupil should tell what his total means. 3. Find the increase in population from 1890-1900 of Cleve- land, Ohio ; of other cities. 4. How many more teachers in 1900 were there in Minneapolis than in Louisville? in Xew York City than in Chicago? in Boston than in St. Louis? in Chicago than in Boston? 5. How many more pupils in 1900 were enrolled in the schools of Boston than in those of St. Louis? than in those of Balti- more? of Cleveland? 6. How much more money was expended in 1900 on schools in Boston than in St. Louis? than in Cleveland? How much more in Cleveland than in Baltimore? than in Buffalo? How much more in Pittsburg than in Cincinnati? 7. How do the combined school expenditures for New York City, Chicago, Philadelphia, and Boston compare with the com- bined school expenditures of all the rest of the cities in the table? How do those of New York City and Chicago compare with the total of all the rest? 8. Find the totals for the five columns of the entire table. What is the increase in the total population of these cities from 1890 to 1900? 37. Subtraction of Literal Numbers. 1. How many five-cent pieces are 7 five-cent pieces 3 five- cent pieces? 2. How many dimes are 12 dimes 5 dimes? 3. How many 9's are 16 9's -9 9's? 4. How many c's are 13c-7c? MULTIPLICATION 5? 5. Write the differences : 183 26y 432 682 6830 1021s 93 8y 37* 39z 97a 879* 6. A lot contains 1603 sq. ft., and the house covers 403 sq. ft. ; how many square feet of the lot are not covered by the house? 7. A boy earned 153 cts. on Friday and spent 103 cts. on Sat- urday ; how many cents did he save? 8. Tell what number x stands for in these problems : (1) 12-a = 7. (2) 65-a; = 20. (3) 3-35=60. (4) 3-19-7. (5) 33-23 = 8. (6) 73-53 = 10. (7) 16^-133 = 15. ' (8) 93-53 = 20. (9) 73=63. (10) 93 = 108. (11) 503 = 100. (12) |3 = 50. (13) |3=27. (14) |3 = 7. (15) f3 = 63. Literal numbers are numbers denoted by letters. MULTIPLICATION \. Definitions. ORAL WORK 1. At 75^ a day, how many dollars does a boy earn in 6 days? This problem may be solved in two ways. We may say he earned the sum of 75^ -f 75^ + 75^ + 75^ + 75$ -I- 75^, which is $4.50. Or, we may say he earned 6 times 75^ (6 x 75#), or $4.50. In either case his earnings are $4.50. 2. Which is the shorter way? How do the addends compare in the first solution? 3. Find in two ways how far a vessel will sail in 4 da., making 22 mi. a day. 4. A factory employee, working by the hour, makes the follow- ing daily record for a week : 8 hr. ; *8 hr. ; 8-J- hr. ; 9 hr. ; 9 hr. ; 9 hr. How many hours does he work during the week? 5. Can this problem be solved in both ways? Give reason for your answer. 6. When the addends are unequal, as in problem 4, what is the only way they can be combined? 7. When the addends are equal, as in the first two problems, in how many ways can they be combined? Which is the shorter 58 RATIONAL GRAMMAR SCHOOL ARITHMETIC way? By what name do we know this shorter way? What then is multiplication? Multiplication of whole numbers is a short way of finding the sum of equal addends when the number of addends and one of them are given. The given addend is the multiplicand. The number of equal addends is the multiplier. The result, or sum, is called the product. The sign of multiplication x is read "multiplied by" when it is written after the multiplicand and "times" when it is written before the multiplicand. Thus 22 mi. x 6 = 132 mi. is read "22 mi. multiplied by G equals 132 mi.," and 0x22 mi. = 132 mi. is read "6 times 22 mi. equals 132 miles." An expression like (5 x 22 mi. = 132 mi. is called an equation. 8. At $3.50 a pair, how many dollars will 12 pairs of shoes cost? $3. 50 = cost of 1 pair; $3.50X 12 = cost of 12 pairs. $3.50 is the multiplicand, 12 is the multiplier , and $42.00 is the product. When a problem is expressed in an equation, the equation is called the statement of the problem. 39. WRITTEN WORK Make statements and solve : . 1. There are 128 cu. ft. in 1 cd. of wood. How many cubic feet in 15 cords? STATEMENT. 15 x 128 cu. ft. = ? Or, 15 X 128 cu. ft. = a; cu. ft. Find the number which should stand in place of x. The number which should stand in place of x in an equation is called the value of x. 2. There are 5280 ft. in a mile. How many feet in 24 miles? 3. 12 Ib. of ham at 170 per Ib. = how many dollars? 4. A cold wave from the northwest traveling at the rate of 33 mi. an hour moves how many miles in 24 hours? MULTIPLICATION 59 5. If galvanized telegraph wire weighs 525 Ib. per mi., how many pounds of wire will it take to stretch 6 wires from Chicago to Milwaukee, a distance of 82 miles? 6. How much will this wire cost at 80 per pound? 7. Find the cost of 275 fence posts at 650. .65 = cost of one post; 275 = number of posts ; $. 65 X 275 = $178.75, cost of 27 posts. In this problem, which number is the multiplicand? 32 5 Which the multiplier? 455 130 $178.75 But it is usually more convenient to use the smaller number as the multiplier. This may be done, noticing that if each post cost 10, 275 posts would cost $2.75; 1375 but as each post costs 050, the cost of all will be 1650 65 x $2. 75, and the work may be arranged as shown here. Or, we may say 275 times $.65 is the same as 65 times $2.75, and multiply as above. 8. In 64 pk. there are how many quarts? 9. How many pounds in 494 bu. of corn, if in 1 bu. there are 56 Ib.? how many pounds in 25 bu.? how many in x bushels? 10. There are 231 cu. in. in 1 gal. How many cubic inches in 587 gall.? in y gall.? in a gallons? 40. Tables. These products must be learned thoroughly : 3X3=9=3X3 3X4= 12 = 4X3 4x3 = 12= 3X 4 4X4=16 = 4X 4 5X3 = 15=3X5 5X4 = 20 = 4X5 6X3 = 18= 3X 6 6X4=24 = 4X 6 7x3 = 21 =3X7 7X4 = 28 = 4X7 8X3 = 24 =3X8 8x4=32 = 4x8 9x3 = 27 =3X9 9x4,= 36 = 4x9 10 X 3 = 30 = 3 X 10 'iO X 4 = 40 = 4 X 10 11 X 3 = 33 = 3 X 11 11 X 4 = 44 = 4 X 11 12 X 3 = 36 = 3 X 12 12 X 4 = 48 = 4 X 12 60 RATIONAL GRAMMAR SCHOOL ARITHMETIC Learn each of these in the same way: 3X6 = 18 3 X 7 = 21 3 X 8 = 24 3X9= 27 4 X 6 = 24 4 X 7 = 28 4 X 8 = 32 4X9= 36 5 X 6 = 30 5 X 7 = 35 5 X 8 = 40 5X9= 45 6 X 6 = 36 6 X 7 = 42 6 X 8 = 48 6X9= 54 7 X 6 = 42 7 X 7 = 49 7 X 8 = 56 7X9= 63 8 X 6 = 48 8 x 7 = 56 8 X 8 = 64 8x9= 72 9 X 6 = 54 9 X 7 = 63 9 X 8 = 72 9X9= 81 10 X 6 = 60 10 X 7 = 70 10 X 8 = 80 10 X 9 = 90 11 X 6 = 66 11 X 7 = 77 11 X 8 = 88 11 X 9 = 99 12 X 6 = 72 12 X 7 = 84 12 X 8 = 96 12 X 9 = 108 ORAL WORK 1. At current prices, find the cost of the following articles: | Ib. Oolong tea. 2. 2 erasers. 3 doz. eggs. 5 Ib. ham. 3 Ib. leaf lard. 2 Ib. Java coffee. 3 Ib. best quality of butter. 3 tablets linen paper. 3 packages white envelopes. 4 doz. pens. 2 lead pencils. 2 bottles writing fluid. 3. 2 doz. oranges. 2 doz. lemons. 1-J- doz. bananas. 1 pk. apples. 4. 8 Ib. rib roast. 2-J- Ib. porterhouse steak. 3 Ib. lamb chops. 2 Ib. sausage. 5. 3 cd. hard wood. 2 T. hard coal. 1^ cd. pine slabs. 4 T. soft coal. 6. 2 T. hay. 3 bu. oats. 2 bu. corn. 4 bales straw. WRITTEN WORK 1. Fill out the vaoant columns and find the totals in the fol lowing table of receipts and expenditures for the fifty-acre oat- field which was shown in Fig. 7, p. 11: MULTIPLICATION 61 In Account with Fifty-Acre Oatfield EXPENDITURES RECEIPTS Mar. 20 Removing old stalks 5 da. Feb. 15 1800 bu. oats @ 22J0 @*1.25 " 20 85 loads straw @ $2.75 " 20 " JiO 100 bu. seed oats @ 350 6i da. work sowing oats Expenditures to be de- $1.25 ducted Aug. 18 Harvesting 50 acres oats Net profit from 50 acres @75 units long and 5J units wide, and dividing it up into square units. The problems just solved show that f x 8 means f of 8, or 3 of the 4 equal parts of 8. 4. Give the meaning of these problems and find the value of the letter in each : f x 9 = x. f x 21 = x. x 56 = y. ft x 108 = y. if x 75 = z. 50. Suggestions for Problems. Make and solve problems based on the following facts : 1. A steel rail weighs 64 Ib. per yd. of length. The distance from Washington to Baltimore is 42 mi. ; to Philadelphia, 138 mi. ; to New York, 227 mi. ; to Boston, 459 miles. NOTE There are 1760 yd. in 1 mile. MULTIPLICATION 67 2. There are double tracks between Washington and each of the cities mentioned. 3. The distance from New York to San Francisco is 3262 miles. 4. The distances in miles from Chicago to 14 railroad centers of the United States are given here : Indianapolis 184 Rochester 605 St. Louis 283 Baltimore 801 Cincinnati 298 Washington 820 Cairo 364 New Orleans 912 St. Paul 410 New York 913 Omaha 490 Denver 1028 Buffalo 536 San Francisco 2349 5. Galvanized telegraph wire weighing 572 Ib. per mi. is used for distances over 400 mi., and for distances under 400 mi., wire weighing 378 Ib. per mi. is used. 6. 8 wires run between Chicago and New York. 7. Galvanized iron telegraph wire costs 6^- per pound. 8. Sound travels in water at the rate of 4708 ft. per second; in air 1130 ft. per second. 9. Silver is worth 50^ an oz., 12 oz. to the pound. 10. Hay costs $23 per T. ; oats 42^ per bu. A horse con- sumes 162 Ib. of hay and 10 bu. of oats a month. Bedding costs $2 a month. 11. Standard silver is f 9 pure silver and T V copper. A silver dollar weighs 412.5 grains. The total number of silver dollars coined on this basis was 378,166,769. 12. The number of grains in an ounce of silver is 480. The amount of silver bought under the Sherman Law by the United States government for coinage purposes was 168,674,682 ounces. 13. Light travels 186,600 mi. per second. It requires 448 sec- onds for light to reach the earth from the sun. 14. It reaches the earth from the moon in 1^\ seconds. 15. An oz. of pure gold is worth $20.67. There are 12 oz. in 1 Ib. of gold. 16. An Alaskan miner can take away 200 Ib, from the mining district. 68 RATIONAL GRAMMAR SCHOOL ARITHMETIC 17. A cu. ft. of granite weighs 170 Ib. A granite step meas- ures y x 2' x 8'. 18. The distance around the^ driving wheel of a locomotive engine is 22 ft. In going a certain distance the driver turned 5280 times. 19. In the year 1901, 82,305,924 Ih. of tea and [511,041,459 Ih. of coffee were imported into the United States. Tea was worth 48^ and coffee 26^ per pound. 20. A prize-winning steer weighed 15.03 cwt. (1 cwt.= 100 Ib.) and sold for $9.00 per hundredweight. 21. Another steer of the same lot weighed 1622 Ib. and sold for $8.85 per hundredweight. 22. A boy^bicyclist rode a miles per da. for ~b days. 23. Pupils should prepare and solve problems based upon price lists obtained from the grocer and the butcher (see p. 8), or from the market reports of the daily papers. 51. Rainfall. How long is a cubic inch (Fig. 26)? how wide? how high? What is a cubic foot? A cubic yard? FIGURE 26 FIGURE 27 A tin box 3 in. square on the bottom and 9 in. high (Fig. 27) was used by a school as a rain gauge. A second tin box 1 in. the bottom arid 9 in. high was used to measure the square on depth of water in the large box. After the water was poured MULTIPLICATION 69 1 foot from the large box into the small box, the depth of water in the small box was measured with a thin stick and a foot rule. 1. The rain gauge was placed one evening where the rain could fall freely into its open top. During the night it rained and the next morning the water was poured from the gauge into the small box. It filled the small box to a depth of 9 in. How deep did the water fill the large box? How many cubic inches of water were caught in the large box? One cu. in. of water for every sq. in. of surface is what is meant by 1 in. of rainfall. What is 6 in. of rainfall? 2. How many cu. in. of water are there in a layer 1 in. deep in the large box (Fig. 27)? 2 in. deep? 5 in. deep? 9 in. deep? 3. How many cu. in. of water fell on 1 sq. . ft. of the ground during a rainfall of 1 in. (Fig. 28)? of 2 in.? of 3 in.? of 6 inches? The number of cubic units (cu. in., cu. ft., cu. yd., etc.), a vessel holds, when full, is called its capacity. 4. What is the capacity of a square- cornered box 3" x 3" x 9"? 5. What was the depth of rainfall during a shower if the large box caught enough water to fill the small box half full? to a depth of 3 in.? of 6 in.? of 1 in.? of 2 in.? of 7 inches? 6. How many cu. in. of water fell on 1 sq. ft. of the ground during a shower giving 1 in. of rainfall? 2 in.? -J inch? 7. During a rainfall of 1 in. how many sq. ft. of ground received enough water to make 1 cu. ft.? 2 cu. ft.? 12 cubic feet? 8. During June, 1902, the rainfall in the vicinity of Chicago was 6 in. During this month how many cu. in. of water fell on 1 sq. in. of ground? on 1 sq. ft.? on 12 sq. ft.? on 1 sq. yd.? on 30 sq. yd.? on \ sq. yd.? on 30^ square yards? 9. Just before a shower set an uncovered bucket where the rain may fall freely into it. After the shower measure the depth of the water in the bucket to find the depth of rainfall. FIGURE 28 70 RATIONAL GRAMMAR SCHOOL ARITHMETIC Find how many cu. in. or cu. ft. of water fell on each sq. ft.? each sq. yd.? each sq. rd. of the ground? NOTE. The bucket used must have the same size at the top and bot- tom, with straight sides. Why? 10. During the first 9 mo. of the year 1902, the region about Chicago received 32 in. of rainfall. How many cu. ft. of water fell during this time on a garden bed 10' x 14'? on a garden 48' x 84'? over a city block 250' x 350'? 11. If the walls of your schoolroom were water-tight, how many cu. ft. of water would it hold if it were filled 1 ft. deep? 2 ft. deep? 5 ft. deep? 8 ft. deep? to the ceiling? | 12. Find the areas of these rectangles : 6"x8", 12"x28", 40'x64', 375' x 486', 9'xz', 9yd. xzyd., a mi. x 1) mi., x units x y units. How can you find the number of square units in any rectangle? NOTE. x X y is written scy and read "a?, y." 13. How many cubic feet are there in 1 cu. yd.? in 18 cu. yd.? 14. What is the capacity of a square-cornered box 3" x 3" x 9"? 3' x 3' x 9'? of a square-cornered room or space, 3 yd.x 3yd. x 9 yd.? 3 rd. x 3 rd. x 9 rd.? 3x3x9? 3 x 3 x a? 15. Find the capacity of a square-cornered box 3" x 4" x 7"; 3"x4"x8"; 3'x4'x8'; 3'x4'xl5'; 3 yd. x 3 yd. x 3 yd. ; 3 units x 4 units x 15 units; 3 x 4 x #; 3 xxxy, a units x b units x c units. NOTE. The product x X y X z is written xyz and read "x, y, z." 16. How can you find the number of cubic units in any square- cornered vessel? 17. Find the total weight of a snow load of 25 Ib. per sq. ft. on a flat rectangular roof 25' x 48'. Find the weight of a load of 15 Ib. per square foot. 18. Find the total weight of the shingles and sheathings for both sides of the roof shown in Fig. 29, if shingles weigh 3 Ib. per sq. ft., and sheathing 5 Ib. per square foot. NOTE. The eaves project 2 ft. over the plate. MULTIPLICATION 71 19. Find the weight of the snow load on both sides of the roof, the weight on each sq. ft. of surface being 12 pounds. 20. Find the cost at 3^ per sq. yd. of lathing and plastering the four walls and the ceiling, no allowances being made. 21. What is the area of the square ABCD of Fig. 29? What is the area of the triangle DEC? 22. A strong gale, giving a pressure of 16 lb. per sq. ft., blows squarely against the end of the building. What is the total FIGUBE 29 wind pressure against the end including the gable? 23. How many cu. ft. of space are inclosed by the walla to the base, DC, of the gables? D 24' C A B 52. Algebraic Problems. 4 x x is written 4#. If 4z = 24, what is the value of x? Notice that the equation 4# = 24 is the statement of this prob- lem. A boy reads 24 pages of a book in 4 days; how many pages does he read a day? What does x itand for in this problem? 1. What number does x stand for in these equations: (l)3z=18. (4)z- 8 = 15. (7)z + 9 = 17. (10) 5x + %x = 70. (2) Qx = 48. (S)a?-18= 4. (8)*+ 6 = 22. (11) 9z- 3x = 36. (3)82=72. (6)z- 9= 8. (9)^ + 16 = 25. (12) Bx + 5x = 26. 2. Write the sum of a and b. 3. Write the difference of a and b. 4. Write the product of a and />; of a and b and c; of x and y and z. 5. Answer the following questions if a = 9, b = 8 and c = 6 : 4rt = ? 12J = ? 9c = ? cib = ? 4ab = ? abc = ? Sabc = ? 6. In the shortest way you can, write eight times or; seven times y ; twenty-five times a times b ; a times x times ?/. 72 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. In the shortest way, write and read a times b\ c times x\ fifteen times a times b times x. DIVISION 53. Division and Subtraction Compared. ORAL WORK 1. A man owes a debt of $12 which he is to discharge by work at $2 per day. How much will he owe at the end of the first day? of the second day? of the third day? of the fourth day? the fifth? the sixth? 2. How long will it take to cancel the debt? 3. How many times may 2 be subtracted from 12, leaving no remainder? How many 2's are there in 12? 4. A man buys a horse for $90 and is to pay $15 a month until the horse is paid for. How much does the man owe after the first payment? after the second? the third? fourth? fifth? sixth? 5. How many months will it take to pay for the horse? 6. How many times may 15 be subtracted from 90, leaving no remainder? How many 15's in 90? 7. What is one of the 6 equal parts of 12? of 90? 8. A rectangular plot of ground 8 yd. wide by 18 yd. long is covered with bluegrass sod. How many square yards of sod does it contain? After a strip of sod 1 yd. wide, extending the length of the plot, has been removed, how many square yards of sod remain? 9. How many square yards remain after the removal of 2 such strips? of 3? of 4? of 5? of 6? of 7? of 8? 10. How many times may 18 sq. yd. be subtracted from 144 sq. yd., leaving no remainder? How many 18's are there in 144? What is one of the 8 equal parts of 144? WRITTEN WORK 1. There are 160 bricks in a pile; find by subtraction how many loads of 20 bricks each there are in the pile. How many 20's are there in 160? DIVISION 73 2. Find by successive subtraction how many 88 's there are in 610. What is one of the 7 equal parts of 016? 3. Find by subtraction how many months of 30 da. there are in 270 da. What is one of the equal parts of 270? 4. What number may be subtracted 4 times in succession from 120, leaving no remainder? 5. Find by subtraction how many times $1203 is contained in $5052. 6. Tell how to find by subtraction how many times one num- ber is contained in another. 7. Find in a shorter way how many 12's there are in 156. S. Find in a shorter way than by subtraction one of the 14 equal parts of 224. By what name do you know this short way? Division is a short way of finding: (1) One of a given number of equal parts of a number. (2) How many equal parts of a given size there are in a given number. When the term "division" has the meaning of (2), the process to which it applies may be called measurement. In the case of whole numbers, division is a short way of sub- tracting one number from another a certain number of times in succession. 54. Division and Multiplication Compared. ORAL WORK 1. A horse traveled 72 mi. in 8 hr. ; find the number of miles traveled per hour. 2. 4 bu. potatoes cost $2.40. What was the price per bushel? 3. A room 4 yd. wide contains 24 sq. yd. ; what is the length of one side? 4. 15 Ib. sugar cost 90^; what is the price per pound? 5. At $0 per ton, how many tons of coal can be bought for $180? 6. A floor containing 132 sq. ft. is 11 ft. wide; what is the length? 7. A train runs 420 mi. in 12 hr. ; find the average number of miles per hour. 8. Find the cost of 8 Ib. veal at IZ

0 12 The first and second are in common use in division. In the third, the line placed between two numbers shows that the number above it is to be divided by the number below it ; as DIVISION 75 in J, 1 is the dividend and 3 the divisor. The fraction itself is the quotient. The line between the two numbers is the division sign. The fourth sign is called the solidus (sol'i-dus). 55. Short Division. 1. How many gallons are there in 296 pints? SOLUTION. As there are 8 pt. in 1 gal. there are as many 37 gallons in 296 pt. as there are 8's in 296. g )~296~ 2. How many days in 120 hours? 3. At $8 per ton, how many tons of coal can be bought for $240? 4. 984 marbles are distributed equally among a certain number of boys. Each boy has 82 marbles. There are how many boys? 5. Selling at G for a cent, how much will a dealer receive for 540 marbles? 6. A man paid $2.60 for 4 bbl. of lime. What did each bbl. cost? 7. I bought 6 Ib. of butter for $1.62. What was the price per pound? 8. A dressmaker used 84 yd. of cloth. for 6 dresses, allowing the same amount for each dress. How many yards in each? 9. A train ran 150 mi. in 6 hr. Not allowing for stops, what was the average number of miles per hour? Find the value of x in problems 10 and 11 : 10. 48 qt. = x gallons. 11. At 6^ I can buy x Ib. of sugar for $1.50. 12. 9 papers of needles cost 72^. One paper costs how many cents? 13. In 1901, the number of trains entering Chicago every 24 hr. was about 1320; what was the average number per hour? ORAL WORK What does x stand for in each of the following equations? 1. 63 + 7 = z 4. 45 + 3 = z 7. 120 + 3 = x 2. 630 -* 9 = x 5. 96 + 8 = x 8. 108 -*- 9 = x 3. 48 *- 4 = x 6. 960 + 8 = x 9. 72 -* 6 = x 76 RATIONAL GRAMMAR SCHOOL ARITHMETIC When dividend and divisor are small numbers, the quotient is readily seen. We say it is obtained by inspection. When the divisor is a large number, the quotient is not readily found by inspectipn. 56. Applications. WRITTEN WORK 1. In 51,448 qt. how many pecks are there? SOLUTION. In 51,448 qt. there are as many pecks as there are 8 qt. in 51,448 quarts. 8 is contained in 51,000, 6000 times, with a remainder of 3000 Write 6 in the thousands place in the quotient. 3 thousands = 30 hundreds; 30 hundreds and 4 hundreds = 3400. 8 is contained in 3400, 400 6431 times, with a remainder of 200. Write 4 in the hundreds 8 ) 51448 P^ce in the quotient. 200 = 20 tens ; 20 tens and 4 tens = 24 tens. 8 is contained in 24 tens 3 tens times, without a remain- der. Write 3 in the tens place in the quotient. 8 is contained in 8 units 1 unit time. Write 1 in the units place in the quotient. In 51,448 qt. there are 6431 pecks. Check: 6431 X 8 = 51,448. 2. There are 5280 ft. in a mile ; how many yards are there in 1 mi.? in 2 miles? 3. If limestone weighs 160 Ib. per cubic foot, how many cubic feet are there in a piece of limestone weighing 0400 pounds? 4. If marble weighs 170 Ib. per cubic foot, find the number of cubic feet in a piece of marble weighing 5100 pounds. 5. If sand weighs 120 Ib. per cubic foot, find the number of cubic feet in a load of sand weighing 4800 pounds. 6. A train of 12 sleeping cars is 840 ft. long. If the cars are all the same length, how long is each car? 7. A steel rail 30 ft. long weighs 720 Ib. ; what is its weight per yard of length? 8. An iron beam 24 ft. long weighs 1080 Ib. ; what is the weight of a piece of the beam 1 ft. long? 9. A steel girder weighs 1728 Ib. Each foot of length weighs 48 Ib. How long is it? 10. There were 487,918 foreign immigrants to the United States in the year 1901. What was the average number per month? per day? (30 da. = 1 month.) DIVISION 77 11. During 1900 there were 448,572 immigrants. Find the- average number per month. 12. Answer the same question for 1891, 1892, 1893, 1895, and 1897, the numbers for these years being 560,319; 623,084; 502,917; 258,530, and 25JO,S32. 13. In Albany, N. Y., there are 30 mi. of street railway, operated by (100 men. What is the average number of employees per mile? 14. From the data here given answer the same question for these cities: CITY MILES EMPLOYEES St Joseph Mo ... . 35 175 Memphis Tenn 70 490 Oakland Cal 80 560 Hartford Conn. . 33 660 Worcester, Mass 43 473 Peoria 111 50 275 15. Find the value of x in each case: (1) 3264- G = ff (4) 89,705+ 5 = x (7) 24,568+ 8 = x (2) 9432 + 12 = x (5) 78,870 + 11 - a (8) 45,900 + 12 = a; (3)2247- 7 = x (0)75,699- 9 = a (9) 1,241,196 - 11 = x 16. How can you prove the correctness of your work in divi- sion? 17. Find what x equals in these equations: !2-io W x 66 4 A = 49 X !E.9o a; , 640 (8)^ = 66 57. Long Division. When dividend and divisor are both large numbers, it becomes necessary to show all the steps of the work. 78 RATIONAL GRAMMTAR SCHOOL ARITHMETIC 1. 14,487 -H 33 = ? 30 V = 439, quotient 400 33)14487 13200 = 400 X 33 1287 990 = 30 297 297 = 9 X 83 SHORTER FORM 439, quotient divisor, 33)14487, dividend 132 128 99 297 SOLUTION. Beginning at the left of the dividend; 33 is not contained in 1, nor in 14, but it is contained in 14,400, 400 times. Write the 400 above the dividend and subtract 400 X 33 = 13, 200 from the divi- dend, leaving 1287. This remainder must also be divided by 33. 33 is not contained a whole number of times in 1, nor in 12, but it is contained 30 times in 1280. Write the 30 above the 400 over the dividend and subtract 30 X 33 = 990 from 1287, leaving 297. 33 is contained in 297, 9 times, leaving no remainder. Thus we see 33 is contained in 14,487 400 -f 30 -f- 9 = 439 times. The work may be shortened a little by omitting the zeros and writing numbers in the shorter form below. Check: 439 X 33 = 14,487. division is probably correct. Since this is the given dividend, the When a zero appears in the quotient proceed as follows : 2. 18,722-46 = ? 407 46)18722 184 322 322 SOLUTION. Begin as above. 46 is contained in 187, 4 times, with the remainder 3. Bring down the 2 in the dividend, giving 32. 46 is not contained in 32 a whole number of times. Write in tens place in the quotient and bring down the next 2 of the dividend, giving 322. 46 is contained in 322, 7 times. Write the 7 in units place in the quotient. Check: 407 X 46 = 18,722, which equals the dividend. Complete these equations and check your work : 3. 3,580-45= 7. 33,768-72 = 4. 15,552 - 64 = 8. 35,096 - 82 = 5. 18,144-56= 9. 62,328-84 = 6. 20,088-72= 10. 44,928-96 = DIVISION 79 >58. Exercises. 1. There are 52 wk. in a year. My friend is 1872 wk. old; how many years old is he? 2. There are 160 sq. rd. in 1 A. and a farmer pays 75^ per acre for cutting and binding wheat. How much will it cost to cut and bind the wheat on a field 68 rd. by 80 rods? 3. A farmer paid a man $21.00 to shock his wheat, wages being $1.50 per day. How many days did the man work? 4. In 1880^25 farm wagons sold for $2250 and in 1900 15 such wagons sold for $855. How much less was the average cost of a farm wagon in 1900 than in 1880? 5. In 1880 a Minnesota farmer paid $3900 for 12 twine bind- ers and in 1900 14 twine binders cost him $1680. How much had the average price of twine binders fallen during these 20 years? 6. A steel rail weighing 72 Ib. per yard is 30 ft. long. How many men are needed to carry it, each man carrying 90 pounds? 7. In 25 da. a man earned $56.25 husking corn at 3^ per bushel. How many bushels per day did he husk? 8. A city lot 175 ft. long, containing 8750 sq. ft., sold for $6000. If the short side fronts the street what was the price per foot of frontage? 9. The force required to draw a street car on a level track is 35 Ib. per ton (2000 Ib.) of the combined weight of the car and its load. What force is needed to draw a car weighing 5600 Ib. when it is loaded with 60 passengers whose average weight is 140 pounds? 10. At the speed of an ordinary horse car a horse can exert about 125 Ib. of force in drawing the car. How many horses will be needed to draw the car of problem 9, no horse to draw more than 125 pounds? 11. At a slow walk a horse can exert about 330 Ib. of force. The force required to draw a loaded wagon on a level pavement is -fy of the weight of the wagon and load. A coal wagon weighing 4860 Ib. is loaded with 4 T. of coal. How many horses will be needed to draw the load over a level pavement, no horse drawing more than 330 pounds? 80 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. A horse can exert 1540 Ib. of force for a few minutes. A. box car weighing 30 T. is loaded with 32 T. The force needed to move the loaded car is -fa of the combined weight of the car and load. How many horses will be needed to start the car on a level track? 59. Larger Numbers. 1. 5,128,672-9272 = ? SOLUTION. The divisor, 9272, being too large to use readily, we first use a trial divisor. For the same reason, we select a trial dividend. 92, the trial divisor, is contained in 512, 5 times; but as the whole divisor con- tains four digits the partial dividend must be enlarged. 9272 is contained in 51,286, 5 times. The quotient figure 5 is of the same order as the last figure of the trial dividend, which is hundreds. We write 5 in hundreds place in the quotient. Multiplying the whole divisor by the quotient figure we have the product 46,360. Subtracting this prod- uct from the trial dividend, 4926 remains. 4926 hundreds = 49,260 tens ; and 49.260 tens + 7 tens = 49,267 tens. Con- tinue in the same manner with each step that follows. The last subtraction gives a remainder of 1256. This remainder must 553 9272)5128672 46360 49267 46360 29072 27816 1256 also be divided by the divisor 9272. 5,128,672 -h 9272 = 9272 553 27816 46360 46360 5127416 1256 5128672 The 553 is the whole, or integral par,; of the quotient and the |ff is the fractional part. Check: Multiply the divisor by the quotient, and to the product add the remainder. The result should equal the dividend. Solve the following problems and check your work : 2.131,320-536 = 4. 630,861-2731 = 3. 195,936-624= 5. 1,057,536-4352 60. Geography. 1. From the table of 30 the area of Massachusetts is seen to be 8315 sq. mi., and that of Illinois is 56,650 sq. mi. How many states the size of Massachusetts could be made from Illinois? DIVISION 81 2. From the same table the area of New England is found to be 66,465 sq. mi. How many states as large as New England could be made from Texas? 3. Make and solve other problems like these, using the table. 4. The same table shows the area of Connecticut to be 4990 sq. mi. and its population for 1900 to be 908,420. How many persons per square mile are there in Connecticut? NOTE. In problems such as this, where the fractional part of the quotient has no meaning, drop the remainder if it is less than half the divisor, and add one unit to the whole part of the quotient if the remain- der is more than half of the divisor. 5. The table of 35 gives the population of Connecticut for 1890 as 740,258. What was the population of Connecticut per square mile in 1890? 6. Answer questions 4 and 5 for your own state. 7. From the same table answer questions 4 and 5 for Okla- homa territory. 8. Make and solve similar problems for any states you are studying in your geography. 9. The area of Switzerland is 15,781 sq. mi. and its popula- tion is 2,933,334. What is the population of Switzerland per square mile? 10. 640 A. = 1 sq. mi. How many square miles in 534,528 acres? 11. The area of the state of Texas is 265,780 sq. mi. ; that of New Jersey is 7815 sq. mi. How many states the size of New Jersey could be made from Texas? How many the size of Delaware, which contains 2050 square miles? 12. Porto Eico has an area of 3531 sq. mi., and a population of 953,243. How many inhabitants does it support to the square mile? 13. The greatest ocean depth found is 31,614 ft. near the island of Guam, in the Pacific ocean. The highest mountain in the world is Mt. Everest in Asia, which rises 29,002 ft. above sea level. Find the difference of level in miles between the greatest ocean depth and the greatest land altitude. RATIONAL GRAMMAR SCHOOL ARITHMETIC 14. The state of New York, with an area of 49,170 sq. mi., supports a population of 7,268,894. How many inhabitants does it average to the square mile? 15. Hawaii has an area of 6449 sq. mi., and a population of 154,001. Find the average per square mile. 16. In 1891 the total wheat area of North and South Dakota was 4,882,157 A.; the yield was 81,819,000 bu. What was the average yield per acre? 17. In the year 1900 Kentucky had 22,488 A. of rye under cultivation. The total yield was 292,344 bushels. What was the average yield per acre? 18. In the year 1890 there were 210,366 persons employed in manufacturing in Chicago. The total wages paid amounted to $123,955,001. What was the average wage paid to each person? 19. In 1880, 3519 factories in Chicago together yielded $249,022,948 worth of products. In 1890, 9977 factories yielded $577,234,446 worth. During which year was the average product per factory greater, and by how much? 20. A comparison of the density of population (population per square mile) may be obtained by finding the density of population of these countries: COUNTRY AREA POPULATION DENSITY German Empire. . ... 208,830 120,979 115,903 204,092 110,646 8,660,395 3,688,110 56,345,014 41,454,578 26,107,304 38,641,333 32,449,754 135,000,000 76,212.168 Great Britain Austria France Italy Russia United States NOTE. Only the whole numbers need be found for these quotients. 61. Division by Multiples of 10. ORAL FORK 1. Multiply each of these numbers by 10: 568 1268 306 $86.50 $8.65 2. Multiply each of the same numbers by 100 ; by 1000. 3. Divide each of these numbers mentally by 10 : 5680 12680 3060 $865.00 $86.50 DIVISION 83 4. Make a rule for dividing any number quickly by 10 ; by 100 ; by 1000 ; by 1 with any number of zeros after it. 5. Name these quotients orally: 60 + 10 = ? 6 + 2 = y 60 + 20 = y 860 + 10 = ? 86 + 2 = ? 860+20 = ? $64.20 + 10 = ? $6.42 + 2 = ? $64.20 + 20 = y Examining your answers to the questions just asked, make a rule for dividing a number quickly by 20; by 200; by 2000. 6. 180 + 10 = ? 18 + 3 = ? 180 - 30 = y 630 + 10 = y 63 + 3 = y 630 + 30 = ? From these answers make a rule for dividing any number quickly by 30; by 300; by 3000. 7. Make a rule for dividing a number quickly by 40; by 400; by 4000; by 50; by 800; by 1200; by 1500. 8. Make a rule for dividing any number quickly by any whole number of tens, as 40, 70, 90, 160; by any whole number of hun- dreds; of thousands. 9. Cutting off zero from the right of a number has what effect on the numbery cutting off 2 zerosy 3 zerosy 66 -f- 10 = 6 1 e , or 6.6. 165 H- 10 = 16&, or 16.5. 75 +-100=,%. or .75. 478 + 100 = 4-^, or 4. 78. 10. Using first 10 and then 100 as a divisor, give and show the quotients of the following : 400 500 2200 3300 460 790 4280 4860 287 439 9647 5732 11. There are 10 pk. in a barrel. How many barrels in 1488 pecksy 12. There are 60 Ib. in a bushel of potatoes. How many bushels in 486 poundsy 13. 100 lb.= 1 cwt. How many hundredweight in 825 poundsy 14. 200 Ib. pork = 1 bbl. How many barrels in 7624 pounds? 84 RATIONAL GRAMMAR SCHOOL ARITHMETIC 15. How many minutes in 42GO seconds? SOLUTION. When there are ciphers at the right of both 71 dividend and divisor, cut off an equal number of ciphers from both and divide. 60)48601 1C. How many barrels will be needed for 4800 Ib. of beef, allowing 200 Ib. to the barrel? 62. Other Methods of Shortening Division. 1. Solve these'problems: 5000)25000 500)2500 50)250 5)25 How do the quotients compare? the dividends? the divisors? 2. Remembering that the dividends stand above the line and the divisors below, solve these problems : 36 12 81 27 ~ 9^ 3 How do the quotients compare? the dividends? the divisors? 3. How do the quotients, the dividends, and the divisors com- pare in these problems : 256 64 = 9 16 4 128 32" 8 2 4. To divide 324 by 81 what smaller numbers may I use to get the same quotient? How can I obtain these smaller numbers from 324 and 81? SOLUTION. We see that, as 324 = 27X12 and 81 = 27x3, we may write 13 X 27 _ 12 _ . 3 X 27 ~~ ~3~ = This can be indicated thus : 12_Xjfr_ sxzr Eemoving these factors is called cancellation. It can often be used to simplify the division of products. 5. Answer the same questions for 25G + 128. Any factor of both dividend and divisor may be dropped or stricken from both and the remaining factors divided. DIVISION 85 6. Solve these problems by cancellation : 34x16 _ 9 6x8x3 = 9 15 x 21 x 846 = 9 2x16 " ^ ' 2 x 8 x 3 ~ 3 x 7 x 846 ~ 18x3x4x67 = ? (5) 625 = ? ( 6 ) ^ = ? (7)- = ? & x 4 x o x 07 <*o loJ o4 To use cancellation effectively, methods of finding factors of numbers are necessary. 63. Tests of Divisibility. 1. Which of these numbers are exactly divisible (can be exactly divided) by 2 : 12 24 36 23 45 18 37 40 59 61 What are the last digits of the numbers which 2 will divide? TEST FOR THE FACTOR 2 : If a number ends in 0, 2, 4, 6, or 8, it can be exactly divided by 2. 2. Which of these numbers are exactly divisible by 10: * 24 30 45 50 700 640 83 765 6400 What is the last digit of the numbers which 10 exactly divides? TEST FOR THE FACTOR 10: If a number ends in 0, 10 exactly divides it. 3. Make a rule for testing whether 100 divides a number. 4. Which of these numbers does 5 exactly divide: 16 18 35 25 60 20 28 65 460 675 1260 What are the last digits of the numbers which 5 will divide? TEST FOR THE FACTOR 5: If a number ends in or 5, 5 exactly divides it. 5. Which of these numbers does 3 exactly divide: 12 17 24 81 27 93 64 75 126 324 185 Of the numbers which 3 exactly divides, will 3 also exactly divide the sum of the digits? Of the numbers 3 does not exactly divide, is the sum of the digits exactly divisible by 3? TEST FOR THE FACTOR 3 : If 3 exactly divides the sum of the digits of a number it divides the number also. 6. Of these numbers what ones does 9 exactly divide: 126 368 453 729 819 639 2358 86 RATIONAL GRAMMAR SCHOOL ARITHMETIC See whether 9 will exactly divide the sum of the digits of the numbers that 9 exactly divides. TEST FOR THE FACTOR 9 : If 9 exactly divides the sum of the digits of a number it also exactly divides the number. 7. Which of these numbers does 4 exactly divide: 113 124 368 560 375 486 1204 See whether of the numbers it exactly divides 4 also exactly divides the number indicated by the last two digits. For example, in the third, 368 can be exactly divided by 4 and so also can 68. TEST FOR THE FACTOR 4 : If the number denoted by the last two digits of a number can be divided by 4, the entire number can be exactly divided by 4. 8. Which of these numbers are divisible by 25 : 60 175 285 625 1350 1275 8645 8675 8625 8650 25 divides 175 exactly and 25 also exactly divides 75, which is the number denoted by its last two digits. Is this true of all numbers 25 exactly divides? Is it true of any numbers 25 does not exactly divide? TEST FOR THE FACTOR 25: If the number denoted by the last 2 digits of any number is divisible by 25 the entire number is divisible by 25. TEST -FOR THE FACTOR 6 : test for both 2 and 3. TEST FOR ANY COMPOSITE FACTOR: test singly for all the factors of the composite factor. 9. TEST FOR DIVISIBILITY by such numbers as 36, 216, 27, 49, etc., which contain some factor two or more times. 10. Pick out the numbers of this list which are exactly divisible by 2 : 6 81 65 72 86 129 9864 8643 7986 16,835 29,860 11. Pick out those which can be exactly divided by 3; by 9; by 4; by 5; by 10; by 12; by 15. 64. Checking Division. Division may be checked by multiplying the divisor by the quotient and adding the remainder to the product. If the sum equals the dividend, the work is checked. DIVISION 8? Another check is to divide by the factors of the divisor suc- cessively and note whether the final quotient is the same as that given by the complete divisor. To check by casting out the nines, add the excess in the product of the excesses of divisor and quotient to the excess of the remainder. If the excess of this sum equals the excess of the dividend, the division is probably correct. ILLUSTRATION. Check the work of problem 1, 59. Cast the 9's out of the dividend, 5,128,672. The excess is 4- Cast the 9's out of the divisor, 9272. The excess is 2. Cast the 9's out of the quotient, 553. The excess is 4. Cast the 9's out of the remainder, 1256. The excess is 5. The product of the excesses of divisor and quotient is 8. The excess of 8 is 8 itself. Add this 8 to the excess of the remainder, giving 13. The excess of this 13 is 4. and as this equals the excess 4 of the divi- dend, the division is probably correct. It is even more important in division than in multiplication to examine a problem carefully before beginning to solve it. Try to foresee about what the answer must be. This often avoids blunders. ILLUSTRATION. 1. If it requires 480 slates to cover a square (1 00 sq. ft.) of roof surface, how many squares are there in a roof which requires 13,680 slates to cover it? Pupil should at once notice that if it required 500 slates to cover a square, there would be a little more than 13,680 -f- 500, or 136 -*- 5 = 27 squares and he might guess 28 squares. The actual division of 13,680 by 480 gives 28.5 squares. First form a rough estimate of the answer and then solve these exercises : 2. $238 was paid for flour @ $3.50; how many bbl. were bought? SUGGESTION. How many bbl. would there have been if the price had been $7.00 a barrel? 3. In still air a hawk flew 375 miles in 2J hr. What was its speed per hour? 4. In 3.9 hr. a crow flew 97.5 m. ; find the rate of flight per hour. 5. From a certain cow -fa f ^ ne m ilk was butter-fat. The cow gave 12 Ib. of milk per day. How many pounds of butter-fat will the milk from this cow yield in 70 days? 88 RATIONAL GRAMMAR SCHOOL ARITHMETIC 65. Applications of Cancellation. In all the problems of this list indicate your divisions and multiplications, then apply the tests of divisibility, and cancel the factors found in both dividend and divisor. Multiply the uncanceled factors in the dividend together and divide this product by the product of the uncanceled factors of the divisor. ILLUSTRATION. A speed of 102 mi. per hour equals how many feet per second? Indicate the work thus: 34 22 102 X 5280 _ V& X #g_ 34 X 22 _ 748 60 X 60 M 5 = 5 = Ans. 149g ft. per second. 1. It took 225,280 Ib. of steel rails to lay 1 mi. of single-track railroad. What was the average weight per yard of the rails? 2. Find the number of cubic feet in a ton of 2000 Ib. for each of the substances given in the table. 3. The average speed of American express trains is about 35 mi. per hour for long dis- tances. How many feet per sec- ond is this? NOTE. 60 sec. = 1 min. ; 60 min. = 1 hour. SOLUTION. Put work in this form: 7 22 35 X 5280 35 X &f 154 r LB. PEK C'u. FT. Cu. FT. PEK T. Granite .... Limestone . . Marble Sandstone. . Slate 170 160 170 140 170 Cast Iron . . . Steel 450 480 60x60 r1 , ., Ans - 51i ft per second - 3 4. In both England and America, the average speed of express trains for distances from 100 to 250 mi. is about 40 mi. per hour. How many feet per second is this? 5. For long distances the average speed of English express trains is about 43 mi. per hour. How many feet per second is this? 6. In 1893 an express train in the United States ran 1 mi. at the rate of 98 mi. per hour. How many feet per second is this? DIVISION 89 7. A railroad train ran for 1 min. at the speed of 130 mi. per hour. Find the number of feet per second which it traveled. 8. Sound travels in air at the rate of about 750 mi. per hour. How many feet per second is this? 9. Cannon balls have been thrown at a speed of 810 mi. per hour for a few seconds. This is how many feet per second? 10. The moon moves around the earth at a speed of about 1,840,000 mi. in 30 da. How many feet is this per second? NOTE. 24 hr. = 1 day. 11. In 365 da. the earth moves around the sun through a distance of about 584,000,000 mi. What is the earth's speed in miles per second? 12. The earth, by turning on its axis, carries a place on its equator about 25,000 mi. in 24 hours. How many feet is it carried per second? 66. The Lever. 1. Rest a foot rule on a support as at F in Fig. 30 and load it with G oz. at W. What weight at P will balance the foot rule? 2. If the rule is first balanced* on the three-inch mark as in Fig. 31 how many ounces will be needed at P to balance 18 oz. at W? P F W I I I I i i I I P F W III!) 11111 1 1 1 FIGURE 30 FIGURE 31 3. If the support were placed at the two-inch mark and the rule balanced how many ounces at P would balance 24 ounces at IT? The point that the (foot rule) bar rests on is called the fulcrum. Call the distance from the fulcrum, F, to the weight, W, to be balanced FW, Fig. 32, and the distance from the fulcrum to the power, P, needed to balance the weight FP. * A piece of brick or other substance should be placed on the short end of the rule or bar to balance it exactly with the long end, in all these problems. 90 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. If FP is just as long as FW> 20 Ibs. at P will balance how many pounds at W? W P F W I FIGURE 32 FIGURE 33 5. If FP is twice as long as FW, 10 Ib. at P will balance how many pounds at Wt 6. A light bar of convenient length, like the one shown in Fig. 32, was supported as at the point F. The bar was then balanced by placing a small piece of brick on the short end. A weight was then placed on the end W of the balanced bar and the weight needed to balance W was hung at P. The distances FW and FP were then measured on a number of different bars similarly balanced and loaded and the following table was made: AtW FW AtP FP 8 oz. 3 in. 3 oz. 8 in. 10 oz. 2 in. 2 oz. 10 in. 12 oz, 4 in. 6 oz. 8 in. 22 oz. 1 in. 2 oz. 11 in. 8 Ib. 6 in. G Ib. 8 in. 12 Ib. 3 in. 4 Ib. 9 in. 7 Ib. 4 in. 2 Ib. 14 in. Compare the products of the numbers in columns 1 and 2 with the products of those of columns 3 and 4. What do you find? 7. When a lever balances, the load multiplied by its distance from the fulcrum equals what other product? 8. In Fig. 33 if IF = 120 Ib., FW= I ft., and FP = 4 ft., what force at P will just balance the stone at W? What force would be needed if FW = G in., and FP = 6 feet? 9. If the load is W Ib., the power P Ib., the distance from F to W is w feet and from F to P is p feet, what equation can you write if the bar balances? NOTE. W X w is written Ww. DIVISION 91 10. By what must 5 be multiplied to equal 6 x 10? SOLUTION. Call x the number by which 5 must be multiplied, then: Statement, 5x = 6 X 10, and = 12. Check: 5X12 = 6x10 = 60. 11. Letting p be the distance from F io P, find what p is in these problems, representing conditions for a balanced lever, can- celing when you can and checking all answers : FW = 2 ft. TF= 80 Ib. P = 16 Ib. FP =p ft. FW=6in. TF=500lb. P = 25 Ib. FP = p in. FW=3it. JF = 420lb P=211b. FP = p ft. 12. The edge of a sheet of tin was placed in the shears as in Fig. 34. If fa = 2 in., fb = 8 in., and a pressure of 22 Ib. was exerted at #, what force was exerted at a to cut the tin? 67. Additional Problems on Town Block and Lots. These problems are based on Fig. 3, page 2. Fig. 35 is a detail drawing of the street and sidewalk crossings at one of the four corners of the block. 1. The city paves all street crossings. If the cost of material is $1.20 per sq. yd. and of labor for excavating and constructing is $1.05 per sq. yd., what will be the expense to the city of paving the 4 rectangular street (not sidewalk) crossings such as 1 (Fig. 35)? Ans. $1006.40. 2. Water mains run along Race and Market streets, and they are connected by a main through the alley. To put in these mains a special water tax of 27|# per foot of street and alley L j *~c^~~* \\V>\\\\\\\\\ ffl 1 Race I 5J?' ^. s UJ FIGURE 35 frontage was assessed by the city against all property abutting on 92 RATIONAL GRAMMAR SCHOOL ARITHMETIC the mains. What tax must the owner of lot A pay? of lot B? 1st. Ans. $27.50. 3. Make and solve problems similar to problem 2 for other lots. 4. Each property holder is required to pay for the material for a concrete sidewalk in front of his premises. The walk is to be 6 ft. wide, and the material costs 27^ per square foot. Omit- ting the corner squares (such as 2, 3, 4, 5, Fig. 35), find the expense to each lot in the block for the sidewalks. 5. If the owner of lot L pays for the material for the square of sidewalk (2, Fig. 35) at his corner of the block, how much will this increase his 'assessment? 6. The cost for labor in making the sidewalks is 13^ per square foot. What will be the entire expense for both labor and material for the sidewalks of the whole block? (Omit strips such as 6, 7, 8 and 9.) 7. An electric lamp is placed at each of 2 opposite corners of the block and a gas lamp at each of the other 2 corners. The cost of maintaining a gas light is $36 a year; an electric light, $172 a year. The entire cost of the 2 gas lights and ' of the cost of the 2 electric lights is assessed against the property holders of this block. What will be the yearly assessment against the whole block to meet the expenses of lighting? 8. Each property holder pays in proportion to his street frontage. Make problems on the cost to individual lots for lighting the streets around this block. 68. Additional Problems on House Plans. These problems refer to Fig. 6, page 7. 1. Masons count 22-j- bricks per cubic foot for chimney and solid work. The cross-section of the kitchen chimney is a rectangle 2 ft. by 4 ft. (Fig. 36) and the chimney is 42 ft. tall. Allowing ^ of the number of cubic feet in the chimney for PIGUBB36 the twQ flueg runn i Dg f rom the bottom to FIGURE w the top, how many bricks are needed to build it? Ans. 7200. 2. The cross-section of the fireplace in the hall is half of a DIVISION 93 6-ft. square (Fig. 37) to a height of 26 ft. from the basement to the garret floor, where it joins the kitchen chimney. Allowing -Jg of the number of cubic feet in the fireplace for the flue and the 2 fireplace openings, how many bricks will be needed for the fireplace? Am. 9945. 3. What is the cost of hauling the brick for foundations, chimneys, and porch piers, at $1.50 per thousand?* (See problems 11 and 14, p. 6.) 4. The walls and ceilings of all downstairs rooms are to be tinted at 20^ per square yard, no deductions of any kind being made. The left side wall of the dining-room is to be considered as straight. The walls downstairs are 9 ft. high. Find the cost of tinting. 5. The kitchen floor is to be covered with linoleum, which comes in 1^-yd. widths, at $1.12 per yard of length. What will it cost? 6. What will it cost to carpet all the bedrooms and the upstairs hall (not including the landing at the head of the stairs) with ingrain carpet 1 yd. wide, at 55^ per yard of length? In rooms not square, strips are to run the long way of the room. 7. What will it cost to cover the bathroom floor with tiles at 45^ per square foot laid? 8. The basement walls are 8 ft. high and the arrangement of the rooms is the same as on the first floor plan. What will it cost to cement the floor and the inside walls of all basement rooms at 190 per square yard, making no allowance for openings? 9. What will be the cost, if allowance is made for the open- ings given in problems 9 and 10, p. 6? 10. The 4 outside walls of a brick store building are to be 3 brick walls, to extend to a depth of 6' and to a height of 36 feet. Two of the walls are to be 40' long and 2 are to be 25' long. There are to be also 3 inside cross-walls each 2 bricks thick and 25' long. What will the brick for all these walls cost at $9.50 per thousand. (Use the nearest thousand). Ans. $2410.50. * See second footnote, p. 6. 94 RATIONAL GRAMMAR SCHOOL ARITHMETIC BILLS AND ACCOUNTS 568. Exercises. MON. Tu. WED. TH. FRI. SAT. WAGES HOURS HOURS HOURS HOURS HOURS HOURS $ CTS. Adams iy* 8 8 sy* 9 8^ Benson 8 8K SX 8 8K 9 Boyd o 4 6 6 6* 7 Claussen 8 8X 8% 8K 9 8M Denning 8K 9 8^ 8K 9 9 Doan 8 8 8 8 8 8 1. Find the number of hours each man worked. 2. Find the number of hours all the men worked. 3. At 30

51.57 Nov. $ 55.28 : DBG. $ 53.15 JAN. $ 3.70 FEB. $ .35 1 MAR. $ 2.86 2 8.94 3.65 .13 51.26 58.10 56.55 3 2.09 7.25 1.68 3.67 1.02 .60 4 7.35 .25 3.75 .55 5.16 3.16 5 6 . 1.50 8.22 1.65 4.78 1.85 5.75 5.90 .12 1.37 2.15 4.60 6.51 7 .05 2.83 1.25 1.60 3.17 2.75 8 .. .. 4 75 3.86 8.21 6.12 1.16 1.28 9 1 77 2 28 .68 3.26 4.18 3.82 10 11 4.13 9.10 6.98 .15 1.86 1.17 1.81 3.60 .22 6.03 .75 2.29 12 . ... 1.19 3.65 4.76 2.31 2.58 1.69 13 3.87 1.63 5.16 .05 1.15 9.60 14 10 .85 1.11 5 86 5.60 .17 15 2 17 6 87 2 68 1 03 1.81 1 25 16 3.98 2.37 .68 4.32 2.81 2.18 17 18.47 8.86 3.27 9.82 .75 .85 18 .. 1.80 12 7.32 2.18 .81 5.63 10 2 55 4.10 1.28 3.28 15.61 3.05 20 21 8.81 .25 1.17 1.68 2.18 11.12 .75 6.29 5.78 4.78 1.06 11.61 22 .25 3.76 16.25 1.18 1.05 4.28 213 3.55 9.26 .22 2.28 3.27 2 57 24 . 55 7.81 3.20 7 60 1.60 .60 25 26 27 28 ,..:.. 29 30 31 1.65 3.70 5.74 .76 2.80 3.86 2.85 .55 3.86 7.52 .28 18.26 1.60 .00 1.68 .54 3.48 6.15 2.97 .21 2.08 3.91 2.83 .25 1.65 5.16 18.21 6.38 2.86 8.07 3.35 4.15 .00 .00 .00 3.05 3.18 1.75 3.10 6.21 .76 .15 Monthly > Totals f 3. Find the daily average per month for each of the 6 months. 4. Find the monthly average for the 6 months from October to March. 5. Add horizontally and find the total expenditure for the first day of the 6 months; for the second day. roa ' RATIONAL GRAMMAR SCHOOL ARITHMETIC 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17 18 19 20, 21 22 23 24 25 26, 27, 28. 29, 30, 31 Monthly } Totals j 52.86 3.87 3.68 6.36 1.86 2.15 .75 1.95 3.28 2.18 4.10 2.15 1.00 .10 1.61 3.21 .57 3.76 .15 4.25 .86 .95 .87 1.16 3.15 1.82 2.78 .60 3.25 .96 .00 MAT JUNE JULY AUG. SEP 53.48 $ 56.75 $ 56.73 $ 55.10 $ .65 2.50 .67 1.28 2.50 54.10 2.18 1.20 2.10 6.82 4.06 1.28 .85 3.50 .04 .60 .25 .75 4.05 .00 2.75 8.12 3.80 1.10 1.28 .25 3.62 2.37 .05 .75 1.52 .67 2.12 .12 4.25 .68 4.10 .55 .15 1.18 2.12 .10 .96 .35 2.60 .39 1.01 2.84 3.60 .00 1.86 .86 3.69 .10 3.10 3.24 1.52 2.50 2.18 .69 3.27 .78 .55 .00 1.58 1.68 2.58 1.86 1.57 3.82 .55 3.65 1.68 .76 .55 1.68 .25 2.75 .56 6.53 2.17 2.51 .99 3.65 .28 .69 .86 6.24 .05 .12 3.00 1.76 3.26 8.29 .17 .12 2.08 .25 .00 2.16 6.81 1.15 4.86 .00 2.60 .05 3.12 .28 1.68 .05 .10 1.28 .83 1.00 8.16 7.06 3.75 1.96 2.28 .15 .19 .15 1.76 .05 .08 2.60 1.67 3.75 6.28 1.20 2.97 2.18 1.28 .00 2.00 3.60 1.01 8.14 3.16 3.60 .10 3.87 1.00 2.15 .25 2.65 .75 .00 1.58 1.68 .00 TOTAL 6. Find the daily average per month for each of the 6 months. 7. Find the monthly average for 6 months from April to September. 8. Find the monthly average for the whole year. 9. Find by adding horizontally the total expenditure for the first day of these 6 months ; for the second day. BILLS AND ACCOUNTS 103 74. The Equation. 1. y Ib. of sugar are placed on the left scale pan, and a weight of 10 Ib. on the right pan balances it (see Fig. 38). How heavy is *? The balance of these two weights is expressed thus : y = 10. (I) ^T^^Q^b 10 Lb This expression is called an equa- tion, and is read il y equals 10." 2. If a 4 Ib. weight is now added to the right pan, how many additional pounds of sugar must be placed upon the left pan to balance the scales? FIGURE 38 Write an equation to express the relation between the weights now in the pans. 3. If x Ib. on the right balance y Ib. on the left, how would you state the fact in an equation? 4. If 20 Ib. are added to the x Ib. already on the right, how many pounds must be added on the left to balance the equation? 5. If 35 is added on the right side of equation (I), what change must be made on the left side to balance the equation? Write the equation thus changed. Just as the horizontal position of the scale beam shows that there is a balance of the weights on the pans, so the equality sign, = , shows that there is a balance of value of the numbers between which it stands. It must never be used between two numbers that do not balance in value. The number on the left of the sign of equality is called ihe first member, or the left side of the equation. The number on the right is called the second member, or the right side. 6. Four equal weights and a 5 Ib. weight just balance 21 Ib. (Fig. 38.). How heavy is one of the equal weights? SOLUTION. 4#4-5 = 21. Subtract 5 from both sides 15 -5 4# =16. If 4x = 16 what is xt What is the answer to the problem? 7. 3#+2 = 14; findy. 8. 5z + 12 = 32; find x. 9. 7z + 8 = 29; find x. 10. 9z + 3 = 75; find z. 11. 15a + 3 = 48; find a. 104 RATIONAL GRAMMAR SCHOOL ARITHMETIC CONSTRUCTIVE GEOMETRY 75. Problems with Ruler and Compass. Problems I. to XL are to be solved with ruler and compass. Keep all the pencil points sharp while drawing and work carefully. The simple instrument shown in Fig. 39 is a form of the compass which will do for these problems. The pencil point of the compass will be called the pencil foot, or pen foot. The other point will be called the pin foot. PROBLEM I. Draw a circle with \ inch FIGURE 39 ' radius. EXPLANATION. First Step : Place the pin foot on an inch mark of your foot rule and spread the compass feet apart until the pencil foot just reaches to the next half inch mark. Second Step: Without changing the distance between the compass feet, put the pin foot down at some point, as A, Fig. 40, of your paper and, with the pencil foot, draw a curve entirely round the point A. A curve drawn in this manner is called a circle. How far is it from A to any point of the curve? Any part of the whole circle, as the part from C to D, or from D to B is an arc of the circle. The point A, where the pin foot stood, is the PIGUKE 40 center. The distance straight across from B, through A to C, is a diameter. BC is a diameter. The distance from A straight out to the circle is a radius. The plural is radii (ra'-di-I). AC, AB and AD are all radii. What part of the diameter equals the radius? EXERCISES 1. Draw circles with these radii: 2. Draw circles with the same center A, and with these radii : i"; i"; i"; i; if- Circles whose centers are all at the same point are called con- centric circles. PROBLEM II. Draw a line equal to a given line. CONSTRUCTIVE GEOMETRY 105 EXPLANATION. Let the given line AB (Fig. 41) have the length a units. With ruler and pencil draw any straight line, as CX, longer than a. Place the pin foot on A and spread the compass feet until the pencil foot just ^ a -g reaches to B. Without changing the dis- \j) tance between tne feet, put the pin foot on ~ X C and with the pencil foot draw a short FIGURE 41 arc across the line CX as at D. Then CD is the desired line; for if we call its length x, we have made x = a. EXERCISES 1. Draw lines equal in length to these given lines: a Z> c 2. Draw lines having these lengths: 1". 11". Ol". O3". QT" 1 .; If ; 4f ,; 2f ; 3$ . PROBLEM III. Draw a line equal to the sum of two or more given lines. , EXPLANATION. Let the two given lines be a and 6, Fig. 42. First step : Draw the indefinite line CX longer than the combined length of a and b, and make CD equal to a as in ? Problem II. b Second Step: Spread the com- pass feet apart as far as the length v of 6. Then put the pin foot on the crossing point (intersection) of the arc and line at D and draw another short arc at E, How long is DEI How long is CEt CE is the desired sum. If we call its length s, we may write this equation : (I.) s = a + b. With ruler and compass how can you find a line equal to the sum of 3 given lines? of 4? of any number of given lines? This is called con- structing the sum of lines. EXERCISES 1. Construct the sum of the lines in each column, denote the constructed line by s, and write an equation like (I.) for each case: cam a den b e p m 106 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Construct these sums : PROBLEM IV. Draw a line equal to the difference of two given lines. EXPLANATION. Let the two given lines be a and b. First step: Draw the indefinite line CX longer than the minuend line a. As above, make the minuend CD, equal ' to a. b Second Step : Spread ~ the compass points apart, U T7" QG V*fYiVi OC "f tJ V Q C! t 1 i i_i C__J as before, as far as the length of the subtra- FIGTJBE 43 hend line b. This is called "taking b as a radius." Place the pin foot on D, swing the pencil foot back toward C, and draw the arc at E across the line CX. This makes DE how long? What line now equals the difference d between a and 6? The equation for this case is : (II.) d = a-b. EXERCISE Construct the differences of these pairs of lines, call each dif- ference d y point it out in the construction and write the equation for each case : m c f a PROBLEM V. Draw a line equal to 2, 3, or 4 times a given line. EXPLANATION. Let a be the given line. We are simply to draw a line equal to the sum of a and a. (See Prob- _ a lemlll.) D E We may write r (III.) p = 2a. How would you con- P struct 3a? 4a? 5a? 7a? FIGUBE 44 oxi &y{ vzc Write and read the equation for each construction. EXERCISE Construct three times these lines and give their equations (like III., above): a c d %a 3m PROBLEM VI. Divide a given line into two equal parts. CONSTRUCTIVE GEOMETRY 107 B DEFINITION. Dividing a line into two equal parts is called bisecting the line. EXPLANATION. Let the given line be AB, Fig. 45, call its length a. First step: Spread the compass feet apart a little farther than the length of AB. Ple^? the pin foot first on A and bring the pencil / foot, by estimate, somewhere above the middle of the line a, and draw the arc 1. Then carry the pencil foot down, by estimate, below the middle of a and draw the arc 2. These arcs should both be drawn long enough to make sure that arc 1 passes over arc 2, under the middle point. Second step : Now change the pin foot to B, and without chang- ing the distance between the feet, draw arc 3 cutting arc 1 and 4 cutting arc 2. Call the crossing points C and D. Third step : Place the edge of the ruler on C and D and draw the straight line CD, crossing a at E. Either AE or BE is equal to of a. Test by measuring with the compass. The equation for this case is: (IV.) q = \a, or q = j. The first is read "q equals one-half a," meaning q equals \ of a, and the second is read "q equals a divided by 2." Do they, therefore, mean the same thing? FIGURE 45 EXERCISES 1. Draw a line f" long and bisect it. 2. Bisect lines of these lengths: J"; If; 3"; 5"; 5i"; 3J". FIGURE 46 FIGURE 47 3. Longer lines may be bisected at the blackboard with crayon and string (Fig. 40) , or on the ground with a cord and a sharp stake (Fig. 47). 108 RATIONAL GRAMMAR SCHOOL ARITHMETIC PROBLEM VII. Construct an equilateral (equal sided) tri- angle with each side equal to a given line. EXPLANATION. Let a Fig. 48 be the given line. Draw CX longer than a and make CD = a, as in Problem II. With length a between the compass feet, place the pin foot on C and draw arc 1, by estimate, above the middle of CD. Without changing the distance between the compass feet, place the pin foot on D and with the pencil foot cut arc 1 with arc 2. Call B the intersection (crossing FIGURE 48 point) of arc 1 and arc 2. With the ruler draw a line from C to B and another from D to B. Then CDB is the desired equilateral triangle. EXERCISES 1. Construct equilateral triangles having sides of these lengths: bed e 2. Construct equilateral triangles having these sides : 1"; 2"; 3"; 2J"; 4". 3. With crayon and string construct these equilateral triangles on the blackboard : 6"; 1'; 14"; 18"; 24". 4. With cord and nail or stake construct these equilateral triangles on the floor or ground : 6'; 10'; 18'; 1 rod; 30'. PROBLEM VIII. Construct an isosceles (i-sos'-see-lees) trian- gle with the base and the two equal sides equal to given lines. DEFINITION. An isosceles tri- ^ angle has at least two sides equal. EXPLANATION. Let b, Fig. 49, . equal the base, and e, one of the equal sides. Make CD = b (Problem II). With C as center and radius equal to e draw arc 1. With the same radius and with D as center draw arc 2. Connect their intersection E with C and with D. Then CDE is the desired isos- celes triangle ; for we made CD = 6, FIGURE 49 CE = e and DE = b. DEFINITION. The side, CD, which is not equal to either of the other two sides is the base. CONSTRUCTIVE GEOMETRY 109 EXERCISES 1. The figures in Fig. 50 represent all forms of the triangle. What is a triangle? FIGURE 50 2. What triangle has all its sides equal? What is an equi- lateral triangle? 3. What triangles have at least 2 sides equal? What is an isosceles triangle? 4. What triangles have no two sides equal? What is a sca- lene (ska-leen') triangle? PROBLEM IX. Draw a scalene triangle with sides equal to given lines (no two being equal). EXPLANATION. On the line CX make CD = a as in Problem I. With C as center and b as radius, draw arc 1. Then with D as center and with c as radius, draw arc 2 across arc 1. Call their inter- section E. With the ruler draw line EC and ED. Then CDE is a scalene triangle (J with the sides equal in length to a, fcandc. FIGURE 51 EXERCISES 1. Draw a scalene triangle having sides of these lengths: a b c 2. Draw a scalene triangle having sides of 3", 4", and 5"j of 1", 1J", and 2". PROBLEM X. With the compass draw a three-lobed figure inside of a circle of I" radius. 110 RATIONAL GRAMMAR SCHOOL ARITHMETIC EXPLANATION. Draw a circle with " radius around some point, as O, as a center. Set the pin foot at any point on the circle, as at 1, Fig. 52, and with- out changing the distance between the compass feet draw the arc from A on the circle through O to B on the circle. With same radius and with pin foot on B, draw a short arc at 2. Put the pin foot on 2 and with the same radius as before draw the arc BOG, C being on the circle first drawn. Put the pin foot on C and draw a short arc at 3. Then place the pin foot on 3 and draw an arc from C through O to A. FIGURE 52 EXERCISES 1. Draw a three-lobed figure using a radius of 1". three-lobed figure using as radius this line FIGURE 53 3. Making the distance between the compass points \" , and using the points 1, 2, 3, 4, 5, and 6 in turn, draw a six-lobed figure like Fig. 53. 4. Color the lobes of your figure with a red lead pencil or with water colors and the spaces a, b, c, d, e, and / with a green or yellow pencil or water colors. PROBLEM XI. Draw a regular C-sided figure (regular hexagon) within a circle of \" radius. EXPLANATION. Draw a circle with center at O and with |" radius. Starting at any point on the circle as at 1, put the pin foot on 1 and draw the short arc 2 across the circle, keeping the radius |". Then put the pin foot on 2 and draw the arc 3. Then with pin foot on 3 draw arc 4 ; and so on around. How many steps do you find reach once round? Now with the ruler and pencil connect 1 with 2, then 2 with 3, 3 with 4, and finally 6 with 1. The figure made by the G straight lines is the regular hexagon desired. FIGURE 54 MEASUREMENT 111 EXERCISES 1. Draw regular hexagons within circles of these radii : 1" ; 2" ; J" 9J." , &\ . 2. How long is one side of each of the hexagons drawn in exercise 1? How long is the sum of all the bounding lines of each hexagon? DEFINITION. The sum of all the bounding lines of any figure is called the perimeter of the figure. MEASUREMENT 76. Measuring Talue. Money is the common measure of the value of all articles that are bought and sold. The unit on which United States money is based is the dollar. The dollar sign is $; thus 5 dollars is written $5 or $5.00. This unit is called the U. S. Standard of value. 1. A dollar is worth as much as how many dimes? nickels? quarters? cents? To measure the value of one amount of money by another is to find how many times one of the amounts is as large as the other. 2. Measure $1 by 50^; by 25^; by 20r/ ; by 40^; by 3. Measure $10.50 by 50^; by 75^; by $1.50; by $5.25. 4. A farm is worth $1000 and a city lot $2000. Measure the value of the farm by the value of the lot. 5. Measure the value of a $150 horse by the value of a $15 pig. 6. Measure $75 by $5; by $25; by $15; by $150; by $225. 7. Measure the value of 160 A. of land worth $75 per acre by $100; by $500; by $1000; by $8000; by $24,000. 8. Which is worth the more, 80 A. of land @ $75, or 50 A. @ $112? how much more? 9. Measure $400 by the value of an $8 calf; of a $25 colt. 112 RATIONAL GRAMMAR SCHOOL ARITHMETIC 77. Measuring Length and Distance. Answer these problems, first, by estimate, recording your esti- mates in a notebook; then answer by actual measurement. Finally, compare your estimates with the results of your measure- ments. 1. How wide is the page of this book? how long? How wide are the margins? 2. How wide is a pane of glass in your schoolroom window? how long is it? 3. How wide is the top of your desk? how long? How high is the top of your desk from the floor? How high is your seat? 4. How high is the bottom of the blackboard from the floor? How wide is your blackboard? how long is it? 5. How tall are you? How far can you reach by stretching both arms as far apart as possible? 6. How far is it around your waist? How far is it around your chest, just below your arms, when as much of the air as possible is exhaled from your lungs? What is your chest measurement when as much air as possible is drawn into your lungs? 7. The difference between these two chest measures is called your chest expansion. What is your chest expansion? How does your chest expansion compare with the average for the pupils of your room? NOTE. You may easily increase your chest expansion by a little practice in deep breathing. 8. How many steps wide is your schoolroom? how many steps long? 9. How many feet long is your room? how many feet wide? 10. How many yards long is it? how many yards wide? 11. How many inches long is your step? how many feet long? how many yards long? 12. How many inches long and wide is your schoolroom? 13. How many feet long is your schoolhouse? how many yards long? MEASUREMENT 113 14. How many steps wide and how many steps long is your school yard? how many feet wide and how many feet long? how many yards? 15. How many steps is it from your home to the school- house? how many feet? how many yards? how many rods (1 rd. = 5* yd.)? 16. How far can you walk in 1 min.? How many miles could you walk in 1 hr. at the same rate? 17. How far is it from your home to a neighboring large city? Answer this by using a map and the scale given with it. How long would it take you to walk to that city at the rate of walking in problem 16? 18. How long will it take a train to run from your nearest station to that city at 24 mi. per hour? 19. Using the scale map of your state (see Geography) find the length and the width of your state; of your county; of the U. S. 20. How long and how wide is a two-cent postage stamp? 21. How long is the diameter (distance across) of a copper cent (see Fig. 40)? of a dime? of a nickel? of a quarter dollar? of a half dollar? of a silver dollar? 22. How long is the radius of each of these coins? 23. Wrap a strip of paper, or a string, around each of these coins and find how far it is around each. 24. What kind of unit do you need to measure and express long distances? medium distances? very short distances? NOTE. Such a number as three-eighths, or f, of an inch means three units each one-eighth of an inch long. Such a unit is called a fractional unit, and such numbers as |, |, , J|, }$, ff, are all said to be numbers expressed in fractional units. Name the unit of each of the six frac- tions just given. 25. How long is the distance around (circumference) a bicycle wheel (wrap a string around the wheel)? How long is its diameter? 26. Answer the same questions for a carriage wheel; for the bottom of a bottle ; of a can ; for the head of a barrel ; for any other circles you can find. 114 RATIONAL GRAMMAR SCHOOL ARITHMETIC 27. Arrange these measures thus: OBJECT DIAMETER CIRCUMFERENCE Silver dollar Half dollar 4 Quarter dollar . Nickel Dime Cent Bicycle wheel Carriage wheel Bottle Can Barrel head Fill out columns 2 and 3 of a table like this with your measurements and keep them for later use. o B 7 * 5 4 I .3 2 '.:: ' 1 i a 3 A .5 e 7 a 9 10 it ie ;78. Measuring Surfaces. A square unit is a square I unit long and 1 unit wide. 1. The side wall of a room is 9 ft. by 12 ft. What will it cost to lath and plaster the wall at 3 To pView -- Side View o o Sieve Scale /4 FIGURE 67 80. Measuring Weight. 1. Fig. 67 shows the top and side views of a sand sieve drawn to a scale of J. Measure the drawing and with the aid of the scale find the length, breadth, and depth of the sieve. Answer for both out- side and inside measures. 2. How thick is the stuff used for the frame of the sieve? Two sieves were made in the manual training shop according to the drawings of Fig. 67. The bottom of one was covered with wire netting of T V in. mesh, and of the other with wire of -fa in. mesh. 3. A 6-in. cube of soil weighed 162 oz. What would a 3-in. cube of the same soil weigh? a 2-in. cube? a 4-in. cube? 4. A 4-in. cube of natural soil, weighing 48 oz., was thoroughly dried and then found to weigh 38 oz. What is the weight of water in the natural soil? How many ounces of water would there have been in a cubic foot of the same soil? 5. 24 oz. of dry sandy loam were broken up and rubbed through a coarse sieve. 8 oz. of coarse gravel remained on the sieve. What part of the loam was coarse gravel? 6. The 16 oz. that passed the coarse sieve were rubbed through the fine sieve. The coarse sand that remained on the fine sieve weighed 7-J- oz. What part of the 24 oz. of loam was coarse sand? 7. A 4-in. cube of dry soil weighed 18 oz. It was then thor- oughly saturated with water and found to weigh 32 oz. How much water would a cubic foot of this dry soil be capable of hold- ing when saturated? 8. Find the difference between your weight and that given in the table, p. 18, for a boy or girl of your age. 9. Estimate the weight of your book, or of a brick, or other object, then weigh it and find the difference between the true weight and your estimate. MEASUREMENT 121 10. Find the cost of 3f Ib. of wire nails @ 4 cents. 11. What is the cost of 1 Ib. of sugar selling 22 Ib. for a dollar? 12. Postage on first-class mail matter is 2^ an ounce. What would the postage be on a 2f Ib. package of first-class matter (16 oz. = 1 pound)? 13. Find the cost of 2f Ib. butter @ 32 cents. 14. New York merchants bought in 1 da. the following : 6324 Ib. butter at an average price of 22 cents ; 1988 Ib. cheese at an average price of 14f cents; 6840 Ib. sugar at an average price of 4f cents; 2780 Ib. sugar at an average price of 4.8 cents. Find the total weight and the total cost. 15. Find the total number of pounds and the total* value of these purchases : 650 Ib. cut loaf sugar @ $5.74 per hundredweight (cwt.); 825 Ib. granulated sugar @ $4.80 per hundredweight; 400 Ib. powdered sugar @ $5.24 per hundredweight; 700 Ib. confectioners sugar @ $4.89 per hundredweight; 350 Ib. extra white sugar @ $4.78 per hundredweight. 16. Find the weight and total value of this shipment: 5400 Ib. plain beeves @ $5.70 per hundredweight; 6375 Ib. choice beeves @ $5.80 per hundredweight; 8450 Ib. fair beeves @ $4.90 per hundredweight; 8625 Ib. medium beeves @ $4.60 per hundredweight; 40,700 Ib. veal calves @ $6.50 per hundredweight; 43,000 Ib. western steers @ $8.50 per hundredweight; 8450 Ib. Texas steers @ $3.70 per hundredweight; 25,200 Ib. beef cows @ $2.85 per hundredweight. 17. How many tons did the purchases of problem 14 weigh (2000 Ib. = 1 T.)? How many tons in the shipment of problem 16? 18. A troy ounce of pure gold is worth $20.67. How much is a troy pound of pure gold worth (12 troy oz. = 1 troy pound)? 122 RATIONAL GRAMMAR SCHOOL ARITHMETIC 19. An avoirdupois pound equals l-f^ of a troy pound. About what is the value of an avoirdupois pound of gold? NOTE. Add to the value of a troy pound of gold 31 times , * 4 of its value. 20. An avoirdupois pound = 10 avoirdupois oz. ; what is the value of an avoirdupois ounce of gold? 21. What is the value of your weight in gold? (Your weight is given in avoirdupois pounds.) 22. A grain dealer received during January 130 carloads of grain, averaging 25^ T. each. Counting GO Ib. to the bushel, how many bushels did he receive during the month? 81. Measuring Temperature. On the Fahrenheit thermometer the point where water begins to freeze is marked 32, and the point where water begins to boil is marked 212. (See Fig. 12, p. 23.) 1. How many degrees are there between the boiling point and the freezing point? 2. At 11 p.m. on a certain date a thermometer read 32. The mercury then fell an hour for 4 hr. What was the reading at 3 a.m. the next day? 3. The mercury then fell 1 an hour for 5 hr. What was the reading at 8 a.m.? 4. At 3 a.m. on a certain day the reading was 26. The mercury fell 2 an hour for 5 hr. What was the reading at 8 a.m.? 5. On a certain date the reading was 10, and the mercury fell on the average lf an hour for 7 hr. What was the reading at the end of the 7 hours? 6. It then fell 1^ an hour for 8 hr. What was the reading then? 7. The mercury fell from the reading 12 above zero to 3 below zero. How many degrees did it fall? 8 We might write readings above zero thus : A. 12; A. 6; A 32; and readings Mow zero thus: B. 3; B. 6; B. 30. MEASUREMENT 123 How many degrees does the mercury fall from the first of these readings to the second? (1) A. 8 to A. 3; (4) A. 5 to B. 7; (7) B. 2 to B. 11; (2) A. 8toB. 3; (5) A. 2 to B. 12; (8) B. 7 to B. 13; (3) A. 30 to B. 2 ; (6) A. 4| to B. 4| ; (9) B. 9^ to B. 12|. 9. How many degrees of rise or fall are there from the first of these readings to the second? If the change is a rise, mark it R; if a, fall, mark it F. : (1) B. 7 to B. 2; (5) A. 9 to B. 9; (9) A. 6|to A. 12; (2) B. 2i to B. 1; (0) A. 2| to B. 2| ; (10) A. 3 to B. 30*; (3) B. 2i to A. 1; (7) B. 15 to B. 6i ; (11) A. 18 to A. 67| ; (4) A. 3 to B. 1 ; (8) B. 15 to A. 6^ ; (12) B. 22 to A. 10. Instead of writing an A. for ''above zero," readings, it is customary to use the sign (+). What sign would you then sug- gest for "below zero" readings? Tell whether a change from the first to the second of these readings denotes a rise or a fall in each case and by how much? (1) + 16 to + 100 ; (4) + 32 to - 3 ; (7) - 18 to - 34 ; (2) +32 to 4-212; (5) -I- 16 to - 17 ; (8) -18to-6|; (3) + 2 to + 161; (6) + 8 to - 30; (9) - 6 to - 2 11. The 12 o'clock (noon) readings for 4 successive days were as follows : + 82| ; + 78f ; + 61^ ; + 534. What is the average of these 12 o'clock readings. 12. Find the average of these 9 a.m. readings for G da. : + 9 ; 4-4; +5; + 12$;+ 14 ; + 9J. 13. Find the average of these 6 readings: - 4; - 6; - 2; -5; - 13; - 12. 14. Find the average of these 2 readings: -r 8 and 2. Show on Fig. 12, p. 23, what point on the thermometer is midway between the readings +8 and - 2. 15. What point is midway between the readings +13 and - 5? between the readings + 4 and - 6? + 12 and - 12? - 4 and - 12? 124 RATIONAL GRAMMAR SCHOOL ARITHMETIC 82. Measuring Time. The primary unit used in measuring time is the mean solar day. This day is the average time interval during which the rotation of the earth carries the meridian of a place eastward from the sun back around to the sun again. It is the average length of the interval from noon to the next noon. 1. If the time piece is running correctly, how many times does the short hand (the hour hand) of a watch or clock turn completely around from noon to the next noon? 2. How much time is measured by one com- plete turn of the hour hand? 3. What part of a day of 24 hr. is measured by the rotation of the hour hand from XII to VI? from XII to III? from XII to I? to II? FIGURE 68 t() IXJ? 4. What name is given to the time interval in which the hour hand moves from XII to I? 5. For measuring shorter periods the motion of the long hand (the minute hand) is used. What part of a day is measured by the movement of the minute hand from XII around to XII again? from XII to VI? from XII to III? from XII to IX? from XII to I? 6. What name is given to the interval of time required for the long hand to make one complete turn? 7. How many times does the minute hand turn around while the hour hand turns around once? 8. The small and rapidly moving hand covering the VI of the watch face is the second hand. How much time is measured by one whole turn of the second hand? 9. Over how much space does the tip of the minute hand move while the second hand moves around once? 10. The second hand circle is divided into how many equal parts? What name is given to the time interval in which the second hand moves over one of these equal spaces? 11. How many times does the minute hand turn around in one MEASUREMENT 125 day? in a week? in a month of 30 da.? in 365 da. (1 common year) ? 12. Answer the same questions for the second hand. 13. There are about 365^ da. in 1 yr. If the hour and the minute hands of a watch start at XII and the second hand at 60 at the beginning of a year, how will the hands point at the instant the year ends, if the watch runs correctly and without stopping? 14. Answer question 13 supposing the length of the year to be 365 da. 5 hr. 48 min. 46 seconds. 83. Measuring Land. 1. A section of land is a tract 320 rd., or 1 mi., square. It is usually divided into halves, quarters, eighths and sixteenths, as shown. How many square rods make a section of land? w 2. An acre many acres in 160 sq. rd. of how land? J Brown I T) CO | WH Dugan "* -\ ? A .S.Park 1 \ J.M5mith SECTION OF LAND FIGURE 69 a section in a quarter section? 3. A section of land is divided up into farms, as shown in the cut; how many acres are there in the farm belonging to A. S. Park? to H. S. Barnes? to J. Brown? H. A. Dryer? J. M. Smith? P. S. Mosier? J. S. Doe? 4. Point out the south half of the section; the east half; the 1ST i; the W 4. 5. Point out the SE ; the NW i; the SW i; the XE . 6. Point out the N of the SE i; the E -J- of the SE ; the W J of the NE i; the S NW i; the W | EJ. 7. Point out the SW i of the SE i; the NE i SW ; the NW i NW i; the NE i SW . 8. The farm of E. Miner would be described as the N -J E -J- E -J- NE i ; describe the farms of the 8 other owners of this section. 9. How many rods of fence would be required to enclose the section and separate it into farms as shown? 10. How many rods long is a ditch starting from the NE corner of H. A. Dryer's farm, thence running due south to H. S. Barnes's 126 RATIONAL GRAMMAR SCHOOL ARITHMETIC N line, thence duo W to Barnes's W line, thence S to J. M. Smith's N line, thence due W across Park's farm to his W line? 11. Each small square in Fig. 70 represents a section. The large square represents a township; how long is a township? how wide? 12. How many square miles in 1 Tp.? how many acres? 13. Point out in Fig. 70 the following: (1) NW i Sec. 33. (2) S i- Sec. 17 ; NE i Sec. 17 ; E ^ NE -j TOWNSHIP FIGURE 70 g ec -^ (3) NW i Sec. 21 ; NE NW }- Sec. 21 ; SE NW i Sec. 21. 14. How many farms of 100 A. each could be made of a township? 15. How long and how wide is a square 160-acre farm? 84. Plotting Observations and Measurements. 1. The hourly temperatures from G a.m. to p.m. Decem- ber 2G, 1902, were: 6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m. 7 7 10 10 12 13 12m. 1p.m. 2p.m. 3p.m. 4p.m. 5p.m. G p.m. 14 14 15 16 15 14 14' Draw 13 equally spaced vertical parallels one for each hour. Draw a horizontal line as OX across the parallels. Using " to repre- sent 1 ; measure off on the first vertical the distance 01 to represent 7; on the second vertical the distance 7-2 to represent the second 7; on ihe third parallel measure off the distance 10, and so on. Mark the top of each measured vertical distance with a dot. FIGURE 71 Draw lines connecting these dots as in Fig. 71. This is called plotting the readings. This line shows the temperature change during the 12 hrs. MEASUREMENT 127 This plotting is very much aided by the use of cross-lined paper, ruled into small squares. A horizontal side of one of the small squares might be used to represent 1 hr. and a vertical side to represent 1 of temperature. NOTE. Pupils in arithmetic should have notebooks containing several pages of squared paper for such work as this. 2. Eead the out-door temperatures from hour to hour at your schoolhouse and plot your readings as above. 3. The average hourly temperatures from 6 a.m. to 6 p.m., December 26 to January 2, were: 6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m. 21.5 22.1 22.2 22.2 22.5 23.2 12 m. 1 p.m. 2 p.m. 3 p.m. 4 p.m. 5pm. G p.m. 23.9 24 24.5 25.2 25 24.7 24.3 Plot these readings on squared paper, or by measurement, as above in figure 71. Plot the tenths by estimate. 4. Does the line for the averages for a week agree in a general way with the line for December 26? What does a comparison of the two lines show? This table gives the heights in feet and weights in pounds of boys and girls of ages given in the first column : HEIGHT IN FEET WEIGHT IN POUNDS AGE BOYS GIRLS BOYS GIRLS 1.6 2 2.6 4 3.0 6 3.4 9 4.0 11 4.4 13 4.7 15 5.1 17 5.4 18 5.4 20 5.5 5. Using squared paper, or making a drawing such as is indi- cated in Fig. 72, plot the numbers in the first column horizontally and those in the second column vertically. Draw free-hand a 1.0 7.1 6.4 2.6 25.0 23.5 3.0 31.4 28.7 3.4 38.8 35.3 3.9 50.0 47.1 4.3 59.8 56.6 4.6 75.8 72.7 4.9 96.4 89.0 5.1 116.6 104.4 5.1 127.6 112.6 5.2 132.5 115.3 128 RATIONAL GRAMMAR SCHOOL ARITHMETIC Ac es f JfU 5 X .346 9 II 13 5 171830 FIGURE 72 smooth line through all the plotted points and obtain the curve for growth in stature of boys. At what age do boys cease growing rapidly in height? 6. Using the numbers of columns 1 and 3 obtain a similar curve for girls. At what age do girls cease growing rapidly in stature? 7. Compare the two curves and note when boys and when girls grow fastest? 8. Using the numbers of columns 1 and 4 draw the curve for growth of boys in weight. 9. Use numbers of columns 1 and 5 similarly for girls. 10. Compare the curves of problems 8 and 9 and note any similarities or differences in the two curves. What do the peculiarities of these curves show? 11. Twelve different rectangles each 12 in. long have the widths given in the first line and the areas in the second line. Widths in inches..!" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" Areas in sq. in.. ..12 24 36 48- CO 72 84 96 108 120 132 144 Two lines, OX and OY, are drawn at right angles. 12 equally spaced lines are drawn par- allel to OY. The horizontal distances 01, 02, 03 and so on denote the heights, 1", 2", 3' and so on. On a scale of -J" (or the vertical side of a small square) to 12 sq. in. of area, mark off the lengths la, 2b, 3c, and so on, to denote the numbers in the second line above. This gives the points a, b, c, and so on. Make the construction to the scale indi- cated or to some other convenient scale, and place the straight edge of a ruler along the points, such as a, b, c, and so on. On what kind of line do the points seem to lie? 12. The bases of 12 triangles are each 16 in. long and the alti- tudes (heights) and the areas in square inches are: V 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 8 16 24 32 40 48 56 64 72 80 88 96 FIGURE 73 MEASUREMENT 129 Using any convenient scale, plot these data. On what kind of line do all the points lie? 13. The lengths of the sides and the areas in square inches of 12 squares are : Sides.. I" 2" 3" 4" 5" 0" 7" 8" 9" 10" 11" 12" Areas.. 1 4 9 1G 25 30 49 G4 81 100 121 144 Using any convenient horizontal and vertical scales plot these observations. Do the plotted points lie on a straight line? 14. Plot these data and draw a smooth free-hand curve through the plotted points : DATE OF CENSUS POPULATION OF U. S. IN MILLIONS DATE OF CENSUS POPULATION OF U. S. IN MILLIONS 1790 3.9 1850 23.2 1800 5.3 1860 31.4 1810 7.2 1870 38.6 1820 a.6 1880 50.2 1830 12.9 1890 02. 6 1840 17.1 1900 76.3 Plot dates on horizontal, and populations on vertical, lines. Draw a smooth free-hand curve through the plotted points. By continuing the curve can you predict about what the popula- tion will be in 1910? Is the line through the plotted points a straight line? 85. Measuring By Hundredths. PERCENTAGE 1. What is T -J o of 500 mi.? of 500 mi.? 2. What is of 200 A.? of 200 A.? of 500 miles? of 200 acres? NOTE. First find ^Q. 3. What is ^A of 100 ft.? T % of 100 ft.? ^j of 100 feet? 4. -J of anything, equals how many hundredths of it? 5. How many hundredths of anything are the following frac- tional parts of it? i; t; i; 1; !; ~s > ro 5 130 RATIONAL GRAMMAR SCHOOL ARITHMETIC DEFINITION. One hundredth is often written \% and read 1 per cent. The sign (% ) is a short way of writing "one one-hundredth. " Per cent is a short way of speaking "one one-hundredth." 6. Kead and give the meaning of the following : 2%; 6%; 8%; 12%; 12*%; 25%; 33*%; 87*%; 100%. 7. Give the simplest fractional equivalents of these per cents : 50%; 25%; 75%; 12*%; 33*%; 66|% ; 20%; 40%; 80%. 8. How many Ib. are 50% of 8 lb.? of 18 lb.? of 24 pounds? 9. How many square feet are 25% of 16 sq. ft.? of 48 sq. ft.? of 88 sq. ft.? of 400 square feet? 10. Heating an iron rod 100 in. long increased its length 2%. How many inches was the length increased? 11. One boy threw a stone 100 ft., and another threw it 12% farther. How many feet farther did the second boy throw the stone? 12. Referring to Fig. 22 (page 45) point out the following per cents of it: 20% ; 40% ; 60% ; 80% ; 2% ; 4% ; 10% ; 100%. 13. Draw a square and divide it by a line so as to show 50% of it; 25% of it; 75% of it; 12*% of it. 14. Similarly, show the same per cBnts using a circle; a rectangle. 15. Draw a line of any length and show 33*% of it; 66f % of it; 100% of it; 12*% of it; 10% of it; 20% of it. 16. Draw a square and divide it into small rec- tangles as shown in the figure. Point out the following per cents of it : FIGURE 74 25%; 33*%; 50%; 66f % ; 75%; 12*%; 100%. 17. A pair of shoes costing $2 was sold at a gain of 50%. What was the amount gained? 18. A boy was 50 in. in height 2 years ago. Since then he has grown 4% higher. How many inches taller is he now? How tall is he now? 19. The height of a 14 year old boy is 60 in., and a girl of the same age is 2% taller. How tall is the girl? MEASUREMENT 131 20. An umbrella was marked $1.50, and was sold at a reduc- tion of 20%. At what price did it sell? 21. Eighty per cent of the cost of a girl's suit is $4.00. What is 1% of the cost? What is the whole cost, or 100% of the cost? 22. An article sold for $4.00 after having been marked down 20%. What was the price before it had been marked down? 23. A lot was sold for $800, which was 0% more than it cost the man who sold it. How much did it cost him? NOTE. The 60% means 60% of what it cost the seller. This cost is how many per cent of itself? $800 is, then, how many per cent of the original cost. 1% of the original cost equals what? 100% of this cost equals what? 24. A thermometer reading was 120, which was .20% higher than the reading taken 10 min. before. What was the previous reading? NOTE. In such questions as this answer first the question, "20% of what?" In this case 20% of the previous reading. 25. A man sold his farm for $12,000, which was 25% more than he paid for it. What did he pay for it? 26. This year a flat rented for $55 a month, which is 10% more than it rented for last year. For what did it rent last year? 27. Out of 80 games played by a championship team, 20 were lost. What per cent of the games were lost? What per cent were won? 28. The chest measure of a boy at the close of the school year was 27 in., which was 10% greater than at the beginning of the year. What was the boy's chest measure at the beginning of the year? 29. A man paid $18 for the use of $300 for a year. What per cent was the sum he paid of the sum he borrowed? 86. Simple Interest. An extensive use of the system of measurement by hnndredths is made with problems in Interest. DEFINITION. Interest is money to be paid for the use of money. Two per cent interest means that 2/ is to be paid every year for the use of 1000 or $1 ; $2 for the use of $100, and so on at this rate. What then would 6% interest mean? 4% interest? 10% interest? 132 RATIONAL GRAMMAR SCHOOL ARITHMETIC The amount of money to be paid each year for the whole sum bor- rowed is called the Interest for 1 year. 1. What is the interest on $200 at 4% for 1 year? 2. What is the interest at 6% on $1 for 1 year? NOTE. In computing interest a year means 12 months of 30 days each. 3. What is the interest at 6% on $1 for ^ yr., or 6 mo.? for 2 mo.? for 1 month? 4. What is the interest at 6% on $1 for 3 mo.? for 5 mo.? for 7 mo.? for 14 mo.? for 16 mo.? for 34 mo.? for any number of months? 5. From the answer to problem 2, find the interest at 6% for 1 yr. on $8;, on $25; on $48; on $85; on $124; on $450. 6. If you know the interest at 6% for 1 yr. on $1, how can you find the interest at 6% for 1 yr. on any number of dollars? 7. From the last answer to problem 5, find the interest at 6% on $450 for 2 yr. ; for 3 yr. ; for 4 yr. ; for 12 yr. ; for any number of years. 8. Using the third answer to problem 4, find the interest at 6% for 7 mo. on $48; on $450; on $75. 9. Tell how to find the interest at 6% on any number of dol- lars for any number of months. ' 10. The interest on a certain sum of money for 1 yr. at 6% is $24. What is the interest on the same sum for the same time at 12%? at 18%? at 30%? at 60%? at 1%? at 3%? at 2%? 11. f of the interest on a sum of money for a given time at 6% equals the interest on the same sum for the same time at what rate per cent? NOTE. | of a number may be easily found by subtracting from it | of itself. How may of a number be found similarly? 12. Knowing the interest on any sum of money at 6%, how can you quickly find the interest on the same sum for the same time at 7%? at 8%? at 11%? at 15%? at any rate per cent? 13. Find the interest on $1200 at 6% for 2 yr. ; for 2f years. 14. A man has $350 in a bank, which pays 3% interest. To how much interest is the man entitled if his money has been in the bank 2 yr. and 4 mo.? 6 yr. and 9 mo.? 8 yr. and 2 months? COMMON USES OF NUMBERS 87. Pressure of Air. ORAL WORK 1. When a glass is filled level with water, covered with a piece of writing paper, and carefully inverted, why does not the water fall out when the glass is held mouth downward? 2. When a soft leather sucker is moistened and spread out on a smooth, flat surface, why does it cling to the surface even when the sucker is raised by being lifted at its middle? 3. When one end of a glass tube is placed in water and you draw with the mouth at the other, why does the water rise in the tube? 4. Fill a bottle with water, leave the cork out, and invert the bottle in a vessel of water. Why does the water not run out? 5. When a glass tube with one end closed is partly filled with mercury and the open end dipped under the surface of mercury in a cup, why does not all the mercury run out of the tube? NOTE. In connection with 5, examine a mercurial barometer. 6. When a sheet of thin rubber, or paper, is held over the mouth of a funnel, and the air is sucked out of the funnel through its neck, why does the rubber curve inward? Will it do this in ali positions of the funnel? Why? These experiments show that the air presses downward, upward, and in all directions upon surfaces. Careful measure- ments have shown that the pressure of the air on every square inch of surface is about 15 pounds. NOTE. This pressure is equal on all sides of surfaces, upper sides and lower sides, outside and inside, toward the right and toward the left. WRITTEN WORK 1. Measure the length and the width of the cover of your book and find the downward pressure in pounds on the upper surface of your book when it lies flat upon the desk. Why is the book so easily, moed about .under, t his pressure? - . , - 133 134 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Measure and compute the downward pressure of the air on the top of your desk. Can you lift this number of pounds? Why can you lift the desk? 3. A room is 10' x 20' x 25'. Find the pressure of the air, in pounds, on the floor, on the ceiling, and on each of the four walls of the room. Why does not this pressure tear the walls apart? 4. The average surface of the human body equals 20.6 sq. ft. What is the total pressure of the air on the outside surface? Why does not this pressure crush the body? 5. Measure the length and the width of the door of your room, and find the air pressure on one side of the door. Why can you open and close the door so easily? 6. Measure and find the air pressure in pounds on the outside surface of a pane of window glass. Why does not this pressure break the glass? 88. Passenger and Freight Trains. 1. A fast train runs from Chicago to a station 356.4 mi. dis- tant in exactly 9 hr. What is the average rate (miles per hour) of the train? 2. The train left Chicago at 6:10 a.m. At what time did it arrive at the station? 3. Another train runs 10 hr. at an average rate of 36 mi. per hour, including stops ; how far does it run? 4. If the train (problem 3) started at 2: 30 a.m., at what time did it reach the end of the run? 5. A traveler went by rail from Chicago to Los Angeles, California, in 4 da. 19 hr., at the average rate of 37.4 mi. an hour, including stops. How far is it from Chicago to Los Angeles by this route? 6. A train ran 361 mi. in 9J hr. What was its average rate? 7. A freight train is running 21 mi. an hour while a brakeman on top of the cars walks toward the engine at the rate of 2-J mi. an hour. How fast does the brakeman actually move forward? how fast, if he walks from front to rear at the rate of 2-J mi. an hour? 8. A conductor walks from the front to the rear of a train at the rate of 3 mi. an hour while the trait* is running 38 mi. an hour. COMMON" USES OF NUMBERS 135 At what rate per hour does the conductor actually move forward? At what rate does he move in the direction of the running train if he walks from the rear to the front at the rate of 3 mi. an hour? 9. How may the conductor (problem 8) suddenly change his rate of motion from 34 mi. to 4H mi. an hour? How many miles an hour does this change of rate amount to? 10. A passenger train running 32 mi. an hour meets a freight train running in the opposite direction on a parallel track 18^ mi. an hour. At what rate do the trains approach each other? 11. At what rate does the passenger train (problem 10) pass the freight if both are running in the same direction on parallel tracks? 12. A boy tries to overtake a street car by running up behind it. If the boy runs 10 ft. a second and the car runs 6 ft. a second, how soon will the boy overtake the car if he is now 20 ft. behind it? 13. Two friends, coming from opposite directions, have arranged to meet at a certain railway station. Their trains are running at 35 mi. and 25 mi. an hour. Not allowing for time lost in stops, how soon will their trains be together at the station, if they are now 30 mi. apart and both trains reach the station at the same time? 14. A passenger train is made up of a postal and baggage car. an express car, 3 common coaches, 2 chair cars, a dining car, and 2 sleepers. The average length of a car is 6 If, and the length of the engine and tender together is 65'. How long is the train? 15. The weight of the engine (problem 14) is 142,780 Ib. ; of the tender, 43,200 Ib. ; and the average weight of a car is 83,480 Ib. What is the total weight of the train, in tons? 16. To draw a train on straight, level track at a speed of 40 mi. an hour, requires a horizontal pull of ?fo of the weight of the train. What force in tons would be needed to draw the train of problem 15 at a speed of 40 mi. an hour? 17. An engine, 62' long and weighing 240,000 Ib., draws a tender, 22' long, weighing 64,500 Ib., and a train of 72 empty freight cars, averaging 36|' in length and 32,700 Ib. in weight. How long is the train? how heavy? 136 RATIONAL GRAMMAR SCHOOL ARITHMETIC 18. If it requires T / 5Tr of the weight of a train to draw it on straight, level track at a speed of 20 mi. an hour, how many pounds of force must the engine of problem 17 exert to draw the train, on such track, 20 mi. an hour? How many tons of force? NOTE. First find j^g of the weight of the train. 19. How much force would be needed to draw 38 cars, each weighing 16 T., and each loaded with 18 T. of coal, on straight, level track at a speed of 20 mi. an hour? (See problem 18.) 20. When a train is moving 5 mi. an hour it takes a horizontal force of about T of the weight of the train to draw it along straight, level track. On such track what force must an engine exert to draw a train of 68 empty freight cars, each weighing 34,300 lb., at a speed of 5 mi. an hour? 21. Under the same conditions as in problem 20, what force would be needed to draw a train of 38 cars, each weighing 34,000 lb. and carrying a load of 58,600 lb. of coal in addition to its own weight? 22. A railroad company purchased 6 locomotive engines, weighing, in pounds, 142,780, 142,630, 158,670, 139,790, 146,890, and 138,960. The tenders weighed, in pounds, 43,750, 44,200, 45,280, 42,920, 43,650, and 44,280. The cost per pound of the combined weight was 13f oCHEDTJijE TION M^ TIME 1ST 2D 4TH 5TH 6TH 7TH STH 1 3:50 3:50 3:491 3:50f 3:50 3:50^ 3:50| 3:51} 2 3.1 3:55 3:55 3-55i 3:56 3:55 3:55 3:55^ 3:56 3 5.5 3:57 3:57 u .wg 3:58 3 :58f 3:57f 3:581 4 7.9 3:59 3:59 4:00 4:00 3:59* 4:00| 4:00 4 4:00 5 .. 12.0 4:02 4:02 4:03f 4 :03f 4:03i 4:04 4:03 4:03 6 17.0 4:06 4:06 v"4 4:073 4 4:07| *- VV 4 4 4:08J 4:07 7 19.9 4:09 4:09 4:10 4 '09? 4-08 1 4:11 4:10 4:10 8.. . 24.5 4:12 4-1H 4:14 4:13 4:12 4:14$ 4:14 4:13| 9 27.6 4:15 2 4:15 4:14 4:17 a 4:16 4 4:15 10 33.8 4:20 4:18 4:211 4:20 4:18 4:22 4:201 4:20| 11 38.7 4:24 4:22 4 4 :25 4:24 2 4:22 4:26i 4:24| 4:23 12 43.5 4:28 4:25| 4 :27i 4:26 ^ V 2 4:30 4:28 4:27 13 50.5 4:33 4:34^ " J 4-321 4:31 4-351 4:33 4:32 14 53.8 4:38 4-331- " 2 4:35 4:34 *"i 4:38 4:36 4:354 15 55.5 4:40 4 :35 4:40 4:37| 4:36 4:40i 4:372 I 4:37 Number of cars 5 7 4 6 5 ^t iV ^ 5 -X. U 9 5 5 Passengers carried 201 441 11 79 107 113 118 Running time Mi. per hour (average) . . PROBLEMS 1. By ''running time" is meant the difference between the time of leaving* station 1 and the time of arriving at station 15. Fill out the blanks in the line "running time." 2. Find the difference between "schedule time" and the time the train actually reached station G on the 2d, 4th, 6th, 7th, 8th. 3. The train is "on time*' if it reaches a station just at "schedule time." Find how much the train was ahead of or behind time in reaching station 12 on each of the 7 da. Mark "ahead of time" results with an "A," thus: A. 2| min., and "behind time" results with a "B," thus: B. 1 minutes. 4. Make and answer similar questions for other stations. 5. How far is it from station 1 to station 15? from 2 to 9? 4 to 11? 3 to 15? Make and answer similar questions for other stations. 6. What is the difference in schedule times between stations 1 and 15? 2 and 9? 4 and 11? 3 and 15? any other two stations? * Stops are so short that leaving and arriving times are regarded as the same. 138 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. From your answers to problems 5 and 6 find the average* rate of running (in miles per minute) between stations 1 and 15, by a train running exactly on schedule time; between stations 2 and 9; 4 and 11; 3 and 15. 8. Find the average rates of running between the same stations (as in problem 7) on the 6th. 9. Find the average rates of running between stations 1 and 15 on each day and write the results in the "mi. per hour" line. 10. What is the total number of cars drawn during the whole period? 11. Find the total number of passengers carried. 12. Make and solve other problems on the table. 90. Areas of Common Forms. ORAL WORK 1. If one side of an inch square represents 80 rd., how many square rods will the inch square represent? how many acres? 2. In a drawing of a rectangular farm to a scale 1 in. = 80 rd., how could you find the number of acres in the farm? 3. If a square field containing 40 A. is cut across diagonally by a railroad, how many acres does each of the triangular posts contain, supposing the railroad itself covers 3 acres? FIGURE 75 4. Read off the areas of the cross-lined portions of the 9 forms of Fig. 75. * "Average rate" here means distance run divided by time of running, COMMON USES OF NUMBERS 139 PROBLEMS Each inch in Fig. 76 represents 80 rods. 1. How many acres are in the tract ABCDEt \p What is the farm worth at $95 per acre? 2. The field is entirely enclosed, except along the river, by a barb- wire fence having 4 lines of wire. Barb wire weighs about 1 Ib. per rod. Find the cost of the wire at $3.50 per bale of 100 pounds. 3. If posts cost 35^ apiece and are set 2 rd. apart, find the cost of the posts. w FIGURE 77 4. Answer similar questions for Fig. 77, the scale being the same as for Fig. 76. 5. If the weight of a certain piece of inch board is 20 oz., and it represents the area of a 20-A. field, how many acres would be represented by a piece of the same board weighing 37 ounces? 6. A farmer stepped the 4 sides and the diag- onal of an irregular field and found them to be 60 rd., 70 rd., 75 rd., 85 rd., and 80 rd. He drew on a board a 4-sided figure (like Fig. 78, but larger) to represent the field. The sides of the drawing were 6 in., 7 in., 7J in., 8J in., and a diagonal AB was 8 in. What was the scale of his drawing? 140 RATIONAL GRAMMAR SCHOOL ARITHMETIC FIGURE 79 7. On the same board and to the same scale as in problem 6 the farmer drew a rectangle 12 in. by 8 in. (like Fig. 79, but larger). What was the area of the rectangle? How many square rods did the rectangle represent? how many acres? 8. Both blocks were then sawed out and weighed. The irregular block weighed 1 T \ Ib. and the rectangle 2J Ib. What was the area of the irregular field? It is desired to find the areas of the outer faces of the separate stones, 1, 3, 3, and so on, of an elliptical arch. The span of the arch is 24' and the rise 9', as shown. A draw- ing of the arch (like Fig. 80, bat larger) was made on heavy, firm cardboard (bristol board) to a scale of 1: 12. The pieces of cardboard representing the separate stones were then carefully cut out and weighed. A square inch of the cardboard was also cut oat and weighed. The weights, in grains, of the several pieces are here tabulated : sq. No. 1 in. WT. 216 No. 7 - 48 gr. WT. 276 2 168 8 156 3 156 9 ' 120 4 150 10 162 5 126 11 168 6 150 12 180 13 210 9. Noticing that 48 gr. is the weight of 1 sq. in. of the card- board and that it represents a square foot on the actual arch ; find from the numbers of the table the areas of the faces of the several stones forming the arch. 10. A block of the stone 1 ft. square and as long as the hori- zontal thickness of the archstones weighed 688 Ib. Noticing that the weight 48 gr. may also be taken to represent this 688 Ib., find the weights of the several archstones. INTRODUCTION TO FRACTIONS 141 INTRODUCTION TO FRACTIONS KATIO AND PROPORTION DEFINITION. The ratio of one number to another is the quotient of the first number divided by the second. 1. What is the ratio of A to 0? of to A? of B to A? of A to ? of B to 0? of to B? of C to B? of C to A? of (7 to 0? of to (7? of D to C? of D to ? of to 0? of to Z>? f FIGURE 81 2. In the second square, compare each of the parts E, F, and G with 0; compare with each of these parts. 3. Compare each division with each of the others, separately. 4. In the third square, J of H is how many times J? J of I is what part of H*? \ of is how many times /? 5. What is the ratio of 3" to 1"? of 1" to 3"? of 1" to J"? of I" to 1"? 6. What is the ratio of 6 to 0? of 50 to 50? of j to J? of a to a? of x to ? 7. What is the ratio of any two equal numbers? , _ x _ N J is a short way of writing the following ' - ' - ' " L^_, e xpr essio n s : f FIGURE 82 The ratio of 1 to 4, 1 :4, i:l- 1:4 is read "1 to 4," and J:l is read " to 1." 8. What is the ratio of 1 ft. to 1 yd.? of 1 hr. to 1 da.? of 1 nickel to 1 dime? of 1 in. to 12 in.? to 1 in.? of 1 mi. to 1 mile? of 1 ft. to 3 ft.? of 12 in. 142 RATIONAL GRAMMAR SCHOOL ARITHMETIC The ratio of a 12-in. line to a 1-in. line is called, also, the measure of the 12-in. line by the 1-in. line. 9. What is the ratio of 1 sq. ft. to a 3-in. square 6 (see Fig. 83)? of 1 sq. ft. to 1 square inch? The ratio of a square foot to a 3-in. square is the measure of a square foot in terms of the 8-in. square. Finding the ratio of the foot to the inch is the same as meas- uring the foot by the inch. Furthermore, to find the ratio of any number, or quantity, to any other number, or quantity, is to find the quotient of the first number, or quantity, divided by the second. The result of measuring one number by another is called the numerical measure of the first by the second. The ratio of one number to another, or the measure of one number by another, can be found only when both numbers are expressed in the same unit. 10. Measure the avoirdupois pound by the ounce ; the foot by the inch; the inch by the foot; the ounce by the pound; the gal- lon by the quart ; the bushel by the quart. 11. Measure the rod by the yard; the mile by the rod; the square yard by the square foot ; the quart by the gallon. 12. Measure 8 ft. by 4 ft. ; 1G ft. by 64 ft. ; a square mile by a square rod; an acre by a square rod; a square mile by an acre. 13. Measure 80 by 8 ; 80 by 4; 3 by 4; 4 by 3; 9 by 18; 18 by 9; 125 by 25. 14. Measure a by #; a by 2; b by a\ %a by a\ 6x by 2z; 2# by 6z. 15. If 5 apples cost 4$, what will 35 apples cost? SOLUTION. How many fives of apples in 35 apples? How much does one five of apples cost? How much then do 35 apples, or 7 fives of apples, cost? This analysis may be thought of in the form of ratios. Cost of 35 apples x 35 x _ . f K F | = 77,, or more briefly -= = -^ (1) Cost of 5 apples 4^ 5 4f and we have to find a number which has such a ratio to 4^ as 35 apples has to 5 apples. This leads us to an equation of ratios. DEFINITION. An equation of ratios is called a proportion. INTRODUCTION TO FRACTIONS 143 91. Proportion. In all these problems use the equation form of statement like (1) in the solution of problem 15 of the preceding section. 1. If 1 doz. oranges cost $0.30, what will 4 oranges cost at the same rate? 2. If a yard of cloth costs $1.50, at the same rate what will J yd. cost? 3. If a 3-in. square of tin costs ^, what will a square foot cost at the same rate? 4. If 6 qt. of oil cost 15^, what will 3 gal. cost at tho same rate? (4 qt. = 1 gal.) 5. Two rectangular flower beds have the same shape. One ia 3 ft. wide and 4 ft. long; the other is ft. wide. How long is it? 6. Two books are of the same shape but of different sizes. One is 5" wide x 7-J-" long. The other is 15" long. How wide is it? 7. In two triangles of the same shape, like those of Fig. 84, 12", and I = 4", how long is B? 8. In triangles of the same shape, like those of Fig. 84, (2), if a = 4", A = 12", and b = 5", how long is B? If a = 6", b = 8", and B = 32", how long is A? If b = 7", B = 35", and A = 30", how long is a? 9. In triangles of the same shape, as those of Fig. 84, (3), if A = 21", b = 9", and B = 27", how long is a? If a = 5", A = 22", and B 30", how long is ? 10. In triangles of the same shape, like those in Fig. 84, (4), if a = 3", A = 9", and C= 6", how long is c? If A = 24", c = 4", and C = 12", how long is a? 92. Fractions as Ratios and as Equal Farts. 1. Into how many equal parts is A (Fig. 85) divided? What is one of the parts called? (i) (4) 144 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Give the number of equal parts and the name of one of the parts in B>, C\ D\ E\ F\ 0; H\ /; /. ABODE FIGURE 85 3. One part of A equals how many of the smallest parts of JB? of (7? of D ? of E? of G ? of /? of /? 4. One part of F equals how many of the smallest parts of 6r? of H? of/? of/? of D? of EV 5. Write the fraction that names one of the equal parts of J, as divided in the illustration. 6. Measure A by one of the equal parts into which it is divided. Ans. 2 halves, written f . 7. Measure the shaded part of B by one of its 4 equal parts. Ans. f, read "2 fourths." 8. Measure the unshaded part of B by the same unit. 9. Using one of the 8 equal parts of C as a unit, express the shaded part of C in this unit; the right half of the shaded part. 10. Express E in terms of the smallest unit it is divided into ; the upper half of E. 11. Express in terms of the smallest unit each figure is divided into: (1) the whole square F; the shaded part; the unshaded part. (2) (3) (4) (5) 0; H; I; J; DEFINITION. | means 3 of the i's (one fourths) of some number, or measured quantity. The is called the fractional unit. The fractional unit of any fraction is one of the equal parts expressed by the fraction. INTRODUCTION" TO FRACTIONS 145 12. What is the fractional unit of each of the following frac- tions: f, |, f, t, 4, fr, f, 1, H, t> i, i, 4, 1, A, A, A, A? 13. What is the fractional unit of a fraction having the num- ber 7 below the line? the number 6? 9? 12? 15? 25? 18? 11? 64? 14. How many fractional units are expressed by the 1st frac- tion of question 12? by the 2d fraction? the 3d? 4th? 5th? 6th? 7th? Draw a square and divide it free-hand by lines showing the meaning of each of the first 8 fractions of problem 12. 15. What does the number written below the line of a fraction express? 16. What does the number written above the line of a fraction express? DEFINITIONS. The number above the line is called the numerator (meaning number er). The number below the line is called the denominator (meaning namer). 17. What are the fractional units of fractions having the fol- lowing denominators : 3? 7? 12? 18? 24? 125? 75? 19? a? x? 18. Point out the fractions of question 12 that have the same fractional units. DEFINITION. The numerator and the denominator are together called the terms of a fraction. 19. If each half of a square is divided into 2 equal parts, into how many equal parts is the whole square divided? 20. Into how many equal parts is a whole divided, if each half of it is divided into 3 equal parts? into 4 equal parts? into 5? 6? 7? 8? 9? 10? 11? 25? 21. Into how many equal parts is a whole divided, if each third of it is divided into 2 equal parts? into 3 equal parts? into 4? 5? 6? 7? 8? 9? 10? 12? 15? 20? 30? 22. Into how many equal parts is a whole divided by dividing each of its fifths into 2 equal parts? 3 equal parts? 4? 5? 6? 7? 8? 9? 10? 20? 30? Give solutions of such as the following and explain : 23. Express as sixths, , f , f , J, }. 24. Express as twelfths, J, J, |, J, f , J, f 146 RATIONAL GRAMMAR SCHOOL ARITHMETIC 25. In question 12 point out pairs of fractions that are not expressed in the same fractional unit, but may easily be so expressed. 26. Express the following pairs of fractions as equivalent fractions having the same fractional unit, or, what is the same thing, having a common denominator: I and f; | and |; f and f ; f and f ; f and | ; J and T 5 T . COMMON FRACTIONS 93. To Reduce Fractions to Higher, Lower, and Lowest Terms. _ % _ 1. Express as 4ths; 8 ths; I . 1.1,1.1,1,1,1,1 IGths. 2. Express J as Gths ; 9ths ; * i_J . L_._J . i im 12ths; 18ths. ^ ^ 3. Express as 3ds f; |; H; H; i; If- 4. Express as 7ths ^ ; f f ; FIGURE 86 24.54 ?6> SS* 5. Compare the values of these fractions after reducing each to its lowest terms (that is, to the smallest possible whole num- bers for numerators and denominators) . i; 1; A; H; W; ifo; ttt- 6. Express f as lOths; 15ths; 20ths; 35ths; 75ths; lOOths. 7. By what must you multiply the numerator of f to make the numerator the same as that of |? By what must you multiply the denominator of f to make the denominator the same as that of |? Draw a square and divide it free-hand to show the com- parative sizes of f and | (see square (7, Fig. 85, 92). 8. By what must you multiply each term of J to obtain T \? f i? It? TO S O ? f U? What, then, is the value of each fraction of problem 5? 9. How then may the value of a fraction be expressed in higher (or larger) terms? COMMON FRACTIONS 147 A fraction may be expressed in higher terms, without chang- ing its value, by multiplying both terms by the same number. 10. Express the following fractions in higher terms: !;!;!; if; if; A; IJ- 11. How can you obtain the terms of f from the terms of f ? of A? offi? of T y ? 12. What fraction do you obtain by dividing each term of s by 4? What, then, is the relation as to value between the fractions / 8 and 4? 13. By using a rectangle, divided as in Fig. 87, show that fa of the rectangle equals \ of it. 14. In what lower terms can f f be expressed? FIGURE s? iVo? Ji? Show that f = f by means of a properly divided rectangle. 15. What effect on the value of a fraction is produced by dividing each of its terms by the same number? 16. How may the value of a fraction be expressed in lower terms? PRINCIPLE I. Multiplying or dividing both terms of a frac- tion by lite same number changes tlte form of the fraction without altering its value. Fractions are most easily used when their numerators and their denominators are the smallest possible whole numbers. DEFINITION. The fractions are then said to be in their lowest terms. Among the many ways in which J-f can be written are these: H; T"*; t; I; f 17. Show how J is obtained from f by the use of Principle I. 18. Give the values of these fractions in their lowest terms: 5 . 6. 21. 8. 35. 32 TO-; f fi; TV; HJ M- To obtain the value of a fraction in the smallest possible terms, it is necessary to divide both terms by the largest exact common divisor of both terms. DEFINITION. This divisor is called the greatest common divisor of the terms. It is indicated by the initial letters G. C. D. 148 RATIONAL GRAMMAR SCHOOL ARITHMETIC Dividing both terms of a fraction by their G. 0. D. reduces the fraction to its lowest terms. 19. These fractions are in their lowest terms: i; i; t; i; I; f; fl; if; if. Among these fractions is there a factor common to any numer- ator and its denominator? DEFINITION. Two numbers that have no common factor, except 1, are said to be prime to each other. 20. How then may we test by means of factors whether a fraction is in its lowest terms? 21. Reduce these fractions to their lowest terms: 16. 27. 60. 25. 39. 74 . 169 4> 8T> THFJ T^J 5 > TIT 5 3i5T Tho problem of finding the common divisors of such numbers as are in the numerators and the denominators of the last two fractions is tedious without a general method of finding the G. C. D. of numbers. 94. Factors, Prime and Composite. NOTE. Review tests of divisibility, 63, pp. 85, 86, and use them through this section and the next, when searching for factors. 1. Write down all the factors, or exact divisors, of 36 and of 48. DEFINITION. By factor is here meant exact divisor, or a divisor that is contained without a remainder. CONVENIENT FORM 2)36, 48 2)18 24 The factors of 36 are 1, 2, 3, 4, 6, 9, 12, -T 18, and 36. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 2 ) 3 > 4 16, 24, and 48. 3, 2 2. What factors are common to both 36 and 48? 3. What is the greatest factor that is common to 36 and 48? 4. Arrange the factors of 42 and 105 as the factors of 36 and 48 are arranged in problem 1, and answer questions like 2 and 3 for the factors of 42 and 105. COMMON FRACTIONS 149 5. Give the factors that are common to the 3 numbers, 18, 24, and 36. Give the greatest factor common to all three numbers. 6. Answer questions like those of problem 5 for these numbers: (1) 12, 72, and 84; (3) 42, 98, and 168; (5) 36, 84, 96, and 108; (2) 27, 36, and 81; (4) 32, 48, and 96; (6) 44, 99, 110, and 121. 7. Are there factors of 5 other than itself and 1? of 7? of 13? of 17? of 23? of 29? of 31? of 37? DEFINITIONS. A number that has no [factors except itself and 1 is a prime number. A number that has factors beside itself and 1 is a com- posite number. 8. Name the prime numbers from 1 to 100; the composite numbers from 1 to 100. QUERY. Is 2 a prime or a composite number? DEFINITIONS. Any number that can be exactly divided by 2 is an even number. All other whole numbers are odd numbers. 9. How can an even number be quickly recognized? (See test of divisibility by 2, p. 85.) 10. Name the even numbers from to 50; the odd numbers from 1 to 50. 11. Tell what numbers of these are (1) even, (2) odd, (3) prime, (4) composite: 5, 8, 9, 2, 21, 15, 19, 26, 27, 38, 41, 42. 12. Mention some numbers that are both odd and composite; even and composite; odd and prime. 13. Factors of 28 are 7 and 4. As 7 is a prime number it is called a prime factor. What is a prime factor? 14. Write the prime factors of 21; of 24; of 25; of 27; of 30; of 32; of 36; of 37. NOTE. Write out the prime factors of these numbers as they are here written out for 96 : 96 = 2X2X2X2X2X3. 15. How many times does 2 occur as a prime factor in 96? This may be indicated by writing 96 thus: 2 6 x 3. The small 5 written to the right and above the 2 is to show how many times 2 is to be used as a factor. 150 RATIONAL GRAMMAR SCHOOL ARITHMETIC 95. Greatest Common Divisor by Prime Factors. 1. What are the prime factors of 36 and 48? (See problem 1, 94.) Write 36 in the form given for 96 in problem 15 of the last section. Write 48 also in this form. 2. Will each of the common prime factors of 36 and 48 divide the G. C. D. of 36 and 48? Will the product of all the common prime factors divide the G. C. D. exactly? 3. Answer questions like 1 and 2 for the numbers 42 and 105 ; for 96 and 216; for 75 and 250. 4. When any common prime factor occurs repeatedly in one or both of the numbers, how often does it occur in the G. C. D. of those numbers? Answer by examining these pairs of numbers : (1) 12 and 48; (3) 75 and 250; (2) 54 and 405; (4) 98 and 343. 5. Make a rule for finding the G. C. D. of two or more numbers from their common prime factors when no common prime factors are repeated in any of the numbers. Test your rule by finding the G. C. D. of 30 and 42; of 105 and 231 ; of 30 and 70. 6. Make a rule that will give the G. C. D. of two numbers when one or more of the common prime factors is repeated in one or both of the numbers, and test your rule by finding the G. C. D. of 72 and 108; of 288 and 648; of 675 and 1125. 7. Find the G. G. D. of 792 and 1080. CONVENIENT FORM 2) 792 1080 2) 396 540 793 = 2 3 X 3 2 X 11 2)198 270 1Q80 = 2 3 X 3 3 X 5 3)99 135" G. C. D. = 2 3 X 8-' = 8 X 9 = 72 3)33 45_ 11 3)15 ~5~ PRINCIPLE II. The G. 0. D. of two or more numbers equals the product of all the prime factors common to all the numbers ^ each common prime factor being used the smallest number of times it occurs in any one of the given numbers. 8. Find the G. C. D. of the sets of numbers in problem 6, 94. COMMON FRACTIONS 151 The method of finding the G. C. D. by factors is tedious when the numbers are large and the factors are not readily detected. We now give a method for large numbers. 96. Greatest Common Divisor by Successive Division. To obtain quickly the G. C. D. of large numbers, the following method is useful : 1. Find the G. C. D. of 851 and 10,952. CONVENIENT FORM 851)10952(12 851 2442 1702 llf>740(6 666 "74)111(1 _74_ 37)74(2 74 The last divisor, 37, which gives the remain- der 0, is the G. C. D. 2. Find the G. C. D. of these numbers: This is called the method of finding ^6 m ' ^* ^7 successive division. To find the G. C. D. of more than two numbers, first find the G. C. D. of two of the numbers, then the G. C. D. of this G. C. D. and a third number, and so on. (1) 1413 and 4710; (2) 432 and 5184; (3) 14,457 and 27,450; (4) 247, 969, and 1235; (5) 272, 357, and 425; (6) 517, 752 and 1034. 3. Eeduce these fractions to their lowest terms by finding the G. G. D. of the numerator and the denominator, and dividing both terms by the G. C. D. : (i)W; (3) MM; (*) VWi; (?) Wt; (4) tttt; (6) K; (8) NOTE. In reducing fractions to their lowest terms it is best to reject (divide out) at once from both numerator and denominator, any common factors that can be readily seen, before seeking the G. C. D. of the terms. This will often be advantageous, if the tests for divisibility. 63, pp. 85, 86, are well known. 152 RATIONAL GRAMMAR SCHOOL ARITHMETIC 97. Greatest Common Divisor of Lines with Compass. PROBLEM. Find the G. C. D. of the lines AB and CD. E F K R EXPLANATION. Take the distance A ' CD between the compass points, and G H I ~ placing the pin-foot on A, mark off AE= CD. Then mark off EF= CD. P, T _. This leaves the remainder FB, which is less than CD. This would be writ- ten FB < CD, and read "FB is less than CD." DEFINITION. The sign < is called the sign of inequality. The point of the < is always turned toward the smaller number. Thus, FB < CD means that FB is less than CD, but CD > FB means that CD is greater than FB. Now take the distance FB between the compass feet, and mark off on CD the spaces CG, GH, and HI, each equal to FB, until a remainder ID is left, such that ID < FB. Then ID is found to be contained just twice in the former remainder FB. ID, the last divisor or measure, is then the greatest common measure, or divisor, of both AB and CD. To see that it is a common measure answer these questions : 1. How many times is ID contained in FB? 2. How many times is FB contained in CI? 3. How many times is ID contained in 77? in CD? 4. How many times is CD contained in AF? 5. How many times is ID contained in AF? in AB? Consequently ID is a common measure of AB and CD. To see that any common measure of AB and CD must measure ID exactly, answer these questions : 6. If any line, as ID, is contained 7 times in CD, how many times is it contained in AF? 7. If ID is contained, say 16 times, in AB and 14 times in AF, how often must it be contained in FB? In the same way it could be seen that each remainder after a number of applications of the preceding remainder would exactly divide this last remainder. 8. Draw two lines of different lengths on the blackboard, and with crayon and string find their greatest common measure or divisor. COMMON FRACTIONS 153 ?98. Problems. In each case when the answer to the problem is a fraction, or ratio, it must be expressed in its lowest terms. 1. A man works 48 da. out of 64 da. What part of 64 da. does he work? 2. A grocer bought 24 boxes of oranges and sold 16. What part of his purchase was sold? 3. Out of 56 bu. of potatoes a huckster sold 49 bu. What part of his potatoes was sold? 4. A street car on one line makes a weekly average of 72 trips. A car on another line makes an average of 144 trips. What is the ratio of the former to the latter? 5. A man pays $50 for wood and $125 for coal in one season. Find the ratio of the cost of the wood to the cost of the coal. 6. Out of 1000 ft. of lumber purchased, 500 ft. were used for the flooring of two rooms. Let x equal the ratio of the number of feet of lumber used to the total number of feet purchased. Find the value of x. 7. Out of 126 bu. of oats, a livery man fed 63 bu. in 1 wk. Let y equal the ratio of the quantity of oats bought to the quan- tity of oats fed. Find y. 8. In 667 Ib. of sandy loam there were 377 Ib. of sand and gravel. What part of the soil by weight was sand and gravel? 9. The human body needs about 94.5 oz. of water and solid food each day, of which 64.8 oz. should be water. The water is what part, by weight, of the total quantity of solid food and water? SUGGESTION. Both terms of the fraction may be multiplied by 10 without changing the value of the fraction. Then find the G. C. D. of the new terms and divide both new terms by it. 10. The total population of a certain city is 36,729, of which 12,243 are colored and 10,017 are foreign. What part of the entire population is colored? What part is foreign? 11. What part of the total population is made up of colored and of foreign persons? The solution of problem 10 will show that J of the population is colored, and that T 3 T of it is foreign. We can solve problem 11 if we can find the value of J + -fa. 154 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. A boy spent f of his money and gave away of it; what part of it did he have left? To solve this problem we must know how to add and subtract fractions. This is what we shall study next. 99. Fractions Having a Common Fractional Unit (a Common Denominator). 1. Complete these equations: fyd.+fyd. = !gal.+fgal.= + = | hr. + J hr. = T 6 -g- -f T 9 7 = f wk. + f wk. = y\ + y 1 ^ = f bu. +ibu. = ^ + ^ = NOTE. -^ is read, "5 divided by a," and -^ is read, "x divided by z." !~ is read, "a plus b divided by c." 2. Make a rule for adding fractions having a common denom- inator. 3. Complete these equations: I - 1 = T\ ~ T s y = if - T 3 s = I! nr - - & hr. = NOTE. ^- is read, "a minus Z> divided by c." 4. Make a rule for finding the difference between two fractions having a common denominator. 5. Denoting any two fractions having common denominators by j and state in symbols : PRINCIPLE III. The sum, or the difference, of any two frac- tions having common denominators equals the sum, or the difference, of their numerators, divided by the common denominator. 100. Fractions Easily Reduced to Common Fractional Unit. 1. i equals how many 12ths? 8ths? IGths? 20ths? 64ths? lOOths? COMMON FRACTIONS 155 2. T \ hr. + i hr. = x hr. What is the value of x? SOLUTION. \ hr. + T \ hr. = { hr. = f hr. x = f. 3. T V br. - i hr. = y hr. What is y? 4. Find these sums and differences: t+t - I -A- A A+A-A 101. Growth of Trees. Twigs, broken from a scrub oak, were taken to the classroom for measurement. Some were taken from the lower, some from the middle, and some from the upper branches. The twigs from the lower branches were grouped into three bunches: one bunch containing those from the north side of the tree; one, those from the east and west sides; and one, those from the south side. The twigs from the middle branches and also those from the top branches were treated in the same way. Measures were made from the tips of the twigs to the first cluster of rings marking the year's growth. In the table below, they are given in inches to the nearest eighth of an inch. Similar measures were made of the distances between the first and the second clusters of rings, then between the second and the third clusters, and finally between the third and the fourth clusters. Pupils are urged to make such measures and to tabu- late and study them as is done in the table and problems below. GROWTH OF OAK TWIGS FOR LAST YEAR, MEASURED IN INCHES LOWER BRANCHES MIDDLE BRANCHES TOP BRANCHES N. E.&W. s. N. E.&W. 's. N. E.&W. s. 5* , A, A, and if; (3) i, A, A, and A; (2) 4, A, A, and A; ( 4 ) * T 5 s, A. and A- 105. Definitions and Principles. Thus far we have had to do with two kinds of number: (1) whole numbers, or Integers, and (2) fractional numbers, or Fractions. DEFINITIONS. A proper fraction is a fraction whose numerator is less than its denominator : as |, , f . An improper fraction is a fraction whose numerator is equal to, or greater than, its denominator : as |, f , I, f. A mixed number is a number, such as 2, 12|, 7|, that is composed of an integer and a fraction. 1. Name the proper and the improper fractions, the mixed and the whole numbers : 4, I, A, s> -Y-, f. !. 11. 3A. 6-75, A, 2- 2. Make and give an example of each class. 3. How many times is 1 contained in 3? in 18? in 25? in any number? 162 RATIONAL GRAMMAR SCHOOL ARITHMETIC We may write this in symbols, thus: Any whole number may be written in the fractional form by writing 1 for Us denominator. 4. Write the following numbers in fractional form: 12; 24; 32; 68; ; x 5. How many halves are there in 1? in 2? in 5? in 2? in 5J? 6. How many 5ths are there in 1? in 2? in 4? in 6? in 6f ? in 7j? 7. Change the following integers into 6ths: 2; 5; 16; 29; 120. 8. Change the following numbers into 12ths: 2; 8; 12; 20; 56; 128. 0, How may any whole number be changed into 12ths? into 9ths? into 24ths? into a fraction having any given denominator? PRINCIPLE V. Any integer may be expressed in the form of a fraction having a given denominator by multiplying the integer by the given denominator and writing the product over that denom- inator. 10. Express 6, 18, 17, 22, 39, 28 as 4ths; as 7ths; as lOths; as lOOths. 11. Change into 6ths, 2; 4J; 24| ; 37f. 12. Change the following numbers into improper fractions having 7 for a denominator: 3|; 24f; 106f; 234?; 648. 13. How can you change any mixed number into an improper fraction whose denominator is the denominator of the given frac- tion? PRINCIPLE VI. A mixed number may be expressed as an improper fraction by multiplying the 'whole number by the denom- inator, adding the numerator to the product, and writing the sum over the given denominator. 14. Eeduce to improper fractions: 2i; 6J; 12i; 18}; 25f; 328| ; COMMON FRACTIONS 163 15. Express Y- as a whole number. Change V to a mixed number. 16. Change to whole, or mixed, numbers the following improper fractions : f; f; i; -; Y; V; V; V; V; W; If; V; V; t- 17. How may any improper fraction be changed to a whole, or mixed, number? PRINCIPLE VII. An improper fraction may be changed to a whole, or mixed, number by performing the indicated division. 18. Reduce to whole, or mixed, numbers the following: I; V; ff ; 7 ^; V; *; HI 1 - 106. Addition of Fractions. 1. If the numbers written on the lines of the drawing are the lengths in feet of the inside walls of the house, how many feet of lumber would there be in a baseboard not over 1" thick, 1 ft. wide extend- ing entirely around the inside of the house, deducting 6 board feet for doors? To solve this problem we add first all the whole numbers and then all the fractions. We must now learn how to add such common fractions as occur here. This problem will be solved later. (See problem 9 below.) 2. A man owns f of an acre in one block and -J of an acre in another block ; how much land does he own? This problem requires us to add f and -J. I. GEOMETRICAL SOLUTION. Fig. 90 (a) is divided by the vertical lines into 3 equal parts and by the hori- zontal lines into 8 equal parts. Into how many equal parts do the 2 sets of parallel lines divide the square (a)? Show by Fig. 90 (a) that | = \\. Show by Fig. 90 (6) that I = |i 11* 16 9 /2 FIGURE 89 (a) i = H (6) i = 11 FlGUKE 90 + n = n = iif. Ans. 1J acres. 164 RATIONAL GRAMMAR SCHOOL ARITHMETIC II. ARITHMETICAL SOLUTION. What is the least common denomi- nator of } and 5? What, then, is the largest fractional unit (therefore having the least denominator) in which both g and % can be exactly expressed? | = how many 24ths? I = how many 24ths? 3 Ac 7 3- Id r } 2 ? T , f , and 2 i ^ - >(* ro . Ans. acres. (a) M FIGURE 91 I. "GEOMETRICAL SOLUTION. Show from Fig. 91 (a), that f = |f ; from Fig. 91 (5), that fr = &; from (c) that | = ff ; from (d) that ft = f$. M + A + IS + II = = if i = HI-- II. ARITHMETICAL SOLUTION. The least common denominator of the fractions is 84. ._ 3X12, 2X4,2X28, 5x7 36 + 8+56 + 35 f "TlT-r ~Ti2 135 7 X 12 ' 21 X 4 ' 3 X 28 ' 12 X 7 84 The last step consists in reducing the result to its simplest form. 4. T \ of the weight of a specimen of soil was gravel, and % was sand; what part of the soil by weight was sand and gravel? Draw a figure and give the geometrical solution. Give also the arithmetical solution. 5. One side of a triangle is 2f in., another is 2f in., and the third is 3| in; how long is the perimeter of (distance around) the triangle? Add first the whole numbers, then the fractions, and finally add the sums. 6. One side of a 4-sided figure is 6f in. long, a second side is 3f in., the third 7f in., and the fourth 7 T 6 ^ in. What is the perimeter of the figure? 7. A coal dealer bought 4 carloads of coal of the following weights: 21J T., 27J T., 29f T., and 30f T. Find the com- bine i weight, in tons. COMMON FRACTIONS 165 8. Solve these problems as rapidly as you can work accurately, using your pencil merely to write the products and the sums : (1) | + * =? (4) 4 + 7 S = ? (7) U + * = (2) | + 1 =? (5) tt + ,V? (8) ^+T = ? (3) T 6 T + T 5 * = ? ()* + -? (9) T + T- 9. Solve problem 1 of this section. 107. Subtraction of Fractions. 1. A boy had $ and spent &J; what part of a dollar did he have left? ARITHMETICAL SOLUTION. The least common denominator is 2 X 5 = 10. 2. William is 4f ft. tall and James is 4^ ft. tall; who is the taller and by how much? 3. A man owned { A. of land and sold 4 A. ; how much land did he then own? 4. From Ib. of loam f Ib. sand was removed ; how much of the loam remained? 5. A man having I4J- paid a debt of $2f; how much money had the man after paying the debt? SOLUTION. J = 2 4 , -5 = $g. As & is less than Jg, write 4^, in the form 3|J. The problem is then 8|$ 2Jg = 1, 9 for 8 2 = 1 and |J 18 = J 9 o- 6. Solve these problems: (1) 134 - 7| -f (4) 128f - 97if = ? (2) 28f -19| = ? (5) 639f -598 T \ = ? (3) 30,V-16J = ? (6) 1217 1 6 f-989i| = ? 7. A tree 72f ft. high is broken off 28 T \ ft. from the top ; how high is the stump? 8. A 5-cent piece weighs 73 J- gr. and a quarter dollar weighs 96^ gr. 5 how much more does a quarter weigh than a 5-cent piece? 9. The dime weighs 38 T \ gr. Before 1853 it weighed 41 J gr. By how much was the weight of the dime reduced in 1853? 1G6 RATIONAL GRAMMAR SCHOOL ARITHMETIC 10. A silver dollar weighs 412-J gr. and a double eagle weighs 516 gr. What is the difference between the weight of a double eagle and that of a silver dollar? 11. What is the difference between the weight of the half dollar (192^ gr.) and that of the quarter dollar (96/ 7 gr-)? 12. Write out rapidly the values of these sums and differences : (i) i+* =? (0) I + t =? (ii) ! +1 =? (2) i - A- ' (?) 4 - T\ = ? (12) A- A - ? (3) t - A - ' (8) T\ + T*J ? (13) A + A = ? (4) t+V =? (9) 4 + T=? (14) f+l- ()V-V- (10)=- -T= ? US) S -7-' 13. Make a rule for finding quickly the sum, or the difference, of any two fractions. NOTE. This latter rule may be formulated thus, 1st num. X 2d den. -\- 2d num. X 1st den. 1st den. X 2d den. and 1st num. X 3d den. 2d num. X 1st den. difference = . 1st den. X 2d den. The result must always be reduced to the simplest form. 14. In three successive runs, a train loses ^, |, and % hr. How much time does it lose in all? 15. f of a man's salary is spent for clothing, for board and lodging, for books and stationery, T * f for traveling expenses. What part of his salary was spent for these purposes? what part remained? 16. In the six days of one week, a man works 8-J- hr., 9f hr., 8f hr., 9 hr., 8f hr., lOf hr. What is the whole number of hours of work for the week? What are his wages at $.30 an hour? 17. Four children are to do a piece of work. Three of them do {, , and | of it. What part is left for the fourth? 18. f , J, and T \ of a man's money are invested in three different enterprises. What part of his money is invested? What part is free? 19. A merchant sold T "V of a gross of buttons to one customer, and of a gross less to another. What part of a gross did the second customer buy? COMMON FRACTIONS 167 20. A owns T \ and B ^ of an estate. How much more of the estate does A own than B? 21. A tank of oil is f full. If of the contents of the tank is drawn off and then -fa of the remainder, what part is left? 22. A owns f } of a mill. B's interest is |f of the mill less than A's. What part of the whole does B own? 23. At $1 a day how much money does a man earn by working 5f da.,6f da., and 8| days? 24. A boy lives where school is taught 1100 hr. a year. He is compelled by sickness and other causes to lose from month to month the following numbers of hours : 115|; 39|; 15f; 6U; 4*5 3|; 5f; 18J; 89 T V How many hours did he lose in all? What part of the whole school year did he lose? 108. Multiplying a Fraction by a Whole Number. 1. 6 times 3 mi. = how many miles. 2. 8 x 4 yd. = ? yards. 3. 8 x 4 fifths = ? fifths. 4. 5 x f = .- ; what is the value of x? 5. 9 x -fa = -^-; what is the value of #? 6. A whole is divided into 11 equal parts, and 5 of them are taken. What fraction represents the part of the whole which is taken? 7. What fraction would represent 3 times this part? 8.. Replace the letter in each of these equations by its correct value; the sign (x) should be read "times" in these problems. (l)6xf = f <5)6xt>f (9)2x T \ = i (2) 4 x 1 - f (G) 3 x | = f (10) 6 x Y- T (3) 3 x i = (7) 4 x * = f (11) 5 x A = (4) 2 x f = f (8) 2 x $ = f (12) 2 x & = In multiplying these fractions by the whole numbers what change was made in the numerator to get the product? Make a rule for multiplying a fraction by a whole number. 168 RATIONAL GRAMMAR SCHOOL ARITHMETIC 9. Apply your rule to these problems, reducing to whole or mixed numbers all the products that are improper fractions : (1) 3 x | = (5) 11 x A = (9) 6 x 1 = (2) 5 x f = (6) 9 x T V = (10) 8 x A = (3) 7 x | = (7) 4 x I = (11) 10 x f = (4) 9 x f = (8) 6x1- (12) 9 x = 2 5 10 NOTE. In such as (7) use cancellation, thus: x ~-= -5 = 3J. p o 3 10. Solve these problems and make a rule for multiplying a fraction quickly by its denominator : (1) 6 x | = ? (5) 12 x A = ? (9) 75 x 4f = ? (2) 9 x | = ? (6) 18 x if = ? (10) b x -JL = ? (3) 5 x f = ? (7) 21 x if = ? (11) m x -- = ? (4) 7x| = ? (8) 48 x W = ? (12) a x -- = ? NOTE. In all such problems use cancellation, thus: /7 x = = 5. 11. Give the values of the letter in each of these problems and make a rule for multiplying a fraction quickly by some factor of 7 *7 its denominator; (use cancellation thus : % x -^ = -^ = l|). 4 (l)3x*=f; (3)12x V = f; (5)25x T W = f; (8)*A = ?5 (4) 9x A = f' (6)13xH=f State the rule. PRINCIPLE VIII. A fraction is multiplied by 1. Multiplying its numerator ~by the multiplier; or, 2. Dividing its denominator by the multiplier. NOTE. The usefulness of the method of cancellation consists in mak- ing these two processes undo each other. (See problem 11). QUERY. When will the second method (Principle VIII, 2), be the more convenient? A fraction is multiplied by its denominator by dropping its denominator. OOMMOST FRACTIONS 169 109. Multiplying a Mixed Number by a Whole Number. 1. What will bo the cost of 12 yd. of muslin at 8f cents? SOLUTION. If 1 yd. costs 8jj f what will 12 yd. cost? 12 X 8| means 12 X 8 -f 12 X = 96 -f 8 = 104. Ans. 1.04. NOTE. It is important to bear in mind in dealing imth such expressions as 12 X 8 -f 12 X |, which contain both signs (X) and (-f), that the indicated multiplications must be performed first, and then the additions. Thin fact is usually expressed by saying, "the multiplication sign (X) takes precedence of 'the addition sign (+)." 2. 13 x 4| - ? 4 13 JT" __ 1Q v A Multiply tlie whole numbers in the usual way. 71 = 18 x I 13 X 8 = = 7|, as in problem H, 108. 59| = 13 X 4* PROBLEMS Solve the following, using the more convenient method in each case, employing cancellation whenever it shortens the work: 1. 5xii = 5. 9x JJ = 9. 2x-i/ = 2. 3 x | = 6. 9 x f = 10. 3 x GJ = 3. 5 x4| = 7. 9 x T V = 11. 9x^8- = 4. 2x2J = 8. 9x -i . 12. 8 x T \ = 13. Find the cost of 25 yd. of cloth at $f per yard. 14. A earns II : {, and B earns 4 times as much. B earns $#. Find x. 15. A grocer sold J bu. of apples to one customer, and 5 times that quantity to another. He sold y bu. to the second customer. Find?/. If apples were $.40 a peck what amount of money did the grocer receive from the first customer? from the second? 16. If a man can cut of a cord of wood in one day, how much can he cut in 6 da., working at the same rate? 17. One boy lives T 5 g- mi. from school and another 4 times as far. What is the distance from school to the second boy's home? 18. Solve problems based upon the following facts : An iron tube weighs f Ib. per foot. f of the weight of water is oxygen. The circumference of a bicycle wheel is 6J-J- feet. A west wind moved 47f mi. per hour. 170 KATIONAL GRAMMAR SCHOOL ARITHMETIC 110. Multiplying a Whole Number by a Fraction. The whole number is here the multiplicand. DEFINITION. To multiply a whole number by a fraction means to divide the multiplicand into as many equal parts as there are units in the denominator, and to take as many of these equal parts as there are units in the numerator of the multiplier. ILLUSTRATION. 12 multiplied by | means that 5 of the 6 equal parts of 12 are wanted. -J times 12= 2, and | times 12 = 5 X 2 = 10. | times 12 and | of 12 mean the same thing. " 1. Solve these problems, reading the sign (x) "times (1) f x 10 = ? (4) f x 15 = ? (7) f x 45 = ? (2) f x 25 = ? (5) f x 28 = ? (8) T \ x 26 = ? (3) 1 x 32 = ? (6) r 6 T x 33 = ? (9) T \ x 47 = ? SUGGESTION. In (9), j> r of 47 = 4^ I( and 7 x4ft = 28?} = 2. Solve the following : (1) T \ x 50 = ? (3) tJ x 105= ? (5) ^ x 110 = ? (2) fj x 39 = ? (4) ff x 220 = ? (6) 6| x 28 = ? SUGGESTION. 6 x 28 means 6 X 28 -f I X 28 = 168 -f 21g = 189. NOTE. See note to problem 1, page 169. 111. Factors May Be Interchanged. In Fig^ 92 (a) the rectangle is supposed to be ^ in. wide and 5 in. high. The area is then 5 times -J sq. in. = 2 square inches. In (b) the rectan- 5 _ . - . gle is 5 in. long and I I i in. high. Its area is % times 5 sq. in., or ^ of 5 sq. in. = 2^ sq. in-., as is plain from a Fig. 92. This exemplifies the truth that when a fraction and a whole number are to be mul- tiplied it makes no difference in the nu- FIGURB 92 merical value of the product which of the factors we regard as the multiplicand. The name of the result is determined by the conditions of the problem. COMMON FRACTIONS 171 Letting f- (read "x divided by i/") stand for any fraction, and a stand for any whole number, state the following principle in. symbols : The product of a whole number by a fraction, or of a fraction by a whole number, equals the product of the whole number by the numerator, divided by the denominator. PROBLEMS 1. Find the cost of 2J yd. of cloth at $3. 2. A man had $75; he spent f of it for a bicycle, and of the remainder for clothing. Find the amount spent, and the amount he had left. 3. A gallon contains 231 cu. in. How many cubic inches in f gal.? in | gallon? 4. A cubic foot of granite weighs 170 Ib. How many pounds in of cubic feet? 5. T \ of a plot of ground containing 918 sq. rd. was fenced for a garden. The garden contained how many square rods? 6. Make and solve problems, based upon items personally obtained, or upon those given below: An avoirdupois pound of gold is worth about $348f . A bale of cotton ordinarily weighs 450 Ib. Price 17|^ per pound. A tank contains 12G gal. linseed oil. Price $.62 per gallon. Corn is quoted and sold @ 50J^ a bu. Wheat is quoted and sold @ 77f ^ a bushel. Oats are quoted and sold @ 32f (2)i 5 ix f -A <>f -|; (4) T 5 T xf = -|of T \; (G) A * ! = H To find -\ of A, we take a fractional unit which is % of A (the fractional unit of A)> an ^ use the same number (5) of them, thus obtaining A- But f of A = 6 x | of A = G x A f f As $ x T 5 r = f ^ A > f x i T = f T- Similar reasoning would show also that A x i = TT- Now we recall that the numerator (30) of the product (A X 4) was obtained by multiplying the numerators (6 and 5) of the factors ( T 5 T and ^). How was the denominator (77) of the product obtained? G. State a rule for quickly multiplying a fraction by a fraction. 7. Solve* these problems as rapidly as you can work accurately: . (1) fxi (5) f x| (9) xf (13) fxf (2) ixf (6) ixf (10) fxf (14) }x| (3) fxf (7) f x f (11) f x | (15) f x f (4) |x| (8) fxf (12) fxf (1G) |x| 8. Letting and -J- denote any two fractions, state the fol- lowing principle in symbols : The product of any two fractions is a fraction whose numer- ator is the product of the numerators and whose denominator is the product of the denominators. 113. Multiplying a Mixed Number by a Mixed Number. 1. How many square feet in the area of a rectangle 4f " x o-|-"? (1) How long is a (Fig. 94 )? how wide? What is its area? (2) Answer similar questions for b, c, and d. (3) How long is the entire rectangle? 5 ' how wide? What is its area? FIGURE 94 Since the area of the entire rectangle equals the sum of the areas of its parts we may write (remember X takes precedence of +; see note, problem 1, page 169): 4f x5i=4xr>+ X5 + 4X + 5 X 5 = 20+ 8f + 2 + f = 26J. Ans. 26| sq. ft. * Cancel whenever it Is possible. 174 RATIONAL GRAMMAR SCHOOL ARITHMETIC Point out on figure the areas that represent all the parts of the product. 2. Find the cost of 9 T. hard coal at $7f . FIRST SOLUTION. If 1 T. costs $7f , 9| T. will cost 9^ x $7f . CONVENIENT FORM EXPLANATION 7| 9J X 7f means- 9 X 7| -f fc X 7f. 9 | X 7| means of 7|. 63 =9x7 9X7| 6 = 9 X = s ^ 7 = \ r X ! 9 X 7| = 73| SECOND SOLUTION. 9J T. = -V*- T. ; $7f = $V J X* = fi | a , or 731. Ans. $73|. Find areas of surfaces having the following dimensions : 3. 44f ft. x 28f feet. 4. 36f ft. x 27f feet. 5. 24f ft. x 18| feet. 6. 45f ft. x 30 feet. 7. 27i ft. x 18f feet. Find the cost of the following items : 8. 24f doz. eggs @ $.16|. 9. 46| gal. of oil @ $.12|. 10. 15^ yd. of cloth @ $.66|. 11. 52} Ib. of sugar @ $.05|. 12. 265f M. of pine flooring @ $35J. Make original problems from the following items : A cubic inch of water weighs 252|| grains. A water tower is 2f J ft. higher than the mound upon which it is built. The mound is 81-J- ft. high. 69-J statute miles - I degree of longitude at the equator. A pine tree 87 ft. high has no branches for f of its height. -|i of a library containing 55,447 volumes was destroyed by fire. COMMON FRACTIONS 175 Flaxseed cost $lf per pound when a certain linseed oil factory bought supplies. A spring furnishes 28 T \ bbl. of water daily. A certain vessel sails llf mi. per hour, on an average. A room is 32 ft. long and 24 f ft. wide. Painting costs $-5^ per square foot. A cubic foot of water weighs 62.5 pounds. Gold is 19^ times as heavy as water. 30J sq. yd. = 1 square rod. 114. Dividing a Fraction by a Whole Number. OEAL WORK 1. What is 4 of 40A.? 40A. divided by 8 equals what? 2. \ of 35 Ib. = ? 35 Ib. + 7 = ? 3. T V of 80 = ? 80 - 10 = ? 4. i of 3 fourths = ? 3 fourths -*- 3 = ? 5. |of|A. = ? I-A. +5=? 6. Jof f ft. = ? ^ ft. +3= ? 7. of f J in. = ? f J in. + 9 = ? 8. Compare these fractional units, or unit fractions: (1) i is what part of J? (7) ^ is what part of ? (2) I is what part of ? (8) T V is what part of ? (3) I is what part of i? (9) J is what part of ? (4) T V is what part of 1? (10) T V is what part of ? (5) | is what part of 4? (11) T V is what part of i? (6) i is what part of J? (12) ^ is what part of J ? 9. How is the size of a unit fraction changed by multiplying its denominator by 2? by 3? by 4? by 10? by 100? by a? by m? 10. By what must you multiply the first fraction in each of these pairs to get the second? (1) and |? (4) A and f ? (7) /,V <~5 7X5 (C) Dividend and divisor both mixed numbers. 4. I paid $49| for coal @ $7f per ton. How many tons did I buy? SOLUTION. 49|=^; 71=^; 49| + 7f = 8 248 ?X)X4 T _ 31 ~ ^ - 6 - T * ^-X 5 X * Ans. 6| T. Let us now seek a general method of dividing any number by a fraction. 5. How many times is each of these fractions contained in 1? i ; i ; i ; T 5 i > TO ; -rs ; A ; ^5 ; 70 ; T^ o ; -^ j v- 180 RATIONAL GRAMMAR SCHOOL ARITHMETIC 6. How many times is each of these fractions contained in 1? I; i; 4; f ; I; A; A; if; H; H; T 3 zrV; 1- 7. "What number should stand in place of the letter x in each of these products? (1) |x|-z; (4) |x f=z; (7) A> forces, each -f -02 T. (4) 18 forces of - 10 T 9 g- lb. each, and 20 forces of + 25J Ib. each. 12. Give the joint effect, with proper sign, of the set of forces in each horizontal line of the table : SET No. No. OF FORCES STRENGTH ob' EACH FORCE No. OF FORCES STRENGTH OF EACH FORCE JOINT EFFECT WITH SIGN 1 1 + 25 oz. 2 16] 07.. 8 2 5 - - 15J3 Hi. - 75 lb. 1 8 186f lb. 4 27 - 28i 3 - lb. 16 624 lb 5 16 - 42 t V T. 7 81f T.' 6 9 - 73| lb. 73 9| lb. 7 36 _ 25- 7 - lb. 25 36^ lb 8 3 4- x lb. 2 xlb. 9 12 -\-xlb. 7 xlb. 10 16 4- 2x lb. 28 xlb. 11 16 -f 5m lb. 23 2m lb. 12 25 + 10m lb. 36 12m lb. 13 12 _L 39 fi T. 16 23 "j 1 14 1 4- x ib. 1 -ylb. ' 15 3 _j_ x lb. 2 -ylb. 16 8 h 2x lb. 5 - 2y lb. 17 a -xlb. a x lb. 18 a - x lb. a 3x lb. 19 a h*lb. a -ylb. 119. Algebraic Phrases. Letting x and y denote any two numbers (x being larger than ?/), represent in symbols: (1) Their sum. (2) Their difference. (3) Their product (4) Their quotient. (Two results.) (5) Their ratio. It is important to note that quotient and ratio mean the same thing. (6) The square of the larger. (7) The sum of their squares. (8) The square of their sum is written thus, (x + y) z . (9) The difference of their squares. (10) The square of their difference. 186 RATIONAL GRAMMAR SCHOOL ARITHMETIC (11) The cube of the smaller is written ?/ 3 , meaning y x y x y. (12) The sum of their cubes. (13) The cube of their sum is written thus, (./: -f y) s . These phrases are frequently used in algebra, and their meanings should be clearly comprehended. 120. Dividing Lines, and Angles. PROBLEM I. Divide the line AB into 2 equal parts, see Problem VI, p. 107. EXERCISES 1. Draw a straight line with a ruler on the blackboard and bisect it with crayon, string and ruler. 2. Bisect each of the 3 sides of a triangle and connect each mid-point with the opposite corner of the triangle. These lines are the medians of the triangle. How do they cross each other? 3. How might a line be divided into 4 equal parts by repeating this method? Divide a line into 4 equal parts by this method. PROBLEM II. Divide the angle BAD into 2 equal parts. EXPLANATION. With any convenient radius and with the pin foot on A (vertex) draw arcs 1 and 2 across AB and AD. Place the pin foot on the crossing point at 2, and with a radius longer than half way from 2 to 1 draw an arc 3. Now place the pin foot on 1 and with the t* ' 'i * radius used for arc 3 draw arc 4 across arc 3. Call the point of crossing C. Angle Bisected With the ruler dra & w the bisector AC FIGURE 99 Then angle BAG = angle CAD. EXERCISES 1. Draw an angle on paper, or on the blackboard, and with pencil, or crayon and string, bisect the angle. 2. Draw a triangle and bisect each of its 3 angles. How do the bisectors of the angles cross each other? 3. The line CD of Fig. 45, p. " 107, drawn from C to D, is called the perpendicular bisector of the line AB. Draw a triangle of 3 unequal sides, and then draw the 3 perpendicular bisectors of its sides. How do these lines cross each other? COMMON FRACTIONS 187 FIGURE 100 121. The Parallel Ruler. A very good parallel ruler may be made by cutting out two strips of cardboard or of very thin wood, exactly alike, as shown at (), Fig. 100. The numbers show the widths and the lengths. To use the ruler place the two parts with their long sides together, the thin ends being in opposite directions as in the cut. Another sort, which the pupil may make for himself or pur- chase for a few cents, is shown at (b) , Fig. 100. The strips S may be of light cardboard or of thin wood of the lengths and the widths shown in the cut. The outside edges of these strips should be made as smooth and as straight as possible. The cross strips, which may be narrower than the strips jS 9 should be of exactly the same length between the pins at (7, />, and at E, F. The distances EC and FD between the pins, should also be made equal. Common pins may be stuck through and bent over W to hold the strips to- -__ m gether at C 9 D, E, and F. The pupil should provide himself with a parallel ruler for the following problems : PROBLEM I. Draw a line parallel to a given line. EXPLANATION. Let AB be the given line. Fig. 101 (a) shows how this is done with the first kind of ruler, by holding the part 2 on the line, A . /? and sliding part 1 along, FIGURE 101 drawing the lines along the upper edge of part 1. Fig. 101 (6) shows how to solve the problem with the second kind of ruler. 188 RATIONAL GRAMMAR SCHOOL ARITHMETIC PROBLEM II. Draw a line parallel to a given line and through a given point. D EXPLANATION. Sup- C - _ _ 2) P? se AB . ( Fi s- 102 ) is the given line, arid P the point. Hold one edge of the parallel ruler along AB, raise the other strip until its edge goes through the point P, and draw a line CD along this edge. CD is the desired line FIGURE 102 PROBLEM III. Solve Problem II with ruler and compass. EXPLANATION. First Step: Place the pin foot on the given point P and spread the feet until the pencil foot reaches some point, as C, on the line AB, Fig. 103. Draw the arc C2. Second Step: Place the pin foot on C and using the same radius as before, draw arc Pi, cutting AB at D. Third Step : Spread the feet of the compass apart as far as from P to D, and placing the pin foot on C, draw the short arc 3. Connect the crossing point E with P. EP is the desired parallel to AB through P. FIGURE 103 EXERCISES 1. Draw a line on paper, mark a point not in the line and with a parallel ruler draw a line through the point and parallel to the first line. 2. Solve Exercise 1 on the blackboard with chalk, string, and ruler. 3. Draw a triangle on the blackboard. Draw a line through each corner of this triangle and parallel to the opposite side. PROBLEM IV. Divide a line AB into 3 equal parts (or trisect AB). EXPLANATION. Draw an indefinite line AC, making any convenient angle with AB. Measure off 3 equal spaces from A to- ward C. Connect D with B . Draw through the points 2 and 1 lines parallel to DB (see Problem II). Then AE = Line Trisected ^F = FB, and E and F are the trisection FIGURE 104 points. COMMON FRACTIONS 189 EXERCISES 1. Draw a line on the blackboard and trisect it. NOTE. Draw the parallels by the method of Exercise 2 under Prob- lem III above. 2. From the suggestion of Fig. 105 draw a line on the blackboard and divide it into 5 equal parts. FIGUBB 105 3. How may a line be divided into 7, or 9, or 13 equal parts? Draw a line on the blackboard and divide it into 7 equal parts. 4. How does the line connecting the opposite corners of a square divide the square? 5. Answer a question like 4 for the rectangle; for the parallelogram. 6. If, then, the area of a square equals the FIGURE ic6 product of its base and its altitude, what is the area of one of the 2 equal triangles into which a diagonal divides the square? FIGURE 107 7. Answer similar questions for the rectangle; for the parallel- ogram. 8. The area (A) of a rectangle of base b ft. and altitude a ft. is how many square feet? 9. What is the area of a right-angled triangle (a right triangle) of base b in. and altitude a inches? 10. The base of a parallelogram is b in. and the altitude is a in. ; what is the area? 11. The base of a triangle is b and the altitude is a\ what is the area? 12. The bases of a rectangle, of a parallelogram and of a tri- angle are b in. and their altitudes are a in. Find the ratio of the area of the rectangle to the area of the parallelogram ; the ratio of the area of the parallelogram to the area of the triangle. 190 RATIONAL GRAMMAR SCHOOL ARITHMETIC 122. Uses of the 30 and the 45 Triangles. PROBLEM V. To make the triangles for use in drawing. EXPLANATION. (a) Fold a piece of smooth heavy paper, having one straight edge (like the piece shown in Fig. 108), over a line near the middle. Bring the straight edges carefully together as shown in Fig. 109. 6in FIGURE 108 FIGURE 109 Crease the paper smoothly with a ruler or a paper knife and paste or glue the two pieces together. When the paper is dry, mark off distances of 4" from the square corner on the crease and on the straight side. Connect the 4 in. marks and cut the paper smoothly along the connecting line. This will give a triangle of the form T, Fig. 110. (b) In the same way, fold, crease, and paste another piece of paper a little larger than before. On the straight side mark off a distance CA equal to 3". With compasses, or with a string or ruler, mark a point, B, on the crease so that AB equals 6v. Draw AB and cut out the tri- angle S, Fig. 111. 3" 4 FIGURE no FIGURE ill If preferred the triangles T and S may be made of thin wood. EXERCISE. Place the side, AC, of S against the long side of T, and Now Simi- holding the triangles with the left hand, draw a line along AB. hold T, slide -S a little (say i") and draw another line along AB. larly, draw a third line. Lines in such positions are called parallel lines. COMMON FRACTIONS 191 PROBLEM VI. Through a given point with a triangle draw a line parallel to a given line. EXPLANATION. AB is the given line and P is the point the parallel is to pass through. Place a ruler CD, Fig 112, in such a position that when the triangle S is placed against it, one of the sides of S will lie along AB. Press the ruler against the paper and hold it with the left hand; with the right, slide the triangle along the ruler until its side just touches the point P. Draw line FE through P and along the edge of the triangle. FE is the desired parallel line. FIGURE 112 FIGURE 113 Fig. 113 shows how this problem is solved with the triangles alone. EXERCISES 1. Solve the first exercise of Problem III with the triangles 8 and T as shown in Fig. 113. 2. Draw a triangle with sides of 1", 1J", and 2", and through each corner draw a line parallel to the opposite side. Use the ruler and the triangles. To draw the triangle see Problem IX, p. 109. 192 RATIONAL GRAMMAR SCHOOL ARITHMETIC PROBLEM VII. At a given point on a line with the triangles draw a perpendicular to the line. EXPLANATION. Let AB be the line and let P be the given point. D IP FIGURE 114 FIGURE 115 Hold one of the triangles, as T, in the position shown in Fig. 114, and, placing one side of the other triangle, S, against the upper side of T, slip S along until the square corner comes to the point P. Hold S firmly and draw the line PD along the side of S. PD is the required perpendicular. PROBLEM VIII. Through a given point not in a line, with the triangles draw a perpendicular to the line. EXPLANATION. Let AB be the line and let P (Fig. 115) be the given point. Hold one of the triangles, as T, so that its side lies along AB and slide the other triangle, S, with its side against the upper side of T until it comes up to P. Then draw PD. PD is the required perpendicular. EXERCISES 1. Draw a line, mark two points on it I" apart and .draw a per- pendicular to the line at each of the two points (by Problem VII) . Mark a point on one of the perpendiculars 1" above the given line and at this point draw a third perpendicular completing a 1" square. 2. Draw a triangle and through each corner draw a perpendic- ular to the opposite side (by Problem VIII). How do these per- pendiculars cross? 3. Draw any circle, also a diameter, marking its ends A and B. Through its center draw a perpendicular radius and prolong it, making a diameter, CD. Connect .4(7, CB, BD and DA, forming an inscribed square. COMMON FRACTIONS 193 ; 123. Scale Drawings of Familiar Objects. 1. Notice the scale of the drawing of the Scale?': 6' - 5 /6 H FIGURE 116 Scale I": 20' playhouse (Fig. 116), and compute the lengths of the following dimensions : (1) The width ; (2) the height of the lower side; of the higher side; (3) the rise (dif- ference of higher and lower sides); (4) the height of the door; the width; (5) the distance from the right side of the door to the right corner of the house. 2. From the dimensions in Fig. 116, with ruler and triangles, make a drawing of the playhouse to a scale 4 times as large as that of the drawing (Fig. 116). 3. From the scale of the drawing (Fig. 117) find the following dimensions of the house : (1) The width; (2) the height of the eaves; (3) the rise (aft); (4) the width and the height of the door; (5) the width and the height of the window. FIGURE m 4. Find the area of the end of the house, including the gable and excluding the areas of the door and the window. 5. Make an enlarged drawing of the house to a scale 8 times as large as the scale of the drawing (Fig. 117). 6. From the scale of Fig. 118, give the length and width of the door ; the length and the width of the panel. (Find the length by measurement). 7. Similarly, -give the length and the width of the door B, also the length and the width of the upper panels, and the widths of the strips enclosing the panels. 8. Make an enlarged drawing of each of the doors to a scale 5 times as large as the scale of the drawings (Fig. 118). /o Xo A i/" , Panel t, ^ __ R 7 e 7 /ao nn "/20 '/ao 5 c ale 1": 60" FIGURE 118 194 RATIONAL GRAMMAR SCHOOL ARITHMETIC 9. How long are the cross-pieces and the strings of the kite represented by the drawing (Fig. 119)? 10. Make an enlarged drawing of the kite to a scale 8 times as large as that of Fig. 119. 11. Notice that the horizontal piece divides the surface of the kite into two trapezoids. The alti- tude of the upper trapezoid in the drawing (Fig. s C a/e r<54" 119) is J", and that of the lower trapezoid is -J-J". FIGURE 119 How many square feet of paper are needed to cover the kite? (Allow 144 sq. in. for folding and pasting over strings.) 12. What is the length of each leg of the chair shown in the drawing (Fig. 120)? What is the height of the back? the depth of the seat to the extreme rear? 13. Make a drawing of the chair to a scale 5 times as large as that of Fig. 120. 14. Find from the drawing of the gate (Fig. 121), (1) the height of the high end of the gate; (2) of the lower end; (3) the length of the long brace; (4) the length of the gate ; (5) the width of the strips. 15. Make a drawing of the gate to a scale 8 times as large as that of Fig. 121. 16. Fig. 122 is a scale drawing of the FIGURE 121 jj side and the end views of a large book. How long is the book? how wide? how thick? 17. Make an enlarged drawing of the book to a scale 4 times as large as that of Fig. 122. 18. 'From your own measurements make a drawing, to any convenient scale, of a thick object, as a block, a brick, a crayon-box, showing two views as in Fig. 122. 19. Make a scale drawing from your own measures of a desk, table, bookcase, or other object in your schoolroom, showing three different views (top, side, and edge views) of it. Jcafe r:40" FIGURE 120 END VIEW Scale I": 16' FIGURE 122 COMMON FRACTIONS 195 124. Schoolhouse and Grounds. 1. Using a foot rule, graduated to 16ths of an inch, and regarding the scale of the drawing (Fig. 123), find the width of the grounds; the length; the area in square rods. (30^ sq. yd.= 1 square rod.) 2. Find the length of the field; the width; the area in square rods. 3. Find the length and the width of the school yard ; the area in square rods. 4. Find the length and the width of the schoolhouse ; the area, in square yards, cov- ered by it. 5. How far is the front door of the schoolhouse from the front fence? from the west front gate? from the sand pile? from the tree? G. How far is it from the back door to the east flower bed? to the back fence? to the west fence? to the coal shed? to the north- east corner of the school yard? to the south end of the pond? to the hill? to the nearest point on the creek bank? to the foot bridge (F)? 7. How wide is the south road? the creek? the branch? 8. How many square rods in the south road in front of the grounds? in the crossing of the roads? 9. How many square rods in the meadow? in the grove? in the pasture? 10. How many square rods are covered by the creek and the branch together, within the fence lines? ROAD Scale I": 160' FIGURE 123 196 RATIONAL GRAMMAR SCHOOL ARITHMETIC 11. How many rods of fence will be needed to enclose the grounds and to run along the lines indicated? 12. From your own measurements make a similar drawing, to a convenient scale, of some tract of ground (school yard or field) near your schoolhouse. Locate any fixed objects on your tract in their proper places on the drawing by measuring their shortest distance from a fence and from a corner. 125. Proportion. ORAL WORK 1. Compare the ratio 3 : 2 with the ratio 6 : 4. What do you find? 2. Compare the ratio 8 d. : 2d. with the ratio 16 men: 4 men. 3. Compare the ratio 10 ft. : 800 ft. with the ratio 100 mi. : 8000 miles. 4. A proportion is an equation of ratios. Thus 6 : 12 = 3 : 6 and T 6 ^ = are two different ways of writing the proportion. DEFINITIONS. The first, second, third, and fourth numbers of the pro- portion are called the first , second, third, and fourth terms of the propor- tion. The first and fourth terms are called the extremes, and the second and third terms are the means. The first two terms are the first couplet, the third and fourth terms are the second couplet. 5. In 6 : 12 = 3 : 6, to what is the product of the extremes equal? the product of the means? 6. Answer the same questions for 3 : 5 = 12 : 20. 7. Which of these pairs of ratios may form proportions: 4 : 9 and 8 : 18? 6 : 11 and 18 : 33? 1:3 and 8 : 21? 3 : 7 and 12 : 28? 6 : 11 and 12 : 22? f and T V^? 1 and A? and -1? 2. and ^L? MM ax ex n 6n 8. Is 4 = ii a proportion? >. Multiply both sides by 21 and we have 4 x 21 = 15. Now multiply both sides of this equation by 7 and we have 5x21 = 7x15. What were the terms 5 and 21 in the proportion called? What were the 7 and 15 in the proportion called? COMMON FRACTIONS 197 9. Compare the product of the 1st and the 4th numbers in these proportions with the product of the 3d and the 3d terms : (i) * = if ; (5) A = M ; 00 A - * ? ; (6) | = &; (3) tt - B ; (?) T = His (4) { .iff.; (8) f-tflf. Can you state the principle problems 8 and 9 illustrate? PRINCIPLE. In a proportion the product of the means equals the product of the extremes. WRITTEN WORK What must x be in each of these expressions to give a propor- tion? Solution of first equation: 2x= 15, or x = 7.5. 1. 2:3 = 5:z 6. a:b = 2a:x 2. 4:3 = 8:2; 7. 18:z = 54:18 3. 6:z = 9:27 8. 4:6= z:9 4. 7:2 = *:14 9. = ^ 5. x: 6 = 8:12 10. ^ - ^ be o 126. Practical Applications. 1. If of a ton of coal is worth $4.81, what are 12 T. worth at the same rate? NOTE. Call the unknown term x. Write the proportion, using x, and then use the Principle, 125, to find x. 2. A spelling class of 52 pupils writes a total of 1352 words; at the same rate how large a class will write a total of 728 words? 3. A girl jumping a rope makes 441 skips in 3 min. How long w,ill it take her to make 392 skips at the same rate? 4. If the average column in a newspaper contains 1600 words, how much should a writer receive for 700 words at the rate of $5 per column? 5. In going 10,725 ft. the front wheel of a bicycle revolves 1430 times. How far would it go in making 1001 revolutions?' 198 RATIONAL GRAMMAR SCHOOL ARITHMETIC 6. it tick 7. A in If BEECH BROC clock ticks 7 times in 5 sec., how many times does 2 da., 6 hours? 60 A. cost $3000, how many acres will cost $2450? 8. The line BC (Fig. 124) represents 20 rods. FE is twice as long, AF four times as long, CD and ED five times as long, and AB six times as long. Find the length of each side. 9. Find the value of x in the following proportions: BC : FE AB : (x); BC : (x) = FE : CD; AF : FE = (x) : BC. FIGURE 124 -T- "" 4 3h 1 FIGURE 125 10. When a foot rule is held 2 ft. (arm's length) in front of the face as shown in Figure 125, 7 in. on the ruler seems just to cover the edge of a door 7 ft. high. If the ruler is held parallel to the edge of the door, how far is the eye from the door? 11. If the shadow cast by a 4-ft. stake is 5 ft. long, how high is a tree which casts a shadow 50 ft. long on the same day and hour? 12. A woodman desires to fell only such trees as will furnish two 10-ft. cuts between the stump and the first limb. He wishes to allow 4 ft. for height of stump and waste in cutting at the bottom and top ends. To test a standing tree, he places a 4-ft. stick QR vertically in the ground 33 ft. from the tree, lies down on his back with his feet against the stake, and sights over the top of the stake to the first limb B. The distance from his eye P to the soles of his feet Q being 5| ft., should he fell the tree? FIGURE 126 13. If KD (Fig. 127)? COMMON FRACTIONS 199 rd., AK= 80 rd., and BC= 56 rd., how long is 14. Make measures on objects in your vicinity and solve such problems as are suggested by those given. 15. How long is x (Fig. 128)? 16. The thumb is held 2 in. from the end of the pencil, and the pencil is held 2 ft. in front of the eye and parallel to the line to be measured. When the end of the pencil is sighted into line with the corner 5 of the table, the FIGURE 128 FIGURE 129 end of the thumb is in line with the corner a. How long is the end, at), of the table, if the eye is 36 ft. from the table? 17. The eye sights past a point A over the point P on the fence (Fig. 130), and sees the upper edge of the moon in line with A and P. Then moving the eye 2 in. up the stake, the lower edge of the moon is in line with B and P. If the stake is 20 ft. from the fence and the moon is 240,000 miles away, how long is the moon's diameter in miles? 18. Make measures like these yourself. 200 RATIONAL GRAMMAR SCHOOL ARITHMETIC 9. The rays of light from the sun passing through a pin-hole in the screen at H (Fig. 131) give a small circular image of the FIGURE 131 sun at /. With distances and diameter of image as shown in the cut, what is the diameter of the sun? 20. A small hole, H, in a window screen, gave a round image of the sun, 1.1 in. in diameter on a sheet of paper held 10 ft. from the pin-hole. If the sun is 93,000,000 mi. away, what is the sun's diameter? 21. Solve this problem from your own measures. DECIMAL FRACTIONS 127. Notation of Decimals. ORAL WORK 1. In $1111 what does the 1st 1 on the right stand for? the 2d? the 3d? the 4th? 2. In $6666 what does the 1st 6 on the left denote?' the 2d? the 3d? the 4th? 3. In $372.68 what is the unit of the 3 (Am. $100)? of the 7? of the 2? the 6? the 8? 4. In $5555 how does the number denoted by each 5 compare with the number denoted by the 5 to its left? to its right? 5. How do the units of the places of a number change as we pass through the number from left to right? 6. Which unit is the fundamental unit out of which the other units are made? How is the place of this fundamental unit indicated in $675.28? DEFINITION. A dot, called the decimal point, or point, is used to show the units' digit. The point always stands just to the right of the units' digit or place. DECIMAL FRACTIONS 201 7. If the law of problem 5 holds in 444.444, the unit of the 1st 4 to the right of the point equals what part of the unit of the 1st 4 to the left of the point? The unit of the 2d 4 to the right equals what part of the unit of the 1st 4 to the right? 8. In any number what part of the unit of the 1st place to the left of the point equals the unit of the 1st place to the right? of the 2d place to the right? of the 3d? of the 4th? DEFINITION. The unit of the 1st place, or digit, to the right is called the tenth; of the 2d place, or digit, the hundredth; of the 3d, the thousandth; of the 4th, the ten-thousandth and so on. 128. Numeration of Decimals. 1. The number 444.444 might be read, "4 hundreds, 4 tens, 4 units and 4 tenths, 4 hundredths, 4 thousandths"; but it is simpler to read it, "Four hundred forty -four and four hundred forty-four thousandths. 11 Read 234.234 both ways. Which is the shorter? 2. The latter mode of reading is the one used in practice. It may be stated thus : Read the integral (whole) part of the number, then read the part of the number to the right of the point just as though it were a whole number standing to the left of the point, then pronounce tke name of the unit of the last digit on the extreme right. The word "and" must be pronounced only at the decimal point. Read 675.328; 236.89; 7.65; 43.6587. 3. Read the following (1) .1; .01; .001; .0101; 1.1; 10.1; (2) .6; 6.06; 66.6; .0606; 600.06; 6000.006; (3) .60; 7.070; 8.50; .0600; 87.087; 10.0008. 4. What is the ratio of 5 to .5; of .5 to .05; of .5 to .005; of 55 to 5.5; of 55 to .55? 5. Find the ratio of 875 to 87.5; of 87.5 to 8.75; of 875 to 8.75; of 875 to .875. 6. How is a number affected by moving the decimal point 1 place toward the left (see problems 4 and 5)? 2 places? 3 places? 6 places? 7. What is a quick way of dividing any number by 10? 100? 1000? 202 RATIONAL GKAMMAR SCHOOL AEITHMETIC 8. Express the ratio of 2.358 to 23.58; to 235.8; to 2358; to 23,580; to .2358; to .02358. QUERY. Where is the point supposed to be in 2358? When the decimal point is not written, where is it supposed to be? 9. How is a number changed by moving the decimal point 1 place to the right? 2 places? 3 places? 6 places? 10. Make a rule for quickly multiplying any number by 10; by 100; by 1000; by 1 followed by any number of zeros. 11. Kef erring to problem 4, can you tell what effect is produced in such a number as .683 or .492 by writing a zero between the decimal point and its first digit? 2 zeros? any number of zeros? 12. Compare .5 with each of these numbers: .50; .500; .50000. What is the effect of writing any number of zeros to the right of the last digit of a decimal? DEFINITIONS. A decimal fraction, or decimal, is a fraction whose denominator is 10, 100, 1000, or some power of ten, in which the denom- inator is not written but is indicated by the position of the decimal point. A power of 10 is a number obtained by using 10 as a factor any number of times. 129. To Reduce a Decimal to a Common Fraction. The fractions we have studied whose denominators are actually written, are called Common Fractions. 1. Express the following decimals as common fractions: .5; .50; .500; .4; .40; .400; .6; .60; .25; .250; .125; .375; .625; .875. 2. Eeduce the results of problem 1 to their lowest terms. 3. Write these mixed decimals as improper fractions : 1.5; 2.50; 2.75; 10.4; 12.5; 6.75; 18.25. 4. Eeduce these improper fractions to their lowest terms. 5. After dropping the decimal point from the following decimals, what numbers must be written beneath them to express them as common fractions : .6; .67; .625; .875; .1275; 12.75; 25.786; 33.333; 6.6666? 6. Make a rule for expressing any decimal as a common frac- tion in its lowest terms. DECIMAL FRACTIONS 203 7. How may 2-J- tenths be expressed as a decimal? 62 hun- dredths? 33 J hundred ths? 14 and 666| thousandths? A pure decimal is a decimal whose value is less than 1; as, 38 thousandths, 66| hundredths. A mixed decimal is a decimal whose value is greater than 1: as, 3.58 or 2.87^. 8. Read and give the meaning of these mixed decimals : .12i; 3.33; 18.66f; 2.1^; .Of; .04f ; .OOf; 36.000J. NOTE. Numbers expressed in both decimals and common fractions are called complex decimals. A simple decimal is expressed without the use of common fractions. To reduce such an expression as 3.44f to a mixed number pro- ceed thus: 3445 _s_ya 2413 313 SOLUTION.-3.44? = -Z = _,_ = __ = 3 ^. 445 313 ai_3. vx 7 Q-jQ Or, thus: 3.44? = 3^ = 3^= 3^ * J= 8 9. Reduce the decimals of problem 8 to mixed numbers or common fractions. 10. Make a rule for expressing any mixed decimal as a common fraction, or a mixed number. PRINCIPLE I. Any decimal may be expressed as a common fraction in its lowest terms or as a mixed number, by dropping the decimal point, iuriting_ the denominator, and reducing the resulting common fraction to its lowest terms. 130. Rain and Snowfall, or Precipitation (ADDITION). DEFINITION. 1 in. of rainfall means a fall of 1 cu. in. of water on each square inch of surface of the ground. (Review 51, pp. 68-9). 1. The following quantities of rain fell from week to week during May, 1902, in Chicago; find the total rainfall for the month: First week, 1.09 in.; second week, 1.11 in.; third week, .45 in., and fourth week, 2.43 inches. SOLUTION. CONVENIENT FORM ExpLANATiON.-It is convenient to write the } n - addends in a column so that the units digits are all in the same vertical column. Then begin on the .45 m. right and add as with whole numbers. In the sum 2. 43 in. th e point should stand directly under the points in Ans. 5.08 in. the addends. DEFINITION. Finding the sum of decimal numbers is called addition of decimals. 204 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. The following numbers denote the monthly precipitations for 1902; what was the total precipitation for the year? .66 1.53 4.16 2.26 5.08 6.45 425 1.44 4.83 1.45 2.03 1.90 5. Without rewriting the numbers, find the total yearly pre- cipitation for the years 1891-1901 from these recorded data: JAN. 1.99 1.99 2.08 1.55 2.15 1.12 4.53 3.54 .58 1.21 1.15 FEB. 1.95 1.57 2.44 2.13 1.60 3.48 2.22 2.59 1.60 3.52 2.05 MAR. 2.13 2.21 1.69 2.66 1.32 1.26 3.56 4.60 2.11 1.58 3.38 APR. 2.14 2.17 4.16 . 65 36 2.79 2.23 .76 .14 1.02 .33 MAY 2.09 6.77 1.93 3.35 1.99 4.16 .84 2.23 4.35 3.59 2.18 JUNE 2.42 10.58 3.59 1.96 1.79 2.82 3.60 5.30 2.71 2.06 2.42 JULY 2.47 2.23 3.08 .60 2.43 3.61 1.47 1.94 6.66 4.64 4.25 AUG. 4.52 1.85 .18 .60 6.49 3.52 1.70 3.03 .91 4.24 2.00 SEPT. .32 1.34 1.98 8.28 1.89 6.70 .84 3.16 2.39 1.56 2.92 OCT. .36 1.54 1.75 .85 .51 1.36 .18 3.26 2.09 1.35 1.29 Nov. 3.83 2.68 2.45 1.18 5.60 2.16 3.06 2.25 1.14 3.30 .85 DEC. TOTAL 1.32 1.63 2.14 1.66 6.76 .16 1.62 1.11 6.81 .58 1.70 4. Foot and average the vertical columns and tell what the footings and averages mean. 131. Other Applications. 1. A coal dealer received 8 carloads of coal of the following tonnages: 24.6, 28.785, 31.25, 24.95, 31.8, 25.125, 28, and 29.25. What was the total tonnage? 2. A farm was divided by its owner into lots of the following acreages: 32.874, 7.124, 68.334, 11.66J, 16.28f, and 21.13J. What was the total area of the farm? 3. The following numbers represent in thousands of feet a lumber dealer's sales in 1 da. : 6.865, 24.245, 16.398. 12.28, 18.2, 6.395, 24, and 18.967. What was the total sales for the day? 4. A quantity of soil contained .125 Ib. gravel; .268 Ib. coarse sand; .175 Ib. fine sand; .0374 Ib. organic matter; .214f Ib. clay, and .275 Ib. water. What was the total weight of the soil? 5. During 8 hr. a freight train made the following mileages: 32.15, 28.375, 15.687, 20.2, 15.63, 17.5, 8.95, and 21.3. Sow far did the train run during the 8 hours? 6. Following are the weights in grains of the U. S. coins: 10-piece, 48; 5^piece, 73.166|; dime, 38.5834; quarter dollar, DECIMAL FRACTIONS 205 96.45; half dollar, 192.9; dollar, 412; quarter eagle, 64; half eagle, 129; eagle, 258; double eagle, 516. Find the total weight of all. 132. Nature Study (SUBTRACTION). 1. 54 cu. in. of soil in its natural state weighed 1.94 Ib. After being thoroughly dried it weighed 1.459 Ib. How much moisture passed off in drying? EXPLANATION. We have seen that 1.94 may be written 1.940. For convenience write the numbers so that units' digits stand in the same column. Be- ginning on the right subtract as though the numbers were whole numbers. In the result the point should stand directly under the points in the minuend and subtrahend. SOLUTION. CONVENIENT FORM 1.940 Ib. 1.459 Ib. Ans. .481 Ib. DEFINITION. Finding the difference of decimal numbers is called subtraction of decimals. 2. After drying, the same soil occupied only 37.125 cu. in. How much did it contract in bulk in drying? 3. 100 green oak leaves weighed .22 Ib. After thorough drying they weighed .087 Ib. What. was the weight of water contained in the green leaves? 4. The organic matter in the leaves was then driven off by burning the dry leaves. The ash weighed .0053 Ib. How much organic matter did the 100 leaves contain? 5. The corresponding numbers for 100 green elm leaves were : Weight of green leaves, .132 Ib. ; weight of dry leaves, .0345 Ib. ; weight of ash, .0035 Ib. Answer questions like 3 and 4 for these leaves. 6. Answer similar questions for these leaves: KIND or LEAVES WEIGHT IN POUNDS FRESH, GREEN DRY ASH 50 Poplar leaves .132 .099 .043 .044 .0043 .0026 35 Compound ash leaves. 7. 1.135 Ib. of dry beans, soaked for 24 hr., weighed 2.212 Ib. What was the amount of water taken up by the beans? 206 RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. The dry beans occupied 34.875 cu. in. and the soaked beans 85.5 cu. in. How much did the beans increase in bulk? 183. Stature and Weight of Persons. The following table contains the average heights and weights of boys, girls, men, and women for the ages indicated by the numbers in the first column. Heights are given in feet and weights in pounds. AGE YR. HEIGHTS Diff. GROWTH IN HEIGHT WEIGHTS Diff. GROWTH IN WEIGHT Males Females Males Females Males Females Males Females 2 2.60 2.56 25.01 23.53 4 3.04 3.00 31.38 28.67 6 3.44 3.38 38.80 35.29 9 4.00 3.92 49.95 47.10 11 4.36 4.26 59.77 56.57 13 4.72 4.60 75.81 72.65 15 5.07 4.92 96.40 89.04 17 536 5.10 116.56 104.43 18 5.44 5.13 127.59 112.55 20 5.49 5.16 132.46 115.30 30 5.52 5.18 140.38 119.82 40 5.52 5.18 140.42 121.81 50 5.49 5.04 139.96 123.86 60 5.38 4.97 136.07 119.76 70 5.32 4.97 131.27 113.60 80 5.29 4.94 127.54 108.80 90 5.29 4.94 127.54 108.81 1. Fill out the vacant columns headed difference (Diff.) ; the first by subtracting the height of the females from that of the males of the same age, and the second by subtracting the weights of females from those of males of the same age. Subtract without rewriting the numbers. 2. There are 4 vacant columns headed growth; two for growth in height and two for growth in weight. Fill out the first of these columns from the column of heights of males by subtracting each number of this column from the one next below it. Tell DECIMAL FRACTIONS 207 what the difference means. Fill out the second similarly from the column of heights of females. 3. In a similar way fill out the last two columns from the columns of weights of males and weights of females. Tell what these differences mean. 4. At what age are boys growing most rapidly in height? in weight? At what age do men begin to decrease in height? in weight? 5. Answer questions similar to 4 for females. 6. Compare your own height and weight with the numbers of this table, for your age. NOTE. If your age is not in the table, it will bs between two _ given there. Use the mean of the numbers for these two ages for your comparison. 7. The following table contains the stature in feet of children of Manchester and of Stockport, (1) who are working in factories, and (2) who are not working in factories. Without rewriting the numbers, fill out the vacant columns of differences between the heights of the two classes for both boys and girls. What effect, if any, of such work can you detect on the growth of the boy or girl? NOTE. When the boy or girl not working in factories is taller than the one working in factories, mark the difference with a plus (-)-) sign before it. In the opposite case mark the difference with the minus ( - ) sign before it. Bo YS Gil ILS AGES Working in Factories Not Working in Factories DlFF. Working in Factories Not Working in Factories DlFF 9 years.. 4.009 4.045 3.996 4036 10 4.167 4.219 4.134 4.114 11 4.272 4.252 4.261 4.341 12 4.446 4.413 4.475 4.472 13 4.537 4.580 4.636 4.590 14 4.715 4.725 4.813 4.852 15 4.971 4.836 4.875 4.928 16 5.134 5.266 4.990 5.003 17 5 223 5.338 5.039 5.059 18 5.276 5.825 5226 5.397 RATIONAL GRAMMAR SCHOOL ARITHMETIC 134. Pointing the Product of Decimals (MULTIPLICATION). ORAL WORK 1. What is the relation between the following pairs of num- bers? (1) 25 and 2.5 (4) 1.28 and 12.8 (7) 2847 and 28.47 (2) 25 and .25 (5) 1.28 and 128 (8) 284.7 and 2.847 (3) 75 and 7.5 (6) 47.8 and 4.78 (9) 28.47 and .2847 2. The product 37 x 25 equals how many times the product 37x2.5? 3. The product 684 x 7.5 equals what part of the product 684x75? 4. What is the product 684 x 75? What, then, is the product 684x7.5? What is the product 684 x. 75? 684 x. 075? 68.4 x. 075? 6.84 x. 075? . 684 x. 075? DEFINITION. By the number of decimal places of a number is meant the number of digits (zero included) on the right of the decimal point. ILLUSTRATION. In 2.005 there are 3 decimal places. WRITTEN WORK 1. Find the product 4862 x 784, and from it write the follow- ing products : 486.2x784; 48.62x784; 48.62 x 78.4; 4.862 x 7.84; 4862 x .784. 2. How many decimal places are there in 486.2? in 48.62? 4.862? 10.03? 3.0060? .0600? 20.0806? 3. How many decimal places are there in each of the prod- ucts of problem 1? 4. Compare the number of decimal places in each of the prod- ucts of problem 1 with the sum of the numbers of decinml places in both the multiplicand and the multiplier. What do you find? 5. Make a rule for finding how many decimal places there must be in the product of two decimals. 6. How, then, can you find where the decimal point belongs in the product of any two decimals? DECIMAL FRACTION'S 209 PRINCIPLE ^11. The member of decimal places in the product equals the sum of the numbers of decimal places in the factors. 7. A field containing 38.75 A. yielded 23.9 bu. of wheat per acre ; what was the total yield? SOLUTION. Since each acre yielded 23.9 bu., 38.75 A. must yield 38.75 X 23.9 bushels. CONVENIENT FORM EXPLANATION. 38. 75 3875 339 x 38. 75 = how many times 23. 9 X 38. 75? 23.9 239 239X3875= " " 239x38.75? ~TT95 "1195 239X3875= " 23.9x38.75? 1673 1673 What is i0 U of 926,125? 1912 1912 Does the rule of problem 5 hold true here? 717 717 926.125 926125 Ans. 926. 125 bu. 8. A steer weighing 16.22 cwt. sold at $8.85' per cwt. ; what price did he bring? 0. A passenger train ran for 12.27 min. at the rate of 65.75 mi. per minute; how far did it run during the time? 135. Force Needed to Draw Loads on Road Wagon. 1. To draw a load in a road wagon at a slow walk over hard, level country roads a horizontal pull of .075 of the total weight of the wagon and load is required. What horizontal pulls will be required to draw the following : WEIGHT WEIGHT TOTAL HORIZONTAL OF WAGON OP LOAD WEIGHT PULL IN LB. (1) 1068 lb. 2168 Ib. (2) 969.5 2408.3 (3) 1580.75 3675.6 (4) 2368 3890 2. Over fresh earth .125 of the total load as a horizontal pull will draw it on a road wagon. Fill out the vacant columns, for these conditions : WEIGHT WEIGHT TOTAL HORIZONTAL OF WAGON OF LOAD WEIGHT PULL IN LB. (1) 980 lb. 1260 lb. (2) 940.8 768.78 (3) 1164 3890.85 (4) 1675 4060.75 .... 210 RATIONAL GRAMMAR SCHOOL ARITHMETIC 3. Over loose sand .258 of the total load will draw it. the forces needed to draw these loads : Find WEIGHT OF WAGON WEIGHT OF LOAD (1) 375 Ib. (2) 1860 (3) 2800 685.9 Ib. 2586.38 3869.25 TOTAL WEIGHT HORIZONTAL PULL IN LB. 4. On good, broken stone pavement the pull is about .0285 of the total load; on wood pavement it is .019 of the total load; and on Macadam pavement it is .0333 of the load. Find the pull in pounds needed for each of these three kinds of pavement for the following : WEIGHT OF WAGON WEIGHT or LOAD (1) 4060 Ib. (2) 5190 (3) 4960 3795.65 Ib. 6340.86 7640.65 TOTAL WEIGHT HORIZONTAL PULL IN LB. 136. Division by an Integer. 1. I paid $941.25 for 15 A. of land; what was the price per acre? CONVENIENT FORM With Decimals With Integers 62.75 15)941.25 90 - 6275 15)94125 90 41 30 41 30 its 10.5 112 105 .75 .75 75 75 EXPLANATION. Compare the steps in the work with decimals with the corre- sponding steps in the work [with integers. How do the decimal points stand through the problem? How does the point stand in the quotient? How may the dividend be found from the divisor and the quotient? How, then, may you check division? Ans. $62.75. 2. I paid $2823.75 for 45 A. of land; what was the price paid per acre; DECIMAL FRACTIONS 211 3. The 36 members of a society were assessed equally to meet a debt of $1341 against the society. What was the amount of the assessment against each member? SOLUTION. 37.25 36)1341.00 108 261 252 9.0 7.2 llJO 1.80 Notice particularly how by writing zeros after the deci- mal point the quotient may be carried out in a decimal form. What effect does writing zeros after the point have on a number? Ans, 137.25. 4. The number of states and territories and the total areas in square miles of the land surface of the principal geographic divisions of continental United States according to the Twelfth Census, are given in the following table. Find the average land surface of a state for each division and for the whole United States. Carry the division to three decimal places.* DIVISION NUMBER LAND SURFACE AVERAGE North Atlantic 9 162,103 South Atlantic 9 168 620 North Central 12 753,550 South Central 9 610,215 Western 11 1,175,742 Continental United States 5. The 10 loads given in pounds in column 2 of the following table required the number of pounds of force given in the third column, to draw them on a common road wagon over a good, level country road. For example, 1,400 Ib. required 98 Ib. to pull it; *If the next decimal place after the last one required is less than 5, write the given quotient as the required decimal. If it is greater than 5, add 1 to the last figure of the decimal required. This rule applies to all cases of division of decimals. 212 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1616 lb. required 112 lb., and so on. Divide each load by the number of pounds of force needed to draw it, and put the quotient to two decimal places in the column headed "Katie." EXPERIMENT LOAD PULL RATIO 1 1400 93 2 1616 112 3 1825 126 4 2236 155 5 2440 170 .... 6 2650 185 7 2863 198 8 3072 216 9 3281 229 10 3662 244 137. Division by a Decimal. 1. What is T V of 55? 2. When no decimal point is written with a number where is it understood to be? 3. Where is the decimal point in 55? 4. How is a number changed by moving its decimal point 1 place toward the left? 3 places toward the left? 1 place toward the right? 2 places toward the right? 5. 55 equals how many times 5.5? .55? .055? 6. If 55 - 11 = 5, to what is 5.5 - 11 equal? ,55 - 11? 7. Name these quotients: 250-25; 25-25; 2.5-25; .25-25. 8. Name these quotients: 250 - 2.5; 25 - 2.5; 25 - .25; 250 -.25. 9. 2176 - 32 = 68; to what number is x equal in each of the following equations : (1) 217.6- 32 = z; (4) 217.6- 3.2 = x; (7) 2176 - 3.2 = x\ (2) 21. 76- 32 = z; (5) 217.6 -.32 = 2; (8) 21.76- .32 = x\ (3) 2.176-32 = ^; (6) 21.76-3.2 = 2; (9) 2.176 - .032 = x? 10. In each case of problem 9 compare the number of decimal places in the quotient with the number of decimal places in the dividend minus the number in the divisor. DECIMAL FRACTIONS 213 11. 101,388-426 = 238; without dividing, write out the numbers to which x is equal in the following equations : (1) 1013S.8-426 = z; (4) 101388 -42.6 = z; (7) 101.388- 42.6 = z; (2) 1013.88-426 = ^; (5) 101388 -4.26=; (8) 10.1388- .426=z; (3) 1.01388-426 = ^; (6) 1013.88-4.26 = ^; (9) 1.01388-.0426 = z. 12. From the results of problems 10 and 11 make a rule for finding the number of decimal places in the quotient. PRINCIPLE III. The number of decimal places in the quotient equals the number of decimal places in the* dividend minus the num- ber of decimal places in the divisor. 138. Problems. 1. In 1891 the total imports of tea into U. S. were 82,395,924 Ib. and of coffee 511,041,459 Ib. If the average cost of tea was 37^ per Ib., and of coffee 18^, what was the total cost of both? 2. On Dec. 6, 1901, a carload of 34 Angus show cattle of weights given in table annexed sold at the prices per cwt. set beside the weights. What was the total weight of the carload? the average weight of the animals? 3. What was the average price per hundredweight? 4. For how much did the entire load sell? 5. What was the average price per head which the owner received for the carload? 6. The previous year on the same occasion, a carload of 26 prize- winning Angus cattle, averaging 1492 Ib., sold at $15.50 per hundredweight. What aver- age price did the cattle bring the owner? 7. How much did he receive for the carload, problem 6? No. WEIGHT PRICE 1 1503 $ 9.00 2 1622 8.85 3 1606 8.60 4 1524 8.50 5 1524 8.50 6 1504 8.10 7 936 8.70 8 1327 6.85 9 1273 8.05 10 1130 8.75 11 1326 7.65 12 1141 8.50 13 1318 8.75 14 1327 8.70 15 1190 7.70 16 1446 7.85 17 1449 7.85 18 1542 7.70 19 980 6.80 20 1450 8.10 21 852 8.30 22 1376 8.30 23 1540 25.00 24 1631 9.30 25 1468 8.65 26 1297 8.50 27 1529 8.20 28 1100 7.60 29 1073 7.60 30 1110 8.10 31 1095 8.15 32 1456 8.75 33 1298 8.00 34 2130 10.75 214 RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. A dairyman finds that during November one of his cows furnished this record : MORNIM; MILKINGS EVENING MILKINGS 1st week 47 2 Ib 40 4 Ib 2d . . . 58 6 " 48 " 3d " 53 8 " 49 8 " 4th " 62 7 " 47 3 " 29th and 30th days 17.8 " 16 4 " What was the total number of pounds of milk given by this cow during the month at the morning milkings? at the evening milkings? at both milkings? 9. If milk was worth 6.25^ a quart (8.6 Ib. per gallon), what was the milk of this one cow worth to the owner during November? 139. Ratio of Circumference of Circle to Diameter. 1. The distance around the rung of a chair was measured and found to be 3.625 in.; the diameter was found to be 1.15.'} in. Divide the distance around the rung by the diameter and find the quotient to 3 decimal places. DEFINITION. The distance around a circle is called the circumference of the circle. 2. The circumference of a circular rod, 1.875 in. in diameter, was measured and found to be 5.884 in. Find the ratio to 3 decimal places, of the circumference to the diameter. 3. Measure the diameters and the circumferences of any circles in your schoolroom and find to 3 decimal places the ratio of their circumferences to their diameters. If no circles are at hand use the measures of this table : OBJECT I'IRCITMFERENCE DIAMETER RATIO Ink bottle .... 5 5 in 1 75 in Tin box r> ir,.j 1 1)5:; Globe of lamp Terrestrial globe Barrel -head . . . 28.588 57.080 71 458 8.444 18.0f>',} 22 750 Iron rincj 18 125 5 675 Avorace DECIMAL FRACTIONS 215 4. Measure the circumferences and the diameters of the fol- lowing objects and find the ratios to three decimal places. If you are unable to make your own measures use those of the table: OBJECT ClBCUMFEKENCE IMAMKTKK RATIO Ii<'V('le wh66l 81 177 26 025 Bicycle wheel 88.055 28.012 Front carriage wheel Rear Locomotive driver. . Average 151.189 176.333 174.762 48.125 56.125 55.625 5. Find the ratio of the circumferences to the diameters of the coins of problem 27, p. 114. G. It is proved in Geometry that the ratio of the circumference of any circle to its diameter is about 3^, or more accurately 3.1416. If then, the diameter of a circle is known, how may the circum- ference be found without measurement? 7. Find the average of all the values of the ratios found in prob- lems 1 to 4 and compare this last average with 3|; with 3.1416. What is the difference in each case? NOTE. For most practical purposes this ratio, denoted by the Greek letter TT, and called pi, may be taken as 3). For greater accuracy use ir = 3.1416. 140. Original Problems. Make and solve problems based on the following facts: 1. 1 cu. ft. is about .8 of a bushel of small grain. A grain bin is 8' x 12' x 22'. 2. 1 bu. of ear corn is about 2.25 cu. ft. A wagon box is 2. 67' x 2.85' x 9.6'. 3. Well settled timothy hay runs about 355.25 cu. ft. to the ton. A hay shed is 24' x 40' and is filled with hay to a height of 18.5'. 4. Loose timothy hay runs about 460 cu. ft. to the ton. A load of hay is 8' x 18' x 22.7'. 5. Stove coal runs about 35.1 cu. ft. to the ton. A coal bin is 6' x 12.5' x 15. 35'. 6. 1 perch of stone is 24.75 cu. ft. A stone wall is 2.75'x 4.385' x 126.8'. 216 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. A man walks about 3.5 mi. per hour. It is 85 mi. from Chicago to Milwaukee. 8. A horse trots about 7.5 mi. per hour. 9. A horse runs about 18 mi. per hour for short distances. 10. A steamboat runs 18 mi. per hour. 11. A slow river flows 3 mi. per hour. 12. A rapid river flows 7 mi. per hour. 13. A crow flies 25 mi. per hour; a falcon, 75 mi.; a wild duck, 90 mi. ; a sparrow, 92 mi. ; and a hawk, 150 mi. per hour. 14. A carrier pigeon flies 80 mi. per hour for long distances. 15. Sound travels through air 1134 ft. per second; through water, 5000 ft. per second; and through iron or steel, 17,000 ft. per second. 16. A rifle ball travels 1460 ft. per second at starting; and a 20-lb. cannon ball, 16,000 ft. per second at starting. 17. Light travels 186,600 mi. per second; electricity, 288,000 mi. per second. The sun is 93,000,000 mi. from the earth; the moon, 240,000 miles. 18. One horse-power raises 33,000 Ib. through a height of 1 ft. in 1 minute. 19. The equatorial diameter of the earth is 7925.6 mi.; the polar diameter, 7899.1 miles. 141. Physical Measurements. The following table gives, for men, of ages from 18 to 26 years, the average lung capacity in cubic inches, the height in inches, and the weight in pounds. AGE LUNG CAPACITY HEIGHT WEIGHT 18 251.4 68.2 134.25 19 251.8 68.2 135.40 20 258.2 68.1 138.65 21 260.4 68.1 140.60 22 264.8 68.2 141.15 23 263.7 68.1 138.60 24 267.1 68.2 143.90 25 267.2 68.2 143.15 26 267.1 68.9 142.30 DECIMAL FRACTIONS 217 1. Find the number of cubic inches of lung capacity per inch of height, for each age, by computing the ratio to 2 decimal places of each number of column 2 to the corresponding number of column 3. Is this ratio the. same for all ages? 2. Similarly find the number of cubic inches, to 2 decimal places, of lung capacity per pound of weight, for each age. 3. For each age find the number of pounds of weight per inch of height to 2 decimal places. Are these numbers the same for all ages? 142. Specific Gravity. The specific gravity of any solid or liquid substance is the ratio of its weight to the weight of an equal bulk of water. The weight of a cubic foot of water is 62.5 pounds. 1. The following table contains the weight in pounds of 1 cu. ft. of the substances mentioned. Find to 3 decimal places the specific gravities of these substances : METAL, WEIGHT OF 1 Cu. FT. SPECIFIC GRAVITY WOOD WEIGHT OF 1 Cu. FT. SPECIFIC GRAVITY LIQUID WEIGHT or 1 Cu. FT. SPECIFIC GRAVITY Aluminum . 166 5 Cork 15.00 Alcohol 50 Zinc 436 5 Spruce 31 25 Turpentine 54 4 Cast iron . . . Tin 450.0 458 3 Pine (yellow) Cedar 34.60 35 06 Petroleum . . Olive oil 55.7 57 Wr't iron. . . Steel 480.0 490 Pine (white) Walnut . . 28.00 41 90 Linseed oil . . Sea water . . . 59.5 64 1 Brass 523 8 Maple . .. . 46 88 Milk 64 5 Copper 552.0 Ash 52 80 Acetic acid . . 66.5 Silver . 655 1 Beech 53 25 Muriatic acid 75 Lead 709 4 Oak 65 00 Nitric acid 95 Gold . . 1200 9 Ebony 76 00 Sulphuric acid 115 1 Platinum . . 1347.0 Lignum vitse 83.30 Mercury 880.0 2. The specific gravity of any gas or vapor, is the ratio of its weight to the weight of an equal volume of air, the gas or vapor and the air being at the same temperature and under the same RATIONAL BKAMMAR SCHOOL ABI1HMETIO pressure. Find to 3 decimal places the specific gravities of the gases and vapors in the following table: GAS WEIGHT, IN POUNDS, OF l CU-BIC FOOT SPKOIFIC UKAVITV Hydrogen 00559 Srnoke (wood) 00727 Smoke (soft coal) .00815 Steam at 212 F 03790 Carbonic oxide 07810 Nitrogen 07860 Air . . .... 08073 Oxygen . . . . 08925 Carbonic acid. . 12344 Chlorine 19700 3. The following familiar substances may be compared with water as to weight. Find their specific gravities to 2 decimal places : SUBSTANCE WEIGHT, IN POUNDS, OF 1 CUBIC FOOT SPECIFIC GRAVITY Glass (average) 175 8 Chalk 174 5 Marble . . . 169 2 Granite 166 4 Stone (common) 158 2 Salt " 133 4 Soil " 124 5 Clay 121 8 Brick 118 3 Sand 113 9 143. To Eeduce a Common Fraction to a Decimal. 1. Express $f decimally. SOLUTION .375 8)3.000 2.4 760 .56 EXPLANATION. Annex zeros to the right of the decimal point after the numerator, and then divide by the denominator. Ans. 1.375 .040 .040 DECIMAL FRACTIONS 211) 2. Express the following common fractions decimally (to 3 decimal places) : f. 2 . 4 . 7 . 5 . 3 . 5.13 ) 3> ~5J 8> 8"> TT> ~5Z> 64* 3. Express the following mixed numbers decimally (to 3 places) : H; 3J; 2i; 6f; 13^; 18Jf 4. Express the following decimally to 5 decimal places : i; A; A; A; H; A- Such decimals as these fractions give rise to, that do not ter- minate, are called non-terminating decimals. 5. Express the following numbers decimally to 6 decimal places : i; !; l-;i; i; I; i; A; A; A; iS; it; f*. DEFINITION. Such non-terminating decimals as these that repeat the same digit or group of digits indefinitely, are called repetends, or circula- ting decimals, or circulates. 6. Express the values of these numbers by the use of integers and decimals only : 1.3$; 10.87*; 5.28| ; 17.UJ; 20.0f; l.UOJ; -OOJ; .0-J; .000^; 30.060|. 7. Express the following fractions decimally to 4 places : 11. I- 7.3. 42 . 3*. M. 8 ' 65 . ^ ' 7*' 7.8' 13.17' 12i' 8. A man sold 37^ A. of his farm of 79f A. ; how many hundredths of his farm did he sell? 9. A man owned a piece of city land 450' square. A strip 37J-' wide was cut from each of its four sides for streets. How- many hundredths of the square were cut away? 10. How many hundredths of the area of a page of this book are in the margins? (Measure to the nearest 16th of an inch). 11. How many hundredths of the area of the surface of your desk is the area of the surface of your book? 12. How many hundredths of the area of the surface of the floor is the area of the surface of your desk? KATlUJSALi lilt AMM Alt SUHUULi 144. Area of a Circle. Draw a circle and diameter AB (see Fig. 132). With an opening of the compasses equal to the radius of the circle and with ^ FIGURE 132 A as center draw short arcs at D and 6r, also, with same radius and with B as center, draw short arcs at E and F. Draw radii OD, OE, OF, and OG. Bisect angle A OD, as in Problem II, p. 186, and draw OS. With an opening of the compasses equal to the distance between D and H and with center D draw an arc at 7; with center B draw arcs at A" and L\ also, with center G draw arcs at M and N. Draw radii O/, OK, OL, OM, and ON. Cut the circle into the twelve equal sectors thus formed and place these sectors as shown in the second part of the figure. How long is the base AB of the approximate parallelogram thus formed? How wide is the approximate parallelogram? If each of the twelve sectors were split into halves and the resulting twenty -four sectors were fitted together as are the twelve sectors, would the wavy base line become more nearly straight? How long and how wide would the new approximate parallelogram be? What would be its area? Having the circumference and the radius of a circle how can you find the area of the circle? Having the radius of any circle how can you find the area of the circle (See 138)? PROBLEMS In problems 1 to 3, inclusive, use tr = 3|. Let r denote the length of the radius of a circle, let c denote its circumference, and A, its area. 1. Find the areas, A, of these circles (1) r = 12.5', c = 78.54'; (3) r - 20", c = 125.664"; (2) r = 6.25', c - 39.27'; (4) r = 60", c = 377.143". DECIMAL FRACTIONS 221 2. The diameter of a drum-head is 2.5' and the circumference is 7.854', how many square inches of skin are in the two heads? 3. The diameter of a circular window is 18.5" and its circum- ference is 58.12", what is its area? In the following problems TT is taken as 3.1416. Results should be correct to 4 decimal places. 4. The wind is blowing squarely against a circular signboard, whose diameter is 32.75 ft., with a pressure of 25.5 Ib. to the square foot. Find the total wind pressure against the board. 5. The circumference of a cylindrical chair rung is 5.5"; find the diameter and area of the right section of the rung. NOTE. A right section is the section that would be made by sawing the rung square across. 6. Steam passes from the boiler of an engine through the passage, P, into the cylinder, and pushes against the circular piston, (7, with a pres- sure of 65 Ib. to the square inch. If the diameter of the circular head, (?, is 12.5", what is the total pressure against the left side FIGURE 133 of the pist(m? 7. Answer similar questions for pistons having these diam- eters with steam pressures per square inch as indicated: DIAMETER OF PRESSURE PER PISTON SQUARE INCH (1) 10.85" 86 (2) 22.35" 120 (3) 14.85" 180 (4) 16.45" 175 (5) 20.75" 125 (6) 15.85" 160 (7) 21.35" 205 8. Find the area, ^4, of a circle whose radius is r ft. long and whose circumference is c ft. long. 9. Find the circumference, c, of a circle whose radius is r rods long. Find the area. 10. A cow is tied to the corner of a corn crib, 18' x 18', with a rope 18' long. Over how many square feet can she graze? RATIONAL GRAMMAR SCHOOL ARITHMETIC 145. Gear of Bicycle. In this section use ?r = s -f-. 1. How many times does the large wheel of a high bicycle revolve while the cranks turn round once? Aa FIGURE 134 2. If the diameter of the driving wheel is 5.5 ft., how far will the high bicycle advance while the cranks turn round once? 3. If there are 7 teeth on the rear sprocket of a safety bicycle and 14 teeth on the front sprocket, how many times will the driv- ing wheel revolve while the cranks turn round once? 4. The diameter of the driving wheel of a safety bicycle is 28 inches. If there are 21 teeth on the front sprocket and 7 on the rear, how far forward will one complete turn of the cranks carry the bicycle? How far if there are 28 and 8 teeth? 5. Answer similar questions if the front and the rear sprockets have 24 and 9 teeth respectively; 30 and 9; 33 and 9. 6. Answer similar questions if the length of the tire of the driving wheel is 6.85', sprockets having teeth as in problem 5. 7. How long is the diameter of a wheel if one revolution car- ries it along 24.09 ft. on the ground? 8. I roll my bicycle along until the driving wheel turns over just once. The distance traversed is 7' 4". What is its diameter? 9. How long is the tire of the driving wheel of a high bicycle (see Fig. 134) if its diameter is 56"? 08"? 72"? 80"? 45"? 10. How far is a high bicycle moved forward by 1 turn of its 84" driving wheel? NOTE A 28"-bicycle means a bicycle whose driving wheel is 28" in diameter. 11. If there are 29 teeth on the front and 7 on the real- sprocket of a 28"-bicycle how far will one turn carry it? DECIMAL FRACTIONS 223 A safety bicycle is said to be "geared" to 72", 84" and so on when one turn of the cranks would carry it just as far forward as would one turn of a wheel having a diameter of 72", 84" and so on. 12. Count the teeth on the front and the rear sprockets of some safety bicycle, measure the diameter of the driving wheel, and find its "gear." 13. Find to 2 decimal places the "gear" of these safety bicycles: NUMBER or TEETH DIAMETEK DRIV. WHEEL GEAR Rear Sprocket Front Sprocket, 18' 7 14 22' 7 24 24' 7 26 26' 8 28 26' 9 32 28' 7 28 28' 8 28 28' 8 34 146. Law Formulated (ALGEBRA). 1. If the diameter of the driving wheel of a safety bicycle is d, and the ratio of the circumference to the diameter of a circle is TT, write an equation showing the length, c, of the cir- cumference. 2. If t is the number of teeth on the rear sprocket, and J'the number on the front sprocket, write an equation showing the number, -w, of times the driving wheel will turn while the cranks turn once. 3. Write an equation showing the distance, d, that one revolu- tion of the cranks will carry the bicycle forward, using the letters of problems 1 and 2. 4. Calling G the gear of a safety bicycle, show that the gear is T given by the equation G = x d, the letters meaning the same as above. 5. Make a rule for finding the gear of a safety bicycle the diameter of whose driving wheel is d feet. 224 RATIONAL GRAMMAR SCHOOL ARITHMETIC 147. Comparison of Sail Areas of Yachts. The sail areas of the four racing yachts, Genesee, Yankee, Illinois, and Milwaukee, were computed from the measurements indicated in the drawing (Fig. 135). L.w.L.aa' Yankee Illinois FIGURE 135 L.W.L 37/84 Milwaukee 1. The small sails in front are called jibs. The base and the altitude of the jibs of each yacht are indicated in Fig. 135. Find the number of square feet in the jib for each of the four yachts. 2. Genesee was cup-winner. Find the ratio, to 1 decimal place, of the sail area of the jib of Yankee to that of Genesee. 3. Find a similar ratio for each of the other two yachts. 4. The jibs were made of a kind of coarse cloth, called duck, which is sold in strips 2 ft. wide and weighs 8 oz. per linear yard (per yard of length). Find the weight of 1 sq. ft. and the weight of the jib of each of the four yachts. 5. To compute the areas of the main sails, each was divided into two triangles, like AEB and ABC of Genesee; the bases and the altitudes were measured and found as indicated in the figures. The altitudes, AF and AD, and bases, EB and BC, of Genesee having the indicated lengths, find the total area of her mainsail. 6. Similarly find the total area of the mainsail of each of the other three yachts. 7. These mainsails weigh 10 oz. per linear yard (strips 2 ft. wide). Find the weight of the mainsail of each yacht. 8. Find to 2 decimals the ratio of the area of the mainsail of the Yankee to that of the Genesee (the winner). 9. Similarly, find the ratio of the mainsail area of each of the other two yachts to the mainsail area of Genesee. 10. Find the total sail areas and sail weights of each yacht. COMPOUND DENOMINATE NUMBERS 225 COMPOUND DENOMINATE NUMBERS 148. Definitions. A denominate number is a number whose unit is concrete; as, 13 mi., 8 hr., 40 A., $125, etc. A concrete unit is a unit having a specific name; as, 1 yd., 1 lb., $1, 1 hat, 1 horse, etc. A compound denominate number is a number expressed in two or more units of the same kind; as, 11 hr. 25 min. 15 sec.; 12 gal. 3 qt. 1 pt. 3 gills. TABLES OF MEASURES NOTE. Read carefully the tables that follow and fix them in mind by solving the problems beginning on page 231. 149. Measures of Value. The standards of value of the United States and of some of the European countries are here given with both their rough and their accurate equivalents in U. S. money. ROUGH ACCURATE COUNTRY STANDARD SYMBOL EQUIVALENT EQUIVALENT United States Dollar $ $1. $1. Great Britain Pound (Sterling) $5.00 $4.8665 Germany Mark (Reichsmark) M. $ .25 $ .2385 France Franc F. $ .20 $ .193 Russia Ruble R. $ .75 $ .772 Austria-Hungary Crown, or Filler C. or F. $ .20 $ .203 Italy Lira L. $ .20 $ .193 TABLE OF U. S. MONEY 10 mills (in. ) = 1 cent (ct. or ^) 10 cents = 1 dime (d.) 10 dimes = 1 dollar 10 dollars = 1 eagle 5 dollars = ^ eagle 2 1 dollars = eagle 20 dollars = 1 double eagle The coins of the United States are bronae, nickel, silver, and gold. TABLE OF ENGLISH MONEY 4 farthings (far.) = 1 penny (d.) 12 pence = 1 shilling (s.) 20 shillings = 1 pound () 21 shillings = 1 guinea 226 RATIONAL GRAMMAR SCHOOL ARITHMETIC TABLE OF MONETARY UNITS OF OTHER NATIONS Germany, 1 mark (M.) =100 pfennige (pf.) France, 1 franc (fr.) = 100 centimes (c.) Russia, 1 ruble (r.) = 100 copecks (c.) Austria-Hungary, 1 crown, or filler = 100 heller Italy, 1 lira =100 centessimi. 150. Measures of Weight. Three systems of weight units are used in the United States, viz. : troy weight, avoirdupois weight, and apothecaries' weight. Troy weight is used in weighing gold, silver, and jewels. Avoirdupois weight is used for weighing all ordinary articles, and apothecaries' weight is used by druggists in mixing medicines. The standard of weight in the U. S. is the troy pound. The grain is the same in all three systems; the pound the same in troy and in apothecaries' weight and different in avoirdupois. TABLE OF TROY WEIGHT TABLE OF EQUIVALENTS 24 grains (gr.) = 1 pennyweight (dwt.) 5760 gr. \ 20 penny weights = 1 ounce (oz.) 240 dwt. / = 1 Ib. 12 ounces = 1 pound (Ib.) 12 oz. ) TABLE OF AVOIRDUPOIS WEIGHT 7000 grains (gr.) = 1 pound (Ib.) 16 ounces = 1 pound (Ib.) 100 pounds = 1 hundredweight (cwt.) 2000 pounds = 1 ton (T.) 2240 pounds = 1 long ton (L.T.) TABLE OF APOTHECARIES' WEIGHT 20 grains (gr.) = 1 scruple (3) 3 scruples = 1 dram (3) 8 drams = 1 ounce (1 ) 12 ounces = 1 pound (Ib.) 151. Measures of Length, or Distance (Linear Measure). The standard unit for measures of length, or distance, is the yard. TABLE OF COMMON LINEAR MEASURE TABLE OF EQUIVALENTS 12 inches (in.) = 1 foot (ft.) 63360 in. 3 feet =1 yard (yd.) 5280ft. , 5$ yards, or 16 ft. = 1 rod (rd.) 1760 yd. C ~~ 320 rods = 1 mile (mi.) 320 rd. COMPOUND DENOMINATE NUMBERS 227 TABLE OP SURVEYORS' LINEAR MEASURE 7. 92 inches = 1 link (li.) 100 links 1 chain (ch.) 80 chains = 1 mile (mi.) 152. Measures of Surface. The unit upon which surface measure is based is the square yard, which is a square each of whose sides equals 1 yard. TABLE OF COMMON SURFACE MEASURE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd. ) 30 square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) TABLE OF SURVEYORS' SURFACE MEASURE 025 square links (sq. li.) = 1 square rod (sq. rd.) Ki square rods = 1 square chain (sq. ch.) 10 square chains = 1 acre (A.) 640 acres (a section) = 1 square mile (sq. mi.) 36 square miles = 1 township (Tp. ) TABLE OF EQUIVALENTS 3686400 sq. rd. \ 23040 sq. ch. I = 1 sq. mi. 36 sq. mi. ) 153. Measures of Volume. TABLE OF CUBIC MEASURE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) TABLE OF EQUIVALENTS 46656 cu. in. , = 1 cu. 27 cu. ft. ) A perch of stone is a square-cornered mass, 1' x !' x 16-J', or 24f cubic feet. Fire wood is measured by the cord. A cord of wood is a straight pile, 4' x 4' x 8', or 128 cubic feet. A cord foot is a straight pile of wood, 4' x 4' x 1'. How many cubic feet are there in a cord foot? 228 RATIONAL GEAMMAR SCHOOL ARITHMETIC 154. Measures of Capacity. Liquid measure is used in measuring liquids, the capacity of cisterns, tanks, etc. TABLE OF LIQUID MEASURE 4 gills (gi.) = l pint (pt.) 2 pints = 1 quart (qt. ) 4 quarts = 1 gallon (gal. ) TABLE OP EQUIVALENTS 32 gi. 8 pt. J- = 1 gal. 4qt. NOTE. A liquid gallon contains 231 cubic inches. Dry measure is used in measuring grain, fruit, vegetables, etc. The standard of dry measure is the Winchester bushel, which is a cylinder 18^ in. in diameter and 8 in. deep, containing 2150.42 cubic inches. , TABLE OF EQUIVALENTS 64 pt. } 32 qt. [ = 1 bu. 4pk. ) NOTE. A dry gallon = 4 qt. or \ pk., or 268.8 cubic inches. AVOIRDUPOIS POUNDS IN A BUSHEL PRESCRIBED BY VARIOUS STATES TABLE OF DRY MEASURE 2 pints (pt. ) = 1 quart (qt.) 8 quarts = 1 peck (pk. ) 4 pecks = 1 bushel (bu. ) COMMODITIES. Barley Beans Blue Grass Seed . Buckwheat Castor Beans Clover Seed Coal (Anthracite) Corn on the Cob . Corn, Shelled Cornmeal Dried Apples Dried Peaches Flax Seed Hqmp Seed Millet Oats Onions Peas Potatoes Rye Sweet Potatoes . . Timothy Seed Turnips Wheat . . 50 48 48 48 48 48 47 48 48 48 48 48 48 60 60 60 60 60 60 60 62 14 40 48 52 46 60 60 60 60 60 60 525656 48 21 80 504850 50 45 5055 60606060 50525056 46 46 46 70 68 70 70 70 33 33 33 33 39 36.. 565656 44 44 44 44 ! 44 46 50 55 45 45 45 45 45 5560 80 80 76 5050 25 24 24 24 48 50 50 45 i o c I ^ 48 48 48 50 52 60 606060 505050 56 54 56 56 56 56 56 56 56 50 56 56 56 56 56 56 56 56 56 56 56 56 5456 45 58 222824 282833 60 60 60 62 60 32 32 32 32 32 32 32 32 32 32 32 32 32 32 30 32 33 30 32 32 32 32 32 32 . . 50 57 48 57 57 57 . . 52 52 5.4 . . 57 . . 57 . . 55 . . 5657 52 57|50 57 . . 60 . . .. 60 60 60 .. 60 .... 60 60 60 60 .. 60 .. 60 60160 . . 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 45 56 56 56 56 56 56 56 56 56 56 56 56 58 56 56 56 56 56 56 56 5S 48 48 48 47 48 48 48 48 48 48 50 48 50 48 64 60 60 62 55 55 56 50 4445 60 60 60 60 60 14 42 48 52 42 48 7070 50.. 2428 ..28 5656 44 5866 50'55 .. 45 45 45 45J40 605550 I! 16060606060 70 70 56 50 282825 322828 56 COMPOUND DENOMINATE NUMBERS 229 155. Measures of Time. The standard unit for measuring time is the mean solar day. The mean solar day is the average time interval from the instant when the sun crosses the meridian of a place (noon) to the next instant of crossing the meridian (the next noon). TABLE OF TIME MEASURE 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr. ) 24 hours = 1 day (da. ) 7 days = 1 week (wk.) 30 days = 1 calendar month (mo.) 12 calendar months = 1 calendar year (yr.) 365 days = 1 common year 366 days = 1 leap year 100 years = 1 century NOTE. One mean solar year = 365 da. 5 hr. 48 min. 46 sec. = 365} da. (nearly). TABLE OF EQUIVALENTS 31536000 sec. 1 525600 min. 8760 hr. 365 da. 52 wk. 1 da. 12 mo. According to the Julian calendar, adopted by Julius Caesar (whence the name), every year whose date number (as 1896, 1904) is divisible by 4 must contain 366 da. and all other years contain 365 da. Years containing 366 da. are called leap years; those containing 365 da. are called common years. According to the Gregorian calendar, which is now used by nearly all civilized nations, every year whose date number is divisible by 4 is a leap year, unless the date number ends in two zeros (as 1600, 1900), in which case the date number must be divisible by 400 to be a leap year. The extra day of the leap year is added to February, giving this month 29 da. in leap years. 156. Measurement by Counting. TABLE FOR COUNTING TABLE FOR MEASURING PAPER 12 things = 1 dozen (doz.) 24 sheets = 1 quire 20 things = 1 score 20 quires = 1 ream 12 dozen = 1 gross 2 reams = 1 bundle 12 gross = 1 great gross 5 bundles = 1 bale 230 RATIONAL GRAMMAR SCHOOL ARITHMETIC The operations to be performed in compound denominate numbers are the following: (1) To change from higher to lower units, or denominations; (2) To change from lower to higher units, or denominations; (3) To add compound denominate numbers; (4) To subtract such numbers; (5) To multiply such numbers; (6) To divide such numbers. These processes will now be illustrated in order. 1. Express 16 bu. 3 pk. 3 qt. 1 pt. as pints. CONVENIENT FORM 16 bu. 3 pk. 3 qt. 1 pt. 4 = No. pk. in 1 bu. 64 pk. = No. pk. in 16 bu. 3 pk. 07 pk. = No. pk. in 16 bu. 7 pk. 8 = No. qt. in 1 pk. 536 qt. = No. qt. in 67 pk v 3qt, 539 qt. = No. qt. in 67 pk. 3 qt. 2 = No. pt. in 1 qt. 1078 pt. = No. pt. in 539 qt. 1 pt. Ans. 1079 pt. = No. pt. in 16 bu. 3 pk. 3 qt. 1 pt. 2. Express (1) 1079 pt. in higher units; (2) 2 pk. 3 qt. 1 pt. in bushels. CONVENIENT FORM (1) 2) 1079 pt. 8) 539 qt. -f- 1 pt- remaining 4) 67 pk. -f 3 qt. remaining 16 bu. -f 3 pk. remaining Ans. 16 bu. 3 pk. 3 qt. 1 pt. - (2) 8 qt. 1 pt. = 3.5 qt. = -~ pk. = .4375 pk. 2 pk. 3 qt. 1 pt. = 2.4375 pk. = ^ - bu. = .609375 bu. Ana. COMPOUND DENOMINATE NUMBERS 231 P>. The three sides of a triangular grass plot were : 8 yd. 2 ft. 10 in.; 12 yd. 1 ft. in. ; and 9 yd. 2 ft. 7 in.; how far is it around the plot? EXPLANATION. First, adding the inches, CONVENIENT FOKM we obtain 26 in ^ 2 ft. 2 in. Write the 2 in. 8 yd. 2 ft. 10 in. in "inches" column, and add the 2 ft. to the 12 1 9 numbers in the "feet" column, giving 7 ft. 9 2 7 = 2 yd. 1 ft. Write 1 ft. in "feet" column 31 yd. 1ft. 2 in. *** nUmberS ln " yards 4. From a vessel containing 25 gal. 3 qt. 1 pt. 2 gi. of oil, 8 gal. 3 qt. 1 pt. 3 gi. were drawn out; how much oil remained in the vessel? EXPLANATION. 3 gi. can not be taken from 2 gi. But the 1 pt. of the minuend CONVENIENT FORM jquals 4 gi. and this added to 2 gi. gives 6 gi. 6 gi. 3 gi. = 3 gi. 1 pt. from pt. 25 gal. 2 qt. 1 pt. 2 gi. can not be taken. But 1 qt. is taken from 8 gal. 3 qt. 1 pt. 3 gi. 2qt., and changed to pints, giving 2 pt. 1fi 0-al 2 nt 1 nt S P-i 3 pt. 1 pt. = 1 pt. 1 gal. = 4 qt. 4 LO gal. 2 qt. 1 pt. 6 gi. ^ + ! qt . = 5 qt< 5 qt _ 3 qt = 2 qt Ans. 16 gal. 2 qt. 1 pt. 3 gi. Finally, 24 gal. 8 gal. = 16 gal. Prove the work by adding the remainder to the subtrahend and comparing the result with the minuend. 5. A man built an average of 45 ft. 8 in. of fence a day for 16 da. ; how much fence_did he build in 16 days? CONVENIENT FORM 45 ft. 8 in. 16 EXPLANATION. 8 in. X 16 = 128 in. = 10 ft. 8 in. 10 ft- 8 in. 45 f t x 16 = 720 ft 720 730 ft. 8 in. 6. An iron rod, 4 yd. 2 ft. 8 in. long, was cut into 5 equal pieces ; how long was each piece? EXPLANATIONS yd. =12 ft. 12 ft. + 2 ft. 5) 4 yd. 2 ft. 8 in. = 14 f t . 14 ft. H- 5 = 2, with a remainder of Ans. 2 ft. 11 ^ in. 4 f t- 4 ft. = 48 in. 48 in. -j- 8 in. = 56 in. 157. Exercises on Tables. 1. Reduce 9 great gross to units. 2. How many great gross are there in 79,630 units? 232 RATIONAL GRAMMAR SCHOOL ARITHMETIC 3. A dealer bought 3600 lead pencils at $5.00 per gross. He sold them at 25 cents per dozen. What was his profit? 4. Bought 2 gr. foot-rules at $.12 a doz. ; 7 gr. Spenceriun pens No. 1 at $.07 a doz.; 8 gr. Eagle pencils No. 3 at $.30 a dozen. Find the amount of my bill. 5. A stationer bought 3 gr. boxes of paper, each box con- taining 6 reams. How many sheets did he buy? 6. A stationer sold 3 quires, 20 sheets of paper to one man, 18 quires to another, 4 reams, 15 quires to another. How many sheets did he sell in all? 7. There are 2 reams, 9 quires of paper in one package, 3 times as much in a second package, and 7 times as much in a third package as in the second package. How much paper is there in the second and third packages each? 8. In 28 there are how many shillings? how many pence? how many farthings? 9. In 18s. there are how many pence? how many farthings? 10. In 28 18s. 9d. 3 far., there are how many farthings? 11. In 342 15s. 6d. there are how many pence? 12. How many pounds, shillings, and pence are there in 28,643 pence? 13. How many feet are there in 78 yd.? how many inches? 14. How many inches are there in 78 yd. 2 ft. 6 inches? 15. How many yards, feet, and inches are there in 2838 inches? 16. How many ounces are there in 12 cwt.? in 10 pounds? 17. How many ounces are there in 12 cwt. 10 Ib. 11 ounces? 18. How many hundredweight, pounds, and ounces in 19,360 ounces? There are two classes of problems to be solved in reducing denominate numbers to their equivalents in different units. In one class numbers expressed in larger units are to be expressed in smaller units. This is called reduction descending, or reduction from higher to lower denominations. In the other class numbers expressed in smaller units are to be expressed in larger units. This is reduction ascending, or reduction from lower to higher denominations. Problems 15 and 18 are examples of the second class, and prob- lems 13, 16, and 17 are examples of the first class. COMPOUND DENOMINATE NUMBERS 19. Reduce 160 yd. 1 ft. 9 in. to inches. 20. Reduce 5781 in. to yards, feet, and inches. 21. Reduce 8 bu. 2 pk. 5 qt. to quarts. 22. Reduce 27 T. 18 cwt. 15 Ib. to pounds. 23. Reduce 23 cu. yd. 14 cu. ft. to cubic feet. 24. Change 395 15s. sterling to dollars and cents. United States gold and silver coins are .9 pure gold or silver and .1 copper. 25. What weight of pure silver is there in a silver dollar weighing 412| grains? 26. The U. S. standard 5 dollar gold piece weighs 129 gr. What is the number of grains of pure gold in the standard five dollar gold piece? 27. The 5-cent piece weighs 73.16 gr. .75 of the weight of the coin is copper and .25 is nickel. What is the weight of copper in the 5-cent piece? of the nickel in the 5-cent piece? 28. The eagle weighs 258 gr. What is the weight of pure gold in the eagle? 29. How many standard gold dollars can be coined from 1 oz. of pure gold? 30. Express the following ratios : 1 franc : $.25; $.25 : 1 franc; 1 mark : $.25; $.25 : 1 mark. 31. If .52 oz. of gold is worth 43s. 4d., how many ounces can be bought for 35 18s.? . 32. A boy laid by a certain sum of money each week. At the end of 1 yr. 3 mo. 2 wk. he had saved $88.50. How much did he save each week? (Take 1 mo. = 4 wk. 2 days.) 33. A man changed $350, half into English and half into German money. How much of each kind of money did he have? 34. How many feet are 4 ch. 30 links? 35. What is the ratio of 1760 yd. to 1 mi. 32 rods? 36. The mast of a ship was 78 ft. 4 in. high. During a storm .3 of it was broken off. How high was the remaining piece? 37. A four-sided field had sides of the following lengths: 63 ch. 2 rd. ; 49 ch. 14 li. ; 53 ch. 1 rd. 16 li. and 38 ch. 24 li. How far is it around the field? 234 RATfONAL GRAMMAR SCHOOL ARITHMETIC 38. A man walked of the lengttfof a breakwater, which was 1 mi. 243 rd. 5 yd. long. How far did he walk? 39. A knot, or geographic mile, equals GOsr, ft. What is the speed in common or statute miles per hour of a vessel that runs 21 knots per hour? 40. A wheel, 12 ft. in circumference, makes how many revolu- tions in 1-J- miles? 41. A telegraph wire is 14 mi. 140 rd. long, and is supported by 38(5 poles, which are placed at equal distances apart, a pole being at each end of the wire. How many feet apart are the poles? 42. What is the cost of 18 A. 120 sq. rd. of land at $52 per acre? 43. A man owned 14 A. of land, and sold 1428 sq. rd. How many acres did he have left? 44. If 1000 shingles are needed to cover 100 sq. ft. of roof, how many shingles are required to cover a roof 40 ft. long and 25 ft. wide, at the same rate? 45. A lawn tennis court 120 ft. long and 85 ft. wide is to be surrounded by a strip of sod 15 ft. wide at each end and S ft. wide at each side. What will the sodding cost at $.35 per square yard? 46. Find the cost of a half section of land at $45 per acre. 47. How many square rods are there in a rectangular field 24 ch. 45 li. long by 16 ch. 34 li. wide? 48. How many cubic inches are there in a tank containing 120 gal. ? how many cubic feet? 49. How many cubic feet are there in a block of stone 5 ft. x 4 ft. x 6 inches? 50. How much must I pay for a board 1C ft. long and 8 in. wide, the board being 1 in. thick and lumber costing $35 per thousand? (A board foot means 144 sq. in. of surface, not over 1 in. thick.) 51. I bought 3 boards 12' long, 6" wide, 8 boards 16' long and 9" wide and 2 boards 10' long, 12" wide and 2" thick; what did the whole cost at $30 per thousand? 52. What is the value of a straight pile of wood 16 ft. long, 8 ft. wide and 6 ft. high, at $7.50 per cord? COMPOUND DENOMINATE NUMBERS 235 53. What is the weight of a pile of oak boards 14 ft. long 8 ft. 4 in. wide and 5 ft. high, at an average weight of 54 Ib. per cubic foot? 54. At $.75 a load of 1 cu. yd. what will be the cost of remov- ing a pile of earth 60 ft. long 24 li. wide and 1 yd. high? 55. In one year the quarries of Minnesota yielded 4,000,000 cu. ft. of sandstone. What was this worth, at $.76 a perch? 56. Find the cost of building a stone wall 150' x 10' x 2-J', at $3.50 a perch? 57. A bin 12' x 8'G" x 5' is filled with wheat, What is the weight of the wheat at 60 Ib. per bushel? 58. What is the value of a straight pile of pine slabs 32'x 7'x 4', at $3.25 per cord? 59. A grocer paid $4.50 for a barrel of vinegar (31^- gal.), and sold it at 50 per quart. What was his profit? 60. A jug contained 214 cu. in. of molasses. How much did it lack of containing 1 gallons? 61. What will 1 mi. of right of way for a railroad cost at $00 per acre, the width of the right of way being 100 feet? 62. What will be the cost per mile for railroad ties at $.45 apiece, the ties being laid one every 2 ft. 63. Find the cost of the rails for 1 mi. of the road, the rails weighing 77 Ib. per linear yard and costing $35 per long ton. 64. Find the cost of fencing 1 mi. on both sides of the track, placing posts costing $.25 apiece 16 ft. from center to center, and using wire weighing 2160 Ib. per mile, and 30 Ib. of staples @ 40. The fence is to be 4 wires high, and labor costs 210 per rod of fence. 65. Reduce .865 gal. to smaller units. SOLUTION. .865 X 4 qt. = 3.46 qt. ; .46 X 2 pt. = .92 pt. ; .93 X 4 gi. = 3.68 gills. Ans. a qt. 3.68 gills. 66. Reduce .168 gal. to smaller units. 67. If oil is $.11 per gallon, what will be the cost of tnree 42- gallon barrels of kerosene? 68. If a certain spring regularly yields 25 gal. daily, how many barrels of the capacity of 31i gal. would it fill in 20 days? 69. A merchant bought 16 gal. 3 qt. of syrup at 38$ a qt., and sold 27 qt. for $12, 18 qt. for $9, and the remainder at 35$ per quart. Did he gain or lose, and how much? 70. Eeduce 2 pk. 7 qt. 1 pt. to the decimal of a bushel. SOLUTION. 1 pt.= .5 qt. 7.5 qt.= ~ pk. = .9375 pk. 2.9375 pk. = bu. = .734375 bushels. 71. Reduce 3 pk. 5 qt. 1 pt. to bushels. 72. How many bushels are there in a bin 14 ft. long 7 ft. 6 in. wide and 5 ft. 8 in. high? 73. A farmer sold 6 loads of corn, each load averaging 36 bu. 2 pk. at 35 cents per bushel. What did he receive for the whole? 74. A boy had a bushel of hickory nuts, and sold 3 pk. 7 qt. 1 pt. What fraction of a bushel had he left? 75. If 9 bu. of potatoes cost $4.80, what is the average cost per peck? 76. A bin contained 5376.25 cu. in. of rye. How much did it lack of containing 3 bushels? 77. How many oz. of quinine will be required to prepare 12 gross of 3-grain capsules? 78. A coal dealer buys two car-loads of coal each weighing 67,200 lb., at $4.50 a long ton, and sells it at $5.75 a short ton. What is his gain? THE METRIC SYSTEM 158. Historical. The metric system is a decimal system of weights and meas- ures adopted by the French Government soon after the French Revolution of 1789. The aim of the system is to base all measures upon an invariable standard, and to secure the simplest possible relations between the different units of the system. The unit of length, which is fundamental to the whole system, is called the meter. It was attempted to make the meter 1 ten- millionth of the length of the part of a meridian of the earth, which reaches from the equator to the pole, called a quadrant of the earth's meridian. The meridian of the earth was measured, and ToWoTnrF f the quadrant was obtained. A platinum bar equal to this length was very accurately cut and stored in the Government archives as the official standard of reference. COMPOUND DENOMINATE NUMBERS 237 Later measurements of the earth's meridian showed the former length of the meridian to be incorrect. The length of the meter as obtained from the erroneous measures was, however, retained, and the length of this bar is the standard meter. From it all the other units of the system are derived. The metric system is used for all purposes in France, and for nearly all scientific purposes in Germany, England, and the United States. The length of the meter is 39.37079 in., or about 1.1 yards. 159. Tables of Metric Measures. Fig. 136 shows a scale graduated along the upper edge to 16ths of an inch, and along the lower edge to lOOOths of a meter, called millimeters. I ,,, I .,,.,, I .,..,. I ,,..,, I ,,. I 2 1 3 1 .ill.lil.hfl.l.l.l.fl.l.l.l.fl FIGURE 136 Decimal parts of the standard units are denoted by the Latin prefixes, abbreviated in each case to a single, small letter: milli, meaning 1000th written m. centi, meaning 100th written c. deci, meaning 10th written d. Multiples of the standard units are denoted by the Greek prefixes, abbreviated in each case to a single, capital letter: deka, meaning 10 times written D hekto, meaning 100 times written H. kilo, meaning 1000 times written K. myria, meaning 10000 times written M. TABLE OF MEASURES OF LENGTH 10 millimeters (mm.) = 1 centimeter (cm.) = about .4 in. 10 centimeters = 1 decimeter (dm.) = about 4.0 in. 10 decimeters = 1 METER (m.) = about 1.1 yd. 10 meters = 1 dekameter (Dm.) = about 32.8 ft. 10 dekarneters = 1 hektometer (Hm.) = about 328 ft. 10 hektometers = 1 kilometer (Km.) = about .62 roi 10 kilometers = 1 myriameter (Mm.) = about 6.21 mi 238 RATIONAL GRAMMAR SCHOOL ARITHMETIC TABLE OF SURFACE MEASURE 100 sq. millimeters (mm 2 .) 100 sq. centimeters 100 sq. decimeters 100 sq. meters 100 sq. dekameters 100 sq. hektometers = 1 sq. centimeter (cm 2 .) = 1 sq. decimeter (dm 2 .) = 1 sg. MKTKR (in-.) = 1 sq. dekameter (Dm-'.) = 1 sq. hektometer (Hm 2 .) = 1 sq. kilometer (Km' 2 . ) SQ.INCH SQ. CENTI- METER FlGUKE 137 The cm 2 . = .155sq. in. Them 2 . = 10. 764 sq.ft. The Km 2 . == 247.114 A. 1 m 2 = 1 cen tare (ca.) TABLE OF LAND MEASURE 100 centares = 1 are (pronounced air) (a.) 100 ares = 1 hektare (Ha.) TABLE OF MEASURES OF VOLUME The standard unit of volume is the cubic meter = 35.314 cu. ft. = about 1.2 cubic yards. 1000 cubic millimeters (mm 3 .) = 1 cu. centimeter (cm 3 .) 1000 cubic centimeters = 1 cu. decimeter (dm 3 .) 1000 cubic decimeters = 1 cu. meter (m 3 . ) etc., etc. TABLE OF MEASURES OF CAPACITY The standard unit of capacity is the liter (leeter). It is equal to 1 cu. decimeter, and equivalent to .908 dry quarts. 10 milliliters (ml.) = 1 centiliter (cl.) 10 centiliters 10 deciliters 10 liters 10 dekaliters 10 hektoliters = 1 deciliter (dl. ) = 1 liter (1.) = 1 dekaliter (Dl.) = 1 hektoliter (HI) = 1 kiloliter (Kl.) TABLE OF MEASURES OF WEIGHT The standard unit of weight is the gram, which is the weight of 1 cu. centimeter of distilled water at its temperature of greatest density (39.1 F.). COMPOUND DENOMINATE NUMBERS 231) 10 milligrams (mg. ) 10 centigrams 10 decigrams 10 grams 10 dekagrams 10 hektograms 10 kilograms 10 myriagrams 10 quintals 1 centigram (eg.) 1 decigram (dg.) 1 gram (g.) 1 dekagram (Dg.) 1 hektogram (Hg.) 1 kilogram (Kg.) 1 myriagram (Mg). 1 quintal (Q.) 1 metric ton (T.) 1 centigram = .15432 grain 1 gram = 15.432 grains 1 kilogram = 2.20462 Ib. 1 metric ton = 2204.621 Ib. 160. Metric and II. S. Equivalents. The equivalents will be of assistance in changing the metric to the common system of measures. 1 m. = 39.37 in. lKm.= .6214 mi. 1m 2 . = 1.196 sq. yd. 1 Km 2 . = .3861sq.ini. 1 are = .0247 A. 1 ra 3. = 1.308 cu^ yd. 1 stere = .2759 cord 1 liter 1H1. = j 1.0567 liquid qt. " ( .9081 dry qt. __ j 2.8376 bu. ( 26.417 liq. gal. 1 g. = 15.432 grains IKg. = 2.2046 Ib. avoir. 1 metric ton = 1.1023 T. 1 mile = 1.6093 Km. 1 yard = .9144 m. 1 square yard = .8361 in 2 . 1 square mile = 2.59 Km 2 . 1 acre = 40. 47 a. 1 cubic yard 1 cord = .7645m s . = 3.624 st. 61.022cu. in. = 11. 1 liquid qt. = .9436 1. 1 dry qt. = 1.101 1. 1 bushel = .3524 HI. 1 grain = .0648 g. 1 Ib. avoir. = .4536 Kg. 1 T. = .9072 T. PROBLEMS WITH THE METRIC UNITS 1. Take a smooth lath, or plane one, lay a straight strip of paper beside the lower edge of the metric rule, Fig. 130, page 237, and with a sharp pencil transfer the graduations from the rule to the strip of paper, and then from the strip of paper to the lath. 240 RATIONAL GRAMMAR SCHOOL ARITHMETIC Continue the centimeter graduation marks along your lath until you get a meter stick (= 100 centimeters). How does this meter stick compare in length with a yard stick? 2. Explain which digit in 6.8752 m. stands for m. ; which for dm. ; which for cm., and which for millimeters. 3. Find by measurement the length and the width of your schoolroom in meters, and write the result of your measurements in meters and decimals of a meter. 4. How many centimeters long, wide, and thick is your arith- metic? 5. Express the length of your pencil in centimeters. 6. A sidewalk is 2112 m. long, how many kilometers long is it? 7. What is the cost of 158 cm. of ribbon at $.45 a meter? 8. How many steel rods, each 16 cm. long, can be cut from a rod 7.68 m. long? 9. From a piece of wire, 98.52 m. long, a piece .047 Km. was cut. How long was the remainder? 10. A bicycle track is .807 Km. long. How many meters would one go in riding around the track 5 times? 11. Express 1 Km 2 , in square dekameters; in square meters. 12. Find the number of square meters in the surface of your schoolroom. Express the same in square dekameters. 13. Express the area of your desk in square decimeters; in square inches. How do the two compare? 14. Using the decimeter as a measure, find the number of square centimeters in the surface of your geography. 15. Express the same in square decimeters; in square inches. 16. Find the area of the door in square meters. What part of an are is this area? 17. A certain plot of ground contains 20 m 2 . How many square feet in the plot? 18. With the aid of your meter stick, lay off a square meter upon the floor. Within this area, lay off a square yard. Note results. 19. With the aid of your decimeter measure, draw a square decimeter. In the corner, draw a square centimeter. How many square centimeters could be drawn within the square decimeter? COMPOUND DENOMINATE NUMBERS 241 20. What is the area of a floor 8.4 m. long and 5 m. wide? 21. A lot is 7.64 m. wide and .033 Km. deep. What is the area? 22. What is the value of a piece of land 8 Dm. long and 6. 5 Dm. wide, at $32 an are? 23. How many ares in a street 1.5 Dm. wide and 2.48 Km. long? 24. How many square meters in a floor 700 cm. long and 500 cm. wide? 25. A man had 3 pieces of land as follows: 8.4 Ha.; 3846 m 2 ., and 2.5 Km 2 . How many hektares of land are there in all? 26. 20 liters equal how many centiliters? what part of a Dl. ? of a KL? Find the difference between 12 1. and 12 quarts. 27. Using your decimeter measure as a basis, model a cubic decimeter or liter. 28. Using your square centimeter as a basis, model a cubic centimeter. What is the relation of a cubic centimeter to a gram? How many cubic centimeters in a liter? 29. How many cubic meters are there in the volume of your schoolroom? 30. What is the difference between an ordinary ton and a metric ton (tonneau)? 31. A gram is the weight of 1 cm 3 , of distilled water. What is the weight in grams of 1 1. of distilled water? 32. A liter of water weighs nearly 2| Ib. What is the weight of 6 1. of water in pounds? in grams? .33. A package of silver weighs 2.58 grams. What is its weight in grains? 34. How many pounds in 1 Q.? in 5 quintals? 35. How many 2 gr. capsules will 5 g. of quinine fill? 36. What would be the cost of the quinine at 10^ per dozen capsules? 37. What is the cost of 5500 Kg. of coal at $6.50 per ton? 38. What is the cost of 2 Q. of sugar at $.08 a kilogram? 39. If a book weighs 3.2 Dg. , how many such books will weigh 1.792 kilograms? JtVAAlUiX ALi \xtlAaLJU. All, 40. A box contains 2500 packages of quinine, each package weighing 1.25 g. How many hektograms does the contents of the box weigh? 41. A vessel, 18 cm. long, 14 cm. wide, and 22 cm. high, is filled with lead, which weighs 11.35 times as much as water. What is the weight of the lead? (1 cu. ft. of water weighs 62.5 Ib.) PEECENTAGE AND INTEREST 161. Percentage. ORAL WOKK 1. $2 equals what part of $8? 9 bu. equals what part of 12 bu.? 8 hr. equals what part of 24 hr.? $6 equals what part of $9? 2. To how many hundredths of a number are the following fractional parts of that number equal : 19 1 9 3 9 1 9 2 9 3 9 1 ? 2 9 3 ? 1V 39 79 19 39 49 % ? -4- ,T- 3- 3- * -' f* TIT- TO- TO- -0V- W h' 3. Express the following as hundredths: A; A; iV -fut i; !; iV> i%> 4; 1; I; IT- DEFINITIONS. The words per ce?i mean hundredth, or hundredths. The sign, " $," is a short way of writing jper cent, or hundredths; thus, 2$, 8$, 38i%, mean -,3, jfo, |^|, and are read, "2 per cent," 8 percent," "33 percent." The number (as, 2, 8, 33), written before the sign, "%," is called the rate per cent. It is well to recall that we have the following ways of writing such numbers as 6 per cent: (1) 6 per cent; (2) 6^ ; (3) 6 hundredths; (4) r $ ff , and (5) .06. The sign, " %," has the numerical value of ifa, or .01. 4. Eeferring to the table (7, p. 15), find what per cent of the total number of animals store food for winter. Answer other similar questions on the table. WRITTEN WORK Review 85, pp. 129-131. NOTE. Whenever you can answer a problem orally, do so. Form the habit of using your pencil only when it is necessary. 1. In a schoolroom containing 40 pupils, f of the pupils were girls. How many girls were there? PERCENTAGE AND INTEREST 243 2. In the last problem, 'what per cent of the pupils were girls? 3. In the month of September, 12 da. were cloudy. What per cent of the clays were cloudy? 4. Express the equivalents of the following fractions as per cents, or hundredths : I; i; tt;*; 4; *; A; T"*; A; H- 5. In a sample of soil weighing 37 oz., 18 oz. were sand. What per cent of the soil was sand? 0. A sample stick of timber, weighing 19 lb., contained 3 Ib. of water. What per cent of the weight of the timber was due to the water it contained? SOLUTION. (1) Annex zeros after the numerator and divide thus: 19)2.00 But .11^ = 11 iV#- Why? 7. What percent of a yard is 1 ft.? 9 in.? 15 in.? 28 in.? 35 in.? 1 inch? 8. The first number of these pairs is what per cent of the second : (1) 8 of 150? (4) 4| of GO? (7) 32.91 of 263.28? (2) 13 of 700? (5) 27J of 600? (8) J of J? (3)7Jof 12? (6) 38.45 of 769? (9) | of 12J? 9. Change the following per cents to their fractional equiva- lents (fractions in their lowest terms) : 5%; 8%; 8J%; 12J% ; 12%; 16%; 16|; 22J% ; 10. Change to their decimal equivalents the following: 7% ; 6i% ; 83i% ; 87^% ; \\% ; 2|% ; |% ; T V of 1% ; T ^ of 1%. 11. At the beginning of a school year, the lung capacity of a boy was 161 cu. in. By the middle of the year it was 7% larger. How many cubic inches had his lung capacity increased? 12. The lung capacity of a boy at the beginning of the year was 160 cu. in., and at the middle of the year it was 166.4 cu. in. What was the per cent of increase? 244 RATIONAL GRAMMAR SCHOOL ARITHMETIC 13. The standing of the several clubs in the National League during one season was determined from this table : CLUB Pittsburg GAMES WON 90 GAMES LOST 49 PER CENT WON 64.7$ Philadelphia .... Brooklyn 83 79 57 57 St. Louis 76 64 69 69 Chicago 53 86 New York Cincinnati 52 52 85 87 Find for each club what per cent of the total number of games played were won. Carry computations to the first decimal place, as indicated for Pittsburg. 14. What per cent of the farm, Fig. 7, p. 11, is the cornfield? the wheatfield? the meadow? the south oat field? the pasture? the lot occupied by the house and grounds? 15. 60% of the value of a mill is $5400; what is the value of the mill? 16. 3% of a school of 1200 children were absent; how many children were absent? 17. I pay 12% of the value of the property I occupy as rent every year. My rent is $240 a year; what is the value of the house? 18. A house was damaged by fire, and an insurance company paid the owner $840 damages, which was 40% of the original cost of the house ; what was the original cost? 162. Algebra. ORAL WORK 1. 50 equals what per cent of 100? of 200? of 150? of 500? 2. 2 equals what per cent of 4? of 8? of 6? of 20? 3. Any number equals what per cent of a number twice as large? 4 times as large? 3 times as large? 10 times as large? 4. x equals what per cent of 2x? of 4#? of 3#? of 10#? 5. 70 equals what per cent of 14? of 28a? of 21a? of 70a? 6. 13 equals what per cent of 17? of 28? of 45? of 55? PERCENTAGE AND INTEREST 245 7. 13# equals what per cent of 17#? of 28z? of 45:r? of 55x? 8. a equals what per cent of ?/? of z? of w? of p? 100- WRITTEN WORK 1. What is 2% of $200? of $375? How is 2% of any number of dollars found? 2. How is 2% of $a found? What is 2% of $a? of $? of $z? of $Gz? 3. How is 5% of any number found? What is 5% of the number a? of ? of a? of s? of 82? 4. How is 12^% of any number found? What is 12% of a? of x? of lOaj? 5. II ow is any per cent of a number found? G. How can you express a as hundredths? x, as hundredths? 9z? 12?/? 452? 7. Express these numbers as hundredths: 16; 20; a; x\ m\ IQx. 8. What is r% of 160? of 350? of ? of W of 6'? of 2wi? DEFINITIONS. The result of finding a given per cent of any amount, or number, is called the percentage. The amount, or number, on which the percentage is computed is often called the base. 9. Calling the percentage, p, the rate per cent, r, and the base, #, show by an equation how to find p, from # and r. 10. Eeplacing the symbols in your equation by the words for which they stand, translate the equation into the common lan- guage of percentage. This translation, properly made, is the fundamental principle of percentage. PRINCIPLE. The percentage equals the product of the base and rate divided ly 100, or more briefly, (i) P - & All of percentage is contained in this equation, called a formula, because it formulates a law. Multiplying both sides of the formula by 100, we have (A) 246 RATIONAL GRAMMAR SCHOOL ARITHMETIC Now divide both sides of this equation by r, and write the second number (see 74, p. 103) first, and obtain (note that br 7 \ -,T = #), (II) I = UL2!. 11. Translate (II) into words. How would yon find the base (b) if the percentage (p) and rate (r) were given? 12. Divide both sides of (A) by Z>, write the second number first, and make a rule for finding r when p and b are given. 163. Gain and Loss. 1. A merchant paid $10 for a suit of clothes. At what price must he sell the suit to gain 25 per cent? 2. A stationer pays $1.80 a dozen for blank books, and retails them at a profit of 66f %. At what price per book does he sell them? 3. A grocer buys eggs at wholesale at 100 a doz., and retails them at a profit of 30%. Supposing there is no loss due to break- age or spoiling, at what price per dozen does he sell them? 4. Allen paid $1 for a sled, and sold it at a loss of 20%. How much did he receive for the sled? 5. From a box containing 200 oranges, 80% were sold in one day. How many oranges were sold? 6. What is the percentage on $640 at 20%? at 28%? at 35%? at 6i%? at 124%? afc 87|%? 7. A farm, costing $4400, was sold at a gain of 18% ; for how much did the farm sell? 8. A farm sold for $5192, which was 18% more than was paid for it. How much was paid for it? SOLUTION. The 18% here means 18% of what was paid for the farm (the cost price). $5192 is then equal to what per cent of the cost? The statement may be written thus : 1.18 times the cost price = $5192, or, more briefly, thus: l.lSa; = $5192. Hence, x = . . Divide and l.lo find the value of x. 9. A center fielder threw a ball 90 ft., which was 12% farther than the third baseman threw it. How far did the third baseman throw it? SUGGESTION. First answer the question, 12% of what? PERCENTAGE AND INTEREST 247 10. A boy bought oranges at the rate of 5 for 3^, and sold them at the rate of 3 for 5

1899 114,000 Guam ............................ J 200 Isle of Pines ..................... 1899 882 10. The area in millions of square miles of the surface of the earth is 148.18; of this surface 108.77 is water, and 39.41 land. Find what per cent of the surface of the earth is water; land. The land surface equals what per cent of the water surface? The following table contains the actual lengths of the coast lines of the continent and also the lengths that would be needed to enclose them if they were solid, with smooth outlines, also the ratio of low land to high land : ACTUAL LENGTH, RATIO Low TO LENGTHS nr SOLID HIGH LAND North America ........ 24,040 10,380 6J : 9 South America ........ 13,600 9,030 9:2f Asia .................. 30,800 13,780 10^:13 Africa.... ............ 14,080 11,760 6J:14| Europe ............... 17,200 0,630 4|:lf Australia .......... 7,600 5,860 3^ : 1 PERCENTAGE AND INTEREST 253 11. Find what per cent of each number in column 2 equals the corresponding number in column 3. What does each per cent mean? Which continent has the longest coast to defend in com- parison with its area? 12. Express the given ratios of column 4 of low land to high land in per cent, and tell what the per cents mean? The table below contains the heights, in feet, of mountains and peaks of the world: Popocatapetl (vole. ) . . 1 7, 784 Mt Wrangell 17,500 Mt. Blanc 15,744 Mt. Shasta 14,350 Longs Peak 14,271 Pikes Peak 14,147 Mt. Etna (volcano) .. 10,875 Rocky Mts 10,000 13. The height of Pikes Peak equals what per cent of that of Mt. Everest? of Mt. Logan? of Chimborazo? of Mt. Shasta? 14. Answer other similar questions on the table. 15. What per cent of North America (16,130,269 sq. mi.) is drained through the Mississippi basin (1,250,000 sq. mi.)? through the St. Lawrence basin (360,000 sq. mi.)? the Columbia basin (290,000 sq. mi.)? the Colorado basin (230,000 sq. mi.)? 16. What per cent of the length of the Mississippi River (4,200 mi.) is the length of the St. Lawrence (2,000 mi.)? of the Columbia River (1,400 mi.)? of the Colorado River (2,000 mi.)? Yukon River (2,000 mi.)? The following table gives the lengths, in miles, and the areas of basins, in square miles, of 24 of the longest rivers of the world: Mt. Everest 29,002 Aconcagua 23,910 Chimborazo (volcano) 20,500 Kilimanjaro Mts 20,000 Mt. Logan 19,500 Karakoram Mts 18,500 Orizaba (volcano) 18.312 Mt. St. Elias 18,100 AREA OF RIVER LENGTH BASIN Mississippi 4,200 1,250,000 Nile 3,900 1,300,000 Amazon 3,600 2,500,000 Yangtzekiang . . 3,300 650,000 Obi 3,000 1,000,000 Yenisei 3,000 1,400,000 Congo 3,000 1,500,000 Niger 2,900 1,000,000 Hoangho 2,800 390,000 Amur 2,700 780,000 Lena 2,700 900,000 La Plata 2,500 1,350,000 AREA OF RIVER LENGTH BASIN Mackenzie 2,400 680,000 Volga 2,400 590,000 St. Lawrence . . 2,000 360,000 Yukon 2,000 440,000 Brahmaputra .. 2,000 426,000 Colorado 2,000 230,000 Indus 2,000 325,000 Euphrates 2,000 490,000 Danube 1,900 320,000 Rio Grande 1,800 225,000 Ganges 1,800 450,000 Orinoco 1,500 400,000 254 RATIONAL GRAMMAR SCHOOL ARITHMETIC 17. What per cent of the length of the Mississippi equals that of the Nile? What per cent of the area of the Mississippi basin equals the Nile basin? 18. Solve other similar problems on the table. The following table contains the areas, in sq. mi., the alti- tudes, in ft. (above sea level), and the greatest depths, in ft., of the great lakes of the world : LAKE AREA ALTITUDE DEPTH Superior .............. 31,200 602 1,008 Huron ............... 23,800 581 700 Michigan ............. 22,450 581 875 Erie .................. 9,950 573 212 Ontario .............. 7,242 248 738 Victoria .............. 22,167 4,000 620 Winnipeg ............. 9,400 710 72 19. The area of Lake Michigan equals what per cent of that of Lake Superior? 20. Similarly compare the altitudes and greatest depths of these lakes. 21. Solve similar problems on the table. 167. Commission. DEFINITION. Commission is a sum of money paid by a person or firm called the principal, to an agent for the transaction of business. It is usually reckoned as some per cent of the amount of money received or expended for the principal. 1. Find the commission on $1850 at 2-J-% ; at 3-j-% ; at 2. A principal sent his agent $2652 to be invested after deduct- ing the agent's commission of 2 % How much money was invested? QUERIES. Of what two amounts is $2652 the sum? What per cent of itself is the amount to be invested? What per cent is the commission of the amount to be invested? What per cent is the total, $2652, of the amount to be invested? SOLUTION. Call the unknown amount to be invested, x. Then 1.02 x = $2652. 3. 5% commission on a certain amount of money was $684.20. What was the amount? (Statement: .05^ = $684.20. Find x.) DEFINITION. A shipment of goods sent to an agent to be sold is called a consignment. PERCENTAGE AND INTEREST '255 4. A consignment of 4560 bu. of wheat was sold by an agent at 78f ^ per bushel. What was the agent's commission at 1^ per cent? 5. A commission agent sold the following consignment of goods: 20 doz. eggs at 14f^-; 40 Ib. creamery butter at 210; 36 Ib. cheese at 13J$; 80 Ib. chickens at 12-J-r/; 8 doz. live chickens at $3. 75; 4 bbl. apples at $3.25; 16 boxes oranges at $2.25; 4 boxes oranges at $2.50; 12 boxes graps fruit at $2.75; 25 bunches bananas at $1.25; 12 boxes lemons at $2.75; 16 bbl. potatoes at $4.25. Find the agent's commission at 6 per cent. 6. An agent's commission at 2-j-% on a certain collection amounted to $97.16. What was the amount of the collection? 7. A consignment of 1200 Ib. cut loaf sugar at $5.75 per hun- dredweight was sold at 12% commission. What amount of money was remitted to the principal? 8. A land agent sold the N-J- NW section 28 at $37.50 per acre. His commission was $75. What was the rate per cent of his commission? 9. An agent charged $120 for selling a $3000 piece of prop- erty. What was the rate per cent of his commission? DEFINITION. The net proceeds of a sale means the amount left after deducting the commission and other expenses. 10. The net proceeds of a sale of real estate were $19,400 and the agent's commission was 3%. How much did the real estate sell for? 11. A piece of land sold for $265. The net proceeds were $240. What was the rate per cent of commission? 168. Trade Discount. DEFINITION. A discount is a certain rate per cent of reduction from the listed prices of articles. The discount is usually allowed for cash pay- ments or for payment within a specified time. 1. A retail merchant buys silk at $1.20 a yard. He is allowed a discount of 10% for cash. He pays cash. How much does the silk cost him? SOLUTION. -1 .20 10# of 1.20 = $1.08. 250 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. A publisher sells 50 books at $1.50 and allows 20% dis- count. How much does he receive for the books? Merchants and manufacturers of ten publish expensive catalogues containing their price lists of articles, whose prices fluctuate rap- idly. When prices fall, instead of publishing new lists, they mark oft' an additional discount. For example, the price of a certain article may be catalogued thus: $60, discount, 20%, 10%, 5%. This means a 20% reduction, then a 10% reduction on the reduced price, and then a 5% reduction on the second reduced price. It would be computed thus : $60 $48 $43.20 .20 .10 .05 $12.00 $4.80 $2.1600 $60 -$12 -$48. $48 -$4.80 = $43. 20. $43.20 -$2.16 =$41.04. NOTE. Notice this is not the same as a discount of 20% -f 10% -f 5% (= 35^ ). Wherein is it different? 3. A New York merchant sells to a customer goods marked thus: Price, $3500; disc't, 10% 60 da. and 5% for cash. What must the customer pay to settle the bill by cash payment? NOTE. The customer gets the benefit of both discounts. What would the customer have paid if he had been given a single discount of 15 per cent? 4. A bill of plumber's supplies was marked thus: Price: $7.50, disc't. 20% and 7% and 5%. How much did the supplies really cost? 5. Compute the amount of money needed to settle the follow- ing bills if paid in cash or within the shortest time mentioned: (1) $45; discount 25% 60 da. and 10% 5 days. (2) $180; discount 16f % 30 da., and 10% 10 da., and 5% cash. (3) $1800; discount 30% and 10% and 7% cash. (4) $54; discount 12|%, and 8%, and 5% 30 days. (5) 46 T. coal at $5.75; discount 10% and 2% cash. (6) 50 men's suits at $18; discount 20% and 5% 10 days. (7) 4 gross tablets at 50^ per doz. ; discount 20% and 7% cash. 6. To what single rate of discount is a discount of 20% and 5% equivalent? SUGGESTION. Take a base of $100. PERCENTAGE AND INTEREST 257 169. Marking Goods. In marking his goods a merchant uses the key word "harmo- nizes." He writes the selliiuj price ahove a horizontal line, and the cost price below it, using the letters h-a-r-m-o-n-i-z-e-x in order for the digits 1-2-3-4-5-6-7-8-9-0. For example a book sells at &2.50 and costs $1.75. The mark would be = r-' . hio 1. Using the key "harmonizes," interpret the following cost marks and find the per cent of profit for each cost mark : ns /A io . . a ao . mrz t . mao (1) - ; (2) - ; (3) -,-r ; (4) - ; (5) - . v ' ma ' or ' v ' /MW ' ' rs-s- ' v ' rrn 2. Complete these marks so that the selling price may be 40% greater than the cost price: 3. Articles bearing the following selling marks are marked to sell at a profit of 33% ; fill in the cost mark, using the same key as above : (i) -; (^) (3) ; (*) ; (5) -; (6) . 4. Using the same key, mark articles costing the following prices to sell at a profit of 37 1% : (1)400; (2) 720; (3) $1.68; (4) $2,88; (5) $6.40; (6) $8.24. 5. Supply complete cost marks for articles sold at 50% profit the selling prices of which are as follows: (1) 900; (2) $1.50; (3) $2.50; (4) $3.66; (5) $7.50; (6) $8.10 6. Solve problems 4 and 5, using the key "black horse." 7. Choose a key word and with it solve problems 4 and 5. 8. A merchant wishes to mark his goods so that he may drop 10% below the marked price and still make 20% of the cost price. Using the key "harmonizes," how must he mark articles costing the following prices : (1) 500? (2) 600? (3) $1.20? (4) $2.50? (5) $6.40? 258 RATIONAL GRAMMAR SCHOOL ARITHMETIC 170. Interest. ORAL WORK Review 86, pp. 131-32. The method of 86 is known as the six per cent method. In reckoning interest the year is regarded as containing 12 mo. of 30 da. each. 1. A man is charged $6 for the use of $100 for 1 yr. What per cent of the sum borrowed ($100) equals the sum ($6) he is charged for its use? 2. A man is charged $21 for the use of $350 for 1 yr. The sum charged equals what per cent of the sum borrowed? DEFINITIONS. Interest is money charged for the use of money. It is reckoned at a certain rate per cent of the sum borrowed for each year it is borrowed. When money earns 3, 6, 7, or 10 cents on the dollar annually (each year) the rate is said to be 3%, 6%, 1%, or 10% per annum (by the year) and the rate per cent is said to be 3, 6, 7, or 10. 3. At 6% per annum, how much interest does $360 earn in 1 yr.? in 3 yr.? in 4| yr.? in 2f yr.? in t years? 4. Make a rule for computing the interest on any sum of money at 6% when the time is in years. 5. At 6% per annum, how much interest does $1 earn in 1 yr.? in 2 mo.? in 1 mo.? in 6 da.? in 12 da.? in 18 days? 6. When the time is in months, how may the interest on any sum of money at 6% per annum be computed? 7. How may the interest at 6% per annum on any sum of money be computed when the time is given in months and days? in years, months, and days? DEFINITIONS. The sum of money on which the interest is computed is called the principal. The principal plus the interest is called the amount. Since the borrower must not only pay the interest on the bor- rowed principal, but also return the principal, the debt he must discharge is the amount. 8. How long will it require any principal (say $1) to amount to twice its value ($2) or to double itself at 6% per annum? 9. Give the reasons for these statements : Any principal at 6% (1) doubles itself in 200 months; (2) earns y-J-Q- of itself in 2 mo. or 60 days. PERCENTAGE AND INTEREST 259 10. Let / denote the interest on $p at r% for t yr. and let i denote the interest on %p at 6% for t yr. Explain the meaning of the equations : *< * (f) WRITTEN WORK To find the Interest. 1. What is .07 of $450? What is the interest on $450 at 7% for 1 yr.? for 2 yr.? for 5 yr.? for 3| yr.? for 4f yr.? for t years? 2. What is .08 of $1250? What is the interest at 8% on $1240 for 1 yr.? for 3 yr ? for 5[f yr.? for x years? 3. $640 was on interest at 5% from July 1, 1896, to July 1, 1900. ^Find the interest. 4. $1800 was on interest at 7% for 3 yr. 7 mo. 21 da. Find the interest. CONVENIENT FORMS I. Interest computed first at the given rate. $1800 principal .07 rate $126.00 int. for 1 yr. 3 whole years $378.00 int. for 3 yr. of 126.00 63.00 int. for 6 mo. i of 63.00 10.50 int. for 1 mo. ' of 10.50 5.25 int. for 15 da. I of 10.50 2.10 int. for 6 da. $458.85 int. for 6 yr. 7 mo. 21 da. II. Interest computed first at 6%. .18 = int. on $1 for 3 yr. at 6% .035 = int. on $1 for 7 mo. at 6% .0035 = int. on $1 for 21 da. at 6% .2185 = int. on $1 for 3 yr. 7 mo. 21 da. at 6% 1800 1748000 2185 393.3000 = int. on 51800 for given time at 6% 65.55 = int. on 1800 for given time at \% $458.85 = int. on $1800 for given time at 1% SUGGESTIONS FOR II. (1) If the interest on $1 for 1 yr. is 6^, what is the interest for 3 years? (2) If the interest on $1 for 2 mo. is IP, what is the interest for 7 mo.? (3) If the interest on $1 for 6 da. is 1 mill, what is the interest for 21 da. ? 200 RATIONAL GRAMMAR SCHOOL ARITHMETIC 5. How much interest must I pay for the use of $600 for 1 yr. 5 mo. 24 da. at 7 per cent? 0. Find the amount of $300 for 2 yr. 4 mo. 25 da. at 4| per cent. 7. Find the interest and the amount under the following con- ditions : (1) $700 for 1 yr. 7 mo. 15 da. at 3 per cent. (2) $400 for 2 yr. 9 mo. 27 da. at 3 per cent. (3) $210 for 2 yr. 5 mo. 28 da. at 6 per cent. (4) $150 for 1 yr. 11 mo. 13 da. at 7 per cent. (5) $280 for 1 yr. G mo. 19 da. at 4| per cent. (G) $3GO for 1 yr. 4 mo. 5 da. at 4 per cent. (7) $260 for 2 yr. 3 mo. 11 da. at 3 per cent. (8) $500 for 1 yr. 11 mo. 14 da. at 7 per cent. (9) $300 for 2 yr. 7 mo. 12 da. at 5 per cent. (10) $625 for 3 yr. 9 mo. 18 da. at 6 per cent. To find the Principal. 8. What principal at 8% will furnish $16 interest in 2 years? SUGGESTION. What interest will $1 produce at 8% in 2 yr.? How many dollars will yield $16 at 8% in 2 years? 9. What principal at 8% will produce $30 interest in 2 years? 10. What principal at 6% will amount to $112 in 1 yr. 6 months? SUGGESTION. What is the amount of $1 at 6% interest for 1 yr. 6 mo.? 11. Find the principal which will yield $61.25 interest in 3 yr. 6 mo. at 7 per cent. 12. Find the principal which will amount to $972.40 in 3 yr. 2 mo. 12 da. at 4| per cent. 13. Make a rule for finding the principal when the rate, time, and interest are given. 14. Make a rule for finding the principal when the rate, time, and amount are given. PERCENTAGE AND INTEREST 261 15. Supply the correct value for the letter in each of the fol- lowing cases: RATE TIME INTEREST PRINCIPAL AMOUNT (1) 6 % 1 yr. 7 mo. 15 da. $19.50 P A (2) 8 % 3 yr. 3 mo. 18 da. $169.02 P A (3) 8% 4 yr. 7 mo. 21 da. / P .$998.50 (4) 7 % 6 yr. 8 mo. 16 da. / P $198.42 (5) 4 % 2 yr. 6 mo. 18 da. $122.50 P A 6) 5 % 1 yr. 10 da. $176.00 P A 7) 4i% 90 da. $32.50 P A To find the Rate. 16. At what rate per cent will $320 yield $34 interest in 2 yr. 9 months? SUGGESTION. How much interest will $320 yield in 2 yr. 9 mo. at \%1 At what rate per cent then will the same sum yield 834 interest in 2 yr. 9 months? 17. At what rate will $780 yield $486.60 in 5 yr. 8 months? 18. A man invested $2000 for 2 yr. 7 mo. 27 da. and received $2638 at the end of this time. What rate per cent of interest did his investment earn for him? 19. A man bought 120 A. of land at $85 and sold it 2 yr. 8 mo. later for $100 per A., after having received $900 in rents from it and having twice paid taxes on it at 75 cents per acre. What was his annual rate per cent of profit? 20. Make a rule for finding the rate when the principal, time, and interest are known. 21. Supply the correct value to the first decimal place for the letter in each of the following problems : (1) $580.00 1 yr. 5 mo. $46.50 r% (2) $1280.00 3 yr. 10 mo. $500.00 r% (3) $798.45 2 yr. 8 mo. 15 da. $258.65 r% (4) $3698.50 1 yr. 5 mo. 19 da. $568.75 r% To find the Time. 22. How long will it take $80 to earn $14 interest at 4% annually? SUGGESTION. How much interest will $80 earn in one year at 4% ? In how many years then will $80 earn $14 at the same rate? 262 RATIONAL GRAMMAR SCHOOL ARITHMETIC 23. How long will it take $125 to earn $57.50 interest at 8% per annum? 24. At 7%, how long will it take $648 to yield $69.84? NOTE. When the time results in decimals of a year the decimal may be reduced to months and days by the method of problem 65, p. 235. 25. How long will it take $750 to yield $750 interest at 8%? 26. How long will it take $10 to double itself at 6%? at 7%? 27. How long will it take $975 to amount to $1225 at 5%? SUGGESTION. What is the total interest? What is the interest on $975 at 5% for 1 year? 28. Make a rule for finding the time, T, required for a given principal, P, to amount to a given sum, A, at a given rate per cent, r? 29. Make a rule for finding how long it will take a given principal to earn a given interest at a given rate per cent. 30. Find the time, T, in years, months and days under the conditions stated in each of the folloAving problems : PRINCIPAL RATE INTEREST TIME (1) $66.00 7 % $28.60 T (2) $460.00 6 % $31.05 T (3) $750.00 5i% $147.475 T (4). $1260.00 5 % $213.15 T (5) $2460.00 4 % $321.44 T 31. Supply the value for which each letter stands in the prob- lems of the following table: PRINCIPAL RATE TIME INTEREST AMOUNT (1) $60.00 6% 3 yr. 3 mo. 8 da. I A (2) $175.00 5% 4 yr. 9 mo. 15 da. I A (3) $800.00 1% T I $926.00 (4) $475.00 8% T $142.50 A (5) $1266.00 \% , T I $1349.85 (6) P 6% 5 yr. 7 mo. 27 da. 8509.25 A (7) $1575.30 r% 1 yr. 4 mo. 18 da. / $1662.46| (8) $728.25 Sft T $209.736 A (9) $364.75 T% 2 yr. 1 mo. 15 da. $69.29 A PERCENTAGE AND INTEREST 32. Complete, to mills, the following interest table : INTEREST TABLE: PRINCIPAL $100. 263 RATE 8# 4# b% 8# 7# 1 da $.008 $.011 .014 $.017 $.019 2 da 3 da 4 da 5 da 6 da 1 mo 2 mo 3 mo Q nio 1 VI* 33. Compute by the table the interest on $758 at 7% for 4 mo. 12 da. ; for 7 mo. 8 da. ; for 1 yr. 3 mo. 10 days. 171. Algebra. 1. Compute the interest on $750 at 5% for each of the following times: (1) 1 yr. ; (2) 2 yr. ; (3) 6^ yr. ; (4) 3f yr. ; (5) 25| F. ; (6) x yr. ; (7) t years. 2. Find the interest and the amount on $1250 for each of the following : (1) 5%, 1 yr. (2) 6%,2fyr. (3) 7%, 6 yr. (4) 6%, 3yr. (5) 4%, 7fyr. (6) 3%, t yr. (8) (9) 3. Denote the interest on a certain principal, P, by /, the rate by r, and the time (in years) by , write an equation showing how to find / from P, r, and t, and translate into words the meaning of the equations. 264 RATIONAL GRAMMAR SCHOOL ARITHMETIC SOLUTION. (I) -I (1) J^PXxe. Translated into words: (1) means, "Interest equals the product of the principal, the rate divided by 100, and the time (in years). ' ; (2) and (3) mean, "Interest equals the product of principal, rate, and time, divided by 100." 4. Calling A the amount, / the interest, and P the principal, write an equation showing how to find A from / and P. 5. Write an equation to show how to find P from /, r, and t, and state in words the meaning of the equation. If we multiply both sides of the equation, I = --^r, by 100 we 1UU have the equation, 100 /= Prt. Now to show how to find P from 7, r, and t, divide both sides by rt, and write the second member on the left. We then have: 6. State in words the meaning of formula (II). 7. Show, hy proper multiplications and divisions of equation (I) (3), that the rate r may be found from P, /, and ^, by the equation, (HI) r = ^. 8. State in words the meaning of formula (III). 9. Show from (I) (3) that the time t may be found from /, P, and r by the formula, 10. State in words the meaning of formula (IV). 11. Solve by formulas I-IV the following problems: (1) P = $64, r = 8, and t = 2; find /. (2) /= $24, r = 6, and t = H; find P. (3) /= $75, P = $850, and t = 2; find r. (4) / = $120, P = $600, and r = 10; find t. (5) /= $230.13|, P = $722, and t = 3f; find r. PERCENTAGE AND INTEREST 265 172. Promissory Notes. DEFINITIONS. A promissory note is a written promise, made by one person or party, called the maker, to pay another person or party, called the payee, a specified sum of money at a stated time. The sum of money for which the note is drawn is called the face value, or the face, of the note. The date on which the note falls due is called the date of maturity, and the time to run is the time yet to elapse before the note falls due. /?J^r KANKAKEE, ILL., fane /, S903. *$mefu dm/4 "after date I promise to pay to the order of S&. ^M. 98a4e9* f Jvo tTCt&ubeiJ ^7i// and wo * f Dollars, for value received, with interest at fix per cent per annum from date. . /, S903. 2^3$. J r ewman. 1. Who is the maker of the above note? the payee? What is the face of the note? the date? the rate of interest? the date of maturity? the time to run? 2. A promissory note, unless otherwise specified in the note, draws interest on its face value at the rate mentioned in the note from the date of the note until it is paid. Compute the interest and the amount on the foregoing note if it was paid Sept. 1, 1903. 3. Find the interest on the following note, paid Feb. 10, 1903: $875. Urbana, 111., May 18, 1897. One year after date I promise to pay to James Black, or order, Eight Hundred Seventy-five and ^ Dollars, at Busey's Bank, for value received, with interest at the rate of seven per cent per annum from date. Due May 18, 1898. HENRY OSBORN. NOTE. To find the time for which interest is to be computed, pro- ceed thus : CONVENIENT FORM EXPLANATION. Date of payment, 1903 2 10 2 mo. 10 da. = 1 mo. 40 da. Date of note 1897 5 18 * mo - 40 da - 18 da - = l mo - 23 da - 1903 1 mo. = 1902 13 mo. . . Time, 5 8 22 !902 13 mo. 5 mo. = 1902 8 mo. yr. mo. da. 1902 1897 = 5 yr. 266 KATIOtfAL GRAMMAR SCHOOL ARITHMETIC 4. Find the amount of each of the following notes : PACE (1) $250 RATE 6 % DATE OF NOTE Mar. 12, 1899 DATE OF PAYMENT Jan. 1, 1902 INTEREST AM'NT (2) $635 7 % Nov. 20, 1896 July 15 1900 (3) $2400 5% Dec. 12, 1899 Jan. 8, 1903 (4) $3865 4i% Oct. 13, 1901 Sept. 7, 1903 (5) $3640 8*<& Aue. 17, 1900 Mar. 3, 1903 173. Discounting Notes. DEFINITION. Discount is. a deduction from the amount due on a note at the date of maturity. In some cases promissory notes do not draw interest. The fol- lowing is an example: John C. Cannon purchased a self -binding harvester from A. R. Crow for $120 and gave him the following note in payment: Peoria, 111., June 20, 1899. Eighteen months after date, for value received, I promise to pay to A. R. Crow, or order, One Hundred Twenty and ffo Dollars, without interest until due. Due Dec. 20, 1900. JOHN C. CANNON. 1. On Sept. 20, 1899, Crow sold the note to Adams at such a price that Adams received his purchase money and 8% interest on it until the date of maturity (Dec. 20, 1900). How much did Adams pay for the note? SUGGESTIONS. Any principal at 8% will amount to 110% of itself in 1 yr. 3 mo. Why? Hence, 120 = 110^ of what number? Or, better, l.lOa? = $120. Find the value of x. DEFINITIONS. The sum of money which, at the specified rate and in the time the note is to run before falling due, will amount to the value of the note, when due, is called the present worth of the note. The difference between the value of the note, when due, and the present worth is called the true discount. The bank discount of a note is the interest upon the value of the note when due, from the date of discount until the date of maturity. 2. If Crow had sold the above note to a banker at a discount of 8%, the banker would have computed the interest at 8% on 1120 from the date of sale (Sept. 20, 1899) until the date of maturity. How much would he have received for the note? Find the difference between the true and the bank discount of the note. PERCENTAGE AND INTEREST 267 3. C. A. Thomas bought a road wagon of J. K. Duncan, giving the following note in payment : $65. Pekin, 111., Sept. 10, 1900. Two years after date I promise to pay J. K. Duncan, or order, Sixty-five Dollars, value received, with interest at 1% per annum. C. A. THOMAS. Due Sept. 10, 1902. On March 10, 1901, Duncan sold this note to a bank at 7% discount. How much did Duncan receive for the note? NOTE. Remember, bank discount is computed on the amount of the note when due, for the time to run from date of sale. 4. Counting money worth 7%, how much did Duncan receive for the wagon? 5. A man bought a horse, giving in payment his note for 1 yr. for $85, dated Feb. 26, 1903, and drawing interest at 1% from date. Two months later the holder of the note discounted it at a bank at 6 % . What was the discount? What did the bank pay for the note? v 6. Find the bank discount and proceeds on the following notes : FACE RATE DATE OF NOTE DATE or MATURITY DATE OF SALE (1) $60 6 (2) $275 (3) $350 (4) $700 8 (5) $858 6 (6) $1260 5 (7) $1800 (8) $2450 7 (9) $3865 6 Apr. 6, 1897 Oct. 6, 1898 June 15, 1897 Aug. 7, 1899 Nov. 7, 1901 , Mar. 13, 1900 6 '% Sept. 10, 1900 Dec. 10, 1902 Jan. 15, 1901 6 % Feb. 15, 1901 Nov. 15, 1903 June 20, 1901 6 % Jan. 1, 1900 July 15, 1903 May 19, 1900 7 % Feb. 28,1896 Aug. 31, 1900 Aug. 8,1896 6 % June 19, 1897 Aug. 19, 1901 Dec. 29, 1898 6 % Nov. 18, 1899 Feb. 28, 1902 Dec. 1, 1900 fy% Dec. 20, 1901 Nov. 20, 1902 Feb. 12, 1902 5 % RATE BANK OF Dis- Disc. COUNT 7 % . (10)18600 4| Oct. 18,1900 Jan. 18,1904 Apr. 2,1901 % 174. Partial Payments. DEFINITION. When a note or bond is paid in part the fact is acknowledged by the holder by his writing the date of payment, the sum paid, and his signature on the back of the note or bond. This is called an indorsement. Partial payments are made only (1) on notes which read, "On or before, etc.," (2) by private agreement between the maker and the holder of the note. 268 RATIONAL GRAMMAR SCHOOL ARITHMETIC For calculating the balance due on a note or bond on which partial payments have been made, nearly all the states have adopted the following rule, known as "The United States Kule of Partial Payments," which has been made the legal rule by a decision of the Supreme Court of the United States: RULE. Find the amount of the principal to the time the payment or the sum of the payments equals or exceeds the interest due; subtract from this amount the payment or the sum of the payments. Treat the remainder as a new principal and proceed as before, ILLUSTRATIVE EXAMPLES 1. Find the balance due on the following note at maturity: $1250. Chicago, 111., May 21, 1900. On or before two years after date I promise to pay to the Order of P. A. Hopper Twelve Hundred Fifty and T o Dollars at the Corn Exchange National Bank, for value received, with interest at 6 per cent per annum. Due May 21, 1902. JOHN P. MILLER. The indorsements on the back of this note were as follows : Nov. 21, 1900 . . $80.00 Feb. 21, 1901 ................... $10.00 May 21, 1901 ................... $150.00 Feb. 21, 1902 ................... $500.00 PERCENTAGE AND INTEREST 269 SOLUTION BY RULE Principal on May 21, 1900 $1250.00 Interest for 6 mo. on 1 .03 Interest due Nov. 21, 1900 I 37.50 Amount Nov. 21, 1900 (date of first payment of 80) $1287.50 First payment 80.00 New principal Nov. 21, 1900 $1207.50 Interest for 3 mo. on $1 .015 Interest to Feb. 21, 1901 (date of second payment of $10) .... $ 18.11 Payment being less than interest no settlement is made. $1207.50 Interest for 6 mo. on $1 .03 Interest to May 21, 1901 (date of third payment of $150) $ 36.23 Amount May 21, 1901 $1243.73 Sum of second and third payments ($10 + $150) 160.00 New principal May 21, 1901 4 $1083.73 Interest for 9 mo. on $1 _ .045 Interest to Feb. 21, 1902 (date of fourth payment of $500) . . . $ 48.77 Amount Feb. 21, 1902 $1132.50 Fourth payment 500.00 New principal Feb. 21, 1902 $ 632.50 Interest for 3 mo., on $1 . . .015 Interest to date of maturity, May 21, 1902 $ 9.49 Balance due at maturity, May 21, 1902 $ 641.99 2. The following payments were made on a $650 note, bearing 7% interest and dated April 20, 1901: July 30, 1901 $75.00 Jan. 15, 1902 $15.00 Aug. 12, 1902 $175.00 Jan. 1, 1903 $50.00 Find the amount due April 20, 1903. 3. A note of $2800, dated Feb. 23, 1900, and bearing 7% interest, carried the following indorsements : Feb. 23, 1901 $100.00 July 16, 1901 50.00 Jan. 1, 1902 $800.00 July 15, 1902 $85.00 Nov. 28, 1902 $380.00 Find the amount due Feb. 23, 1903. APPLICATIONS TO TRANSPORTATION PROBLEMS FIGURE 138 175. Locomotive Engine. 1. The small wheels under the front of the engine are called pilot wheels, or leaders. How many leaders are there under the engine (on both sides)? The large wheels are called drivers. The smaller wheels just behind the drivers are called trailers. Answer the following questions by referring to Fig. 138: 2. How high is the center line of the boiler above the top sur- face of the track? 3. Give the following distances in feet: (1) Between the centers of the leaders; (2) Between the centers of the rear leader and of the front driver ; (3) Between the centers of the drivers ; (4) Between the centers of the rear driver and of the trailer ; (5) Between the centers of the trailer and of the front wheel of the tender; (6) Between the centers of the front two wheels of the tender; (7) Between the centers of the front leader and of the rear tender wheel ; (8) Between the centers of the front leader and of the trailer. 270 APPLICATIONS TO TRANSPORTATION PROBLEMS 271 4. Give the distance in inches (1) between the nearest points on the rims of the drivers ; (2) between the nearest points on the rims of the leaders; (3) between the nearest points on the rims of the front tender wheels. 5. How long are the radii of the leaders shown in the cut? How long are the circumferences of these wheels? 6. Compute the radii and the circumferences of the tender wheels. 7. Find the circumference of the drivers ; of the trailers. 8. When the drivers turn over 240 times a minute, how fast does the engine go? 9. How many times do the leaders turn while the drivers turn round once? 10. The weights written beneath indicate the number of pounds of the weight of the engine which is borne by the different pairs of wheels. Find the total weight of the engine. 11. The weight of the empty tender is 43,000 Ib. When the bender is loaded it carries 10 T. coal and 7000 gal. of water. A cubic foot of water weighs 62-J Ib., and contains 7-J- gal. Find the total weight of the loaded tender. 12. The engine shown in the cut drew a train of 3 sleepers, averaging 93,000 Ib. ; 5 passenger coaches, averaging 78,000 Ib. ; an express car, weighing 34,000 Ib. ; and a mail car, weighing 75,000 Ib. What was the total weight of the train, including both engine and tender? 13. The force exerted by an engine to draw a train is dif- ferent for different speeds. For a speed of 10 mi. an hour it has been found that on straight, level track an engine must exert a force of 4f Ib. for each ton of weight of the train, including the weight of both engine and tender. Find the force required to draw the train of problem 12 under these conditions. 14. For a speed of 15 mi. an hour a force of 5 Ib. per ton of train weight is needed. What force will draw the train of problem 12 at this speed? 15. Following are the forces for different speeds from 20 to 75 mi. an hour. Find the pulling (tractive) force to be exerted by RATIONAL GRAMMAR SCHOOL ARITHMETIC the engine to draw the train of problem 12 at each indicated speed : SPEED FORCE IN LB. PERT. TRACTIVE FORCE SPEED FORCE IN LB. PER T. TRACTIVE FORCE 20 6i 50 ui 25 n 55 124 30 8 60 13 35 8| 65 181 40 l 70 14} 45 10 75 15] 16. Find the difference between each number and the number next above it in the table. What do you find? y 17. In the equation F= + 3, let F stand for the force in pounds per ton and let V stand for the speed in miles per hour. Let V = 20 in the equation. Find F by dividing 20 by 6 and adding 3 to the quotient. Compare your result with the number of column 2, and in line with 20. What do you find? 18. Let V = 25. Find F and compare with the number of the table in line with 25. 19. Let V equal other numbers of columns 1 or 4 and find F for each speed. DEFINITIONS. Replacing V in this way by numbers like 20, 25, and so on is called substituting for V the numbers 20, 25, and so on. Performing the operations indicated in the equation and obtaining the number for F is called find- ing the value of F. 20. How does the equa- tion say that the force in Ib. per ton (F) needed to draw '%% J4? the train can be obtained when the speed (F) of the train is known? 21. The speeds of the table (problem 15) are plotted to scale on the horizontal line of Fig. 139, and the forces, in Ib. per ton of load, are plotted to a different scale on the vertical parallels. The points 1 to 12 are 30 25 30 35 T40 45 50 55 60 65 70 75 80 85 90 Speed (V) in mi.perhr. Scale/a"5mi.perhr. FIGURE 139 APPLICATIONS TO TRANSPORTATION PROBLEMS 273 the upper ends of the vertical lines, whose lengths represent the successive numbers of columns 2 and 5 of the table. Place the edge of a ruler along these points, or stretch a thread taut just over them. How do the points seem to lie? 22. Notice the horizontal and the vertical scales and make a drawing like that of Fig. 139 to a scale 4 times as large. y 23. Assuming that the same law, F= + 3, relating force and speed, holds also for a speed of 80 mi. per hour, substitute V- 80 in the equation and compute F, the force in pounds per tonneeded to draw the train 80 mi. per hour. Make a similar computation for F= 85 and for F=90. 24. Plot to scale on the '80, 85, and 90 lines of your enlarged drawing the computed values of F, these lines becoming 13, 14 and 15. Be careful to get each computed, F, on the proper vertical line. 25. Stretch a string, or place a ruler, along the points 1 to 15 of your drawing. Do the points added from your computed values seem to lie on the straight line through the points 1 to 12? With a ruler draw a single straight line through all the points of your drawing. 26. Mark a point on the horizontal line midway between the 35 and 40 points. What speed does this point represent? Draw a vertical from this point up to the line through the points 1 to 12. Mark the upper end of this vertical a. What force in pounds per ton does the line from a to 37| represent? 27. How could you find from the drawing the number of pounds per ton needed to draw the train 42J mi. per hour? 67 mi. per hour? 22-J- mi. per hour? 15 mi. per hour? 17 mi. per hour? 10 mi. per hour? 5 mi. per hour? 28. How could you find from the equation, F = 4- 3, the numbers of pounds per ton needed to draw the train at the speeds 18, 36, 42, 57, and 69 mi. per hour? Compute these forces and compare them with the numbers found from measurements on the drawing of Fig. 139. REMARK. The straight line through the points 1 to 12 is said to represent the equation, F= -f- 3. 274 RATIONAL GRAMMAR SCHOOL ARITHMETIC 176. Laws of Tractive Force. 1. The force (F), in pounds, needed to draw a load (L), in pounds, on a common road wagon over loose sand is given by 4F = L. Find the forces (F) needed to draw these loads: (1) L = 864 Ib. ; (2) L = 1280 Ib. ; (3) L = 2648 Ib. ; (4) L = 3268 Ib. ; (5) L = 4893 Ib. 2. Over fresh earth, the law is F= .125 L. What forces (F, in pounds) are needed to draw the following loads, in pounds, over fresh earth : (1) L = 680? (2) L = 1624? (3) L = 2160? (4) L = 3840? (5) L = 4580? 3. With common road vehicles on dry level highways the law of pulling (tractive) force (in pounds) is F = .025 L. L denotes the combined weight of the wagon and load in pounds. What forces will be needed to draw the following loads : (1) 45 bu. wheat on a 1200-lb. wagon? (2) 2 T. coal on a 2200- Ib. wagon? (3) 1-J- T. hay on a 1680-lb. wagon? (4) A traction engine weighing 8^ T.? (5) A thresher, weighing 4J T. 4. On well packed gravel roads the law is F = .052 L. To draw a certain load on gravel road a tractive force of 44.72 Ib. was exerted. What was the load L? SOLUTION. Dividing both sides of the equation by .052, we have 77T 771 -Trr^ = L, or, what is the same thing, L = -7^. We have then, by sub- .Uo/e .uo^ A A 70 stituting, L = =%- = 860. Ans. L = 860 Ib. .UO/3 5. Under the conditions expressed in problem 4, find the loads the following forces will draw: (1) ^ T =54.08lb.; (2) ^=66.56 Ib. ; (3)^=137.8 Ib. ; (4) 201.76 Ib.; (5) 234 pounds. 6. The law of tractive force, on good, straight, level railroad track is F= .0035 L\ on fair track, straight and level, it is F = .0059 L. Weights of cars are given in problems 15 and 17, p. 135, and in problem 12, p. 271. Make and solve problems based on these facts. APPLICATIONS TO TRANSPORTATION PROBLEMS 275 7. The tractive force (F) that can be exerted by a locomotive on dry track, straight and level, is about .3 of the part of the weight of the locomotive which rests on the drivers. The loads that can just be moved along by a locomotive are given by the equations of the last problem. Make and solve problems based on the following actual weights upon the drivers of certain loco- motives : (1) 80,890 Ib. ; (2) 85,850 Ib. ; (3) 86,030 Ib. ; (4) 106,875 Ib. ; (5) 112,190 lb.; (6) 131,225 Ib. ; (7) 141,320 Ib.; (8) 202,232 Ib. (See also Fig. 139). 8. Problems may also be made on the following laws for road wagons : (1) Broken stone (fair) . . F=.Q%SL; (2) Broken stone (good) . . F= .015 L\ (3) Worn macadam . . . . F = .033 L\ (4) Nicholson pavement . . F= .019 L\ (5) Asphalt pavement . . . F- .012 L\ (6) Stone pavement ... ^=.019Z; (7) Granite pavement . . . F= .008 L\ (8) Plank road F= .010 L. 9. Find the value of x in these equations : (1) .35z = 7; (2).58z = 11.6; (3) 1.28* = 3.84; (4) .092z = 3.68; (5) .019z = 133; (6) 7.51z = 302.8; (7) .175z = 10.5; (8) 1.093z = 13.116. SOLUTION of (1): x=-^-=20. .00 10. Writing law (1) of problem 8 in the form .028Z = F, find the loads (L) to 2 decimals, the following forces (F) will draw under the conditions of problem 8 (1) : (1) 50 lb.; (3) 56.28 lb.; (5) 68.58 lb. ; (2) 135 lb.; (4) 165 lb. ; (6) 110.35 lb. 11. To find the loads (L) that given forces (F) will draw, the laws of problem 8 are more conveniently writtenjthus : L = ^-g F= 35.71 F. Change other laws of problem 8 to this form. 276 RATIONAL GRAMMAR SCHOOL ARITHMETIC CONSTRUCTIVE GEOMETRY 177. Problems. PROBLEM I. Draw a perpendicular to a given line from a point upon the given line. EXPLANATION. Let AB denote the given line, and let P be a point upon AB. We wish to construct a perpendicular to AB at P. Place the pin foot on P and close the compass feet until their distance apart is less than either PA or PB. Pi is such 34. a distance. Then with Pi as /risin by measuring the lengths of its edges? 4. How many square inches in the whole surface of the flat prism of Fig. 14G? 181. Comparison of Prisms. (a) A right prism. Using the lengths of lines as shown in Fig. 147, draw a pattern on heavy paper and construct the model of a right prism as was done in Fig. 145. Provide the edges with flaps and paste up the model, leav- ing one end open, so that it may be filled with sand. How many cubic inches of sand will just fill the model? How can you find the capacity of the model by using the lengths of the edges? FIGURE 147 Development of Right Prism CONSTRUCTIVE GEOMETRY 281 (b) An oblique prism with parallelograms for bases. Similarly draw a pattern, cut, fold, and paste up a model of the oblique prism, as shown in Fig. 148, leaving an end unpasted for sand. When the model of the right prism is just full of sand, the sides not being bulged out with the sand, pour the sand into the model of the oblique prism. Does it fill the second model? What is the ratio of the capac- ities of the two models? What is the ratio of the areas of the bases (ends)? (c) A triangular prism. FIGURE 148 Development of Parallelogram Prism X Model a triangular prism like the one shown in Fig. 149. Compare the capacity of this model with that of the model of a square prism I"xl"x4". What is their ratio? Model a triangular prism such as that of Fig. 149, but use for lengths 3" instead of 1" and 7" instead of 4". How does its capacity compare with that of the prism of Fig. 147? of Fig. 148? Give the ratio in each case. Compare the ratios of the bases. 182. Volume of an Oblique Prism. The volume of a figure is the number of cubical units in the space enclosed by its bounding surfaces. 1. How many cubic inches are there in a straight pile of visiting cards 2" high, if each card is 2" x 3" ? FIGURE 149 Development of Triangular Prism FIGURE iso P Right HAT1UJNAL liKAMMAR SCHOOL ARITHMETIC 2. How many cubic inches would there be in the pile if it were 5" high? 9" high? a in. high? 3. Push the straight pile of problem 1 over as in Fig. 151. How many cubic inches of paper are there in this oblique pile? . 4. Has the height of the pile been changed in Fig. 151? has the area of the base? has the volume? 5. How can you find the number of cubic units in a right prism from the FIGURE i5i area of its base and its height? in an Oblique Pile, or Oblique Prism oblique prism? 6. Find the volumes of square prisms having edges of the following lengths : (1) 3" x 5" x 6" ; (4) 4f ' x 6' x 12' ; (7) a in. x b in. x c in. ; (2) 6" x 5" x 9" ; (5) 16' x lOf ' x 20' ; (8) x ft. x v ft. x z ft. ; (8) 15"x9"x8"; (6) 45" x 16f" x 21"; (9) -myd. x nyd. xp yd. 183. Paper-Folding, For the following exercises in paper-folding any moderately thick, glazed paper will do. Tinted or colored paper, without lines, will however show the creases more clearly. It is con- venient to have the paper cut into pieces, about 4" square. Such paper is inexpensive and may be had of any stationery dealer. PROBLEM I. At a chosen point on a line, make a perpendicular to the line, by folding paper. EXPLANATION. Fold one part of a piece of paper over upon the other and crease the paper along the fold, as at AB, by drawing the finger along the fold. Taking D to denote the chosen point, fold the paper over the point D, and bring the two parts, DA and DB, of the crease AB exactly together. Hold the paper firmly in this position and crease the paper along the line DC. Compare the portion of the paper between the creases DB and DC with the portion between the creases DA and DC. How do the angles BDC and CDA compare in size? DEFINITIONS. When two lines meet in this way making the angles at their point of meeting (inter- section) equal, the lines are said to be perpendicular to each other, and each is called a perpendicular to the other. The angles thus formed are called right angles. FIGURE 153 CONSTRUCTIVE GEOMETRY 283 PROBLEM II. Bisect an angle, by folding paper. FIGURE 153 EXPLANATION. Crease two lines, as OA and OB, lying across each other. They make the angle AOB. Now fold the paper over, and bring the crease OA down on OB. Holding the creases firmly to- gether crease the bisector OC. How do the angles AOC and BOC compare in size? Do they fit? What is the ratio of angle AOC to angle BOC? of AOB to AOC? How could the angle AOB be divided by creases into 4 equal parts? '. !,. :.':!. -''."Tig PROBLEM III. Crease three non-parallel lines. EXPLANATION. Crease AB in any position. Crease CD in any posi- tion, not parallel to AB. Finally, crease EF in any position not parallel to either AB or CD, Fig. 154. In general, in how many points do three non-parallel lines cross each other? Can you crease three lines, no two of which are parallel, in such a way as to obtain just two crossing points (intersec- tions)? Try it. Crease three non-parallel lines in such positions as to give but one intersection (see Fig. 155). Lines which go through the same point are called concurrent lines. FIGURE 154 FIGURE 155 PROBLEM IV. Crease the bisectors of the 3 angles of a triangle. EXPLANATION. Crease out a triangle such as ABC. Then crease the bisector of each angle, as in Problem II. Work carefully. 1. How do the bisectors cross each other? 2. Since the three bisectors of the three angles of a triangle all go through 0, what name would be applied to them FIGURE 156 284 RATIONAL GRAMMAR SCHOOL ARITHMETIC PROBLEM V. Bisect a given line, by paper-folding. EXPLANATION. Crease <* line and stick the point of a pin through the crease at A and at B. AB is the line to be bisected. Fold the paper over and bring the pin hole at B down on the pin hole at A. Press the paper down and crease a line across AB, as at C. CD is the perpendicular bisector of AB. Crease a line from D to B and another from D to A. When the paper is folded over the line CD, how do these two creases seem to lie? Compare the lengths of DB and of DA. 1. Crease the perpendicular bisectors of the 3 sides of a triangle and find how they cross each other. 2. What kind of lines are the perpendicular bisectors of the sides of a triangle? PROBLEM VI. Crease a square and its diagonals. EXPLANATION. Crease 2 lines, as AB and AX, perpendicular to each other. (See Problem I.) Fold the paper over the point B and when the two parts of the crease AB fit, crease the line BY. Now fold the paper over so that crease BY conies down along crease BA, and crease the line BC. Fold the paper over a line through C, bringing CX down along CA, and crease CD. ABCD is the required square. FIGURE 158 1. When the square is folded over the diagonal BC, where does D fall? Along what crease does BD lie? CD? 2. How does the diagonal divide the area of the square? 3. Fold over and crease the diagonal AD? How does it divide the square? 4. Compare OB with OC\ OD with OA. 5. How do the diagonals of a square divide each other? 6. How do the diagonals divide the area of the square? 7. How does AD divide the angle BA C? the angle JSDC? 8. Fold a second square and crease its diagonals (Fig. 159). Fold over 0, bringing D down on HO and crease HG. Similarly crease EF, CONSTRUCTIVE GEOMETRY 285 9. When the paper is folded over OF along what line does Onfall? Offl 10. How does OB compare with OD in length? How then do AD md B C compare? 11. How do EF and HG divide the square? 12. Crease the following lines : GfF, FH, HE, and EG. How does the area of the square GFHE compare with that of ABDC? FIGUR K ir/j PROBLEM VII. Crease the three perpendiculars from the vertices of a triangle to the opposite sides. EXPLANATION. Crease the triangle ABC, Fig. 160. Fold the paper over the vertex C, and bring the crease DB down along DA, and crease the per- pendicular CD. Similarly crease AE and BF. Work carefully. How do the creases CD, AE and BF cross each other? What kind of lines are the three perpen- diculars from the vertices to the opposite sides of a triangle? FIGURE 160 PROBLEM VIII. Crease the three perpen- dicular bisectors of the sides of a triangle, Fig. 161. How do they cross? What kind of lines are they? PROBLEM IX. Crease a rectangle and its diagonals. FIGURE 161 EXPLANATION. Crease AB, Fig. 162, making the distance from A to B two inches. By Problem I, crease the perpendiculars AD and BC. Make AD and BC each 1" long and crease CD. Crease the diagonals, one through A and C, and the other through B and D. Call their crossing point O. Fold the paper over the perpendicular FE, bring- ing B down on A. Where does C fall? What other line equals 2J5? OB? OC? Fold the aper over the point O so that B falls on C, and cfeas- the perpendicular HG. What FIGURE 162 ot her line equals AB1 OG1 OD1 How does the intersection, O, of the diagonals divide the diagonals? /COD KAT1UJNAL, WKAMMAK SCHOOL ABITHMETIC 184. Perimeters. The perimeter of any figure is the sum of the lines bounding the figure. Thus, in form (6), Fig. 163, if p de- note the perimeter, (i) p=x+y+ z- 1. What does x + 2x + x mean? How may it be more briefly written? 2. What is the coefficient of x in the answer to problem 1? 3. Write the perimeter, p, of (1), Fig. 1G3, in two ways. 4. Write the perimeter, p, of (13) in three ways. 5. Write the perimeter, p, of (12) in three ways. 6. Write -J of the perimeter, p, of (13) in two ways. 7. Write an expression showing that the two answers to 6 are equal. FIGURE 163 8. In (12), if x = 80 rods and y = 40 rods, how many feet in the perimeter? 9. In (1) and (2), if x and y each = 20 rods and one side of the triangle rests on the square, making a new figure, omitting the common line, what is the perimeter of this new form, in feet? How many sides has the new figure thus formed? 10. Forms (1) and (13) are combined into a single figure. Write p for the new figure in two ways, supposing x the same in both. CONSTRUCTIVE GEOMETRY 287 185. auadrilaterals. 1. Forms (2), (7), (8), (10), (11), (12), and (13), Fig. 163, are different kinds of quadrilaterals. What is a quadrilateral ? 2. What quadrilaterals have their opposite sides parallel? These figures are parallelograms. Define a parallelogram. 3. What parallelograms have all their sides equal? What is a rhombusf 4. What rhombus has all its angles equal? What is a square? 5. What quadrilaterals have their opposite sides equal but con- secutive angles not equal? Define a rhomboid. 6. What parallelograms have their angles all equal? Define a rectangle. 7. What quadrilateral has only one pair of sides parallel? De- fine a trapezoicl. 8. Is a trapezoid a parallelogram? Is it a quadrilateral? 9. Is a rectangle necessarily a quadrilateral? Is it a parallel- ogram? a square? May a rectangle be a square? 10. Is a square necessarily a quadrilateral? Is it a parallel- ogram? a rectangle? a rhombus? 186. Perimeters of Miscellaneous Figures. 1. Denote the perimeter of each of the forms in Fig. 163 by p and write an equation like (I) in 184, showing the value of p for each figure. 2. Omitting the lines on which the forms join, write an equa- tion showing the value of p when (2) and (6), Fig. 163, are joined on y, which has the same value in both. 3. In the same way join (3) and (10) on , which has the same value in each, and write an equation showing the value of p. What is the name of the figure thus formed? 4. Join (2) and (7) in which x and y are equal. If the perimeter of the figure thus formed is 240 rods, find the value of x in feet. 5. What is the length of the perimeter of a figure like (6), Fig. 163, whose sides are 2 in., a in., and 1) in. long? whose sides are x in., 8 in., and z in. long? like (10), whose sides are c ft., 2c ft., d ft., and x ft.? like (12), whose sides arc 2z rd., 4y rd., 2# rd., and 4=y rd. long? Write the perimeter (j;) of each of the figures below. y y (0) o - -- ^ i X X 1 (7) y y o _j ^ X (9} FIGURE 164 ' d d n (10) i 6 In such forms as those from '5) to (10), Fig. 164, some line, or lines, must be found by subtracting others. In (G) for example, note that the ends are each x y. The perimeter is then x + x-\- (x-y)+y +y + (x-y) = x -y -y + y + y = x- % + 2y= 4rc. All sides not lettered must be expressed without using other letters than those given on the figure. The perimeter means the sum of all the lines that bound the strip, or surface, of the figure. NOTE. One-half the difference of a line ra and a line n is written m n ^ or \ (m n). 1. In (8), a = 15', I = 12', x = 25' and v = 30'. Find the length of the perimeter of the figure. 2. Find the area of (8) enclosed by the solid lines. 3. Make and solve other similar problems. CONSTRUCTIVE GEOMETRY 289 FIGURE 165 : 187. Measuring Angles and Arcs. ORAL WORK We may measure the amount of turning of each clock hand in either of two ways : (1) By the length of the circular arc passed over by the tip of the rotating hand ; (2) By the wedge-shaped space having its point at the hand-post, A, over which the stem, A-III, A-VI, etc., of the rotating hand moves. While the clock hand turns from XII to III its tip moves over 1 quadrant of arc and its stem moves over a right angle. 1. While the hand turns from XII to VI over how many quadrants does its tip move? Over how many right angles does its stem turn? 2. Answer same questions for a turn of the hand from XII to IX ; from XII to XII again ; from XII around through XII to III. 3. Over what part of a right angle does the stem of the hand move while the hand is passing from XII to I? from XII to II? from III to IV? from VI to VIII? 4. Over what part of a quadrant does the tip of the htind pass in each case of problem 3? 5. Over what part of a right angle does the stem of the hand move while the tip moves 1 min. along the arc? In the same case over what part of a quadrant does the tip move? DEFINITIONS. Any wedge-shaped part of the face m< yed over by the stem of a clock hand as it turns around the hand-post is called an angle. The curve passed over by the tip of the hand, while the stem of the hand moves over the angle, is called the arc of the angle. ILLUSTRATIONS. The wedge-shaped spaces, having their points at A, and included between any two positions of the hand, as A-XII and A-I, A-I and A-III, A-I and A-VI, are all angles ; the space swept over by the hand, A-I, as it moves around through II, III, IV, V, VI, etc., to IX is also an angle. 6. As the hand moves from A-XII to A-VI, that is, so that the two positions of the hand are in the same straight line, the hand moves over a straight angle. A straight angle equals how many right angles? 290 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. The arc of a straight angle equals "how many quadrants? How many quadrants make a complete circle? 8. How many 5-min. spaces make the circumference of a complete circle? How many 1-min. spaces make the circumfer- ence of a complete circle? 9. What part of a right angle is passed over by the stem of the hand as its tip moves from one end to the other of a 1-min. space? 10. If lines were drawn from the hand-post to the ends of all the 1-min. spaces, the whole face of the clock would be divided up into how many small angles? The instrument in common use for meas- uring angles and arcs is the protractor. (See Fig. 166.) 11. Instead of divid- ing the right angle up FIGURE t66 by radiating lines into 15 equal parts, the protractor divides the right angle into 90 equal parts, one of which is called the angular degree. These same lines would divide the quadrant up into how many equal parts? Each of these parts is a degree of arc. 12. How many angular degrees are there in a straight angle? How many degrees of arc in the arc of a straight angle? 13. How many angular degrees are swept over by a clock hand while moving entirely around once? How many degrees of arc in the circumference of a circle? 14. How many degrees of angle are passed over by the stem of the minute hand in two hours? In the same time how many degrees of arc are passed over by the tip of the hand? 15. A sextant is J of a circumference ; how many degrees of arc are there in a sextant? 16. An octant is -J of a circumference; how many degrees of arc are there in an octant? 17. Study the protractor, Figs. 166 and 167; notice how its marks are numbered. How many degrees of arc are there between CONSTRUCTIVE GEOMETRY 291 the closest lines on the outer edge? How many degrees of angle are there between the lines which converge toward the center? 18. For smaller angles a shorter unit is -fa of a degree of angle and the smaller unit is called the minute of angle. V of the FIGURE 167 minute, called the second, is a still smaller unit. How many minutes of angle in a right angle? in a straight angle? in a com- plete revolution, or perigon? How many seconds, in each case? 19. The arcs between the sides of the minute and the second angles are the minute and the second of arc. How many minutes of arc in a quadrant, in a sextant, in an octant, in a circumfer- ence? How many seconds, in each case? A paper protractor can be purchased at any bookstore, and each pupil should supply himself with one. To measure an angle the center of the protractor is placed on the vertex A, and the line through A is placed along one side AD (see Fig. 168) . The reading of the mark below which the other FIGURE 168 side, AF, falls is the number of degrees in the angle, or in the arc of the protractor included between AD and AF. 292 RATIONAL GRAMMAR SCHOOL ARITHMETIC WRITTEN WORK 20. Draw a triangle, and, with a protractor, measure its angles. To what is the sum of all three equal? 21. With ruler draw a half a dozen triangles of different shapes, carefully measure each angle, and find the sum of the three for each triangle. 22. Measure with the protractor the number of degrees in the angle at one corner of the page of this book. 23. Draw two straight lines crossing each other as in Fig. 169. With the protractor measure and compare a and c ; b and d. / The angles a and c, or 5 and d, lying opposite each other, are called opposite angles, or vertical FIGURE 169 angles. 24. Draw two crossing lines in different positions, and in each case compare the measures of a pair of opposite angles. What do you find to be true? 25. Draw a pair of parallel lines, as a and Z, Fig. 170, and draw a third straight line c cutting across the parallels. Measure with a protractor and compare the four angles, in which 1 is written in Fig. 170. Measure and compare those in which 2 is written. What do FIGURE 170 you find to be true? 26. Draw a line perpendicular to another, see 177, and measure all the angles formed. What do you find? DEFINITIONS. The lines which include the angle are sides of the angle. The point where the sides meet is the vertex of the angle. 27. Draw a quadrilateral of any irregular shape and measure each of its 4 angles. To what is the sum equal? Try another quadrilateral and see whether you obtain the same sum. 28. Draw a given angle at a point on a gi^en line. EXPLANATION. Let the given line be ED, the given angle 35, and the given point A, Fig. 168. Place the protractor with its center at A, and with the diameter of the protractor along the line ED. Make a dot opposite the 35 mark on the protractor. With the ruler draw a straight line through A and this dot to F. The angle DAF is the required angle. 29. Draw a straight line, mark a point on the line and draw the angle 75; 95; 100; 135; 5530'. CONSTRUCTIVE GEOMETRY 293 TABLE OF UNITS OF ANGLE AND ARC MEASUREMENT 60 seconds (") = 1 minute (') 60 minutes = 1 degree () 360 degrees = 1 circumference (or perigon) 90 degrees = 1 quadrant (or right angle) TABLE OF EQUIVALENTS 1,296,000" ] 21,600' 360 4 quadrants = 1 circle 188. The Sum and Difference of Angles. EXERCISE 1. With ruler and compasses, draw AB and CD perpendicular to each other. 1. Suppose a protractor supplied with a pointer as shown in Fig. 171, the center of the protractor heing placed at the point, 0, over how many degrees of arc would the tip of the pointer turn while the stem of the hand turns from line OB to line 0(7? from OB through 00 to 0^4? 2. How many degrees are there in the angle BOC? in angle #0^4? in GOD1 in A OD? in the angle from OA around through OD to OB? from OA around to 00? from OA entirely around to OA again? 3. How many degrees of angle fill the space around a point, as 0, on one side of a straight line, as AS? on both sides? DEFINITIONS. An angle that is smaller than a right angle is called an acute angle (see Fig. 172). An angle that is larger than a right angle is called an obtuse angle. Acute Angles Obtuse Angles FIGURE 172 294 RATIONAL GRAMMAR SCHOOL ARITHMETIC FIGURE 173 EXERCISE 2. Find the sum of two angles 1. Draw two acute angles like those at the top of Fig. 173, care- fully cut them out, and place them as in the lower part of the figure. Push their vertices and nearer sides en- tirely together. Draw two lines along the other sides of the angle. Eemove the angles and have an angle like that on the left, which is the sum of the two [angles. 2. Calling the larger angle ic, and the smaller y, what denotes the last angle? 3. Draw an angle equal to the sum of an acute and an ohtuse angle. EXERCISE 3. Find the difference of two angles. 1. Draw two angles like those at the top of Fig. 174, cut them out, and place them as in the lower, left-hand part of the figure. Mark along the lower side of the upper angle and cut along the mark. The lower part of the larger angle (shown on the right) equals the difference of the two angles. FIGURE 174 FIGURE 175 FIGURE 176 2. If the larger angle equals x degrees and the smaller equals y degrees, how many degrees are there in the diiference? CONSTRUCTIVE GEOMETRY 295 FIGURE 177 DEFINITIONS. Two angles whose sum equals a right angle, or 90, are called complemental angles (Fig. 175). Two angles whose sum equals 2 right angles, or 180, are called sup- plemental angles (Fig. 176). The sum of all the angles that just cover the plane on one side of a straight line is equal . to how many right angles (Fig. 177)? To how many degrees is this sum equal? EXERCISE 4. Draw an equilateral triangle (Problem VII, p. 108). Tear off a corner and fit it over each of the other corners in turn. How do the three angles compare in size. Tear off the other corners and place them as in Fig. 178. To what is the sum of all three angles equal? If one of the angles is x degrees, how many degrees are there in the sum of all three angles? As the sum of the three angles FIGURE 178 FIGURE 179 equals both 3x degrees and also 180, we may write the equation 3x = 180. If 3x = 180, to what is x equal? How many degrees are there in one of the angles of an equilateral triangle? EXERCISE 5. Draw any scalene triangle. Tear off the corners and place them as in Fig. 179. To how many right angles is the sum of all the angles of the triangle equal? To how many degrees? 1. Letting #, y, and z denote the numbers of degrees in the respective angles, in what other way may we write the sum of all three? What equation may we then write? Tell what the equation # + ?/ + = 180 means. 2. Draw other scalene triangles, tear off the corners, place them as in the figure, and find whether x + y + z = 180 in all cases. 3. Crease a right triangle and find whether the equation, x + y + z= 180, is true for it. 4. Cut along the creases and tear off the two acute angles of a carefully creased right triangle and fit them over a carefully creased right angle? What seems to be true? If this is true what equation may you write for the sum of the two acute angles (x and y) of a right triangle? EXERCISE G. Draw a quadrilateral and cut it along a straight line from one corner to the opposite corner as in Fig. 180. Such a line as is indicated by the cut is called a diagonal. 1. The cut divides the quadrilateral into figures of what shape? 2. To what is the sum of the three angles of each part equal? 3. To what is the sum of all six FIGURE 180 angles of the two triangles equal? 4. To what is the sum of all 4 angles of the quadrilateral equal? 5. Do your answers hold true for the parallelogram of Fig. 180? 6. Do they hold good for any shape of quadrilateral you can draw? 7. To what then is the sum of the four angles, #, y, z, and w, of any quadrilateral equal? EXERCISE 7. Draw a hexagon and cut it along diagonals as shown in Fig. 181. 1. Into figures of what shape is the hexagon divided? 2. Into how many such figures do the diagonal cuts divide the hexagon? 3. To how many degrees is the sum of all the angles of all the triangles equal? 4. To how many right angles is the sum of all the angles of the hexagon equal? 5. Do all your answers hold true for the regular hexagon (sides equal and angles equal) of Fig. 181 also? 6. In each of these hexagons how many less triangles than sides of the uncut hexagons are there? FIGURE 181 CONSTRUCTIVE GEOMETRY %\)7 7. Draw an 8-sided figure and find how many triangles the diagonal cuts from any vertex would give. 8. How many less triangles than sides wero given by the quadrilaterals of Fig. 180? 9. To how many right angles is the sum of all the angles of any 15-sided figure equal? of an ^-sided figure? 10. The angles of a regular hexagon are all equal. How many degrees does each contain? EXERCISE 8. Draw a rectangle and cut it out. Cut it along a diagonal, and denote the angles by letters, as shown in Fig. 182. 1. Turn the lower right-hand triangle around and place it on the upper piece so that e may fall at , / at c, and d at I. Can you make the parts fit? 2. How does a diagonal seem to divide a rectangle? 3. Show the relative length of sides m and n by an equation ; of k and /; of the angles a and e\ c and /; b and d-, b + e and a + d. 4. From these equations point out those which show that the opposite sides of a rectangle are equal. 5. Show from your equations how the opposite angles compare in size. 6. With parallel rulers draw a parallelogram and cut it as sug- gested by Fig. 182. Answer questions 1-5 for the parallelogram. 7. Write the perimeters of the parallelogram and of the rect- angle of Figs. 180 and 182. 189. Products of Sums and Differences of Lines. x y 1. What is the area of A? of B* of C? q l A NOTE. The product of a and oc + y iswrit- _ ten a(x-\-y) and is read "a times the sum x + 9 2. Read the equation ax + ay = a (% + y) and explain its meaning from Fig. 183. 3. Draw a figure and show that z). A tviixiM C/HU 4. What is the area of each of the parts of Fig. 184? NOTE. a -f- b times x-\-y is written x + y FIGURE 184 5. From Fig. 184 show the meaning of each product in the equation (a -\-b) (x + y) - ax + ay + bx + by. Also show from Fig. 184 why the equation is true. 6. What is the area of each part of Fig. 185? 7. Write an equation showing how to mul- tiply x + y by itself, or how to square x + y. 8. Point out from Fig. 185 the meaning of (x + y) 2 = x 2 + Zxy + y 2 . 9. Point out from Fig. 186 the meaning of b (a c) = ab be. x+y FIGURE 185 I _ * - I LU EE b b FIGURE 186 a+b * a FIGURE 187 10. Draw a figure and show that b (a + d c) = ab + bd be. 11. Show from Fig. 187 that (a + b)(a-b)=a z -b\ 1.-A 12. Show from Fig. 188 that (a - b) (a - b) = a 2 - 2ab -f < a-b FIGURE 188 13. What is the areaof the rectangle ABCD (Fig. 189)? of S? 14. What is the length of the dotted part of BCt 15. What is the area of E? 16. What is the area of the cross-ruled part? 17. Write the areas of the cross-lined figures below (Fig. 190). FIGURE 189 STUDY OF THE SUN S BAYS 299 Call the difference of the two areas d in each case, and answer with an equation. (1) (2) (3) (4) FlGUBB 190 18. A lot, 50' x 175', is occupied by a house which, with its porch, covers a 40' square. A cement walk 3^' wide runs from the front line of the lot to the porch, and from the back door of the house to the rear line of the lot. The rest of the lot is covered with grass. How many square feet of grass are there? 19. A man cuts an 18" strip of grass entirely around a rectan- gular lawn, 75' x 175'; how many square feet of grass does he cut? 20. He then cuts another 18" strip around just inside of the strip mentioned in problem 19; how many square feet of grass does he cut the second time round? 190. Distribution of the Sun's Light and Heat. Fig. 191 shows a simple instrument, easily made in a manual training shop, with which the varying slant of the sun's rays and the law of their distribution may be studied. The in- strument is called a skiam- eter. DESCRIPTION OF THE IN- STRUMENT. The instru- ment consists of a smooth, straight stick, 4 in. (or 10 cm.) square by 12 in. (or 30 cm.) long (a square prism), hinged at its lower end to a baseboard, as shown. The prism may be held at any in- clination by means of a slide which works along a slot and may be clamped by a thumb-nut at the rear. An essential part of the apparatus is the brass, or paper, protractor at whose center a short plumbline is so suspended as to swing freely just in front of its FIGURE 191 300 RATIONAL GRAMMAR SCHOOL ARITHMETIC outer surface. The protractor should be tacked in such a position that the plumbline may swing exactly over the zero (0) when the prism lies in a level plane. Then as it is raised the line swings past the successive graduations of the protractor so that the reading of the position of the line on the arc of the protractor gives directly the inclination, or slant , of the prism. The plumbline may be made of a thread and a split bullet. If desired, instead of the solid prism, a prism-shaped box open at both bottom and top may be used. When exposed to the sun- shine the rectangular spot at gf would then be bright. This form of the instrument is called a helios. It is also convenient to have the baseboard marked off into inches and quarter inches (or into centimeters) along the edge, #/', of the shadowed, or illuminated, spot. This permits the direct reading of the length of the rectangular spot. The work and description which follow refer to the skiameter. The modifica- tions to adapt them to the helios are obvious. USE OF THE INSTRUMENT. On a sunny day the baseboard is placed in a level position and turned so that the prism may be > pointed directly toward the sun. The base- A board may be leveled by pouring a little water upon it and sliding thin wedges under one edge or another until the water shows no tendency to run in any direction. A smooth marble may be used instead of water. A home-made level like that shown in Fig. 192 is better still. The _ ,, mark on the cross-bar of the A, Fig. 192, should DCd.le I 1 j,j e ma( j e by setting the A on a level surface in FIGURE 192 ^he manua l training shop and marking it just behind the plumbline. The A may now be set 011 the baseboard of the skiameter and the baseboard tipped by the thin wedges until the plumbline swings freely over the mark. Make such a level from the scale drawing of Fig. 192. Now raise or lower the prism of the skiameter until the shortest possible length, gf (Fig. 191), of the rectangular spot is obtained. Clamp the thumb-nut behind the groove and read on the protractor the angle over which the plumbline has swept. This angle is the slant of the sun's rays. Measure and record also the length, #/, of the rectangular spot. The skiameter will enable us to learn two things . 1. The way the slant of the sun's rays changes during the day and from day to day during the year. 2. The law of distribution of these rays over a definite area of the earth's surface at any place. HOUB r3J-jAiN 1 \JE RAYS ijj^mjrJ-H \JE SPOT T J A U J. -U SPO 8 a.m. . . 22.7 10.37" 4" 9 a.m. . . 33.8 7.20" 4" 10 a.m.. . 42.0 5.98" 4" 11 a.m.. . 48.2 5.36" 4" 12 a.m.. . 50.2 5.20" 4" 1 p.m. . . 48.0 5.38" 4" 2 p.m.. . 41.2 6.08" 4" 3 p.m. . . 32.5 7.45" 4" 4 p.m. . . 21.0 11.16" 4" STUDY OF THE SUN'S RATS 301 191. Problems with the Skiameter. On a certain day that promised to be clear, the observations described above were made at 8 a.m., 9 a.m., and so on hourly to 4 p.m., with results as here tabulated: WIDTH OF LENGTHS TO SCALE FOB SLANT FOB SPOT 2.3" 5.2" 3.4" 3.6" 4.2" 3.0" 4.8" 2.7" 5.0" 2.6" 4.8" 2.7" 4.1" 3.0" 3.2" 3.7" 2.1" 5.6" The angles were estimated as closely as possible from the pro- tractor to tenths of a degree. The spot lengths were measured to the nearest sixteenth of an inch and these measures were reduced to hundredths of an inch. 1. Why does the width of the spot remain always the same? 2. Draw a horizontal, line and a set of 9 equally spaced parallels, all perpendicular to the horizontal line as shown in Fig. 194, p. 304. Call the first parallel on the left the 8 o'clock line, the second the 9 o'clock line, and so on, and, letting 1 in. represent 10, measure off distances from the horizontal on the parallels to represent the successive values of the slant of rays (column 2). Draw as smooth a curve as possible through the points at the upper ends of the plotted distances. This curve gives a picture of the way the slant changes from hour to hour on this day. 3. Draw a similar set of lines, or use the same set, and measure off to the scale 1 : 2 the given lengths of the spot on the corre- sponding hour lines. Draw a smooth curve through the points as before. In what respects are the two curves different? 4. Make a set of these hourly measures some clear day and draw the curves from your own measures. 5. At what hour is the slant of the sun's rays greatest? least? G. At what hour is the shadow's length the least? the greatest? 302 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. Make sets of measures on days two weeks or a month apart, and plot them, preferably on the same sets of parallels. How does the shape of the curve of slant change from fortnight to fort- night, or from month to month. Following is a set of noon (12 o'clock) measures separated by an interval of 30 days through the year for latitude 41.9 (Chicago) : DATE SLANT SHADOW LENGTH Jan. 1 26.8 8.87" Jan. 31 32.4 7.46" Mar. 2 42.4 5.93" April 1 54.1 4.94" May 1 64.7 4.42" May 31 71.6 4.21" June 30 73.3 4.17" July 30 68.7 4.29" Aug. 29 59.7 4.63" Sept. 28 48.4 5.35" Oct. 28 37.3 6.60" Nov. 27 29.1 8.22" Dec. 27 26.8 . 8.87" 8. Using convenient scales for slant and for shadow lengths plot the data of columns 2 and 3 on a set of equally-spaced parallels representing the dates of column 1. This curve shows the law of change of noon slant of the sun's rays through the entire year. Compare your curve with the curve of Fig. 194, p. 304. 9. On what date is the noon slant the greatest? the least? On what date is the shadow's length the least? the greatest? 10. Look at the curve and tell when the slant is increasing most rapidly, least rapidly. When is it decreasing most rapidly? 11. Answer similar questions for the lengths of the shadowed rectangle. 12 If the inclined prism were not present a prism of the sun's rays 4" square would be spread over the shadowed rectangle. When would this rectangle be most intensely heated and lighted by this prism of rays, when the rectangle has the least or the STUDY OF THE SUN'S RAYS 303 greatest area? Give a reason why the earth is warmer in summer than in fall, winter, or spring. 13. If the rectangle had an area of 18 sq. in. at one time and 36 sq. in. at another, at which time would it be most strongly heated? 14. Recalling that the width of the rectangle is always 4" and the lengths are as given in the third column of the last table above, find all the areas and the ratio of the area for Jan. 1 to the area for each date of the table. 15. How do the areas of rectangles, having the same width, change as the lengths change? Answer by finding the ratios of two areas, the ratios of their lengths, and comparing the ratios. 16. The strength of the sun's heat and light on a given area of the earth's surface for any date of the table is how many times as great as for Jan. 1? 17. Draw a set of parallels, or use cross-lined paper, and plot, to a convenient scale, the ratios found in problem 14. Do the rises and falls in this curve agree with the seasonal changes in temperature? A still simpler device for obtaining more accurately the slant of the sun's rays for any time is the one shown in Fig. 193. CONSTRUCTION OF APPARATUS. Cut and surface, from inch lumber, two boards of dimensions shown in the cut. Square up one edge, PN, of the square board to an accurate right angle with the adja- cent face MNPO. USE OF THE APPARATUS. By the aid of the wedges place the rectangular board horizontal (indicated by pouring a little water or placing a marble upon it), and then set the square board edgewise upon it and edgewise to the sun in such posi- tion as to make the shadow at 8 as narrow as possible. Stick a pin in the face of the vertical board at a about 2 inches from the edge PN. Stick a second pin at b so that its shadow shall fall upon that of pin a. Turn the board about so that edge OP shall be toward the sun, and make the shadow S its narrowest again. Stick a third pin at e, as the FIGURE 193 304 RATIONAL GRAMMAR SCHOOL ARITHMETIC pin I was placed formerly. Through the pin-marks draw the lines ab and c, forming the angle bac. Bisect (see Problem II, p. 186) the angle bac with the line az and measure baz with a protractor. This gives the angle between the sun and the zenith (the point on the sky directly overhead). Point to the zenith. Move the extended arm downward until it points toward the horizon (the earth and sky line). Through how many degrees of angle does your arm move? The slant of the sun's rays is the difference between baz (or caz) and 90. Why? One half of cad would be the slant directly. Why? 18. Solve problems 1-7 with this apparatus. 19. Measure and plot the noon slant from month to month during the school year. 20. With a carefully drawn curve made hourly from measures (Fig. 194) with the boards and pins, the time of apparent noon (sun noon) may be found. Draw a line as AB parallel to EF and cutting the curve at two points, A and B. With compasses bisect A B with the perpendicular ON. The highest point of the curve being the noon point, noon occurs such a fractional part of an hour before 12 as the distance JV-12 is of the distance 11-12. From a curve of your own, measure these distances and compute the clock time of apparent noon. From problem 15 it is clear that instead of comparing the areas of the rectangles we may compare their lengths. The shorter the rectangle the greater the amount of heat and light on any given surface at a place. In latitude 50 north (meaning 50 north of the equator) the / \ M ^T -^ A / 10 cvi k>^ ^ \ M \ \ CVJ \ cvi w (VI 00 o in 5 i E N F 8 9 10 II IE 1 2 3 4 FIGURE 194 STUDY OF THE SUN'S RAYS 305 lengths of the rectangles would be found to be as in this table for the dates given : DATE LENGTH HEAT DATE LENGTH HEAT Jan. 1.. 8.84" 1 heat unit July 1.. 4.18" Jan. 15.. 8.33" 1.06 July 15. . 4.21" Feb. 1.. 7.40" 1.19 Aug. 1.. 4.31" Feb. 15 .. 6.65" Aug. 15.. 4.44" Mar. 1.. 5.97" Sept. 1.. 4.69" Mar. 15 . . 5.39" Sept. 15. . 4.98" Apr. 1 .. 5.01" Oct. 1.. 5.47" Apr. 15.. 4.52" Oct. 15.. 6.00" May 1 .. 4.42" Nov. 1.. 6.82" May 15.. 4.29" Nov. 15.. 7.61" June 1 .. 4.21" Dec. 1.. 8.44" JunelS.. 4.17" Dec. 15.. 8.87" 21. Using a scale of 1 : 2, plot the lengths of the rectangles for the 24 dates given and draw a smooth curve through the points. When does the rectangle shorten most rapidly? least rapidly? 22. Suppose a given surface (say 1 sq. ft.) is heated by 1 heat unit on Jan. 1. Compute the number of heat units the same sur- face receives on Jan. 15; on Feb. 1; on the remaining dates. SOLUTION. Call the required number of heat units, JET. TT Q QJ Then, -^ = ~ ~ = 1.06 (to 2 decimals). 1 o.Ou For Feb. 1, -^ = ~ = 1.19 (tp 2 decimals). 23. Using the scale V to 1 heat unit, plot for the given dates the computed values to be filled into the columns in the above table. Do the rise and fall in the curve agree with the seasonal changes of heat? 24. How does this curve (which belongs to the latitude of Van- couver, Winnipeg, Newfoundland, Land's End, and Prague) compare with the corresponding curve for Chicago? (Use the values to be filled into the heat columns of the table of problem 20, above.) 306 RATIONAL GRAMMAR SCHOOL ARITHMETIC 192. Longitude and Time. The earth may be regarded as an immense sphere turning, like* a top, round one of its diameters as an axis from west to east once every 24 hr. This carries the surface of the earth and all objects fixed upon it (as a schoolhouse) round through 360 in 24 hours. Measuring one complete turn (rotation) of the earth in degrees, it may be said to be equivalent to 360; measuring it in time, it may be said to be equivalent to 24 hours. This gives us the following tables of equivalent measures : 360 correspond to 24 hr. ; 1 corresponds to ^ of 24 hr. = ^ hr. = 4 min. of time; 1' corresponds to F V of 4 min. = T ^ min. = 4 sec. of time ; V corresponds to ^ ff of 4 sec. = y 1 - sec. of time. 24 hr. correspond to 360 ; 1 hr. corresponds to 15; 1 min. of time corresponds to -fa of 15 = t = 15'; 1 sec. of time corresponds to fa of 15' = J' = 15". All objects seen on the sky, as the sun, may be regarded as stationary, while the turning of the earth carries us past them from the west toward the east. This makes the sun appear to rise in the east, move over, and set in the west. 1. Will the sun pass over eastern or western places earlier? Which places have later local* times, those over which the sun passes earlier, or later? Which places have earlier local times, eastern places, or western places? 2. What time is it at a place 45 west of Washington when it is 10 o'clock at Washington? At the same instant, what time is it at a place 30 east of Washington? DEFINITION. Longitude is the distance in degrees, minutes and seconds (of arc) due eastward or westward from a chosen meridian, called the prime meridian. Astronomers and navigators have agreed that the prime meridian shall be the meridian of the Royal Observatory at Green- wich, England. 3. The difference between the local times of two places is 4hr. 3 min. ; what is the difference in longitude between them? * Local sun time is obtained by setting timepieces at XII as the sun crosses the meridian. LONGITUDE AND TIME 307 4. The difference of longitude between two places is 105 45'; what is the difference of their local times? 5. It is 4: 50 (4 hr. 50 min.) p.m. at a certain place and 1 : 48 p.m. at another; which place is east of the other and what is the difference of their longitudes? 6. Two men met in Chicago with their watches keeping the correct local times of the places whence they came. On comparing their times one watch showed 10:47 p.m., and the other 3:18 p.m., when it was 9: 26 p.m. in Chicago. From which direction did each man come? 7. If the watches (problem 6) were keeping the correct local times of their places, what were the differences of longitude between the places and Chicago? 8. Explain how it happened that the announcement of Queen Victoria's death was read in the Chicago dailies at an earlier hour than that borne by the announcement itself. 9. How may it happen that a cable message sent Wednesday forenoon may be received at a remote place Tuesday? 10. The local times of two ships at sea differ by 4 hr. 18 min. 15.6 sec. ; what is the difference of their longitudes? 11. One ship is in longitude 186 40' 12" west and another is in longitude 20 16' 48" west; what is the difference of their longi- tudes? of their times? 12. The longitude of one ship is 3 28' 10" west and that of another is 18' 38" east. What is the difference of their longitudes? of their times? 13. The longitude of the Observatory of Madrid, Spain, is 3 41' 17" west and that of the Berlin Observatory is 13 23' 43" east. What is the difference of their longitudes? of their times? 14. The longitudes of the Cambridge (Eng.) and of the Paris Observatories are 5' 41.25" east and 2 21' 14.55" east respec- tively. What are the differences of their longitudes? of their times? 15. When the sun is on your meridian, that is, when it is noon at your place, at what places on the earth is it forenoon? after- noon? night? midnight? 308 RATIONAL GRAMMAR SCHOOL ARITHMETIC The longitudes from Greenwich, Eng., of places are also often given (as below) in hours, minutes and seconds of time. The plus (+) sign means that the place is west of the meridian of Greenwich, and the minus ( ), that the place is east of this meridian. p T . __ LONGITUDE FROM _ . LONGITUDE FROM GREENWICH GREENWICH H. M. S. H. M. S. Albany + 4 54 59.99 Denver +6 59 47.63 Algiers -0 12 08.55 Edinburgh +0 12 44.2 Allegheny, Penn .. + 5 20 02.93 Glasgow +0 17 10.55 Ann Arbor, Mich. + 5 34 55.19 Madison, Wis +5 67 37.93 Berkeley, Calif. . . + 8 09 02.72 Madrid -1-0 14 45.12 Berlin -0 53 34.85 Mexico +6 36 26.73 Bombay -4 51 15.74 New York.. ...... +4 55 53.64 Cambridge (Eng.) -0 00 22.75 Paris -0 09 20.97 Cambridge (Mass.) + 4 44 31.05 Philadelphia +5 00 38.51 Cape of Good Hope -1 13 54.76 St. Petersburg ...- 2 01 13.46 Chicago +5 50 26.84 Washington +5 08 15.78 16. Give from the table the difference of local times of Green- wich, England, and of each of the following places and state whether the time is earlier or later than Greenwich time : Albany ; Algiers; Ann Arbor; Berlin; Bombay; Chicago; New York; Paris; St. Petersburg. 17. What is the longitude in degrees (), minutes (') and sec- onds (") of arc of Albany? of Algiers? of Cambridge (Mass.)? of Washington? 18. Which of each of these pairs of places has the earlier local time and how much earlier is this time : Albany and Allegheny? Chicago and Paris? Ann Arbor and Berkeley? Edinburgh and Madison? Ann Arbor and Bombay? Cape of Good Hope and Bombay? Chicago and Cambridge, Mass.? Paris and Bombay? 19. Give the differences of longitude in each case of problem 18. 20. Solve other similar problems on the table. LONGITUDE AND TIME 309 193. Standard Time. For convenience of railway traffic a uniform system of time- keeping, known as Standard Time, was agreed upon in 1883 by the principal railroad companies of North America. It was decided that places within a belt of 15 extending (roughly) 7 on each side of the 75th meridian west of Greenwich should use the time of the 75th meridian. All places in similar belts extending about 7^ on each side of the 90th, of the 105th, and of the 120th meridian should take the times of those meridians respectively, and hence should have times just 1 hr., 2 hr., and 3 hr. earlier (less) than the time of the 75th meridian time belt. This system has now been generally adopted by most civilized coun- tries. In practice, however, the dividing lines of the time belts are irregular lines running through railroad terminals. The fol- lowing map shows the time belts and the names used to dis- tinguish the times of the several belts which cover continental United States. In this system the times at all places in the United States differ only by whole hours. PACIFIC TIME MOUNTAIN TIME CENTRAL TIME EASTERN TIME 120 105 90 75 FIGURE 195 1. When it is 8 o'clock a.m. (Standard Time) at Chicago, what is the time at each of the following places : New York? Pittsburg? 310 RATIONAL GRAMMAR SCHOOL ARITHMETIC St. Louis? Kansas City? Denver? Spokane? San Francisco? (See the map, Fig. 195.) 2. When it is 9 hr. 10 min. 45 sec. p.m. at Omaha, what is the time at New Orleans? Philadelphia? Buffalo? Washington? Austin (Tex.)? Boise City? Los Angeles? El Paso? Salt Lake City? (Refer to your Geography.) 3. Answer the questions of problem 2 for 2 : 15 p.m. at St. Paul ; for 3: 25 a.m. at Dodge City, Kan. 4. Answer for other places marked on the map such questions as are asked in problems 2 and 3 for Omaha, for St. Paul, and for Dodge City. The circle SqNr (Fig. 196) represents the earth and the curved lines represent the hourly meridians running from the equator, <2T, and converging toward the poles S and N. If represents F some place in the northern hemi- sphere, SqNr is the meridian of the place and BQPR may be im- agined to represent the meridian of the sky (hour circle of the sun) , on which the sun, T, is situated. The hour circle of the sun in the apparatus is held in place at A. If now the crank F is turned so as to carry q forward and downward through E to r, the sun's meridian standing stationary, the motions which cause the differences of time and the changes of day and night may be understood, by recalling that only the half of the globe which is turned to the sun is light (in day) . SGN denotes the prime meridian. 5. As the globe is standing in Fig. 196, what time is it at 0? at places on the 105th meridian? on the 60th meridian? on the 15th meridian east* of Greenwich? on the 45th east? 6. If the meridian 8GN is continued around on the other side of the globe, what will its number be? 7. Remembering that any place on the earth, as 0, turns com- pletely round through 360 in 24 hr., how long does it require c s r u * Remember that east is the direction toward which the globe is turning, that is east- ward means from E toward G through r and around to E again. LONGITUDE AND TIME 311 the space between any two adjacent meridians shown in Fig. 196 to pass under the sun's hour circle? 8. If a man should start from some place on the prime meridian (say London) on Friday noon and move westward just as fast as the globe turns eastward, what time (by the sun) would it be to him during his journey all the way round the globe? What hour and day would a Londoner call it when the traveler returned 24 hr. later? The problem raises the question, "Where should the traveler have changed his date so that his date might agree with that of his starting place when he returns?" The answer is, "It has been agreed that the date should change at the 180th meridian." When vessels cross this meridian from the east toward the west they add a day to their reckoning. If they cross at noon on Friday, Friday noon instantly becomes Saturday noon. Crossing from the west toward the east, they repeat a day. In the case mentioned, Friday would "be done over again." The 180th meridian is for this reason called the Date Line. Trace it round the earth in your Geography. MENSURATION 194. Roofing and Brick Work. 1. Give the rule for computing the area of a square from its measured sides ; a rectangle ; a parallelogram ; a triangle ; a trape- zoid. (See pp. 114-118 and 189.) DEFINITION. A square of roofing means a 10' square of roof surface or 100 sq. ft. A shingle is said to be laid 4", 4", or 5" to the weather when the lower end of each course of shingles on the roof extends 4", 4|", or 5" below the course next above it. 2. Draw two perpendicular center lines, and with the aid of the dimen- sions given in the left part of Fig. 197, complete an enlarged drawing to a con- venient scale, of the development of the roof, (shown on the right.) FIGURE 197 312 RATIONAL GRAMMAR SCHOOL ARITHMETIC 3. If 1000 shingles laid 4" to the weather cover a square of roofing, how many shingles will be needed to cover a square, if laid 5" to the weather? 4" to the weather? 3" to the weather? a*"? 4. The dimensions on the development (Fig. 197) being in feet, find the cost of the shingles, at $1.10 a bunch of 250, needed to cover the four sides of the deck roof (Fig. 197). Find the FIGURE 198 cost of enough tin to cover the 12' x 16' flat deck at 15^- a 20"x 28" sheet, the long sides of the sheets being laid parallel to the long side of the deck, and allowing 10% loss for joints and overlap at edges. 5. When shingles are laid 4" to the weather, 1000 shingles are estimated to cover a square. Find the number of shingles laid 4" to the weather, needed to cover the roof, including the gables, dimensions being as given in Fig. 198, and supposing the other side and the end to require the same number of shingles as do the side and the end shown in the figure. (The upper part MENSUKATION 313 of the figure represents the house facing toward the right. The lower part represents the front end of the house.) Find the cost of the shingles at 90^ a bunch of 250. 6. Ten years after building this house (Fig. 198) a new roof was put on it. It cost $1.25 per M. to remove the old shingles, $3 per M. to dip the new ones, and $2.75 per M. for the labor of putting on the shingles. How much did it cost to remove the old shingles and to re-roof with dipped shingles? 7. The roofs of my neigh- bor's house and porches are as shown in Fig. 199. It is covered with slates, 240 to the square (100 sq. ft.) of roof. The towers are octagonal (8-sided), the sides all being equal triangles. The dimensions are given in feet. Find the number of slates needed for the front of the roof as shown. 8. Draw the development of the tower as shown in Fig. 199. 9. If it takes 14 brick per square foot of outside surface to lay a 2 -brick wall, how many brick will be needed to lay the side wall of the build- ing shown in Fig. 200, deducting for 5 windows each 3^' x 6-^' and for 5 windows each 3f x 9f. 10. Make other similar problems from your own measurements or from dimensions obtained from an architect or builder. FIGURE 200 FIGURE 199 314 RATIONAL GRAMMAR SCHOOL ARITHMETIC 195, Land Measure. Keview 83, pp. 125 and 126. The law requires land to be marked out or surveyed in divisions of the form of squares and rectangles. In the western states the land has been surveyed in accordance with this law. To mark out the largest squares, north and south lines, called meridians, are first run 24 mi. apart and marked with corner- stones, or by trees, or other permanent objects. East and west lines, called base lines, are then run at right angles to these meridians at distances 24 mi. apart. This would divide the land up into 24-mi. squares were it not for the convergence of the meridians toward the poles of the earth. Notice this on a map in your Geography and on the map of Fig. 195. Each 24-mi. tract is then divided into 16 nearly equal squares, called townships, by running north and south, and east and west lines through the quarter points of the sides of the large tract. How long is a township? how wide? how many square miles does it contain? Certain meridians, called principal meridians, are run with great care, and these principal meridians govern the surveys of lands lying along them for considerable distances both toward the east and toward the west. The tiers, or rows, of townships running north and south along the principal meridians are called ranges. The first tier on the east is called range No. 1 east, and is written K1E; the second range is No. 2 east, written I12E, and so on. Point out on the drawing, Fig. 201, R1W; E2W; R3W; R2E; E4W. Certain base lines are run with great care and are called standard base lines. The rows of townships running east and * * * . * tf> \j oe.te.cc.ac. Stands Ul Ul (M CC IX rd Line 1 4 4 4 4 4 ^ 3 3 (*> Meridia 3 3 Z * Z | 2 Z 1 ' I Ba s 1 se Lin 1 3 1 I ' 1 1 i 1 west are numbered with refer- FiGUBE2i)i ence t thggg standard base lines. A township in the first row north of a base line is town- ship No. 1 north, and is written TIN; one in the second row south is called township No. 2 south, written T2S, and so on. Interpret the following symbols and point out on the drawing, Fig. 201, the townships indicated: T3N; T4N; T1S; T2N. MENSURATION 315 A township is identified by giving its number and range from some standard base and principal meridian. Point out these townships on the drawing, Fig. 201 : TIN, R2W; T3N, E1W; T4N, K2E; TlS, K3W. The law also requires townships to be sub- divided into smaller squares, called sections. Sections are numbered, beginning at the north- east corner and run- ning toward the west to 6, then 7 is just south of 6, and so on, as in Fig. 202. Sections are then sub- divided into quarters (see section 16), half-quarters (see sections 13 and 26), and quarter-quarters (see section 29). 1. Eef erring to Fig. 202, read and write the descriptions of the divi- sions of section 16; of section 26; of 13; of 29. Whatever deviations there may be from exactly 640 A., in the sections of any township, due to convergence of meridians or other TownshD Lin* causes > are required by law to be added to, or sub- tracted from, the north and west rows of half -sections. These tracts are then not called fractional sections, but are called lots, and are numbered in regular order as shown in Fig. 202. A section then always means exactly 640 A. Any fractional part of a section means the corresponding fractional part of 640 A. The lot, on the contrary, must always be measured FIGURE 203 before its area is known. Lot Ib Lot 15 321 A Lot i4r JLot 13 Lot IZ Lot II 318 A $ 6 5 4 3 Z 1 00 7 8 9 10 II IZ Cj"^ 18 17 i 15 14 i -\ SE 4 i j p s 19 ZO Zl ZZ Z3 Z4 >ro 30 Z8 Z7 Z5 a c\ ? f\ v J t. O ro 31 3Z 33 34 35 36 FIGURE 202 L48 L47 L 46 L 45 1 L44 L43 38 J.CH 41 URCH 79 A W. BROWN 79 A 39A J. BRADEN P 38A L49 1 o 4-IA 40 80 80 40 240 4 C f M . EVANS H. BRADEN }g L50 V j j_ 82 A 160 160 leo Sao- \ Sect on Line i m i LSI % *S 3? 8IA 80. 160 160 80 c_ 80 S.PERRY m L52 C. PAI KS t m 1 8IA 80 160 160 ?80 o <80 H.BE VENS 316 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Following is an assessment list of farm property. Fill the blanks from Fig. 203 : OWNER DESCRIPTION No. ACRES H. Peabody E SEJ Sec. 8 H. Peabody EJ NEJ Sec. 8 J. White E SEi Sec. 5 J. James SEJ NEJ Sec. 5 J. James Lot 43 .... 0. Gibson Lot 44 0. Gibson SWJ NEJ Sec. 5 .... 3. Make out an assessment list for all owners in Sec. 6. 4. Make and solve a similar problem for owners in Sec. 7. 5. Correct the mistakes in this erroneous assessment list: OWNER DESCRIPTION No. ACRES S. Perry Wi NWJ Sec. 7 81 J. Hay WJ- SWJ Sec. 6 82 J. Hay Wi SWi Sec. 7 80 H. Ochiltree SWJ NWJ Sec. 6 41 H. Ochiltree SEJ NEJ Sec. 6 40 H. Ochiltree EJ SEJ Sec. 6 80 J. Church NWi NWJ Sec. 6 38 J. Church NEi NWi Sec. 6 41 196. Volumes. DEFINITION. The volume of any figure is the number of cubical units within its bounding surfaces. 1. Give the rule for finding the volume of a square prism (called also a rectangular parallelepiped) . (See pp. 118, 119.) 2. Give a rule for finding the volume of an oblique parallele- piped (Fig. 148, p. 281) having the same base and the same altitude as a given rectangular parallelepiped. (See Figs. 150 and 151, pp. 281 and 282.) 3. How would you find the volume of a hollow beam 12 ft. long, having a cross section like (1) Fig. 75, p. 138? (2)? (3)? (5)? (6)? (4)? (9)? MENSURATION 317 4. Giv-e the volumes of the beams of problem 3, if the length of the beam in each case is I feet. 5. A model of a square prism (like Fig. 145) having a base 2 in. square and an altitude of 8 in. will contain how many cubic inches of sand? 6. The model of a right circular cylinder, made as shown in the scale drawing of Fig. 204 and pasted along the flap, DF, and around one end with a strip of paper, was filled with sand Scale 1:8 and poured into the empty model of the square G prism of problem 5. It filled the square prism a little more than f full. About how many cubic FIGURE 204 inches are there in the model of the cylinder? 7. It is shown in geometry that the volume of any right cir- cular cylinder is .7854 (= -^) times the volume of a square prism of the same altitude and having for one side of its base the diameter of the cylinder. Find the volumes of these circular cylinders : (1) Diameter 4", altitude 6" (4) Diameter 18", altitude 8" (2) " 7" 7" (5) " 6' " 10.5' (3) " 10" " 8J" (G) " 2r " a 8. How long is AB? How many square inches in the rectangle ADFG (Fig. 204)? How many square inches are there in the en- tire outside surface of the cylinder? 9. The inside diameter of the cylindrical water tank of a street sprinkler is 3.0 ft., and its length is 10 ft. ; how many liquid gallons does the tank hold? 10. The tank of the sprinkler is filled through a hose of 2-J-" inside diameter from a hydrant from which the water flows at the rate of 300 linear feet per minute. How long will it take to fill the tank? SUGGESTION. In 1 min. a cylinder of water as large as the inside of the hose and 300 ' long flows into the tank. 11. Find the weight of an iron rod " in diameter and 24' long, if iron weighs 450 Ib. per cubic foot? 318 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. How many cubic inches of air are there in the hollow tire of a bicycle wheel 28" in diameter (from center line to center line of tubes), the hollow having a diameter of If"? 13. How many cubic inches of rubber are there in the hollow tire of a 32" automobile wheel if the inside diameter of the cylin- drical tube is 3" and the outside diameter is 4 inches? 14. Fig. 205 is a scale drawing of a pattern for the paper model of a cone whose base is to be a circle 2" in diameter and whose sloping side from the apex to the base is to be 8". The arc A LIB is just as long as the circumference of the base. Compute the length of the cir- cumference of a circle whose radius is 2", and of another whose radius is 8", and find the ratio of the first to the second. FIGURE 205 15> Tfth a t par t o f the whole circumference (with center C) is the arc AHB? What part of 360 is the angle ACB? 16. Make a right angle, as HCX, and bisect it (See Problem II, p. 186). CD is the bisector. Bisect angle HOD and obtain CB. 17. With C as center and with a convenient radius, as CE, draw the indefinite arc PNME. Put the pin-foot on G and mark an arc across arc GP at N. Draw ON and prolong it. This makes angle GCN equal to angle GCM. Now with (7 as a center and with 8" as a radius, draw the arc AHB. Prolong CH, make HO = 1", and draw the lower circle. Provide the flap and paste up the model of the cone. If such a model is carefully made and the height is measured and if the model of a cylinder has the same height and the same base, the model of the cone will be found to hold ^ as much sand as does the model of the cylinder. It is proved in geometry that the volume of any cone equals ^ of the volume of a cylinder having an equal base and an equal altitude. 18. Find the volumes of these circular cones: (1) Diameter of base 3", altitude, 7" (2) " " 6" " 20" (3) " " 6.8" " 2.25' (4) " " 3.95" 10.25" MENSURATION 319 19. How many cubic inches of water are there in a funnel- shaped vessel, if the water is 4" deep and the diameter of the surface of the water is 4.5"? (See Fig. 206). 20. How many cubic inches of water will ifc take to fill the same vessel to twice the depth, or 8"? (See Fig. 206.) 21. How many cubic inches had to be poured yt * into the vessel to fill it to 8" depth if the depth of the water was 4" at the beginning? FIGURE 206 22. A mountain peak has the shape of a circular cone whose altitude is 1.25 mi. and the diameter of whose base is 2.35 miles. Find its volume in cubic miles. 23. How much water will it take to fill a conical vase 10" deep and 3.225" across at the top. 24. The conical tower of a building is 24.75' across at the base and 36.8' high. How many cubic feet of space does it occupy? 25. If a cord or waxed tape (bicycle repair tape) be wrapped around the curved outside surface of a half croquet ball, as a top is wound, until the surface is covered, and then if the same cord, or tape, be wrapped around on the flat circular base beginning at the center until the circle is covered, the length of the former cord will be found to be just twice the latter. The area of the surface of the whole ball is then how many times the area of the circular section of the ball? What radius has this circular section? It is proved in geometry that the area of the surface of a sphere equals 4 times the area of a circular section going through the center of the sphere. 26. The radius of a ball is 1|"; what is the area of the greatest circular section of the ball? What is the area of the surface of the ball? 27. Measure the circumference of a baseball and find how many square inches of leather there are in the cover of the ball. 28. How many square inches of paint will it take to cover the surface of a globe of 10" radius? of 10" diameter? 29. The average diameter of the earth is 7918 mi. ; how many square miles are there in its surface? 320 RATIONAL GRAMMAR SCHOOL ARITHMETIC 30. Calling s the surface and r the radius of a sphere, write an equation showing the relation between s and r. It is seen on p. 281 that the volume of a triangular right prism equals ^ the volume of a square prism whose base and alti- tude are equal to the base and the altitude of the square prism. 31. Find the volume of a right triangular prism whose altitude is 18" and whose base is a triangle having a base of 8" and an altitude of 5" inches. FIGURE T FIGUBE 208 ff ft ^^ pyramid be ^ fully modeled (Fig. 207) and filled with sand it will be found that just 3 times the volume of the pyramid is equal to the volume of the model of a triangular prism (see Fig. 208) of equal base and equal altitude. It is proved in geometry that the volume of any pyramid equals of the volume of a prism having an equal base and an equal altitude. Notice in Fig. 208 how a triangular prism may be completed on a triangular pyramid having the same base and the same altitude as the prism. 32. Calling V the volume, B the area of the base, and a the altitude of a triangular pyramid, write an equation, showing the way V would be computed from B and a. 33. Find the volume of a pyramid whose base contains 16 sq. in. and whose altitude is 12 inches. 34. The Great Pyramid of Egypt is 481 ft. high and its base is a 756' square. If it were solid and had smooth faces, how many cubic feet of masonry would it contain? 35. At the close of the nineteenth century the United States had 195,887 mi. of railroad. How many times would these rail- roads, if placed end to end, encircle the earth? (Use TT = 3|, and the radius of the earth = 3959 miles). 36. The ties used for these roads would contain wood enough to make a pyramid of the same shape 1395' high with a 2192' square for its base. How many cubic feet of wood were used for the railroad ties? 37. It has been computed that the materials used for the road beds for these railroads would make a solid pyramid 2470 ft. high CONSTRUCTIVE GEOMETRY 321 and having a 3870 ft. square for its base. How many cubic feet would this make? If the entire surface of a globe, or sphere, were divided up into small triangles like the one shown in Fig. 209, and the sphere were cut up by planes cutting along the curved sides of the triangles and passing through the center, 0, the volume of the sphere would be divided up approx- imately into small triangular pyramids, having their vertices at the center. The volume of each pyramid would be the product of its triangular base by of the FIGURE 209 the radius of the sphere. The sum of the a*- eas of all triangular pyramids would equal the surface of the whole sphere and the sum of all the volumes of the pyramids would equal the sur- face of the whole sphere multiplied by | of the radius of the sphere. 38. Calling Fthe volume and r the radius of a sphere, give the meaning of the formula : 39. Find the number of cubic inches in a sphere of 2" radius. 40. How many cubic inches in a croquet ball 4" in diameter? in a tennis ball 1.75" in diameter? in a baseball 2.1" in diameter? in a globe 10.15" in diameter? 41. Calling the sun, the moon and the planets all spheres with diameters in miles as in the following table, compute their cir- cumferences in miles, their surfaces in square miles and their volumes in cubic miles: Moon ........ 2,160; Mars ......... 4,230; Uranus ...... 31,900; Mercury ..... 3, 030 ; Jupiter ...... 86,500 ; Neptune ..... 34, 800 ; Venus ........ 7,700; Saturn ...... 73,000; Sun ........ 866,400. Earth ........ 7,918; 197. Constructive Geometry. PROBLEM I. To find the center of a give"n arc. EXPLANATION. Let AB, Fig. 210, be the given arc whose center is to be found. Mark any three points, as C, E, and D, on the arc. Draw the straight lines CE and ED. Bisect each of these lines as in Problem VI, pp. 106 and 107, and prolong these bisectors until they intersect as at O. O is the required center. DEFINITION. The lines CE&ud ED, each of which connects two points of the arc, are called chords of FIGURE 216" the arc. 322 RATIONAL GRAMMAR SCHOOL ARITHMETIC To solve this problem is it necessary actually to draw the chords? How could you find the center of a circle that would go through any three points not in a straight line? Mark 3 points not all in the same straight line and draw a circle through them. PROBLEM II. To bisect a given arc. X % EXPLANATION. Let AB, Fig. 211, be the given arc. ' \ With A as a center and then with B as a center and with a radius greater than the distance from A to the middle of the arc AB, draw the dotted arcs as indicated. Lay a ruler on the intersections of the two arcs and draw a short line across the arc, as at C. C is the mid-point of the arc, and arc AC = arc CB. y FIGURE 211 PROBLEM III. To circumscribe a circle around an equilateral triangle. EXPLANATION. Draw an equilateral triangle as in Problem VII, p. 108, and bisect two of its angles as shown in Fig. 212. With the intersec- tion of the bisectors as a center and with a radius equal to the distance from this intersection to any vertex, draw a circle. This circle is said to be circumscribed around the triangle. PROBLEM IV. To draw a trefoil. FIGURE 212 FIGURE 213 EXPLANATION. Draw an equilateral tri- angle and bisect one of its sides, as shown in Fig. 213. With the upper vertex as center and with a radius equal to the distance from this vertex to the middle of the bisected side draw an arc of a circle around until it touches the sides of the triangle both ways. Draw arcs around the other vertices in the same way. If desired, other circles, with slightly longer radii, may be drawn just outside of these until the arcs come together but do not cross. PROBLEM V. To construct a square and to draw a quatrefoil (a four-foil) upon it. EXPLANATION. Prolong BA through A far enough to draw a perpendicular, as AC, to AB at A. Draw this perpendicular and make AC = AB. With AB as a radius and (1) with C as a center, then (2) with B as a center, draw two intersecting arcs at D. Draw CD and BD and complete the drawing as shown in Fig. 214. FIGURE 214 CONSTRUCTIVE GEOMETRY 323 PROBLEM VI. To draw the designs of Fig. 215. EXPLANATION. Draw a square like the dotted squares'of Fig. 215. Study the two draw- ings, decide where the centers of the arcs are and complete the designs. FIGURE 215 PROBLEM VII. To draw a sixfoil. EXPLANATION. Draw a circle, like the dotted one in Fig. 216, with a radius as long as one side of the regular hexagon is to be. Draw the hexa- gon (see Problem XI, p. 110.) Bisect a side of the hexagon, and, using the vertices of the hexagon as centers, complete the sixfoil, as shown. Outside arcs may be added if desired. FIGURE 216 PROBLEM VIII. To draw a five-point star within a circle. EXPLANATION. Draw a circle and open the feet of the compasses by trial until 5 steps will just reach round the circumference. Complete the drawing as shown in Fig. 217. The strips need not be interlaced unless desired. In this case the inside lines need not be drawn. FIGURE 217 PROBLEM IX. To draw a regular pentagon (five-sided figure) on a given line as side. EXPLANATION. Let the lower side of the regular pentagon, Fig. 218, be the given side. With this side as a radius draw the dotted half -circum- ference as shown. Spread the points of the compasses by trial until 5 steps of the compasses will just reach round this half- circumference. Draw a radius of the dot- ted half-circumference to the end of the third step. Bisect this radius and also the given side with perpendiculars and prolong the perpendiculars until they cross. Using the crossing point as a center and the distance from it to either end of the given side as a radius, draw a circle. With the| given side as distance between the feet of the compasses, and with the intersection of the two circumferences as a center, mark off a point on the full circum- ference, and, with this latter point as a center, mark another point. Con- nect the points as shown, thus completing the regular pentagon. FIGURE 218 324 RATIONAL GRAMMAR SCHOOL ARITHMETIC PROBLEM X. To draw a cinquefoil (a five-foil). EXPLANATION. Draw a regular pentagon, bisect one of its sides, and complete the drawing, as shown in Fig. 219. FIGURE 219 PROBLEM XI. To draw the design shown in Fig. 220. EXPLANATION. First draw the three circles touching each other two and two as in the left part of Fig. 220. Noticing the thin center lines in the design draw the part of the figure on the right. FIGURE 220 PROBLEM XII. To draw the design of Fig 221. EXPLANATION. First, draw the large circle of the part of the figure on the left. Then draw two perpendicular diameters and at the end Tmake AT perpendicular to OT and as long as OT. Draw OA and bisect the angle OAT thus locating C. With O as a center and OC as a radius mark the centers of the other three circles. Draw the four small circles. Now notice the fine center lines of the part of the figure on the right and draw it. FIGURE 221 PROBLEM XIII. To draw the designs of Figs. 222 and 223. EXPLANATION. Study the thin center lines and make the sides of the dotted square half as long as the sides of the large square. In both figures be careful to have the lines just touch but not cross. FIGURE 222 FIGURE CONSTRUCTIVE GEOMETRY 325 PROBLEM XIV. To draw the designs for moldings of Figs. 224 and 225. EXPLANATION. Make enlarged drawings of the patterns, being careful to draw the proper lines to locate the centers of all the circular arcs. The last pattern is a drawing of an elliptical molding. The figure will explain how to find the points through which the elliptical curve is to be drawn free-hand. FIGURE 224 FIGURE 225 1. Torus. 2. Scotia. 3. Ovolo, or quarter round. 4. Cavetto. 5. Cyma recta. 6. Cyma recta. 7. Echinus, or ovolo. PROBLEM XV. To construct a right triangle. The symbol i means "perpendicular," "perpendicular to," or "is perpendicular to." CONSTRUCTION. Draw the line CD LAB, Fig. 226. Then draw the line OF connecting any point, as 67, of the line CD with any point of AB, as F. DEFINITIONS. The longest side, that is, the side opposite the right angle of a right triangle, is called the hypothenuse. What side of the triangle, GOF, is the hypothenuse? What angle is the right angle? PROBLEM XVI. To construct a right triangle having a given hypothenuse. CONSTRUCTION. Let the line AB, Fig. 227, denote the given hypothenuse. Bisect AB as at O and with O as a center and OA as a radius, draw a semicircle. Connect any point of the \ / semi -circumference, as E, D, or C, with A and. with B. Any such triangle is a right triangle FIGURE 227 and the line AB is its hypothenuse. RATIONAL GRAMMAR SCHOOL ARITHMETIC 237. Point out the right angle of each of the three triangles of Fig. FIGURE 228 PROBLEM XVII. To construct a right triangle, having given the two sides which include the right angle. CONSTRUCTION. Let the two given sides be a and b, Fig. 228. Draw BC i DE, and make OA = a and OB = b. Connect A with B. BOA is the re- quired triangle. How may an isosceles right triangle be constructed? PROBLEM XVIII. To find the relation of the squares of the sides of an isosceles right triangle. CONSTRUCTION. Construct an isosceles right trian- gle and on each of its three sides draw a square. Draw the dotted lines and cut the side squares as shown in ig. ^9. Fit the pieces over the large square on the hypothenuse. If the area of each side square were 9 sq. m. what would be the area of the large square? FIGURE 229 PROBLEM XIX. To find the relation of the squares of the three sides of any right triangle. FIGURE 230 CONSTRUCTION. Construct any right triangle and then construct a square on each of its three sides. Cut the side squares as shown. Fit the pieces over the square drawn on the hypothenuse, as indicated by the dotted lines in Fig. 230. What single square has an area that equals the sum of the areas of the squares on the two shorter sides of any right triangle? SQUARES AND SQUARE ROOTS 327 1. Denoting the length of either short side of Fig. 229 by a, what denotes the area of the square drawn upon this side? 2. Denoting the length of the hypothenuse of Fig 229 by &, what denotes the area of the square drawn on the hypothenuse? 3. Write an equation from Fig. 229, showing the relation between a? and A 2 . 4. Denote the lengths of the three sides of any right triangle (Fig. 230) by a, ft, and li (li being the hypothenuse). What will denote the areas of each of the squares on the three sides? 5. Write an equation showing the relation of the squares of the sides of any right triangle. PROBLEMS 1. The sides of a right triangle are 3" and 4"; what is the length of the hypothenuse? SUGGESTION. 7i 2 = 3 2 -f 4 2 = 9 + 16 = 25 ; what is the value of h? 2. The hypothenuse of a right triangle is 10", and one of the sides is 6"; what is the other side? SUGGESTION. 10 2 = a 2 -f- 6- ; find the value of a? 3. The sides of a right triangle are denoted by , ft, and the hypothenuse by h find the unknown side in each of the following right triangles : (1) a = 9, ft = 12; (4) a = 32, ft = 18; (2) a = 12, li = 20; (5) ft = 21, li = 35; (3) ft = 15, U = 25; (6) ft = 27, li = 45. Before the hypothenuse of a right triangle, whose sides are 34 and 26, can be computed, it is necessary to know how to find the square roots of given numbers. 198. Squares and Square Roots. DEFINITION. The product obtained by using any number twice as a factor is called the square of that number Thus, 36 is the square of 6, 'because 6, used twice as a factor gives 36 (6x6 = 36). The square of a number as 6 is often written thus, 6 2 . What does the small 2 show? RATIONAL GRAMMAR SCHOOL ARITHMETIC The following squares should be committed to memory : SQUARES OF UNITS SQUARES OF TENS SQUARES OF HUNDREDS I 2 = 1; 10 2 = 100; 100* =-10000; 2 2 = 4; 20 2 = 400; 200 2 = 40000; 3 2 = 9; 30 2 = 900; 300 2 = 90000; 4 2 = 16; 40 2 = 1600; 400 2 = 160000; 5 2 = 25; 50 2 = 2500; 500 2 - 250000; 6 2 = 36; 60 2 = 3600; 600 2 = 360000; 7 2 = 49; 70 2 = 4900; 700 2 = 490000; 8 2 = 64; 80 2 =6400; 800 3 - 640000; 9 2 = 81 ; 90 2 = 8100; 900 2 = 810000. 1. Write all the pairs of numbers which, multiplied together, give the product 36; the product 16; the product 64; the product 49. NOTE. Write the pairs of factors of 36 thus: 1 and 36; 2 and 18 ; 3 and 12; 4 and 9; 6 and 6. Proceed similarly with the rest. DEFINITION. The square root of a number is one of the two equal factors of it. The sign of square root is >/, called the radical sign. Thus, \/25 means the square root of 25, which is 5. 2. Give the square roots of the following numbers: 9; 16; 49; 64; 81; 400; 2500; 3600; 160000; 490000; 810000. To find the square root of a number not in the table above, it is necessary first to learn how the square of a number is formed form the number. From the table answer the following questions : 8. How many digits are there in the square of any number of units? of tens? of hundreds? 4. Find the square of each of the folio wing decimals: .1, .3, .5, .7, .9, .01, .03, .04, .06, .07, .09, .001, .003, .005, .006, .007, .009. 5. How many decimal places are there in the square of any number of tenths? of hundredths? of thousandths? 6. If, then, the square of any number of units contains only units and tens, what places of any number that is a square must contain the square of the units of its square root? 7. What places must contain the square of the tens? of the hundreds? of the tenths? of the hundredths? of the thousandths? SQUARES AND SQUARE ROOTS 329 8. If, then, we separate a number, whose square root is desired, into two-digit groups, beginning at the decimal point and proceed- ing both toward the left and toward the right, what one of these groups must contain the square of the units of the square root? the square of the tens? of the hundreds? of the tenths? of the hundred ths? 9 Find the square of 46. 40 COMMON METHOD MEANING OF COMMON METHOD 46 = 40+6; 46 2 = (40 + 6) 2 = (40 + 6) (40 + 6) = 40X40 + 40X6+6X40 + 6X6 = 40 2 +2 X (40 X 6) + 6 2 = 1600 + 480+36 = 2116. Show the meaning of the parts of this sum in Fig. 231. Thus it is seen that the square _ ire number is the sum of (1) the square of the tens, (2) twice the product of the tens by the units, and (3) the square of the units. This shows that if any two-figure number be denoted by t + u, where t denotes the tens and u the units, the square is formed thus : , L ml t + u * R s 46 46 276 184 o S T 2116 46 2 = 2116. 40 6 of a two-fi| xniia.r nf t, FIGURE ssi 40+6 40 + 6 240 + 36 1600 + 240 1600 + 480 + 36 p + 2tu +u* = (t + uf <- R sl: S T t II FIGURE 232 Show the meaning of the equation by Fig. 233. 10. Find the square root of 2116. CONVENIENT FORM 46 = required root ; 2116 denoted by t 2 + 2tu + u 2 ; 1600 = greatest square (of the table) in 2116; 2t = 80 I 516 contains 2tu + u 2 , where t = 40 ; 2t + u = 86 | 516 denoted by 2tu + w 2 , where u = 6 ; Check: 46 X 46 = 2116. SHORTENED FORM 21 '16 I 46 = required root 80 16 Check: 46 X 46 = 2116. 516 ,516 330 RATIONAL GRAMMAR SCHOOL ARITHMETIC 11 Find the square roots of the following numbers : (1)484; (3)1156; (5)1296; (7)7569; (9)9604; (2) 729; (4)1225; (6)5625; (8) 9409; (10) 110224. 12. Find the square root of 2079.36. 20'79.36'|45.6 16'00 HO 90. 4'79. 4'25. 54.36 54.36 v/2079.36 = 45.6 Check: 45.6X45.6 = 2079.36. CONVENIEI^T FORM EXPLANATION. Separate the number into two-digit groups. Beginning on the extreme left, subtract the greatest square of tens in 20 hundreds, viz., 16 hundreds (=40 2 ), and write 4 tens on the right as the first root digit. Double the 4 tens and use the result 80 as a trial divisor. 80 is contained in the remainder, 479, 5 times. Write 5 as the 2d root digit and also add it to the 80, giving 85 as the complete divisor. Double the part of the root found, 45, giving 90 for the next trial divisor and complete the steps as before. 13. Find the square root of 1578 to 3 decimal places. CONVENIENT FORM 60 78.0 .7 78.7 79.40 .02 79.42 79.440 .004 79.444 15'78.00'00'00'139.724-f- 9 1678. 1621. 57.00 55.09 1.9100 1.5884 .321600 .317776 .003824 remainder. EXPLANATION. Annex zeros and proceed as before. Check: 39.724 X 39.724=1577.996176. rem. = .003824. 1578. In actual practice the decimal point is needed only in the root. 14. Find the square roots of the following numbers : (1) 1900.96; (2) 4719.69; (3) 5055.21; (4) 61.1524; (5) 75.8641; (6) 79.9236; (7) .5476; (8) .458329; (9) 1.216609. SQUARES AND SQUARE ROOTS 331 15. Find to 3 decimal places the square roots of the following numbers : (1) 5; (4) .85; (7) 1683; (10) 1.85; (2) 7; (5) .25; (8) 6875; (11) 26.79; (3) 15; (6) 125; (9) 7328; (12) 64.893. 16. Find by multiplication the values of the following expres- sions: (1) (i) 2 ; (3) (I) 2 ; (5) ( T \) 2 ; (7) (|-) 2 ; (2) (f) 2 ; (4) (if) 2 ; (G) (B) 2 ; (8) (f) 2 . 17. Make a rule for finding the square root of a common fraction. 18. Find, without reducing the common fractions to decimals, the values of the following expressions and prove them by multi- plication : (i) ,/i; (*) (2) y/f; (5) (3) ,/; ( 6 ) 19. The sides of a right triangle are respectively 34" and 26" long, how long is the hypothenuse? 20. The center pole of a circus tent is 35' high, and a guy rope is stretched from the top of the pole to a stake 56' from the bottom. How long is the rope, supposing the ground level and the rope straight, allowing 4' for tying? 21. 50' of the top of a tree standing on level ground is broken by the wind and remains fastened to the stump. If the top strikes the ground 30' from the stump, how high was the tree? 22. A horse is staked out by a rope 40' long to the top of a stake 15" high. Over what area can the horse graze? 23. The vertical mast of a hoisting derrick 35' high is held in position by four guy ropes staked to the ground at the four corner points of a square. The stakes are 75' from the bottom of the mast. How much will the rope for the 4 guys cost at 2.5^ a foot, 20' being allowed for knots? 24. What is the area of the square whose corners are at the stakes? 332 RATIONAL GRAMMAR SCHOOL ARITHMETIC 199. Square Root of Numbers and Products Geometrically. 1. Find the square root of 2 geometrically. CONSTRUCTION. To any convenient scale draw a line 2-f- 1 (=3) units long. Bisect the line and draw a semicircle upon it as a diameter (Fig. 283). At the point of division between the second and third unit, draw a perpendicu- lar to the diameter. The part of this perpendicular between the diameter and the circumference is FIGURE 233 v/a (= 1.414 _j_) un it s long to the scale used. 2. Draw the square root of 7. SUGGESTION. Proceed as before, taking a diameter 7 + 1 (= 8) units long and drawing the perpendicular at the point of division at the end of the 7th unit. 3. Draw the square root of each of the following numbers : 5; 11; 13; 17; 19. 4. Find the square root of 4 x 3, or 12, geometrically. CONSTRUCTION. Draw, to a convenient scale, a line AB (Fig. 234) 7 units long. Draw a semicircle on AB as a diameter, and at C, the end of AC ( = 4), draw a perpendicular CD. The part of the perpen- dicular between the circumference and the diameter represents the square root to the scale used. FIGURE 234 5. Find the square roots of these products geometrically: (1) 3 x 2, or 6; (4) 2 x 4; (7) 28; (10) 18; (2)3x5, or 15; (5)7x3; (8)32; (11)26; (3) 4 x 5, or 20; (6) 6 x 5; (9) 40; (12) 27. 200. Cubes and Cube Roots. 1. What is the volume of a cube whose edge is 5"? 6"? 7"? 11"? 18"? 21"? 24"? DEFINITION. The cube of a number is the product obtained by using the number 3 times as a factor. Thus the cube of 4 is 4 X 4 X 4 = 4 3 = 64- Table of cubes of numbers to be learned : CUBES OF UNITS CUBES or TENS I 3 = 1; 10 s = 1000; 2 3 = 8; 20 3 = 8000; 3 3 = 27; 30 3 = 27000; 4 3 = 64; 40 3 = 64000; 5 s = 125; 50 3 = 125000; 6 3 = 216; 60 3 = 216000; 7 s = 343; W = 343000; 8 3 = 512; 80 3 = 512000; 9 3 = 729; 90 3 = 729000. CUBES AND CUBE ROOTS 333 DEFINITION. The cube root of a number is one of its 3 equal factors. Cube root is indicated by the sign i/ 7 . Thus, t/ 7 729 means one of the three equal factors of 729, which is 9. The sign i/ 7 7s called a radical sign. 2. Between what two whole numbers are the following cube roots : iXIC? i/ x 35? iX?8? i/450? ^075? /895? v 7 58000? i/ 480000? 3. Find by multiplication the values of the following cubes : 4. Make a rule for finding the cube of any common fraction, o. Prove the following relations by multiplication: K / 2107 = 13 ; 5/4096 16; PHtt-5 G. Make a rule for finding the cube root of any common fraction. 7. Find the cube roots of the following: -1 27.. 12.6 . 7_2_9_ _3_4_3_ UL9__ J 64J 41? > 8000J 1000J TS 5 * 201. Triangles Haying the Same Shape (Similar Triangles) . 1. Write the numerical values of the follow- ing ratios from Fig. 235 : ^ ~T6' ; ^ ' '** FIGURE 235 2. In Fig. 236 the triangles have the same shape. Write the values of the ratios : 3. In Fig. 237 the triangles have the same shape, and A C = 7 x ac. Find the sides of the triangle ABC, if ac= t V'; * = i", and Jc-iJ"- FIGURE 236 FIGURE 237 334 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. Supposing that ac ( = } AC) represents 7' (Fig. 237), ab, 10' and be, 11', what lengths do AC, AB, and BC represent? 5. In Figs. 238 and 239, AB = 4J; if A B represents 1 mi., BC, 3 mi., and AC, 3| mi., what distances do ab, be, and ac represent? b b i^ ^-^. c y- ,--: *-=-^ AL A/ ^-^ B p FIGURE 238 FIGURE 239 6. Draw a triangle having sides of 1", }", and J", and another having sides of 3", 2J", and 1|". Call the angles opposite the sides 1", |", and |", a, #, and c, respectively, and the angles opposite the sides 3", 2J", and 1J", ^4, .5, and (7, respectively. Do these triangles have the same shape? Carefully cut out the triangles and place angle a over angle A ; then angle b over angle B-, and, last, angle c over C? What do you find to be true in each case? DEFINITION. In triangles having the same shape, angles lying oppo- site proportional sides in different triangles are called corresponding angles. 7. Read the pairs of corresponding angles in Figs. 235, 236, 237, and 239. 8. In triangles having the same shape, how do corresponding angles compare in size (see Problem 6) ? 9. In Fig. 235 find the ratio of the side lying opposite the angle B in triangle ABC to the side lying opposite the corresponding angle in triangle abc. Find a similar ratio between any other pair of such sides. How do the ratios compare? 10. Answer similar questions for the triangles of Figs. 236, 237, and 239. DEFINITION. In triangles haying the same shape, sides, as AB and ab (Fig. 239) or BC and be, that lie opposite the equal angles are called corresponding sides. 11. In triangles having the same shape state a law connecting the corresponding sides. USES OF SIMILAR TRIANGLES 335 FIGURE 240 B 12. Triangles ABC and ade (Fig. 240) have the same shape. How do their corresponding angles compare? What relation holds for their cor- responding sides, i.e., how do these ratios compare: AB-.adl BC:de? ACiae? With a protractor measure each pair of corresponding angles. How do they compare in size? 13. What is the ratio of the areas of triangles AB and ade (Fig. 240)? How may this ratio be found from the ratio of a pair of corresponding sides, as AB and ad? 14. Draw a pair of triangles, one having sides of 1", 1-J-", and If", and the other having sides of 4", 6", and 7". Give the ratio of each pair of corresponding sides; the ratio of the areas of the triangles? How may the ratio of the areas be computed from the ratio of a pair of corresponding sides? 15. State a law connecting the ratio of the areas of triangles having the same shape with the ratio of any pair of corresponding sides. 202. Uses of Similar Triangles. In Fig. 241 MNQR represents an B 18"x20" board, provided with a mov- ing arm, OE, and a protractor. The arm is made of light wood. A slot E8 is cut away and a thread is stretched on its under side and fastened at the ends. The ends of the thread are sunk into the wood so as to permit the thread to move smoothly over the protractor as the arm CO is turned round 0. A common pin stuck ver- tically at C and at completes the apparatus. The pin at should be driven through far enough to fasten the arm to the board at as a pivot. The protractor is held by thumb tacks at T, T. The board maybe placed on a window sill, or on a post in a fixed position, while measure- ments are being made. It is desired to measure the distance from A to B, A and B being on opposite sides of a large building. Use the proportion form of statement in all problems, calling the length of the unknown line x. Then find the value of x. Q FIGURE 211 336 RATIONAL GRAMMAR SCHOOL ARITHMETIC Solve some problems like this from your own measures. Build- ings, steeples, hills, trees, furnish good problems. 1. The board was placed on a window sill and the arm set in such position, FO, that when the eye sighted along the pins, and Z>, these pins were in line with B. The mark on the pro- tractor at G was 52. The board being held firmly in place, the arm was swung around until the pins at and at C were sighted into line with A and the reading on the protractor at H was 114. How many degrees were there in the angle EOF? 2. The distances from ^1 and from B to the window at were measured and found to be 484' and 520'. A triangle O'ef (shown on the left) was drawn having angle e(9'/'=G2 , and O'e = 4.84", O'f - 5.20". The side ef was then drawn, measured, and found to be 5.18". How long is the line ABt 3. Fig. 242 shows a crude apparatus for finding distances that cannot be measured directly. It consists of a board about 25" long and 6" wide with end pieces about 6" high. The wood is cut ElGURE 242 away from the end pieces so that the eye may sight through between the straight edges of two visiting cards at E, past two parallel threads at T, set carefully " apart and 25" in front of E. If 8 is in line with the upper thread of the instrument and D with the lower thread, how far is it from the stake at SD to the instrument, if 8D = 8'6"? SUGGESTION. * : 25 = 8| : x. By the Principle on p. 197, \x =8J X 25, or ^ = 2121. Find x, o 4. A pupil sighted through E at a stake held by another pupil at 8D. When the pupil at E sighted over the top thread, the USES OF SIMILAR TRIANGLES 337 pupil had to put his hand at # to line in with the slit at E and upper thread at T. The pupil at E then beckoned him to slide his hand down the pole until it lined in with the lower thread at T. The distance SD was 7.52'. How far was it from the instrument to the flagpole SD? 5. The distance from E to the bottom of the building, Fig. 243 FIGURE 243 is 288', the distance EB = 8' and AB = 18"; how high is the building? 6. In Fig. 244 it is desired to find the distance across the lake from 'O to D. A surveyor stuck a flagpole at a place from which he could see both C and D. He measured the distances from D and from C to the pole he is holding, and found them to be 4563' FIGURE 244 and 5481' respectively. He then set a pole at B, T ^ of the dis- tance from the first pole to Z>, and another pole at A, T ^ of the distance from the first pole to C. The distance from A to B was measured and found to be 84.64'. How far is it from C to Z>? 338 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. It is desired to make a scale drawing of the tract of ground ABODEF (Fig. 245). The parts of the apparatus to be used, as shown at the right, are a square board about 16"xl6", with a block containing a 1" hole to fit over the top pin of a sharp stake to be stuck in the ground to support the board. A small level (which may be a phial of water) and a foot rule with a vertical pin at each end for sights to be used for sighting complete the apparatus. The board is set up in the field as shown, a sheet of paper is pinned on it with thumb tacks, and the foot rule is placed upon the paper. A third pin is stuck near the, center of the board at a point o (not shown in Fig. 245). Holding the edge of the foot rule against this center pin, sighting along the pins toward a pole at A the observer turns the front of the foot rule until the two pins of the ruler are in line with A. He holds the ruler and draws a line along its edge on the paper. He now holds the edge of the ruler against the center pin, and carefully sights the two ruler pins into line with B, and, hold- ing the ruler in place, draws a line on the paper toward B. He proceeds in the same way with each of the points (7, />, E, and F, being careful not to turn the board around on the stake pin. 8. The lines from the stake supporting the board to A, to B, to (7, and so on to F were measured and found as follows : to A , 460'; to B, 452'; to (7, 378'; to Z>, 527'; to E, 535' and to F, 832'. Using a scale of 1": 100', the distance to A (460') was laid USES OF SIMILAR TRIAXGLES 339 off from the center pin on the line drawn toward A giving a (on the board), the distance to B (452') was laid off from the center pin on the line drawn toward B giving I (on the board), and so on around to F. Calling the center pin 0, how long is oa? ob? oc? od? oe? of? 9. With a ruler the points , #, c, d, e and / were connected as shown in the figure. The lines were ab = 2.8" ; be = 4.5" ; cd = 5.12"; de = 2.95"; ef= 6.25"; / = 4.48". How long are the lines , EC, CD, DE, EF, and FA? The board may be supported by a light camera tripod or by a home-made tripod and the board may be held to the flat top of the tripod by a thumb nut. (See Fig 246.) 10. The distances from the apparatus to the corners of the field (Fig. 246) were: to A, 678'; to B, 612'; to <7, 683'; to D, 738'; to E, 698'; to F, 625'; to , 679'. Using a scale of 1":200', how long should the distances be made from the center pin to a? to M to c? to d? to e? to/? to g? 11. With a ruler a, I, c, d, e, /, #, and a were then connected and lines measured. The measures were : 0^ = 3.81"; c = 4.12"; c^ = 2.86"; ^e = 2.75"; e/=5.86"; /# = 6.18" and ^ = 5.98". How long are the lines AB? BC? CD? DE? EF? FG? OA? RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. Having made the scale drawings, perpendiculars may be drawn from the center pin to each of the sides ab, b'i and so on. From the measured lengths of these perpendiculars and the bases AB, BC, and so on of the triangular parts of the figures, the areas of the triangles aob, boc, and so on (calling o the point where the center pin stands) may be computed. The sum of these areas gives the total area of the figure abcdefga. 13. Suppose the area of triangle aob were 4.94 sq. in. what would be the area of A OB (0 being the point on the ground just under o), if the scale were 1":200'? (See Problems 13 and 15, 201). 14. If the areas of the triangles aob, boc, cod, doe, eof, and foa (Fig. 245 prob. 7) in square inches were: 6.073, 7.740, 9.172, 7.490, 16.725 and 7.563, respectively, and the scale were 1": 100'; what were the areas of the triangles AOB, BOC, COD, DOE, EOF, and FOA? APPLICATIONS OF PERCENTAGE 203. Insurance. 1. What will it cost to insure $680 worth of household furni- ture, at the rate of If %? DEFINITION. The amount paid for insurance is called the premium. 2. An art gallery, valued at $500,000, is insured at the rate of li%- What is the premium? DEFINITIONS. The written agreement between an insurance company and the insured is called a policy. The amount for which the property is insured is the face of the policy. 3. A vessel, valued at $16,000, was insured for of its value. Find the face of the policy. 4. A growing crop was insured at 5%. The premium was $140. What was the face of the policy? 5. If it costs $420 to insure a house for f of its value, at 3J-%, what is the house worth? APPLICATIONS OF PERCENTAGE 341 6. A stock of hardware was insured for $7000, insurance cost- ing $110.75. What was the rate of insurance;' 7. If a shipment of grain is worth $840, and the premium amounts to $17.60, what is the rate of insurance? 8. A machinist insured his tools, valued at $300, for J of their value, paying $8.19. What rate did he pay? 9. A farmer paid a premium of $10.50 for insuring his stock at l-J-%. For what amount was the stock insured? 10. A man paid $67.50 for the insurance of a steam launch at 1-J%. For what amount was the launch insured? 11. What is the rate paid for insuring a bridge, valued at $15,000, for | of its value, the premium being $240? 12. A library, worth $28,000, was insured for f of its value, the premium being $720. What was the rate of insurance? 13. The contents of a grain elevator were insured at a rate of -f%. What was the amount of the policy, the premium paid being $272.40? 14. The premium paid for insuring a quantity of lumber, for two-thirds of its entire value, at 3% , was $36. What was the value of the lumber? 15. A piece of property, valued at $27,200, was insured for g of its value. The premium was $212.50. What was the rate? 16. A tank of oil, holding 2592 gal., worth I8

/ 2 s ..... 88M( 10000 Wabash R. R. 5.s ...... 115K115} 10000 Manhattan 4s ........... 105* 1. A certain express company has a capital stock of $500,000, divided among its stockholders in shares of the par value of $100 each. In 6 mo. the net profits amount to $10,500, which the company distributes among its stockholders as a dividend. What is the rate of dividend? How much does a holder of 200 shares receive as dividend ? 2. What must a man have paid for 400 shares Amal. Copper stock on the date of the table, including brokerage, if he bought at the opening price? at the highest price? at the lowest? at the noon price? 3. Answer similar questions for IT. S. Steel pfd. (preferred). DEFINITIONS. Preferred stocks are stocks which pay a fixed dividend (say of 1%) before any dividends are paid on common stocks ; Common stocks pay dividends dependent on the net earnings of the company after expenses and dividends on preferred stock have been paid. 4. How much did a man gain or lose by buying 2000 Am. Sugar at the opening price and selling at the highest price, paying brokerage of % for buying and also for selling? 5. Answer similar questions for B. & 0. com. ; for B. & 0. pfd. 6. How much does a man make or lose who buvs 2500 shares 111. Cen. R. R. stock at the lowest quotation of the table and sells at the highest, paying brokerage of -J for buying and -J for selling? 7. If a man invests in St. L. and S. F. pfd. stock, paying 7% annual dividend, at the lowest quotation of the table, what interest does he receive on his investment? SUGGESTION. The investor must pay $79* -f- $ per share (of $100), and he receives $7 per year as interest. 8. Answer a similar question on C. & A. pfd. 9. What interest does a man receive on an investment in Gov- ernment New 2s if he invests at the price "bid," brokerage ? at the price "asked"? NOTE. Government 2s, 8s etc., are government bonds paying 2%, 3%, etc., annually. 10. Answer similar questions for New 3s; for New 3s cou- pon ; for Registered 4s ; for Registered 4s new. 11. What rate of interest does the investor who bought the 1000 B. & 0. 3As receive? APPLICATIONS OF PERCENTAGE 347 12. Answer similar questions for the investor who bought the 9000 C. & A. 3s, if he paid the lowest quoted price; the highest. 13. Which pays the higher rate of interest on the investment, and by bow much, U. P. 4s as quoted, or Wabash E. R. 5s at the lowest quotation? at the highest? 14. Which is the more profitable investment, and by how much, B. & 0. coupon 4's at the lowest quotation, or a straight loan at 4%? 15. Answer other similar questions on the table. 10. An investor in Chi. & Alton com. stock at the highest quotation received two 2% and one 3% dividend, and sold the stock 18 mo. later at 38-J-. What rate of interest did he receive on his investment? How much profit did he make if he bought 1000 shares? 17. Answer similar questions on the table. 207. Compound Interest. With certain classes of notes, if the interest is not paid when due, it is added to the principal and this amount becomes a new principal, which draws interest. Savings banks add the interest on savings deposits at each interest-paying period and pay interest on the entire amount. This interest on interest is called com- pound interest. Bankers also collect their interest on loans and then reloan the interest, and in this way virtually receive com- pound interest on their money. Compound interest is usually payable annually, or semi- annually. 1. Find the compound interest on a note of $200 at 6% for 5 yr., interest payable annually. SOLUTION. Interest on $200 for 1 yr. at 6% = 200 X $ .06 = $ 12.00 Amount of $200 for 1 yr. at 6% = 200 X $ 1.06 = $212.00 Amount of $212 for 1 yr. at % = 200 X $(1.06) 2 = $224.72 Amount of $224.72 for 1 yr. at Q% = 200 X $(1-06) 3 = $238.23 Amount of $238.23 for 1 yr. at 6% = 200 X $(1.06)* = $252 52 Amount of $252.52 for 1 yr. at 6% = 200 X $(1.06) 5 = $267.68 The compound interest=$267. 68 $200=$67.68. NOTE. The $267.68 is called the compound amount. The expressions (1.06) 3 , (1.06)*. and (1.06) 5 mean 1.06 X 1.06 X 1.06, 1.06 X 1-06 X 1.06 X 1.06, and 1.06 X 1.06 X 1.06 X 1.06 X 1.06, and are read "1.06 cube," "1.06 to the 4th power," and "1.06 to the 5th power." 348 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Find the simple interest on a note for the same face at the same rate and for the same time as in problem 1. By how much does the compound interest exceed the simple interest? 3. Find the compound interest on $180 at 6% for 3 yr., interest compounded semi-annually. SOLUTION. Amount on $180 for J yr. at 6% = 180 X $ 1.03 = 185.40 Amount on $185.40 for yr. at 6% = 180 X $(1.03) 2 = $190.96 Amount on $190.96 for yr. at Q% = 180 X $(1.03) 3 = $196.69 Amount on $196.69 for yr. at 6% = 180 X (1.03)* = $202.59 Amount on $202.59 for \ yr. at 6% = 180 X $(1.03) 5 = $208.67 Amount on $208.67 for \ yr. at 6% = 180 X $(1.03) 6 = $214.93 Compound interest = $214.93 $180 = $34.93. 4. If P denotes any principal at /% for n yr., interest being compounded annually, show that if A denotes the compound amount, 5. Show that if the interest is compounded semi-annually, ( r \2n 1+ x -f- 10, which is read "?/ plus 15 is greater than 9 plus 10." If the balance, or equality, is destroyed by adding a heavier weight to x than to y, the fact is stated in symbols thus: y -f~ 10 < x -\- 15 ; read "y plus 10 is less than x plus 15." Notice in each case that the vertex of the horizontal V always points toward the smaller number. DEFINITIONS. An expression in which the sign < or > stands between two numbers is called an expression of inequality. 354 RATIONAL GRAMMAR SCHOOL ARITHMETIC We now have signs for writing briefly the three possible relations which may exist between any two numbers, as a and b. They are called relation signs. 8. Bead and give the meanings of (1) a > b, (2) a = b, (3) a < b, (4) x + 20 < x 4- 15 -f 10. 9. If y = x, and a Ib. are added to y lb. and b lb. to x lb., how will the knot G move for case (1) Problem 8? for case (2)? for case (3) ? State in symbols the fact shown by the knot C in each case after adding the weights, using the correct relation signs. 10. If any given weight, a lb., be added to y, what must be true of the weight b lb. if, when added to x, we may write y + a = x + b? 11. If z lb. be added to both the y lb. and the x lb., when y - x, what equation may we write? PRINCIPLE I (FOR ADDITION OF EQUATIONS). If the same number, or equal numbers, be added to both sides of an equation, the sums are equal. ILLUSTRATION. If y = x, and a = b, then we may write y -\- a = x -\- a, and y -\- b = x -\-b, and y -\- a = x -\- b, and y -\- b = x + a. 12. If 8 lb. be removed from the left scale pan (Fig. 249), how many 4-lb. weights must be removed from the right pan to restore the balance ? NOTE. This is written in signs y 8 = x 4 4, and is read "y 8 equals x minus 4 minus 4." 13. How many pounds in all were removed from the right scale pan? 14. If y = x, and if a lb. be removed from the left, and b from the right, to what must the value of b be equal to enable us to write y a = x b? PRINCIPLE II (FOR THE SUBTRACTION or EQUATIONS). If the same numjter, or equal members, are subtracted from equal num- bers, the differences are equal. ILLUSTRATION. If y = x and a = b, then we may write: y a = x a, and y b = x b, and y a = x b, and y 6=0? a. USE OF LETTERS TO REPRESENT NUMBERS 355 15. If in Fig. 249 y be doubled, what corresponding change in x will restore the balance? NOTE. The double of y is written 2y and read "two ?/." 16. What equation states that there is a balance? 17. If y = x, and if 4 weights, each equal to y, are put on the right of the apparatus in Fig. 249, how many weights., each equal to x, must be put on the left to keep the balance? 18. If a weights, each equal to y, are on the right, how many weights, each equal to x, must go on the left to secure balance? NOTE. The equation is ay = ax. 19. If a = 5, and a weights, each equal to y, are put on the right, and I weights, each equal to #, are .put on the left, what will be shown by the scales if y = x. (Answer with an equation.) PRINCIPLE III (FOR THE MULTIPLICATION OF EQUATIONS). If equal numbers are multiplied by the same number, or by equal numbers, the products are equal. ILLUSTRATION. If y = x and a b, then ay = ax, and by = bx, and ay = bx, and ax = by. 20. If y = x, and half of the weight on the left be taken off, what fractional part of the weight on the right must be taken off to restore the balance? * 21. If only of y be kept on the right (Fig. 249), what frac- tional part of x must remain on the left? 22. If only ^ Ib. be kept on the right (Fig. 249), what frac- tional part of x Ib. must remain on the left for balance? 23. Suppose y = x and a = #, and that Ib. are on the right, x and -7 Ib. are on the left; what equation would be shown to be true by the apparatus? NOTE. The equation is = -r, read "y divided by a equals x divided by 6." 356 RATIONAL GRAMMAR SCHOOL ARITHMETIC PRINCIPLE IV (FOR THE DIVISION OF EQUATIONS). If equal numbers are divided ~by the same number, or by equal numb&rs, the quotients are equal. ILLUSTRATION. If y == x and a = b, then and =, b b b and|=*. 6 a In Fig. 183, p. 297, a (x + y) represents the area of the whole rectangle, while ax and ay represent the areas of its two parts. As the two parts of the large rectangle, taken together, must equal the whole rectangle, we may write : a (x + y) = ax + ay In Fig. 184, p. 298, the area of the whole rectangle is given by (a + b) (x -f ?/), while the areas of the several parts are ax, ay, bx, and by. Since the parts, taken together, make up the whole rectangle, we may write: (a -t- b) (x + y) = ax + ay + bx + by These two equations illustrate the meaning of a fifth principle of the equation, viz. : PRINCIPLE V. Any whole equals the sum of all its parts. These five fundamental principles or laws, exemplified by the scales, p. 103, and the pulley device (Fig. 249), p. 353, must not be violated in using the equation. 211. Problems. 1. Find the value of the. letter #, y, or z, in each of the first nine problems: (1) 3x = 15 Ib. ; (4) 9* = 36

18 in each case). 2. State in words what these expressions mean. For example, (1) means "double a certain number, diminished by 9": (1) 33-9; (6) (x + l) ( + !); (11)2^ + 6 = 12; (2) 16 + a;; (7) *(&-4); (12) 7z-2 + 16; (3) 282; + 17; (8) 3(9 -a); (13) 5z+7 = 42; (4) z* + x\ (9) 12(^ 2 -1); (14) (a - 4) (z + 4) =20; (5) x z -x^ (10) (a-1) (z-1); (15) (a + b)(a-b) = a*-b\ 3. Translate into symbols these verbal phrases and statements, using a and &, or x and y, for the two numbers: (1) The sum of two numbers equals 25. (2) The difference of two numbers equals 15. (3) The sum of the squares of two numbers is less than 27. (4) The square of the sum of two numbers equals 100. USE OF LETTERS TO REPRESENT NUMBERS 359 (5) The difference of the squares of two numbers equals 9. (6) The sum of the squares of two numbers equals seven times the difference of the numbers. (7) The product of two numbers equals their sum. (8) The quotient of two numbers equals their difference. (9) A certain number increased by 1 equals another number diminished by 3. 4. Translate into words these symbolic expressions. For example, (1) means "one-ninth of the difference between 6 times a certain number and its square": " y (3) 3 -^; (8) = =a + 6; (13) 4 9*.-; (14) y x ' ' 5. Find the number which may be put in place of the letter in each of these equations to furnish true equations: (1) 2 + 3 = 7; SOLUTION. -3 2x = 4 by Principle II, 2ic_4_ i 2 ~2 x= 2 by Principle IV. Check : 205 + 3 = 2x2-1-3 = 44-3 = 7, which is correct. (2)3^-1=5; (4)t + l = 2; (6) 6 | + f = V 4 5 (3) 8a-6 = 18; (5) - 3 |--2 = 1; (7) - 6 f-f=V 6 . NOTE. First multiply both sides of (7) by 21. 6. Find the value of the numbers of problem 1 (8), (9), and (10). RATIONAL GRAMMAR SCHOOL ARITHMETIC 213. Problems. FOR EITHER ARITHMETIC OR ALGEBRA. 1. The mercury column in a thermometer rose a certain num- ber of degrees one day, and 3 times as many degrees the next day. It rose 12 during the 2 days. How many degrees did it rise each day? ARITHMETICAL SOLUTION. A certain number denotes the rise the first day. 3 times this number denotes the rise the second day. Hence 4 times a certain number denotes the rise in two days. 4 times a certain number equals 12 (by the given problem). Once the number equals 3 , the rise the first day (Principle IV). 3 times the number equals 9, the rise the second day (Principle III). Check 3 + 9 = 12, the rise in two days. ALGEBRAIC SOLUTION. Let x denote the first day's rise. Then, 3.t denotes the second day's rise. x -\- 3x denotes the rise in 2 days. 4x = 12. x = 3, the first day's rise (Principle IV). 3x = 9, the second day's rise (Principle III). Check: 3 -f 9 = 12. 2. A man bought 4 times as many hogs as cows, and after selling 5 hogs he had 23 hogs left. How many cows did he buy? ARITHMETICAL SOLUTION. A certain number represents the number of cows bought. 4 times this number represents the number of hogs bought. 4 times this number minus 5 denotes the number of hogs left. Then 4 times this number, minus 5, equals 23 (by the problem). 4 times this number equals 23 plus 5 (Principle I). 4 times this number = 28. This number = 7, the number of cows (Principle IV). Check: 4 X 7 5 = 23. ALGEBRAIC SOLUTION. Let x denote the number of cows bought. Then, 4x denotes the number of hogs bought. 4x 5 denotes the number of hogs left. Then 4x 5 = 23 (by the problem). 4x = 28 (Principle 1). x = 7 (Principle IV). Ans. 7 cows. Check: 4 X 7 5 = 23. 3. Two masses were placed on one scale pan of a balance and found to weigh 18 Ib. One of the masses was then placed in each pan, and it required 4 Ib. additional on the light pan to balance the scales. What was the weight of each mass? USE OF LETTERS TO REPRESENT NUMBERS 361 ARITHMETICAL SOLUTION. A certain number of pounds denotes the weight of the heavier mass. Another number of pounds denotes the, weight of the lighter mass. The first number plus the second number denotes the combined weight (18 pounds). The first number minus the second denotes the difference of the weights, or the additional weight, which equals 4 pounds. 2 times the first number equals 18 plus 4 equals 22. The first number equals 11. 11 plus the second number = 18. (Principle IV). The second number = 7. (Principle II). The weights are, then, 7 Ib. and 11 Ib. Check: 11 Ib. + 7 Ib. = 18 Ib. 11 Ib. 7 Ib. = 4 Ib. ALGEBRAIC SOLUTION. Let x denote the number of pounds in the weight of the heavier mass. Let y denote the number of pounds in the weight of the lighter mass. Then, z-f- y denotes the number of pounds in the combined weight,=l& x y denotes the number of pounds in the additional weight, = 4. x + y = 18 x y= 4 2x = 22 (Principle I). x =11 (Principle IV). 11 -f y = 18 y = 18 11 == 7 (Principle II). Check: 11 -f 7 = 18 11 7= 4 Solve the following problems by both the algebraic and tho arithmetical method, and state which of the solutions is the shorter : 4. A man cut for me 3 times as many ash trees as oak trees, and as many hickory trees as ash trees and oak trees together. In all he cut 32 trees. How many of each kind did he cut? 5. A man bought 4 times as many 2(f stamps as 5^ stamps, and - as many 10$ stamps as % 2 ; S= 17.0127) 2 . For steel wire rope, JF= 3.940/.> a ; = 4.441Z> 2 ; #=27.6307> 2 . Find the weight per yd., the working load and the breaking load of a hemp cable of 1" diameter ; of 2-J-" diameter. 22. Solve similar problems for tarred hemp rope ; for manila rope. 23. Find W, L, and 8 for an iron wire rope for which D = 1 J" ; D = 2i". 24. For the same values of 7), find TF, Z, and for steel wire rope. USES OF THE EQUATION 215. Sliding and Static Friction. EXPERIMENT I. With appliances arranged as suggested in the cut, the empty carriage, B, was weighed and placed upon a smooth surface of the substance whose friction was JH, to be studied. Known weights were then placed upon the pan 8 (which was also weighed) until, after pushing the block loose from the surface, it would just slide slowly along it. Known loads were then placed, one after another, upon the block at 7/, and, for each load, weights were put upon 8 until the carriage would just slide along after being started. The loads put at L are tabulated in the first column below, and the weights at 8 in the second. The weight needed at 8 in each case, together with the weight of the pan, is the force of friction, or friction, simply. EXPERIMENT II. In the above experiment it is found that the surfaces of carriage-block and substance cling together at starting, and that a slight thrust on the block is needed to start it to avoid USES OF THE EQUATION 365 overloading at S. When the weight at S is heavy enough of its own account to start the load, it moves off with increasing speed; and for Experiment I this must not be allowed. This is because friction at starting is greater than after motion has begun. The first is called static (standing) friction, and the second, sliding friction. The static friction was also found for this apparatus, and is tabulated in the third column for each load. LOAD INCLUD- FORCE OF FRICTION IN LB. ING CARRIAGE IN LB. Sliding Static 7.6 1.0 3.2 11.6 1.7 4.4 14.6 2.0 5.7 18.6 2.8 6.8 21.6 3.0 7.5 25.6 3.6 9.2 28.6 4.0 10.0 "35.6 4.9 12.2 42.6 5.9 14.7 49.6 6.9 16.8 566 7.8 20.0 Starting at any point as 0, Fig. 251, on a page of cross-ruled paper, draw through a horizontal line, and also a line perpendic- ular to the horizontal. Letting a horizontal side of one of the small squares represent 5 lb., measure off from 0, toward the right, dis- tances to represent the numbers standing in the first column of the in ., J 1 2 5 z P 2)4 H i J I B ( : I )E > I 2 . Show the correctness of the following rule : EULE. Square the tens, double the product of the tens by the units and square the units, then add the three results. The sum is the square of the number. 4. Square these numbers by the rule : (1) 36. (3) 64. (5) 19. (7) 125. (2) 47. (4) 58. (6) 95. (8) 148. NOTE. Call the 12 in (7) 12 tens; also call the 14 in (8) 14 tens. 36 might be written 40 4. Then 36 2 = (40 4) 2 , (40 4) 2 = 40 2 2 X 4 X 40 -(- 4 2 . Algebraic form : (a 6) 2 = a 2 2 ab + b' 2 . Make a rule for squaring 36 in this form. When long decimals are to be multiplied and the product is required to only a few decimal places, contracted multiplication is of great advantage. The next problem illustrates this sort of multiplication. 5. The radius of a circle is 238.36 ft. What is the length of the circumference to the second decimal place, or to the nearest .01 foot? NOTE. The length of the diameter is 476.72 feet. COMMON FORM SHORTENED FORM 476.72 476.72 3.1416 6141.3 digits reversed 28 47 1906 4767 143016 1497.66 6032 1430.16 672 47.67 88 19.07 2 48 29 3552 1497.67 SHORTENING AND CHECKING CALCULATIONS 377 EXPLANATION. In the shortened form the digits of the multiplier are written in reverse order, and the units digit is always written under that decimal place in the multiplicand which is to be the last one retained in the product. Multiply by units digit first, then by tens, and so on; in each case begin the multiplication by any digit with the digit just above it in the multiplicand. Begin the writing of each partial product in the same vertical line on the right. NOTE. It is necessary on beginning to multiply by any digit to glance at the product by the preceding digit of the multiplicand to see how many units are to be added into the product by the digit just above. Thus, the multiplication by 4 would begin with 6, but 4 times the pre- ceding digit (7) is 28, and this being nearly 3, the product 4x6 would be increased by 3, giving 27. Expert computers use the shortened form altogether. 3. Eind the following products to the second decimal place by the method of shortened multiplication: (1) 30.428 x 3.1416. (2) 186.086 x 108.336. (3) 7.8843 x 1.0863. (4) 168.7431 x 28.329. 4. Find these products to .001 by shortened multiplication: (1) 36.1872 x 6.8734. (2) 128.63 x 3.8629. (3) 629.3865 x 3.1416. (4) 1284.683x3.1416. $25. Shortened Division. 1. Divide 648.7863 by 68.372 to the nearest .01. CONVENIENT FORM 9.49 Quotient 68.37)648.7863 61533 3345 610 6 15 EXPLANATION. Find the units digit of the quotient in the usual way. Then cut off one digit from the right of the divisor and find the next digit of the quotient, then cut off another digit from the divisor, etc. A dot is sometimes placed over each digit in the divisor as it is set aside. 2. Find the following quotients to two decimal places: (1) 1786.786-3.1416. (2) 632.068-8.6249. (3) 1206.3862-28.3762. (4) 865.28476 + 361.2946. 378 ' RATIONAL GRAMMAR SCHOOL ARITHMETIC 226. Shortened Square Boot. Square root may be found by subtraction, after noticing that 1 = 1 2 ; l + 3 = 2 3 ; l + 3 + 5 = 3 2 ; H-3 + 5 + 7 = 4 a ; and so on. It is seen that the square root of the sum of the odd numbers in order from 1 upward is equal to the number of odd numbers added. The method* of using this can be best understood from an example. 1. Extract the square root of 104976. 104976 ( 324 1 9 3 01 641 149 61 88 63 2576 641 3 subtractions 2 subtractions 1935 643 1292 645 647 4 subtractions 324 = square root. EXPLANATION. Place a dot over each alternate digit beginning on the right, thus separating the number into groups of 2 digits each. From the first group on the left subtract 1, then 3, then 5, and so on as shown, until the remainder is less than the next odd number. The number of subtractions is the first root digit. In the present case it is 3. Bring down the next two-digit group. Double the root digit found and annex 1 to it, and subtract, then replace the 1 by 3 and subtract, etc. The number of sub- tractions indicates the next root digit. Study the remaining steps and learn how to proceed further. This method gives the exact square root and may be used to check results found in the ordinary way. Some computing machines are based on this method of obtaining the square root. 2. Find the square roots, by subtraction, of the following: (1) 1156. (3) 174.24. (5) 54,756. (2) 44,944. (4) 49,284. (6) 289444. SYNOPSIS OF DEFINITIONS. 4. A one-brick wall is a wall one brick thick, the bricks lying on the largest surfaces, the sides being exposed. 10. The average of two or more numbers is their sum divided by the number of them. 12. A board foot is a board one foot long, one foot wide, and not more than one inch thick. 14. Cash rent of farming land is a stated amount of cash per acre. Grain rent of farming land is a stated part of all the crop. The tenant farmer is one who raises his crop on another man's farm. 15. What per cent means "how many in a hundred" or "how many hundredths." (Cf. p. 130.) 17. Normal means average here. The mean of two numbers is half their sum. 22. To average means to find the average. 23. The range of temperature is the difference between the highest and the lowest temperatures. 26. The reading of a barometer is the difference between the lengths of the mercury columns. 28. The range of the barometer is the difference between the greatest and the least readings. 31. The digits (or figures) are the ten characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The name value of a digit is the value depending only upon the name of the digit. ThepZac-e value of a digit is the value depending only upon the place of the digit. 34. The index notation is such a form as 675xl0 12 , meaning 675,000,- 000,000,000. 36. Addition is combining numbers into a single number. The sum or amount is the result of the addition. The addends are the numbers to be combined or added. 47. Subtraction means either of two things : (1) The way of finding the difference, or remainder, of two numbers. (2) The way of finding either one of two addends when their sum and the other addend are known. With the first meaning, the number from which we subtract is the minuend. The number to be subtracted is the subtrahend. The result is called the difference or remainder. With the second meaning, the known sum is the minuend, 379 380 RATIONAL GRAMMAR SCHOOL ARITHMETIC PAGE The known addend is the subtrahend, and the unknown addend is the difference or remainder. 57. Literal numbers are those that are denoted by letters, 58. Multiplication of whole numbers is a short way of finding the sum of equal addends when the number of addends and one of them are given. The given addend is the multiplicand. The number of equal addends is the multiplier. The result or sum is the product. When a problem is expressed in an equation, the equation is called the statement of the problem. The number which should stand in place of x in an equation is called the value of x. 63. A factor of a given whole number is one of two or more whole num- bers, which, multiplied together, produce the given number. (Cf. p. 148.) 64. A number which has factors other than itself and 1 is called a com- posite number. A number which has no factors other than itself and 1 is a prime number. 69. One inch of rainfall means one cubic inch of water for each square inch of horizontal exposed surface. The number of cubic units (cu. in., cu. ft., cu. yd., etc.) a vessel holds, when full, is called its capacity. 73. Division is a short way of finding: (1) One of a given number of equal parts of a number. (2) How many equal parts of given size there are in a given number. When the term "division" has the meaning of (2), the process to which it applies may be called measurement. Division of whole numbers is a short way of subtracting one number from another a certain number of times in succession. (Cf. p. 74.) 74. Division is a way of finding one of two numbers when their product and the other number are given The product is called the dividend. The given number is the divisor. The required number is the quotient. We may say also that the dividend is the number to be divided, the divisor is the number by which the dividend is meas- ured or divided, and the quotient is the measure. (Cf. p. 73.) 89. The point on which a lever rests is called the fulcrum. 103. Such an expression as y=10 is called an equation. The number on the left of the sign of equality is called the first SYNOPSIS OP DEFINITIONS 381 PAGE member, or the left side, of the equation. The number on the right is called the second member, or the right side, of the equa- tion. 104. The pencil point of the compasses is called the pencil-foot or the pen- foot. The other point is called the pin-foot. A circle is a curve such as is drawn with compasses. An arc of a circle is a part of a circle. The center of a circle is the point where the pin-foot of the compasses was placed to draw the circle. A diameter of a circle is a straight line joining two points of the circle and passing through the center. A radius of a circle is a straight line joining its center and a point of the curve. Circles whose centers are at the same point are called concentric circles. 107. Dividing a line into two equal parts is called bisecting the line. 108. An equilateral triangle is an equal-sided triangle. An isosceles triangle has at least two sides equal. That side of an isosceles triangle not equal to either of the other two sides, is called the base of the isosceles triangle. 109. A scalene triangle has no two sides equal. 110. A regular hexagon is a regular six-sided figure. 111. The sum of all the bounding lines of a figure is called the perimeter of the figure. Money is the common measure of the value of all articles that are bought and sold. 113. One of a whole number of equal parts of a given magnitude is called a fractional unit. (Cf. p. 144.) 114. A square unit is a square one unit long and one unit wide. 115. A representation of an object, showing the various parts as folded back and spread out on a flat surface, is called a development. A quadrilateral is a four-sided figure. A parallelogram is a quadrilateral whose opposite sides are parallel. A rectangle is a parallelogram whose angles are (equal) right angles. A rhombus is a parallelogram whose sides are equal. A square is both a rectangle and a rhombus. . 116. The altitude of a parallelogram is the distance square across. 117. The altitude of a triangle is the shortest distance to the base from the opposite corner. With all except isosceles triangles any side may be regarded as the base. The altitude of a trapezoid is the distance square across between two parallel sides called the bases of the trapezoid. 118. The lengths of the bases and of the altitude of a trapezoid are called, its dimensions. o8-> RATIONAL GRAMMAR SCHOOL ARITHMETIC PAGE 124. The mean solar day is the average time interval during which the rotation of the earth carries the meridian of a place eastward from the sun back around to the sun again. It is the average length of the interval from noon to the next noon. (Of. p. 229.) 125. A section of land is a square tract containing 640 acres. 126. A township is a tract of land six miles square. 130. Per cent means hundredth or hundredths. (Of. p. 15.) 131. Interest is money to be paid for the use of money. (Cf. p. 258.) 138. The average rate of running is the distance divided by the time. 139. A diagonal of a figure is a straight line, not a side, joining two corners. 1 12. The ratio of one number, (or quantity), to a second number, (or quantity), is the quotient of the first number, (or quantity), divided by the second. The ratio of one magnitude to a second magnitude is called also the measure of the first by the second. The result of measuring one number by another is called the numer- ical measure of the first by the second. An equation of ratios is called a proportion. 144. The fractional unit of a fraction is one of the equal parts expressed by the fraction. (Cf. p. 113.) 145. The number above the fraction line is called the numerator (mean- ing numberer}. The number below the fraction line is called tne denominator (meaning namer). The numerator and the denominator are together called the terms of a fraction. 147. A fraction is said to be in its lowest terms when the numerator and the denominator are the smallest possible whole numbers without changing the value of the fraction. The greatest common divisor (G. C. D.) of two numbers is their greatest exact common divisor. 148. Two numbers that have no common factor, except 1, are said to be prime to each other. A factor of a number is an exact divisor of the number, or a divisor that is contained without a remainder. (Cf. p. 63.) 149. A number that has no factors except itself and 1 is called a prime number. A number that has factors beside itself and 1 is called a composite number. Any number that can be exactly divided by the number 2 is called an even number. All other whole numbers are called odd numbers. SYNOPSIS OF DEFINITION'S 383 PAGE 156. The average growth from both lower and middle branches of a tree is called the lateral growth. The average from the top branches, is called the terminal growth. 157. A denominator which is common to two or more fractions is called a common denominator. When a common denominator is the least number that can be found which may be used as a common denominator of the fractions, it is called the least common denominator (L. C. D.). 158. A number that can be exactly divided by another number is called a multiple of the latter number. A number that can be exactly divided by two or more numbers is called a common multiple of those numbers. 159. The least common multiple (L. C. M.) of two or more numbers is the least whole number that is exactly divisible by each of the num- bers. 161. A proper fraction is a fraction whose numerator is less than its denominator. An improper fraction is a fraction whose numerator is equal to, or greater than, its denominator. A mixed number is a number that is composed of an integer and a fraction. 170. To multiply a whole number by & fraction means to divide the multi- plicand into as many equal parts as there are units in the denominator, and to take as many of these equal parts as there are units in the numerator, of the multiplier. 180. Dividing 1 by any number, whole or fractional, is called inverting the number. The reciprocal of any number is the number inverted. 182. Fractions containing fractions in one or both terms are called com- plex fractions. The outside terms of a complex fraction (such as-?-) are called the 1 5 T extremes and the inside terms are called the means. 186. A straight line connecting the mid-point of a side of a triangle with the opposite corner is called a median of the triangle. The vertex of an angle is the corner. (The plural of vertex is vertices (ver'-tises.) (Cf. pp. 278, 292.) 188. To trisect a magnitude is to divide it into three equal parts. 189. A right-angled triangle is called a right triangle. 190. Parallel lines are lines running in the same direction. 192. A square is inscribed in a circle if the vertices of the square are all on the curve. 196. The first, second, third, and fourth numbers of a proportion are called the first , second, third, and fourth terms of the proportion. 384 RATIONAL GRAMMAR SCHOOL ARITHMETIC PAGE The first and the fourth terms of a proportion are called the extremes and the second and the third terms, the means. The first two terms of a proportion are together called the first couplet ; and the third and the fourth terms, the second couplet. 200. A dot, called the decimal point, or point, is used to show the units' digit. The point always stands just to the right of the units' digit or place. 201. The unit of the 1st place, or digit, to the right of the decimal point is called the tenth; of the 2d place, or digit, the hundredth; of the 3d, the thousandth; of the 4th, the ten-thousandth; and so on. 202. A decimal fraction, or decimal, is a fraction whose denominator is 10, 100, 1000, or some power of ten, in which the denominator is not written but is indicated by the position of the decimal point. A power of 10 here means a number obtained by using 10 as a factor any whole number of times. 203. A pure decimal is a decimal whose value is less than one. A mixed decimal is a decimal whose value is greater than one. Numbers expressed in both decimals and common fractions are called complex decimals. A simple decimal is expressed without the use of common fractions. Finding the sum of decimal numbers is called addition of decimals. 205. Finding the difference of decimal numbers is called subtraction of decimals. 208. By the number of decimal places of a number is meant the number of digits (zero included) on the right of the decimal point. 214. The distance round a circle is sometimes called the circumference of the circle. 217. The specific gravity of any solid or liquid substance is the ratio of its weight to the weight of an equal bulk of water. 219. Decimals that do not terminate are called non-terminating decimals. Non-terminating decimals that repeat a digit or group of digits indefinitely are called repetends, or circulating decimals, or cir- culates. 221. A right section is a section made by cutting squarely across. 223. A safety bicycle is said to be geared to 72", 84", and so on, when one turn of the cranks would carry it just as far forward as would one turn of a wheel having a diameter of 72", 84", and so on. 225. A denominate number is a number whose unit is concrete. A concrete unit is a unit having a specific name. A compound denominate number is a number expressed in two or more unite of one kind. 227. A perch of stone is a square-cornered mass, l'Xl>2'Xl6^' cubic feet. A cord of firewood is a straight pile, 4'X4'X8'=128 cubic feet. SYNOPSIS OF DEFINITIONS 385 PAGE A cord foot is a straight pile of wood, 4'x4'Xl'=16 cubic feet. 229. A leap year is a year of 366 days. A common year is a year of 365 days. 232. Reduction from higher to lower denoniinations is called reduction descending. Reduction from lower to higher denominations is called reduction ascending. 236. A meter is approximately one ten-millionth of the length of the part of a meridian of the earth, that reaches from the equator to the pole, called a quadrant of the earth's meridian. 238. An are is one square dekameter. A liter is one cubic decimeter. A gram is the weight of one cubic centimeter of distilled water at the temperature of its greatest density (39.1 Fahrenheit). 242. The number written before the sign "%" is called the rate per cent. The number, together with the sign, "%," is called the rate. (Cf. p. 258.) 245. The result of finding a given per cent of any amount, or number, is called the percentage. The amount, or number, on which the percentage is computed is called the base. 249. Elevation means height above mean sea level. 254. Commission is a sum of money paid by a person or firm, called the principal, to an agent for the transaction of business. It is usually reckoned as some per cent of the amount of money received or expended for the principal. A shipment of goods sent to an agent to be sold is called a consign- ment. 255. The net proceeds of a sale means the amount left after deducting the commission and other expenses. A commercial (or a trade) discount is a certain rate per cent of reduction from the listed prices of articles. The discount is usually allowed for cash payments 'or for payment within a specified time. 258. Interest is money charged for the use of money. It is reckoned at a certain rate per cent of the sum borrowed for each year it is bor- rowed. (Cf. p. 258.) When money earns 3, 6, 7, or 10 cents on the dollar annually (each year) the rate is said to be 3 %, 6%, 7%, or 10% per annum (by the year) and the rate per cent is said to be 3, 6, 7, or 10. (cf. p. 242. ) The sum of money on which interest is computed is called the principal. The principal plus the interest is called the amount. 265. A promissory note is a written promise, made by one person or 386 RATIONAL GRAMMAR SCHOOL ARITHMETIC PAGE party, called the maker, to pay another person or party, called the payee, a specified sum of money at a stated time. The sum of money for which the note is drawn is called the face value, or the face, of the note. The date on which the note falls due is called the date of maturity, and the time to run from any given date is the time yet to elapse before the note falls due. 266. Discount is a deduction from the amount due on a note at the date of maturity. The sum of money which, at the specified rate and in the time the note is to run before falling due, will, with interest, amount to the value of the note when due, is called the present worth of the note. The difference between the value of the note when due, and the present worth is called the true discount. The bank discount of a note is the interest upon the value of the note when due, from the date of discount until the date of maturity. 267. When a note or bond is paid in part the fact is acknowledged by the holder by his writing the date of payment, the sum paid, and his signature on the back of the note or bond. This is called an indorsement. 270. The small wheels under the front of a locomotive engine are called pilot wheels, or leaders. The large wheels are called drivers. The smaller wheels just behind the drivers are called trailers. 271. Tractive force is pulling (or drawing) force. 272. Replacing a letter (in an equation) by a number is called substituting the number for the letter. Performing the operations indicated in an equation and obtaining the number-value of a letter is called finding the value of that letter. 278. The surfaces of the cube meet each other in edges, thus forming lines. The corners are called vertices. A single corner is a vertex. (Cf. pp. 186, 292.) The edges meet each other in corners of the cube, thus forming points. 282. "When two lines meet making the angles at their point of meeting (intersection) equal, the lines are said to be perpendicular to each other, and each is called a perpendicular to the other. The angles thus formed are called right angles. 283. Lines which go through the same point are called concurrent lines. 287. A rhomboid is a parallelogram that is not a rectangle. A trapezoid is a quadrilateral having at least one pair of opposite sides parallel. SYNOPSIS OF DEFINITIONS 387 PAGE 289. A quadrant of arc is one quarter of a complete circle. An angle is the amount of turning of a line about a point as a pivot. It may also be regarded as the difference of direction of two lines. A straight angle is the sum of two right angles. 290. An angular degree is one of the ninety equal parts of a right angle. A degree of arc is one of the ninety equal parts of a quadrant. A sextant is one-sixth of a complete circle. An octant is one-eighth of a complete circle. 291. A perigon is an angle equal to one complete revolution. 292. If two lines intersect each other, two angles lying opposite to each other are called opposite or vertical angles. The lines which include an angle are called the sides of the angle. The point where the -sides meet is called the vertex of the angle. (Of. pp. 186, 278.) 293. An angle that is smaller than a right angle is called an acute angle. An angle that is larger than a right angle and less -than a straight angle is called an obtuse angle. 295. Two angles whose sum equals a right angle, or 90, are called comple- mental angles. Two angles whose sum equals two right angles, or 180, are called supplemental angles. 304. The zenith is an imaginary point in the sky, directly overhead. HOC. Longitude is the distance in degrees, minutes, and seconds (of arc) due eastward or westward from a chosen meridian called the prime meridian. Local sun time is obtained by setting timepieces at XII as the sun crosses the meridian. 311. The date line is the 180th meridian. A square of roofing is a ten-foot square (100 sq. ft.). Shingles are said to be laid so many inches to the weather when the lower end of each course of shingles extends so many inches below the course next above it. 314. A range is a tier (or row) of townships running north and south. 315. Such of the north and the west rows of half-sections of a township as do not have exactly 320 acres each are called lots. 316. The volume of any figure is the number of cubical units within its bounding surfaces 321. A straight line connecting two points of an arc is called a chord of the arc. 322. A circle is said to be circumscribed around a triangle and the triangle is said to be inscribed in the circle if all the vertices of the tri- angle are on the curve. 325. The longest side of a right triangle, that is, the side opposite the right angle, is called the hypothenuse. 388 RATIONAL GRAMMAR SCHOOL ARITHMETIC PAGE 327. The product obtained by using any number twice as a factor is called the square of that number. 328. The square root of a number is one of its two equal factors. 332. The cube of a number is the product obtained by using the number three times as a factor. 333. The cube root of a number is one of its three equal factors. 334. In triangles having the same shape, angles lying opposite propor tional sides in different triangles are called corresponding angles. In triangles having the same shape, sides lying opposite the equal angles are called corresponding sides. 340. The amount paid for insurance is called the premium. The written agreement between an insurance company and the insured is called a policy. The amount for which the property is insured is called the face of the pc'icy. 342. A tax is $ sum of money levied by the proper officers to defray the expen: :s of national, state, county, and city governments, and for pu -lie schools and public improvements. Assessed 1 aluation means the estimated value of the property that is assessed. 345. A stock (called also a stock certificate) is a written agreement made by a company to pay the holder a certain part of the earnings of the company. When a stock company pays to the holders of its stock #2 on each 100 of its capital stock, or 2fo on its stock, the company is said to be paying a $3 dividend, or a 2% dividend. At par means at its face value. Above par means more than its face value. Below par means less than its face value. A bond is a written agreement made by a national, state, county, or city government, or by a company, to pay the holder interest at a stated rate on a stated sum of money, called the face of the bond. A broker is a man that makes a business of buying and selling stocks and bonds for other people. A broker's charge for his services is called brokerage. 346. Preferred stocks are stocks which pay a fixed dividend before any dividends are paid on common stocks. Common stocks pay dividends dependent on the net earnings of the company after expenses and dividends on preferred stock have been paid. Government 2s are government bonds paying 2% interest. 347. Compound interest is interest on interest. The compound amount is the principal plus the compound interest. SYNOPSIS OF DEFINITIONS 389 349. Coupon notes are notes, attached to interest-bearing notes, for the amount of interest due at each interest-paying period. 353. An expression in which the sign < or > stands between two numbers is called an expression of inequality. 354. The signs =, < , and > are called relation signs. 364. For static and sliding friction, see pp. 364-5. 367. For freezing and boilings points, see p. 367. General Definitions Number is the result of the measurement of quantity. Number may also be defined as the ratio of quantities of the same kind. Quantity is limited magnitude. Arithmetic is the science of numbers and the art of using them. GENERAL INDEX EXPLANATORY. The following index has been prepared for daily use by young students of various ages and the principal purpose underlying its construction has been to enable all students to find readily and quickly the thing desired, that it may be in use not occasionally but continually. To this end the entries have been arranged not merely for the noun (or the principal noun) but also, in many cases, for other words of the phrase, thus making the index usable also by students to whom grammatical dis- tinctions are not entirely familiar. In reading, the phrase after the comma (where one occurs) is to be read first; thus "addition, short methods for" is read "short methods for addition." The references are to pages. Where two page references are sepa- rated by a dash the meaning is "and included pages;" thus 27-30 means pages 27, 38, 29, 30. Above par, 345 Account, forms of, 9, 28-30. 61, 62. 94-102 Accounts, bills and, 94-102 Accounts, farm, 28-30, 61, 62 Accounts rendered. 97 Acquired territory of United States, area of, 252 Acre, 227 Acres covered by one mile of furrow of given width, number of, 171 Acute angle (and figure), 293 Addend, 36 Addition, checking, 38, 42. 374 Addition of angles, 293-297 Addition (of common numbers), 35-40 Addition of decimals, see also decimals, 203, 204 Addition of fractions, sec fractions Addition of fractions having common fractional unit. 154 Addition, short methods for. 374 Additional problems on town block and lots, 91, 92 Admission of states, territories, etc., dates of, 53 Air, pressure of, 133. 134 Algebraic phrases. 185. 186 Almanac, 250. 251 Altitude, see also elevation and height Altitude of parallelogram. 116 Altitude of trapezoid. 117 Altitude of triangle, 117 Amount, 36. 258 Amount, compound, 347 Angles, 289 Angle (and figure), acute, 293 Angle (and figure), obtuse, 293 Angle (and figure), right, 282, 293 Angle (and figure), straight, 289 Angle and of arc, degree of, 290, 293 Angle and of arc, minute of, 291, 293 Angle and of arc, second of, 291, 293 Angle. s'.deg of, 292 Angle, vertex of, 186. 292 Angles, addition of, 293-297 Angles and arcs, measuring. 289-293 Angles (and figure), opposite or ver- tical, 292 Angles, cpmplemental, 295 Angles, difference and sum of, 293-297 Angles, division of, 186 Angles formed by parallel lines and . a line cutting them, 292 Angles of hexagon, sum of. 296 Angles of u-gon, sum of. 297 Angles of octagon, sum of. 297 Angles of parallelogram, sum of. 296 Angles of quadrilateral, sum of. 292, 296 Angles of right triangle, sum of acute, 296 Angles of similar triangle, correspond- ing. 334 Angles of triangle, sum of. 292, 295 Angles, subtraction of. 294 Angles, sum and difference of. 293-297 Angles, supplemental. 295 Angular degree. 290. 293 Angus cattle, weights and prices of, 213 Animals, habits of, 15 390 GENERAL INDEX 391 Annually, 258 Apothecaries' weight, 226 Apparent (sun) noon, clocktime of, 304 Applications of percentage, 340-349 Applications of proportion, practical, 197-200 Applications to transportation problems, see also tractive force, 270-275 Arabic notation, 34 Arabic numeral, 34, 35 Arc, 104, 289 Arc, degree of angle and of, 290, 293 Arc, minute of angle and of, 291, 298 Arc, second of angle and of, 291, 293 Arch, figure of stone, 140 Arcs, measuring angles and, 289-293 Are (unit of land measure), 238 Area, see also measuring surface and mensuration, Areas of common forms, 138-140 Areas of countries, 82 Ascending, reduction, 230, 232 Assessed valuation, 342 Assessment, lighting, 92 Assessment, paving, 304 Assessment, sidewalk, 92 Assessment, special water, 91, 92 Average, 10, 19, 22, 156 Avoirdupois weight, 226 Bale (unit of paper measure), 229 Bale of cotton, weight of, 171 Ball, velocity of cannon, 89, 216 Ball, velocity of rifle, 216 Bank discount, 266 Barometer (and figure), sec also weath- er, 26-28 Barometer, range of, 28 Barometer, reading of, 26 Barometer readings, table of, 27 Barrel, number of gallons in, 48, 235 Base (in percentage), 245 Base line, 314 Base of isosceles triangle, 108 Bases of trapezoid, 117 Basins, areas of river, 253 Battleships, data of, 40 Beans, capacity for absorbing water, 205, 206 Below par, 345 Bicycle (and figure), gear of, 222, 223 Billion, 32 and footnote Bills and accounts, 94-102 Birds, speeds of, 87, 216 Bisect, 107 Bisector, figure of perpendicular, 107 Bisector, perpendicular, 186 Block and lots (and figures), town, 2-4, 91, 92 Blocks, figure of city streets and, 116 Board foot, 12, 234 Board measure, see measuring wood, 12 Body, average surface of human, 134 Body, daily requirement of water and of solid food for human, 153 Body, normal temperature of a man's, 372 Boiling point, 367 Boiling temperature, 368, 372 Bond, 345 Bond, face of, 345 Bond quotations and transactions, stock and, 345 Bonds, stocks and, 345-347 Book, figure of, 194 Breeze, kinds of, 19 Brick wall, figure of, 313 Brick wall, kinds of, 4, footnote, Brick work, 3-6, 92, 93, 313 Broker. 345 Brokerage, 345 Bulk, see volume Bundle (unit of paper measure), 229 Bushel, 228 Bushel, number of cubic feet of grain in, 118, 215 Bushel of various articles, table of weights of, 228 Bushel, Winchester, 228 Butcher's price list, 8 Cables, data concerning, 364 Calculations, methods of shortening and checking, 373-378 Calendar, Gregorian, 229 Calendar, Julian, 229 Calendar month, 229 Calendar year, 229 Cancellation, 84, 88, 89, 168 and note. Cannon ball, velocity of, 89, 216 Capacity, 69. Capacity, measuring, 118, 119, 228, 238 Carrier pigeon, speed of, 216 Cars, engines and tenders, heights, lengths and weights of railroad, 20, 76, 135, 136, 271 Cash rent, 14 Casting out the nines, 42, 65, 66, 87 Cattle at Chicago Stock Yards, receipts of, 49. Cattle, weights and prices of Angus, 213 Cattle, weights and values of, 68 Cavetto molding, 825 Cellar excavation, 5 Cent, 225 Center of circle, 104 Centesimi, 226 Centi-, 237 Centigrade thermometer, 367-373 Centime, 226 Central time, 309 Centre. See center Century. 229 Certificate, stock, 845 Chain ; aZso square chain, 227 Chair, figure of, 194 Change of notation, 35 Checking calculations, methods of short- ening and, 373-378 Checking calculations : Addition, 38, 42, 374 Casting out the nines, 42, 65, 66, 87 Division, 86, 87, 377 Multiplication, 65, 66, 375-377 Square root, 378 Subtraction, 375 Chest expansion, 112 Chest measure, 17 392 RATIONAL GRAMMAR SCHOOL ARITHMETIC Chicago ^tock Yards, receipts of cattle Chord (of a circle), 321 Circle, area of, 220, 221 Circle, center of, 104" Circles, concentric, 104 Circulate, 219 Circulating decimal, 219 .^. <*M.civni, jj_o, ^J.i, ^;TO ' Cl 21 1 5 erenCe t0 diameter - rat io of, 214, Circumferences of various things. ,SYe olso diameters, etc 68 114 ifin 214 215 221 AOW, Circumscribed, 322 Cities, populations of, 42 55 City streets and blocks, figure of 116 (leaning upon death rate, effect of street, Coal, hard vs. soft, 8 of cubic feet of of continents, lengths of, Coffee imports of United States, 41, 68, *-_LO Colors, rates of vibration for 34 Commerce, 52-55 Commission, 254, 255 Common denominator, 146 157 Common fraction. See fraction Common fraction, reduction of' decimal Common^fra'cfhm to decimal, reduction Common multiple, 158 Common stock, 346 Common uses of numbers, 133-140 Common year, 229 Comparison of prisms, 280, 281 Comparison of sail areas of yachts, 224 Compasses (and figure), pen-foot and pin-foot of, 104 Complemental angles, 295 Complex decimal, 203 Complex fractions, 182, 183 Composite number, 149 Compound amount, 347 Compound denominate numbers, 25-24'> Compound interest, 347-349 Concentric circles, 104 Concrete unit, 225 Concurrent lines, 283-285 Cone, model and development of right circular, 318 Cone, volume of right circular, 318 Consignment 254 Constructions. (See also constructive geometry and models and develop- ments) : Dra icings : I. Circle with given radius, 104 TT'T L T ne equal to given Hue, KM*] Tl "" ' "- sum of two 105 V. Line equal to two, three or four times given line, 106 VI and I. Divide given line into 107, iff Pai ' tS (bisect it} ' 1 <* VII Equilateral triangle with each side equal to given line, 108 viii. Isosceles triangle with sirip equal to two given lines 108 * *H lene * rlan gle with sides equal to three given lines, 109 X. Three-lobed figure in circle of given radius, 109, 110 in circie f II. Divide given angle into two equal parts (bisect it), 186 I. Line parallel to given line, 187 Y lne P arallel to given line and TT1 through given point, 188 [II. Line parallel to given line and Tv^rM? glven P int < 188 IV. Divide given line into equal parts (trisect it), 188 3 ' an6 ' and 4 , line Through given point on given VTY| ie VrE erpei ? dicular ^ lin e, 192 VI 11. Through given point out of gmm line, perpendicular to line, I. Perpendicular to given line from given point on it, 276 II. Perpendicular to given line from given point out of it, 276; ^qu^-e in circle of given radius-, Greasings (paper-folding) : I. Perpendicular to given line through given point on it 282 II. Bisect an angle 283 TV' Sh l ' e l n n -P ara el lines, 283 ooo three angles of triangle ^OO V. Bisect given line. 284 VTT rft, iare and its dia gnals, 284 VII. Three perpendiculars from of triangle to opposite VIII. Three perpendicular bisectors of sides of triangle, 285 . 285 Find sum of two given agles, 294 294 eren e f tW given GENERAL INDEX 393 Find sum of angles of scalene tri- angle, 295 Find sum of angles of right triangle, 29G Find sum of acute angles of right triangle, 296 Find sum of angles of quadrilateral, 296 Find sum of angles of hexagon, 296 Find sum of angles of octagon, 297 Find sum of angles of n-gon, 297 Find relations of parts of rectangle with a diagonal, 297 Find relations of parts of parallelo- gram with a diago.nal, 297 I)r is less than ; is greater than, K2, 35S iperpendicular to, 325 V s_quare root of. 328 iK cube root of, 333 Silver coins, fineness of United States gold and, value of one pound of, 67, 233 Similar triangles. See also ratio and proportion, 143, 333-340 Similar triangles, uses of, 198-200, 335- 340 Simple decimal, 203 Simple interest, 131, 132 Six per cent method. 131. 132. 258 Skiameter (and figure), 299-305 Slant of sun's rays, changes in. 300-305 Sliding and static friction, 364-367 Sliding friction, law of. 367 Solar day, mean, 124. 229 Solar year, mean, 229 Solidus, 75 Sound, velocity of. 67, 89, 216 Sparrow, speed of, 216 Specific gravity, 217, 218 Sphere, area of surface of, 319 Sphere, volume of, 321 Spirometer (and figure), 16, 17 Sprocket wheel. 222, 223 Square (and figure), 115. 287, 327 Square, area of, 114, 189 Square cornered box, volume of, 69, 70, 118 Square, inscribed, 192 Square of number, 327 Square of roofing, 311 Square prism, model and development of. 279 Square prism, volume of, 118, 119, 281, 282, 316 Square root, 328 Square root by subtraction. 378 Square root, checking, 378 Square root of numbers and of products geometrically, 332 Square root, short method for, 378 Square roots, squares and, 327-332 Square unit, 45, 114 GENERAL INDEX 403 Squares and square roots, 327-832 Squares of units, tens, and hundreds, table of. 328 Standard base line, 314 Standard time, 309-311 Star, distance from sun to nearest, rf4 Statement of the problem. 58 Statements in words and in symbols, 358-364 States, territories, etc., areas of, 44, 4 States, territories, etc., dates of admis- sion and populations (1890) of. oo States, territories. etc., populations (1890, 1900) of, 44, 53 States, territories, etc., school data tor, Static friction, sliding and, 364-867 Stature of persons. See also heights, etc., 206, 207 Statute mile, 234 Steamboat, speed of, 216 Steel rail, weight of one yard of, bb, Steel, weight of one cubic inch of, 119 Stere, 239 Stock' and bond quotations and transac- tions, 345 Stock certificate, 345 Stock, common, 346 Stock, preferred, 346 Stock, sbare of, 345 Stocks and bonds, 345-347 Stone, measuring, 215, 227 Straight angle (and figure), 289 Straight line law, 366 Street cleaning, effect upon death rate, 19 Street crossing, figure of, 91 Street railroad cars, tractive force for 79, 80 Street railroad data, 77 Street railroad mileage and numbers ot employees, 77 Streets and blocks, figure of city, 116 Study of sun's rays, 299-365 Substituting, 272 Subtraction, checking, 375 Subtraction compared, division and, i 73 Subtraction of angles, 294 Subtraction (of common numbers), 46-o Subtraction of decimals. Sec also deci mals, 205, 206 Subtraction of fractions. See fraction Subtraction of fractions with commoi fractional unit, 154 Subtraction of literal numbers, 56 Subtraction, short methods for, 375 Subtrahend, 47 Successive division, greatest common di visor by, 151 Sum' and difference of angles. 296-29 Sum of acute angles of right .jtnangle 296 Sum of angles of hexagon. 296 Sum of angles of n-gon, 297 Sum of angles of octagon, 297 Sum of angles of parallelogram, 296 um of angles of quadrilateral, 292, 296 um of angles of triangle, 292-295 urns and differences of lines, products of, 297-299 Hm, diameter of, 321 mn, mass of, 34 >un, mean distance from earth to, dd Sun, mean radius of, 33 un noon, finding time of, 304 un time, local, 306 Sun's light and heat, distribution of, 299-805 Sun's rays, changes in slant of, 300-305 ^'s rays, study of, 299-305 plemental angles, 295 urface area, 114-118. 227. 238 Surface, measuring. 114-118. 227, 238 Surveying land. 314-316, 335-340 Surveyors' measure, 227 Switzerland, population and area of, 81 Symbol. See sign Symbols, statements in words and in, 358-364 Taxes. See also assessment, 342, 343 Tea into United States, imports of, 41, ro oi O Telegra'ph wire, cost and weight of, 59, 67 Temperature. See also thermometer end weather, 22-25, 250 Temperature, measuring, 122, 123 Temperature of a man's body, normal, 372 Temperature, range of, 23 Temperatures, boiling, 122, 368, 372 Temperatures, freezing, 122, 368 Temperatures, mean, 368 Temperatures, melting, 368, 871 Temperatures of gases, critical, 372 Ten-thousandth, 201 Tenant farmer, 14 Tenders, weights, lengths, and weights of railroad cars and engines, 20, 7b, 135, 136, 271 Tenders, water and coal capacity of locomotive. 271 Tenth (decimal), 201 Terminal growth of trees. 156, Iu7 Terms of fraction. See also lowest terms, 145 Terms of proportion, 196 Territorial growth of United States, 252 Territory of United States, area of ac- quired. 252 Tests of divisibility of numbers, 80, 86 Thermometer -(and figure), centigrade. See also weather, 367-373 Thermometer (and figure), Fahrenheit, 22-25. 122, 367-373 Thermometer (and figure), Reaumur, Thermometer, graphically, laws of, 372, ^73 Thermometer, positive and negative read- ings of, 369, 371 Thermometer, range of, 23 Thermometer, reading of, 367 404 RATIONAL GRAMMAR SCHOOL ARITHMETIC Thermometer. See also weather, 22-25, 367-373 Thermometers, equivalent readings of, 371-373 Thermometers, graduation of, 367-370 Thousand, 32 Thousandth, 201 Time belts of United States, map of, 301) Time, central, 309 Time, eastern, 309 Time, equivalents of longitude and, 306 Time, local (sun), 306 Time, longitude and. 306-311 Time, measuring, 124, 125, 229 Time, minute of, 229 Time, mountain, 309 Time, Pacific, 309 Time, second of, 229 Time, standard, 309-311 Time, sun, 306 Time to run (phrase), 265 To the weather (phrase), 311 Ton, 226 Ton, long. 226 Ton, metric, 239 Ton, number of cubic feet of hay and of coal in, 215 Torus molding, 325 Tower, model and development of, 313 Town block and lots (and figure), 2-4, 91, 92 Township (and figure). 126, 227, 314, 315 Tract of land, figure of, 11. 38, 91, 116. 125, 130, 195, 198, 199, 314, 315, 336-339 Tractive force for railroad trains, 135, 136, 271-275 Tractive force for road wagon, 79, 209- 212, 274, 275, 352 Tractive force for street railroad cars, 79, 80 Trade discount, 255, 256. 343, 344 Trade, 1891, 1901, United States ex- port, 54 Trailers of locomotive, 270 Train despatched report, 136-138 Train schedule, 137 Trains at Chicago, number of daily, 75 Trains, freight and passenger, 134-136 Trains, problems on railroad, 134-136 Trains, speed of express, 88, 89 Trains, tractive force for railroad, 135, 136, 271-275 Transportation problems, applications to. See also tractive force. 270-275 Trapezium (and figure), 115 Trapezoid, altitude of, 117 Trapezoid (and figure), 115, 287 Area of, 117, 118 Bases of, 117 Dimensions of, 118 Trees, growth of, 155-157 Trees, lateral growth of, 156, 157 Lateral growth of. 156, 157 Leaves of, 205, 249 Terminal growth of, 156, 157 Triangle, altitude of. 117 Triangle (and figure), equilateral, 108, 109 Triangle (and figure), isosceles, 108, 109 Triangle (and figure), right, 189 Triangle (and figure), scalene, 109 Triangle, area of, 117, 189 Triangle, base of isosceles, 108 Triangle, relation of sides of right, 326, 327, 331 Triangle, sum of angles of, 292, 295 Triangles (and figures) and their uses, similar, 143. 198-200, 333-340 Triangles (and figures), uses of the 30 and 60, and of the 45 right, 190- 192 Triangular prism (and figure), volume of, 320 Triangular prism, model and develop- ment of, 281 Triangular pyramid (and figure), volume of, 320 Triangular pyramid, model and develop- ment of, 320 Trillion, 32 and footnote Trisect, 188 Troy weight, 226 True discount, 266 Unit, 32 Unit, concrete, 225 Unit, fractional, 133, 144 Unit, square, 45, 114 United States, land areas of divisions of, 211 United States, maps of. 43, 809 United States notation, 32 and note United States Rule for Partial Payments, 268 United States, territorial t growth of, 252 United- States, 1790-1900, population of, 129 United States. 1900. distribution of pop- ulation of, 43-45 Uranus, diameter of, 321 Use of letters to represent numbers, 349-364 Uses of numbers, common, 133-140 Uses of similar triangles, 198-200, 335- 340 Uses of the equation. 352. 364-373 Uses of the 30 and 60. and the 45, right triangles, 190-192 Valuation, assessed, 342 Value, face, 265 Value, measuring, 225, 226 Value of digit, name, 31 Value of digit, place, 31 Value of x, 58 Values of cattle, weights and, 68 Venus, diameter of, 321 Vertex, 186, 278. 292 Vertex of angle, 186, 292 Vertical angles, opposite or, 292 Vertices. Plural of vertex Vibration for colors, rates of, 34 Vital statistics, 19 Volume, 316 Volume, measuring, 118. 119, 227, 238 Volumes, 316-321 GENERAL INDEX 405 Wagon, tractive force for road, 79, 209- 212, 274, 275, 352 Wall, figure of brick, 313 Walls of house, 4-6 Watch, figure of, 124, 289 Water area of earth, 252 Water, weight of one cubic foot of, 175, 271 Water, weight of one cubic inch of, 174 Weather, tiec also barometer, heat, light, meteorology, precipitation, rainfall, temperature, thermometer, wind, 250 Weather Bureau stations, elevations of, 249 Weather, to the (phrase), 311 Week, 229 Weight, apothecaries', 226 Weight, areas and volumes of irregular forms by proportion and, 139, 140 Weight, avoirdupois, 226 Weight, growth of boys and girls in height and, 18, 127'. 128. 206, 207 Weight, measuring, 120-122, 226, 238. 239 Weight of one cubic foot of various things, 88, 119. 217. 218 Weight of one cubic inch of steel, 119 Weight, troy, 226 Well lighted (room), 7, 114 Wheat yield and area of Dakotas, 82 Wheel, sprocket, 222, 223 Wild duck, speed of, 216 Wind movement. See also weather, 248, 249 Wind pressure, 19, 20 Wind, velocity of, 19, 20, 248 Winds, names of, 19 Wire, cost and weight of fence, 11, 12 Wire, cost and weight of telegraph, 59, 67 Wood (not fire-wood), measuring, 12 Worth, present, 266 Writing numbers, 34 Yachts, sail areas of, 224 Yard ; also square and cubic yard, 226, 227 Year, 229 Year, calendar, 229 Year, common. 229 Year, leap, 229 Year, mean solar, 229 Zenith, 304 Zero, 31 YC 49548 VERSITY OF CALIFORNIA LIBRARY