Anderson's Historical Series, A Junior Class History of the United States. Illustrated with hundreds of portraits, views, maps, etc. 272 pages. i6mo. A Grammar School History of the United States. Annotated; and illustrated with numerous portraits and views, and with more than forty maps, many of which are colored. 340 pp. i6mo. A Pictorial School Historr of fh#» TTm'i-Prl c^faf-o Tr^n* •,^^,,. SAMPLE COPY. Complete Graded Aritl:\nietic. PRICE. Fop Introduction, _-____- --q ^^t^g In Exchange for book in use on same subject of similar grade, - - - -.- - - ^5 *^^®* Books ordered for introduction wUl he delivered at above named rate in any part of the United States. A sample copy for examination, witli a view to introduction, 'mil be sent hy ma'l to teacher or school officer on receipt of the introduction price. Address, Clark & Maynard, New York. J. nc jniijLuiicai jxeauer, em uracing seiecnons m prose ana verse, irom standard writers of Ancient and Modem History ; with a Vocabulary of Difficult Words, and Biographical and Geographical Indexes. 544 pp. l2mo. The United States Reader, embracing selections from eminent Ameri- can historians, orators, statesmen, and poets, with explanatory observations, notes, etc. Arranged so as to form a Class-manual of United States His- tory. Illustrated with colored historical maps. 414 pp. lamo. CLARK & MAYNARD, Publishers, 734 BROADWAY, NEW YORK. ^ TREATISE ON PHYSIOLOGY AND HYGIENE. FOR EDUCATIONAL INSTITUTIONS AND THE GENERAL READER. By Joseph C. Hutchison, M.D., President of the New York Pathological Society; Vice-President of the New York Academy of Medicine; Surgeon to the Brooklyn City Hospital; and late President of the Medical Society of the State of New York. Fully Illustrated with ITumerous Elegant Engravings. 12aio. 300 pages. 1. Tlie Plan of tlie Work is to present the leading facts and principles of human Physiology and Hygiene in language so clear and concise as to be readily comprehended by pupils in schools and colleges, as well as by general readers not f amilar with the "subject. 2. The Style is terse and concise, yet intelligible and clear; and all useless technicalities have been avoided. 3. The Bange of Subjects Treated includes those on which it is believed all persons should be informed, and that are proper in a work of this class. 4. Ihe Subject-moMer. — The attempt has been made to bring the subject-matter up to date, and to include the results of the most valuable of recent researches to the exclusion of exploded notions and theories. Neither subject — Physiology or Hygiene — has been elabo- rated at the expense of the other, but each rather has been accorded its due weight, consideration, and space. The subject of Anatomy is in- cidentally treated with all the fullness the author believes necessary in a work of this class. 5. The Engravings are numerous, of great artistic merit, and are far superior to those in any other work of the kind, among them being tv>^o elegant colored plates, one showing the Viscera in Position, the other, the Circulation of the Blood. 6. The Size of the work will commend itself to teachers. It contains about 800 pages, and can therefore be easily completed in one or two school terms. The publishers are confident that teachers will find this work full of valuable matter, much of which cannot be found elsewhere in a class manual, and so pre- sented and arranged that the book can be used both with pleasiu-e and success in the schoolroom. " Many of the popular works on Physiology now in use in schools, academies, and colleges, do not reflect the present state of the science, and some of them abound in absolute errors. The work which Dr. Hutchison has given to the public is free from these objectionable features. I give it my hearty commendation." — Samuel G. Armor, M.D., late Professor in Ifichiijan University. "This book is one of the very few school books on these sub jects which can be imconditionally recommended. It is accurate, free from needless technicalities, and judicious m the practical advice it gives on Hygienic topics. The illustrations are excellent, and the book is well printed and bound. "—Boston Journal of Chemistry. "Just the thing for schools, and I sincerely hope that it maybe appreciated for what it is worth, for we are certainly in need of books of this kind." — Prof. Austin Flint, Jr., Professor of Physiology in Bellevue Hospital Kedical College, JS^ew York City, and author of " Physiology of il/a?!," etc., etc. "I have read it from preface to colophon, and find it a most desirable text-book for schools. Its matter is judiciously selected, lucidly presented, attractively treated, and pointedly illustrated by memorable facts: and, as to the plates and diagrams, they are not only clear and intelUgible to beginners, but beautiful speci- mens of engraving. I do not see that any better presentation of the subject of physiology could be given ■v\'ithin the same compass." — Prof. John Ordronaux, Professor of Physiology in the University of Vermont, and also in the 2^'ational Medical College, Washington, D. C. The above work is the viost popular loork on the above subjects yet published. It is used in thousands of schools tvith marked success. PnbHshed bj CLAEK & MAYNABD, New York. EDUCATION DEPT. Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/completegradedarOOthomrich THOMSON'S ARITHMETIC^A^K\ \SERIES IN TWO BOOKSi , ^ ''\'''. ' ■ , COMPLETE GRADED ARITH M ETIC, Oral and Written, UPON THE INDUCTIVE METHOD OF INSTRUCTION^ FOR SCHOOLS AND ACADEMIES. James B. Thomson, LL. D., AUTHOR OF MATHEMATICAL SERIES. NEW YORK: Clark k Maynard, Publishers, 734 Broadway. CHICAGO : 46 Madison Street. 1884. THOMSON'S MEW ARITHMETICAL SERIES IN T^A^O BOOKS. Q f\ 10 3. 1-^* .. I. First Lessons in Arithmetic, Oral and Written. Illustrated. ^^ (For Primary Schools.) fo^-^^-'VoAr . II. Complete Graded Arithmetic, Oral and Written. In one Volume. (For Schools and Academies.) Key to Complete Graded Arithmetic. (For Teachers.) THOMSON'S MATHEMATICAL SERIES. illustrated table-book. NEW rudiments OF ARITHMETIC. complete INTELLECTUAL ARITHMETIC. NEW PRACTICAL ARITHMETIC. KEY TO PRACTICAL ARITHMETIC. HIGHER ARITHMETIC. KEY TO HIGHER ARITHMETIC, PRACTICAL ALGEBRA. KEY TO PRACTICAL ALGEBRA. COLLEGIATE ALGEBRA. KEY TO COLLEGIATE ALGEBRA. GEOMETRY AND TRIGONOMETRY. (In preparation,) Copyright^ 1882, by James B. Thofnson. Smith k McDougal, Electrotyper.s. 82 BeekmaD St., N. Y. TT/n * rPTrVNT mTOT ■t [^ E E F A O E rpHE book now offered to the public, unites in one volume Oral and Written Arithmetic upon the inductive method of instruction. Its aim is two-fold : to develop the intellect of the pupil, and to prepare him for the actual business of life. In securing these objects, it takes the most direct road to a practical knowledge of Arithmetic. The pupil is led by a few simple appropriate examples to infer for him- self the general principles upon which the operations and rules depend, instead of taking them upon the authority of the author without explana- tion. He is thus taught to put the steps of particular solutions into a concise statement, or general formula. This method of developing princi- ples is an important feature. It has been a cardinal point to make the explanations simple, the steps in the reasoning short and logical, and the definitions and rules brief, clear, and comprehensive. The discussion of topics which belong exclusively to the higher depart- ments of the science is avoided ; while subjects deemed too difficult to be appreciated by beginners, but important for them when more advanced, are placed in the Appendix, to be used at the discretion of the teacher. Arithmetical puzzles and paradoxes, and problems relating to subjects having a demoralizing tendency, as gambling, etc., are excluded. All that is obsolete in the former Tables of Weights and Measures is eliminated, and the part retained is corrected in accordance with present law and usage. Examples for Practice, Problems for Review, and Test Questions are abundant in number and variety, and all are different from those in the Practical Arithmetic. iviS4*J34B iv Preface, The arrangement of subjects Is systematic; no principle is anticipated, or used in the explanation of another, until it has itself been explained. Subjects intimately connected are grouped together in the order of their dependence. In connection with the Notation of Integers, that of Decimals is taught to three places below units, corresponding to dimes, cents, and mills. Decimal or Metric Weights and Measures are placed next after Decimal Currency. Percentage is followed by its applications in their proper order, as Profit and Loss, Commission, etc. General Analysis, covering the several departments of Commercial Arithmetic, has received special attention. The articles devoted to Test Questions, and to the Entrance Problems of different Colleges, will be found a valuable addition and excellent practice. Thanks are due to the College OflBcers who have kindly furnished copies of their Examination Papers. In the preparation of the work the author has carefully weighed tlie discussions in the various journals of education respecting the present wants of our schools, and has endeavored to provide for them. He has availed himself of many valuable suggestions from business men, practical teachers, and educators, all of whom he desires to thank most cordially for the aid they have rendered. He cheerfully submits the result of his labors to his friends, the teach- ers, and the public, for whose favorable verdict upon his former efforts he desires to express renewed obligations. J. B. T. Brooklyn, April, 1882. :^Bz ONTENTS. 286 Involution 289 Evolution 292 Square Root 293 Right-angled Triangles .... 299 Cube Root 303 Illustration by Cu. Blocks. . . 305 Progression 311 Arithmetical Progression. . . . 311 Geometrical " 316 Mensuration 319 Triangles 320 Quadrilaterals 323 Circles 326 Solids 328 Gauging of Casks 333 Tonnage of Vessels 334 Test Questions 335 College Entrance Papers. . . . 353 Appendix 359 Roman Notation 359 Contractions 361 Demonstration of g. c. d. . . 363 Circulating Decimals 365 Surveyors' Measure 368 Government Lands 369 Apothecaries' Fluid Measure. 372 Annual Interest 373 Partial Payts. Ct., Vt., N. H. 374 12% Method of Casting Int. . 378 Average of Mixtures 378 QJ < " RITHMETIC. Definitions. Art. 1. A Unit is one or any single thing; as, one, one book, one chair. 2. A Number is a imif or a collection of units. Thus, one, or one book, is a unit ; five, or five books, is a collection of units. 3. The Unit of a Number is one of the collection forming that number. Thus, the unit of four books is one book, of seven is one. 4. An Abstract Number is one that is 7iot applied to any object. Thus, four, five, thirteen, etc., are abstract numbers. 5. A Concrete Number is one that is applied to some object. Thus, five boys, seven apples, etc., are concrete numbers. 6. Like Numbers are those which express units of the same ki7id. Thus, eight pears and five pears, four and seven, are like numbers. 7. Unlike Numbers are those which express units of different kinds. Thus, seven, six peaches, nine days, are unlike numbers. 8. Numbers are expressed by words, by figures, or by letters. 9. Arithmetic is the science which treats of numhers and their applications. r^ •>>.. o i \ - < ♦ OTATIO]^ AND NUMERATION. 10. Notation is expressing numbers by figures or letters. 11. Numeration is reading numbers expressed by figures or letters. Note. — There are two methods of Notation, called the Arabic and the Romnn. The former is the method in general use, and is so called be- cause it was introduced into Europe in the loth century by the Arabians. 12. The Arabic Notation expresses numbers by teyi different characters, called Figures ; viz., /, 2, 3, J,, 5, 6, 7, 8, (j, 0. One, Two, Tliree, Four, Five, Six, Seven, Figlit, Nine, Naught. 13. The first nine are called Significant figures, because they always express some value. They are also called Digits. 14. The last one is called Naught, because when standing alone it has no value. It is also called Cipher or Zero. 15. The Value of a figure is the numher it represents. 16. Nine is the largest number expressed by one figure. 17. The significant figures standing alone, express single things or ones ; as, 4 apples, 5, 7. 18. To express the numbers from 7ii7ie to one liundred requires two figures written side by side. 19. The first figure at the right denotes Ones, wdiich are called Units of the First Order. 20. The figure in the second place denotes ten ones, which are called Tens, or Units of the Second Order. Thus, the figures 35, denote 5 ones and 3 tens, and are read, "Thirty- five." Notation and Numeration. Write the following numbers in figures : 1. Twenty-five. 5. Thirty-six. 2. Tliirtj-eiglit. 6. Forty-nine. 3. Fifty-six. 7. Fifty-four. 4. Forty-two. 8. Sixty-eight. Eead the following numbers : 13. 63. 17. 43. 21. 64. 14. 54. 18. 38. 22. 57. 15. 49. 19. 69. 23. 76. 16. 78. 20. 84. 24. 92. 9. Seventy- three. 10. Fifty-nine. 11. Eighty-eight. 12. Ninety-nine. 25. 75. 26. 88. 27. 93. 28. 99. 21. Ninety-nine is the largest number which can be ex- pressed by two figures. 22. To express the numbers from ninety-nine to 07ie thou- sand, requires tliree figures written side by side. 23. The figure in the third place denotes ten tens, which are called Hundreds, or Units of the Third Order. Thus, the figures 436, denote 4 hundred, 3 tens, and 6 units, and are read, " Four hundred thirty-six." Write the following numbers in figures : 29. Two hundred forty-six. 33. Five hundred eight. 30. Three hundred fifty-four. 34. Six hundred seventy. 31. Five hundred thirty-two. 35. Eight Iiundred three. 32. Four hundred fifty. 36. K'ine hundred ninety-nine. Read the following numbers :* 37. 243. 41. 632. 45. 38. 420. 42. 567. 46. 39. 364. 43. 740. 47. 40. 419. 44. 321. 48. 24. Nine Hundred Ninety-nine is the largest number ex- pressed by three figures. * In reading numbers expressed by three or more figures, omit the word and after hundreds. 407. 49. 830. 536. 50. 604. 249. 51. 783. 680. 52. 999. 10 Notation and Numeration, 25. To express larger numbers, other orders of units are formed, called thousands, ten-thousands, hundred-thousands, millmis, etc. 26. A figure in the fourth place denotes Thousands, which are called Units of the Fourth Order. 27c A figure in the fifth place denotes Ten-thousands, which are called Units of the Fifth Order. 28. A figure in the sixth place denotes Hundred-thousands, which are called Units of the Sixth Order. 29. A figure in the seventh place denotes Millions, which are called Units of the Seventh Order. 30. If any orders are omitted, ciphers must be written in their places. Thus, four tliousand three hundred five, is written 4305. The figures 5046, denote 5 thousands, hundreds, 4 tens, and 6 units, and are read, " Five thousand forty-six." Write the following in figures : 53. Three thousand two hundred sixty-eight. 54. Five thousand seventy-five. 55. Six thousand three hundred ten. 56. Seven thousand fifty-three. 57. Eight thousand seven hundred five. 58. Nine thousand nine hundred, ninety-nine. Read the following numbers : 59. 1265. 63. 3420. 67. 5101. 60. 1503. 64. 3051. 68. 5049. 61. 2034. 65. 4036. 69. 6008. 62. 2105. 66. 4003. 70. 7059. 31. The different values of units expressed by the significant figures, are determined by the place they occupy, and are called simple and local values. Notation and Nameration, 11 These values are illustrated by the following diagram 1000. 100. 10. 1. 32. The Simple Value of the units represented by the sig- nifieaDt figures is the number which they represent when standing alone or in units place. 33. The Local Value of these units is the number which they represent wdien standing oq either side of units place. Thus, 2 standing alone, or in the jirst place, denotes 2 simple units ; in the second place, it denotes 2 tens, as in 25 ; in the third place, it denotes 2 hundreds, as in 246, etc. Note. — These different orders of units correspond to dollars, dimes, and cents. Thus, 10 cents make 1 dime, 10 dimes 1 dollar. Xow a cent is a unit, a dime is a unit, and a dollar is a unit ; but these miits have diiferent values, corresponding to the orders of units. 34. From the above illustrations w^e derive the following Principles. j?°. Ten units of any order mahe a unit of the next higher order. 2°. Moving a figitre one place to tlie left, increases its value ten times. 3°. Moving a figiire one place to the right, diminishes its value ten times. 35. Hence, the great law of the Arabic Notation, viz.: The Orders of Units increase and decrease by the uniform scale of Ten. The Arabic Notation is therefore called the Decimal System, from the Latin word decern, wiiich means ten^ 12 Notation and Numeration, 36. To Express Decimal Parts of a Unit. By the law of the decimal notation a unit of the tliird order is ten times a unit of the second order ; a unit of the second order is ten times a simple unit or one. By extending this law heloiu units, a simple unit is ten times a unit of the first decimal order ; a unit of the first decimal order is ten times a unit of the second decimal order, and so on. In this way a series of orders is formed below units which regu- larly decrease by the scale of ten. 37. The first order on the right of units is called tenths ; the second, hundredths ; the third, thousandths ; etc. 38. These lower orders are separated from units by a period (.) called the Decimal Point. 39. The orders on the left of the decimal point are called Whole Numbers or Integers ; those on the rigid, Decimals. Thus, seven and five tenths are written 7.5; nine and fifty-three hundredths are written 9.53 ; sixty-five and two hundred seventy-three thousandths are written 65.273. The figures 4.7 denote four ones and seven tenths of one, and are read, " Four and seven tenths." The figures 6.35 denote six ones and thirty-five hundredths of one, etc. Write the following in figures : 1. Eive tenths. 7. 63 and 7 hundredths. 2. Four hundredths. 8. 3 units and 5 thousandths. 3. Sixty-five hundredths. 9. 245 and 25 hundredths. 4. Seventeen thousandtlis. 10. 7 and 62 hundredths. 5. Forty-two thousandths. 11. 456 and 273 thousandths. 6. Fifty-four thousandths. 12. 503 and 6 tliousandths. Bead the following : 13. 3.7. 17. 62.3. 21. 0.25. 25. 42.365. 14. 5.24. 18. 75.21. 22. 0.7. 26. 125.034. 15. 23.9. 19. 36.45. 23. 0.253. 27. 245.007. 16. 31.25. 20. 68.4. 24. 0.45. 28. 360.248. Notation and Nmneration, 13 40. The French Method of writing and reading large num- bers, is shown in the following Numeration Ta b le Names of Periods. Trillions. Billions. Millions. Thousands. Units. Thou- sandths. ~ a ;-, Orders of '^ . m .Q 'i o 7^ ^ o Units, ;-i i; _o '^3 2 is oqual to 3 tens and 2 units ; now 3 tens from 7 tons leave 4 tens or 40, and 2 units from 5 units, leave 3 units, wliieh added to 40 makes 43. Ans. 45] cts. Note. — When the numbers in subtraction are large, it is advisable, iu mental operations, to be^n at the hi^diest orders, as in addition. Subtract ion. 33 7. If the price of a history is 90 cts., and that of a reader is 70 cts., what is the difference in their price ? 8. A fiu'mer raised SO bu. com and sold 50 bu. : how many bushels had he left ? 9. The united ages of two persons is ^ years, and the younger is '^'^ : what is the age of the older ? 10. William and Chiirles together caught 5S fish, and William caught "27 ; which caught the more, and how many? 11. In a school of So pupils. -IS ai-e girls : how many Ixiys are there, which department is the larger, and by how many ? 12. A lady haviug "2 ten-dollar bills, paid 19 for a hat. -$4 for lace, and $'2 for gloves : how much money had she left ? 13. Which is the greater. '24 -f 10. or 5"2 — 9 ? 14. A gentleman paid $1*2 for pants, 19 for a Test, and #7 for boots ; he paid for them with "2 ten-dollar bills and 2 fives : how much change should he have ? 15. A merchant paid #7S for a case of goods, and ^5 freight : for how much must he sell them to make #10 ? 16. If you have $1:7 and pay |17 for a bicycle and $"2 for a cap, how much mouey will you have left ? 17. A lad had 51 uuirbles : he gave away •2S and found 5 : how many marbles had he then ? Oral Drill in Adding and Subtracting. 1. To 5 add 6, subtract 3. add 7. subtract S, add 4, subtract 7, add S, subtract 3 : what is the result? XOTE. — ^YMle the teacher dictares the example. "* To 5 add 6. subtract 3," etc.. the pupils thi .k 11. S. 15. etc. The roiswer mav be given in con- cert, or bv some individual designated by the teacher. 2. From 15 take 6, add 7, take S, add 5, take 6, add 1. take 9, add 10 : result ? 3. To 11 add 5, take 7, add 1, take 3, add S, take 5, add 6, take 4, add 9, take 6 : result ? 4. How many are *23 — 7 — 3— 1-flO— S-1-5 — 7-f 6 ? 5. How many are 7 + 9 — 10-f 0—1-^7 — S-4-9? 6. How many are 1'2 -^ — S -h 1—3 — '20—10 -f 5 ? 7. How many are •2?— S — 9— 10 -[-7—0— S-r 9 — 7 — 5 ? 34 Subtraction, 8. How many are 23 — 6 + 11 — 8 + 9 — 6-1-4—7 + 8 ? 9. How many are 32 — 5 + 3 — 7 + 6 — 8 + 9 — 7 + 4—6? 10. How many are 35 + 8 — 7 + 6 — 4 + 8 — 7 + 5 — 8 + 12? 11. How many are 38+7 — 4 + 5 — 8 + 6 + 2 — 9 + 6? 12. How many are 28 + 4—7 + 6 — 9 + 8 — 9 + 10 ? Written Problems for Review. 74. 1. The minnend is 3642.05, and the difference is 3202.8 \ what is the subtrahend ? 2. Two brothers commenced business at the same time ; one gained $3678 in five years, the other gained 12387. How much more did one gain than the other ? 3. The subtrahend is 48206.5 and the difference is 35206.2 ; what is the minuend ? 4. A ship having a cargo valued at $100000, was overtaken by a storm, and $27680 worth of goods were thrown overboard. How much of the cargo was saved ? 5. A gentleman having $1768 on deposit, gave a check for $175 to one man, to another for $238.25, and to another for $369.50. How much remained on deposit ? 6. An orchard contained 120 apple trees,«47 peach trees, and 28 pear trees. Of the apple trees 26 were cut down, 18 of the peach trees died, and 5 of the pear trees were blown down. How many trees wTre left ? 7. A gentleman had $2700 to distribute among his three sons. To the eldest he gave $825, to the second $785, and the" remainder to the youngest. How much did the youngest son receive ? 8. A merchant had in his storehouse 6384 bushels of wheat, 3752 bushels of corn, 4564 bushels of oats, and 1384 bushels of rye ; it was broken open and 3564 bushels of grain taken out. How many bushels remained ? 9. If a man's income is $4586 a year, and he spends $384.86 for clothing, $568 for house rent, $784.75 for provisions, $568.50 for servants, and $369 for traveling, how much will he have left at the end of the year? Subtraction. 35 10. A gentleman left a fortune of $18864 to his two sons and one daughter; to one son he gave $6389, to the other $6984. How much did the daughter receive ? 11. A man having $7689, invested $689 in railroad stock, $500 in a woolen factory, and $1250 in bank stock. How much had he left ? 12. What number added to 3645 makes 630712 ? 13. A man worth $30000, lost a store by fire worth $5000, and goods to the amount of $3578. How much had he left ? 14. From twenty-five thousand, twenty-five, take 28 hundred. 15. From 16 millions, 16 thousand, take 16 hundred. 16. The difference between 185 billions, and 185 millions ? 17. What number mast be added to 836.25 to make 2323 ? 18. How many times can 563 be subtracted from 2815 before the latter will be exhausted ? 19. What number is that, from which if you take 42371, the remainder will be 19289 less 176.05 ? 20. What number is that, from which if you take 18268, the remainder will be 26017 — 17312? 21. What number is that, from which if 27239 be taken, the remainder will be 9897—3076.5 ? 22. A says to B, " I have 2675 sheep " ; B replies, '' I have 763 less than you " ; C adds, " I have as many as both lacking 105." How many sheep had B and C ? 23. The sum of 3 numbers is 23257 ; the first is 9277, the second is 1283 less than the first ; what is the third number ? 24. The population of the U. S. in 1840 was 17069453, in 1880 it was 50155783 ; what was the increase in 40 years ? Questions. 62. What is Subtraction? 63. The Subtrahend? 64 Minuend? 65. The Answer? 66. The Sign of Subtraction? What called ? How read? 67. For what are the Parenthesis and Vinculum used? 68. What num- bers only can be subtracted ? What orders ? 68. If the difference of two numbers is added to the less, to what is the sum equal? If two numbers are equally increased, how is their difference affected? 71. How subtract decimals and dollars and cents? 72. Give the general rule. How is subtraction proved? Ul/riPLICATION. -[Sy -K- Mental Exercises. 75. 1. What will 3 pencils cost, at 4 cents apiece ? Analysis. — At 4 cts. apiece, 3 pencils will cost the sum of 4 cts. + 4 cts. + 4 cts., or 4 cts. taken 3 times, whicli are 12 cts. Or, more briefly, 3 pencils will cost 3 times as much as 1 pencil, and 3 times 4 cts. are 13 cts. 2. At 15 each, what w^ill 4 hats cost ? 3. At 5 cts. apiece, what will 3 bananas come to ? 4. In 1 gallon there are 4 qts.; how many qts. are in 5 gallons ? 5. At 6 cts. a lb., what will 4 lbs. of rice cost ? 6. If 1 qt. of cherries cost 6 cts., what will 3 qts. cost ? 7. What will 5 vests cost, at $7 apiece ? 8. If it takes 6 yds. of cloth to make 1 cloak, how many yards will it take to make 5 cloaks ? 9. In 1 week there are T days; how many days in 4 weeks? 10. How many units in five 8's united in one number ? Definitions. 76. Multiplication is finding the amount of one number taken as many times as there are units in another. 77. The Multiplicand is the number to be multiplied. 78. The Multiplier is the number by which we multiply. It shows how many times the multiplicand is to be taken. 79. The Ansioer, or number found by multiplication, is called the Product. Thus, when it is said that 4 times 6 are 24, 6 is the mutiplicand, 4 the multiplier, and 24 the product. Multiplication, 37 80. The multiplicand and multiplier which produce the product, are called its Factors. 81. The Sign of Multiplication is x . It shows that the numbers between which it is placed are to be multiplied together, and is read '' times/' or " multiplied by." Thus, 7 X 4 = 28, is read, " 7 times 4," or " 7 multiplied by 4 equals 28." Note. — Multiplication is similar in principle to addition, and may be performed by it. Thus, the product of three times 4, is 12, which is the same as the sum of 4 + 4 + 4, Multiplication Table. 2 times 1 are 2 2 '^ 4 3 " 6 4 5 6 7 9 10 1 1 12 a a a a 10 12 14 16 18 20 22 24 3 times 1 are 3 2 '' 3 " 4 5 6 7 8 9 10 II 12 6 9 12 15 18 21 24 27 30 4 times I are 4 2 " 8 3 " 12 4 S ^^ 16 '' 20 6 7 - 24 '' 28 8 " 32 9 " 36 ^^ 40 I 2 " 44 - 48 5 times 1 are 5 2 '^ 10 20 25 30 35 8 '' 40 9 " 45 10 '' 50 11 " 55 12 '^ 60 6ti] Qies 7ti I are 6 I ai 2 '' 12 2 ' 3 " 18 3 ' 4 " 24 4 ' 5 ' 6 ^ 30 36 5 ' 6 ^ r: 42 - 48 7 ' 8 ' 9 ' 10 ^ 54 ' 60 9 ' 10 ' II ' 66 II ' 12 ^ 72 12 ' 14 21 28 35 42 49 56 63 70 77 84 8 times I are 2 3 4 5 6 7 8 9 10 II 12 16 24 32 40 48 56 64 72 80 88 96 9 times I 2 3 4 5 6 7 8 9 10 1 1 12 are a a a a a a a a 9 18 27 3^ 45 54 63 72 81 90 99 108 10 times I 2 3 4 5 6 7 9 10 II 12 are 10 '' 20 " 30 " 40 " 50 60 70 80 90 100 2 3 4 5 6 7 8 9 TO I 10 I II 120 12 times are II iC 22 ii ii 33 44 ii 55 66 ii ii 77 88 i( ii 99 no ii 121 a 132 12 times I 2 3 4 5 6 7 8 9 10 II 12 are ii ii ii ii a ii ii I 2 24 36 48 60 72 84 96 108 120 132 144 Note. — Promiscuous exercises upon the table should be repeated till any combinations within its limits can be answered instantly. 38 Multvplication, 11. Count by 3's to 30 and back to 0. 12. Name the products by 4 to 40 and back. 13. Name the products by 5 to 50 and back. 14. Name the products by 6 to 60 and back. 15. Name the products by 7 to 70 and back. 16. Name the products by 8 to 80 and back. 17. Name the products by 9 to 90 and back. 18. How many times 7 are 28 ? 22. Times 5 are 30 ? 19. How many times 6 are 42 ? 23. Times 7 are 56 ? 20. How many times 8 are 48 ? 24. Times 9 are 54 ? 21. How many times 7 are 63 ? 25. Times 9 are 72 ? 26. 6x7—5x6=? 29. 7x8—4x6? 32. 8x9 — 7x5? 27. 7x5—4x8=? 30. 8x6 — 6x8? 33. 7x8 — 6x9? 28. 5x9 — 6x7=? 31. 9x7 — 6x9? 34. 9x9-8x8? DeveTjOpment of Principtes. 82. 1. What is the product of 17 multiplied by 4 ? 2. What is the product of 9 multiplied by 5 ? 3. What kind of numbers are these miiUipUcands 9 Ans. The first is concrete, the second is abstracts 4. What kind of a number is the ynultiplier f Ans. It is an abstract number in both examples. 5. What kind of a number is the product ? A71S. The same in each as the multiplicand. 6. What is the product of 7 days multiplied by 9 pounds ? Ans. Pounds are concrete numbers and 7 days cannot be taken 4 pounds times. 7. Which is the greater number, 3 times 4, or 4 times 3 ? 83. From the above examples we deduce these Principles. i°. The multiiMcand may be either abstract, or coyicrete. 2°. The multiplier ynust be considered an abstract number. S°. The midtiplicand and product are like numbers. Jf°. The product is the same in ivhatever order the factors are taken. Multiplicatioiu 39 Written Exercises. 84. When the multiplier has but one figure. 1. If a rail-car goes 538 miles a day, how far will it go in 4 days ? Analysis. — The car ^vill go 4 times as far in operation. 4 days as in 1 day. Write the multiplier under 538 Multiplicand, the multiplicand, and beginning at units say, ^ Multiolier "4 times 8 units are 32 units, or 8 tens and 3 units." We set the 2 in units place and ^^^^' ^^^^ miles, add the 8 tens to the product of tens, as in addition. Next, 4 times 3 tens are 12 tens, and 8 tens added make 15 tens, or 1 hundred and 5 tens. We write the 5 in tens place, and add the 1 hundred to the product of hundreds. Finally, 4 times 5 hundreds are 20 hundreds, and 1 hundred added makes 21 lumdreds, or 2 thousand and 1 hundred. We write the 1 in hundreds place, and the 2 in thousands place. The product is 2152 miles, Aiu. 2. If 1 bale of cotton weighs 250 pounds, what will 7 bales weigh ? 3. A drover bought 6 flocks of sheep, the average number of which was 735 ; how many sheep did he buy in all ? (8.) 5178 in. 4 85. When the multiplicand or multiplier contains Decimals. Decimals and dollars and cents are multiplied like integers, as many figures being pointed off for decimals in the product, as are found in both factors. 9. What is the product of $64,375 multiphed by 7 ? Explanation. — U. S. Money as well as Decimals is operation. multiplied like whole numbers ; from the rio-ht of the $64,375 product, as many decimal figures are pointed off as there f^ are decimal places in hoth factors. $450,625 (10.) (11.) (12.) (13.) (14.) 384.9 67.02 54.37 8.603 87.46 5 6 8 7 9 (4.) (5.) (6.) (7.) 574 lbs. 725 ft. 869 yds. 4256 bu, 3 4 5 3 40 Midtiplication. (15.) (16.) $4352.67 $676,238 (17.) $7283.04 (18.) $9280.23 (19.) $807,206 6 5 7 8 9 (20.) Multiply $34.56 By 4 (21.) $242.63 6 (22.) $0.96 8 (23.) $0,873 9 24. What cost 8 barrels of flour, at $7.50 a barrel ? 25. What will 458 hats cost, at $6 apiece ? Note.— In tliis example the true multiplicand is $6. But it is more convenient to use 6 as tbe multiplier and 458 as the multiplicand, as follows: (Art. 83, 4 .) 458 Analysis. — At $1 each, 458 hats would cost $458, and at $6 each, they will cost 6 times $458, or $2748. . - Ans. $2748 26. What cost 375 tons of hay, at $8 per ton ? 27. What cost 5265 bales of cotton, at $8 a bale ? 28. At $4 a barrel, what cost 1500 barrels of apples ? 29. At $5.67 a yard, what cost 8 yds. of cloth? 30. What cost 2350 clocks, at $9 each ? Mental Exercises. 86. 1. What will 4 vests cost, at $7 apiece ? 1st. Analysis. — 4 vests will cost 4 times as much as 1 vest, anc* 4 times $7 are $28, Ans. 2d. Analysis.— Since 1 vest costs $7, 4 vests will cost 4 times $7, or $28. Therefore, 4 vests will cost $28. Note, — The essentials of a good iVnalysis are clearness, brevity, appro- printe language, and a logical conclusion. Sameness in form should be avoided. 2. If you write 8 lines a day, how many lines will you write in 6 days ? 3. If there are 5 school days in 1 week, how many school days are there in 12 weeks ? 4. What is the cost of 9 bananas, at 6 cents each ? 5. A grocer sold a turkey weighing 8 lbs. at 11 cts. apound ; what did it come to ? Multiplication. 41 6. What will 9 qts. of cherries come to, at 12 cts. a quart ? 7. At 8 cts. a mile, what will it cost to ride 9 miles ? 8. Carrie made 9 bouquets, each having 10 flowei'S ; how many flowers had they all ? 9. At $12 a hundred, what will 4 hundred melons cost? 10. What cost 8 dozen eggs, at 12 cts. a dozen ? 11. If a man walk 47 miles a day, how many miles will he walk in 6 days ? Analysis. — 47 equals 4 tens and 7 units. Now 6 times 4 tens are 24 tens, or 240, and 6 times 7 units are 42 units, which added to 240 make 282. Ans. 282 miles in 6 days. Note, — When the multiplicand is large, it is advisable in mental opera- tions to begin with the liigest order, as in addition. (Art. 54.) 12. What cost 4 acres of land, at $36 an acre? 13. At $75 a share, what will 7 shares of Bank Stock come to ? 14. A furniture dealer sold 10 sofas at $87 apiece ; what did he get for all of them ? 15. If a railroad car goes at the rate of 57 miles an hour, how far will it go in 8 hours ? 16. How many are 9 times 73 ? 8 times 63 ? 17. How many are 7 times 86 ? 6 times 97? 18. How many are 8 times 94 ? 9 times 89 ? 19. If melodeons are $75 apiece, what will 6 cost ? Written Exercises. 87. When the multiplier has two or more figures. 1. What is the product of 324 multiplied by 132 ? Explanation. — We write the multiplier under 324 Multiplicand, the multiplicand, as in the margin, and beginning at ^^^2 Multiclier tlie right, multiply by each order successively. Thus, multiplying 824 by 2 units, the first partial ^^^ P^^- P^o beginning at the right, multiply each order of the multi- plicand by each order of the multiplier, placing the right hand' figure of each partial product under the order which produced it. II. Add the partial products together, and from the right point off as many figures for decimals as there are places of decimals in the multiplicand and multi- plier ; the result will be the product required. Proof. — Multiply the multiplier by the multiplicand ; if the two results a.gree the luorh is correct. (For proof by casting out the 9's, see Art. 876, Appendix. ) A.PJPT.ICATIONS. 1. Allowing 365 days to a year, how many days has a man lived who is 45 years old ? 2. If a garrison consume 725 pounds of beef in one day, how many pounds will they consume in 126 days? 3. What cost 243 chests of tea, at 137 per chest? 4. A man bought 268 horses, at $63 apiece ; what did they come to ? 5. If sound moves 1142 feet in a second, how far will it move in 60 seconds ? 6. If a cattle train has 23 cars and each car contains 68 sheep, how many sheep in the train? Multiplication. 43 7. How much can a man earn in 48 months at $125 a month ? 8. At $32 each, how much will it cost to furnish the outfit for 560 policemen ? 9. How many bushels of corn may be raised on 485 acres which average 37 bu. to the acre ? 10. There are 640 acres in a square mile ; how many acres are there in 75 square miles ? What is the product of 11. 8623 by 24? 23. 2734 by 234? 12. 2538 by 39 ? 24. 4803 by .325 ? 13. 4752 by 43 ? 25. 6578 by 467 ? 14. 5843 by 63 ? 26. 5967 by 504 ? 15. $32.45 by 57? 27. 43672 by 564 ? 16. $47.08 by 68? 28. 54865 by 647 ? 17. 6264 by 70? 29. 60435 by 704? 18. $29,451 by 49 ? 30. 74321 by 839 ? 19. 420643 by 76 ? 31. 543267 by 1563 ? 20. 572062 by 84? 32. 684039 by 1783? 21. 398025 by 87 ? 33. 709564 by 2803 ? 22. 703270 by 93 ? 34. 894037 by 3085 ? 35. Tf a clerk has $36 a month for the first 4 months ; $48 a month for the next 4 ; and $60 a month for the next 4; what will he receive for the year ? 36. If I receive $350 a month, how much shall I have at the end of the year, after deducting $38 a month for board ? 37. If it takes 385 laborers 18 months to build a railroad, how long would it take 1 man to build it ? ^ 38. A ship of war has provisions to last a crew of 645 men 90 days ; how long would they last 1 man ? 39. If I sell 29 bbls. of flour at $8 a barrel, and 50 bbls. of beef at $18 a barrel, and receive 5 hundred-dollar bills in pay- ment, how much will be due me for both ? 40. A farmer sold 32 sheep at $8 a head, and 9 cows for $45 apiece ; how much more did he receive for the cows than the *heep ? 41. What cost 169 chairs, at $3.25 apiece ? 44 Midtiplication. 42. What cost 279 barrels of salt, at 11.75 a barrel? 43. What cost 1565 acres of land, at 127 per acre ? 44. What cost 758 baskets of peaches, at 12.50 a basket? 45. If a hall Avill seat 1250 persons, and each seat is occupied by a person weighing 135 lbs. what weight is sustained by the floor ? 46. Bought 2 farms; one contained 327 acres at $83 an acre, the other 526 acres at $58 an acre. What did they both cost ? AVhat was the difference in their cost ? 47. What is the value of 56 railway cars, at $9550.75 each ? 48. A man bought 31 colts at $28 apiece, and paid 10 tons of hay at 125 a ton ; how much did he ow^e for the colts? 49. A grocer bought 21 barrels of flour at |5 per barrel, and sold 16 barrels of it at 17; finding the rest damaged, he put it at 13 a barrel. How much did he make or lose, by the operation ? 50. A farmer having 75 turkeys, sold 50 of them at 86 cents apiece, and the rest at 54 cts. apiece ; what did they come to ? 51. A man owns 7 orchards ; in each orchard there are 8 rows of apple-trees, and 29 trees in each row; how many apple-trees has he? 52. In a certain school there are 3 departments, in each department there are 11 classes, and in each class there are 48 pupils ; how many pupils are there in the school ? 53. If you multiply 58 by 37, and this product by 29, what will be the result ? 54. If 450 is multiplied by 254, and this product by 178, what will be the result ? 55. A man sent 37 loads of wheat to market; every load contained 16 bags, and each bag 3 bushels ; how many bushels did he send? 56. What is the product of 378 + 342 by (763 — 251) ? 57. What is the product of 254 + 451 by (836 — 434)? 58. Required the product of 823 — 567 by (827 + 230). 59. Multiply 267 + 75 + 430 by (468 — 324). 60. Multiply 869 — 675 by (300 + 87 + 90). 61. What is the product of (843 — 478) x (973 + 379) ? Multiplication. 45 89. To multiply by the Factors of the Multiplier. 1. What will 15 tables cost at $7 apiece ? OPERATION. Analysis. — The factors of 15 are 5 and 3. ^'^ Cost of i Table, (Art. 80.) Now as 1 table costs $7,5 tables k will cost 5 times $7 or $35. Again, since 15 = 3 X 5 it follows that 15 tables will cost 3 $35 " " 5 " times as much as 5 tables, and 3 times $35 are 3 $105. Hence, the 777^ ^lUo '• '• 15 '• Rule. — Jlicltiply the niulUplicaiid by one of the factors of the inultiplier, then this product by another, and so on, till all the factors have been used. The last product will be the answer. 2. What will 27 sofas cost, at $85 apiece ? 3. What will 24 wagons cost, at S3 7 apiece? 4. What will 36 cows cost, at $19 per head ? 5. If a man travels at the rate of 42 miles a day, how far can he travel in 205 days? 6. What cost 45 acres of land, at 110 dollars -per acre ? 7. At $6 per week, how much will it cost a person to board 52 weeks ? 8. At the rate of 56 bushels per acre, how much corn can be raised on 460 acres of land ? 9. What cost 672 pieces of cashmere, at $24 apiece ? 10. What cost 1265 yoke of oxen, at $72 per yoke ? 90. When the Multiplier has ciphers on the right. 1. What is the product of 56 multiplied by 10 ? Solution. — When a figure is moved one 'place to the left, its value is increased ten times. (Art. 34, 3'.) Hence, if we annex a cipher to 56 we mnltiply it by 10 and it becomes 560. Ans. 560. 2. Multiply 64 by 100. Solution. — Annexing two ciphers to 64 increases its value 100 times, and therefore multiplies it by 100. Ans. 6400. 3. Multiply 87 by 1000. A7is. 87000. 46 Multiplication. 4. Multiply 316 by 40. Explanation.— Multiplying 316 by 4 ones, it becomes 1364. But we are required to multiply by 4 tens instead of 4 ones; therefore the true product is ten times 1264. To correct this result, we annex a cipher to it, which mul- tiplies it by 10. OPERATION. 316 40 Ans. 12640 5. Multiply 345 by 700. Solution.— Multiplying by 7 ones only, gives 2415, which must be multiplied by 100 for the true product. Tliis is done by annexing two ciphers. Hence, 91. To multiply by 10, 100, 1000, etc. OPEEATION. 345 700 Ans. 241500 Annex as many ciphers to the inultiplicand as there are ciphers in the multiplier. When the significant figures have ciphers on the right. Multiply by the significant figures, and to the result annex as many ciphers as are on the right of both fac- tors. (See Art. 864, Appendix.) 6. What will 100 bales of cotton weigh, at 468 lbs. to abate ? 7. How many pages in 2300 books, of 352 pages each ? 21. 56300000 X 64 = ? 22. 62300000 X 890 = ? 8. 476 X 1000 = ? 9. 53486 X 10000 = ? 10. 12046708 X 100000 = ? 11. 26900785x1000000=? 12. 89063457x10000000=? 13. 9460305068x100000=? 14. 1920 x 2000 = ? 15. 4376 X 2500 = ? 16. 50634 X 41000 = ? 17. 630125 X 620000 = ? 18. 12000 X 31 = ? 19. 370000 X 32 = ? 20. 8120000 X 46 = ? 23. 54000000 X 700 = ? 24. 43000000 X 600 = ? 25. 563800 X 7200 = ? 26. 1230000 X 12000 = ? 27. 310200 X 20000 = ? 28. 2065000 X 810000 = ? 29. 2109090 X 510000 = ? 30. 6084201 X 740000 = ? 31. 7283900 x 958300 = ? 32. 86007400 x 9700 = ? 33. 90690000 X 8600 = ? Multiplication. 47 Oral Problems for Review. 92. 1. If 4 men can do a job of work in 6 days, how long will it take 1 man to do it ? Analtsis. — It will take 1 man 4 times as many days as it takes 4 men ; and 4 times 6 days are 24 days, Ans. 2. In 1 peck are 8 quarts ; how many quarts in 6 pecks ? 3. How many quarts are 7 pecks and 3 quarts ? 4. In 1 bushel are 4 pecks ; how many pecks in 15 bushels ? 5. How many pecks in 20 bushels and 3 pecks ? 6. If 9 men can build a wall in 20 days, how long will it take 1 man to do it ? 7. In 1 pound are 16 ounces ; how many ounces in 4 pounds? 8. How many ounces in 5 lbs. 7 ounces ? 9. If a barrel of flour will last 8 persons 12 days, how long will it last 1 person ? 10. AVhat will 8 lbs. of maple sugar cost, at 9 cts. a pound ? 11. How many inches in 11 feet? 12. How many feet in 12 yds. and 2 feet? 13. How many quarts in 25 gallons of milk? 14. How many quarts in 30 gallons and 5 quarts ? 15. If two men start from the same place and travel in opposite directions, one at the rate of 4 miles per hour, the other, 3 miles, how far apart will they be in 6 hours ? Written Problems for Review. 93. 17. George has 27 cents, and Henry has 3 times as many cents as George lacking 5 ; how many cents has Henry ? How many have both ? 18. At a military parade there were 5 regiments, in each regiment 8 companies, in each company 9 platoons, and in each platoon 10 soldiers ; how many soldiers were on parade ? 19. If 325 men can grade a street in 28 days, how long will it take 1 man to do it ? How much will he receive for it, if he has $2 per day ? 20. A and B are 20 miles apart, and travel in opposite directions, K goes 4 miles an hour and B 5 miles ; how far apart will they be in 48 hours ? 48 Multiplication, 21. If I hire a carpenter at 128 a month, and his appren- tice at $14, how much will be due them in 12 months ? 22. A man bought a drove of 1560 sheep, at $4 a head; it cost him $68 to send them to market, and they brought him $5 apiece ; how much did he make on them ? 23. A drover bought 360 head of cattle and 96 horses ; he afterwards sold the former at a profit of $19 a head, and the latter at a loss of 23 dollars a head ; did he gain or lose by the operation, and how much ? 24. A grocer bought 585 barrels of flour at $6 a barrel, and 117 barrels at $7 ; he then sold the whole at $6.50. What was the result of his speculation ? 25. In music, two minims equal a semibreve ; two crotchets a minim ; two quavers a crotchet ; two semi-quavers a quaver ; and two demi-semiquavers a semi-quaver; how many demi- semiquavers are equal to 259 semi-breves ? 26. Two persons start from the same place, and travel in the same direction ; one at the rate of 33 miles per day, and the other at the rate of 37 miles per day ; how far apart will they be at the end of a year ? 27. Multiply two thousand seven, by one thousand four. 28. Multiply four thousand forty, by two thousand one hun- dred three. 29. Multiply forty thousand, four hundred four, by ten thousand ten. 30. Multiply one hundred five thousand seven, by sixty thousand, four hundred three. 31. Multiply five millions, two hundred six, by seventy thousand, two hundred five. Questions. 76. What is Multiplication ? 77. The Multiplicand ? 78. The Multi- plier ? What does the Multiplier show ? 79. What is the Answer called ? 80. The numbers which produce the product called ? 81. Make the Sign of Multiplication. What does it show ? How read ? 85. How proceed when the multiplicand or multiplier has decimals ? 88. Give the general rule ? How prove multiplication? 89. How multi- ply by the factors of a number ? 91. How multiply by 10, 100, 1000, etc.? When there are ciphers on the right, how proceed ? i^ IVISION ■ I - ^ Oral Exercises, 94. 1. How many times can 4 cents be taken from a purse containing 12 cents? Solution.— 13 cents — 4 cts. = 8 cts. ; 8 cts. — 4 cts. r= 4 cts. ; and 4 cts. — 4 cts. = 0. Ajis., 3 times. 2. How many times are 4 cents contained in 12 cents ? 3. How many 3's in 15 ? How many 5's ? 4. How many oranges at 4 cts. apiece, can I buy for 20 cts.? Analysis. — I can buy as many oranges as 4 cents are contained times in 20 cents. Ans., 5 oranges. 5. How many times 3 in 18? How many times 6 ? 6. In 1 gallon are 4 quarts ; how many gallons in 24 quarts ? 7. In 1 week are 7 days ; how many weeks in 28 days ? 8. \Yhat is one of the 5 equal parts of 30 ? Analysis.— Since 30 is 6 times 5, one of the 5 equal parts of 30 is 6. 9. On Christmas day a father divided $25 equally among his 5 children ; how many dollars did each receive ? 10. How many boxes, each holding 12 lbs., will a dairyman require to pack 36 lbs. of butter ? 11. A teacher having 45 pupils, formed them into classes of 9 each ; how many classes did she have ? Definitions. 95. Division is finding hoiv 7nany times one number is con- tained in another; or finding one of the equal i)arts of a number. 96. The Dividend is the number to be divided. 97. The Divisor is the number by which we divide. 98. The Answer, or number found by division, is called the Quotient. It shows how many times the divisor is contained in the dividend. 50 Division, 99. The Remainder is the part of the dividend left when the divisor is not contained in it an e.ract number of times, and is always less than the divisor. 100. The Sign of Division is -^. It is read *' divided by." Thus, 8 -4- 4 is read, " 8 divided by 4." 101. Division is also denoted by writing the cUviso?' under the dividend, with a line between them. Thus, I is read, "8 divided by 4." Notes. — 1. Division is the reverse of multiplication ; the former sepa- rates numbers into equal parts ; the latter unites equal parts in one num- ber. The dividend corresponds to the product, and the divisor and quo- tient to the multiplier and multiplicand. (Art. 79.) 2. Division is also similar in principle to subtraction, and may be performed by it. Thus, 5 is contained in 15, 3 times, and 5 can be sub- tracted from 15, 3 times. 12. How many times 3 in 36 ? 13. 4 in 28? 16. . 5 in 45 ? 19. 8 in 56 ? 14. 4 in 32 ? 17, . 6 in 54 ? 20. 9 in 54? 15. 6 in 48 ? 18. . 7 in 63 ? 21. 9 in 72? 22. 18-f-3 = ? 30. 32^4=? 38. 36- ^3=:? 23. 27^3 = ? 31. 28-^4=? 39. 48- =-4=? 24. 35-^5 = ? 32. 42 -=-7=? 40. 54- =-9r:r? 25. 45-^5 = ? 33. 56-=-8=? 41. 56- =-8=? 26. 30-^6=? 34. 35^7=? 42. 72- =-9=? 27. 42-^6=? 35. 48-^8=? 43. 88- l-llrr^? 28; 60-^5r=? 36. 64^8=? 44. 72- 1-8=? 2^. 54-f-6==? 37. 63-^7=? 45. 96- 1-12=? 46. -v-=? 49. _5_4 — 6 — :? 52. -¥-= = p 55. W=? 47. 48. — '? 6 50. 63 — 7 — :? 53. ^■- — 9 56. 1 20 9 1^" 48. 4f=? 51. ¥= :? 54. M= -9 57. 1 .3 3. — 9 -T3 — • 102. The Name of the equal parts into which a number or thing is divided depends upon the Jtumher of parts. Thus,^ Division. 51 One of ttuo equal parts is called One-half, written J. One of three equal parts is called One-third, written J. One of four equal parts is called One-fourth, written J. Two of three equal parts are called Two-thirds, written f . Three of four equal parts are called Three-fourths, written f. 103. Write the following in figures : 1. Two-ninths ; three-fifths ; four-fifths ; five-eighths. 2. One-sixth ; two-sixths ; four-sixths ; five-sixths. 3. Two-sevenths; three-sevenths; fiye-sevenths. 4. Three-tenths ; five-ninths ; three-elevenths ; six-twelfths. 104. AVhen a unit is divided into equal parts, the parts are called Fractions. 5. What part of 3 is 1 ? Is 2 ? Analysis. — If 3 is divided into 3 equal parts, one of these parts is l-third of 3 ; 2 of the parts are 2-tliirds of 3. 6. What part of 4 is 1 ? What part of 3 ? 7. What part of 5 is 2 ? Is 3 ? Is 4 ? 8. How find a half of a number ? A third, a fourth, etc. Ans., By dividing it respectively by 2, 3, 4, etc. 9. What is 1 third of 6 ? Of 9 ? Of 12 ? Of 18 ? 10. What is 1 fifth of 10 ? Of 15 ? Of 30 ? Of 45 ? 11. What is 1 eighth of 24 ? Of 32 ? Of 40 ? Of 48 ? 12. If $40 are distributed among 8 laborers, what part of the money and how many dollars will each receiye ? Analysis.— One is l-eighth of 8. Therefore, 1 man will receive 1 -eighth of $40, which is $5, Ans. 13. How many tons of coal, at $7 a ton, can be bought for $63 ? 14. A farmer sold 5 tons of hay, at $12 a ton, and took his pay in flour at $6 a barrel; how many barrels did be receive? 15. An express traveled 108 miles in 9 hours ; at what rate was that per hour ? 16. How many times is \ of 42 contained in 54? 17. How many times is \ of 63 contained in 84 ? 53 Division, jyErELOPMENT OF I^RINCIPLES. 105. 1. How many times 4 cents in 20 cents ? 2. What kind of a number is the quotient ? Ans. It is an ahdract number. 3. If 30 apples are divided into 5 equal parts, how many apples will there be in one part ? 4. What kind of number is the quotient ? Ans. A concrete number, the same as the dividend. 5. If 7 is the divisor and 4 the quotient, what is the dividend ? Ans. Their product, 7x4, or 28. 106. From the above examples we derive the following Principles. i°. When the divisor and dividend are lihe numbers, the quotient is an abstract number. 2°. When the divisor is an abstract number, the quotient and dividend are like numbers. 3°. The product of the divisor and quotient is equal to the dividend. Written Exercises. 107. 1. Divide 952 by 4. Explanation.— Write the divisor on the left Divisor. Divid. Quotient, of the dividend, witli a curved line between them. 4 ) 952 ( 238 First. — Beginning at the left to divide, we find g 4 is contained in 9 hundreds 2 (hundreds) times, — and place the 2 (in hundreds place) at the right of -'■" the dividend for the first quotient figure. 12 Second. — We multiply the divisor by the quo- oo tient 2, and set the remainder under the order divided. ?? Third. — Subtract the product from the part of the dividend used. Fourth. — To the remainder 1 (hundred) we annex the 5 tens, making 15 tens, for a second partial dividend. Now 4 is contained in 15 tens, 3 (tens) times ; write the 3 in tens place in the quotient, and multiplying the divisor by it, subtract the ])roduct from the second partial dividend. Division, 53 To the remainder 3 tens, we annex the 3 units, making 32 units, a third partial dividend. 4 is contained in 32 units, 8 times. Write the 8 in units place in the quotient. Multiplyhig the divisor by it and subtract- ing the product, there is no remainder. Ans. 238. Proof— Quotient 238 x 4 (Divisor) = 952 (Dividend). (Art. 106, 3°) 2. Divide 6570 by 5, and explain the operation in like man- •ner. Ans, 1314. 3. Divide 7650 by 6. 6. Divide 9219 by 7. 4. Divide 8211 by 7. 7. Divide 68696 by 6. 5. Divide 9872 by 4. 8. Divide 89240 by 8. 9. What is the quotient of 6272 divided by 4 ? Explanation. — We draw a line under operation. the dividend, and begin to divide at the Divisor. 4 ) 6272 Dividend, left as before. Dividing 6 (thousands) by 4, T^c the quotient is 1 (thousand), which we write ^^° below the line, under the order divided. Subtracting, the remainder is 2 (thousands). To this we annex, mentally, the 2 hundreds, making 22 (hundreds) for the next partial dividend. Divide as before, and proceed in this way till all the orders are divided, carrying the multiplications and subtractions in the mind, simply setting down the quotient figures. The quotient is 1568. Proof.— The quotient 1568 x 4 (divisor) = 6272 (dividend). Divide and prove the following in like manner : (10.) (11.) (12.) (13.) 3 ) 1134 5 ) 1230 6 ) 2562 7 ) 8638 (14.) (15.) (16.) (17.) 6 ) 3276 8 ) 9872 7 ) 2345 9 ) 3141 108. The method of dividing in which the results of the several steps are set down, as illustrated by Ex. 1, is called Long Division. 109. The method in which the quotient only is set down, the results of the several steps being carried in the mind, as illustrated by Ex. 9, is called Short Division. When the divisor does not exceed 12, this method is preferable. 54 Division. 18. Divide 13875 by 4. Notes. — 1. To indicate the division of the final operation. remainder, if any, it must be written over the divi- 4 ) 13875 sor and placed at the riffht of the quotient as part . iTT^oo ^ ., Ans. 34681- OI it. * 2, In proving the work, the remainder, if any, must be added to the product of the divisor and quotient. (Art. 99.) * (19.) (20.) (21.) (22.) 5 ) 4567 6 ) 3971 8 ) 6567 9 ) 41756 Solve tlie following, both by Long and Short Division : (23.) (24.) (25.) (26.) 5 ) 76453 4 ) 8235 4 6 ) 52387 7 ) 63874 (27.) (28.) (29.) (30.) 7 ) 842952 6 ) 42846 3 8 ) 768345_ 9 ) 783952 (31.) (32.) (33.) (34.) 8 )_6592^ 9 ) 75327 10 ) 562340^ 12 ) 18396 110. Decimals are divided like integers, and from the i^ight 'of the quotient as many figures must be pointed off for deci- mals, as the decimal places in the dividend exceed those in the 'divisor. OPERATION. 9 ) 391.86 35. Divide 391.86 by 9. Explanation. — We divide as in whole numbers and point off two figures for decimals in the quotient. Ans. 43.54 36. Divide 198.752 by 8. Ans. 112.344. 37. Divide 2563.48 by 4. 38. Divide 645.328 by 8. 39. At 14 a yard, how many yards of cloth can be bought for 19850 ? 40. If '1^48.78 are divided into 9 equal parts, what is the Talue of each part ? Division, 55 Oral Exercises. 111. 1. If the price of 7 huts is 128, what is the price of 1 hat ? Analysis. — 1 is 1 seventh of 7 ; therefore 1 hat will cost 1 seventh of $28, and 1 seventh of $28 is $4, Ans. 2. If 1 man can hoe a field of corn in 40 days, how long will it take 5 men to hoe it ? 3. A grocer bought 8 barrels of flour for $56 ; what must he sell it for per barrel to gain $12 ? 4. A hardware merchant paid 169 for 7 kegs of nails, and $15 freight ; what did each keg cost bim ? 5. How many times 9 in 8 times 12? 6. How many times 11 in 74 plus 13 ? 7. How many cords of wood at $4 a cord, will pay for 8 pairs of boots at 16 a pair ? 8. A farmer gave 5 tons of hay for 2 cows, worth $30 apiece ; what was the value of the liay per ton ? 9. How many tons of coal, worth $6 a ton, must I give for 5 suits of clot lies worth $9 a suit ? 10. If I pay $12 apiece for 7 barrels of beef, and sell it so as to lose $24, what shaU I get a barrel ? Written Exercises. 112. Ex. 1. Divide 172859 by 34. Explanation.— 34 is contained in 172, operation. 5 (thousands) times and 2 (thousands) rem. 34 ) 172859 ( 5084-3^j Setting the 5 at the right of the dividend, '[^Q we annex the 8, the next figure of the divi- dend, to the remainder 2, making 28 for the '^^'^ second partial dividend. 272 As 33 is not contained in 28, we write a -. oq cipher in the quotient, and annex the 5 to 28, making 285 (tens). Now 34 is in 285 H!? (tens), 8 (tens) times. Writing 8 in the 3 quotient, we multiply, subtract, etc., and proceed as before. Finally, we write the divisor under the remainder, and place it at the right as part of the quotient. 56 Division. (2.) 16 ) 45807 ( (5.) 19 ) 560372 ( (8.) 48 ) 9230.56 ( (11.) 47 ) 159.85 ( (3.) 28 ) 348072 ( (6.) 36 ) 39245 ( (9.) 29 ) 702345 ( (12.) 56 ) 175.845 ( (4.) 37 ) 516780 ( (7.) 37 ) 45567 ( (10.) 38 ) 145.84 ( (13.) 48 ) 196.354 ( 113. From the preceding illustrations we derive the following General Rule. /. Write the divisor at the left of the dividend, and find how many times it is contained in the feiuest orders that will contain it, setting the quotient at the right. II. Multiply the divisor hy this quotient, and sub- tract the product from the orders divided. To the r^inainder anneoc the succeeding figure of the divi- dend, and divide as before. III. If there is a remainder after dividing the last order, write it over the divisor, and place the result at the right as part of the quotient. Finally , point off as many decimal figures at the right of the quotient as the decimal places in the divi- dend exceed those in the divisor. Proof. — Multiply the divisor and quotient together, and to the product add the remainder. If -the result is equal to the dividend, the worh is right. Note. — The quotient figure both in short and long division, is always the same order as the right hand order divided. 114. To prove Multiplication by Division. Divide the product by one of the factors, and if the quotient is equal to the other factor, the worh is right Division. 57 Appi^ications, 115. 1. A man wishes to invest $2562 in railroad stock ; how many shares can he buy, at $42 per share ? 2. In 1 year there are 52 weeks ; how many years are there in 1640 weeks? 3. In one hogshead there are 63 gallons ; how many hogs- heads are there in 3065 gallons? 4. If a man can earn 175 in a month, how many months will it take him to earn -S3280 ? 5. If it takes 18 yards of silk to make a dress, how many dresses can be made from 1350 yards? 6. If a 3'onng man's expenses are $83 a month, how long will $4265 support him ? 7. A man bought a drove of 95 horses for $4750 ; how much did he give apiece ? 8. A farmer having $1840, laid it out in land, at $25 per acre ; how many acres did he buy ? 9. In a cask there are 93 gallons ; how many casks in 4260 gallons ? 10. If a man travels 45 miles a day, how long will it take him to travel 1215 miles ? Divide and prove the following : 11. $467.2 by 15. 19. $84.53 by 62. 12. $56.84 by 18. 20. $73.56 by 48, 13. 786.3 by 21. 21. 6893 by 82. 14. 48.27 by 33. 22. 9721 by 65. 15. 6972 by 35. 23. 23456 by 28. 16. 7842 by 23. 24. 72350 by 45. 17. 8253 by 47. 25. 80854 by 84. 18. 21.08 by 32. 26. 92635 by 92. 27. A garrison had 5580 pounds of beef, which the com-« mander wished to have last 62 days ; how many pounds could be used per day ? 28. A man paid $9565 for a farm, at $64 an acre; how many acres were there ? 58 Division, 29. A grocer packed 18144 eggs in boxes holding 144 eggs each ; how many boxes did he use ? 30. If he had packed the same eggs in 63 equal boxes, how many eggs would he have put in a box ? Note. — When the divisor is large, find how many times its first figure is contained in the first or first two figures of the dividend, allowing for the addition of tens from the product of the second figure of the divisor. 31. Divide 1814G by 683. A^is. 20||f. 32. 62346 by 254. 40. 89256.48 by 732. 33. 70893 by 532. 41. 2439.2642 by 765. 34. 294763 by 306. 42. 592348.276 by 879. 35. 375426 by 521. 43. 569389.175 by 1247. 36. 2445224 by 812. 44. 8679538.46 by 3238. 37. 3560325 by 904. 45. 134259.8640 by 56813. 38. 4256348 by 638. 46. 396478.9523 by 75436. 39. 5437502 by 743. 47. 425367.805 by 83247. Dictation Exercises.* 116. 1. Subtract 4 from 11, add 3, multiply by 6, divide by 10, add 6, subtract 4, multiply by 5, and divide by 8; what is the result ? 2. Multiply 9 by 7, subtract 3, divide by 5, add 10, divide by 11, multiply by 8, subtract 4, and add 12 ; result ? 3. Divide 54 by 9, multiply by 3, subtract 8, add 4, divide by 7, multiply by 9, add 10, subtract 7, divide by 3 ; result? 4. If from 39 you take 7, divide by 8, multiply by 9, sub- tract 6, divide by 5, add 12, divide by 9, multiply by 11, add 3, divide by 5, add 7, and multiply by 4, what is the result ? 5. 7 + 8 — 3x4^6 + 10 -^3x7 — 2; result ? f 6. 30 — 6-^8x9 + 5-^4 + 14 -f- 11 X 12 + 6-^5x 7 + 6 -^ 8 + 21 ; what is the result ? 7. 8x7 — 8-^4x2^6 + 16 -^4x8 — 12 -^7 + 6 X 6 ; what is the result ? * The object of these exercises is three-fold ; First, to give facility in mental com- binations of numbers ; Second, to cultivate the habit of fixing the attention ; Third, to drill the whole class at the same time. t Perform the successive operations indicated by the signs. Division, 59 8. 48 -7- 6 X 4 + 10 — G -M) + 30 — 7 H- 3 X 12 — 5 + 7 -f- 10 X 3 ; what is the result ? 9. 25 + 7->8xll — 8^9 + 23-^9x7 + 12-^3 — 5 X 8 + 6 -^ 9 ; what is the result ? 10. 27 -^ 9 + 15 — 10 X 7 + 7 -4- 9 X 5 — 8 -^ 3 + 12 _^_ 7 X 20 — 12 -^ 6 4- 25 = how many ? 117. When the Divisor has Ciphers on the right. 1. Divide 3563 by 100. Explanation.— Cutting off the right- operation. hand figure of a number, removes each of 1|00 ) 35|63 its other figures one place to the right, i q" ftQ and therefore divides it bj 10. Cutting off two figures divides it by 100 ; cutting off three figures divides it I y 1000, etc. (Art. 90.) 2. Divide 345231 by 100. 4. Divide 6423544 by 10000. 3. Divide 672487 by 1000. 5. Divide 7364159 by 100000. 6. Divide 937643 by 4000. Analysis. — By cutting off three figures operation. at the right of the divisor and dividend. 41000 ) 937 1 643 we divide each by 1000; the quotient is . TTTi ^TTT^o no-r ^ ^1 ■ ^ c^Ao ^ * a- a AllS. 234-1643 Rem. 937 and the remainder 643. Next, divid- ing by 4, the quotient is 234, and 1 remainder, which we prefix to the figures cut off, making the true remainder 1643. Hence, 118. To Divide by 10, 100, 1000. Cut off as many figures at the right of the dividend as there are ciphers in the divisor; the remaining figures will he the quotient, and those cut off' the remainder. When the divisor is greater than I, with ciphers on the right. Cut off the ciphers from the divisor and as many fig- ures from the right of the dividend. For the quotient, divide the remaining part of the dividend by the remaining part of the divisor. To the figures cut off, prefix the remainder , and the result will be the true remainder. (Art. 870, Appendix.) 60 Divmon, 7. Divide 4885970 by 6000. Ans. 814 and 1970 rem. 8. Allowing 200 lbs. to a barrel, how many barrels will 68000 lbs. of beef make .^ 9. In $1 there are 100 cents ; how many dollars are in 45650 cents ? 10. How many bales of cotton, each weighing 450 lbs., are in 36000 lbs. ? 11. If $96000 are divided equally among 2400 soldiers, how much will each receive ? 12. A pound of cotton has been spun into a thread 76 miles long, and a pound of wool into a thread 95 miles long ; how many pounds of both together will spin a thread which will reach round the world, a distance of 25000 miles ? 13. If 600 steam engines can do the work of 2 million 496 thousand men, to how many men is 1 engine equivalent ? 119. From the relations of the Divisor, Divide^id, and Quotient, we deduce the following General Principles of Division, First. — Let 24 be a dividend and 6 a divisor. The quotient is 4. Then (24 X 2) - 6 = 8 I _ And 24 -- (6 -^ 2) = 8 S - * ^ ''• ^^"^^^^ i°. Mnltiplying the dividend, or Dividing the divisor, 8econd.-{U -^ 2) - 6 = 2 | _ And 24 -- (6 X 2) = 2 ( - ^ * ^- ^''''^^' [ Multiplies the quotient. 2°. Dividinq the dividend, or ^. ., ,, ,. , ,^ , . y . ,- ,. . \ Divides ihQ (moiiQui. Multiplying the divisor, ) raM-rf.-(24 X 2) - (0 X 2) ( Or, (24 -^ 2) -^ (6 - 2) f - *• ^'^"°*' 5°. Multiplyinq or dividinq both ) ^ , . ,, ^. : ^ "^ -, ,. ., , , ( Does «o/ change the quo- di visor and dividend by /■ . the same number, ' Division. 61 119, a, 1st. When the jirodiict of two factors and one of them are given, the other is found by dividing the product by the given factor. 2d. When the product of three or more factors and all but one of them are given, the other factor is found by dividing the given product by the product of the given factors. 3d. When the sum and difference of two numbers are given, the less number is found by subtracting the difference from the sum and dividing the remainder by 2. 4th. The average of two unequal numbers is half their sum. The average of three unequal numbers is one-third the sum. Oral Problems for Review. 120. 1. The dividend is 63, the quotient 9; what is the divisor ? 2. When the dividend and quotient are given, how ^nd the divisor ? 3. The divisor being 11 and the quotient 10, what is the dividend ? 4. When the divisor and quotient are given, how find the dividend ? 5. The quotient being 9, the divisor 20, and the remainder 7, what is the dividend ? 6. When the divisor, quotient, and remainder are given, how find the dividend ? 7. At 17 a week, how many weeks can you board for $84 ? 8. How long will it take a printer to earn $132, if he gets $11 a week? 9. A farmer bought 12 yards of cloth, at $4, and paid for it in hay, at $8 a ton; how many tons did it take? 10. In 7 times 11, less 5, how many times 9 ? 11. In 9 times 12, less 8, how many times 5 ? 12. How many tons of coal, at $6 a ton, will pay for 8 bar- rels of flour, at $9 a barrel ? 13. If 12 men can earn $100 in a week, how much can 1 man earn in the same time ? 62 Division. 14. When wood is 14 a cord and coal is $9 a ton, how much wood is equal in value to 8 tons of coal ? 15. If eggs are worth 9 cents a dozen, and butter 12 cents a pound, how many eggs are worth 6 lb. of butter? 16. A man being on a journey, finds he can reach home in 9 days by traveling 20 miles a day; but becoming lame, he traveled onlv 12 miles a day ; in how many days did he reach home ? 17. A man bought 6 hats at |4 apiece, and 5 caps at 12, and paid in apples at 16 a barrel ; how many barrels and what part of a barrel did it take to pay the bill ? 18. John has 12 marbles and William has 9 times as many as John, minus 11 ; how many marbles has William? 19. When peaches are sold at the rate of 5 for 8 cents, how many will 56 cents buy ? 20. What cost 60 apples, at the rate of 10 for 7 cents ? 21. George bought 12 oranges, at 4 cents apiece, and after eating 3 of them, sold the rest at 6 cents apiece ; did he make or lose by his bargain, and how much ? Written Problems for Review. 121. 1. The product of two numbers being 252, and the multiplier 18, what is the multiplicand? 2. The product of two numbers is 576, the multiplicand 48 ; what is the multiplier? 3. When the product of two factors and one of the factors are given, how find the other factor ? 4. The sum of two numbers is 250, their difference 50 ; what is the smaller number ? The greater ? 5. At an election A and B together received 273 votes, and A had 37 more than B ; how many had each ? 6. A grocer mixed two kinds of tea in equal quantities, worth 63 and 75 cts. a pound respectively; what is the average price of the mixture a pound ? 7. What is the average age of 3 brothers, who are respectively 76, 81, and 89 years old? Division. 63 8. What is the average price of 4 horses, worth respectively $180, $273, $804, and $375 ? 9. The ship America of Boston, sailed 56 hours at the rate of 11 miles per hour, when she encountered a storm of 10 hours duration which drove her back at the rate of 14 miles per hour ; how far from port was she at the end of 72 hours? 10. A thief fled from New York, at the rate of 85 miles a day ; 5 days after an officer started in pursuit of him at the rate of 138 miles a day ; how far from the thief was the officer at the end of 8 days from the time the latter started ? 11. A is worth $1265, B is worth 4 times as much as A, and $183, and C is worth three times as much as A and B lacking $2348 ; how much are B and C worth respectively ; and how much are they all worth ? 12. If a man's salary is $3176 a year, and he spends $7 a day, how much can he lay up ? 13. In a single city, $2170 are spent daily for cigars ; how many free schools will this support, at $1085 each per annum ? 14. A man bought 467 acres of land, at $16 per acre, and sold it for $9340 ; how much did he get per acre ; and how much did he gain or lose by his bargain ? 15. A man bought 563 horses, at $65 apiece, and sold them so as to make $860 ; how much did he get apiece ? 16. Which are worth more, 863 cows at $38 apiece, or 356 horses at $75 apiece ? How much ? 17. A owns 1368 acres of wild land, which is 6 times as much as B owns, and B owns twice as much as C ; how much land do B and C own ; and how much do all own ? 18. The smaller of two numbers is contained 14 times in 252, the greater is 49 times the smaller ; what are the numbers ? 19. A man bought a drove of oxen for $18130, and after selling 84 of them at $51 apiece, the rest stood him in $43 apiece ; how many did he buy? 20. What is the difference between 9313702853 divided by 1987, and 46481 multiplied by 936? 21. A man sold 155 acres of land at $34 per acre, and took in payment for it, 19 horses at $65 apiece, and 15 cows at $17 apiece ; how much was still due him ? A. 64 Division, 22. AYhat number besides 137 will exactly divide 11371 ? 23. The quotient being 275, the divisor 383, and the remain- der 49, what is the dividend ? 24. If the dividend is 2756, the quotient 184, and the re- mainder 180, what is the divisor? 25. What must 5376 be multiplied by, to make 6521088 ? 26. How many times can 437 be subtracted from 18791 ? 27. If the sum of 14350 and 7845 is divided by 965, the quotient multiplied by 386, and the product diminished by 761, what will the remainder be? 28. The sum of 250 and 173, being multiplied by their differ- ence, and the product divided by 45, what is the quotient ? 29. How many men will it take to do as much work in 1 day, as 368 men can do in 134 clays ? 30. How many men would be required to do the same work in 16 days? 31. Four men. A, B, 0, and D, bought a ship together for 116256 ; A paid 14756, B paid 1763 more than A, and C $256 less than B ; how much did D pay ? 32. Bought sofas for 19212 and selling them at 167 gained 120 on each ; how many were bought ? Q U ESTI O N S. 95. What is Division? 96. The Dividend? 97. Divisor? 98. What is the answer called ? What does it show ? 99. Remainder ? 100. Make the sign of division. How is it read ? 101. How else is division denoted ? 102. When a number is divided into two equal parts, what is one of the parts called? 104. When a unit is divided into equal parts, what are the parts called ? 106. When the divisor and dividend are like numbers, what is the quotient ? When the divisor is an abstract number, \vhat are the dividend and quotient ? To what is the product of the divisor and quotient equal ? 110. How divide when the dividend has decimals? 113. What is the general rule ? How prove division ? 118. How divide by 10, 100, 1000, etc. ? When the divisor is greater than 1, with ciphers on the right, how proceed ? 119. What is the effect of multiplying the dividend or dividing the divisor? Of dividing the dividend or multiplying the divisor? Of multi- plying or dividing both by the same numl^er? them thus, --^ — ^— ; what factors are common to both ? ^CANCELLATION. Devjsloi^ment of PbINCIPIjES. 122. 1. What is the quotient of 24 divided by 6 ? Ans. 4. 2. Separate the dividend and divisor into factors, and write 2x3x4 2x3 3. If you cancel the factor 2, w^hich is common to both, what is the quotient ? A ns. 4. Note. — To cancel means to cross out or reject. 4. If you cancel both the 2's and the 3's, what is the effect ? Ans. The quotient is not altered. Hence, the following Principles. 123. i°. CancelU7ig a factor of a number divides the nuniber hy that factor. 2°, Cancelling equal factors of the divisor and dividend does not change the quotient. (Art. 119, 5°.) 124. Cancellation is the method of shortening Division, by rejecting equal factors from the divisor and dividend. The Sign of Cancellation is an oblique mark drawn across the face of a figure ; as, $, ^, ^, etc. 125. To divide by Cancellation. 5. Divide the product of 14 x 15 x 56 by 8 x 45 x 7. 1st form. 2d FOEItt. 14 = 4f, Ans. Explanation. — Since 8 in the divisor is a factor of 56 in the dividend. 2 1 P $x0xti - 3 -^^' ^'''* . * 3 3 66 Cancellation. cancel the 8 in both, retaining 7, the other factor of 56. Also cancel 15, a factor of 45, and 7 a factor of 14, retaining the prime factor 2 in the dividend, and 3 in the divisor ; then (7 x 3) -r- 3 = 4|, Ans. Hence, the EuLE. — Cancel all the factors comnioii to the divisor and dividend, and divide the product of those remain- ing in the dividend by the product of those remaining in the divisor. (Art. 123, 2°.) Note. — When a factor cancelled is equal to the number itself, the unit 1 always remains. If the 1 is in the dimdend it must be retained ; if in the divisor, it may be disregarded. What is the quotient of 6. 28x56x15-^14x5x3? 10. 1365-^21x5? 7. 112x40x18-^56x3x4? 11. 2850-^125? 8. 48x72x20-^48x15x7? 12. 3236-^256? 9. 54x36x25-f-45x7x30? 13. 1728-^576 ? 14. 120 X 24 X 35 X 9-f-42 x 15 x 54 x 7 ? 15. An agent sold 176 boxes of starch, of 15 lbs. each, at 12 cts.; how many loads of corn, having 9 sacks of 5 bu. each, worth 44 cts. a bushel, will it require to pay for the starch ? 16 "a 4 The val. of starch = 176x 15x12) ^lUxUxU ,. , ^ ,, > and — - — ^ — -—:=lh,An§. '' '' corn = 9x5x44j ^x$xM ~a lis Note. — Practical Problems, should first be analyzed, and the o[3erations indicated. Then cancel as before. 16. A farmer bought 9 cows at 125 apiece, and paid for them in hay at 115 a ton ; how many tons of hay did it require ? 17. How many bags of coffee containing 5G lbs., at 28 cts. a pound, must be given for 8 pieces of muslin, each containing 40 yards, at 8 cts. a yard ? 18. How many barrels of flour worth 18 a barrel, must be given for 45 tons of coal at |6 a ton ? 19. A miller bought 7 loads of wheat, each containing 28 bags of 3 bushels each, worth $1.50 a bushel, and paid for it m flour at $7 a barrel ; how much flour was required ? * (tJ ■v'.S^ -^ EOPEETIES OF \ D ^•'isiv ^ Definitions. 126. Numbers are diyided into Odd, Even, Prime, and Composite. 127. An Even Number is one that can be exactly divided by 2. 128. An Odd Number is one that cannot be exactly divided by 2 ; as 3, 5, 1, etc. 129. A Prime Number is one that cannot be exactly divided by any number, except a unit and itself; as 5, 7, 11, etc. Note. — All prime numbers except 2 are odd. 130. Two numbers are Prime to each other when the only number by which both can be exactly divided is a unit or one ; as 5 and 6. 131. A Composite Number is the product of two or more factors, each of which is greater than 1 ; as 21 = 3 x 7. Note. — The least divisor of a Composite Number is a prime number. 132. An Exact Divisor of a number is one which will divide it without a remainder. One number is said to be divisiUe by another when there is no remainder. 133. The Factors of a number are the numbers whose product equals that number. (Art. 80.) Thus, 7 and 9 are the factors of 63 ; 8, 4 and 5, of 60. 134. A Prime Factor is a prime number used as a factor. Note. — The prime factors of a number are also exact divisors of it. 135. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 4 is 1 -^ 4, or \. 68 Properties of Numhers, Oral Ex e rcises. 136. 1. Name the eyen numbers up to 31. 2. Name the odd numbers less than 30. 3. Name the prime numbers less thin 30. 4. Name the composite numbers up to 30. 5. Name an exact divisor of 18, 27, 42. 6. Name all the exact divisors of 24 ; of 36. 7. What is the smallest number except 1, that will exactly divide 10? 15? 25? 35? 49? 8. What is the largest number, except itself, that will exactly divide 18? 22? 24? 30? 36? 9. AVhat numbers multiplied together j^roduce 21 ? 35 ? 42? 27? 45? 48? 10. What will produce 33 ? 54 ? 63 ? 36 ? 72 ? DeVEIjOPMENT of JPltlNCirLES. 137. First. — Take any number, as 12, and separate it into the factors 3 and 4. If we multiply 12 by 2 the product is 24, if we multiply it by 3 the product is 36, etc. Now each of these products is divisible by 3 and by 4. Hence, i°. If one 7imy}her is a factor of another, the former is also a factor of any Product or Multiple of the latter. Second. — Take any number, as 2, which is a common factor of 4 and 12. The sum of 4 + 12 = 16; their difference 12— 4 =r 8, and their product 12 x 4 = 48. By inspection we see that 2 is a factor of 16, of 8, and of 48. Hence, ^°. A factor common to two or more numhers, is also a factor of their Sum, their Difference, and their I*roduct. Third. — Take any composite number, as 30. 30 is divisible by 2, 3 and 5 ; also by 2 x 3, or 6 ; by 2 x 5, or 10 ; by 3 X 5, or 15 ; and by no other number. But 2, 3 and 5 are its prime factors ; 6, 10 and 15 are the different products of them. Hence, «5°. Every composite number is divisible by each of its Prime factors; and by the Product of any ttvo or more of them. <1 ^ „ ■ I ■ )l( . (g 1 * ACTORING. Definitions. 138. Factoring a number is separating it into factors. Thus, the factors of 21 are 3 and 7 ; the factors of 32 are 4 and 8. 139. A Composite Number is separated into ttvo factors by- dividing it by any exact divisor. Note. — It is not customary to consider the unit 1 and the number itself as factors ; if they were, all numbers would be composite. (Art. 131.) 140. A number that is a factor or divisor of two or more numbers is called a Common Factor or Common Measure of those numbers. 141. The following facts will assist the learner in separating large numbers into factors : All numbers are divisible i°. By 2, which end with a cipher, or a digit divisible by 2. 2°. By 3, when the sum of the digits is divisible by 3. 3°. By 4, when the number expressed by the two right hand figures is divisible by 4. Jf°. By 5, which end with a cipher or 5. 5°. By 6, when divisible by 2 and 3. 6°. By 8, W'hen the three right hand figures are ciphers, or when the number expressed by them is divisible by 8. 7°. By 9, when the sum of the digits is divisible by 9. (See Art. 875, Appendix.) ^°. By 10, 100 or 1000, which end with an equal number of ciphers. Note. — For 7, no convenient rule can be given 70 Properties of Numhers, 2 2310 Given Namber. 3 1155 1st Quotient. 5 385 2d Quotient. 7 77 3d Quotient. 11 4th Quotient. Oral Exercises. 142. 1. What prime factors will exactly divide 12 ? 18 ? 26 ? 2. What prime factors will exactly divide 30 ? 36 ? 40 ? 3. What prime factors are common to 18, 24, and 36 ? 4. Name the prime factors common to 45, 27, and 60 ? Written Exercises. 143. To Separate a Number into Prime Factors. 1. What are the prime factors of 2310 ? Explanation. — We divide tlie given number by any prime factor, as 2, and the successive quotients by the prime factors 3, 5 and 7, and the last quotient 11, is a prime number. Therefore, the several divisors with the last quotient are the prime factors required. Proof.— 2 x 3 x 5 x 7 x 11=2310. Hence, the Rule. — Divide the given munhev hy any prime facto?' ; then divide this quotient by another prime factor ; and so on until the quotient obtained is a prime number. The several divisors, with the last quotient, are the prime factors required. Find the prime factors of 2. 225. e. 672. 10. 3420. 14. 10376. 3. 376. 7. 796. 11. 18500. 15. 25600. 4. 344. 8. 864. 12. 46096. 16. 64384. 5. 576. 9. 945. 13. 96464. 17. 98816. 144. To find the Prime Factors common to two or more numbers. 18. Find the prime factors common to 168, 42, and 210 ? 2 ) 168, 42, 21 3 ) 84, 21, 10 5 7 ) 28, 7, 3 5 4, 1, 5 Explanation. — Dividing by the prime factor 2, the quotients are 84, 21, and 105. Dividing these by 3, we have 28, 7, and 35. Dividing by 7, the quotients are prime to each other (Art. 130). The divisors 2, 3, and 7, are the prime factors required. Hence, the FactoriiKj. 71 EuLE. — Divide the given ninnhers hy any common prime factor, and the quotients thence arising in like manner, till they have no common factor ; the several divisors will he the primps factors required. 19. What are the prime factors common to 24, 76, and 32 ? 20. 28, 54, and 48 ? 24. 120, 96, and 384 ? 21. 58, 64, and 84 ? 25. 168, 320, and 256 ? 22. 436, 308, and 506 ? 26. 225, 350, and 475 ? 23. 252, 126, and 210 ? 27. 144, 276, and 524 ? Common Divisors. 145. A Common Divisor is one that will divide ttvo or more numbers without a remainder. It is often called a Common Measure. 146. The Greatest Common Divisor or Greatest Common Measure of two or more numbers is the greatest number that w^ill divide each of them without a remainder,* Thus, the greatest common divisor of 18 and 30 is 6. Note. — Numbers which are prime to each other have no common divisor or measure greater than 1. Oral Exercises. 147. 1. What divisor is common to 15 and 27 ? 2. What divisor is common to 16 and 20 ? 3. Find a common factor of 15, 18, and 24. 4. What is the greatest number that will divide 21 and 35 without a remainder ? 5. What is the greatest divisor common to 30 and 48 ? Hence, 148. The greatest common divisor of two or more numbers is the iwoduct of all their common inime factors. Illustration.— Take any two numbers, as 30 and 42, and separate them into their prime factors ; thus, 30 = 2 X 3 X 5 ; 42 = 2 x 3 x 7. Now 2 and 3 are the only prime factors common to both numbers, and their product, 6, is the greatest di^^sor common to both. * The letters g. e.d. etand for Greatest Common Divisor. 72 Properties of Niimhers. Written Exercises, 149. To find the f/. c, d, of two or more numbers by Prime Factors. 1. What is the f/. c, d. of 45, 30, and 105 ? 3 ) 45, 30, 105 Or, 45 = 5 x 3 x 3 5 ) 15, 10, 35 30 = 5 X 3 X 2 3, 2, 7 105 == 5 X 3 X 7 5 X 3 = 15, 13^0? ToT* 110 203 326 5 3 3T? TBT? T¥3? aT4* 170. An Integer may be expressed in the form of a fraction by writing 1 under it for a denominator. Thus, 3 may be written \, and read " 3 ones." 171. A Proper Fraction is one whose numerator is less than the denominator ; as, J, f , f. 172. An Improper Fraction is one whose numerator equals or exceeds the cienominator ; as, f, f. 1. Two-thirds. 7. 2. Three-fourths. 8. 3. Two-fifths. 9. 4. Five-sevenths. 10. 5. Five-eighths. 11. 6. Six-sevenths. 12. Copy and read the following : 13. t'i. iV T^. 11- 15 14. H, 4il+.'ff. 16 Fractions, 79 173. A Simple Fraction is one whose terms are integers, and may be proper or improper. 174. A Compound Fraction is a fraction of a fraction ; as, 1 of }. 175. A Mixed Number is a ivliole number and a fraction expressed together ; as, of, 34-|^. .176. Fractions arise from division, the numerator being the dividend, and the denominator the divisor. (Art. 101.) Hence, 177. The Value of a Fraction is the quotiejit of the numera- tor divided by the denominator. Thus, the value of 1 fourth is l-f-4, or | ; of 6 halves is Q-r-2, or 3 ; of 3 thirds is 3-r-3, or 1. Note. — This value depends upon the size of the number or thing divided, and upon the number of parts into which it is divided. 178. The Reciprocal of a Fraction is 1 divided by the fraction, or the fraction inverted. (Art. 135. ) Thus, the reciprocal of | is 1 h- f = | ; of y^ is Y-. General Principles of Fractions. 179. Since fractions arise from division, the numerator being a dividend and the denominator a divisor, the general principles of division are true of fractions. (Art. 119.) That is, r. Mtdtiplying the numevatoY, or ) T,r i,- t .i /• /• -r. . ;. ^ > Multiplies the fraction. Dividing the denominator, ) ^°. Dividing the numerator, or) ^. ., ^, ^ ■nr ij- 1 ■ ii -. . i > Divides the fraction. Multiplying the denominator, ) 3°. Multiplyinq or Dividing both ) ^ , , ^ "^ / „ ,. , ,, ( Does not change its terms oi a fraction by the r , , '' \ value. same number, ) 80 Fractions. Reduction of Fractions. Oral Exercises. 180. 1. How many halves in a whole apple ? How many fourths ? 2. How many fourths in \ of an apple ? Analysis. — The required denominator 4, is twice the given denomina- tor 3. Multiplying both terms of \ by 2, it becomes |, An8. 3. How many sixths in f ? How many ninths ? 4. How many eighths in f ? How many twelfths? 5. Change \, \ to tenths. 6. Change f, J, f , -J-, f to sixteenths. 7. Change \, \, f, \, J, \, | to twenty-fourths. 8. Change f to twenty-eighths ; f to forty-fifths. 9. Change -^^ to thirty-ninths ; -f-^ to sixtieths. 181. Reduction of Fractions is changing their terms without altering the value of the fractions. (Art. 179, 5°.) Written Exercises. 182. To Reduce a Fraction to Larger Terms. 1. Change J4 to a fraction whose denominator is 81 ? Analysis. — The given denominator 27 is contained 81 -r- 27 = 3 in the required denominator 81, 3 times. Multiplying 14x3 42 both terms of the given fraction by 3, we have |f, the -^ -^ = -^ fraction required. (Art. 179, r.) Hence, the ^7 X ^ bl Rule. — Divide the required denojivinator by the denmii- inator of the given fraction, and multiply both terms by the quotient. 2. Change ^ to 104ths. 6. Change -Jf to 196ths. 3. Change || to 120ths. 7. Change f| to 288ths. 4. Change ^ to 176ths. 8. Change -| to 192ds. 5. Change f| to 144ths. 9. Change ff to 57Cths. Reduction. 81 Oral Exercises. 183. Reduce the following at sight to their lowed terms : 1. 2. 3. 4. 2. 3. 2 _2_ _2_ 6^ 6> 8? 10' 16' 3 3 3 3 3 T^> T'g^? ¥¥> "JF? "&¥• _4_ 4_ _4 4 4 12? f 07 3¥' 18? 6 8* ■J'B' "55? "56? "63? "sr* An f = M- Now to sub- 246| = 246^ tract U from ^»o, we take 1 = (|g) from 6, and add it 132| — .132||- to ^^^, making |f, then subtract, and take 2 from 5, TTqTT etc. Hence, the ^^^^' ^^^^ Rule. — /. Reduce the given fractions to a com-mon denominator, and ovei' it write the difference of the numerators. (Art. 195.) II, If there are mixed numbers, subtract the frac- tional and integral parts separately, and ivniUy the results. (Ex. 2.) Note. — In most cases it is better to reduce the fractions to the least common denominator. Wliat is the difference between 3. H and i^A- 8. M and ^. 13. 2 and |. 4. ■h and ,\. 9. 2 and ^2^. 14. 65 and 25f . 5. 2 1 qnrl 2 3 3T ana 4 8 . 10. +4 and if. 15. 21f and 9f 6. S^ and of. 11. 12| and 8|. 16. 25| and 17f. 7. 121- and 7^. 12. 15f and 9J. 17. 37f aud 19i. 4 ? 18. From 385J rods take 67J rods. 19. From 573| tons take 21 6| tons. 20. From 563J pounds take 260|^ pounds. 21. From 1673-1 bushels take 356f bushels. 22. A man bought a wagon for -$851, and a sleigh for $69 how much more did he pay for one than the other ? 23. A man having 246i^g acres of land, sold 195f acres ; how many acres did he have left ? 24. If from a piece of cloth containing 125|-§^ yards, you cut 87^ yards, how many yards will be left ? 90 JFraotions, Multiplication of Fractions. Oral Exercises. 205. 1. What is the cost of 5 books, at $| apiece ? ANA1.TSIS. — Since 1 book costs $|, 5 books will cost 5 times $f, which are $V- = $1|» ^''^^^ 2. What cost 6 bushels of apples, at $y\- a bushel ? 3. At %-^^ a pound, what will 10 pounds of butter come to ? 4. Multiply f by 8. 5. Multiply J by 12. 6. How many units in 7 times 4- ? 7. At 1^ a pound, what will 4 pounds of tea come to ? Analysis. — Dividing the denominator 12 by 4, multiplies the fraction (Art. 179), the result is $V = $3f , Ans. 8. Multiply I by 3. li. Multiply /-j by 7. 9. " -^ by 5. 12. " -^ by 10. 10. « iVl^y^. 13, « if by 15. 14. AYhat will 4 yds. of braid come to, at 5| cts. a yard? Analysis. — Since 1 yd. is worth 5| cents, 4 yards are worth 4 times 5| cents. Now 4 times 5 are 20 cts. and 4 times 2 thirds are 8 thirds equal to 2| cents, which added to 20 make 22f cents. Therefore, etc, 15. At 6 J cents each, what must I pay for 8 oranges ? 16. At $5f a yard, what is the cost of 7 yds. of cloth ? 17. What must a lady pay for 8 yds. of silk at |3| a yard ? 18. How many are 7 times 8|-? 19. Multiply lOf by 8. 20. What is the product of 12| by 9 ? Written Exercises. 206. Multiplying a Fraction by an Integer. A Fractio7i is multijMecl hy nniltiplyinci Us numerator or hy dividing its denominator. (Art. 179, 1°.) Multiplication. 91 1. Multiply J^ by 9. Explanation. — Multiplying 1st opekation. the numerator 13 by 9, the result ^| ^ 9 r= -iji/ — 2|, Ans, equals 3|. Or, M X ^ = ¥ = -i Cancelling the factor 9, which g is common to both terms, the 2d operation. result is the same. j^ jLJ. 9|. Ans» Or, dividing the denominator 45-^9 5 1' 45 b,y 9, the result is 3|, as before. Note. — In the 1st operation, the number of parts is increased, while their size is unchanged. In the 2d operation, the size of the parts is increased, while their number is unchanged. 2. Multiply ^ by 14. ^^5. 3|. 3. ^x9 = ? 6. Ax45==? 9. 1^0^x10=:? 4. ^Vxl2z=? 7. AVx48=? 10. f/^x41=? 5. ifX^l = ? 8. -r*^x86==? 11. 22_.5.^ X 54 =: ? 12. Multiply 15f by 7. 152- ExPLANATiON. — Multiply the fractional and integral * parts of 15| separately, and uniting the results, we have ' 110}, the product required. Ans. 110 J Note. — When the multiplicand is a mixed number, the fractional and integral part should be multiplied separately, and the results be united. 13. 87ix8=? 15. 205|x24=:? 17. 256f x3 = ? 14. 165|xl2=? 16. 196yVxl8 = ? 18. 575yVx48=r? Oral Exercis es. 207. When the Multiplier is a Fraction. 1. If a barrel of flour is worth $6, what is J barrel worth ? Analysis. — 1 half barrel is worth 1 half as much as a whole barrel, and 1 half of $6 is $3. Therefore, etc. 2. If a stage goes 9 miles an hour, how far will it go in \ of an hour? 3. What is A of 14 apples ? 1 of 15 pounds ? ^ of 28 days ? 4. At $5 a yard, what Avill J of a yard of cloth cost ? Analysis. — | of a yard will cost f times $5, or 3 times \ of $5. Now \ of $5 — $1^, and 3 times $1^ are %^. 92 fractions, 5. Whatis Jof I? Analysis. — | of f are equal to 3 times \ of |. Now |^ of f is /^, and 3 fourtlis are 3 times /g ^^ if = T2> ^^*- 6. At If a pound, what will f pound of tea cost ? 7. What costs "I of a box of lemons, at $G a box ? 8. At $8 a barrel, what will | of a barrel of flour cost ? 9. fof 12=? 12. fof 42feet = ? 15. |of 60=? 10. -f of 13=? 13. 1 of 40 yds. = ? 16. T^of 72 = ? 11. 1 of 16 = ? 14. y6TOf25lbs. = ? 17. /o of 200 = : 18. At 8 cts. a yd. what will be the cost of 5f yds. of muslin ; Analysis. — 5| yds. will cost 5| times 8 cts. Now 5 times 8 cts. are 40 cts. ; 1 fourth of 8 cts. is 3 cts. and 3 fourths are 3 times 2 cts. or 6 cts. which added to 40 cts. make 46 cents. Therefore, etc. 19. At 7 cts. a pound, what will 5 J pounds of sal soda cost ? 20. How many are 8f times 9 ? 21. How many are 7f times 12 ? 22. At 6 shillings a pound, what cost 5f pounds of tea. 23. What cost 7f acres of land, at $10 per acre ? Written Exercises. 208. Multiply i7ig hy a fraction is tahing a certain part of the tmdtiplicand as many times, as there are like parts of a unit in the multiplier. Thus, Multiplying by ^, is taking 1 half of the multiplicand once. Multiplying by ^, is taking 1 third of the multiplicand once. Multiplying by f , is taking 1 third of the multiplicand tivicc. Note. — 1. To find a halfoi a number, divide it hy 2. To find a if!iird of a number, divide it hy 3. To find ix fourth of a number, divide it hy 4, etc. 1. Multiply 63 by f Analysis.— Multiplying 63 by |, is find- ^- — ^^ = 36, Ans. ing I of 63. Now -| of 03 = 4 times ^ of 63 ; 9 I of 63 is 9 and j is 4 times 9 = 36. ' Or, ^^-^ = 36, A71S. Or, cancelling tire factor 7, common to both terms, we have 9 x 4 = 36. Midti'plication. 93 Note. — 2. A fraction is multiplied by a number equal to its denomi- nator \>j cancelling its denominator. (Arts. 123, r ; 179, 1°.) In like manner a fraction is multiplied by any factor of its denomi- nator by cancelling that factor. Find the product of 2. 60 by /_ 5. 112 by ^^, 8. 39 by if. 3. 63 by ^§j.. 6. 168 by l^. 9. 896 by |^. 4. 70 by T^. 7. 105 by V- 10- 572 by J-Q. 11. Multiply 160 by 5|. * OPERATION. Note. — 3. When the multiplier is a mixed IgQ x — = 120 number, multiply by the fractional and integral i p^ p- o^rj parts separately and unite the results. ^ ^?i5. 920 Multiply the following : 12. 93 by 12f . 16. 256 by 17^- 20. 107 by 47|J. 13. 184byl8|. 17. 196by41ii. 21. 510 by 85i|. 14. 125 by 10^6^. 18. 341 by 30yV 22. 834 by 89^1^. 15. 268 by 12Jf . 19. 457 by 12ff 23. 963 by 951^. Oral Exercises. 209. 1. If I cut J sheet of paper into 2 equal parts, what part of 1 sheet will there be in each piece ? Ans. ^ of a /m//* sheet, which is equal to J sheet. 2. If a bushel of apples costs %^, what will J bushel cost? Analysis. — i^ bushel will cost \ as much as a whole bushel ; and \ of $i is $i. (Art. 179, 2° .) 3. What part of 1 is | of | ? 4ofi? iofi? 4. Whicli is greater J or ^ ? JrOrJj? |or-,V? 5. What is 1 off? |of 1 ? |of-ft? 4 of A? 6. If a pound of tea is worth $|, what is f pound wortk ? 7. What cost -J yard of ribbon, at ^f a yard ? 8. If a yard of cashmere is worth -^f , what is f yd. worth ? 9. A man owning f of a yacht, sold J of it to his neighbor ; what part of the yacht did each then own ? 94 Fractions, Written Exercises. 210. Multiplying a Fraction by a Fraction. 1. At $f a yard, what will J yd. of silk cost ? Explanation. — One-fourtli of a yard operation. will cost \ as much as 1 yard, and \ ot $-| X | = f^ or If. $/g, and 3 fourths yard will cost 3 ^ times %i^, or $ff = ||, ^7^«. Or, | X f =r $f , Ans. Or, indicating the operation, and cancel- 3 ling the factors common to the terms of the fractions, we have $| the same as before. Note. — The above solution is the same in effect as multiplying the numerators together for the numerator, and the denominators for the denominator of the required product. 2. Multiply i of I of I by i of ^. Explanation. — The product of the numerators f , |, f , ^, y^, is 120 ; the product of the denominators is 1440 ; and y 4V0 = tV» Ans. Or, cancelling the factors common to the |.xfxl-Xi^XT\ ^^ numerators and denominators, the result is Xs/^v/'5.viv_4 _i_ 3*2, the Ans. required. 211. The word of, iu Compound Fractions, has the force of the sign of multiplication x . Multiplying compound fractions together reduces them to 2i_ simple fraction. Thus, f of f of I is a compound fraction, and is equivalent to f x | x |, which is equal to l^, a simple fraction. 3. Multiply I of 4 by | of -| of 14. Ans. f| = 2||. 4. At |6|- a barrel, what are 5^ barrels of flour worth ? Note. — If either factor is a mixed operation. or whole number it may be reduced 6|- = -^ and 5-^ = ^^ to an improper fraction, and the 5 1 v^ 1 6_ _- 432 __ ^3^ AnS operation becomes the same as mul- 9 4 tiplying a fraction by a fraction. Or, ^^ X -^^ == $36, Ans, 5. -Axi} = ? 9. fof|offo=:? 6. «xH = ? 10. tfofifofH^? 7- t\xH = ? 11. f of 25xf of J=z:? 8. |x|x| = ? 12. fof 30x-Hof | = ? i3. IIow many ^re j^ of 45 x |^ of | ? Multiplication, 95 14. If a quart of chestnuts costs |^ of j of 40 cents what will -J of ^ of a quart cost ? 15. What cost 15-| tons of coal, at $6f a ton ? 212. The preceding principles may be summed up in the following General Rule. Reduce whole and mixed ninnhers to improper frac- tions, compound fractions to simple ones, and cancelling the common factors, lurite the product of the numerators over the product of the denominojtors. A-PPLICATIONS. 213. 1. At l-l a cord, how much will the sawing of 20J cords of wood amount to ? 2. What cost 1 6 pounds of cheese, at 8^ cents a pound ? 3. What cost 9 dozen of eggs, at 12-|- cents per dozen ? 4. What cost lof yards of cambric, at 15 cents per yard? 5. What cost 111 cords of wood, at |3|- per cord ? 6. At 12 J cents a pound, what cost 2f pounds of pepper? 7. What cost 18 ounces of nutmegs, at 16 J- cts. an ounce ? 8. At 12f cents a yard, what will 27 yards of cotton cost? 9. At $3^|- a yard, what cost 15^ yards of broadcloth ? 10. What cost 15} yards of ribbon, at 40 cents per yard ? 11. What cost 22 penknives, at |^| apiece? 12. At %^-Q a yard, what cost 8} yards of silk ? 13. At $1 a yard, what will 9^ yards of muslin cost ? 14. At If a bushels what cost TyV bushels of wheat? 15. What will 8-f pounds of tea cost, at $f a pound ? 16. What cost 66 bushels of apples, at 18| cents a bushel ? 17. At 32^ cents a yard, what cost 12| yards of gingham ? 18. What cost 18-| yards of lace, at 16^^ cents per yard ? 19. What cost 43 bushels of oats, at 18} cents a bushel? 20. What cost 31J yards of sheeting, at $f per yard? 21. At $y\ a quart, what cost 18-|- quarts of cherries? 22. What cost 14| bushels of potatoes, at 18} cents a bushel? 23. At $-| a yard, what cost 8| yards of velvet ? 24. At 8-J a bushel, what costs 47|- bushels of pears ? 96 Fractions. 25. What cost 63f pounds of sugar, at 9f cents per pound ? 26. What cost 22|- yards of velvet, at |3f a yard ? 27. What cost 25^ pounds of figs, at Ib^ cents a pound ? 28. What cost 35| cords of wood, at $3| per cord ? 29. What cost 175 J bushels of corn, at If a bushel ? 30. What cost 38| tons of hay, at 115-J a ton ? 31. At 42|^ miles a day, hoAV far can you travel in 17^ days ■ 32. Mult. 126 by | of 33. 37. Mult. -||f by ^ of |f|. 33. Mult, f of 9 by f of 7. 38. Mult, ff by l^. 34. Mult, f of 184- by f of 241-. 39. Mult. -| of | by f of f. 35. Mult. 217i by | of f of 8. 40. Mult. 16f by f of 6. 36. Mult. Ill by^l of HI- 41. Mult. 468t5j. by j of f^-o 42. Multiply f of I of -j^ of if of 11 by 1 of f of 45. 43. Multiply I of A of If of i of 29 by H of y^o of A- 44. Multiply I of If of -^ of 16i by || of || of | of 49, Division of Fractions. Oral Exercises. 214. 1. If 2 melons cost If, what will 1 melon cost ? Analysis. — 1 melon is | of 3 melons ; therefore, 1 melon will cost ^ of $1, and I of $4 is $f, Ans. 2. If 3 knives are worth %-^-q, what is 1 knife worth ? 3. K I pay 1^ for 4 slates, what do I pay for 1 slate ? 4. If 2 pears cost -| of a dime, how much will 1 pear cost? 5. If J of a yard of cloth is divided into 3 equal pieces, what part of a yard will 1 piece contain ? 6. By what do you divide to find 1-half a number? To find 1-third? 1-fourth? 1-fifth? 7. How do you multiply by |^ ? By J ? I^J f ? 8. What is the difference between multiplying by J and dividing a number by 2? Between multiplying by | and dividing by 3 ? 9. If 5 fans are worth i{-|, what is 1 fan worth? 10. If 4 melons cost $i\-, what will 1 melon cost? Division, 97 11. If 5 apples are worth | dime, what is 1 apple worth? Analysis. — A Fraction is divided bv dividing its numerator or multi- plying its denominator. Since the numerator 3 cannot be divided by 5 without a remainder, we multiply the denominator 4 by it, and 5 times 4 are 20. Therefore, 1 apple is worth o% dime, Ans. (Art. 179, 2".) 12. A grocer divided f of a cocoanut among 6 boys ; what part of a cocoanut did each receive ? 13. Paid If for 5 Table-books ; what was the price of each ? 14. How many ways can you divide a fraction ? 15. Divide f by 3. ^ by 4. -if by 6. f| by 9. 16. Divide | by 3. -J by 4. ^V by 6. H by 5. ^ by H. 17. What is the quotient of -^-^9 ? Of f^^S ? 18. What is the quotient of -?,6- divided by 9 ? ^^..^n p 19. A man had ^ of a dollar, and gave all for 9 hats ; how mnch did each hat cost him ? 20. At $4 a bushel, how many bushels of quinces can be had for llOf ? Analysis. — As many busbels, at $4, may be had, as $4 are contained times in $10|. Now |10| = %^-, and %2.-^4 = s^ or 2| bushels, Ans. 21. Divide 6| by 4. ^ by 5. TJ by 8. 11| by 9. 22. If 8| pounds of candy are divided equally among 5 chil- dren, what part and how much will each receive ? Written Exercises. 215. Dividing a Fraction by an Integer. 1. If 4 yds. of muslin cost $^, what will 1 yd. cost ? First. — Dividing tbe numerator by 8-^-4 4, we have %f^ = %\. (Art. 179, 2\) -T^' = ^TW = H^ -1^^'''' Second. — Multiplying the denom- :i inator by 4, we have — r — - = :^^, or x/c- x ■± 1^ X 4: iQ $t. Ans. ^ Or, cancelling the factors common to — := $^, Aus. both terms, we have %\, as before. « Notes. — 1. It is better to divide the numerator when it can be done without a remainder. 5 98 Frdctions, 2. By the first operation the number of parts is diminished, but their size remains the same. Bj the second operation the number of parts remains the same, but their size is diminished. 2. Divide -V^ by 9. 6. 2||-^70=? lO. fo|-^-120=:? 4. H-12=:? 8. J,V--^5:r.? 12. iMf-75=? 5. f 22=? 9. 3^-^-93 = ? 13. -i^V/--^14^=? 14. At $7 a barrel, how many barrels of cranberries can be bought for $25| ? Note. — When the dividend is a mix- operation. ed number, it should be reduced to an $25-| = %^-^ imj)roper fraction; then proceed as |v?, _l. 7 := |1J._ ^ $3%:. above. Ans. $3f. 15. If 2|| of a ton of hay were fed to 6 horses, what part and how much would each receive ? 16. Paid $7^ for 12 books ; what was the price of each ? Oral Exercises. 216. 1. How long will it take a lad to earn $5, if he earns $1 a day ? Analysis. — At $| a day, it will take as many days as f are contained times in 5. In 1 there are 3 thirds, and in 5, 5 times 3, or 15 thirds. Now 2 thirds are contained in 15 thirds, 7^ times. Aiis. 71 days. 2. How many times is J contained in 4 ? In 5 ? In 9 ? 3. How many times 1 in 5 ? |- in 7 ? -J- in 8 ? |^ in 9 ? 4. If you earn $J in 1 day, how long will it take you to earn 112 ? 5. How many times f in 7 ? In 8? In 10 ? 6. At If apiece, how many books can you buy for $12 ? 7. If a boy saws | of a cord of wood in 1 day, how long will it take him to saw 8 cords ? 8. How many times are f contained in 5 ? In G ? In 10 ? 9. At $f a bushel, how much corn can you buy for 110? 10. If I burn f ton of coal in 1 day, how long will G tons last me ? 11. How many times are f contained in f of IG ? Division. .99 12. How many times | in | of 32 ? In f of 40 ? 13. If 2|- yards of cloth will make a coat, how many coats can be made from 20 yds. of cloth ? Analysis. — In 2i yards there are 5 half -yards, and in 20 yds. there are 40 half-yards. Now 5 is contained in 40, 8 times. Ayis. 8 coats. 14. How much wood at $3^ a cord can be had for 15. How many barrels of potatoes at $2f can you buy for $22? 16. At $6f a week, how long can a man board for $100 ? Written Exercises. 217. Dividing an Integer by a Fraction. 1. How many times are f contained in 21 ? Explanation. — Reducing 21 to fourths opekation. we have 21 = V- ^^ow | and ^- are like 21 X 4 = 84 fractions, and one numerator is divided by ^_ _i_ |. ;:^ 28 Ans. the other like integers. r), *,-• ^ 4 no Or, Multiply the integer by the fraction ' inverted. 2. 56 by ^\. 5. 240 by A- 8. 384 by if. 3. 72 by A. 6. 256 by ^-. 9. 576 by if. 4. 132 by 11 7. 110 by Y- 10. 1880 bv /t 11. At $^ a yard, how many j^ards of silk can be had for $37 ? 12. If you pay $J a day for board, how many days can you board for $126 ? 13. How many cloaks can be made from 72 yds. of cloth, allowing 4 J 3'ds. for a cloak ? Note.— When the divisor is a mixed operation. number, it should be reduced to an im- 4|^ = |- proper fraction before dividing ; then 72 -h # = 16 multiply the integer by the fraction in- Qj. tvo y 2 _- ;[g Alls verted. (Art. 217. Ex. \.) ' "5 — ^ 14. 120-^12^=:? 16. 240-^yV=? 18. 785-^62izIr? 15. 192-^10-|=r? 17. 552-^ff:=? 19. 2000-r-87i=? 20. At $3^ apiece, how many sheep can be had for $1500 ? 100 Fractions. 21. How many yards of silk, at %Z^ can be had for $185 ? 22. Allowing 4| yards of cloth for a cloak, how many cloaks can be made from 154 yards ? 23. At $4J each, how many chairs can be bought for 1250 and what remainder. 24. If a stage coach travels at the rate of lOf miles per hour, how long will it be in going 320 miles ? Oral Exer cises. 218. 1. How many slates at %-^ can be bought for $\^ ? Analysis. — Since these fractions express like parts of like units, it is plain that as many slates can be bought as j\ are contained times in ^f, or 3. Ans. 3 slates. (Art. 197.) 2. If a vest can be made from f yd. of velvet, how many vests can be made from ^ yards ? 3. How many times are | contained in ^-£- ? In ^-^- ? In -^^ ? 4. Divide A by A. ^ by ^. i| by j%. J| by ^V 5. If pen-knives are || apiece, how many can you buy for 6. How many melons at $| apiece can a person buy for If ? Analysis, — He can buy as many as | are contained times in f . Now I = f , and I are cc-ntained in |, 2 times. Ans. 2 melons. 7. How many books, at $f , can be bought with $| ? 8. At $f a yard, how much fringe will If buy ? 9. At if a pound, how many pounds of spice can be had for$^? Forlfl? 10. Divide f by A- r\ by f . {% by ^. U by i- 11. At $J a yard, how many vards of flannel can you buy for 12. How many pounds of tea at $f, can be bought for $-|f ? 13. At $1, how many yards of calico can be had for S-j^ ? 14. If cinnamon is $f a pound, how much can be bought for 15. How much coffee can be bought for If J when the price is If a pound ? - Division. v^'",'^ \j i(^i ' J > '»©" •• jj Written ExrR'cfscs. 219. Dividing a Fraction by a Fraction. 1. How much tea, at $f a pound, can be had for $| ? 1st Method. — Reducing the given frac- 1st operation. tions to a c. d,, | = j^, and f = j% Now if | = -^ ; f = i^^ $^% will buy 1 pound, $j% will buy as many ^9^ _^ _8_ — 9 _i_ g pounds as y% are contained times in y\, and g _j_ g __ -j^ 1 2|)_ AflS^ 9h-8 = 1|. "^ws. Impounds. (Art. 191.) ' 8 •? * 2d Method.— The above process may be ~° opekation. shortened by inverting the divisor and mul- $|- -^ §-| =: f X ^ tiplying the two fractions together as in the |- x | = -§, or 1^ lb. margin. (Art. 210.) Note. — 1. It will be seen by inspection that the 2d method in effect reduces the fraction to a c. d. and divides one numerator by the other at the same time, the numerators only being used, as in 1st method. 2. Divide 10| by 6|. Solution. — Reducing the operation. mixed numbers, to improper 10|- = ^^- ; 6|- == -^ fractions and dividing, the result 8j_ _^ 2_i =: JUL x -A- = 14, Ans. is 14. 3. Divide J of f by 4 of -jV 4 Solution.— I X | X | X -^^ = f, or IJ, Ans. 3 220. The preceding principles may be summed up in the following General Rule. Reduce iclwle and mixed numbers to improper frac- tions, and multiply the dividend by the divisor inverted. Or, Reduce the fractions to a commoii denominator and divide the numerator of the dividend by that of the divisor. Note. — The object of inverting the divisor is convenience in multiply- ing. After inverting the divisor, cancel the common factors. 4. Divide -f^ by \. 6. Divide 81| by 45|. 5. Divide 75 by Sf. 7. Divide i^j of f by 30. 102 J^r actions. \^r PLICATIONS. 221. 1. At 16J cents per pound, how many pounds of figs can you buy for 87|- cents ? 2. How many cords of wood, at $6J per cord, will it take to pay a debt of I67J- ? 3. How many barrels of pork, at $llf per barrel, can be obtained for $95J ? 4. A man bought 15J barrels of beef for $124| ; how much did he give per barrel ? 5. A man bought 13 1 pounds of sugar for 94r|- cents ; how much did his sugar cost him a pound ? 6. A lady bought 15|- yards of silk for IIS^^^ shillings ; how much did she pay per yard? 7. Bought 15^ baskets of peaches for |24| ; how much was the cost per basket ? 8. Bought 30:^ yards of broadcloth for |181|- ; what was the price per yard ? 9. Paid $375 for l'2b\ pounds of indigo ; what was the cost per pound ? 10. How many tons of hay, at IIGJ per ton, can be bought for $1961? 11. How many sacks of wool, at $17^ per sack, can be pur- chased for $1500 ? 12. How many bales of cotton, at II 5| per bale, can be bought for 12500 ? Divide ^ of 16 by | of |. Divide f of | by 21. 21. Divide y\ of -^ by I of f Divide 223-J by f o/si. Divide | of | by 48. Divide 42J by | of 53^ 25. Divide J of } of -^ of f of -Jf by | of if. 26. Divide y^ o^ f o^ tI of ^ by -|f of \\ of 18. 27. Divide -|f of if of 67 by |J of f| of 25. 28. Divide ff of |^ of 4U by ff of |i of 31. 29. Divide fj of |4 of |f "of 82f by \\ of f| of 42f. 13. Divide | of y^ by 6f 19. 14. Divide ^V of 30 by 19. 20. 15. Divide i\ of 1% by 31. 21. 16. Divide ^ by | of 12. 22. 17. Divide ^ by ISf 23. 18. Divide 42^ by | of 7. 24. Division. * 103 222. To Reduce Complex Fractions to Simple Ones. Expressions which have a Fraction in the numerator or denominator or in both, are called Complex Fractions. ing Division of Fractions. Thus, if ; 57 ; if ; f > are complex fractions, and are a form of indicat- 1. What is the yalue of T-f- OPEBATION. Analysis. — Reducing the mixed numbers to 31 -— 10 improper fractions, we divide the numerator by 03 40 the denominator according to the rule. (Art. 220.) 10.48 5_o The result is AOx = #1, Ans. Hence, the -X" ~ ^" — m EuLE. — Treat the numerator as a dividend and the denominator as a divisor, and divide one by the other according to the rule for division of fractions. 2. Reduce ^ to a simple fraction. Ans. ff. Reduce the following to their simplest form : 6 ^ 121 94 251 3. — ■• 6. — -• 9. — -. 12. H 6i 7J f 4. -• 7. -^- 10. — -. 13. — . ■I 6 12J li 8 44 20| « Note. — Complex Fractions, when reduced to Simple Fractions, are added, subtracted, multiplied, and divided like other fractions. 15. Find the sum of the 2d and 3d. 16. Find the difference of the 4th and 5th. 17. What is the product of the 6th by the 7th ? 18. What is the quotient of the 10th divided by the 9th ? 19. What is the product of the 7th and 8th ? 20. What is the quotient of the 12th diWded by the 13th ? 104 - Fractions, 223. Finding what Part one Number is of Another. 1. What part of 9 inches is 2 inches ? 4 in. ? 7 in. ? 2. What part of a yard is 1 foot ? Analysis. — In 1 yard there are 3 feet, and 1 foot is \ of 3 feet, Ans. 3. What part of a week is 1 day ? 2 days ? 5 days ? 4. What part of 3 days is 1 foot ? Ans. Days and feet are unlihe numbers, and therefore one cannot be compared with the other. 5. What part of -f is f ? Analysis. — Since these fractions have a c, d, they are like fractions, and their numerators are compared like integers. Ans. f . 224. From the examples above are derived the following Principles. 1°. Only like mimhers, or those wliich are so far of the same hind that one may he said to he a part of the other, can he compared. 2°. When fractions have a comynon de7iominator, their nu^ merators are compared lihe integers. Oral Exercises. 225. 1. What part of 30 cents are 5 cents ? Ans. -^q or -J-. 2. What part of 21 yards are 7 yards? Of 45 days are 9 days ? 3. $7 are what part of $15-? Of $45 ? Of $63 ? Find what part one of the following numbers is of the other, expressed in lowest terms : 4. Of 30 is 12 ? 6. Of 96 is 48 ? 8. Of 65 is 100 ? 5. Of 63 is 14 ? 7. Of 120 is 30 ? 9. Of 108 is 144 ? 10. If an acre of land is worth $63, what part of an acre will $9 buy ? 11. If a piece of carpeting can be bought for $120, what part of a piece can be bought for $12 ? 12. 23 is what part of 69 ? 48 of 72 ? 84 of 99 ? Division. 105 13. I is what part of ^ ? t% of H ? H ^f fj ? 14. AYhat part of ^ is y^^- ? Suggestion.— I = -f^. Ans. |. 15. What part of -H is f ? Of|is-|^? 16. What part of ^ is iV ? Written Exercises. 226. To find what part one number is of another. 1. What part of 49 is 28 ? Analysis. — 28 is ff of 49, or | of 49, Ans. 2. What part of tV is J-g- ? Analysis. — Reduced to a c. (J, the given frac- tions become |f and f f , which are like fractions. 22 Now 22 is II of 35, Ans. (Art. 224.) Hence, the so — 6 22 -^ 35 = II Rule. — Mahe the Jiimider denoting the pa?'t the numerator, and that with luhieh it is compared the denominator. Note. — If either or both the given numbers are fractional, they should be reduced to a c, d, : their numerators are then compared like integers. OPERATIOK. T¥ — 6 5 3. What part of 36 is -| ? 9. lOOf is what part of 175f 4. What part of 62 is -J ? 10. 6 J is what part of 45 ? 5. What part of 86 i^ |i ? 11. 40 is what part of 954 ? 6. AYhat part of 58 is 7f ? 12. if is what part of -Jg. ? 7. What part of 112 is | ? 13. |f is what part of ^ ? 8. What part of 325 is I ? 14. 18|- is what part of 46f ? 15. At S23 per acre, how much land will $17 buy ? 16. A man paid $185 for a horse, and sold it for $150 ; what part of the cost did he get ? 17. A man 76 years old has a son whose age is 54 years ; what part of the father's age is that of his son ? 18. If from a piece of silk coutaining 27j- yds., you cut 11|^ yds., what part of the pie,ce will be left ? 106 Fractions. 19. If a man can perform a journey in 24 days, what part of it can he go in 9 days ? 20. What part of |268| is Il75f ? 21. If A can do a job of work in 20 days, and B in 10 days, what part will each do in 1 day ? What part will both do ? Oral Exercises. 227. 1. 4 is 4" of what number ? Analysis.— 4 is -| of 4 times 7, or 28. Therefore, 4 is | of 28, Ans. 2. 36 is f of what number ? Analysis.— Since 36 is | of a number, | of tliat number is ^of36, which is 12, and 4 fourths are 4 times 12, or 48. Therefore, etc. Note. — If the learner is at a loss which term of the fraction to take for the divisor, let him substitute the word i^arts for the denominator, and his difficulty will vanish. 3. 15 is I of what ? 7. 15| is f of what ? 4. 16 is f of what ? 8. 10| is 4 of what ? 5. 45 is 4 of what ? 9. 45 == |^f of what ? 6. 210 is I of what ? 10. 48 = i| of what ? Written Exercises. 228. To find a Number when a Part of it is given. 1. 56 is J of what number? . CI- 1 £ \ • t;a OPERATION. Analysis. — Since f of a number is 56, 1 ninth is \ of 56, which is 8, and 9 ninths are Ob -r- / = o 9 times 8, or 72, A7is, Hence, the 8 X 9 r= 72, Ans. KuLE. — Divide the numher denoting the part hy the ninneratoi% and multiply the quotient by the denomi- nator. (Art. 208.) 2. 48 is I of what ? 6. 132 is f| of what ? 3. 56 is I of what ? 7. 257 is f of what ? 4. 75 is f of what ? 8. 394 is ^ of what ? 5. 96 is T^o of what ? 9. 859 is \l of what ? 10. A merchant lost $4368, which was ^ of his capital ; what was his capital ? Division. 107 11. If f of a farm is worth $2360, what is the whole worth ? 12. A drover being asked how many sheep he had replied that 147 was equal to ^^ of them ; how many sheep had he ? 13. A man lost f of his money and had $260 left ; how much had he at first V Oral Problems for Review. 229. 1. A lad having $5, paid $2|^ for a pair of skates and $1^ for a sled ; how much did he have left ? 2. A lady went shopping with $15 in her purse ; she paid $f for a handkerchief, S2^ for a pair of gloves, and the rest for a shawl ; what did the shawl cost her ? 3. A laborer earned $1J one day, $1^ the next, and paid $1|- for board ; how much had he left ? 4. A grocer bought a load of apples at $| a bushel, and sold them at $f ; how much did he make on a bushel ? 5. A man owning ^ of a ship, sold f of her ; what part had he left? 6. The sum of two fractions is ^^ and one of them is f ; what is the other ? What is their difference ? 7. The greater of two numbers is 6f, and their difference is 2J ; what is the less number ? 8. The less of two numbers is 5|, and their difference 2-|- ; what is the greater number ? 9. The product of two fractions is \\, and one of the frac- tions is J ; what is the other fraction ? 10. If the dividend is ^, and the quotient is -f, what is the divisor ? 11. What number divided by | will give a quotient of 7^ ? 12. A teacher spends f of his salary for board and -^^ for clothing ; what part of his salary is left ? 13. If a man earns $60 a month and spends f of it, how much can he lay up ? 14. I sold I of my farm and had 48 acres left ; how many acres did my farm contain ? 15. What is the difference between 4|^ and o| ? 16. What number subtracted from 15|- will leave 10^%? 108 ^'r actions. 17. The sum of two fractions is 17|, and one of them is 12f ; what is the other ? 18. At %VZ^ a sack, what are 5 sacks of coffee worth ? 19. At $5f a barrel, what will 10 barrels of flour cost? 20. At $6 a ton, what will 15 1 tons of coal ceme to? 21. What will 9 cords of wood cost, at 1^3 J a cord ? 22. Bought a horse and sleigh for 1175, and the sleigh was worth f as much as the horse ; what was the value of each ? 23. A lady bought 6 neck-ties at S| each, aud has $15 left ; how much money had she at first ? Written Problems for Review. 230. 1. Reduce ^ to the denominator 243. 2. Reduce |||| to lowest terms. 3. Find the prime factors of 486, 576, and 972. 4. What is the I, c, d» of ~-f-^, |, and J|- ? 5. What is the sum of f of f , i |, and 5|- ? 6. What is the difference between 14i + 25f, and 25f + 19i? 7. The greater of two numbers is 375|, and their difference 273f ; what is the less ? 8. If I buy H ^f ^ ship, and sell | of what I buy, how much shall I then own ? 9. Required the sum and difference of } and ^ ? 10. A railroad car goes 225J miles in a day and a steamer 185J miles ; how far do both go in a day, and what is the dif- ference m the distance they go ? 11. A grocer bought a quantity of apples for $162|- and sold them for |210f ; what was his profit? Find the sum of the following : (24| + 12i)-(llJ + 2f). (28-2f) + (16~2A). (140 + l|-)-(8A-li). 145 + ^V+(112|-8t). 20. A farmer sold a cow for $26f, 15 sheep for $52J, and the buyer handed him a $100 bill ; how much change should he return ? 12. 8i-f6^-3i. 16. 13. 14|4-6i-7f. 17. 14. 20f-8i- + 4|. 18. 15. 26f + (4}-2i)+3i. 19. Division. 109 21. Paid $275f for a quantity of wheat, $320^ for a quan- tity of corn, and sold the former for I316|, and the latter for $41 0|- ; what w\as my profit ? 22. If t of a factory cost 123245, what is the whole worth ? 23. What number multiplied by 7f will produce 872|? 24. If the divisor is f^, and the quotient ^, what must be the dividend? 25. The dividend is 42f, the quotient 8^, what is the divisor? 26. A father bequeathed J, \, and ^ of his property to his 3 children, and had 14800 left for his wife ; what was the amount of his property ? 27. A merchant lost f of his capital by one creditor, and f by another, and had 1500 left ; what was his capital ? 28. My neighbor having 356| acres of land, sold ^ of it to one man, and f of it to another"; what was the value of the remainder at I25f per acre ? 29. A man gave his check for $1675J, which was | of what he had on deposit ; how much had he in bank ? 30. A man had 6f acres of land, which he divided into building lots each containing ■^\ acres ; how many lots did he have ? 31. Bought a horse for $160J, and sold it for f of the cost ; how much did I lose ? 32. How many books, at $-| apiece, can be bought for 810-^ ? 33. At 13 J a day, how much can a man earn in 25|- days ? 34. What cost 34J bushels of flaxseed, at $2^ a bushel ? 35. A market-woman being asked how many eggs she had, replied, " 244 equals f of them ; " how many had she ? 36. A man paid $5250 for a house, which was -f- of all his property ; how much was he worth ? 37. If T^ of a ship cost S8360, what is the whole ship worth ? 38. A lady teacher paid $750 for a piano, which was f of her salary for a year ; what was her salary ? 39. A tree casts a shadow^ of 48 feet, which is f of its height ; how" high is the tree ? 40. Nine feet of a flag-staff stands in the ground, which is ■^ of its w^hole length ; what is its length ? 110 Fractions. 41. A lad being asked how many marbles he had, said he had f as many as his friend, and that both together had 255 ; how many had he ? 42. A goldsmith paid 175 for a watch, which was f of what he got for it ; how mnch did he get for the watch ? 43. A can build a school-house in 90 days, which is f of the time it would take C ; how long would it take C to build it ? 44. An army lost \ of its men in battle and \ by sickness, and had 9600 left ; what was its whole number ? 45. 16| is -i- of what ? 48. f of G| is f of what ? 46. 18| is ^ of what ? 49. | of ff is ^^ of what ? 47. 25f is f of what ? 50. f of 48 is how many times 10 ? 51. A man bought a buggy for §185, which was f the price of his horses ; what did his horses cost ? 52. A whale-ship lost -^-^ of the bread, and the men were allowed 12 ounces per day a2:)iece ; what had each at first ? 53. A man sold his farm for $4760, and thereby gained J of the cost ; what did he pay for it ? 54. A man bequeathed to his son 17600, which was If of what he gave his daughter; what was his daughter's portion ? Questions. *^ 160. What is a fraction? 161. The unit of a fraction? 162. A frac- tional unit? 164. What is the denominator ? 165. The numerator ? 166. What are the terms of a fraction? 171. What is a proper frac- tion? 172. Improper? 173. Simple? 174. A Compound? 175. A mixed number ? 176. From what do fractious arise? 177. What is the value of a frac- tion ? 179. Name the three general principles of fractions. 181. What is reduction of fractions? 182. How is a fraction reduced to higher terms? 185. How to the lowest tei-ms ? 187. Improper fractions to mixed numbers ? 189. Mixed numbers to improper fractions? 191. What is a common denominator? 192. The least common denom- inator? 196. How found ? 197. What are like fractions? 198. Unlike? 200. What fractions can be added? 201. Rule for adding fractions? 204. Rule for subtracting fractions ? 211. The force of the word "of" in compound fractions ? 212. General rule for multiplying fractions? 220. General rule for dividing fractions? 222. What are complex fractions ? How reduce complex fractions to simple ones? 226. How find what part one number is of another? 228. How find ^ number when a part of it is given ? f I » ... 9 1 — ■ — I g) ! ^^"' i— ECIMAL FkACTIOIsTS. -i=^.\~/l=r Oral Exercises. 231. 1. If a unit is divided into 10 equal parts, what is each part called ? 2. If one of these tentlis is subdivided into 10 equal parts, what part of the unit is one of them ? Ans. TS" ^^ iV — Too"? or one hundredth. 3. What part of the unit is 2 of these parts ? 3 of them ? 6 of them? 11 of them? 4. If 1 hundredth of a dollar is divided into 10 equal parts, what part of a dollar is one of these parts ? Ans. iV of To-o = roVo? or one-thousandth. 5. What part of a dollar is 2 of these parts ? 4 parts ? 6. What is meant by a tenth ? 3 tenths ? 7 tenths ? 7. What is meant by a hundredth 9 4 hundredths ? 8. What is meant by a thousandth 9 5 thousandths ? Definitions. 232. A Decimal Fraction is one or more of the equal parts of a unit divided into tenths, hundredths, thousandths, etc. Note. — They are called Decimals from the Latin decern, ten, which indicates their origin and scale of decrease. 233. A Mixed Decimal is an integer and decimal expressed together. Thus, 34.153, and 42.65 are mixed decimals. 234. Decimals are expressed by writing the numerator only, with a decimal point ( . ) on the left. 235. The Denominator of a decimal is always 10, 100, 1000; etc. ; or 1 with as many ciphers annexed to it as there are decimal places in the given numerator. 112 Decimal Fractions, 236. The Notation of Decimals is an extension of the Notation of Integers. (Art. 36.) Table. '!3 • m fl X! ^ (3 aj -M OQ _o OB aj -a J3 Names s ) prefixed to them, and the decimal point placed after them. Cents occupy hundredths place on the right, and mills the place of thousandths. Notes. — 1. Eao^les and dimes are seldom used in business calculations j the former are re?..d as dollars, the latter as cents. Thus, 15 eagles are read as $150, and 6 dimes as 60 cents. * The United States adopted the decimal syptem of currency in 1789. Since then it has been adopted by France. Beliiium, Brazil, Bolivia. Canada, Chili, Denmark, Ecuador, Greece, Germany, Italy, Japan, Mexico, Norway, Pera, Portugal, Spain, Sweden, Swit- zerland, Turkey, U. S. of Colombia, and Venezuela. TI, S. Money, 127 2. Cents occupy two places, lience if the number to be expressed is less than 10, a di^lier must be prefixed to the figure denoting them. 3. In business calculations, if the mills in the result are 5 or more, thej are considered a cent ; if less than 5, they are omitted. 276. To reduce dollars to cents, multiply them by 100. To reduce dollars to mills, multiply them by 1000. To reduce cents to mills, multiply them by 10. 277. To reduce cents to dollars divide them by 100. To reduce mills to dollars divide them by 1000. To reduce mills to cents, divide them by 10. 278. Dollars, cents, and mills correspond to the orders of integers and decimals, and are expressed in the same manner. Thus, 78 dollars 47 cents 5 mills are vmtten, $78,475. Write the following in like manner: 1. 50 dols. 10 cts. 5 mills. 4. 372-3^ dollars. 2. • 75 dols. 5 cts. 8 mills. 5. 407 dols. Vl\ cts. 3. 627 cents 5 mills. 6. 5260^% dols. Oral Exe r cises. 1. Change 5 cents to mills. 7. Eeduce $4 to mills. 2. Change 8 cents to mills. 8. Keduce 16.10 to mills. 3. Change 40 mills to cents. 9. Reduce 600 cents to dollars. 4. Change 65 mills to cents. 10. Eeduce 7000 mills to dollars. 5. Change 83 to cents. ii. Reduce 460 cents to dollars. 6. Change $5.20 to cents. 12. Reduce 5420 mills to dollars. 13. A lad bought a History for $1.10, and gave a two-dollar bill in payment; what change did he receive ? 14. A dealer paid $4.30 for a pair of boots, and sold them for $5.25 ; how much did he make ? 15. If a laborer earns $1.25 in one day, how much can he earn in 4 days ? 16. What cost 5 barrels of flour, at $6.50 a barrel ? 17. If 5 caps cost $3.25, what will 1 cap cost ? 18. At 20 cents apiece, how many citrons can be bought for $2.40 ? 128 Decimals, Written Exercises. 279. United States Money is added, subtracted, multi- plied, and divided in all respects like Decimal Fractions. (Arts. 252-261.) 1. A man bought a cow for 115.75, a calf for $2,375, a sheep for 13.875, and a load of hay for 18.68 ; how much did he pay for all ? 2. A farmer sold a firkin of butter for 19.28, a cheese for $1.17, a quarter of veal for $.56, and a bushel of wheat for $1.12 ; how much did he receive for the whole ? 3. A man bought a hat for $5,375, a cloak for $35.68, and a pair of boots for $4. 75 ; how much did he pay for all ? 4. What is the sum of 63 dols. and 4 cts., 86 dols. and 10 cts., and 47 dols. and 37 cts. ? 5. If I pay $217 for a horse and $145.50 for a buggy ; what is the cost of both ? What is the difference in cost ? 6. What is the difference between $137.25 + $65.07 and $126,121- + $93.06? 7. A man paid $63.87^ for a sleigh and $27.50 for a robe, and sold them both for $185 ; how much did he make ? 8.. What will 145 loads of wood cost, at $3.25 a load r 9. Bought 115 barrels of apples, at $3 a barrel, and sold 20 barrels at $2.50 and the remainder at $4.25 a barrel ; did I gain or lose by the operation ? How much ? 10. A paid $15 per acre for his farm of 365 acres, and B paid $23 per acre for his farm of 285 acres ; required the dif- ference in tlie cost of their farms ? 11. A farmer bought 165 sheep at $6 a head, 16 cows at 134, and 27 tons of hay at $21 a ton, and paid $500 down ; how much did he then owe for them ? 12. If a man has a salary of $1800 a year, and pays $225 for his board, and spends $175 for clothes and $220 for inci- dentals, how much will he lay up in a year ? 13. A grocer bought 1365 sacks of coffee at $20 per sack ; he sold 563 sacks at $25 and the balance at $27 a sack ; how much did he gain or lose ? U, S. Money. 129 14. A butcher bought 116 head of cattle at $47 a head, and 3 times as many sheep at |6 a head ; how much did he pay for his cattle and sheep ? 15. How many hats at $3.75 apiece can you buy for $18.75 ? 16. If a man pays $7.25 a week for board, how long can he board for $258. 50? 17. A mason received $194,375 for doing a job, which took liim 75|^ days ; how much did he receive per day ? 18. At $1.12|- per bushel, how many bushels of wheat can be bought for $523.75 ? 19. If $1285.25 were divided equally among 125 men, what would each receive ? 20. The salary of the President of the United States is $50000 a year ; how much does he receive per day ? 21. A man paid $66.51 for broadcloth, which was $7.39 per yard ; how many yards did he buy ? 22. If flour is $8. 12 J per barrel, how many barrels can be bought for $2047.50 ? 23. If 556.25 lbs of tobacco cost $69,532, how much is that a pound ? 24. At $47,184 per ton, how many tons of railroad iron can be bought for $28310.40 ? Smout Methods. 280. An Aliquot Part of a number is an exact divisor of that number. Thus, 2, 2i, 3^, and 5, are aliquot parts of 10. Aliquot Parts of a Dollar. 50 cents = $i. 12^ cents = $-|-. 33J cents = %\. 10 cents = $-1^. 25 cents = $J. 8| cents = %^. 20 cents = %\. 6J cents = $yV 16| cents = $|. 5 cents = $^. 130 Decimals. Oral Exercises. 281. 1. What part of II is 50 cts. ? 25 cts.? 20 cts.? 2. What part of $1 is \%\ cts. ? 10 cts. ? 8-^ cts. ? 6} cts. ? 3. Wiiat will 27 yds. of delaine cost at 50 cts. a yard ? Analysis. — 50 cents are %\ ; therefore 27 yds. will cost 37 times %\, or %^-, wMcli are equal to $13i, or $13.50, Ans. 4. At 25 cts. a pair, what cost 75 pairs of mittens ? 5. At 12|- cts. each, what will be the cost of 72 slates? 6. If you pay 20 cts. a day for car-fare, what will be your fare for 60 days ? 7. At 33J cts. a bushel, what will be the cost of 31 bushels of apples ? Of 36 bushels ? Of 45 bushels ? Of 63 bushels ? 8. At 16| cts. a pound, what cost 30 pounds of butter? 9. What cost 64 qts. of milk at 6|- cts. a quart ? 80 quarts ? 10. How many melons, at \%\ cts. each, can be had for 16 ? Analysis. — Since 124 cts. are $|, $6 will buy as many melons as %\ is contained times in $6, or 48 melons, Ans. 11. At $.50 a pound, how many pounds of tea can be bought for 111? For 1184? For 125 ? For $50 ? 12. A farmer sold 36 bushels of oats at 1.33^ a bushel, and took his pay in raisins at 12 J cts. a pound ; how many pounds of raisins did he receive ? Written Exercises. 282. Price is the money value of a unit of like things. 283. Cost is the sum paid for a given number of like things. 284. To find the Cost of a number of like things, when the Price of one is an Aliquot Part of $1. 1. What is the cost of 675 Histories, at 33^ cts. each ? Analysis.— At $1 apiece, they would cost $675. But 3 ) 675 the price is only \ of $1 each ; therefore, the cost is -^^ of - r~ $675, which is $225. J «5. Hence, the Ans. IZb EuLE. — Multiply the given ninriber of things hy the fractional part of ^1 which expresses the price of One : the result is the cost. (Art. 208. ) TI. S. Money, 131 2. At 10.50 a bushel, what cost 876 bu. of potatoes? 3. At 25 cts. a yard, what will 1200 yards of ribbon cost ? 4. If I pay 20 cts. a bu. for apples, what must I pay for 688 bu.? 5. What cost 898 Spellers, at 12|- cts. each ? 6. At 33-|- cents a pound, what cost 750 pounds of butter ? 7. What cost 450 boxes of lemons, at $1.25 a box ? Analysis. — At $1 a box, they would cost $450. 4 ) $450 at $1. But the price is $1 + $J ; therefore, the cost is $450 $112.50 at ^K + 1 of $450, which is $562.50, Ans. *— ** ' $562.50, Ans. 8. At $1.33^, what cost 796 Geographies? 9. K a man saves $1.1 6f each week, how much will he save in 312 weeks ? 10. A shoe dealer sold at wholesale 250 pairs of slippers for $1.20 a pair ; what was the amount of his bill ? 285. To find the Number of Things when their Cost is given, and the Price of One is an Aliquot Part of $1. 11. How many gallons of milk, at $.33^ a gallon, can be bought for $175 ? Analysis. — Since $1 will pay for 3 gallons, operation. $175 will pay for 175 times 3 gallons, or 525 $.33-J^ = $J gallons. Or, at $i a gallon, $175 will pay for |]^75 x 3 == 525 as many gallons as %\ is contained times in r\ a-j 7K _^ we have Dec'imeter, -}^ meter. '' Cen'ti, yio. " Cen'timeter, -^^ '' - Milli, ^^Vo. " MirUmeter, ^\^ - Note. — The numeral prefixes are the Key to the whole system, and should be thoroughly committed to memory. Measures of Length. 306. The principal unit of each table is printed in capital letters ; those in common use in full-faced Eoman. Ta b l e. 10 mil'li-yne'ters {rmn.) = 1 cen'ti-me'ter, - - cm, 10 cen'ti-me'ters = 1 dec'i-me'ter, - - dm. 10 dec'i-me'ters = 1 meter, - - m. 10 me'ters = 1 dek'a-me'ter, - - Dm. 10 dek'a-me'ters = 1 hek'to-me'ter, - - Hm. 10 hek'to-me'ters = 1 kilo-me'ter, - - Km. 10 kil'o-me'ters = 1 myr'ia-me'ter,- - Mm. Notes. — 1. The Accent of each unit and 'prefix is on the first syllable, and remains so in the compound words. 2. Abbreviations of the higher denominations begin with a capital, those of the lower begin with a small letter. CoMMoi^ Equivalents. 1 cen'timeter = 0.3937 inches. 1 dec'imeter = 3.937 1 me'ter = 39.37* '' 1 kil'ometer — 0.6214 mile. * EstabliBhed by Act of Congress in 1866. Metric System. 139 OyE DECIMETER, llllllll 1 I I ll| , il I II I I I llllll 11 liiiil II I ill II1I1111 li inl 1111 111 llliii I I III il 1 II I I I II I li II 1 ll II I III 11 100 MiUinietcrs. 307. The Meter is the Standard Unit of length, and, Hke the yard, is used in measuring cloths, laces, short distances, etc.* 308. Tlie Kilometer, like the mile, is used in measuring long distances. 309. The Centimeter and Millimeter are used for minute measurements, as the thickness of glass, paper, etc. Note. — The compound words may be abbreviated by using only the prefix and the first syllable or letter of tbe unit ; thus, centimeter, milli- meter, centiliter, milliliter, centigram, decigram, may be called cen- tim, millim, centil, decig, etc. 310. The ap2?roximafe length of 1 meter is 40 inches ; of 1 decimeter, 4 inches ; of 5 meters, 1 rod ; of 1 kilometer, I mile. Note. — Decimeters, dekameters, hektometers, like dimes and eagles, are seldom used. 311. Since meters, centimeters, and millimeters, correspond to dollars, cents, and mills, it follows that metric numbers may be read like U. S. Money. Thus, 128.375 is read, "28 and 375 thousandths dollars," or "28 dollars, 37 cents, 5 mills." • In like manner, 28.375 meters are read, "28 and 375 thou- sandths meters," or " 28 meters, 37 centimeters, 5 millimeters. 312. Eead the following : 1. 14.5 m. 5. 47.3 Dm. 9. 89.63 Hm. 2. 236.4 m. 6. 83.25 Dm. 10. 434.5 Km. 3. 78.35 m. 7. 568 Hm. 11. 65.48 Km. 4. 23.7 Dm. 8. 648.8 Hm. 12. 9.237 Km. * It is important for the teacher to show the class a meter stick, with its subdivi- sions marked on one side, and halves, quarters, etc., on the other. 140 Decimals. 313. To write Metric Numbers decimally in terms of a given Unit. 1. Write 7 Hm. 9 m. 3 dm. 5 cm. in terms of a meter. Explanation. — We write meters in units operation. place, on the left of the decimal point, the Dm. in 709.35 m., Ans. tens place, the Hm. in hundreds place, etc., and the decims. in tenths place, centims. in hundredths, etc., as we write the orders of integers and decimals in simple numbers. Hence, the Rule. — Write the given unit and the higher denoini- nations in their order, on the left of a decimal point, as integers, and those helow the unit, on the right, as decimals. Note. — If any intervening denominations are omitted in the given number, their places must be supplied by ciphers. Write the following as meters and decimals : 2. 256 millimeters. A7is. 0.256 m. 3. 8 decimeters 4 centimeters. 4. 25 meters 3 centimeters. 5. 348 dekameters 43 centimeters. 6. 465 hektometers 48 millimeters. 7. 4725 meters 25 centimeters. 8. 4 Km. 8 Hm. 6 Dm. 4 dm. 5 cm. 3 mm. 9. 23 Km. 6 Hm. 8 dm. 6 cm. 314. To reduce Metric Numbers from higher denominations to lower, and from lower to higher. 1. Eeduce 45 meters to millimeters. 45 m Solution.— Since 1 m. = 1000 mm., 45 meters ^^ ' must equal 45 x 1000, or 45000 mm., Ans. -2??_ Ans. 45000 mm. 2. Eeduce 64000 millimeters to meters. Solution.— In 1000 mm. there is 1 m., and in 64000 1000 ) 64000 mm. there are as many meters as 1000 is contained . ~~r. times in 64000, or 64 meters. Hence, the ^^^^- "^ ^• EuLE. — Move the decimal point one place to the right or left, as the case may require, for each denomination to which the given number is to he reduced. Metric System. 141 3. Reduce 25.7 Km. to meters. Ans. 25700 m. 4. Eeduce 43.4 m. to millimeters. 5. Eeduce 65.3 Dm. to decimeters. 6. Eeduce 84.25 Km. to centimeters. 7. Eeduce 4823.6 meters to Hektometers."^M5. 48.236 Hm. 8. Eeduce 36482.9 m. to kilometers. Ans. 36.4829 Km. 9. Eeduce 28526 mm. to meters and decimals. 10. Eeduce 48639 cm. to meters and decimals. 11. Eeduce 438.6 m. to millimeters. 12. Eeduce 738.4 Dm. to centimeters. Measures of Surface. 315. A Surface is that which has length and breadth only. 316. The Measuring Unit of Surfaces is a Square, each side of which is a Linear Unit. 317. A Square is a figure which has four equal sides and four equal angles, called right angles. Tabl e. 100 sq. mil'li-me'ters (sq. mm.) = 1 sq. cen'ti-me'ter, sq. cm. 100 sq. cen'ti-me'ters = 1 sq. dec'i-me'ter, sq. dm. 100 sq. dec'i-me'ters =1 ' , , ' q- - ^ t or cent ar, ca. 100 sq. me'ters ' = ] ^ '\ ^^'^^'^^'^^' *?• Dm. ^ ( or Ar, A. 100 sq. dek'a-me'ters = | ^ '^^ ['f^'to-'^e'ter, .?. Hm. ^ f or hek tar, Ha. 100 sq. hek'to-me'ters = 1 sq. kH'o-me'ter, sq. Km. COMMOIis" EqUIVALEIs'TS. 1 sq. centim. = 0.1550 sq. in. 1 sq. decim. r= 0.1076 sq. ft. 1 sq. meter = 1.196 sq. yd. 1 ar — 3.954 sq. rods. 1 hektar = 2.471 acres. 1 sq. kilo = 0.3861 sq. mile. 142 Decimals, 318. The sq. meter is used in measuring ordinary surfaces, as floors, ceilings, etc. ; the ar and hektar in measuring land; and the sq. kilometer in measuring States and Territories. Note. — The term ar is from the Latin araa, a surface. 319. The approximate area of a sq. meter is lOf sq. ft., or \\ sq. yd., and of the hektar about %\ acres. a 320. The scale of surface measure is 100 (10 x 10). That is, 100 units of a lower denomination make a unit of the next higher; hence, each denomination must haye two places of figures. ^^" ^®"t^™- Thus, 23 Ha. 19 a. 25 ca., written as ars, is 2319.25 a., and may be read "2319 ars and 25 centars." If written as hektars, it is 23.1925 Ha., and may be read " 23 hektars and 1925 centars." 1. Write 78.29 a. as centars, also as hektars. 2. Write 9 sq. m. as sq. dm. Write 7 sq. cm. as sq. mm. 3. In 3246 ca., how many ars ? In 63.42 ars, how many Ha. ? Measures of Solids. 321. A Solid is that which has length, breadth, and thickness. Ta b le. 1000 cu. mirii-me'ters {cu, mm.) = 1 cu. cen'ti=me'ter, cu. cm. 1000 cu. cen'ti-me'ters = 1 cu. dec'i-me'ter, cu. dm. 1000 cu. dec'i-me'ters = 1 cu. meter, cu. m. 10 dec'i-sters = 1 ster, st 10 sters = 1 dek'a-ster. Dst. Common" Equivalents. 1 cu. centimeter = 0.061 cu. in. 1 cu. decimeter = 61.022 cu. in. 1 cu. meter = 1.308 cu. yds. Note.— The ster = .2759 cord is seldom used. 322. The Measuring Unit of solids is a Cube, the edge of which is a Linear Unit. Metric System, 143 323. A Cube is a regular solid bounded by six equal squares called its faces. Hence, its length, breadth, and thickness are equal. A Cubic Centimeter is a cube, each side of cu- cm. which is a square cejitinieter. 324. The cubic meter is used in measuring ordinary solids, as timber, excavations, embankments, etc. When applied to fire- wood, it is sometimes called a Ster, atid is equal to about 35-|- cubic feet. Note. — The ciibic decimeter when used as a unit of dry or liquid measure is called a Liter. 325. The units of cubic measure increase by the scale of 1000 (10 X 10 X 10) ; hence, each denomination must have three places of figures. 1. Express 6000 cu. mm. as cu. centimeters. 2. Express 8000 cu. dm. as cubic meters. 3. Express 86.005 cu. dm. as cu. meters ; as cii. cm. 4. Write 0.6235 cu. m. as cu. dm. ; as cu. cm. 5. In 862 cu. dm., how many cu. meters ? In 250 cu. m. how many cubic decimeters ? Measures of Capacity. 326. The Liter is the principal unit of Dry and Liquid Measure, and is equal in volume to a cubic decimeter. Table. 10 mil'li-li'ters (ml.) = 1 cen'ti-li'ter - - - cl. 10 cen'ti-h'ters = 1 dec'i-li'ter - - - dl 10 dec'i-li'ters = 1 liter - . . . l, 10 li'ters = 1 dek'a-li'ter - - - Dl 10 dek'a-li'ters = 1 hek'to-li'ter - - HI 10 hek'to-h'ters = 1 kil'o-li'ter - - - Kl. 10 kil'o-li'ters = 1 myr'ia-li'ter - - Ml. 144 Decimals. ^g % ««^ ^ •^SUm ^ 1 cubic centimeler = 1 milliliter of water. CoMMOi^ Equivalents. 1 liter - 61.022 cu. inches. 1 liter = 1.056"/ liquid quarts 1 liter : 0.908 dry quarts. 1 hektoliter = 3.531 cu. feet. 1 hektoliter = 26.417 gallons. 1 hektoliter — 2.837 bushels. 327. The Centiliter is a little less than \ gill, and is used for measuring liquids in small quantities. The Liter is used in measuring milk, wine, and small fruits, and is about equal to a quart. The Hektoliter is used in measur- ing grain and liquids in casks, and is equal to about 26-|- gal., or 2|- bushels. 1. Express 8.53 1. as centiliters. As deciliters. 2. Express 4. 640 kiloliters as liters. As hektoliters. 3. How many deciliters in 8 liters ? In 9.35 liters? 4. How many liters in 6.358 centiliters ? In 800 cl. ? 5. In 8500 liters how many kiloliters ? How many HI. ? Weight. 328. The Gram is the 'principal unit of weight, and is equal to a cubic centimeter of distilled water at its greatest density, viz., at 4° Centigrade, or 39.2° Fahrenheit. Metric System, 145 Ta b l e 10 mil'li-grams {mg^ = 10 cen'ti-grams = 10 dec'i-grams =: 10 grams = 10 dek'a-grams = 10 hek'to-grams = 10 kiro-grams = 100 m}T'ia-grams == 1 cen'ti-gram - 1 dee'i-grani 1 CRAM - - 1 dek'a-gram 1 liek'to-gram - 1 kiro-gram 1 myr'ia-gram - 1 tonneau or Ton eg. clg. 9- Dg. Hg. Kg. Mg. T. IDg. 1 gram. COMMOI^ idg. 1 ds. leg Ics 1 gram 1 kilogram = 1 metric ton = 1 gram = 1 gram z=z 1 kilogram =: 1 metric ton = Equiyalents. 1 cu. centim., or 1 millil. of water. 1 cu. decim., or 1 liter of water. 1 cu. meter, or 1 kiloliter of water. 15.432 grs. Troy. 0.03527 oz. Av. 2.2046 lbs. Av. 1.1023 tons. © Img. 329. The Gram is used in weighing gold, silver, jewels, and letters, and in mixing medicines, 7 146 Decimals. 330. The Kilogm7n, (often called hilo) is used in weighing common articles ; as sugar, tea, butter, etc. The Metric ton is used in weighing heavy articles ; as hay, coal, etc. Notes. — 1. The kilo is equal to 2 1 lbs., nearly ; the metric ton about 2300 pounds.* 2. The nickel 5-cent piece weighs 5 grams. The silver I dollar 12i grams. The silver dime weighs 2i grams. The silver |- dollar Q\ grams. 3. The weight of a letter for single postage must not exceed 15 grams, or 3 nickels. 1. Express 6.354 g. as decigrams. As centigrams. 2. Write 5834 mg. as dg. As eg. As grams. 3. How many grams in 78.45 Dg. ? How many Kg. ? 4. How many kilos in 3.54 T. ? How many Dg. ? 5. Express 1 g. in the decimal part of a kilo. 6. Express a kilo in the decimal part of a ton. 7. Express 2.0005 T. as grams. 331. To Add, Subtract, Multiply, and Divide IVIetric numbers. A2)ply the correspo?idmg rules of decimals or U. S. money. (Art. 279.) 1. What is the sum of 45.68 Dm., 63.4 Hm., and 6845 cm.? ,,r ... 1 V. . 456.8 Solution. — Writing the numbers as meters and decimals of a meter, the principal unit of ooiu.u the table, and adding, we have 6865.25 meters. 68. 45 Ans. 6865.25 m. 2. Find the sum of 24.35 m., 6.425 m., 32.7 m., and 42.26 m. 3. What is the difference between 8.5 kilograms and 976 grams ? Solution.— 8.5 kilos — .976 kilos = 7.524 kilos, Ans. 4. From 1 hektoliter of oil, 36 liters were drawn out ; how many liters remained ? 5. How much silk is there in 12J pieces, each containing 48.75 meters ? Solution.— 48.75 m. x 12.5 = 609.375 m., Ans. Metric System. 147 6. It is 285 meters around my garden ; how many Km. shall I walk in a week by going twice around it every day ? 7. At 16.50 a meter, what will 37 meters of silk cost? 8. What cost 24 meters of fringe, at 12.25 a meter? 9. How many cloaks, each containing 5.68 meters, can be made from 426 meters of cloth ? Solution. — 426 m. -7- 5.68 m. = 75 cloaks, Ans. 10. If a car goes 160 Km. in 6 hours, how far does it go in 1 hour ? 11. How many Km. in 85.72 m. multiplied by 2036 ? 12. If the price of 1 liter of milk is 6 cents, what cost 75 liters ? 13. At 12 cents a liter, what cost 4.5 liters blackberries ? 14. If 1 hektoliter of wheat costs $3.50, what will 234 hekto- liters cost ? 15. A man paid !i5281.75 for 245 hektoliters of oats; what was the price of 1 hektoliter ? 16. What cost 46.25 kilos of butter, at 10.50 per kilo ? 17. At $1.28 per kilo, what will 82.5 kilos of tea come to ? 18. At $16 a ton, what will the coal cost to supply a factory a week, if 25 kilos are burned each day ? 19. If 735 kilos of flour are distributed among 35 persons, how many kilos 'vvill each person receive ?* 332. The contents of Rectangular Surfaces are found by multiplying the length by the Ireadtli. 20. A garden is 18 meters long and 12.5 meters wide ; how many square meters does it contain ? Solution.— The product of 18 x 13,5 = 225 sq. m., Ans. 21. How many sq. meters in a blackboard 2.5 meters long and 1.2 meters wide ? 22. If a room is 8.4 meters long and 4.5 meters wide, how many square meters of carpeting will it take to cover the floor ? 23. How many sq. meters of flagging in a side-walk 35.5 meters long, and 2.4 meters wide? * For reducing Metric to common Weights and Measures, etc., see Ai't. 405, 148 Decimals 24. How many centars in a piece of land 45 meters long, and 23.2 meters wide? 333. The contents of Rectangular Solids are found by multi- plying the length, breadth, and thickness together. 25. How many cu. meters of earth in a mound whose length, breadth, and height are each 6.4 meters. Solution. — 6.4 x 6.4 x 6.4 = 262.144 cu. meters, Ans. 26. How many cu. meters of earth must be removed in dig- ging a cellar 23.4 meters long, 15.2 m. wide, and 2.4 m. deep? 27. How many loads of earth each equal to a cu. meter, will it take to fill an excavation 4 dekameters long, 8 meters wide, and 2. 4 meters deep ? 28. At $1.45 a cu. meter, what will be the cost of digging a trench 2 dekameters long, 2 meters wide, and 1.5 meters deep? 29. At 12.50 a ster, what is the cost of a pile of wood 3 meters long, 1.5 m. wide, and 1.1 m. high ? 30. What is the value of a nugget of gold 2.6 cm. long, 2.3 cm. wide, and 0.65 cm. thick, at $15.40 a cu. centimeter? QU ESTIO N S. 300. What are Metric weights and measures ? 301. What is the Base? 304. How are the names of the higher denominations formed? 305. The lower ? 306. Repeat the table of measures of length. 307. What is the standard unit of length ? For what used ? 308. The kilometer? 311. How read metric numbers ? 318. How write them ? 314. How reduce metric numbers from higher to lower denominations ? From lower to higher? 317. Repeat the table of measures of surface. 318. For what is the square meter used? The sq. kilometer? The ar and hektar? 321. Repeat the measures of solids. 324 For what is the cu. meter used ? When called a ster ? 826. Repeat the table of measures of capac- ity. 327. For what is the liter used? The hektoliter ? 828. Repeat the table of weight. 329. For what is the gram used ? 830. The kilogram? The metric ton? 381. How are metric nimibers added, subtracted, etc. ? ^i =-€^ — ' ■ — ^)) C) M P O U ^' I) U M B E R S . -\ ^ i'^'in >~ Definitions. 334. A Simple Number is one wliicli expresses one or more units of the same navie or (lenoniination j as five, 4 feet, etc. 335. A Compound Number expresses units of two or more denominations of the same kind, which increase and decrease by varying scales ; as, 3 yards 2 feet 4 inches. But 2 feet and 4 pounds is not a compound number, for the units are unlike. Note. — Compound Numbers are often called Denominate Numhers. The term denomination is a name given to the different units of weights and measures. Linear Measure. 336. A Measure is a standard unit established by law or custom, by which the length, surface, capacity, and weight of things are estimated. 337. Linear Measure is used in measuring lines and dis- tances. 338. A Line is that which has length only. Ta b l e. 12 inches {in.) = 1 foot, - ■ ■ - fi- 3 feet =r 1 yard, - - ■ - yd- 5i yds., or IGJ ft. = 1 rod, - - ■ - rd. 40 rods =: 1 furlong, ■ ■ - fur 320 rods, or 5280 ft. = 1 mile, - ■ ■ - mi. 3 miles = 1 league. ■ ' I 339. The Standard Unit of length is the Yard, which is used in measuring cloths, laces, ribbons, etc. (Art. 900, App.) 150 Compoimd Numbers, Oral Exer cises. 340. 1. Draw a line 4 inches long. A foot. A yard. 2. How long is this book ? Your slate ? How wide ? 3. How long is this table ? How wide ? How high ? 4. In 6 feet how many inches ? In 8 ft. ? In 9 ft. ? 5. How many feet in 7 yards ? In 15 yds. ? In 20 yds. ? 6. In 120 inches how many feet ? How many yards ? 7. How many feet in 4 rods ? In 5 rods ? Square Measure. 341. Square Measure is used in measuring surfaces ; as, flooring, land, etc. 342. A Surface is that which has lengtli and hreadth only. 343. An Angle is the opening between two lines which meet at a point, as BAG. The Lines AB and AC are called the sides ; and the Point A, at which they meet, the Vertex of the angle, 344. When two straiglit lines meet so as to make the tivo adjacent angles equal, the lines are Perpendicular to each other, and the two angles thus formed are called Right Angles ; as, ABC, ABD. 345. A Square is a rectilinear figure which has four equal sides, and four right angles. 346. The yneasuring unit of sur- faces is a Square, each side of which is a linear unit. -.A,, ir.' g 1 1. ^ D 347. The Area of a figure is the quantity of surface it contains. 1 !!' II '!| il il 11 |ii| ill ii hliiiiiiil l\,l 1 1 1 1 III 'li '■mi |i ii Il il 1 II II nil ml III II i 1 III T. ":: III' 3 ft. X 3 ft. = 1 sq. yd. Cubic Measure, 151 Ta b le . 144 square inches {sq, in.) = 1 square foot, - - sq. ft. 9 square feet = 1 square yard, - - sq. yd. 30J sq. yds., or 272^ sq. ft. = 1 square rod, - - sq. rd. 160 square rods := 1 acre, - - - - A. 640 acres = 1 square mile, - - sq. ml. (For Surveyor's Measure, see Art. 889, Appendix.) Oral Exercises. 348. 1. How many sq. inches in 2 sq. feet? 2. In 8 sq. yds. how many sq. ft. ? In 15 sq. yds. ? 3. In 2 acres how many sq. rods ? In 3 acres aud 5 sq. rd. ? 4. How many sq. ft. in 288 sq. inches ? 5. In 320 sq. rods how many acres ? Cubic Measure. 349. Cubic Measure is used in measuring solids or volume. 350. A Solid is that which has length, breadth, and thickness; as, timber, boxes of goods, etc. 351. A Cube is a regular solid bounded by six equal squares called its faces. Hence, its length, breadth, and thickness are equal to each other. 352. The measuring unit of solids is a Cube the edge of which is a linear unit. Ta b l e. 1728 cubic inches (6'?^ in.) = 1 cubic foot. - cu. ft. 27 cubic feet = 1 cubic yard, - - cu. yd 128 cubic feet — - 1 cord of wood. - a 152 Compound Numhers. 353. A Cord of wood is a pile 8 ft. long, 4 ft. wide, and 4 ft. high ; f or 8 x 4 x 4 = 128. 354. A Cord Foot is one foot in length of such a pile ; hence, 1 cord foot = 16 cu. feet; 8 cord ft. == 1 cord. Oral Exercises. 355. 1. How many cubic inches in 2 cubic feet ? 2. How many cu. feet in 2 cubic yards ? In 3 cu. yards ? 3. How many cubic feet in 2 cords ? In 3 cords ? 4. In 54 cu. feet, how many cu. yards ? In 126 cubic feet ? 5. How many cord feet in 3 cords of wood? In 5 cords ? 6. In 32 cord feet how many cords ? In 72 cord feet ? Liquid Measure. 356. Liquid Measure is used in measuring millc, oil^ ioine, etc. Ta b le 4 gills {(ji.) == 1 pint, - - - pL 2 pints = 1 quart, - - - qt 4 quarts ^ 1 gallon, - - - gal 31|- gallons = 1 barrel, - - - lav, or libl 63 gallons = 1 hogshead, - - liM. pt. gl. 357. The Standard Unit of Liquid Measure is the gallon, which contains 231 cubic inches. Note.— The harrel and hogshead, as units of measure, are chiefly used in estimating the contents of cisterns, reservoirs, etc. Dry Measure. 153 Oral Exercises. 358. 1. How many quarts in 20 pints ? In 36 pts. ? 2. In 24 pints how many quarts ? In 40 pts. ? 3. How many gallons in 2 hogsheads ? In 5 hhd. ? 4. How many qts. in 12 gal. of milk ? In 15 gallons ? 5. What is the cost of 5 gal. of syrup at 60 cts. a gal. ? Dry Measure. 359. Dry Measure is used in measuring grciiyi, fruit, salt, etc. Tab le. 2 pints (pt.) = 1 quart, - 8 quarts =: 1 peck, - 4 pecks, or 32 qts. = 1 bushel, qt. hii. 360. The Standard Unit of Dry Measure is the huslml, which contains 2150.4 cubic inches.* Note. — The dry quart is equal 1| liquid quart nearly. Oral Exercises. 361. 1. How many pints in 12 quarts ? In 25 quarts? 2. How many pecks in 40 qts. of chestnuts ? 3. How many bushels in 72 pecks ? 4. If you pay 40 cts. for J bushel of apples, what must you pay for 5 bushels ? 5. If I buy a bushel of walnuts for $3, and sell them at 5 cts. a pt., how much shall I make ? 6. How many bu. in 36 pecks ? In 96 quarts ? * For the standard weight of a hushel of different grains, see Art. 896, Appendix. 154 Compound Number's. Troy Weight. 362. Troy Weight is used in weighing gold, silver, etc. Ta b le . 24 grains {gr.) z= 1 pennyweight, - - 2^^^' 20 pennyweights = 1 ounce, - - •■ - oz. 12 ounces = 1 pound, - - - - Ik lb. ; oz. pwt. gr. 363. The Standard Unit of weight in the United States, is the Troy Pound. 1. How many grains in 6 pennyweights ? 2. How many pwt. in 6 lb. 7 oz. ? In 8 lb. 5 oz. ? 3. How many ounces in 8 pounds of silver ? 4. Change 4 pwt. to grains. 25 oz. to pwt. Avoirdupois Weight. 364. Avoirdupois Weight is used in weighing coarse articles; as liay, cotton, groceries, etc., and all metals except gold and silver. Ta ble. 16 ounces (oz.) = 1 pound, - - - lb. ^^^ , , ( cental, or - - ctl. 100 pounds = 1 i , 1 1 • 1 i. ^ ■^ ( hundredweight, ciut. 2000 lb., or 20 cwt. = 1 ton, - - - - T. Note.— In calculating duties, etc., 112 lb. are called a act., and 2240 lb. a long ton. 365. Gross Weight is the weight of goods including the boxes, etc., which contain them. Net Weight is the weight of goods after deducting all allow- ances. Time. 155 366. Comparison of Avoirdupois and Troy Weight. 7000 grains Troy 5760 grains '' 43 7|- grains " 480 grains ^^ 1 lb. Avoirdupois. 1 lb. Troy. 1 oz. Avoirdupois. 1 oz. Troy. Apothecaries Weight. 367. Apothecaries Weight is used by Apothecaries in mixing medicines. (Art. 898, Appendix.) Tab l e. 20 grains {gr.) = 1 scruple, - - sc, or 3. 3 scruples = 1 dram, - - - dr., or 3. 8 drams = 1 ounce, - - - oz., or 3 . 12 ounces = 1 pound, - - lb., or lb. Note. — The pound, ounce, and grain are the same as Troy weight. Oral Exerci s€s. 1. How many ounces in 5 pounds ? In 100 pounds? In 6200 lbs. ? In ^ ton ? 368. 2. How many tons in 4000 lbs. ? 3. How many pounds in ^ ton ? 4. Ac 90 cts., what will J lb, of tea cost ? 5. At $20 a ton, what wiU J ton of hay cost ? Time. 369. Time is a measured portion of duration, are shown in the following Tab le. Its divisions 60 seconds {sec.) 60 minutes 24 hours 7 days 365 days 366 days 12 calendar months {mo.) 100 years 1 minute, - - min, 1 hour, - - - Jir. 1 day, - - - d. 1 week - - - ivh. 1 common year, c. yr, 1 leap year, - I. yr. 1 civil year, - yr. 1 century, - - C. 156 Compound Niimnbers. 370. A Civil Day is the day adopted by government for business purposes. It begins and ends at midnight, and is divided into two parts of 12 hours each ; the former being designated a. m., the latter p. m. 371. The Solar Year is equal to 365 d. 5 hr. 48 min. 49.7 sec, or 365;} d. nearly. In 4 years this fraction amounts nearly to 1 day. To provide for this excess, 1 day " is added to the mo. of Feb. every 4th year, which is called Leap Year.* 372. The Civil year includes both common and leai^ years, and is divided into 12 Calendar months, viz: January (Jan.) 31 days. July (July) 31 days February (Feb.) 28 " August (Aug.) 31 " March (Mar.) 31 " September (Sept.) 30 " April (Apr.) 30 " October (Oct.) 31 " May (May) 31 " November (Nov.) 30 " June (June) 30 " December (Dec.) 31 " Note. — Tlie following couplet will aid the learner in remembering the mouths that have 30 days each : " Thirty days hath September, April, June, and November." All the rest have 31 days, except Fehfuary, which in common years has 28 days ; in leap years, 29. Oral Exercises. 373. 1. How many days in 7 weeks ? In 9 weeks ? 2. How many weeks in 42 days ? In 63 days ? In 90 days ? 3. How many months in 6 years ? In 8 years ? In 11 years? 4. In 48 months how many years ? In 72 months ? 5. How many centuries in 500 years ? In 1800 years ? 6. At $9 a week, how much will a man e[irn in 6 weeks? 7. If you pay ^3 a week, how long can you board for $60 ? 8. How many days has a person lived who is 12 years old? 9. If you count 60 a minute, how long will it take to count 1800 ? * For an explanation of the mean Solar days, leap years, etc., see Art. 901, Appendix- Cf/rcular Measure, 157 Circular Measure. 374. Circular Measure is used in measuring angles, latitude and lo7igitude, lieavejily bodies, etc. 375. A Circle is a plane figure bounded by a curve line every part of which is equally distant from a point within, called the center. 376. The Circumference of a circle is the curve line by which it is bounded ; as ADEBF. 377. The Diameter is a straight line drawn through the center, terminating at each end in the circumference ; as AB. 378. The Radius is a straight line drawn from the center to the circumference, and is equal to liaJf the diameter ; as AC, DO. 379. An Arc is any part of the circumference ; as AD Table. 60 seconds (") = 1 minute, - - '. 60 minutes = 1 degree, - - °, or deg. 30 degrees = 1 sign, - - - S. 12 signs, or 360°= 1 circumference, Cir. 380. The Measure of an angle is the arc of a circle included between its two sides, as the arc DE. The Standard Unit for measuring angles is the Degree. 381. A Degree is the angle measured by the arc of -g^ part of the circumference of a circle. The length of the arc which measures an angle of 1°, varies according to the size of different circles, while the angle remains the same. A degree at the equator, also the average degree of latitude, adopted by the U. S. Coast Survey, is equal 69.16 miles, or 69^ miles, nearly. 158 Compouhd Numbers. 382. A Semi-circumference {lialf a circumference) is an arc of 180°, as AFB. 383. A Quadrant, or one-fourth of a circumference, is an arc of 90°, as EB. A right angle contains 90° ; for the quadrant, which meas- ures it, is an arc of 90°. Oral Exercises. 384. 1. How many degrees in J a cir. ? In J cir. ? 2. How many degrees in a quadrant? In a right angle? 3. How many miles in 2° ? 3° ? 5° ? 4. Through how many deg. does the hour-hand of a clock moA e in 12 hours ? In 3 hrs. ? In 6 hrs. ? In 1 hr. ? 5. Through how many degrees does the minute-hand of a clock pass in 1 hour ? In \ hr. ? In J hr. ? In 1 minute ? 6. In making a voyage around the world, through how many degrees would you sail ? FoREiG-N Moneys. 385. English or Sterling Money is the currency of Great Britain. Table. 4 farthings {qr. ov far.) = 1 penny, - . . . d. 12 pence = 1 shilling, - - - - s. 20 shillings = 1 pound or sovereign, £. 10 florins {fl.) = 1 pound, ----£. 386. The Unit of English Money is the Pound Sterling, which is represented by a gold Sovereign equal in value to 14.8665. 387. Canada Money is expressed in dollars, cents, and mills, which have the same nominal value as the corresponding denominations of U. S. money. Foreign Moneys. 159 388. French Money is the currency of France. Table. 10 centimes = 1 decime. 10 decimes =: 1 franc. 389. The Unit of French Money is the Franc, the value of which in U. S. money is 19.3 cts., or about i- of a dollar. Note. — The system is founded upon the decimal notation ; hence, all operations in it are the same as those in U. S. money. 390. The Money Unit of the German Empire is the Mark, whicli is divided into 100 pennies. The value of a Mark is $0,238, or %\ nearly.* Miscellaneous Tables. 12 things = 1 dozen. 12 dozen = 1 gross. 12 gross = 1 great gross. 20 things r= 1 score. 24 sheets •= 1 quire of paper. 20 quires = 1 ream. 2 reams = 1 bundle. 5 bundles = 1 bale. 2 leaves = 1 folio. 4 leaves = 1 quarto, or 4to. 8 leaves = 1 octavo, or 8vo. 12 leaves = 1 duodecimo, or 12mo. Note. — The terras folio, quarto, octavo, etc., denote the number of leaves into which a sheet of paper is folded in making books. * For Table of Foreign Coins, see Art. 631. 160 Compound Numbers. Oral Exercises. 391. 1. How many farthings in 5 shillings ? 2. How many pence in £3 ? 3. What cost 8 meters of lace, at 12 francs a meter ? 4. How many Sovereigns will 12 yards of silk cost, at 10s. d yard ? 5. What will 8 doz. eggs cost, at a cent apiece ? 6. What will 18 quires of paper cost, at 20 cts. a quire? 7. What will a gross of buttons cost, at 15 cts. a dozen ? Questions. 334. What is a simple number? 335. Compound? 336. What is a measure? 337. For what is linear measure used? 338. What is a line? Recite the table. 339. What is the standard unit of length? 341. For what is square measure used? 342. What is a surface ? 343. An angle ? The vertex ? 344. A right angle ? 345. What is a square ? 347. What is the area of a figure ? Repeat the table. 349. For what is cubic measure used ? 350. What is a solid ? 351. What is a cube? Recite the table. 356. For what is liquid measure used ? Recite the table. 359. For what is dry measure used ? Repeat the table ? 362. For what is Troy weight used? Recite the table. 363. The standard unit of weight ? 364. For what is Avoirdupois weight used ? Recite the table. What is a long ton ? 367. For what is Apothecaries weight used ? 369. What is Time? Recite the table. 370. What is a civil day? Themeaningof A.M. ? Of p.m. ? 371. Length of a Solar year ? 372. How many calendar months in a civil year? Name them. 374. For what is circular measure used? 375. What is a circle? 376. The circumference? 377. Diameter? 378. Radius? 379. An arc? Table? 380. The measure of an angle? 381. What is a degree? 382. A semi-circumference ? 383. A quadrant ? How many degrees in a right angle ? 385. What is English or Sterling money ? Repeat the table. 386. The iinit of English money? Its value? 387. How is Canada money ex- pressed? 388. What is French money? Recite the table. 389. The unit ? Its value ? 390. What is the money unit of the German Empire ? Its value ? Recite the miscellaneous tables. EDUCTION. Oral Exercises. 392. 1. How many pints in 3 gallons ? Analysis.— In 1 gal. there are 4 qt„ and in 3 gal., 3 times 4, or 13 qt. In 1 qt. there are 2 pints, and in 13 qts., 13 times 3, or 34 pints, Ans. 2. How many feet in 4 yd. ? In 8 yd. ? 3. In 6 sq. yd. how many square feet ? 4. How many gills in 9 quarts ? In 12 quarts ? 5. In 10 bushels how many pints ? 6. Id 5 days how many minutes ? 393. Keduction is changing Compound ^Numbers from one denomination to another without altering their values. It is of two kinds, Descending and Ascending, 394. Reduction Descending is changing higher denomina- tions to loioer ; as, yards to feet, etc. 395. To reduce Higher Denominations to Loiver, 1, Reduce 34 rods 4 yds. 2 ft. to feet. Analysis. — As 5| yds. make 1 rod, there must 34 r. 4 yd. 2 ft. be 5|^ times as many yards as rods ; and (34 x 5^) 5 1 + 4 (the given yds.) = 191 yds. (Art. 208.) — ^ Again, as 3 ft, make 1 yd. there must be 3 times ^^^ J^®" as many feet as yards; and (191 x 3) + 3 (the 3 given ft.) = 575 feet. Hence, the 51^5 f^^ Ans Rule. — Multiply the highest clenoinination hy the nuviber required of the next lower to make a unit of the higher, ctncl to the product acid the lower denojiiination. Proceed in this manner with the successive denomina- tions, till the one required is reached. 162 Com/pound Numhers. 2. In 5 mi. 12 rd. 4 yd. 2 ft. liow many feet ? 3. Eeduce 143 lb. 3 oz. 6 pwt. to grains. 4. Reduce 217 tons 35 lb. to pounds. 5. Reduce 106 tons 68 lb. to ounces. 396. Reduce the following : 6. 23 mi. 5 rd. 6 ft. to feet. 13. 32 A. 6 sq. rd. to sq. feet 7. 24 lb. 4 oz. 6 pwt. to gr. 14. 26 C. 7 cu. ft. to cu. ft. 8. 48 T. 2 cwt. 36 lb. to oz. 15. 36 wk. 1 d. 5 hr. to min. 9. 328 gal. 3 qt. 1 pt. to gills. 16. 21 yr. 26 d. to hours. 10. 85 hhd. 15 gal. to pints. 17. 145° 28" to seconds. 11. 45 bu. 3 pk. 4 qt. to pints. 18. £68 3s. 6d. to pence. 12. 124 sq. yd. 8 sq. ft. to sq. in. 19. £205 7s. ^(\. to far. 20. How many sec. in 3 yr. 42 wk. 5 d. 9 hr. 17 min. ? 21. What will 7 bu. 3 pk. of cranberries cost at 8 cts. a quart ? 22. Bought 84 gal. syrup at 75 cts. a gal., and sold it at 22 cts. a quart ; what was the gain ? 23. What is the value of 12 lb. 5 oz. 6 pwt. of gold, at 87 cts. a pwt. ? Oral Exercises. 397. 1. In 64 pints how many quarts ? How many gallons ? Analysis. — Since in 2 pints there is 1 qt.,in 64 pints there are 32 quarts. In 4 qts. there is 1 gallon, and in 32 qts. there are 8 gallons, Ans. 2. How many feet in 120 inches ? How many yards ? 3. In 60 ounces Troy, how many pounds ? In 168 ounces ? 4. In 72 hr. how many days ? In 96 hours ? 5. How many cords of wood in 72 cord feet ? 6. Change 120s. to pounds, and 240 pence to shillings. 398. Reduction Ascending is changing lower denominations to liiglier ; as, feet to yards, etc. Reduction. 163 399. To reduce Loiver denominations to Higher, 1. Reduce 6900 inches to rods, etc. Analysis.— Since 12 in. make 1 ft., 6900 in. 12 ) 6900 in. = as many feet as 12 is contained times in G900, "7 or 575 ft.' As 3 ft. make 1 yd., 575 ft. = as ^ lATr. -'■'^* many yd. as 3 is contained times in 575, or 54 ) 191 yd. 2 ft. 191 yd. and 2 ft. over. Finally, as 5|yd. make o 1 rod, 191 yd. = as many rods as 5i is contained times in 191, or 34 rd. and 8 half yards, or 4 yd. 11 ) 382 over. (Art. 217.) Am. 34 rd. 4 yd. 2 ft. ^ ^^ ^ ^ Hence, the ' *^ Rule. — Divide the given denomination by the ninnher reqnii^ed to make one of the Jiext higher. Proceed in this manner with the successive denomina- tions, till the one required is reached. Tlie last quotient, with the several remainders annexed, ivill he the answer. Note. — The remainders are the same denomination as the respective dividends from which they arise. 400. Proof. — Reduction Ascending and Descending prove each other ; for, one is the reverse of the other. 2. In 245640 ft. how many miles, rods, etc. ? Ans. 46 mi. 4 fur. 7 rd. 1 yd. 1 ft. 6 in. Reduce the following to the denominations indicated : 3. 34248 gills to bbl. 11. 85264 sq. ft. to sq. rods. 4. 46840 pt. to hhd. 12. 2118165^ sq. yd. to acres. 5. 653674 pwt. to lb. 13. 16568 cu. ft. to cords. 6. 426508 gr. to lb. 14. 43228 qt. to bushels. 7. 35624 oz. to cwt. 15. 28956 pt. to bushels. 8. 8420724 oz. to tons. 16. 5685720 hr. to com. yr. 9. 29728 in. to rods. 17. 856700 d. to weeks. 10. 48400 ft. to miles. 18. 4683248 far. to pounds. 19. What will a can of milk containing 28 gal. 3 qt. cost, at 6 cts. a quart ? 20. At $0.75 a yd., what will it cost to build a wall 182 r. long ? 21. If a grocer buys 3 bu. of cranberries at $2.25 a bu. and sells them at 9 cts. a quart, how much does he make ? 164 Denominate Fractions. Denominate Fractions. 401. Denominate Fractions are fractions of denominate Integers, and may be common or decimal. 402. To reduce Denominate Fractions, Common or Decimal, of higher denominations, to Integers of lower denominations. 1. Eeduce -J yard to integers of lower denominations. Solution. — 1 yd. = 8 ft., and -| yd. X 3 = -V-, or 2| ft. I yd X 3 = -V- ft- or 2f ft. Again, | ft. X 12 = ^«S or 7| in. f ft. X 12 = «/ in., or 7i in. j^^^^ 2 ft. 7^ in. 2. Eednce .875 yard to integers of lower denominations. Solution.— 1 yd. =.3 ft., and .875 yd, x 3 = 2.625 ft. .875 yd. Again, .625 x 12 = 7.500 inches. g The answer is 2 ft. 7.5 inches, the same as above. — 2.625 ft. Note. — Pointing off 3 figures in the several products -. « is equivalent to dividing them by 1000, the denominator of the given decimal. Hence, the 7.500 in. Rule. — Multiply the given iiinnerator, whether couzmon or decimal, and the remainder, if aihy, by the successive numbers which will reduce a unit of the given fraction to the denomination required, and divide the several pro- ducts by the given denominator. 3. In ^ day, how many hours and minutes ? 4. In 4:1 week, liow many days, hours, etc. 5. Eeduce ff mile to furlongs, etc. 6. Eeduce f J bu. to pecks, quarts, etc. 7. Eeduce f sq. mile to acres, rods, and yards. 8. Eeduce -j^-g- gal. to the fraction of a gill. 9. What part of a pint is 2-0T ^^ ^ bushel ? 10. Eeduce £.4625 to shillings and pence. 11. Eeduce .756 gallons to quarts and pints. 12. Eeduce .6254 days to hours, minutes, and seconds. 13. Eeduce .856 cwt. to ounces. 14. Eeduce .7582 of a bushel to pecks, etc. 15. Eeduce 0.98 rod to yards, feet, and inches. 1.5 pt. 3.75 qt. Reduction. 165 403. To Reduce Denominate Integers or Fractions of lower, to Fractions, either Common or Decimal, of higher denominations. 16. Reduce 7s. 6d. to the common fraction of a pound. Solution.— 7s. 6d. = 90d., and £1 = 210d. Now, £^^^ = £|, Ans. 17. Reduce 3 quarts 1 pint 2 gills to the decimal of a gallon. Solution. — Writing the numbers under each '* ■^* ^* other, the lowest denomination at the top, we divide 2 the 2 gi. by 4, and place the quotient .5 below, at the right of the next higher denomination. Thus, ^ 1.5 pt. -T- 2 = .75 qt., and so on. Hence, the AnS. .9375 ffal. Rule. — Reduce the giveiv compound jzumher to the lowest denoiTbination mentioned for the numerator, and a unit of the required fraction to the same denomina- tion for the denominator. For decimcds, divide the given numhers as in reducing integers to higher denominations. (Art. 399. ) Note. — If the lowest denomination of the given number contains a fraction, the number must be reduced to the ])arts indicated by the denominator of the fraction. 18. Reduce f pint to the fraction of a bu. (Art. 179, 2°.) 19. What part of a bushel is 3 pk. 5 qt. 1 pt. ? 20. What part of a gallon is 3 qt. 1 pt. 3 gills ? 21. Reduce 9 hr. 15 miu. 12 sec. to the fraction of a week. 22. Reduce 15f gr. to the fraction of a pound Troj. 23. What part of an acre is 18|^ square feet ? 24. Reduce 3 pk. 2 qt. 1 pt. to the decimal of a bushel. 25. Change 18 hr. 9 min. to the decimal of a day. 26. Change 2 ft. 8 in. to the decimal of a yard. 27. Change 8 oz. 7 pwt. 12 gr. to the decimal of a lb. Troy. 28. Change .4 of a pt. to the decimal of a gallon. 29. Change .25 lb. to the decimal of a ton. 30. Reduce 2 yr. 3 mo. 18 d. to the decimal of a year. 166 Ootnpound Numbers, 404. To find what part one Compound Number is of another: Reduce tlie numhers to the same denomination, and make the numher denoting the imrt the numerator, and tliat loitli luhich it is compared the denominator. (Arts. 226, 249.) 31. What part of 2 gal. 3 qt. 1 pt. is 1 gal. 2 qt. ? 32. What part of 4 wk. 2 d. 6 hr. is 3 d. 12 hours ? 33. What part of 15 miles 40 rd. is 6 mi. 30 rods ? 34. What decimal of 4 lb. 2 oz. 12 pwt. is 6 oz. 8 pwt. ? 35. What decimal of 10 bu. 3 pk. 4 qt. is 4 bu. 1 pk. 5 qt. 405. To Reduce Metric to Common Weights and Measures. 1. Reduce 84 decimeters to feet. OPERATION. ANALYSis.—Taking 39.37 in., the value of the 39.37 m. principal metric unit, as the standard, we multiply §4 -^^ it by the given metric number expressed in the same metric unit ; and 84 dm. = 8.4 m. ^*^ ' '*" Since 1 m. is equal to 39.37 in., 8.4 m. are 31496 equal to 8.4 times 39.37 in., or 330.708 in., and -.^^ qon wao -^ 330.708 in. = 27.559 ft., Aiis. Hence, the j 66K).jKm m. Ans. 27.559 ft. Rule. — Multiply the value of the prineipal iivetrie unit of the Table by the given metric number expressed in the same unit, and reduce the product to the denomination required. (Art. 399.) 2. In 45 kilos, how many pounds? Ans. 99.207 lb. 3. In 63 kilometers, how many miles ? 4. Reduce 75 liters to gallons. 5. Reduce 56 dekaliters to bushels. 6. Reduce 120 grams to ounces. 7. Reduce 137.75 kilos to pounds. 8. In 36 ars, how many square rods? Analysis. — In 1 ar there are 119.6 sq. yd. ; hence in 36 ars there are 36 times as many. Now 119.6 x 36 = 4305.6 sq. yd., and 4305.6 sq. yd.-r- 30^ = 142.33 sq. rods, Ans. 9. In 60.25 hektars, how many acres ? 10. In 120 cu. meters, how many cu. feet ? Addition. 167 406. To reduce Common to Metric Weights and Measures, 11. Reduce 2190 yds. 2 ft. 11 in. to kilometers. OPERATION. Explanation. — Reducing the omn i no. -• -i • , ^ . , , 2190 yd. 2 ft. 11 in. given number to inches we have "^ 7887o in. Dividing this number £ by 89.37, the number of inches in 6572 ft. a meter, we have 2003.429+ m. ..« To reduce meters to Km. we remove the decimal point 3 places 39.37 ) 78875 in. to the left. Ans. 2.003429 + Km. Hence, the 2003.429+ m. A71S. 2.003429+ Km. Rule. — Divide the given number by the value of the principal jnetrie unit of the Table, and reduce the quo- tient to^the denomination required. Note. — Before dividing, the given number should be reduced to the denomination in which the Dalue of the principal unit is expressed. 12. In 63f yards, how many meters ? 13. Reduce 13750 pounds to kilograms. 14. Reduce 250 liquid quarts to liters. 15. Reduce 2056 bu. 3 pecks to kiloliters. 16. In 3 cwt. 15 lb. 12 oz., how many kilos? 17. In 7176 sq. yards, how many sq. meters ? 18. In 40.471 acres, how many hektars ? 19. In 14506 cu. feet, how many cu. meters ? 20. In 36570 cu. yards, how many cu. meters? Addition. 407. The method of Adding, Subtracting, Multiplying, and Dividing Compound Numbers is the same as the correspond- ing operations in simple numbers and special rules are unnec- essary. Note — 1. The apparent difference arises from their scales of increase, one being variable and irre^lar, the other decimal and uniform. 168 Compound Nuinbers, 1. What is the sum of 18 bu. 3 pk. 5 qt. 1 pt., 24 bu. 2 pk. 6 qt., 6 bu. 2 pk. 7 qt. 1 pt, 8 bu. 3 pk. 4 qt. 1 pt. ? Explanation. — The sum of the right- hand col. is 3 pt. = 1 qt. 1 ])t. Set the 1 pt. under the col. of pints, and adding the 1 qt. to the col. of qt., the sum is 23 qt. = 2 pk. 7 qt. Write the 7 qt. in the col. of quarts, and adding the 2 pk. to the col. of pk., proceed as before. Ans. 59 bu. pk. 7 qt. 1 pt. bu. 18 24 6 8 OPEKATION. pk. qt. 3 5 59 pt. 1 1 1 1 (2.) £ 5 6 5 12 far. 3 2 1 2 (3.) gal. qt. pt. 2 1 3 1 3 1 2 1 2 (4.) wk. da. hr. min. 2 3 8 40 4 6 5 10 2 5 20 35 6 4 18 23 5. What is the sum of 5 rd. 4 yd. 2 ft. 7 in., 6 rd. 5 yd. 2 ft. 6 in., 4 rd. 4 yd. ft. 4 in., 3 rd. 3 yd. 2 ft. 8 in. ? Note. — 2. When a fraction occurs in the amount in any denomination except the lowest, it should be reduced to integers of lower de- nominations, and uuited with like integers. Thus, in Ex. 5 the \ yd. = 1 ft. 6 in., which added to 2 ft. 1 in. make 3 ft. 7 in. ; and 1 yd. plus 3 ft. plus 7 in. equals 2 yd. ft. 7 in. rd. yd- ft. in. 5 4 2 7 6 5 2 6 4 4 4 3 3 2 8 21 H 2 1 i = : 1 G Ans. 21 7 6. What is the capacity of 3 bins holding respectively 35 bu. 3 pk. 4 qt., 42 bu. 1 pk. 6 qt., and 56 bu. 2 pk. 5 qt. ? 7. How mucli land in 3 farms containing 87 A. 48 sq. rd., 97 A. 67 sq. rd., and 65 A. 42 sq. rd. ? 8. Bought 3 casks of oil ; holding 2 hhd. 30 gal. 2 qt. ; 3 hhd. 10 gal.; 1 hhd. 13 gal. 1 qt.; how much did all hold ? 9. Add together 23 yr. 2 mo. 3 wk. 5 d., 68 yr. 3 mo. 2 wk. 3 da., 60 yr. 4 mo. 1 wk. 6 d., 49 }t. and 4 d. 10. Required the number of miles, etc. in 3 roads, measuring 23 mi. 67 rd. ; 32 mi. 65 rd. ; and 46 mi. 28 rods. Subtraction, 169 11. A mason plastered one room containing 45 sqtiare yards 7 ft. 6 in., another 25 sq. yd. 6 ft. 95 in., another 38 sq. yd. 4 ft. 41 in.; Avhat was the amount of his phistering? 12. One pile of wood contains 10 C. 38 ft. 39 in., another 15 C. 56 ft. 73 in., another 30 C. 19 ft. 44 in., another 17 0. 84 ft. 21 in.; how much do they all contain? 13. Find the sum of 45 mi. 17 rd. 5 yd. % tt. 9 in., 43 mi. 44 yd. 1 ft. 8 in., 89 mi. 216 rd. 3 yd. 2 ft. 5 in. 14. What is the sum of £ J, \^. , and ^d. ? OPERATION. Note.— 3. Denominate Fractions should ^ ~ ^* ' ^ ^^^' be reduced to integers of lower denomina- i^^* -— ^^* IQ-* -^ l^-r. tions, then added as above. (Art. 402.) Jd. =: Os. Od. 1^ far. Ans. 3s. ^ id. 3^ far. 15. Add 1 hu. H pk. |qt. i pt.) ■A bu. \ pk. f qt. ■ipt. 16. Add 1 of A day, f of A h I--, A of M min. , and f of 2^ sec. 17. Add |lb. 1 to| : oz. 1 pwt. 19. f gal. to iqt. lipt. 18. Add fwk. to 1 d. 1| hr. 20. £-1 to f s. 2|d. Subtraction. 408. 1. From 35 rd. 2 yd. 1 ft. 8 in., take 22 rd. 2 yd. 2 ft. 6 in. Explanation. — Write the numbers and proceed as in sim- ple subtraction. Taking 6 in. from 8 in. leaves 2 inches. As 2 ft. cannot be taken from 1 ft., AUS. SVC take 1 yd. = 8 ft. from 2 yd., and adding it to 1 ft., we have Qj. 4 ft., and 2 ft. from 4 ft. leave 2 ft. Again, 2 yd. cannot be taken from the 1 yd. remaining, yd., added to 1 yd. make 6| yd., from which subtract 2 yd., and A.\ yd. remain. Finally, 22 rd. from 34 rd. leave 12 rd. The 4 yd. = 1 ft. 6 in., which added to the above make 12 rd. 5 yd. ft. 8 in., Ans. 2. From 121 hhd. 28 gal. 1 qt., take 63 hhd. 21 gal. 3 qt. 8 OPERATION. 35 rd. 2 yd. 1 ft. 22 2 2 8 in. 6 12 ^ 2 i - 1 2 6 12 5 'd. remaining. But 1 re 8 1. =5^ 170 Compoimd Namhers. 3. Bought 2 silver pitchers, one weighing 2 lb. 10 oz. 10 pwt. 1 gr., the other 2 lb. 3 oz. 12 pwt. 5 gr. ; what is the difference in their weight ? 4. A merchant had 228| yards of cloth, and sold 115| yards; how much had he left ? 5. From 25 mi. 7 fur. 8 rd. 12 ft. 6 in., take 16 mi. 6 fur. 30 rd. 4 ft. 8 in. 6. A man owning 95 A. 75 rd. 67 sq. ft of land, sold 40 A. 86 rd. 29 ft. ; how much had lie left ? 7. A tanner built two cubical vats, one containing 116 ft. 149 in., the other 245 ft. 73 in.; what is the difference be- tween them ? 8. A man having 65 0. 95 ft. 123 in. of wood in his shed, sold 16 C. 117 ft. 65 in. ; how much had he left? 409. To find the Exact Number of Years, Months, and Days, between two dates. 9. What is the difference of time between July 4th, 1879, and Nov. 15th, 1882 ? Analysis.— The time from July 4th, 1879 to July 4th, 1882 = 3 yr. The time from July 4tli to Nov. 4th — 4 mo The time from Nov. 4th to Nov. 15th. = 11 d. Ans. 3 yr. 4 mo. 11 d. Hence, the Rule. — First find the nmiiher of entire years, next the ninnher of entire months, then the days in the parts of a month. Note. — 1. The day on which a note or draft is dated, and that on which it becomes due, must not both be reckoned. It is customary to omit the former and count the latter. 10. A ship started on a trading voyage round the world Mar. 3d, 1875, and arrived back Aug. 24th, 1878 ; how long was she gone ? 11. What is the time from Oct. 15th, 1875, to March 10th, 1882? la. A note dated Oct. 2d, 1870, was paid Dec. 25th, 1882 ; how long was it from its date to its payment ? 13. A ship sailed on a whaling voyage, Aug. 25th, 1880, and returned April 15th, 1882 ; how long was she gone ? Oct. = 31 d, Nov. = 30d, Dec. = 31d. Jan. = 15 d. Ans. 119 d. Subtraction. 171 14. A mortgage was dated April 10th, 1875, and was paid Aug. 25, 1880 ; how long did it run ? 15. How many days did a note run which was dated Sept. 18th, 1879, and paid Jan. 15th, 1880 ? Analysis.— In Sept. it had 30-18=12 operation. days; in Oct., 31 d. ; in Nov.. 30 d. ; in Sept. 30 — 18 = 12 d. Dec, 31 d. ; in Jan., 15 d. Hence, Note. — 2. To find the number of days between two dates, write in a col. the num- ber of days remaining in the first mo., and the number in each succeeding month, including those in the last ; the sum will be the number of days required. 16. A note dated May 21st, 1879, was paid Nov. 28th, 1879 ; how many days did it run? 17. What is the number of days between Oct. 5th, 1879, and March 3d, 1880 ? 18. A person started on a journey Aug. 19th, 1869, and returned Nov. 1st, 1869 ; how long was he absent ? 19. A note dated Jan. 31st, 1870, was paid June 30th, 1870 ; how many days did it run ? 20. How many days from May 23d, 1868, to Dec. 31st, fol- lowing ? 21. The latitude of New York is 40° 42' 43'' N., that of St. Augustine, Fla., is 29° 48' 30" ; what is the difference of their latitude ? Note.— 3. When two places are on opposite sides of the Equator, the difference of latitude is found by adding their latitudes, 22. The latitude of Cape Horn is 55° 59' S., that of Cape Cod is 42° 1' 57" N. ; what is the difference of their latitude ? 23. The longitude of Cambridge, Mass., is 71° 7' 22", that of St. Louis is 90° 15' 16" ; what is the difference of their longitude ? 24. The Ion. of Paris is 2° 20' E., that of Washington D. C. is 77° 0' 15" W. ; what is the difference of their longitude ? 172 Compound Numhers, Multiplication. 410. 1. If a man can build a fence 12 rd. 1 yd. 2 ft. 5 in. long in one day, how long a fence can he build in 6 days ? Analysis.— In 6 d. he can build 12 rd. 1 yd. 2 ft. 5 in. 6 times as much as in 1 d. 6 times g 5 in. are 30 in. =2 ft, 6 in. Write -; -^ — -^r~ the G under the in., and add the '^^ ^<^- "^2" 1^' ^ ^^' ^ ^^• 2 ft. to the next product. Proceed (-g-) = 1 6 in this way till all the denomina- ^^ic. 73 y^ 5 \i^ \ ft in tions are multiplied. Note. — If a fraction occurs in the product of any denomination except the lowest, it should be reduced to lower denominations , and be united to those of the same name as in Compound Addition. 2. Multiply 8 lb. 6 oz. 3 pwt. by 8. 3. Multiply 27 gal. 3 qt. 1 pt. 3 gi. by 7. 4. Multiply 26 mi. 87 rd. 4 yd. 2 ft. by 9. 5. What is the weight of 12 silver cups, each weighing 8 oz. 17 pwt. 6 gr. ? 6. How much water in 28 casks, each containing 54 gal. 3 qt. 1 pt. 2 gi. ? 7. If a railroad car goes 21 mi. 2 fur. 10 rd. per hour, how far will it go in 25 hours ? Division. 411. 1. A grocer paid £5 2s. 9d. for 4 boxes of sugar ; how much was that a box ? Analysis.— Since 4 boxes cost £5 2s. 9d. , operation. 1 box will cost \ as much, and £5-^4 = £1 and 4 ) £5 2s. 9d. £1 over. Reducing the remainder to the next j o-i ^k7 ^X(\ lower denomination, and adding the 2s., we * • 4 • have 22s., which divided by 4 = 5s. and 2s. over. Reducing 2s. as before, continue the division till each denomination is divided. 2. A silversmith melted up 2 lb. 8 oz. 10 pwt. of silver, which he made into 6 spoons ; what was the weight of each ? Longitude and Time. 173 3. If 8 persons consume 85 lb. 12 oz. of meat in a month, how much is that apiece ? 4. A man traveled 50 mi. and 32 rd. in 11 hours; at what rate did he travel per hour ? 5. A man had 285 bu. 3 pk. 6 qt. of grain, which he wished to carry to market in 15 equal loads ; how much must he carry at a load ? Longitude and Time. 412. The Earth turns on its axis once in 24 hours ; hence, ^ part of 360°, or 15° of longitude, passes under the sun in 1 hour. Again, -^ of IS'^ Ion., or 15', passes under the sun in 1 min. of time. And -^ of 15', or 15'' Ion., passes under the sun in 1 sec. of time, as seen in the following 413. Comparison of Longitude and Time. 360° Ion. make a difference of 24 hrs. of time. 15° " '' " 1 hr. '' 1° " " '' 4 min. '' V '' '' '' 4 sec. " 1" '' " " -^sec. '' 414. The Longitude of a place is the number of deg., min., and sec, reckoned on the equator, between a standard meridian (marked 0°) and the meridian of the given place. All places are in East or West longitude, according as they are East or West of the Standard Meridian, until 180°, or half the circumference of the Earth is reached. Notes. — 1. The Enprlish reckon Ion. from the meridian of Greenwich ; the French from that of Paris. Americans generally reckon it from the meridian of Greenwich ; sometimes from that of Washington. 2. When two places are on opposite sides of the Standard Meridian, the difference of Ion, is found by adding their longitudes. (Art. 409, N 3.) 174 Compound Numhers. 415. To find the Difference of Longitude between two places, the Difference of Time being known. 1. The difference of time between New York and Chicago is 54 min. 19 sec. What is the difference of longitude ? Analysis. — Since 15' of Ion. make a operation. difference of 1 min. of time, tliere must be 54 m. 19 sec. 15 times as many min. of Ion. as tliere are 15 min. and sec. of time, and (54 min. 19 sec.) -— -"- —,, xl5 = 13°84'45", ^/?«. Hence, the ^^ ^^ 45 , ^/^S. Rule. — Multiply the difference of time, expressed in hours, minutes, and seconds, hy 15 ; the product will he the diff'erence of longitude in degrees, minutes, and seconds. (Art. 412.) 2. The difference of tiaie between Boston and Albany is 9 min. 2 sec. ; what is the difference of longitude ? 3. The difference of time between Savannah, Ga., and Port- land, Me., is 43 min. 32.13 sec. ; what is the dif. of longitude ? 4. The difference of time between Boston and Detroit is 47 min. 56 sec. ; w^hat is the difference of longitude ? 5. The difference of time between Philadelphia and Cincin- nati is 37 min. 8.4 sec; what is the difference of longitude ? 6. The difference of time between Louisville, Ky., and Bur- lington, Vt., is 49 min. 20 sec; what is the dif. of longitude ? 416. To find the DiflTerence of Time between two places, the Difference of Longitude being known. 1. The difference of longitude between Chicago and Boston is IQ'^ 34' 15" ; what is the difference of time ? Analysis. — Since 15° Ion. make a operation. difference of 1 hour of time, there must 15 ) 16° 34' 15" be A as many hours minutes, and J,,,, l h^. 6 min. 17 sec seconds, as there are deg., mm., and sec. of Ion., and (16° 34' 15") -4- 15 == 1 hr. 6 min. 17 sec. Hence, the Rule. — Divide the difference of longitude, in degrees, minutes, and seconds, hy 15 ; the quotient will he the difference of time in hours, minutes, and seconds. Longitude and lime. 175 2. The difference of longitude between Cambridge, Mass., and Charlottesville, Va., is T 23' 49" ; what is the difference of time ? 3. The Ion. of St. Louis is 90° 15' 15", that of Charleston, S. C, is 79° 55' 38" ; what is the difference of time ? 4. The Ion. of Berlin is 13° 23' 45" E., that of :N^ew Haven, Ct, is 72° 55' 24" W. ; what is the difference of time ? 5. The Ion. of Montreal is 73° 25' W., that of New Orleans is 90° 2' 30" W. ; what is the difference of time? 6. The Ion. of Paris is 2° 20' E., Kome is 12° 27' E. ; what is the difference of time ? 7. The Ion. of West Point is 73^ 57' W., that of Washington, D. C, 77^ 0' 15" W. ; what is the difference of time? 8. How much earlier does the sun rise in Albany, Ion. 73° 44' 50", than in St. Paul, Min., Ion. 93° 4' 55"? Than in Astoria, Oregon, Ion. 124° ? 9. When it is 9 a.m. in New York, Ion. 74° 3', what is the time in Richmond, Va., Ion. 77° 25' 45" ? In San Francisco, Ion. 122° 26' 45" ? Q U ESTI ON S. 393. What is reduction ? 393. Descending? 896. Rule? 397. Reduc- tion Ascending: ? 399. Rule ? How proved ? 401. What is a denominate fraction ? 402. How reduce them from higher denominations to integers of lower ? 403. How reduce denominate integers to fractions of higher denominations ? 404. How find what part one number is of another ? 405. How reduce metric to common weights and measures ? 406. How reduce common to metric weights and measures ? 407. How are compound numbers added, subtracted, multiplied, and divided ? From what does the apparent difference atise ? 409. How find the difference between two dates in years, months, and days ? How find the difference of latitude between two places on opposite sides of the equator ? 414. What is the longitude of a place ? When is a place in East longitude ? When in West ? From what meridian do the English reckon longitude ? The French ? Americans ? 415. How find the difference of longitude when the difference of time is given ? 416. How find the difference of time when the difference of longitude is given ? 176 Coinpound Number So Measurement of Surfaces. Oral Exercises. 417. 1. How many sq. feet in the surface of a blackboard 4 ft. long and 3 ft. wide ? Analysis. — Let the sides of the black- board be divided into 4 equal parts, and the ends into 3 equal parts, each denoting a linear foot. The blackboard contains as many sq. feet as there are squares in the figure. Since there are 4 squares in 1 row, in 3 rows there are 3 times 4, or 12 squares. Ans. 12 sq. feet. 2. How many sq. feet in a flagging stone 8 ft. long and 4 feet wide ? 3. How many sq. feet in a strawberry bed 20 ft. long and 5 ft. wide? 4. How many sq. yards in a lawn whose length is 9 yards and its breadth 7 yards ? 5. If a meadow is 12 rods long and 8 rods wide, how many sq. rods does it contain ? 6. A house lot containing 84 sq. rods is 7 rods wide ; what is its length ? 7. I wish to lay out an orchard 12 rods in width ; what must be its length to contain 240 sq. rods ? 8. What is the difference between 4 square feet and 4 feet square ? Written Exercises. 418. A Plane Figure is one which repre- sents a plane or flat surface. 419. The Perimeter of a plane figure is the line which hounds it. 420. The Area of a plane figure is the quantity of surface it contains. Measurement of Surfaces. 177 421. A Rectangle is a plane figure haying four sides and four right-angles. (Art. 418.) 422. When all the sides of a rectangle are equal it is called a Square. 423. The Dimensions of a rectangular figure are its length and Ireadth. 424. To find the Area of Rectangular Surfaces. 1. How many square rods in a garden 18 rods long and 12 rods wide ? ^ . , -I -.o 1 1 1 H J OPERATION. Solution. — A rectangle 18 rods long and 1 rod wide will contain 18 sq. rods. And a garden 18 rods roas. long and 13 rods wide will contain 12 times 18, or 12 216 sq. rods, Ans. Hence, tlie j^^^^ ^ ^^ ^^^^^^ EuLE.- — Multiply the length dy the hreaclth. Notes. — 1. Both dimensions sliould be reduced to the same denomina- tion before they are multiplied. 2. One line is said to be multiplied by another, when the rivmber of units in the former are taken as many times as there are like units in the latter. (Art. 83, i°.) 3. The area and one side of a rectangular surface being given, the other side is found by dividing the area by the gimn side. (Art. 119a.) 2. How many yards of carpeting 1 yd. wide will it take to cover a floor 22 ft. long and 15 ft. wide ? 3. How many yards of carpeting 27 in. wide will it take to cover the same^oor. 4. In a meadow 68 rd. long and 43 rd. wide, how many acres ? 5. A building lot is 50 ft. front, and contains half an acre ; how far back does it extend ? 6. At 25 cts. per sq. foot, what is the cost of an acre of land ? 7. Bought a rectangular farm 210 rods long and 88 rods wide, at $15 per acre ; what was the cost ? 8. The length of a pasture is 231 meters, and its breadth is 87 meters : what is its area in sq. meters ? 9; The area of a meadow is 210.6 sq. meters, and its length is 64.8 meters ; what ^s its width ? 178 Compound Nunribers, 10. If I pay 1276 for 92 meters of broadcloth 1.5 meters wide, what is that per square meter ? 11. How many acres in a field 800 rods long, and 128 rods wide ? 12. Find the area of a square field whose sides are 65 rods in length. 13. A man fenced off a rectangular field containing 3750 sq. rods, the length of which was 75 rods ; what was its breadth? 14. How many hektars in a rectangular field 475.5 meters long and 246 meters wide ? 15. The length of the Capitol at Washington is 751 ft., its width 348 ft. ; how many sq. rods, and how many acres does it cover ? 16. What is the difference between two asparagus beds one of which is 2 rods square, and the other contains 2 sq. rods ? 17. The length of the main Centennial building in Philadel- phia was 1880 ft., and the width 464 ft. ; how many acres did it cover ? 18. A speculator bought 50 acres of land at 150 per acre, and sold it in villa lots of 5 rods by 4 rods, at 1150 a lot; what did he make by the operation ? 19. A garden 27 yd. long and 15 yd. wide has a gravel walk round it 6 feet wide ; what did the walk cost, at 50 cts. per square yard ? 20. What will it cost to carpet a floor 18 by 16 ft., the carpet being 27 in. wide, and its cost $1.12 a yard ? 21. What is the cost of paving a street 628 ft. long and 60|- ft. wide, at 12.25 a sq. yard ? 22. How many tiles 10 in. square are required to lay a side walk 168 ft. long and h\ ft. wide ? 23. What will it cost to concrete a court 168 ft. square, at $3.75 per sq. yard ? 24. A farm containing 150 acres, is 200 rods long ; what is its width ? What will it cost to build a wall around it, at $4 a rod? 25. How many planks 15 ft. long and 6 in. wide will it take to floor a room 20 ft. long and 15 J ft. wide ? Measurement of Solids, 179 Measurement of Solids. 425. A Rectangular Body is one bounded by six rectangular sides, etich opposite pair being equal and 'parallel ; as, boxes of goods, blocks of hewn stone, etc. 426. When all the sides are equal, it is a Cube ; when the oiJliosite sides only are equal, it is a Parallelepiped. 427. The Contents or Volume of a body is the quantity of matter or siiace it contains. 428. The Dimensions of a rectangular body are its length, breadth, and thickness. 429. To find the contents or volume of Rectangular Bodies. 1. How many cubic feet in a block of granite 4 ft. long, 3 ft. wide, and 2 ft. thick ? Illustration. — Let the block be repre- sented by the adjoining figure, the length of which is divided into 4 equal parts, the width into 3, and the thickness into 2 pnrts, each of which is a linear foot. Since the block is 4 ft. long and 3 ft. wide, in the upper face there are 3 times 4, or 13 sq. feet. Now, if the block were 1 foot thick it must have as many cu. feet as there are sq. feet in the upper face. But the given block is 2 ft. thick; therefore, it contains 2 times (4 X 3), or 24 cu, feet, Ans. Hence, the Rule. — Multiply the length, hreaclth, and thiclcness together. (Art. 424.) Notes. — 1. When the contents and two dimensions are given, the other dimension may be found by dividing the contents by the product of the two given dimensions. (Art. 119a,) 2. Excavations and embankments are estimated by the cubic yard. In removing earth, a cu. yard is called a load. 2. How many cu. feet of air in a school-room 20 ft. square and lOi ft. high ? 180 Comjpound Numbers. 3. How mauy cu. feet in a mound 54 ft. long, 36 ft. wide, and 12 ft. high ? 4. How many loads of earth must be removed in digging a cellar 48 ft. long, 25 ft. wide, and 8|- ft. deep ? 5. What will it cost to dig such a cellar, at 33-| cts. a cu. yard ? 6. What will it cost to fill in a street 55 feet wide, GOO ft. long, and 5| ft. below grade, at 42 cents a cu. yard ? 7. What is the volume of a cube whose edge is 5 yd. 2 ft. 6 in. ? 8. Find the volume of a cube whose edge is 15 J ft. ? 9. The width of a reservoir is 24 ft., its depth 8 ft., and its volume 5760 cu. ft. ; what is its length ? Wood Measure, , 430. A Cord of Wood is a pile 8 feet long, 4 feet wide, and 4 feet high. (Art. 353.) 431. A Cord Foot is 1 foot in length of such a pile. Hence, 1 cord foot ^16 cubic feet 8 cord feet = 1 cord. 1. How many cords of wood in a pile 35 ft. long, 6 ft. high, and 4 ft. wide ? 2. Find the number of cords in a pile of wood 42 ft. long, 5-| ft. high, and 8 ft. wide. 3. At $4.25 a cord, what will a pile of wood 26 ft. long, 4 ft. wide, and 4 ft. high cost ? 4. In 3 cord feet, how many cu. feet ? In 5 cord feet ? Masonry. 181 5. How many cubic feet in 12 cords ? In 24 cords ? 6. If. a pile of wood is 28 ft. long^ 4 ft. wide, how high must it be to contain 112 cords ? 7. How many cord feet in a load of wood 8 ft. long, 4 ft. high, and 3 ft. wide ? 8. What is the worth of a pile of wood 4 ft. in height, 6 ft. in length, and o^ in width, at $4.50 per cord ? 9. What must be the height of a load of wood that is 6 ft. long and 4 ft. wide, to contain a cord ? Masonry. 432. Stone Masonry is sometimes estimated by the percli. Brickwork is estimated by the thousand bricks. Notes. — 1. A perch of stone masonry is 164 ft. long, li ft. wide, and 1 ft. liigli, which is equal to 24J cu. ft. It is customary, however, to call 25 cu. ft. a perch. 2. The average size of bricks is 8 in. long, 4 in. wide, and 2 in. thick. In eBtimating the labor of brickwork by cu. feet, it is customary to measure the length of each wall on the outside ; no allowance being made for windows, doors, or corners. But a deduction of -^^ the solid contents is made for the mortar. 1. How many perch (25 cu. ft.) in the walls of a cellar, the thickness of which is 1 ft. 6 in., the height 8 ft., each side wall being 42 ft., and each end wall 24 feet ? 2. At $4.75 a perch, what will it cost to build the walls of the above cellar? 3. How many bricks will it take to build the walls of a house 50 ft. long, 25 ft. wide, 21 ft. high, and 1 ft. thick, deducting -^ of the contents for the mortar, but making no allowance for windows and doors ? 4. How many bricks will be required to build a house, the walls of which are 48 ft. long, 24 ft. wide, 42 ft. high, and 1 ft. thick, making no allowance for windows, doors, or corners ? 5. At 83.50 per M. for bricks, deducting -^ for mortar, and $4.25 per M. for laying them, what will the walls of such a house cost? 182 Compound Nuinbers, Board Measure. 433. A Board Foot is 1 ft. long, 1 ft. wide, and 1 in. thick; that is, a square foot 1 inch thick. 434. A Board Inch is ^^ of a board foot ; that is, 1 inch long by 12 inches wide and 1 inch thick. Hence, Twelve board feet are equal to 1 cubic foot. 435. Sawed timber, as plank, joists, etc., is estimated by cu. feet ; heivn timher, as beams, etc., either by board feet or cu. feet; round timber, as masts, etc., by cu. feet. Written Exercises. 436. To find the Contents of Boards, Planks, etc. 1. How many board feet in a board 11 ft. long, 18 in. wide, and 1 inch thick ? Explanation.— Multiplying the length operatiok. in feet by tlie width and thickness expressed 11 X 18 X 1 = 198 in. in inches, we have 198 board inches. Di- 198-^-12 = 16^ ft. viding this product by 12, the result is 16| Ans IQ^ ft board feet, Arts. 2 2. How many board feet in a scantling 14 ft. long, 4 in. wide, and 2^ in. thick ? Solution. — Multiplying the length in feet by the width and thickness expressed in inches, we have 14 ft. x 4 x 2|^ = 140, and 140 -f- 12 = llf board ft., Ans. Hence, the EuLE. — Multiply the length in feet hy the width and thichness expressed in inches, and divide the product hy 12 ; the quotient will he in hoard feet. Notes. — 1. The standard thickness of a board is 1 inch. If?^5sthan 1 inch, it is disregarded ; if more than 1 inch, it becomes a factor in find- ing the contents of plank, scantling, etc. 2. If aboard is ^a^enVi^, multiply the length by half the sum of the two ends. Board Measure. 183 3. The approximate contents of round timher or logs may be found by multiplying \ of the mean circumference by itself, and this product by the length. 3. What is the number of feet in a board 14 ft. long and 17 in. wide ? 4. Find the contents of a tapering board 15 ft. long, 17 in. wide at one end and 11 in. at the other ? 5. Required the contents of 8 boards, 11 ft. long and 15 in. wide ? 6. What is the worth of 120 boards of the above size at 4 cents a board foot ? 7. Find the contents of a board 16 ft. long, 15 in. Avide, and J in. thick ? 8. What are the contents of 8 scantlings 15 ft. long, 4 in. wide, and 3 in. thick, board measure ? 9. How many feet in a beam 16 ft. long, 8 in. wide, and 4 in. thick, board measure ? Cubic measure ? 10. What cost 24 joists whose dimensions are 4 in. by 3 in. and 11 ft. long, at 25 cts. a cu. foot ? 11. Wbat must be the length of a piece of timber 16 in. by 15 in., to contain 20 cu. feet ? 12. How many cu. feet in a log 65 ft. long, whose mean circumference is 8 ft. ? 13. How many cords of wood in such a log ? 14. How many feet of inch boards will it take to build a fence 4 ft. high and 125 ft. long ? 15. At $2.25 per 100 ft., what will the boards cost for such a fence ? 16. What amount of inch boards would be required to make a box 4 ft. long, 3 J ft. wide, and 2J ft. deep ? 17. AVhat is the cost of a stock of 9 boards 14 ft. long 15 in. wide, at $23.50 per 1000 ft. 18. How many cu. feet in a mast 54 ft. long, the circumfer- ence of which is 9 ft. ; and what will it cost at $1.09 a cu. foot ? 19. What cost 12 planks 14 ft. long, 12|^ in. wide, and 2|- in. thick ; at $18 per M. ? 20. How many cu. feet in a log 62 ft. long and 28 ft. in cir- cumference ? i84 Compound Numbers. Rectangular Cisterns, Bins, Etc. 437. The Capacity of rectangular cisterns, bins, etc., is measured by cubic measure, but tlie results are commonly expressed in units of Liquid and Dry Measure. 438. To find the Number of Gallons in Rectangular Cisterns, etc. 1. How many gal. of water will a rectangular cistern 6 ft. long, 4 ft. wide, and 3 ft. deep contain ? Analysis.— The product of 6 ft. x 4 x 3 = 72 cu. feet in the cistern ; and 73 x 1728 = 124416 cu. inches. Again, in 1 gallon there are 231 cu, inches, and 124416-T-231 :ir 538f? gal., Ans. (Ai-t. 357.) 2. How many bushels in a bin 11 ft. long, 4 ft. wide, and 3 ft. high ? Analysis.— 11 fl. x 4 x 3 = 132 cu. feet, and 132 cu. ft. x 1728 = 228096 cu. inches. Now 1 bu. contains 2150.4 cu. in. and 228096 cu. in. -^ 2150.4 = 106 i^j bu., Ans. (Art. 360.) Hence, the Rule. — Find the iiiunber of cubic inches in the thing measured, and reduce them to liquid or dry measure, as may he requij^ed. (Arts. 356, 359.) 3. Find the number of gallons in a cistern 8 ft. long by 6 ft. wide and 5 ft. deep. 4. How many hogsheads in a tank 12 ft. square and 8 feet deep ? 5. In a reservoir 40 ft. long, 30 ft. wide, and 15 ft. high, how many hogsheads ? 6. I wish to build a cistern containing 5000 gal., whose base is 12 ft. by 8 ; what must be its height ? 7. If a reservoir 45 ft. long, 28 ft. wide, contains 40000 hhd., how high must it be ? 8. At $1.12J a bushel, what is the value of a bin of wheat 9 ft. long, 5 ft. wide, and 4 ft. deep ? 9. How many cu. feet in a bin which will contain 300 bushels of grain ? Mectangidar Cisterns, Bins, JEtc. i85 . 439. Shorter Methods.— Since 2150.4 cu. inches -r- 1728 cu. in. = 1:1, it follows that a bushel must contain 1^ cu. ft. nearly. (x\rt. 360.) Hence, we have the following methods : 1st. Divide the number of cu. feet in a bin by 1^ and the quotient will be the approximate number of bu. in the bin. 2d. Multiply the number of bu. in a bin by 1^, and the pro- duct will be the approximate number of cu. feet in the bin. 3rd. A ton (2000 lbs.) of Lehigh white ash, ^gg size, coal in bins measures 34|^ cu. ft. A ton of white ash Schuylkill, Qgg size, measures 35 cu. ft. A ton of pink, gray, and red ash, egg size, measures 3G cu. ft. 4th. A ton of hay upon a scaffold measures about 500 cu. ft. ; when in a mow, 400 cu. feet ; and in well settled stacks, 10 cubic yards. 10. How many bushels of corn can be put into a bin 6 ft. long, 5 ft. wide, and 4 ft. deep ? A7is. 96 bushels. 11. A farmer has a bin 10 ft. long, GJ ft. wide, and 4 ft. deep ; how many bushels does it hold ? 12. A bin holding 150 bu. is 6 ft. wide and 4 ft. deep ; what is its length ? 13. A bin containing 280 bushels is 10 ft. long and 7 ft. wide ; what is its depth ? 14. "What must be the length of a bin 8 ft. wide, 5 ft. deep, to contain 320 bushels ? 15. At ll-J a bushel, what is the value of a bin of wheat 12.5 ft. long, 6 ft. wide, and 4 ft. deep ? 16. A farmer filled a bin 8 ft. long, 7 ft. wide, and 5 ft. deep, with the corn raised on 5 acres ; how many bushels was that per acre ? 17. How many tons of Lehigh white ash, egg size coal, "g^ill fill a bin 12 ft. long, 8 ft. wide,^ 6 ft. high ? 18. How many tons of hay in a mow 20 ft. long, 18 ft. wide, and 14 ft. high ? 186 Compound Ntiynhers. Oral Problems for Review, 440. 1. At 3 cts. a yd., what will 5 mi. of telegraph wire cost? 2. My neighbor's farm is f mile square ; how many acres did it contain ? 3. Bought 40 acres of land at 75 cts. per sq. rod, and sold it so as to double my money ; required my gain ? 4. At 25 cts. a gallon, what is a family's milk bill for 60 days, taking 2 qts. daily ? 5. If a man lives 2| miles from the City Hall, how many miles will he travel in 6 days, making 1 trip a day ? 6. The length of a blackboard is 6 ft., its width 4 ft. ; how many sq. yards does it contain ? 7. If a garden is 5 rods long and 4 rods wide, how many rods in its perimeter ? 8. If 1 oz. of spice costs 8 cents, what will 2J pounds cost ? 9. At II a sq= yard, what will it cost to carpet a room 18 feet long and 15 ft. wide ? 10. A stationer paid 11.25 a gross for pencils, and sold them for a cent apiece ; how much did he gain on 5 gross ? 11. In a certain school are 72 girls, and f of the pupils are boys; how many pupils in the school ? 12. The surface of a cube is 150 sq. inches ; what is the sur- face of one side ? 13. What fraction of a semi-circumference is 45 degrees ? 14. How many writing books of 36 pages each can be made from a half ream of pajoer ? 15. How many days in 7 of the longest months ? 16. A can do a job in 2 days, B in 3 days ; what part will each do in one day ? 17. How long will it take both to do the same job working together ? 18. How many yards of carpeting j yd. wide, will carpet a room 18 ft. square ? 19. How many sq. yards in the pavement of a street 60 ft. wide and 800 ft. long ? 20. How many suits of clothes can be made from 648 yards, allowing 4 yds. to a suit ? Review, 187 Written Problems for Review. 441. 1. How many acres in a piece of land 18i rods long and 96 rods wide ? 2. What will 16568 cu. feet of wood cost, at 13^ a cord ? 3. How many dollars can be made out of 50 lb. 9 oz. of silver, allowing ^Vl\ grains to a dollar ? 4. How many cubic inches in a box whose length is 30 inches, its breadth 18, and its depth 15 inches ? 5. How many cubic inches in a block of marble 43 inches long, 18 inches broad, and 12 inches thick ? 6. How many cubic feet of air in a school-room 16 feet long, 15 feet wide, and 9 feet high ? 7. How many cubic feet in a pile of wood 16 feet long, 6 feet wide, and 5 feet high ? How many cords ? y 8. How many cords of wood in a pile 140 feet long, 4|- feet wide, and ^\ feet high ? 9. At 50 cts. per decister, what will a ster of wood cost ? 10. What will a metric ton of h^mp cost, at 25 cts. per kilo? 11. At 6 cts. per liter, what cost a hektoliter of milk? X 12. How many square yards in the four sides of a room 18 feet long, Vl\ feet wide, and 14J feet high ? N 13. How many square yards of plastering will it take to cover the four sides and the ceiling of a room 18 feet square, and 15 feet high ? 14. How many yards of muslin 3 qrs. wide, are equal to 36 yds. brocatelle, which is \\ yard wide ? 15. How many yards of silk 3 qrs. wide, will 51 yds. of cambric line, which is IJ yd. wide ? 16. What will it cost to pave a street 3 mi. 115 rods long, and 2 rods wide, at %\h\ a square rod ? 17. A man having 15 acres and 60 rods of land, laid it out ^ in lots each containing 12 sq. rods, and sold the lots at $150 apiece ; how much did he realize for his land ? 18. What is the worth of a pile of wood 18 ft. long, 10-J ft. high, and 9J wide, at %'6\ a cord ? 19. How many times will a wheel of a railroad car, 9 ft. in circumference, turn round in going 1500 miles ? 188 Compound Numbers. 20. How long would it take a cannon ball, flying at the rate of 8 miles per minute, to reach the moon, a distance of 240000 miles ? 21. The velocity of light is 11875000 miles per minute, and it takes 8 minutes for it to pass from the sun to the earth ; how far from the sun is the earth ; and how many weeks would it take to travel this distance, 30 miles an hour ? ^yC^ 22. How many bricks will it take to pave a sidewalk 75 feet long and 8 feet wide, each brick being 8 inches long and 4 inches wide ? 23. Required to reduce 5 mi. 6 fur. 23 rods 5 yd. and 8 in. to inches, and prove the operation. Y 24. Allowing 1 shingle to cover 24 sq. inches, how many shingles will be required to cover the roof of a house 50 feet long, the rafters on each side being 29 feet long ? 25. How many farms of 160 A. in a township 6 miles square ? 26. How many bricks will it take to build a prison 60 feet long, 25 feet wide, and 48 feet high, whose walls are 1 foot thick, the bricks 8 in. long,, 4 in. wide, and 2 in. thick ? 27. If the pendulum of a clock vibrates 65 times per minute, how much time will it gain in a common year ? 28. How many years would it take to count a billion, count- ing 60 a minute, working 10 hours a day, and allowing 365 days to a year ? Questions. 418. What is a plane figure ? 419. The perimeter of a plane figure ? 430. The area ? 421. What is a rectangle ? 428. The dimensions of a rectangular fig- ure ? 424. How find the area of rectangular surfaces ? When the area and one side are given, how find the other? 425. What is a rectangular solid? 427. The contents ? 428. The dimen- sions ? 430. What are the dimensions of a cord of wood? 431. How many cubic feet does it contain ? Cord feet ? 432. How is stone masonry estimated? Brickwork ? 483. What is a board foot ? 434. A board inch ? How many board feet in a cu. foot ? 435. How are sawed and liewn timber estimated ? Round timber? 436. How find the contents of boards, plank, etc. ? 438. How find the contents of cubical bins, cisterns, etc. ? EKCENTAGE. Oral Exercises. 442. 1. When a number is divided into a hundred equ&! parts, what is one of tlie parts called? Two of the parts? Five ? Ten ? 2. A man paid $100 for a horse and sold it for $105 ; how many dollars did he gain ? How many hundredths of the cost did he gain ? 3. What part of $100 is $5 ? (Art. 226.) 4. If I pay $100 for a sofa and sell it for $94, how many dollars shall I lose ? How many hundredths of the cost? 443. The number of hundredths gained or lost is called the Rate j^er cent. 444. Per Cent, means by the hundred, or simply hun- dredths. Thus, 8 per cent is 3 hundredtlis of a number ; 5 per cent is 5 hun- dredths, etc. 445. The Sign of Per Cent is %. Thus, 4:% means 4 per cent. 446. The process of calculating by hundredths is called Percentage. 5. A farmer lost 8 sheep out of every 100 of his flock ; what per cent of them did he lose ? 6. A man gave away 810 out of every $100 of his income; what per cent of his income did he give away ? 7. A teacher having a class of 150 pupils, promoted 10^ of them ; how many were promoted ? 190 Percentage. 447. Per cent is expressed by decimals, by %, or by fractions. Tab le Sign. Decimal. Fraction. Sign. Decimal, Fraction H .01 — 100 \% .005 = 200 H .05 — 1 — 20 H% .035 = 1 "20¥ 10^ .10 — 1 — 10 i% .0025 =^ 40 25%^ .25 i H% .0625 = 1 50^ .50 — 1 2 m% .1875 zzz A 75^ .75 t 3H% .33i — i 100^ 1.00 — i — 1 mi% 1.125 ^; H 448. Since liwidredtlis occupy two decimal places, every per cent requires, at least, two decimal figures. Hence, if the given per cent is less than 10, a cipher must be prefixed to the figure denoting it. Thus, 2^ is written .02; 6^, .06, etc. Notes. — 1. A hundred per cent of a number is equal to the number itself; for {^^ is equal to 1. 2. In expressing per cent, when the decimal point is used, the words per cent and the sign {%) must be omitted, and mce versa. Thus, .05 de- but .05 per cent or .05% notes 5 per cent, and is equal to jf ^ or ^ denotes ^f o of j^o. and is equal to y^f^yo or ^oVo- 449. To read any given Per Cent, expressed Decimally. Call the first two decimal figures per cent ; and those on the right, ydecimal parts of 1 per cent. Note. — Parts of 1 per cent, when easily reduced to a common fraction, are often read as such. Thus, .105 is read 10 and a half per cent ; .0125 is iread one and a quarter per cent. Eead the following as rates per cent: 8. .06; .052; .085; .094. .012; .174; .0836; .154. .1857; .2352; .1685;. 7225. 9. 10, 11. 12. 1.07; 2.53; 4.65; 2.338. 13. 5.33^; 4.125; 8.0623; 6.73f .12i; .08i; .161; .5775. Percentage, 191 450. Express the following by Com. Frac. in lowest terms : 14. 4:%. 17. 20^. 20. 75^. 23. 150^. 15. Q%. 18. 25^. 21. 100^. 24. 200^. 16. 10^. 19. 50^. 22. 125^. 25. 500^. 26. To what common fraction is 8J-^^ equal ? ANALYSis.-8i% = ^^i , or 8i -H 100 ; and 8i -=- 100 = ^^ ^ ^ = _2J^^. or jV, ^^«- (-^rt. 220.) 27. To what common fraction is ^% equal ? Analysis.— i^% = .005; and .005 = yofo> or ¥¥o> ^^'''- (^t. 185.) 28. ^% = what fraction ? 37i^ ? 18|^ ? 2f % ? Mental Exercises. 451. 1. What per cent of a number is -J- of it ? Analysis. — Since any number equals 100% of itself, i of a number must equal i of 100%, or SS^^i, Ans. 2. What per cent of a number is|-? Ts|? f? |? 3. What per cent of a number isf? J? f? f? 4. What per cent of a number is^? ^''g-? ^? f? Written Exercises. 452. To change a Common Fraction to an equivalent per cent. 1. What per cent of a number is -§-§- of it ? • Analysis. — Every number is equal to 100 % operation. of itself; hence, f^ of a number = f^ of 100%, |A = 24 -r- 60 or eV of 2400%. Therefore, annexing ciphers QQ ) 24.00 ( .40, A71S. to the numerator and dividing by the denom- oaq inator, we have .40 or 40%. Hence, the KuLE. — A7vne.v ciphers to the numerator, and divide it by the denominator. (Art. 249.) 2. What per cent of a number is ff of it ? Ans. .625, or 62-1-;;. 3. What per cent of a number is |4 oJ^ i^ ^ Is -|f ? 192 Percentage, 4. What per cent of a number is -^-^ of it ? Is ^ ? 5. What per cent of a number is ff ? Is -^^q-? 6. What per cent of a number is ^^o of it ? Is f f f ? 453. The Parti or Elements employed in calculating per- centage are the Base, the Rate per cent, the Percentage, and the Amount or Difference. 454. The Base is the number on which the percentage is calculated. 455. The Rate is the number of Imndredtlis of the Mse taken. 456. The Percentage is the part of the base indicated by the rate per cent. Thus, when it ia said that 4% of $50 is $2, the rate is .04, the base $50, and the percentage $2. 457. Tlie Amount is the stim of the base and percentage. 458. The Difference is the base less the percentage. Thus, if the base is $75 and the percentage $4, the amount is $75 + 4 = $79 ; the difference is $75- $4 = $71. The relation between these parts is such, that if any two of them are given, the other three may be found. PROBLEM I. Oral Exercises. 459. 1. What is h% of $60 ? Analysis. — 5% of a number equals y^^, or ^^ of the number, and 4x of $G0 is $3. Therefore, ^% of $60 is $3. How much is How much is 2. 4^ of $80 ? 7. 121% of 320 rods ? 3. Q% of 1100 ? 8. 20^ of 275 gallons? 4. 1% of 1200 ? 9. 25^ of 260 acres ? 5. 8^ of $400 ? 10. 50%' of 700 men ? 6. 10% of $250 ? 11. 100/^ of $2000 ? Percentage, 193 12. A teacher who received $30 a month, had her salary increased V)% ; what was the increase per month ? 13. From a cistern of water holding a hogshead Z?>\% leaked out \ how many gallons remained ? Written Exercises. 460. To find the Percentage when the Base and Rate are given. 1. What is 9^ of 13465 ? OPKEATION. Analysis. — 9% of a number equals yf^, or .09 of it ; $3465 B. therefore, the percentage must be .09 times $3465, whicli qq -p is equal to $311.85, Ann. Hence, the ' — $311.85 P. EuLE. — Multiply the hase by the i^ate, expressed in decimals. Formula. — Percentage = Base x Rate. Notes. — 1. When the rate is an aliquot part of 100, i\ie percentage may- be found by taking a like part of the base. (Art. 447.) Thus, for 20%, take i ; for 35 % , take \, etc. 2. When the hase is a compound number, the lower denominations should be reduced to a decimal of the highest ; or the whole number to the lowest denomination mentioned ; then apply the rule. (Ex. 5.) 3. Finding a per cent of a number is the same as finding z. fractional part of it. (Art. 226.) 2. What is 31% of 1546 pounds 8 ounces ? A71S. 572.205 pounds. Find the percentage of the following : 3. 25^ of $5068. 9. 50^ of £2436. 4. 42^ of £6248. 10. 12^^ of $2874. 5. 75^ of 8675 bu. 3 pk. ii. 22^5^ of 865 acres. 6. 100^ of 2240 pounds. 12. 4:2^% of 84820. 7. 61^ of $1000. 13. 62^^ of 4360 feet 6 in. 8. 371^ of $1568. 14. 33^^ of $564175. 15. Which is greater, 7 per cent of $6300, or 6 p^r cent of $7200 ? 9 194 Percentage, 16. Which is less, 9 per cent of 182000, or 6 per cent of $93000 ? 17. A man had 18750 in bank and drew out 8^ of it at one time, and then 10^ of the remainder ; how much had he left on deposit ? 18. A man who owed 19584 failed in business and paid 40^ of his debts ; how much did he pay ? 19. A land speculator paid $6075 for a farm, and sold it at \h% less than cost ; how much did he lose ? 461. The Amount is found by adding the percentage to the base. 462. The Difference by subtradmg the percentage from the base. ,-, j Amomit = Base + Percentage. I Difference = Base — Percentage. 20. A began business with $4200 capital, and increased it 7 per cent the first year ; what amount of caj^ital did he then have ? Solution.— $4200 X .07 = $394.00, and $4200 + $294 = $4494, Ans. 21. B commenced business with 16500 capital, and lost 6 per cent of it the first year ; how much capital had he then ? Solution.— $6500 x .06 = $390.00, and $6500- $390 := $6110, Ans. 22. A man sold his house, which cost liim $5760, at 12^^ above cost ; what amount did he receive for his house ? 23. A farmer raised 4256 HI. of grain, and sold 12^^^' of it ; how many hektoliters did he have left ? 24. What is the amount of $252500 increased by 20^ of itself ? 25. A commander having an army of 16293 men, lost 33^^^ of them by sickness and desertion ; how many soldiers remained ? 26. A farmer owning 3500 sheep, lost 50 per cent of them by disease ; how many had lie left ? 27. If my annual income is 13560, and I spend 25^ of it each year, how much shall I save in 4 years ? Percentage, 195 PROBLEM II. Oral Exercises. 463. 1. A farmer had 100 sheep and lost 50 of them ; what part of them did he lose ? How many hundredths ? How many per cent ? Analysis. — 50 is equal to -^§i^, or 1 half ; and since per cent means hundredths, -^^^ equals 50 per cent. 2. A man spent $25 for a suit of clothes, which was |- of his money ; what per cent of his money did he spend ? 3. A pupil missed \ of his questions; how^ many hundredths did he miss? How many per cent ? 4. What part of $12 is 13 ? What per cent ? 5. What per cent of 20 is 7 ? Analysis.— 7 is ^ of 20 ; and 20 is 100;^ of itself. Now J^ of 100 /o is 5%, and ^^ of 100;:^ is 7 times 5, or 35^. What per cent What per cent 6. Of $25 are $8 ? 13. Of $24 are $18 ? 7. Of $10 are S9 ? 14. Of 20 pears are 12 pears ? 8. Of 5 is 3 ? 15. Of 25 gal. are 16 gal. ? 9. Of 16 is 4 ? 16. Of 50 lb. are 45 lb. ? 10. Of 36 is 9 ? 17. Of $50 are $12i ? 11. Of 63 is 31i ? 18. Of $1 are 6^ cents ? 12. Of 16f is 81 ? 19. Of $1 are 33i cents ? 20. If you pay $5 for the use of $50 for a year, what per cent do you pay ? 21. What per cent of 30 kilograms are 6 kilograms ? 22. If a pint of water is added to a gallon of milk, what per cent of it is water ? 23. If a man earns $80 a month and spends $30, what per cent does he spend? 196 Percentage. Written Exercises. 464. To find the Hate when the Base and Percentage are given. 1. What per cent of $63 is 142 ? Analysis.— Percentage is the product of the hase operation. and the rate ; therefore, the percentage $42, divided by 63 ) 42.00 the base S63, gives .66 1, or 60 1 ;^, the rate. Hence, the Ans, .^^^ Rule. — Divide the percentage hy the hase. Formula. — Rate = Percentage -^ Base. 2. What % of £18 is 15s. Ans. 4^%- 3. What % of 96 meters is 28 meters ? 4. What % of $18 is 12 cts. ? 5. iVhat % of 168 is 15 ? 6. What % of 275 is 18 ? 7. What % of 15 is 5} dimes ? 8. What % of 4 ton is ^ ton and 16 ponnds ? 9. Henr}^ spelled 225 words out of 250, and his sister 235 ; what per cent of the words did each spell correctly ? 10. From a cask of kerosene containing 52 gal., 6 gal. 2 qts. leaked ont; what per cent of it was lost ? 11. A farmer haying 250 bu. of wheat, sold | of it ; how many bnshels and what per cent did he sell ? 12. A man worth $12500, beqneathed $3125 to bis wife and the rest to his 3 children ; what per cent of it did his wife have, and how much had each child ? 13. What per cent of 365 days are 30 days ? 14. Of 1880 years are 4000 years ? 15. Of 27 lb. Avoir, are 12 oz. ? 16. Of 125 miles are 250 rods ? 17. Of 88 kilograms are- 75 grams ? 18. If a man owns f of a ship and sells | of her, what per cent of his part does he sell ? 19. What per cent of 75 bu. 3 pk. are 50 bu. 2 pk. ? 20. A man gave 19863 to 3 charities ; to the first $2500, to the second $4500 ; how much was left for the third and what per cent did each receive ? Percentage, 197 PROBLEM III. Oral Exercises. 465. 1. $48 are %% of what number ? Analysis. — Since $48 are 6 5^ of the number, 1% is ^ of $48, which is 8, and 100^ is 100 times 8, or $800, Am. 2. 24 is 4^ of what ? 7. 40 gal. are 20^ of what ? 3. 32 is h% of what ? 8. $68 are 12^ of what ? 4. 48 is 20g of what ? 9. 25 yd. are 40%' of what ? 5. 12^^ is 10^ of what ? 10. 12d. are 30^ of what ? 6. 6} is 25^ of what ? 11. 25 doz. are 12^^ of what ? Written Exercises. 466. To find the Base when the Rate and Percentage are given. 1. 192 is 25^ of wliat number ? Analysis. — Percentage is the product of the .25 ) 192.00 P. base by the rate. The base 192-^.25 = 768, the . — T base required. Hence, the ^^^*'- "^^ ^' Rule. — Divide the percentage hy the rate. Formula. — Base = Percentage -^ Rate. 2. 84 is 12|-/^^ of what number? Ans. 672. (Art. 447.) 3. 96 = ^d\% of Avhat ? 9. 31.25 = 12^% of what ? 4. 234 = 10^ of what ? 10. 60 cts. = 1% of what ? 5. £240 = 7% of what ? 11. $100 = 1% of what ? 6. 62.5 = 61^ of what ? 12. $42.30 = \% of what ? 7. 60 yd. = 1% of what ? 13. 94 = Vd{)% of what ? 8. 78 = 25;:^ of what ? 14. 58^ = 125% of what ? 15. The number of children of age to attend school is 862, which is 20;^ of the population ; what is the whole population ? 16. A man sold a house, making $360, which was b% more than it cost him ; what did he pay for the house ? 17. 4.% of $230 is b% of what ? "^ 12|-/^^' of $530 is 6^% of what ? 18. A man paid a war tax of $73.50, which was 2% on the value of his property ; what was he worth ? 198 Percentage, PROBLEM IV. Oral Exercises. 467. 1. A man sold a cow for 140, which was 25^ more than she cost him ; what did he pay for her ? Analysis. — $40 is the cost increased by 25% of itself : and since the cost is \%% of itself, $40 must be \%%, or f of the cost. Now, as $40 — | of the cost, \ is \ of $40, which is $8, and f are 4 times 8, or $32, Ans. 2. What number increased by 25^ of itself, is 100 ? 3. A furniture dealer sold a bureau for $20, which was 10^ more than it cost him ; how much did it cost him ? 4. What number plus 12^^ of itself amounts to 96 ? . A grocer sold a barrel of apples for $5.50, and gained 20^ on the sum it cost him ; what did he pay for it ? 6. A jeweller sold a watch for $150, which was 50^ more than it cost him ; what did he pay for it ? 7. What number diminished by 25^ of itself is 60 ? Analysis. — As 60 is the number after it is diminished, 60 must be 100% —25fo = iVo> or f of the number. Now if 60 is f of the number, ^ of it is 60-^3 = 20, and 4 fourths are 4 times 20, or 80, Ans. 8. What number diminished by 20;^ of itself is 48 ? 9. A pupil answered on examination 45 questions correctly, which was 10^ less than the number asked him ; how many were asked him, and how many did he miss ? Written Exercises. 468. To find the Base when the Amount or Difference, and the Rate are given. 1. What number increased by 25^ of itself is 3500 ? Analysis. — Since 3500 is the number after it is 1 -j- .25 = 1.25 increased by 25% of itself, 3500 must be 125% of 1.25 ) 3500.00 the number, or 1.25 times the number, and 3500 — -125 = 2800, Ans. ^^^- ^^00 2. What number diminished by 20^ of itself is 2560 ? Analysis.— Since 2560 is the number after it is ■'• .-vO =: .80 diminished by 20% of itself, 2560 must be 80%, or .80 ) 2560. 00 .80 times the number, and 2560-T-.80 = 3200, Ans. Ans 3200 Percentage, 199 469. From- the operations above, we derive the following Rule. — Divide the amount hy 1 increased hy the rate. Or, Divide the difference hy 1 diminished hy the rate, ^ „ ( Amount -^ (1 + Rate), Formulas. — Base = \ ^..^, ' . ^ , . ( Difference -^ (1 — Rate). What number increased What number diminished 3. By !()% of itself is 5342 ? 9. By 25% of itself is 3900 ? 4. By %% of itself =2418 ? 10. By Q% of itself =2100 ? 5. By 105^ of itself =28600 ? ii. By 12% of itself =1200 ? 6. By 16% of itself =2552 ? 12. By 15% of itself =2300 bu.? 7. By 20% of itself = .$3720? 13. By 7|% of itself =$6475 ? 8. By 28i% of itself =18995 ? 14. By 12^% of itself =13125 ? 15. At thQ end of the year, a merchant's stock was 18400, which was 17% more than his capital ; what was his capital? 16. A man sold his house for $2700 and lost 12 J % ; what did the house cost him ? 17. A grocer sold 950 barrels of flour for $5760, which was 20% advance on the cost ; Avhat was the entire cost, and the cost per barrel ? 18. A provision dealer sold 800 barrels of beef for 112000, which was a loss of 25% ; what Avas the whole cost, and how much per barrel ? - 470. Percentage is applied to two classes of problems. First. — Those which are independent of Time ; as. Profit and Loss, Commission and Brokerage, Insurance, Taxes, Duties. Second. — Tliose in which Time is an element ; as, Interest, Discount, Equation of Payments, Averaging Accounts, Stocks and Exchange. Note. — lu applying the Principles of Percentage to these subjects, the pupil should carefully observe what elements or parts are given and what required in each example, and then apply the corresponding rule or formula. 200 Percentage. Profit and Loss. Oral Exercises. 471. 1. A man paid $60 for a watch, and sold it at V)% above the cost ; how much did he gain ? Analysis.— He gained 10% of $60. Now 10% of a number is yVo. or yV ; and iV of $60 is $6. Therefore, etc. 2. If a man pays $40 for a cow, and sells her at 20^ advance, what will be his profit ? 3. A jockey bought a horse for $80, and sold it at a loss of h%\ how much did he lose? 4. A man having 120 acres of land, bought 25^ more ; how many acres did he buy ? 5. What part of a number is 12J;^ of it? 6. What is \2\% of 32 ? Of 48 ? Of 96 ? 7. What is 6^^; of 32 ? Of 64 ? Of 80 ? 8. What is 33^;^ of $15 ? Of $50 ? Of $60 ? 9. A merchant sells flannel at a profit of 10 cts. on a yard, and gains 12-|-.^ ; what is the cost ? Analysis. — V^\fo — \ ; lience, 10 cts. = \ the cost ; and | are 8 times 10 cts., or 80 cts., Ans. 10. A farmer lost $32.40 on a reaping machine, which w^as 33 J-^ of the cost ; what was the cost ? 11. A goldsmith sold a watch at 25% profit, and made $26 ; what was the cost ? 12. A tradesman sold out his stock of goods for $2760, which was 8^^ less than he paid ; what did they cost him ? 13. A grocer sold strawberries at 15 cts. a liter and made 20% ; what did he pay for them ? 14. A fruit dealer sold a barrel of apples for $1.50, which was a loss of 50% ; what did he pay for them ? 15. A newsboy sells papers at 5 cts. apiece, and makes 100% ; what does he pay for them ? 16. A man sold his house for $7500, which was 33^% more than he paid for it ; required the cost? Profit and Loss. 201 Written Exercises. 472. Profit and Loss denote the gain or loss in business transactions. They are calculated by percentage. The cost is the iase ; the per cent of gain or loss, the rate ; the gain or loss, the 2)ercentage ; the selling price, the amount or difference. 473. To find the Profit op Loss. (Art. 460.) Formula. — Profit or Loss = Cost x Bate. 1. A house bought for 15860 was sold for 23% above cost ; what was the gain ? 2. A grocer bought a cask of oil for $96.50, and retailed it at a profit of 6 per c&nt ; how much did he make on his oil ? 3. A pedlar bought a lot of goods for $2150, and retailed them at 25 per cent advance ; how much was his profit ? 4. A merchant bought a cargo of coal for $450, which he sold for 12-J- per cent less than cost ; what was his loss ? 5. What is the loss on a piano that cost $1260, and sold at 20^ loss ? 6. What was the gain on a form that cost $3585, and sold at a profit of 12^/^' ? 7. What is the profit on wool which cost $2538 and sold at an advance of 15^? 8. A dealer bought a quantity of grain for $1375, and sold it for S% profit ; what amount did he receive ? (Art. 461.) 9. A young man having $2750, lost 35^ of it in speculation ; how much had he left ? 10. Bought a quantity of produce for $989.33, which I sold at 20% loss ; how much did I receive for it ? 11. A drover bought a flock of sheep for $2275, and sold them at 25*^ advance ; for how much did he sell them ? 12. A merchant had a quantity of groceries on hand, which cost him $367.13 ; to close up his business he sold them at 15^ less than cost ; how much did he get for them ? 13.- A man bought a farm for $875, and was offered 33^ advance for his bargain ; how much was he offered ? 202 Percentage. 14. A merchant bought a cargo of cotton for 130000 ; the price declming, he sold it at %Y/o less than cost ; for how much did he sell it ? 474. To find the Rate of Profit or Loss. (Art. 464.) FoEMULA. — Rate = Profit or Loss -^ Cost. 15. A dealer bought a span of horses for 1450, and sold them for IGOO ; what per cent was his profit ? 16. A mowing machine was sold for $175, which cost $225 ; what per cent was the loss ? What is the rate per cent profit 17. On coffee bought at 25 cts. and sold at 30 cts. ? 18. On tea bought at 55 cts. and sold at 67 cts. ? 19. On starch bought at 10 cts. and sold at 13 cts. ? 20. On goods sold at double the cost ? 21. On goods sold at H the cost ? 22. A merchant bought a quantity of goods for $155.63, and sold them for $148.28 ; what per cent did he lose ? 23. A gentleman bought a house for $3500, and sold it for $150 more than he* gave ; what per cent was his profit ? 24. A speculator laid out $7500 in land, and afterwards sold for $10000 ; what per cent did he make ? 25. A merchant bought $10000 worth of wool, and sold it for $12362 ; what per cent, and how much was his profit ? 475. To find the Cost. (Art. 466.) Formula. — Cost = Gain or Loss -^ Rate. 26. The loss on a cargo of lumber was $1260, which was 23;^ of the cost ; what was the cost ? 27. A speculator gained $3748 in land, which was 22;^ of the cost ; required the cost ? 28. An importer made $3900 on a cargo of goods, wdiich was 16^^ of the cost ; required the cost ? 29. If a grocer pays $3584 for a cargo of flour, for how much must he sell it to gain IQ^';^ ? (Arts. 460, 461.) Profit and Loss, 203 30. A mercliant paid 18500 for a case of silks ; at what price must he sell it to lose 18^ ? 31. A merchant bought butter for $322.75 ; for how much must he sell it to gain 15^ by his bargain ? 32. Bought tea for $437.50; for how much must I sell it, to make 1S% by the operation ? 33. What is the selling price of hay bought for $845 and sold at 1Q% gain ? 34. What is the selling price of land costing I18G8.25 and sold at 12|-^ loss ? 35. What is the selling price of goods costing 82576.40 and sold at 331^ profit? 36. What is the selling price of furniture costing $1848.75 and sold at a loss of 8^%' ? 476. To Find the Cost from the Selling Price and the Rate per cent of Profit or Loss. 37. A manufacturer sold a carriage for $432, which was 20^ above cost ; what was the cost ? Analysis.— $432 is the cost, plus 20% of itself; hence, the cost was $432 H- (1 + .20) = $360, Ans. (Art. 461.) 38. Another carriage sold for $432, which was 20^ less than cost ; Avhat was the cost ? Analysis. — $432 is the cost, minus 20 % of itself ; hence, the cost was $432 -^ (1 - .20) = $540, Ans. (Art. 462.) Hence, the ^ ^ , ( SelUnq Price -^ (1 -}- Rate of Gain). Formulas. — Cost — { ^ ^-,. ^ . )^ -n ± \c r \ \ Setting Price -f- (1 — Kate of Loss). 39. A farmer sold land for $86.50 a hektar, and made V2% ; what was the cost ? 40. A merchant sold a bill of goods for $675}, and made lOJ^ profit ; what did he pay for the goods ? 41. A drover sold cattle for $1750, which was a profit of 12^^ ; what did they cost him ? 42. A dealer sold 525 hektoliters of gTain for $2750, which was a loss of 15^ ; what was the cost ? 204 Percentage. Commission and Brokerage. 477. A Commission Merchant, Agent, or Factor is a person who buys or sells goods or transacts business for another. 478. Commission is the Percentage allowed the agent on the money invested or collected. 479. A Broker is one who buys and sells Stocks, Bills of Exchange, etc., and his commission is called Brokerage. 480. A Consignment is Goods sent to an agent to sell. The Consignor is the person sending them. The Consignee is the person to whom they are sent. 481. The Net Proceeds are the gross amount of a business transaction, mimes the commission and other charges. 482. The computation of commission and brokerage is the same as Percentage ; the money employed being the dase ; the per cent for services, the rate ; the commission, the percentage. 483. 1. Find 4^% com. on sales for 13468. (Art. 460.) 2. An agent sold a house for 17265 ; what was his commis- sion at 1^% ? 3. Find b^% com. on 375 bbl. apples, sold at $2.25 a barrel. 4. Find ^Y/o com. on a ton of wool, at 87|^ cts. a pound. 5. A commission merchant sold goods amounting to l>7468, at b% for commission and guaranty. How much did he re- ceive, and how much did he pay the owner ? 6. An auctioneer sold a farm for 1^12482, and charged 3|-^ com., and $50 for advertising it. What was his whole bill, and what the net proceeds ? 7. When a commission of 1150, at QY/'c is received for goods sold, what is the amount of sales ? (Art. 466.) 8. When the commission, at ^V/o is $294 ? 9. When the commission, at 6^' is $105 ? 10. When the commission, at lY/o is $270 ? 11. An auctioneer charged $405 for selling a saw-mill, which was 1^%', for what did he sell it and what did the owner receive? Commission and Brokerage. 205 12. A commission merchant charged \\% com., and ?>\% for guaranty ; he received 1105.30. What were the net proceeds ? 484. To Find the Amt. of Sales from the Net Proceeds and Rate. Formula. — Amount of Sales == Net Proceeds h- (1 — Rate). 13. The net proceeds of goods sold were $4845, and the agent charged 2J% commission and 2^% for guaranty. What was the amount of sales ? Ans. $5100. 14. When the net proceeds are 1229,80 and the rate 3^, what is the amount of sales ? 15. My agent charged 1^% commission and $62.40 expenses for selling my house, and sent me $15250. For how much did the house sell ? 485. To find the sum to be invested, after deducting the per cent commission from the amount remitted. 16. If $7098 are remitted to an agent to buy cotton, after deducting 4^^ com., how much will be left to be invested ? Analysis.— The money remitted includes both the operation. commission and the investment. The money invested 1.04 ) $7098 is 100 % of itself, and 100 % + 4 % =104 % . Therefore, ^ ~$6825 $7098 ^ 1.04 = $6825, the money to be invested. Hence, the FoEMULA. — Su7?i Invested = Remittance ^ (1 + Rate). 17. When the remittance is $1623.10, and the commis- sion 2|-5^, how much remains to be invested ? 18. When the remittance is $4454, and the commission 2^^? 19. When $4908 are sent, and the commission is ^% ? 20. How many apples at $2 a barrel can be bought for $6720.80, after deducting b% commission ? 21. Sent an agent $50000 to buy a ship. How much did the owner receive after deducting 1^% commission ? 22. How many buffalo robes at $5 each can be bought with a remittance of $2575, after deducting 3^ commission ? 23. A college sent an agent $10250 to be invested in a library ; how much remained after deducting 2^% com- mission ? 206 Percentage, Insurance. 486. Insurance is security against loss. 487. Fire Insurance is security against the loss of property by tire. 488. Marine Insurance is security against the loss of property at sea. Note. — Risks of transportation partly by land and partly by water, are called Transit Insurance. The same policy often covers both Marine and Transit Insurance. 489. Accident Insurance is security against loss by accidents. 490. Health Insurance secures a certain sum during sickness. 491. Life Insurance secures a stated sum to the heirs and assigns of the insured in case of death. 492. The parties who agree to make good the loss, are called Insurance Companies or Underwriters. Note. — When only a part of the property insured is destroyed, the underwriters are required to make good only the estimated loss. 493. The Premium is the sum paid for insurance. 494. The Policy is the ivritten contract between the insurers and the insured. 495. Insurance Companies are of two kinds : Stock Com- panies and Mtitual Com2)anies. 496. A Stock Company is one which has a paid up capital, and divides tlie profit and loss among its stockholders. 497. A Mutual Company is one in which the losses are shared by the parties insured. 498. Insurance is calculated by Percentage ; the sum msured being the base ; the 2)67' cent premium, the rate; the 7;r^;;?/«?7Z itself, the percentage. Insurance. 207 Mental Exercises. 499. 1. How much must be paid for insuring a house for $5000, at ^% premium ? Analysis. — Since the premium is i%, the sum paid must be ^% oi $5000. Now 1 % of $5000 is $50, and |-% is i of $50, or $25, Ans. 2. Find the annual premium of insurance, at 1\% on a store and goods valued at $800. 3. I paid $8 for insuring $400 ; what was the rate ? Analysis. — As the premium on $400 is $8, the premium on $1 is ^-^ of $8 which is $.02, or 2% , Ans. (Art. 464.) 4. Paid $15 for insuring $1000 ; required the rate? 5. What amount of insurance, at %% can be obtained for $40 ? Analysis. — Since 2^ is jf „ or 5^ of the amount insured, the premium is 5V of t^^s amount ; and $40 is 5V of 50 times $40 or $2000. (Art. 466.) 6. What amount of insurance, at 4^ can be obtained on a vessel for $100 ? Written Exercises. 500. 1. What is the premium at 2^^^ for insuring $16000 ? 2. What is the premium for insuring a store and goods valued at $7500, at l^% ? 3. What is the premium for insuring a house and furniture valued at $65000, at i% ? 4. If $72 are paid for insuring $4800, what is the rate ? 5. If $420 are paid for insurance on $18000, what is the rate ? 6. If $860 are paid for insurance of $1720, what is the rate ? 7. A merchant paid $157.80 to insure his store, at 1^%', what amount did he insure ? 8. Paid $187 to insure half the value of a ship at 2f^; what was the total value of the ship ? 501. To find the sum to be insured to cover the value of the goods and premium. 9. Bought goods in London for $7194. What sum insured, at 'd^% will cover the value of the goods and the premium ? 208 Percentage, Analysis. — The bill is 100% of itself, and the premium is Z\% of that sum; therefore, $7194 - 100% -S] % - 96| %, or .9675 times the sum; now $7194-f-.9675 = $7435.658, the sum required. (Art. 468.) FoEMULA.— /S'zfw insured == Value -f- (1 — Rate). 10. If a store and goods are worth $16625, what sum must be insured, at ^% to cover the property and premium ? 11. What sum must be insured, at 2|% on a consignment of tea which cost $352.50 to cover property and premium? 12. A dealer shipped 1000 bbls. flour worth $6-|- a bbl. ; for what sum must he take out a policy, at 2-|%' to cover the value of the flour and the premium ? Life Insurance. 502. Life Insurance Policies are of cliff ere7it kinds, and the premium varies according to the ex])ectation of life. 503. Life Policies, are payable at the death of the party named in the policy, the annual premium continuing through life. 504. Term Policies are payable at the death of the insured, if he dies during a given term of years, the annual premium continuing till the policy expires. 505. Endowment Policies are payable to the insured at a given age, or to his heirs if he dies before that age, the annual premium continuing till the policy expires. Note. — The expectation of life is the average duration of the life of individuals after any specified age. 13. What premium must a man, at the age of 27, pay annually for a life policy of $4500, at 4J^ ? 14. What is the annual premium on $5000, at h^%, and what Avill it amount to in 20 years ? 15. A man took an endowment policy of $25000 for 20 yrs., at b%. Which was the greater, the sum paid or the sum insured? Taxes, 209 Taxes. 506. A Tax is a sum assessed upon the person, property, or income of citizens, for public purposes. 507. A Property Tax is a tax upon property. 508. A Personal Tax is a tax upon the person, and is called ixp)oll or capitation tax. Note. — The term poll is from the German polle, the head ; capitation, from the Latin caput, the head. 509. Property is of two kinds, personal and real estate. 510. Personal Property is that which is movable ; as, money, stocks, etc. 511. Real Estate is that which is fixed ; as, houses and lands. 512. Assessors are persons appointed to make a list of taxable property and estimate its value for the purpose of taxation. 513. Property taxes are computed by Percentage. The valuation of the property is the Base, The tax on $1 is the Rate. The net sum to be raised, the Percentage. 514. To assess a Property Tax, when the sum to be raised and the valuation of the property are given. 1. A tax of $12500 is to be raised in a town the property of which is valued at $1500000, and there are 250 polls, each taxed at 12; what is the rate of the tax, and what is A's tax whose real estate is valued at $6000, and personal at 13000 ? Analysis.— The sum to be raised is $12500 operation. less $500 on the polls, which is equal to |12000, Town tax $12500 and $12000 -=- $1500000 = $.008, ^j^'/c, or 8 ^q\\ ' ii 599 mills. A's property is $6000 + |3000 = |9000. As 1500000 ) $12000 he pays 8 mills on $1, on $9000 he pays 9000 Rate .008 X . 008 = $72, and $72 + $2 (his poll tax) = $74. Ans. The rate is 8 mills or xV/^> ^^^^ ^^ *^^ $'^'^- Hence, the 210 Percentage, Rule. — /. From tlw sum to he raised subtract the poll tax and divide the remainder by the amount of taxable property ; the quotient will be the rate. II. Multiply the valuation of each man's property by the rate, and the product plus his poll tax will be his entire tax. Notes. — 1. If a poll tax is included, the sum arising from the polls must be subtracted from the sum to be raised, before it is divided by the value of taxable property. 2. The computation of taxes may be shortened by finding the rate, and giving the tax on $1 to $10, etc., as in the following ) Tax Ta b l e. 515. Showing the tax on various sums at the rate of 8 mills on Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. $1 $0,008 $7 $0,056 140 10.32 $100 $0.80 2 0.016 8 0.064 50 0.40 200 1.60 3 0.024 9 0.072 60 0.48 300 2.40 4 0.032 10 0.08 70 0.56 400 3.20 5 0.040 20 0.16 80 0.64 500 4.00 6 0.048 30 0.24 90 0.72 1000 8.00 2. Find by the table B's tax whose property is valued at $7256, and who pays for 3 polls at $1.50. 3. Find O's tax on property valued at $9480, who pays for 3 polls at $1.25. 4. AVhat is D's tax on a valuation of $15676, and pays for 2 polls at $1.50 ? 5. A tax of $250000 is levied on a County whose real estate is valued at $3000000, and has 500 polls taxed at $2 each. Required the rate of tax, a tax table for that rate, and a per- son's tax whose property is valued at $5250, and who pays for 3 polls at $2 each. Duties or Oiisto7ns. 211 Duties or Customs. 516. Duties or Customs are taxes levied upon imported goods for revenue, or the encouragement of home industry. 517. An Invoice or Manifest containing a description of the goods and their cost in the country from which they are im- ported, is required by law to be exhibited to the Collector of the Port on the arrival of the ship. 518. Duties are either Ad valorem or Specific. 519. An Ad valorem Duty is a certain per cent laid on the cost of goods in the country from which they are imported. 520. A Specific Duty is a fixed sum laid on a given article or quantity, without regard to its value. 521. Before calculating specific duties, certain allowances are made called Tare, Leakage, and Breakage, Tare is an allowance for weight of box, bag, cask, etc. Leakage is an allowance for loss of liquids in casks. Breakage is an allowance for loss of liquids in bottles. 522. 1. What is the Specific didy on 75 hogsheads of alco- hol, at Is. per gallon, i.% leakage ? SOLUTION. The number of gal. = 63 x 75 = 4735 gal. The leakage at 4% = 4725 x .04 = 189 gal. The net gallons = 4725 - 189 = 4536 gal. The duty at Is. = 4536s. = $1103.7222, Arts. 2. What is the specific duty, at $0.75 a meter, on 150 pieces of broadcloth, each containing 45 meters ? 3. What is the specific duty on 182 cases of shawls, containing 75 each, at $1.50 per shawl ? 4. What is the duty, at 5 cts. a pound, on 400 sacks of coffee, each containing 63 lb., the tare being 2^ ? 212 Percentage, 5. What is the Ad Valorem duty on 2^ doz. clocks, invoiced at $32.75, and 5 doz. watches invoiced at $45J-, at 25;?^ ? Analysis.— The cost of 30 clocks = $32.75 x 30 = $982.50 The cost of 60 watches = $45.25 x 60 = $2715.00 The cost of both = $.B697.50 The ad valorem duty at 25% = $8697.50 x .25 = $924,375 6. What is the ad valorem duty, at 33|^^, on 150 chests of tea, each weighing 60 lb., and invoiced at 48 cts. a pound, the tare being 5 lbs. a chest ? 7. Find the ad valorem duty, at ^\%i on 110 boxes of raisins, 25 lb. in a box, invoiced at $0.12 a pound, the tare being 3^ lb. a box ? 8. At Vl\% what is the ad valorem duty on 5250 kilograms of Eussia iron, invoiced at 75 cts. a kilogram. Qu EST! ONS. 444. What does per cent mean ? 447. How expressed ? 452. How change a common fraction to a per cent ? 454. What is the base ? 455. The rate per cent ? 456. The percentage ? 457. The amount ? 458. The difference ? 460. How find the percentage when base and rate are given ? 461. How find amount ? 462. Difference ? 464. How find rate from base and per- centage ? 468. How find base from the rate and amount or difference ? 472. What are profit and loss ? What are the corresponding parts ? 473. How find the profit or loss ? 474. The rate ? 475. The cost ? 476. How find cost from selling price and rate ? 478. What is commission? 479. Brokerage? 482. How calculated? Corresponding parts ? 485. How find the sum to be invested after deducting commission ? 486. What is insurance? 493. The x^reraium ? 494. Policy? 498. How calculated ? 501. How find sum to be insured to cover loss and premium ? 506. What are Taxes ? 513. How computed ? Corresponding parts ? 516. What are duties or customs? 519. Ad valorem? 520. Specific? 521. What deductions are made in specific duties ? What is tare ? Leakage ? Breakage ? .•:t — o - -- i' <— — - NT EREST (S-i^ ■^^ 523. 1. How much must I pay you for the use of 1100 for 1 year at Q^c, and what shall I owe you at the end of the year ? Analysis.— 6% is jf o ; lieiice, I must pay you jf ^^ of $100, or $6, for Its use. Again, I shall owe you at the end of 1 year the sum borrowed together with $6 for its use, and $100 + $6 = $106, the amount due. Note, — In this solution four elements or parts are considered, called the Interest, the Principal, the Per cent, and the Amount. Definitions. 524. Interest is the money paid for the use of money. 525. The Principal is the money for which interest is paid. 526. Tlie Rate is the per cent of the principal, paid for its use 1 year, or a specified time. 527. The Amount is the stmt of the principal and interest. 528. Simple Interest is the interest on the principal only. 529. Legal Interest is the rate established by law. 530. Usury is a Jiigher than the legal rate. 531. Interest differs from the preceding applications of Percentage only by introducing time as an element in connec- tion with the 7'ate per cent. 532. The Principal is the Base ; the Per cent per annum is the Rate ; the Interest is the Percentage ; the Sum of principal and interest, the Amount. 214 Percentage. Ta b le. 533. Showing the legal rates of interest in the several States, compiled from the latest official sources. states. Rate %. States. Bate %. States. Rate %. States. Rate i. Ala Ark.... Arizona Cal Conn. . . Colo. . . . Dakota. Del Fla Ga Idaho. . . Ill 8 6 10 Any*: 7 'Any. 6 1 10 'Any. 7 ; 12 6 ' 8 Any. 7 Any. 10 24 6 8 Ind. .. Iowa. . Kan Ky La Maine . . Md.. .. Mass . . . 1 Mich Minn. . . Miss.. . . Mo 6 6 7 6 5 6 6 6 7 7 6 6 8 10 12 8 8 Any. Any. to 12 10 10 Montana N. H. . . . N.J N. Mex.. N. Y.... N. C... Neb Nev Ohio.... Oregon . Penn R.I 10 6 6 6 6 6 7 10 6 10 6 6 Any. 12 8 10 Any, 8 12 Any. S.C Tenn . . . ! Texas . . Utah. . . . Vt Va W. Va.. W. T. . . . Wis Wy D.C.... 7 6 8 10 6 6 6 10 7 12 6 Any, 12 Any. 8 Any. 10 Any. 10 1 534. In computing interest, a legal year is 12 calendar months. Oral Exercises. 535. 1. What is the interest of 140 for 1 year at 5^ ? Analysis. — At 5%, the interest for 1 yr. is -^%q of the principal, and yfo of $40 = $2, Ans. 2. What is the int. of 150 for 1 yr. at 5^ ? 2 yr. ? 5 yr. ? 3. Of 1100 for 1 yr. at % ? At 8^ ? At 7% ? 4. Of $200 for 2 yr. at 7%' ? At 4% ? At 8% ? 5. Of 1500 for 2i yr. at 6% ? At 6^^? At 10^ ? 6. Of $400 for 3 yr. at 6% ? At 6^ ? At lOf^ ? 7. What part of a year is 6 months ? 4 mo. ? 3 mo. ? 2 mo. ? 1 mo. ? 8 mo. ? 7 mo. ? 9 mo. ? 10 mo. ? 11 mo. ? 12 mo. ? 8. What part of 1 year's interest is tlie interest on the same, sum for 6 mo. ? For 3 mo.? For 4 mo. ? For 2 mo. ? 9. At 4:%, what is the interest of $600 for 1 yr. and 6 mo. ? 10. Calling a month 30 days, what part of 1 mo. is 15 days ?' Is 10 days ? 6 days ? 5 days ? 3 days ? 2 days ? 1 day ? 11. If the interest on a sum for 1 year is $48, what is it for 1 month ? For 3 months? 5 months? 7 months? * By special agreement. no. 6 d. , at 1% ? $250 Prin. 17.50 Int. 1 yr. 3.1 Yr. $54.25 Int. $304.25 Amt. Interest 215 PROBLEM I. General Method. 536. To find the Interest and Amount, when the Principal, Rate, and Time are given. I. By the time expressed decimally in years. (Art. 403.) 1. What is the interest of $250 for 3 yr. 1 mo. 6 d., at What is the amount ? EXFLAKATION. The given principal The int. of $250, at 1% for 1 yr. is $250 x .07 1 mo. 6 d. = .1 yr. (Art. 403) ; hence, the time Int. for 1 yr. $17.50 x 3.1 gives int. for 3.1 yr. The amount = prin. $250 + $54.25 int. Hence, the Rule. — I. Multiply the principal by the given rate, and this product by the tii7ve expressed in years. II. vddd the interest to the principal for the amount. 2. Find the interest of $75.36 for 1 yr. 7 mo. 18 d. at 5%. What is the amount ? S0LUTI0N.-I75.36 X .05 x 1.63^ (time)= $6,154, Int. And $6,154+ $75.36 = $81,514, Amt. 3. What is the int. of $340.20, at 6%, for 2 yr. 8 mo. 12 d.? What is the amount ? 11. By Aliquot Parts. (Art. 280.) EXPLANATION. Taking example first, the given Principal is $250 Prin. For 1 yr. the int. at 7% is $250 x .07 = 17.50 Int. 1 yr. For 3 yr. the int. is $17.50 x 3 = $52.50 Int. 3 yr. For 1 month the int. is $17.50 -4-12 =^ 1.4583^ Int. 1 mo. For 6 d. (4 of 30 d.) the int. is $1.4583 V ^5 = .2916 | Int. 6 d. The entire int. = S52.50 + 1.4583^ + .2916| = $54.2500 Int. The prin. $250 + $54.25 interest = $304.25 Amt. 216 Percentage. 537. From the above illustrations we derive the following EuLE. — Foe oke year. — Multiply the principal hy the rate. For two or more years. — Multiply the interest for 1 year hy the number of years. . For months. — Tahe the aliquot part of 1 year's interest. For days. — Tahe the aliquot part of 1 month's interest. The entire interest is the sum of the partial interests. For the Amoui^t. — Add the interest to the principal. Notes. — 1. For 1 montli take ^^ of the interest for 1 year ; for 2 montlis, ^ ; for 3 months, |, etc. 2. For 1 day take 3^^ ^f the interest for 1 month ; for 2 days, ^^ ; for 6 days, \ ; for 10 days, i, etc. 3. In computing interest 30 days are commonly considered a month. Solve the following by either or both methods : 4. What is the interest of 1684 for 1 yr. 9 mo. 10 d. at 6^? 5. At ^%, what is the amt. of 11125 for 1 yr. 2 mo. 3d.? 6. At 5^, what is the amt. of $1056 for 10 mo. 24 d. ? 7. At 6^, what is the int. of $1340 for 1 mo. 15 d. ? 8. At 7^, what is the int. of $815 for 3 yr. 2 mo. 21 d. ? 9. At m, what is the amt. of $961 for 2 yr. 4 mo. 10 d. ? 10. AVhat is the int. of $3500 for 11 mo. 20 d., at 10^? 11. What is the amt. of $39,275 for 2 yr. 6 mo., at 12^^? 12. What is the int. of $113.61 for 5 yr. 5 mo., at 5%' ? 13. What is the int. of $1000 for 2 yr!^ 3 mo. 10 d., at 4J^? 14. What is the int. of $1260.34 for 10 yr., at ?>% ? 15. What is the int. of $234.56 for 2 yr. 4 mo. 5 d., at 6;^ ? 16. What is the amt. of $600 for 1 yr. 6 mo. 10 d., at h% ? 17. Find the amount of $60 for 7 mo., at %%. 18. What is the interest of $96 for 10 months, at ^% ? 19. At ^%, what IS the amt. of $700 for 1 yr. 2 mo. 12 d.? 20. At 4^, what is the amt. of $470 for 10 days? 21. Find the int. of $1000 for 1 yr. 1 mo. 1 d., at %%. 22. At 6^, what is the amt. of $4565.61 for 4 mo. 7 days ? Interest. 217 23. What is the interest of 15625.43 for 4 mo. 18 d., at 6J^ ? 24. At 54^, what is the int. of $624,625 for 7 mo. 3 days? 25. At S%, what is the int. of $11261.18f for 3 mo. 3 days? 26. At 7^0 what is the amt. of 89208.95 for 11 mo. 5 days ? 27. What is the amt. of $15206.843, at 1\%, for 1 year 8 months 25 days? 28. The amtf of $10050.69, at ^%, for 2 yr. 9 mo. 5 d. ? 29. What is the amt. of 811607.858, at 7^, for 3 years 6 months 9 days ? 30. The amt. of $41361.18, at 6^, for 5 yr. 7 mo. 3 d. ? 31. What is the interest on $1145 from July 20th, 1881, to Dec. 7th, 1881, at 7^^? (Art. 409, K 2.) Note.— The time is 4 mo. and 11 d. (July) + 7 d. (Dec.) = 4 mo. 18 d. 32. What is the interest on a note of $568.45 from May 21st, 1881, to March 25th, 1882, at b% ? 33. Required the amount of $2576.81 from Jan. 21st, 1881, to Dec. 18th, 1881, at 1%. Six Per Cent Method. Develofment of Pbincip l es, 538. The interest of $1 at 6% For 1 yr., or 12 mo., is 6 cts., = .06 of the principal. For -J- yr., or 2 mo., is 1 cent, ■— .01 of the principal. For ^ yr., or 1 mo., is 5 m., = .005 of the principal. For \ mo., or 6 d., is 1 m., = .001 of the principal. For ^Q mo., or 1 d., is -J m., = .000|- of the principal. Hence, we derive the following Principles. i°. Tlie interest of SI at 6%, is half as many cents as there are montJis m the given time. 2°, The interest of $1 at 6%, is one-sixth as 7nany mills as there are days in the given time. 10 « 218 Percentage. 539. To find the Interest, when the Principal, Rate, and Time are given. 1. What is the interest of $250.26 for 1 yr. 3 mo. 21 d., at 6^ ? What is the amount ? ExPLA. — The interest of $1 for 15 mo. = .075 operation. By 2°, int. of $1 for 21 d. = .0035 1250.26 Prin. Int. of $1 for 1 yr. 3 mo. 21 d. = .0785 .0785 Int. $1. As tlie interest of $1 for the given time and 125130 rate is $.0785, the interest of $250.26 must be o 00208 $250.26 X .0785 = $19.04541 interest. The prin. $250.26 + $19.64541 = $269.90541, 17.5182 Amount. Hence, the $19.645410, Ans. Rule. — Multiply the principal hy the interest of $1 for the ^iven time, and rate. Notes. — 1. When the rate is greater or less than 6%, find the interest of the principal at 6% for the given time ; then add to or subtract from it such a part of itself, as the given rate exceeds or falls short of 6 per cent. 2. If the mills are 5 or more, it is customary to add 1 to the cents ; if less than 5, they are disregarded. 3. Only three decimals are retained in the following Answers, and each answer is found by the rule under which the Example is placed. 4. In finding the interest of $1 for days, it is sufficient for ordinary purposes to carry the decimals to four places. 2. What is the amt. of 1350.60 for 1 yr. 5 mo. 15 d,, at 6% ? 3. What is the int. of $56.19 for 4 mo. 3 d., at '7% ? 4. What is the int. of $242.83 for 7 mos. 18 d., at o% ? 5. Find the int. of $781.13 for 11 mo. 21 d., at 6^. 6. Find the int of $968.84 for 2 yr. 10 mo. 26 d., at 6%. 7. What is the int. of $639 for 18 mo. 29 d., at 1%? 8. Wliat is the int. of $745.13 for 17 d., at 5% ? 9. What is the int. of $1237.63 for 8 mo. 3 d., at S% ? 10. What is the int. of $2046^ for 25 d., at 4:% ? 11. Find the amount of $640.37^ for 9 mo. 15 d., at 10^. 12. Find the amount of $2835.20 for 2 mo. 3 d., at 9^^ 13. Find the amount of $4356.81 for 3 mo. 10 d., at 6^%. 14. What is the int. of $12240 for 63 d., at ^% ? 15. What is the int. of $350000 for 10 d., at di% ? Interest. 219 Method by Days. 540. 1. What is the interest of I248.G0 for 93 days, at Q% ? OPERATION. Analysis.— Since the interest for 30 1248.60 Priu. days is 5 mills, or -^^^ of the principal, 93 No. d. for 1 day it is 3V of ^^^, or « oV ' lience, 74580 tlie interest for 93 days is gfgo of tlie 99q'*'4.0 principal. And ^ff^ of $248. 60 =($248.60 j^^rf^U_ X 93)-^6000 = $3.85. Hence, the 6|000 ) 23|119.80 $3,853, Ans, EuLE. — Multiply the principal hy the number of clays, and divide the product hy 6000. 2. Find the interest of $360 for 95 d., at 1%. Ans. $6.65. What is the interest of 3. $450 for 63 d. at 6^ ? 7. $600 for 63 days at b% ? 4. $245.50 for 33 d. at Q% ? 8. $735 for 45 days at 7%? 5. $278.68 for 75 days at Q% ? 9. $1200 for 60 d. at b% ? 6. $500.75 for 130 days at Q% ? 10. $1500 for 93 d. at 8%? Exact Inter est. 541. The methods based upon the supposition that 360 days make a year and 30 days a month, though common, are not strictly accurate. As a year contains 365 days, the int. found by these methods is ^-|-g^, or -^ part of itself too large. Hence, 542. To compute exact interest for months and days, find, the interest hy the 6% method and subtract from it yV part of itself. (An. Int., Art. 905, App.) 1. What is the exact interest, at 6^, of $248.60 for 3 mo. 3 d. ? Ans. The interest at Q% is $3,853, J^ part of which is $.053, and $3.853— $.053 = $3.80. 2. What is the exact interest of $2568 for 93 d., at 6^? 3. What is the exact interest 0^ $5000 for 12 d., at 1% ? 220 Percentage, Partial Payments. 543. Partial Payments are parts of a note paid at different times. 544. A Promissory Note is a written promise to pay a speci- fied sum at a given time. 545. The Maker is the person who signs the note. 546. The Payee is the person to whom it is to be paid. 547. The Holder is the person who has the note in liis possession. 548. Indorsements are partial payments, the amount and date of which are written upon the back of notes and bonds. 549. The Face of a note is the sum named in it. 550. A Negotiable Note is one payable to the bearer, or to the order of the person named in it. Notes. — 1. A note payable to A. B., or "order," is transferable by indorsement ; if to A. B., or " bearer," it is transferable by delivery. Treasury notes and bank bills belong to this class. 2. If the words " order " and •' bearer " are both omitted, the note can be collected only by the party named in it. 551. An Indorser is a person who writes his name on the back of a note as security for its payment. 552. The Maturity of a note is the day it becomes legally due. In most States a note does not mature until 3 days after the time named for its payment. These three days are called Days of Grace. 553. To compute Interest on notes and bonds, when JPartial l\ujnient.s have been made. United States Rule. Find the amount of the piineipal to the tUne of the first payment, and suhtracting the payment from it, find the amount of the remainder as a new principal, to the time of the next paymenl. Partial Payments. 221 // the payment is less than the interest, find tlie amount of the principal to the time when the sum of the payments equals or exceeds the interest due; and subtract the sum of the payments from this amount. Proceed in this manner to the tUi%e of settlement. Notes. — 1. The principles upon which the preceding rule is founded are, 1st. That payments must be applied first to discharge accrued interest, and then the remainder, if any, toward the discharge of the principal. 2d. That only unpaid principal can draw interest. 3. The following examples show the common forms of promissory notes. The first is negotiable by indorsement ; the second by delivery ; the third is o, joint note, but not negotiable. ^'^50. Washington, Jan. 1st, 1880. 1. On demand I promise to pay to the order of Alexander Hunter, eight hundred fifty dollars, loith interest at 6 per cent, value received. John FRAXKLiiT. The following payments were endorsed on this note : July 1st, 1880, received $100.62. Dec. 1st, 1880, received $15.28. Aug. 13th, 1881, received 1175.75. What was due on taking up the note Jan 1st, 1882 ? SOLUTION. Principal, dated Jan. 1st, 1880, $850.00 Int. to 1st payt. July 1st, 1880 (6 mo.) (Art. 539), _^^^ Amoimt, = 875.50 1st payment, July 1st, 1880, 100.62 Remainder, or new principal, = 774.88 Int. from 1st payt. to Dec. 1st (5 mo.) 19.37 2d payt. less than int. due, $15.28 Int. on same priu. to 3d payt., Aug. 18 (8 mo. 12 d.) 32,54 Amou7it, = 826.79 3d payt., to be added to 2d, |175.75 = 19103 Remainder, or new principal, = 635.76 Int. to Jan. 1st, 1882 (4 mo. 18 d.) 14.62 Balance due Jan. 1st, 1882, = $650.38 222 Percentage, $692j%% . Boston, Aug. 15th, 1879. 2. Three months after date, I pro7nise to imy John War- ner, or hearer, six hundred and ninety-two dollars and thirty- five cents, with interest at 6 yer cent, value received. Samuel Johj^son". Endorsed Nov. loth, 1879, $250,375. Endorsed March 1st, 1880, $65,625. How much was due on the note, July 4th, 1881 ? ?^_: New York, May 10th, 1878. 3. For value received, we jointly and severally 'promise to pay James Monroe & Sons, five hundred dollars on demand, with interest at 7 per ce?it. George Johkson. Henry Smith. The following sums were endorsed upon it : Received, Nov. 10th, 1878, 175. Received, March 22d, 1879, $100. "What was due on taking up the note, Sept. 28th, 1879 ? ^^0^^' Philadelphia, June 20th, 1878. 4. Six months after date, I promise to pay 3fessrs. Caret, Hart S Co., or order, one thousand dollars, luith interest at 5 per cent, value received. Horace Preston. Endorsed Jan. 10th, 1879, $125. Endorsed June 16th, 1879, $93. Endorsed Feb. 20th, 1880, $200. "What was the balance due on the note, Aug. 1st, 1880 ? Note. — Massachusetts, New York, Pennsylvania, Ohio, Illinois, and most of the other States have adopte 1 this rule. (For Connecticut, Ver- mont, and New Hampshire methods, see Art 906-908, Appendix.) Fartial Payments, 223 Mercantile Method. 554. When Partial Payments are made on sliort notes or interest accounts, business men commonly employ the follow- ing method : Find the amount of the whole debt to the time of set- tlement ; cdso find the amount of each payment fi^om the time it was made to the time of settlement. Subtract the amount of the payments from the amount of the debt ; the remainder will be the balance due. ^i^ Albany, March 31st, 1880. 5. On demand, I 'promise to pay to the order of Henry Pattox, four hundred and sixteen dollars, with interest at 7 ])er cent, value received. JoHiT Marshall. Eeceived on the aboye note the following sums : June 15th, 1880, $35.00. Oct. 9th, 1880, 123.00. Jan. 12th, 1881, $68.00. What was due on the note, Sept. 21st, 1881 ? SOLUTION. Principal, dated March 21st, 1880, $416,000 Int. to settlement (1 yr. 6 mo.), o.tl'/o, 43.680 Amount, Sept. 21st, 1881, = 459.680 1st payt, $35.00, Time (1 yr. 3 mo. 6 d.), Amount = $38,103 2d payt., $23.00, Time (11 mo. 12 d.), Amount = 24.530 3d payt., $68.00, Time (8 mo. 9 d.). Amount = 71.29 2 Amount of the payments, = 133.925 Balance due Sept. 21st, 1881, . $325,755 6. A bill of goods amounting to $750, was to be paid Jan. 1st, 1880. Received June 10th, $145 ; Sept. 23d, $465 ; Oct. 3d, $23 ; what was due on the bill Dec. 31st, 1880, int. Q% ? 7. An account of $1200 due March 3d, received the follow- ing payments: June 1st, $310 ; Aug. 7th, $219; Oct. 17th, $200 ; what was due on the 27th of the following Dec, allowing 7^ interest ? 224 Percentage, PROBLEM II. 555. To find the Hate, when the Principal, Interest, and Time are given. 1. At what rate of interest must 1236 be loaned, to gain $17.70 in 1 year and 3 months? Analysis.— The int. of $236 for 1 yr. at 1 % = $236 x .01 = $3.36 The int. for 3 mo. (| yr.) = $2.36 x i = .59 The int. for 1 yr. 3 mo. at Ifc = $2.95 Now as $2.95 gain requires 1%, $17.70 gain requires as many per cent as $2.95 are contained times in $17.70, or 6%, Ans. Hence, the EuLE. — Divide the given interest hy the interest of the principal, at 1 per cent for the time. Formula. — Rate = Interest -f- {Prin. x 1% x Time). Note. — When the amount is given the principal and interest may be said to be given. For, the amt. = the prin. + int. ; hence, amt. —int. = the prin. ; and amt. —prin. = the int. 2. At what rate per cent, must 1450 be loaned, to gain $56.50 interest in 1 year and 6 months ? 3. At what per cent must $750 be loaned, to gain 1225 in 4 years ? 4. A man has $8000 which he wishes to loan for $500 per annum ; at what per cent must he loan it ? 5. A gentleman deposited $1250 in a savings bank, for which he received $31.25 every 6 months; what per cent interest did he receive on his money ? 6. A capitalist invested $9260 in railroad stock, and drew a semi-annual dividend of $416.70 ; what rate per cent interest did he receive on his money ? 7. A man built a hotel costing 1175000, and rented it for- 18750 per year ; what per cent int. did his money yield him ? 8. A man gave his note payable in 1 year and 3 montlis for $640, and at its maturity paid 1688 ; what was the rate of interest ? 9. At what rate must 1865 be loaned for 2 years to yield $129.75 interest? Interest, 325 PROBLEM III. 556. To find the Time when the Principal, Interest, and Rate are given. 1. In what time will $500 gain 145 at 6% ? Analysis.— The interest of $500 for 1 yr. at 6% is $30. opbbation. Therefore, to gain $45 will require the same principal as 30 ) $45.00 many years as $30 are contained times in $45 ; and $45 h- < ~ 7"^" $30 =: 1.5 or 1| years, Ans. Hence, the EuLE. — Divide the given interest by the interest of the principal for 1 year, at the given rate. Formula. — Time = hit. -^ {Prin. x Rate). Notes. — 1. If the quotient contains decimdls, reduce them to months and days. (Art. 402.) 2. If the amount is given instead of the principal or the interest, iSnd the part omitted, and proceed as above. 3. At 100,^^, any sum will double itself in 1 year; therefore, any per cent will require as many years to double the principal, as the given per cent is contained times in 100%! 2. In what time will $4500 gain $430 at b% ? ■ 3. How long will it take $5000 to earn $5000 at Q>% ? 4. How long will it take any sum to double itself at 4^? 5^? 6^? 7^? 10^? PROBLEM IV. 557. To find the Principal, when the Interest, Rate, and Time are given. 1. What principal at Q% will yield $225 interest in 2 yr. 6 mo. ? Analysis.— At 6 % , the interest of |1 for 2 yr. 6 mo. operation. is $.15, therefore, $225 must be the int. of as many .15 ) 225.00 dollars as $.15 are contained times in $225, and . ^i^nr $225 --$.15 = $1500, ^;?«. Hence, the Ans. ^l0^(} Rule. — Divide the given interest hij the interest of $1 for the given time and rate, expressed deeiniaUy. Formula. — Principal = Interest h- {Pate x Time). 226 Percentage. 2. What principal at '7% will yield 1500 in 1 year ? 3. At 6% what principal will yield 1350 in 6 months ? 4. What principal at 5% will yield $400 in 7 mo. 15 d. ? 5. What sum must a father invest at 6%, that his son, now 18 yr. old may have 1^5000 when he is 21 ? (Art. 556, N. 2.) 6. What sum loaned at 1% a mo. will amount to $500 in 1 yr.? 7. AYhat sum must be loaned at 4:% a year to amount to 11200 in 8 months ? Compound Interest. 558. Compound Interest is the interest of the jyrmcipal and of the unpaid interest after it becomes due. 559. To compute Compound Interest, when the Principal, Rate and time of compounding it are given. I. What is the compound interest of 1500 for 3 years at 6%? Principal, ' = $500 Int. for 1st year, $500 x .06, 30 Amt. for 1 yr., or 2d prin., = 530 Int. for 2d year, $530 x .06, 31.80 Amt. for 2 yr., or 3d prin., = 561.80 Int. for 3d year, $561.80 x .06, 33.71 Amt. for 3 years, = 595.51 Original principal to be subtracted, 500 Compound int. for 3 years, = 95.51 Hence, the EULE. — I. Find the amount of the principal for the first period. Treat this amount as a new principal, and find the amount due on it for the next period, and so on through the juhole time II. Subtract the given principal from tlie last amount, and the rem^ainder will he the com pound interest. Note. — If there are months or days after the last regular period at which the interest is compounded, find the interest on the amount last obtained for them, and add it to the same, before subtracting the principal. Compound Interest. 227 2. What is the compound int. of S450 for 3 yr. 6 mo., at 6^ ? 3. What is the compound int. of 1550 for 3 yr. 4 mo., at 1% ? 4. What is the compound int. of 1850 for 4 yr. 6 mo., at b% ? 5. What is the com. int. of 1865 for 5 yr., at 1% ? 6. What is the amt. of 1950 for 6 yr. 3 mo., at b%, com. int.? 560. Table shoiving the amount of II, at 3, 3 J, 4, 5, and Q% compound interest, for any number of years from 1 to 20. Yrs 3%. 3i%- 4%. 5%. 6%. I 1.030 000 I 035 000 1.040 000 1.050 000 1.060 000 ^ 1.060 900 I 071 225 1. 08 1 600 1.102 500 1. 123 600 3 1.092 727 I 108 718 I. 124 864 1.157 625 I. 191 016 4 I. 125 509 I 147 523 1-169859 1. 215 506 1.262 477 5 1. 159 274 I 187 686 1. 216 653 1.276 282 1.338 226 6 1. 194052 I .229255 1.265 319 1.340096 1.418519 7 1.229 ^74 I .272 279 ^•315 932 1.407 100 1-503630 8 1.266 770 I 316 809 1.368569 1-477 455 1.593848 9 1.304773 I 362 897 1.423 312 1-551 Z^^ 1.689479 lO 1.343 916 I 410599 1.480 244 1.628895 1.790 848 II 1.384234 I 459970 1-539 451 1. 710 339 1.898 299 12 1.425 761 I 511 069 1. 601 032 1-795 856 2.012 196 13 1.468534 I 563956 1.665 074 1.885649 2.132 928 14 1. 512 590 I 618695 1. 731 676 1.979932 2. 260 904 ^5 1-557967 r 675 349 1.800 944 2.078 928 2-396558 16 1.604 706 I 733 986 1.872 981 2.182 875 2-540352 17 1.652848 I 794676 1.947 900 2. 292 018 2.692 773 ' 18 1.702433 I. 857489 2.025 8^7 2.406 619 2.854339 , 19 1-753506 ; I 922 501 2.106 849 2.526 950 3.025 600 20 1.806 in 1 I. i 989 789 2. 191 123 2.653 298 3.207 135 Note. — Compound interest cannot be collected by Imo ; but a creditor may receive it, without incurring the penalty of usury. Savings Banks pay it to all depositors who do not draw their interest when due. 228 Percentage, 561. 1. What is the int. and' ami of 12000 for 10 yr. at ^6% ? Solution.— Tabular amt. of $1 for 10 yr. at 3%, $1.34^916x2000 = $2687.832, amt. for 10 yr. And $2687. 832 -$2000 prin. = $687,832, Com. Int. for 10 years. Hence, the EuLE. — ^I. Multiply the tabular amount of $1 for the given time and rate by the pidncipal ; the product will he the amount. II. From the amount subtract the principal, and the remainder will be the compound interest. Notes. — 1. If the given number of years exceed that in the Table, find the amount for any convenient period, as half the given years; then on this amount for the remaining period. 3. If interest is compounded semi-annually take | the given rate and twice the number of years ; if compounded quarterly, take ^ the given rate and 4 times the number of years. 2. What is the amt. of 13500 for G yr., at 5^ com. interest ? 3. What is the amount of $350 for 12 years, at 4^ ? 4. What is the com. int. of $469 for 15 years, at 3% ? 5. What is the com. int. of $500 for 24 years, at 6% ? 6. What is the com. int. of $650 for 30 years, at 3^%? 7. What is the amount of $1000 for 3 yr., at (]% compound interest, payable semi-annually ? 8. What is the amount of $1200 for 2 years, at 12^ componnd interest, payable quarterly ? 9. What is the amt. of $1500 for 5 yr. 3 mo., at 5% com. int. ? Questions. 524. What is interest ? 525. Principal ? 526. Rate ? 527. Amount V 528. What is simple int. ? 529. Legal interest ? 530. Usury? 536. The general method of computing interest ? 539. The Q^i method ? 540. The method by days ? 542. Exact interest ? 543. What are Partial payments? 544. Promissory note? 549. The face of a note ? 550. A negotiable note ? 552. The maturity of a note ? 548. Indorsements? 553. U. S. Rule for partial payments? 555. How find the rate ? 556. The time ? 557. The principal ? 558. Compound Interest ? 559. How computed ? ISCOUNT. -^^- Oral Exercises. 562. 1. The price of a watch was 150, but for cash it was sold at 10^ off ; how much was the deduction ? Analysis. — 1% of $50 is 50 cents, and 10;^ is 10 times .50, or $5; hence the deduction was $5, Ans. 2. A man asked 1200 fpr a horse, but for cash would take 6% off ; liow^ much was deducted ? 3. A merchant sold a bill of goods amounting to $500, and for cash deducted 6^}' ; how much was deducted ? 4. A man owed $800 on Acct., and settled it for cash at 4^ off ; what was the deduction ? 5. If you borrow 1300 and pay G^ in advance for its use, how much is deducted from the loan ? True Discount. 563. Discount is a deduction from a stated price, or from a debt paid before it is due. 564. True Discount is the difference between the face of a debt and its present wortli. The Present Worth of a debt, due at some future time with- out interest, is the sum which put at interest at the legal rate will amount to the debt when it becomes due. 565. To find the Present Worth and True Discount. 1. What is the present worth and true discount of $378, due in 1 year and 8 months, at G;^ ? 230 Percentage, Analysis. — The amount of $1, at 6%, for 1 yr. 8 mo. = $1.10. Since $1.10 is the amt. of $1, at 6^ for the given time, $378 is the amt. of as many dollars for the same time and rate, as $1.10 is contained times in $378. and $378^$1.10 = $343.64, present worth. Then $378- $343.64 = $34.36, the true discount. Hence, the EuLE. — I. Divide the debt by the cunount of $1 for the given time and rate; the quotient will be the present worth, II. Subtract the present worth from the debt, and the reinainder will be the true discount. Find the present worth and true discount of 2. $850.25, due in \\ years, at 6^. 3. $1272.50, due in 1 yr. 3 mo., at 7^. 4. $2895, payable in 2 years, at h%, 5. $5650.75, payable in 3|- years, at ^\%, 6. $10000, due in 1 yr. 5 mo., at '^%. 7. What is the difference between the interest and true dis- count of $12250, for 1 year, at ?)% ? 8. Bought a farm for $4822, payable in 2-|- years without interest, but for cash 20^ discount ; what was the true discount ? 9. When money .is worth 5*^, which is preferable, $12000 cash, or $13000 payable in 1 year ? Bank Discount. 566. Bank Discount is simple interest, paid in advance. 567. The Proceeds of a note are the part paid to the owner ; the Discount is the part deducted. 568. The Maturity of a note is on its last day of grace. Note. — If the last day of grace occurs on Sunday or a legal holiday, the note matures on the preceding day. 569. The Term of Discount is the time from the date of discount to the maturity of the note. Baiik Discount 231 570. To find the Bank Discount and Proceeds, when the Face of a note, Rate, and Time are given. 1. What is the bank discount of $368 for 3 mo. at 6^? What are the proceeds ? Solution. — The face of the note = $368 Int. of $1 for 3 mo. and grace at 6% = .0155 Discount — $5,704 Proceeds, $368- $5. 704 = $363,296. Hence, the EuLE. — Find the interest of the note at the given rate for three days more than the specified time ; the result is the discount. Subtract the discount from the face of the ?iote; the remainder will be the proceeds. Note. — If a note is on interest, find its amount at maturity, and taking this as the face of the note, cast the interest on it as above. • 2. Find the proceeds of a note of S650, due in 3 mo., at 6%, 3. Find the proceeds of a draft of $825, on 60 days, at 6%. 4. Find the maturity and term of discount of a note of $1250, at 5% int., on 60 days, dated July 1st, 1880, and dis- counted Aug. 21st, 1880, at 6%. What were the proceeds? 5. Find the difference between the true and bank discount on $4000 for 1 year, allowing each 3 days grace, at 7% ? 6. A merchant bought $6500 worth of goods for cash, sold them on 4 months, at 15^ advance, and got the note dis- counted at 6^c to pay the bill. How much did he make ? 571. To find the Face of a note, when the Proceeds, Rate, and Time are given. 1. For what sum must a note be made on 4 months, that the proceeds may be $640, discounted at 6^ ? Solution.— The bank discount of $1 for 4 mo. 3 d. =: $.0205 The proceeds of $1 = $l-$.0205 = $.9795 Therefore, The face of the note is $640 ^$.9795 =$653,394 Hence, the Rule. — Divide the given proceeds by the proceeds of $1 for the given time and rate. 232 Percentage. 2. What must be the face of a note on 6 months, discounted at 1%, that the proceeds may be 1500 ? 3. The avails of a note were $4350.90, the term 4 months, and the rate of discount 8^ ; what was the face of the note ? 4. How large a note on 3 months, must I have discounted at 6^, to reaUze 15260 ready money ? Commercial Discount. 572. Commercial Discount is a per cent deducted from the face of bills, the list price of goods, etc. 573. The Net Price of goods is the sum received for them. 574. To find Coniiiiercial Discount, when the rate is given. 1. What is the commercial discount on goods, the list price of which is 1235, sold at b% off ? Solution.— 5% is .05, and $335 x. 05 = $11.75, Ans. 2. What was the net price received for a parlor organ, whose list price was $450 on 3 mo., at 7^^ off for cash ? 3. What is the net value of a bill of books, amounting to $568.50, on 60 days, at 10;^ off for cash ? 4. After 0% had been deducted from the list price, a bill of goods was sold for $625 ; what was the list price ? 5. Sold a bill of goods amounting to' $850, on 4 mo., at 8^ discount, and deducted b% for cash ; what was the net price ? Solution.— 1850 x .08 = $68. And $850 -$68 = $782. Again, $782 x .05 = $39.10, and $782-$39.10 = $742.90. Hence, the EuLE. — Deduct the discount from the marhed price, and from the remainder tahe the discount for cash. 6. What is the net value of a bill of goods amounting to $2560, sold at 10$ discount and 4$ off for cash ? 7. What is the net value of a cargo of flour invoiced at $3765, at 12$ discount and 5$ off for cash ? 8. Find the net value of a bill amounting to $4372, at 15$ discount and 2|$ off for cash ? Commercial Discount. 233 9. Find the sum received for a sale of goods marked at $6500, at S% discount and ^1% off for cash ? 10. What is the cash value of a bill of $10000, at 7^ discount and '\:^% off for cash ? 11. Find the net value of the following : 63 lb. tea, at 88 cts., sold on 3 mo., 8^ off ; 95 boxes of starch, at 68 cts., 4^ off ; 54 drums of figs, at 75 cts., ^c off; 85 bbl. flour, at $7.50, 10% off ; allowing ^% discount for cash. 575. To Mark goods so that a given per cent may be deducted and leave a given per cent profit. 1. Bought ladies' hats at 15.10 ; what price must they be marked, that 15^ may be deducted and leave 20^ profit ? Analysis. —The selling price is 120% of $5.10, and $5.10 x 1.20 =$6.13. But the marked price is to be diminished by 15% of itself, and 100% — 15% = 85% ; hence, $6.12 is 85% of the marked price. Now .|6.12 ^ .85 = $7.20, the marked price, (Art. 466.) Hence, the KuLE. — Find the selling -price and divide it hy 1 minus the given per cent to he deducted; the quotient will be the marhed price. 2. Paid %oQ for a sewing machine ; what must I ask for it that I may abate 5^ and sell it at a gain of 25% ? 3. A shoe dealer paid $3.60 a pair for boots; what must he ask for them that he may deduct 1%'^% and make 16f^ ? 4. A jeweller bought diamond rings at 1120 ; what must he ask for them that he may abate 4^ and still make 20^ ? 5. Bought a piano for $250 ; what must I ask for it that I may deduct 20^ and leave a profit of 20^ ? Questions. 563. What is discount ? 564. True discount ? The present worth of a debt? 565. How found? 566. What is bank discount ? 567. The proceeds of a note ? 568. When does a note mature ? 569. Wliat is the term of disconxit ? 570. How find the discount of a note ? The proceeds ? 571. How find the face of a note that the proceeds may be a given sum ? 572. What is commercial discount ? 573. What is the net price of goods ? W' / .\fc . .. ln»1i>, ,., r;Txv^^-P iQUATION O F (PAYMENTS. DevetjOip^ient of I^rinciples. 576. 1. How long must $1 be kept on interest to equal the interest of $2 for 3 mo. at tlie same rate per cent ? Analysis. — As $2 are twice $1, at the same rate $1 must be kept on interest twice as long as $2, and 2 times 3 mo. are 6 months, Ans. 2. How long must 12 be kept on interest to equal the interest of 18 for 3 months ? 3. How long must 13 be kept on interest to balance the interest of $9 for 4 months ? 4. How long must 110 be kept to balance the interest of $5 for 4 months ? Analysis.— |10 is twice $5 ; therefore, $10 must be kept half as long as $5, and ^ of 4 mo. is 3 months, Ans. 5. How long will it take 130 to balance the interest of 110 for 6 months ? 6. In what time will the interest of 1200 balance the interest of $50 for 8 months ? 577. From the examples above we deriye the following Principles. 1°. The rate and time remaining the same, Double the principal producer twice the interest. Half the ^jrincipal produces half the i^iterest, etc. 2°. The rate and principal remaining the same. Double the time produces twice the interest. Half the time produces half the interest, etc. Hence, 578. The interest of any given principal for 1 year, 1 month, or 1 day, is the same as the interest of 1 dollar for as many years, months, or days, as there are dollars in the given principal. Equation of Payments, ^35 579. Eq[uation of Payments is the method of finding the average time for the payment of several debts, due at different times, without loss of interest to either party. 580. The Average or Eq[uated time is the date when the several payments m.ay be made at one time. 581. The Term of Credit is the time before a debt becomes due. 582. The Average Term of Credit is the time at which debts due at different times may be equitably paid. Written Exercises. 583. To find the Averdge Tune, when the items have the same date, but different terms of credit. 1. Bought Oct. 10th, 1880, the following bills of goods, for which I was to pay $485 cash, $200 in 2 mo. ; $275 in 4 mo. ; and $360 in 5 mo.; what is the average time and the date, when these bills may be paid without loss to either party? ExPLANATiox.— The first bill is cash and has no interest. The int. of |200 for 3 mo. is the same as the int. of $1 for 400 mo. (Prin. r) The int. of $275 for 4 mo. is the same as that of $1 for 1100 mo. The int. of $300 for 5 mo. is the same as that of $1 for 1800 months. Therefore, the amount of interest due on the whole debt, is equal to the interest on $1 for 3800 mo. Now as $1 is entitled to int. for 3300 months, the whole debt $1320 is entitled to interest for ygV^ of 3800 mo., and 8300 -f- 1320 = 2^ months, the average term of credit. And 21 mo added to Oct. 10th, 1880 = Dec. 25th, 1880, the date of payment. Hence, the Rule. — Multiply each item hy its term of credit, and divide tlie sum of the products hy the sum of the items. The quotient luilt he the average term of credit. Adding the average term of credit to the date of the Bill, will give the date of payment. Notes. — 1. When an item contains cents, if less than 50, they are rejected ; if 50 or more, $1 is added OPEKATION. $485 X = 200 X 2 = 400 275 X 4 m 1100 360 X 5 = 1800 $1320 3300 236 Percentage, 2. In the quotient, a fraction less than | d., is rejected ; if 4 d. or more, 1 day is added. 2. A merchant buys goods, and agrees to pay 1400 down, $400 in 4 months, and 1400 in 8 months ; what is the average time of the whole ? 3. A man borrows 1600, and agrees to pay $100 in 2 months, $200 in 5 months, and the balance in 8 months; when can he justly pay the whole at once? 4. A man buys a house for $1600, and agrees to pay $400 down, and the rest in 3 equal annual instalments ; what is the average term of credit ? 5. I have $1200 owing to me, \ of which is now due ; |- of it will be due in 4 montlis, and the remainder in 8 months ; what is the average term of credit ? 6. A grocer bought goods amounting to $1500, for which he was to pay $250 down, $300 in 4 months, and $950 in 9 months ; when may he pay the whole at once ? 7. A young man bought a farm for $2000, and agrees to pay $500 down, and the balance in 5 equal annual instalments; what is the average term of credit ? 584. To find the Average Time, when the terms of credit are different, and begin at different dates. 8. Bought goods as follows : March 1st, 1880, $200 on 2 mo.; April 6th, $800 on 4 mo.; June 17th, $1000 on 3 mo.; what is the average time and date of payment ? OPERATION. $200 due May 1, 00 d. x 200 = 00 800 " Aug. 6, 97 d. X 800 = 77600 1000 " Sept. 17, 139 d. x ^000 = 139000 2000 ) 2 16600 The average time is 108 d. (Art. 583, N.) 108 Date of payment 108 d. from May 1st, or Aug. 17th, 1880. Explanation. — Taking as the standard the earliest date at which either of the items becomes due (May Istj, the term of credit to Aug. 6, is 97 d., to Sept. 17tli, 139 days. The average term of credit is therefore 108 days, and the date of payment is Aug. 17th, 1880. Hence, the Equation of Payments. 237 Rule. — I. Find the date when each item matures. Take the first day of the month in which the earliest item becomes due as a standard, and find the number of days from this to the maturity of each of the other items. II. Multiply each item by its number of days, and divide the sum of the products by the sum of the items The quotient will be the average term of credit. III. Add the average time to the standard date, and the result will be the equitable date of payment. Note. — Any date may be assumed as the standard, but it is most convenient to take the first day of the month in which the earliest item falls due. 9. Bought the following amount of goods on 4 months' credit : March 10th, 1879, $200 ; April 15th, 1160 ; May 1st, $440; at what time is the amount payable ? 10. Bought the following bills on 8 months : July 5th, 1879, 1620.25 ; Aug. 11th, $240.56 ; Sept. 20th, $321.64 ; Oct. 12th, $510.38; Nov. 1st, $308.17 ; when ought a note for the whole amount to be dated ? 11. A merchant bouglit the following bills of goods : March 19th, $350 on 4 mo.; April 1st, $430 on 130 days; May 16th, $540 on 95 days ; June 10th, $730 on 3 mo. ; what is the average time for payment of the whole ? 12. Bought the followiug bills of goods on 90 days' credit : May 10th, $375.63; May 18th, $738.45; June 3d, $860.40; June 17th, $692.38 ; July 3d, $379.68 ; July 12th, $417.13; at what time will the whole be due at once ? 13. A grocer sold the following amount of goods : June 3d; $380 on 90 days' credit ; June 10th, $485 on 30 d. ; July 21st, $834 on 60 d. ; July 27th, $573 on 110 d. ; Aag. 2d, $485 on 80 d. ; when will the whole be due ? 14. Sold the following bills of goods on 3 months ; Sept. 5th, 1880, $1163.25; Sept. 20th, $2368.41 ; Oct. 7th, $3561.34; Oct. 23d, $840.90; Nov. 13th, $1307.63; at what time must a note for the whole amount be dated to give the buyer the specified credit ? 338 Percentage. Averaging- Accounts. 585. Averaging an Account is finding tlie equated time at which the balance may be joaid. 586. To find the Average Tune for settling an account. 1. Find the equated time and date of paying the balance of the following account : Dr, John Hamiltoi^ in acct. with Henry Morgan. Cr. 1881. \ 1881. Jan. 5 For Mdse. 2 mo. 1300 Jan. 25 By Draft 90 d. 1200 Feb. 26 " " 3 mo. 200 March 28 '' Cash. 300 March 28 " 1 mo. 500 May 25 " Cash. 100 OPERATION. Due. Maicli 5 Amt. $300 Time. 4d. Prod. 1200 Due. Apr. 28 Amt. $200 Time. 58 d. Prod. 11600 Mav 26 200 86 17200 March 28 300 27 8100 April 28 500 58 29000 May 25 100 85 8500 $1000 600 Bal. $400 47400 28200 600 28200 ) 19200 ( 48 days. Ans. Bal. $400, due in 48 days from March 1st, or April 18th. Explanation. — Having found when each item of debt and credit becomes due, by adding its term of credit to its date, we assume as the standard date the first day of the month in wliich the earliest item on either side of the account matures, viz.: March 1st. Multiply each item on both sides by the number of days between the standard date and the maturity of each item, and divide the difference between the sums of the products (19200), by the difference between the sums of the items (400). The quotient is the average time of payment. Since the equated time requires the interest of $1 for 19200 days, it will require $400, ^^^ part as long, and 19200-^400.= 48 ; and 48 days added to March 1st gives April 18th. Hence, the Averaging Accounts. 239 Rule. — I. Write^ the date at which each item on both sides matures, and assume the first day of the montli in which the earliest item on either side becomes due, as the standard date. Find the number of days from this standard to the matuj^ity of the respective items. (Art. 583, N.) II. Multiply each item by its Jiumber of days, and divide the difference between the sums of products by the difference between the sums of items ; the quotient will be the average time. III. // tJte grea,ter sum of items and the greater sum of products are both on the same side, add the average time to the assumed date; if on opposite sides, subtract it ; and the result will be the date when the balance of the account is equitably due. Notes. — 1. In finding the maturity of notes and drafts, 3 days grace should be added to the specified time of payment. 3. When no time of credit is mentioned, the transaction is understood to be for cash, and its payment due at once. 2. Find the average time of papng the following account : Dr. George Hadley. Or. 1880. 1880. March 1 To Mdse. 1500 Apr. 12 By Draft, 20 d. $300 Apr. 5 " " 2 mo. 700 May 10 '-' Cash. 540 May 20 " " 4 mo. 650 June 4 a a 500 3. At what date can the balance of the followins: account be equitably paid ? Dr. W. H. Hendersoi^^, Cr. 1881. 1881. Apr. 7 To Mdse., 2 mo. $300 May 1 To Mdse., 60 d. 1350 July 5 " 1 mo. 500 June 10 a a 30 d. 500 Aug. 10 " " 1 mo. 400 Aug. 30 " Cash. 200 240 P&rcentage. 4. Average the following account : Dr. James Brown & Co. Cr 1882. 1882. Jan. 10 To Mdse., 3 mo. $400 Jan. 1 By Bal. of Acct. $485 " 25 " " 30 d. 265 Feb. 10 " Note, 3 mo. 2500 Apr. 20 " " 2 mo. 850 March 1 " Draft, 30 d. 260 5. Balance the following account : Dr, C. J. Hammoi^d. CV. 1880. 1880. Jan. 20 To Sundries ,30d. $500 Jan. 20 By real estate 60 d. $400 Feb. 12 a a 60 d. 340 March 1 " Draft 60 d. 200 March 1 a i( 30 d. 300 " 20 " Cash. 400 6. Average the following account : Dr. He:n^ry Eatmond & Co. Or 1881. 1881. Aug. 10 To Mdse., 60 d. $150 x\ug. 25 By Mdse., 30 d. $500 Oct. 1 '^ Cash. 350 Sept. 20 a a 20 d. 300 '' 18 " Dft. 30 d. 200 7. Find when the balance of the following account becomes due : A. B. bought of C. D., July 16th, 1882, merchandise $350 ; Aug. 11th, $460 ; Sept. 9th, $570; Sept. 14th, $840; Oct. 18th, $780. The former paid August 1st, $260 ; Sept. 30th, in grain $340 ; Oct. 5th, cash $500 ; Oct. 21st, $625. Questions. 577. When the time and rate of interest remain the same, what is the effect of doubling the principal ? The principal and rate remaining- the same, what is the effect of doubling the time ? 579. What is equation of payments? 580. What is average or equated time? 581. The term of credit ? 586. Describe the process of averaging accounts. * T O C K S . Definitions 587. A Corporation is a company authorized by law to transact business as a single individual, having the same rights and ohligations. 588. Stock is the Capital or money used by a corporation in carrying on its business. 589. A Share is one of the equal parts into which the stock is divided. Note. — The 'oalue of a share varies in different companies. It is usu- ally $100, and will be so regarded in this work, unless otherwise stated. 590. A Certificate of Stock is a written instrument issued by a corporation, stating the number of shares to which the holder is entitled, and the original value of each share. 591. The Par Value of stock is the sum named in the certificate. 592. The Market Value is the sum for wiiich it sells. Notes. — 1. When shares sell for their nominal value, they are a.t par ; when they sell for more, they are above par, or at a premium ; when they sell for less, they are below par, or at a discount. 2. When stocks sell at par they are often quoted at 100 ; when at 7% above par, they are quoted at 107, or at 7 5?^ premium ; when at 15% below par, they are quoted at 85, or at 15% discount 593. An Assessment is a percentage required of stockholders to replace losses, etc. 594. The Gross Earnings of a company are its entire receipts. 11 242 Percentage. 595. The Net Earnings are the remainder after all expenses are deducted. 596. A Dividend is a percentage divided among the stock-holders. 597. A Bond is a written agreement to pay a sum of money at or before a specified time. Notes. — 1. U. S. Bonds are generally designated according to the rates of interest tliey bear. Thus, U. S. 5's denote bonds issued by the United States bearing 5% interest ; U. S. 4's, those bearing 4%, etc. 2. Bonds of States, cities, corporations, etc., are named by combining the rate of interest they bear with the name of the State, corporation, etc., by which they are issued j as, Ohio 6's, N. Y. Central 5's, etc. 598. A Coupon is a certifica.te of interest due on a bond, to be cut off when paid, as a receipt. 599. The term Stocks is applied to government, state, city, and railroad bonds, to the capital, of banks, etc. 600. Premiums, discounts, dividends, and assessments are calculated by Percentage. The jpar value of the stock is the hase ; the 'per cent of pre- mium or discount is the rate ; the premium or discount is the percentage; the par value phis the premium i^ ^t\\Q amount ; and the par value minus the discount is the difference. Written Exercises. 601. To find the Premium, Discount, Dividend, etc., from the Par Value and Rate. Formula. — Premium, etc. == Par Value x Rate. 1. What is the premium, at 7^, on 40 shares of bank stock ? 2. What is the discount, at 15 5"^, on 50 shares of railroad stock ? 3. What is the dividend, at 6^, on 85 shares of telegraph stock ? 4. Find the assessment, at 10%, on 42 shares of oil stock ? Stocks. 243 602. To find the Market Value of Stock from the Par Value and the Premium or Discount. ^ ^^ , ^ ^^ ^ ( Par Value + Premium. Formula. — Marlcet Value = ' „ rr 7 r^- ( Far value — Discount. 5. Required the market value of 23 shares of bank stock, at 7^ premium ? 6. Find the market value of 28 shares of telegraph stock, at 7^ discount ? 7. What cost 87 shares of iron mountain stock, at \h% pre- mium and brokerage \% ? 8. Find the cost of 150 shares of insurance stock, at '^\% dis- count, brokerage \% ? 9. What is the cost of 100 shares of N. Y. and New Haven R. R. stock, at 125, brokerage \% ? Oral Exercises. 603. 1. A premium of 120 was paid on 4 shares of bank stock ; what was the rate per cent ? Analysis. — Since 4 sliares pay $20, one share ($100) pays \ of $20, or $5. Therefore, the rate was jf^ , or 5 % . 2. Bought 10 shares of stock for which a premium of %{ was paid ; what was the rate of premium ? 3. Paid a premium of 150 on 20 shares of oil stock ; what was the rate per cent ? 4. Sold 15 shares of mining stock for $75 ; what was the rate of discount ? Written Exercises. 604. To find the Rate from the Par Value, the Premium, Discount, Dividend, etc. 1. The gross receipts of a manufacturing company are $17250, the expenses are $6250, and its capital $50000 ; what per cent dividend can it make ? 244 Percentage. Analysis.— The receipts less expenses are $17250 — $6250 = $11000. Now as $50000 are entitled to $11000, $1 is entitled to $1 1000 h- $50000 = .22, or 22%. Hence, the ^ T, J i Freynium, Discount, ] „ ^^ , FOKMULA.— i^fif^fe = \ r>--77^ \ ^ P^L'T VolUG. ( Dividend, etc. \ 2. A premium of $375 was paid for 25 shares of E. E. stock ; at what rate was the premium ? 3. The discount on 50 shares of the Pacific Eaih-oad was $625 ; what was the rate of the discount ? 4. If the income on $2356 is $268.50, what is the rate % ? 5. What per cent of 3648 acres is 456 acres ? 605. To find the Cost of a given number of shares, the market value of one share and the rate of brokerage being given. 6. What cost 15 shares of E. E. stock, at 120, brokerage \% ? Analysis.— The cost of 1 share is 129% +i% brokerage, or 120^% of $100 = $120.25, and 15 shares will cost $1803.75, Ans. Hence, the -r^ ^ , i Market Value of 1 sliare -\- Brolceraqe Formula. — Cod = ^ ,^ ^ ; , ^ { X Number of shares. 7. What is the cost of 78 shares of E. E. stock at 124}, and brokerage at \% ? 8. Find the cost of 121 shares, at 89|^, and brokerage |^? 9. Sold 250 shares of bank stock at 87f , and paid ^% broker- age ; how much did I receive for it ? 10. What is the cost of 375 shares of National Express stock, at 25^ premium and brokerage -|% ? 606. To find the Number of Shares, when the investment and the cost of I share are given. 11. How many shares of bank stock at h% discount and brokerage 1%, can be bought for $7620 ? Analysis. — Since the discount is 5 % and brokerage \ % , the cost of 1 share is 95 %+\%,ot 95^ % of $100 - $95.25. As $95.25 will buy 1 share, $7620 will buy as many shares as $95 25 are contained times in $7620, and $7620 -^$95.25 = 80 shares, Ans. Hence, the FoKMULA. — Numher of Shares = luvestmenf ~ Cost of 1 Share. Stocks. 245 12. How many shares of telegraph stocky at 7|^ premium and brokerage 1%, can you buy for $13500? 13. Fmd the number of shares of mining stock at 102|, that can be bought for $5150, and brokerage ^%. 14. What number of raih'oad shares at 125, brokerage i%, will $7515 pay for ? 15. How many shares of express stock, at 10^ premium, can be bought for 18030 ? 16. Find the number of shares, at 20^ discount, that can be bought for $3200 ? 607. To find how stock must be bought which pays a given pep cent dividend, to realize a given per cent on the investment. 17. At what price must I bny Western K. R. stock which pays 6^ dividend, so as to realize 8^o on the investment? Analysis. — Dividend .06-^.08 income = .75, or 75%, price of stock. FoEMULA. — Price = Dividend -^ Rate of Income. 18. What must be paid for 4^ bonds that the investment may yield Q%^ 19. What must be paid for U. S. 5's that 8^ may be receiyed on the investment ? 20. W^hat must be paid for stock that yields 10^ dividends, so as to realize lY/o on the investment ? 608. To find what sum to invest to yield a given income, the cost of I share, rate of interest, or dividend being given. 21. What sum must be invested in N. Y. 5's, at 108-J-, to produce an annual income of $1500 ? Analysis. — The income $1500-^|o (int. on 1 share) = 300 shares, and 108| (price of 1 share) x 300 = $32550. Hence, the FoEMULA. — Investment = Cost 1 Share x Number of Shares. 22. What sum must be invested in U. S. 4's, at 105, to yield $3000 annually ? 23. What sum must be invested in Nebraska 8's, at 75, to yield an income of $1540 annually ? 246 Percentage. 24. Wbat sum must be invested in stock at 112, which pays V)% annually, to obtain an income of 12200 ? 25. What sum must be invested in Alabama 6's, at 85, to realize $2000 a year ? 26. How much must be invested in stock at 106, to yield an income of 1600, the stock paying 10^ dividend annually ? 609. To find the rate per cent of income from bonds paying a given rate of interest, and bought at a given premium or discount without regard to their maturity. 27. What is the rate of income on bonds paying 8^ interest; bought at 112 ? Solution.— Interest on 1 share $8-r-112, cost per share = 7^:^, Ans, 28. Bought bonds paying 6^ interest, at 75 ; what was the rate per cent of income ? Solution. — Interest of 1 share |6-5-75 cost per share = 8%, Ans. Interest per Share Formula. — Bate % Income — , ^ , [ -^ (Jost per bfiare. 29. Find the per cent of income on U. S. 5's, bought at 110. 30. What is the per cent of income on Iowa 6's, bought at 108, brokerage \% ? 31. Which is the more profitable, $10000 invested in 3|- per cents at 75, or in 7 per cents at 105 ? 610. To find the rate per cent of income from bonds paying a given rate of interest, bought at a given premium or discount and payable at par in a given time. 32. What rate per cent income will be realized from N. Y. 5's, bought at a premium of 8^, and paid at par in 10 years ? Analysis. — Since the bond matures in 10 years, the premium on 1 share ($8) decreases -fj^, or || each year. Now the interest $5— $|==$41, annual income on 1 share. And $4|^ -i- 108, cost of 1 share = .OS^, or 3y %, the rate required. Stocks. 247 33. What rate per cent income will be realized from North Carolina 8's, bought at 90, if paid at par in 20 years ? Analysis. — Since the bond matures in 20 years, the average decrease of the discount on 1 share is $10 -i- $20 = $| each year. Now the interest $8 + $i = |8|, the annual income on 1 share. And $8.50 -4- $90 (cost of 1 share) = $.09|, or 9|%, the rate required. Hence, the KuLE. — First find the average annual decrease of the premiiun or discount. If the bonds are a,t a premium, subtract it from the given rate of interest ; if at a discount, add it to the interest; the result will he the average income of one share. Divide the average income of one share by the cost of one share, and the quotient will be the rate per cent of income. Notes. — 1. When bonds are at a 'premium, the longer the time before maturity, the greater will be the rate per cent of income. 2. When bonds are at a discount, the longer the time before maturity, the less will be the rate per cent of income. 34. What rate per cent of income will be received on U. S. 4's at 106, and payable at par in 15 years ? 35. Bought Milwaukee and St. Paul bonds at 90, due at par in 30 years, drawing 10^ interest ; what is the rate per cent of income ? Q U ESTION S. 587. What is a corporation ? 588. Stock ? 589. A share ? 590. A cer- tificate of stock ? 591. The par value ? 592. Market value ? 593. An assessment ? 594. Gross earnings ? 595. Net earnings? 596. A dividend ? 597. A bond V 598. A coupon ? 599. To what is the term stocks applied ? 600. How are premiums, etc. , computed ? 601. What does the premium equal? 602. The market value? 604. The rate? 605. The cost? 606. The number of shares? 607. How find the price ? 609. How find the rate of income without regard to maturity ? 610. On bonds payable at par at maturity ? IXCHANGE. 611. Exchange is a method of making payments between distant places without sending the money. 612. A Draft or Bill of Exchange is a written order direct- ing one j^erson to pay another a certain sum, at a specified time. 613. The Drawer is the person who signs the draft. 614. The Drawee is the joerson to whom it is addressed. 615. The Payee is the person to whom the money is to be paid. 616. A Sight Draft is one payable on its idvesentation, 617 A Time Draft is one payable at a specified time after date or presentation. Note. — Drafts or Bills of Exchange are negotiable like promissory notes, and the laws respecting them are essentially the same. 618. An Acceptance of a draft is an engagement to pay it. As evidence, the drawee writes the word accepted across the face of the draft, with the date and his name. 619. The Par of Exchange is the standard by which the Talue of the currency of different countries is compared, and is either intrinsic or commercial. 620. Intrinsic Par is a standard having a reed and fixed value represented by gold or silver coin. 621. Commercial Par is a conventional standard, having any assumed value which convenience may suggest. Note. — The fluctuation in the price of bills from their par value, is called the Course of Exchange. Domestic Exchange, 249 Domestic Exchange. 622. Domestic Exchange is a method of making payments between distant places in the same conntry. 623. To find the Cost of a Draft, when the Face and Rate of Exchange are given. 1. What cost the following aiglit draft, at '%\% premium ? $2500. New York, June 30tli, 1881. At sight, pay to the order of James Clark, twenty-five hundred dollars, vsilue received, and charge the same to the account of To S. Bareett & Co., New Orleans, La. Smith Bros., & Co. Analysis. — Since exchange on operation. N. O. is 2i% prera., the cost of $1 1 + .025 =r $1,025 draft is ^1.025, and $2500 will cost $1,025 X 2500 = $2562.50 $1,025 X 2500 = $2562.50, Ans. 2. What is the cost of a sight draft on San Francisco for 13000, at 2i% discount ? Solution.— A draft of $1 at $2|% discount will cost $0,975, and $3000 X .975 = $2925.00, Ans. Hence, the Rule. — Multiply the face of the draft by the cost of $1. 624. On time drafts, both the rate of exchange and bank discount are commonly included in the rate, which in quotations for time drafts is enough less than for sight drafts, to allow for bank discount. Required the cost of a sight draft 3. On St. Louis, at 1^% premium, for $850 ? 4. On Buffalo, at \% discount, for $975 ? 5. On Savannah, Ga., for $2000, at 1|^ premium ? 250 Percentage. 6. What is the cost of the following time drafts at \^% pre- mium, and interest at 6^ ? ^Jfi^O^ " Philadelphia, Julj 5tli, 1881. Sixty days after sight, pay to the order of George Wilcox, four thousand dollars, value received, and charge the same to the account of H. Adams & Co. To S. Parkhurst, Trenton, N. J. Analysis. — At \\% premium, tlie opeeation. cost of $1 draft at sifflit is $1,015. %1 -f $0,015 — $1,015 But the draft is subject to interest ^ ^ $.0105 = $0.0105 for 60 d. + 8 d. grace. The int. of n ^ ^ .e $1 for 63 d., at 6% is $0.0105, and ^^^^ ^1 <^^^ft = $1.0045 $1,015-0.0105 = $1.0045 the cost of $1.0045 X 4000 = $4018 $1 draft, and a draft of $4000 will cost 4000 times $1.0045, or $4018, Ans. 7. Find the cost in Omaha of a draft on New York at 90 days sight, for $5265, at %% premium, interest being Q\%. 8. Required the worth in Memphis of a draft on Boston for $3500, at 30 days sight, at 1% discount and interest Q>%. 9. What is the worth of a draft of $5000 on St. Louis, at 30 days sight, premium 1^%, including interest ? 625. To find the Face of a Draft, when the Cost and Rate of Exchange are given. 10. How large a draft on Philadelphia can he bought in Charleston, at 60 days sight, for $3000, the premium being lY/c, and interest 6^ ? Analysis — Since the premium is opekatton. li%, the cost of $1 sight draft $1 -|- .015 = $1,015 would be $1,015. But the bank Bank dis. 63 d. = .0105 discount on $1 for 63 d. is .0105, f a,^ a «- _ ^^ hence, the cost of $1 draft is $1.0045. ^^'^^ ^^ ^^ \% premium, interest %% ; what was the face of the draft ? 16. What is the face of a draft on Cincinnati at 90 days sight, bought for $3000, exchange 2^% premium, interest Q% ? Foreign Exchange. 626. Foreign Exchange is the method of making payments between different countries. 627. A Set of Exchange consists of three bills of the same date and tenor, First, Second, and Third of exchange. They are sent by different mails in order to save time in case of mis- carriage. W^hen one is paid, the others are void. 628. Exchange with Europe is chiefly done through large commercial centers, as London, Paris, Geneva, Amsterdam, Antwerp, Hamburg, Frankfort, and Berlin. 629. Bills drawn on England, Scotland, or Ireland, are called Sterling Bills, and the value of a Pound Sterling is quoted in U. S. money. 630. The present Par of Exchange on Great Britain is $4.8665 gold to the pound sterling, which is the intrinsic value of a Sovereign, as estimated at the U. S. Mint. 252 Percentage. 631. In quoting exchange on a foreign country, it is cus- tomary to quote the value of the money unit of that country in U. S. money. Note. — These values are published annually by the Secretary of th© Treasury. Those given on the 1 st day of Jan. , ] 882, are as follows : Country. Austria Belgium Bolivia Brazil British N. A Chili Cuba Denmark Ecuador Egypt France Great Britain. . . . Greece German Empire.. Hay ti India Italy Japan Liberia Mexico Netherlands Norway Peru Portugal Kussia Sandwich Islands Spain Sweden Switzerland Tripoli Turkey U. S. of Colombia Venezuela Monetary Unit. Florin Franc Boliviano Milreisof 1000 reis... Dollar Peso Peso , Crown Peso Piaster „ . Franc Pound sterling Drachma Mark Gourde Rupee of 16 annas. . . . Lira Yen Dollar Dollar Florin Crown Sol Mil reis of 1000 reis. . . Rouble of 100 copecks Dollar Peseta of 100 centimes Crown Franc Mahbub of 20 piasters Piaster Peso Bolivar Standard. Silver Gold and silver.. Silver Gold Gold Gold and silver,. Gold and silver.. Gold Silver Gold Gold and silver.. Gold Gold and silver. . Gold Gold and silver. . Silver Gold and silver. . Silver Gold Silver Gold and silver. . , Gold Silver Gold Silver Gold Gold and silver. . Gold Gold and silver. .. Silver Gold Silver Gold and silver. . U Value in . S. Money. .40,7 .19,3 .82,3 .54,6 p. 00 .91,2 .93,2 .26,8 .82,3 .04,9 .19,3 4.86.61 .19,3 .23,8 .96,5 .89 .19,3 .88,8 1.00 .89,4 .40,2 .26,8 .82,3 1.08 .65,8 1.00 .19,3 .26,8 .19,3 .74,3 .04,4 .82,3 .19,3 Foreign Exchange. 253 632. The method of finding the cost of foreign bills is essentially the same as that of domestic bills. (Art. 624. ) 1. What is the cost of the following bill on London, at $4.8665 to the £ sterling ? £35 Jf 1 2s. Ne^ York, July 4th, 1880. At sight of tliis first of exchange (the second and third of the sayne date and tenor unpaid), pay to the order of Henry Crosby, three hundred fifty four pounds, twelve shillings sterling, value received, and charge the sa?ne to the account of J. Kii^G & Co. To Geokge Peabody, Esq., London. Analysis.— £354 12s. = £354.6. (Art. 403.) As 4.8665 £1 is worth $4.8665, £354.6 are worth 354.6 times as 354.6 much ; and $4.8665 x 354.6 ■-= $1725,661, the cost. j^^^^^ $1725 661 2. What is the cost of a bill on Liverpool for £345 5s. 6d., at $4,875 to the pound sterling? 3. What is the cost in currency of a bill on Edinburgh for £360.5, exchange being at par and gold 6% premium ? 633. Bills on Paris, Anttverp, and Geneva, are quoted by the number of francs and centimes to a dollar in gold. Note. — Centimes are commonly written as decimals of a Franc. 4. What is the cost of a bill on Paris for 575 francs, at 5.16 francs to a dollar in gold ? Analysis. — As 5.16 francs cost $1, 575 francs will cost as many dollars as 5.16 is contained times in 575, and 575 -^ 5.16 = $111.43, Ans. 5. Find the cost of a bill on Geneva for 750.25 francs, at 5.15-|- fr. to the dollar in gold. 6. Find the cost of a bill for 1000 francs on Antwerp, at 5.17J fr. to a dollar, gold at 1<}( premium ? 354 Percentage. 634. Bills on Bremen, Frankfort, Hamburg, and Berlin, are quoted by the value in U. S. Money of four marks (reichs- marks) in gold. % 7. What cost a bill on Frankfort for 540 marks, at |. 94|- ? Analysis. — Since 4 marks are worth $.945. the worth of 540 marks is 540 times i of $.945, or $127.58, Ans. (Art. 632.) 8. What cost a bill on Berlin for 2800 marks at 1. 96 J in gold ? 635. The method of finding the face of a foreign bill of exchange is essentially the same as that of domestic bills. 9. What is the face of a bill of exchange on London, bought for $4500 at $4.87^ in gold ? Analysis.— Since $4,875 will buy a bill of £1, $4500 will buy as many pounds as $4,875 are contained times in $4500, and $4500 -j- 4.875 = £923.076, or £923 Is. G^d., Ans. 10. What is the face of a bill on Dublin for which $6500 was paid in gold, at $4.86 ? 11. What is the face of a bill on Paris for $2400, exchange being 5.15 fr. to a dollar ? 12. Find the face of a bill on Geneva, which cost $1500 gold, exchange 5.16. 13. Find the face of a bill on Frankfort costing $762 in gold, exchange at 95J. 14. Paid $2000 for a bill on Berlin, exchange 93f ; what was the amount of the bill ? Qu ESTI ONS. 611. What is Exchange ? 612. A draft or bill of exchange ? 613. The drawer of a bill? 614. The drawee? 615. The payee? 616. A sight draft or bill ? 617. A time draft or bill ? 618. What is the acceptance of a draft ? 619. The par of exchange ? 622. What ia domestic exchange? 623. How find the cost of a draft? 625. How find the face ? 626. What is foreign exchange ? 627. A set of exchange ? 620. What are sterling bills? 631. How is foreign exchange quoted? 632. How find the cost of foreign bills ? 635. How find the face ? ^- • 4 h^ 636. Business men have a method of solving practical ques- tions, which is frequently shorter and more expeditious^ than that of arithmeticians fresh from the schools. 637. Their method consists in Analysis, and may, with propriety, be called the Common Sense Method. 638. No specific directions can be given for the analysis of problems. The learner must be taught to depend on his judgynent as a guide. 639. He may, however, be aided by the following : General Principles. 1°. We reason from that which is self-evident, or Tcnoimi, to that which is unknown, or required. 2°. We reason from a part to the whole j as, iohe7i the value of one is given, and the value oftiuo or more of the same hind is required. S°. We reason from the whole to a part j as, when the value of two or more is given, and that of a part is required. Ji°. We reason from a given cause to its effect, or from agfven effect to its cause ; as, when different comhinations of numhers are given, to find the result. Thus, If 3 men can mow 6 acres in 1 day, how many acres can they mow in 5 days ? Or, If to draw 4 tons requires 6 horses, how many horses will be required to draw 8 tons ? 256 General Analysis. Oral Exer cises. 640. 1. If 8 tons of coal cost 140, how much, will 6 tons cost? Analysis. — 1 is | of 8; therefore, 1 ton will cost 1 eighth of $40, which is $5. As 1 ton costs $5, 6 tons will cost 6 times $5, or $30, Ans. Or, thus : 6 tons are f of 8 tons ; therefore, 6 tons will cost f of $40 Now 1 eighth of $40 is $5, and 6 eighths are 6 times $5, or $30, the same as before. 2. If 7 lb. of tea cost 42 shillings, what will 10 lb. cost 3. If 9 sheep are worth $27, how much are 15 sheep worth ? 4. If 10 barrels of flour cost $60, what will 12 barrels cost ? 5. If a man walks 54 kilometers in 6 days, how far does he walk in 15 days ? 6. If 12 men can build 48 rods of wall in a day, how many rods can 20 men build in the same time ? 7. Suppose 75 kilos of butter last a family 25 days, how many kilos will supply them 12 days? 8. If 7 meters of cloth cost $30, how mucli will 9 meters cost ? 9. If 10 bbl. of beef cost $72, how much will 8 bbl. cost ? 10. If 7 acres of land cost $50, what will 12 acres cost? 11. If f ton coal cost $6, what will 5 tons cost? Analysis. — Since 3 fourths ton cost $6, 1 fourth will cost \ of $6, or |2, and 4 fourths, or 1 ton will cost 4 times $2, or $8. Now at $8 a ton, 5 tons will cost 5 times $8, or $40, Ans. 12. If f lb. tea cost 40 cts., what will 12 lb. cost ? 13. If 5 J bbl. apples cost $22, what will 9 bbl. cost ? 14. If -J acre land cost $28, what will 10 acres cost ? 15. If 16 cords of wood are worth $48, how much is | cord worth ? 16. If I of a citron cost 28 cents, what must you pay for 12 citrons ? 17. If I yd. cloth cost $2, what is that a yard ? 18. If 4 lb. ginger cost $f , what will 11 lb. cost ? 19. If 3 melons cost $yV, what will 20 melons cost ? 20. Paid $^0 for 4 slates ; what must I pay for 18 slates ? General Analysis, 257 Written Exercises. 641. The pupil should be required to Analyze each of the following examples, giving results as he proceeds, and be encouraged to invent different solutions. 1. If 60 bbl. flour cost $300, what will 42 bbl. cost ? Analysis. — Since 60 barrels cost $300, 1 barrel costs gV ^^ $300, and $300 4- 60 = $5.00. Again, 42 barrels will cost 42 times as mncli as 1 barrel. Therefore, $5 X 42 = $210, Alls. Note. — Mucli labor may often be saved by indicating the operations required, and cancelling the common factors before the multiplications and divisions are performed. 5 Thus, %^-§- X 42 = %^- X 42 = S210, Ans, 2. A man bought 30 cords of wood for $76.80 ; how much must he pay for 195 cords ? 3. A gentleman bought 85 meters of carpeting for $106.25 ; how much would 38 meters cost ? 4. A drover bought 350 sheep for $525 ; how much would 65 cost, at the same rate ? 5. If 12|^ pounds of coffee cost $1.25, how much will 245 pounds cost ? 6. If 126 bushels of corn are worth $52.92, how much are 84 bushels worth ? 7. Paid $20 for 60 pounds of tea ; how much would 112| pounds cost, at the same rate ? 8. Bought 41 meters of flannel for $16.40; how much would 18|- meters cost ? 9. Bought 18 pounds of ginger for $4.50 ; how much will 20| pounds cost? 10. If a stage goes 84 miles in 12 hours, how far will it go in 108 hours ? 11. If 16 horses eat 72 bushels of oats in a week, how many bushels will 125 horses eat in the same time ? 12. If a railroad car goes 120 miles in 5 hours, how far will it go in 212J hours ? 358 General Analysis, 13. If a steamboat goes 189 kilometers in 12 hours, how far will it go in 5| hours ? 14. If -^ of a cord of wood costs f of a dollar, how much will } of a cord cost ? ANA1.TSIS. — Since y\ cord costs $|, ^^ cord will cost \ of $|, or $|-, and }f or 1 cord will cost, $^ x 13 = %--i~, or 1|. Again, since 1 cord costs $\^-, I cord will cost | of $V- = fi or $li. Or, $1 X-1/-XI zz: Ifx^Xf = If nr %1\, AuS. 15. If f of a yard of cloth cost £-f, how much will | of a yard cost ? 16. If -^ of a ship cost 116000, how much is | of her worth ? 17. If a man pays $47 for building 23|- rods of fence, how much would it cost him to build 213f rods ? 18. A farmer paid $45.42 for building 36| rods of stone wall ; how much will it cost him to build QO-^ rods ? 19. If 7|- meters of satinet cost $9|, how much will 18J meters cost? 20. A ship's company of 30 men have 4500 pounds of flour ; how long will it last them, allowing each man 2J lb. per day ? 21. How long will 56700 pounds of meat last a garrison of 756 men, allowing 1^ lb. apiece per day ? 22. A can chop a cord of wood in 4 hours, and B in 6 hours ; how long will it take both to chop a cord ? Analysis. — Since A can chop a cord in 4 hours, in 1 hour he can chop ^ of a cord ; and since B can chop a cord in 6 hr., he can chop |^ of a cord ; hence both can chop ^ + ^ cord = fW cord in 1 hr. Again, if to chop /^ cord takes both 1 hr., to chop J^ cord will take I hour, and || or a whole cord will take them 12 times i hr., or 2| hours. 23. If a man can plant a field in 8 days, and a boy in 12 days ; how long will it take both to plant it ? 24. A can do a piece of work in 20, B in 40, and C in 60 minutes ; how long would it take all together to do it ? 25. A cistern has 3 faucets, one of which will empty it in 5 hr., another in 10 hr., and the other in 15 hr. ; how long will it take all 3 to empty it ? General Analysis. 259 26. If 10 men require 8f days to finish a piece of work, how long will it take 11 men to finish it ? 27. A water-tank has 3 pipes; the first will empty it in 12 hr., the second will fill it in 6 hr., the third in 8 hours ; in how many hours will the tank be filled if all run together? 28. A deer starting 150 rods before a dog, runs 30 rods a minute ; the dog follows at the rate of 42 rods a minute. In what time will the dog overtake the deer ? Oral Exercises. 642. 1. A grocer sold 8 lb. of sugar at 12 cents a pound, and took his pay in lard, at 10 cents a pound ; how much lard did it take to pay for the sugar ? Analysis. — Since 1 lb. of sugar is worth 12 cents, 8 lb. are wortli 8 times 12, or 96 cents. Again, since 10 cents will pay for 1 lb. of lard, 96 cents will pay for as many pounds as 10 cents are contained times in 96 cents, or 9| pounds. Or, thus : 13 cents are jf of 10 cents ; therefore, 1 lb. of sugar is worth ^ lb. of lard ; and 8 lb. of sugar are worth 8 times {§ lb. of lard, which is fl lb. = 9i% or 9| lb., Ans. 2. How many dozen eggs, at 15 cents a dozen, will pay for 12^ yards of muslin, worth 8 cents a yard ? 3. A farmer exchanged 8 tons of hay, worth $20 per ton, for flour worth $6 a barrel ; how many barrels of flour did he receive for his hay ? 4. A man exchanged 50 lb. of wool, valued at 37^ cents a pound, for flannel worth 87J cents a yard ; how many yards did he obtain ? 5. A lad bought 75 apples, at the rate of 3 for a cent, and exchanged them for oranges worth 5 cents apiece ; how many oranges did he receive ? 6. How many slate pencils worth -J- cent apiece, can you buy for 150 steel pens worth 4 cents per dozen ? 7. How many acres of farming land worth $12|- per acre, must be given in exchange for 4 building lots in the city, valued at $75 per lot ? 260 General Analysis, 8. How much lard at 27 cents a kilo, will pay for 153 Kg. of rice, worth 9 cents a kilo ? 9. How many oranges at 7J cents apiece, can you buy for \ of 35 quarts of strawberries, at Vl\ cents a quart ? 10. A lad bought 12 peaches, at the rate of 3 for 4 cts., and afterwards exchanged them for oranges which were 3 for 8 cents ; how many oranges did he obtain ? 11. Frank sold 10 apples, which was f of all he had ; he then divided the remainder equally among 5 companions; how many did each receive ? 12. Lincoln spent 60 cents for a book, which was \\ of his money ; with the remainder he bought oranges, at 4 cents apiece ; how many oranges did he buy ? 13. A man paid away $35, which Avas f of all he had ; he then spent the rest in cloth at 12 per yard ; how many yards did he obtain ? 14. A farmer bought a quantity of goods, and paid 120 down, which was f of the bill ; how many cords of wood, at $3 per cord, will it take to pay the balance ? Written Exercises. 643. 1. A merchant sold 75 yd. silk, at 10.84 a yd., and took his pay in corn, at $0.60 a bu. ; how many bu. did he receive? Analysis.— At $.84 a yd. 75 yards are worth $.84 x 75 = 63. Again, to pay $63.00 will require as many bushels of corn as $.60 are contained times in $63.00, and $63.00 -^ .60 = 105 bu., Ans. 2. How many pounds of butter, at 35 cents, must be given in exchange for 186^ yards of calico, at 18J cents per yard ? 3. How many pounds of tobacco, at 16J cents, must be given for 256 pounds of sugar, at 6J cents a pound? 4. A farmer bought 325 sheep at $2J- apiece, and paid for them, in hay, at $10|- per ton ; how many tons did it take ? 5. A man bought a hogshead of vinegar, worth 37-|- cents per gallon, and gave 331^ pounds of cheese in exchange ; how much was the cheese a pound ? General Analysis. 261 6. Bought 274 bushels of salt, at 42^ cents per bushel, and paid in wheat at $1.25 per bushel ; how many bushels of wheat did it require ? 7. A bookseller exchanges 400 dictionaries worth $1.87|- cents apiece, for 900 grammars ; how much did the grammars cost apiece ? 8. How many meters of silk, worth llf per meter, will pay for 249-|- meters of cloth worth 15:^ per yard ? 9. Bought 263| yds. of satinet, at $lf per yard, and paid for it in cheese, at $9-|- per hundred ; how much cheese did it take ? 10. Bought 25 hhds. 22 gals. 3 qts. of molasses at 37^ cents per gallon, and paid for it in wool, at 62|^ cents a pound ; how much wool did it take ? 11. Bought 432 sheep at $2 J apiece, for which I paid 144 barrels of flour ; what was the flour per barrel ? 12. If 15 yards of flannel are worth 25 yards of muslin, how many yards of flannel are worth 315 yards of muslin ? 13. A market-woman bought 10 dozen oranges, at the rate of 3 for 4 cents, and then exchanged them for eggs, at the rate of 4 for 5 cents ; how many eggs did she receive ? 14. If 15 lbs. of pepper are worth 25 lbs. of ginger, how many pounds of ginger must be given for 195 lbs. of pepper? Oral Exercises. 644. 1. What cost 36 bushels of oats, at 33| cts. a bushel ? Analysis. — At $1 a bushel 36 bu. would cost $36 ; but the price is 33| cents, or %\ ; lience, at %\, 36 bu. will cost I of $36, wliicli is $12, Ans. (Art. 280.) 2. At 12|- cts. a pound, what cost 72 lbs. of maple sugar ? 3. At 20 cts. apiece, what will 150 melons come to ? 4. What cost 60 rolls of tape, at Q\ cts. a roll ? 5. What cost 72 yd. delaine, at 16f cts. a yard ? 6. At 25 cts. a yard, what must I pay for 64 yards of ribbon ? 7. At 50 cts. a bushel, what will 250 bushels of corn cost ? 8. At 33^ cts. apiece, what cost 12 doz. Grammars? 262 General Analysis, Written Exercises. 645. lo What will 1268 bushels of apples come to, at 25 cts. a bushel ? Analysis.— 25 cts. = %\ ; therefore the apples will cost \ as many dol- lars as there are bushels, and 1268-7-4 = 317. Ans. $317. 2. At 8 J cts. apiece, what cost a gross of slates ? 3. What cost 480 yards of ribbon, at 16f cts. a yard ? 4. At 33-J cts. a hektoliter, what must I pay for 750 hekto- liters of potatoes ? 5. What cost 1250 melons, at 20 cents each ? 6. At 50 cents apiece, how much will 1745 Readers cost? Oral Exercises. 646. 1. Two boys formed a partnership in selling news- papers ; A put in 30 cts. and B 50 cts. They gained 40 cts. the first day ; what was the share of each ? Analysis. — Their capital was 30 cts. + 50 cts. = 80 cts. A's part of it was f §, or | ; and B's part was |§, or |. Now I of 40 is 5 cts., and | are 3 times 5, or 15 cts., A's share. And f are 5 times 5, or 25 cts., B's share. 2. A and B bought a pony together for $100 ; A put in $60 and B $40 ; they sold it so as to gain 130 ; what was each one's share of the gain ? 3. Two men buy a carriage together for $500 ; A put in $300 and B $200 ; they sold it at a loss of $150. What was the share of each in the loss ? 4. C and D joined in a speculation and cleared $90 ; C put in 1400 and D 1800 ; what share of the gain had each ? 5. A man failed in business, owing A $700 and B $400; his property was valued at $880 ; how much would each creditor receive ? 6. B and C engaged in business; B furnished $900 and C $600 ; they made $300 ; what was the share of each ? General Analysis. 263 Written Exercises. 647. 1. A, B, and 0, formed a partnership ; A put in $2000, B 13000, and $4000 ; they gained $2700 ; what was each man's share of the gain ? Ai^ALYSis.— The capital was $2000 + $3000 + $4000 = $9000. Since A's part of the capital was |§^§, or |, his share of the gain was I of $2700, and f of |2700 = $600 A's gain. B's part was f g^g, or f of $2700, and | of $2700= $900 B's gain. C's part was f§§§, or | of $2700, and f of $2700= $1200 C's gain. Proof, $2700 Whole gain. 2. A, B, and C hired a farm together, for which they paid $175 rent ; A advanced $75, B $60, and C $40. They raised 250 bushels of wheat ; what was each man's share ? 3. A, B, and C together spent $1000 in mining stocks. A put in $400, B $250, and C $350. They gained $1500 ; how much was each man's share ? 4. A, B, C, and D fitted out a whale ship ; A advanced $10000, B $12000, $15000, and D $8000. The ship brought home 3000 bbls. of oil ; what was each man's share ? 5. A, B, and formed a partnership ; A furnished $900, B $1500, and $1200. They lost $1260 ; what was each man's share of the loss ? 6. X, Y, and Z entered into a joint speculation, on a capital of $20000, of which X furnished $5000, Y $7000, and Z the balance. Their net profits were $5000 per annum ; what was the share of each ? 7. A bankrupt owes one of his creditors $300, another $400, and a third $500. His property amounts to $800 ; how much can he pay on a dollar, and how^ much will each of his creditors receive ? 8. A bankrupt owes $2000, and his property is appraised at $1600 ; how much can he pay on a dollar? 9. A man failing in business, owes A $156.45, B $256.40, and C $360.40 ; and his effects are valued at $317 ; how much will each man receive ? 264 Gi^ueral jLualyals. 10. The assets of a man failing in business amonnted to $3560; he owed $35600; how much can he pay on a dollar, and how much will B receive, who has a claim of $5000 ? 11. A man died insolvent, owing $55645, and his property was sold at auction for $2350 ; how much will his estate pay on a dollar ? 12. A, B, and C sent flour by sloop from New York to Bos- ton. A had 600 bbl.*, B 400 bbl., and C 200 bbl. In a gale 200 bbl. were thrown overboard ; what was the loss of each ? 13. A and B formed a partnership ; A put in $300 for 2 months and B $200 for 6 months. They gained $150 ; what was each man's just share of the gain ? Suggestion. — The gain of each depends both upon the capital he fur- nished, and the time it was employed. (Art. 583.) A's capital $300 x 3 = $600, the same as $600 for 1 mo. B's " 200 X 6 = 1200, " " 1200 Whole capital, $1800 A's share must therefore be -^^^^ = i of $150, or $50. B's " " " '* i|^^ = f of $150, or $100. Pkoof.— $50 + $100 = $150, the gain. 14. A, B, and C enter into partnership ; A puts in $500 for 4 mo., B $400 for 6 mo., and C $800 for 3 mo. ; they gain $340 ; what is each man's share of the gain ? 15. A and B hire a pasture together for $60 ; A put in 120 sheep for 6 months, and B put in 180 sheep for 4 months ; what should each pay ? 16. The firm A, B, and C lost $246 ; A had put in $85 for 8 mo., B $250 for 6 mo., and C $500 for 4 mo.; what is each man's share of the loss ? 17. Smith and Jones graded a street for $857.50. S. fur- nished 5 men for 20 days and 6 men for 15 days ; J. furnished 10 men for 12 days and 9 men for 20 days ; what was the share of each contractor ? 18. Three men hire a farm of 250 A., at $8|- an acre ; A put in 244 sheep, B 325, and C 450 ; what rent ought each to pay? General Analysis, 265 Oral Exercises. 648. 1. If f ton of hay costs 115, what will a ton cost ? Analysis. — lu this example we have a 'part of a number given, to find the whole. Since 15 is f of the number. \ of it is \ of 15, which is 5, and I are 4 times 5, or 20. Ans, $20. Note. — In solving examples of this kind a difficulty often arises from supposing that if f of a certain number is 15, \ of it must be \ of 15. This mistake will be easily avoided by substituting the word imrts for the given denominator. Tlius, if 3 parts cost $15, 1 part will cost \ of $15, which is $5. But this part is a fourth. Now if 1 fourth cost $5, 4 fourths will cost 4 times as much. 2. A builder paid 120 for | of an acre of land ; what was that per acre ? 3. A boy being asked how many pears he had, replied that he had 50 apples, which was f the number of his pears ; how many pears had he ? 4. Henry lost 42 yards of his kite line, which was f of his whole line ; what was its length ? 5. 50 is f of what number ? |^ of what ? -| ? y\ ? 6. 75 is J of what number ? f of what ? -^ ? -^\ ? 7. 100 is I of what number ? | of what ? f ? H ? 8. A man bought a yoke of oxen, and paid 156 in cash, which was ^ of the price of them ; what did they cost ? 9. A merchant bought a quantity of wood and paid $45 in goods, which was f of the whole cost ; how much did he pay for the wood ? 10. A man bought a buggy and paid S45 down, which was ^ of the price of it , what was the price, and how much did he owe ? 11. The crew of a whale ship having been out 24 months, found they had consumed ^ of their provisions ; how many months' proyisions had they when they embarked, and how much longer would their provisions last ? 12. A man bought a meadow and paid $75 cash which was I of the price, and gave his note for the balance ; how large was the note ? 266 General Aoialysis. Written Exercises. 649. 1. A man being asked how far he had traveled, replied that 140 miles equaled ^\ of the distance ; how far had he traveled? Analysis.— Since 140 mi. is /g, 4^ is f of 140 mi., or 20 miles. As 20 mi. is aV of the distance, || is 25 times 20 mi., or 500 miles, Ans. 2. 560 is f J of what number ? 3. 1500 is I J of what number ? 4. 2000 is If of what number ? 5. A man paid $150 for a carriage, which was ff of what he sold it for ; what did he get for his carriage ? 6. A builder paid 1145 for -| A. of laud ; what was that an acre ? 7. A man pays $0.96 for | bu. of wheat, what is that a bu. ? 8. A lad being asked how many pears he had, replied that he had 150 apples, which was -^ the number of his pears ; how many pears had he ? 9. 680 is I of what number ? 10. 1260 is If of what number ? 11. A man traveled 240 miles by railroad, which was f the distance he traveled by steamboat ; how far did he go by boat ? 12. If 4 times | of $32 is f the price paid for a cow, what did she cost ? 13. 7 times \ of 28, is ^ of what number ? 14. 7 times f of 36 cts. is ^ of the price of a Dictionary ; what is the price ? 15. A man spent $560 for books which was |-§- of his money, and bought hay with the remainder, at $16 a ton ; how much hay did he buy ? 16. A tailor bought a horse and paid $120 in cash, which was ^\ of the price ; how many coats at $24 apiece will it take to pay the balance ? 17. A grocer sold 2205 lbs. butter, which was }| of all he had ; how many tubs would hold the rem-ainder, allowing 42 lb. to a tub ? General Analysis. 267 18. A lad being asked how many peaches he had in his basket, replied that J, J, and ^ of them made 104 ; how many had he ? Analysis.— The sum of \, a, and \ = if. (Art. 195.) Now if 104 is 13^ ^^ is _i_ of 104, vvliicli is 8 ; and [f is 8 x 12 = 96. Ans. 96 peaches. 19. A farmer lost J his sheep by sickness, \ by wolves, and he had 72 sheep left ; how many had he at first ? 20. A person having spent \ and -J of his money, finds he has 148 left ; what had he at first ? 21. After a battle a general found that \ of his army had been taken prisoners, \ were killed, -^^ had deserted, and he had 900 left ; how many had he before the battle. 22. A certain post stands -J in the mud, J in the water, and 20 feet above the water ; how long is the post ? 23. Suppose I pay r^l85 for -| of an acre of land ; what is that per acre ? 24. A man paid $2700 for f^ of a vessel ; what is the whole vessel worth ? 25. A gentleman spent ^ of his life in Boston, J of it in New York, and the rest of it, which was 30 years, in Philadel- phia ; how old was he ? 26. AVhat number is that, -| of which exceeds -^ of it by 10? 27. In a school \ were studying arithmetic, \ algebra, \ geometry, and the remaining 18 were studying grammar ; how many pupils were in the school ? 28. A owns -J and B -^ig of a ship ; A's part is worth $650 more than B's ; what is the value of the ship ? 29. In a certain orchard ^ are apple trees, \ peach trees, \ plum trees, and the remaining 15 were cherry trees ; how many trees did the orchard contain ? 650. 1. A merchant paid $1165.25 for a case of goods and sold them at 15^^ advance ; what was the profit ? Analysis.— The profit was JjV of !^1165.25. Now ^ = |11.6525, and fl^ = $11.6525 X 15 = $174.7875, Aus. 2. A man sold a house for $2969.50, which was 25^ more than it cost him ; what did it cost him ? Analysis. — Since he gained 35%, he received $1.25 for each dollar of coet. Now $2969. 50 -r- $1.25 = $2375.60, Ans. 268 General Analysis, 3. Eeceived $4100 to buy stock, after deducting 2^% com- mission ; how many shares at par can I buy ? 4. What is the premium at If^ for insuring $3560 on a house and furniture ? 5. What sum must be insured on goods worth $4760, at ^^%, to cover both the goods and premium ? 6. What is the specific duty on 175 pieces of silk, each con^ taining 50 yd., at 25 cents a yard ? 7. What is the int. of $765.50 for 3 yr. 8 mo., at Q% ? 8. If $850 at simple interest amounts to $986 in 2 years, what is the rate per cent ? 9. When money is at 6^, what is the present worth of $4218, due in 1 yi\ 6 months? 10. What is the bank discount on a note of $1640.50 for 90 days, at b% ? 11. What must be the face of a note on 60 d., at %%, to yield $1000, if discounted at a bank ? 12. What is the equated time for the payment of $400, due in 3 mo., and $600, due in 5 months ? 13. In a mercantile house, A's capital is $4500, B's $5200, and C's $5300 ; they make $3000 ; what is the profit of each ? 14. What cost a sight draft on New Orleans for $750, at 2f ^ premium ? 15. What cost a bill of exchange on Liverpool for £800 10s., at$4.86i? 651. 1. A father divided $2700 among his 3 children in the proportion of 2, 3, and 4 ; how much did each receive ? Analysis.— The sum of 2 + 3 + 4 = 9. Hence, the first received $2, the second $3, and the third .$4, as often as 9 is contained in $3700 ; and 9 is contained in $2700, 300 times. Therefore, the first received $800 x 2 = $600 ; the second $300 x 3 = $900 ; the third $300 x 4 = $1200. Pkoof.— $600 + $900 + $1200 = $2700. 2. A man had 756 sheep which he divided into 2 flocks in the proportion of 3 to 4 ; how many were in each flock ? General Analysis. 269 3. Divide 1248 into 2 sucli parts that one shall be to the other as 2 to 6. 4. Divide 435 into 2 such parts that one shall be 3 times the other. 5. Two farmers have 755 acres of land, one having 4 times as many acres as the other ; hovt^ many had each ? 6. Two families, one containing 4 persons the other 5, hired board together for $2954 a year ; what proportion ought each family to pay ? 7. Divide the number 720 into 3 parts in proportion to 3, 4, and 5. 8. Divide $650 among 4 persons so that their shares shall be to each other in the proportion of J, \, |, and y\. Analysis. — Since one part is \ or VV share, another \ or -^^ share, another f or ^^, and the other /^ of a share, the whole is \% share, and 1 share equals $650 -^|| = $300. Hence, \ share is $150, \ share is $100, I, $225, yV, $175, Ans. Peoof.— $150 + $100 + $235 + $175 = $650. 9. Divide 945 into 3 parts which shall be to each other in the proportion of J, -J, -f^. 10. What number added to 5 times itself will make 576 ? Analysis. — A number added to 5 times itself will make 6 times that number. Since 576 is the product of two factors, one of which is 6, the other factor must be 576-f-6 = 96, Ans 11. What number added to \ of itself will make 369 ? 12. What number added to 4|- times itself will make 60-^ ? 13. A man being asked how far he had walked, replied that the number of kilometers he had traveled was 364, and he had ridden twice as far as he had walked ; how many kilometers had he walked ? 14. A lad bought apples, pears, and peaches, in all 280 ; the number of his apples was twice that of his pears, and the num- ber of his pears was twice that of his peaches ; how many of each did he buy ? 15. Divide 192 into three such parts that the first shall be twice the second^ and the third three times the second. 270 General Analysis. 652. 16. If 4 men can mow 48 acres of grass in 5 days, how long will it take 9 men to mow 60 acres ? Analysis. — Since 4 men can mow 48 acres in 5 d., 1 man can mow \ of 48 A., or 12 A. Now if 1 man can mow 12 A. in 5 d., in 1 d. lie can mow \ of 12, or 2| A. Again, since to mow 2| A. requires 1 man 1 d., 60 A. will require liira as many days as 2| are contained times in 60 = 25 d. ; and since it takes 1 man 25 d., it will take 9 men i of 25 = 2| d,, Ans. 17. If 14 men can build 84 rods of wall in 3 days, how long will it take 20 men to build 300 rods ? 18. If 1000 hektoliters of provisions will support a garrison of 75 men for 3 months, how long will 3000 hektoliters support a garrison of 300 ? 19. If 7 men can reap 42 acres in 6 days, how many men will it take to reap 100 acres in 5 days ? 20. If a man travels 320 miles in 10 days, traveling 8 hours per day, how far can he go in 15 days, traveling 12 hours per day ? 21. If 24 horses eat 126 bushels of oats in 36 days, how many bushels will 32 horses eat in 48 days ? 653. 22. A farmer wishes to mix a quantity of corn worth 75 cts. a bushel, with oats worth 37J- cts. a bushel, so that the mixture may be worth 50 cts. a bushel ; what part of each must he take ? Analysis. — Since the mixture is worth 50 cts. a bushel, on every bushel of corn he puts in, the loss is 25 cts., and on every bushel of oats, the gain is 12| cts. Since it requires 1 bu. oats to gain 12^ cts., to gain 25 cts. will require as many bu. of oats as 124 cts. are contained times in 25 cts., and 25-4-12^ = 2. Hence, he must take 2 bu. oats to 1 bu. corn. Proof.— A mixture of 3 bushels is worth 37| cts. + 37i cts. + 75 cts. — $1.50 ; hence, 1 bu, mixture is worth 50 cents. Note. — The principle by which this and similar examples are solved, is that the excess of one article alove the mean price of the mixture, coun- terbalances the deficiency of another article which is below it. 23. A tea merchant has two kinds of tea worth 40 cts. and 90 cts. a pound, and wishes to make a mixture worth 60 cts. a pound ; what part of each must he take ? 24. How much ginger at 24 cts. and 30 cts. a pound, will form a mixture worth 25 cts, a pound ? Definitions. 654. Ratio is the relation of one number to another. It is found by dividing one by the other. Thus, tlie ratio of 6 to 3 is 6 h- 3, and is equal to 3. 655. The Terms of a Ratio are the numbers compared. 656. The Antecedent of a ratio is the Jirst term. 657. The Consequent is the second term. 658. The two terms together are called a Couplet. Thus, in the ratio 9:3, 9 is the antecedent, 3 the consequent, and 9 and 3 together form a couplet. 659. Ratio is commonly denoted by a colon ( : ), whicli is a contraction of the sign of division. Thus, the ratio of 6 to 3 is written " 6 : 3," and is equivalent to 6 -=-3. 660. Ratio is also denoted by writing the consequent under the antecendent in the form of a fraction. Thus, the ratio of 8 to 4 is written | , and is equivalent to 8 : 4. Oral Exercises. 661. 1. What is the ratio of 48 : 6 ? Of 63 : 7 ? Of 72 : 8 ? 2. What is the ratio of 21 : 42 ? Of 15 : 45 ? Of 25 to 100 ? 3. What is the ratio of 8 lb. to 40 lb. ? Of 54 yd. to 6 yd. ? 4. What is the ratio of 150 : llO ? Of $25 : $100 ? 5. Find the ratio of 6 ft. : 3 hr. Ans. The ratio cannot be found, because one of these num- bers is neither equal to nor a part of the other. Hence, the Principle. 662. Only like numbers can he compared with each other. 272 Ratio, 663. A Simple Ratio is the ratio of two numbers, as 8 : 4. 664. A Compound Ratio is the product of two or more sim- ple ratios. They are commonly denoted by placing the simple ratios under each other. Thus, 4:2) ^noo- j^- ^ > or, 4 X 9 : 2 X d, IS a compound ratio. 9 : o ) 665. A Compound Ratio is reduced to a simple one by mak- ing the product of the antecedents a new antecedent^ and the product of the consequents a new consequent. 666. A Reciprocal of a Ratio is a simple ratio inverted, and is the same as the ratio of the reciprocals of the two numbers compared. Thus, the reciprocal of 8 to 4 is | to |- = 4 : 8, or |. Note. — 1. Reciprocal Ratio is sometimes called Inverse Ratio. 2. The reciprocal of a ratio, when a fraction is used, is expressed by- inverting the terms of the fraction which expresses the simple ratio. Wlien the colon is used, the order of the terms is inverted. 667. The ratio between two fractions which have a common denominator, is the same as the ratio of their numerators. Thus, the ratio of f : f is the same as 6 : 3. Note. — When the fractions have different denominators, reduce them to a common denominator ; then compare their numerators. Compound numbers must be reduced to the swme denomination. Written Exercises. 668. Find the following ratios in the lowest terms : 1. 95 to 25. 5. 65 to 180. 9. f| to f^. 2. 110 to 48. 6. 84 to 132. 10. « to If. 3. 135 to 51. 7. 108 to 256. 11. M to ^. 4. 186 to 84. 8. 220 to 500. 12. ^ to 4^. 13. 96 gal. to 24 qt. 17. 15s. to 4s. 6d. 14. 75 bu. to 160 pk. 18. 10 ounces to 95 pounds. 15. 140 rd. to 20 ft. 19. 8 yards to 9 inches. 16. 175s. to 130d. 20. 3 pnits to 4 gallons. Ratio. 273 669. Since the antecedent corresponds to the numerator, and the consequent to the denominator, changes on the terms of a ratio, have the same effect upon its value as like changes have upon the terms of a fraction. Hence, the Principles. i°. Mtdtipluinq the antecedent, or) ,, ,,. ,. T^. .t jX j_ r Multiplies the ratio. Dividing the consequent. ) ^ 2°. Dividinq the antecedent, or \ t^. . . ,-, ,^ ,,. , . ^, , \ Divides the ratio. MuUijnying the consequent, ) S°. Multiplying or dividing hoth \ Does not alter the value terms iy the same quantity, ) of the ratio. 670. The ratio, antecedent, and consequent are so related to each other, that if any two of them are given the other may be found. Hence, the i 1. The Ratio = Antecedejit -^ Consequent. Formulas. I 2. The Consequent = Antecedent -i- Ratio. ( 3. The Antecedent = Consequent x Ratio. 21. What is the ratio, when the antecedent is 63 and the consequent 9 ? 22. When the antecedent is 25 and the consequent 60, what is the ratio ? 23. If the antecedent is 8 and the ratio 14, w^hat is the con- sequent ? 24. When the consequent is 16 and the ratio 7, what is the antecedent ? 25. When the antecedent is 6J and the consequent 8, what is the ratio ? 26. If the antecedent is 5^ and the ratio 9f, what is the consequent ? 27. When the consequent is 24 and ratio 8, what is the antecedent ? 28. If the consequent is 36 and ratio 12, what is the antecedent ? 29. When the antecedent is 9| and ratio 8|, what is the consequent ? ROPORTION. Oral Exercises. 671. 1. What is the ratio of 12 to 60 ? Of 9 to 36 ? 2. Which of the above ratios is the larger ? 3. How does the ratio of 30 to 6 compare with 15 to 3 ? 4. How does the ratio of 5 to 25 compare with the ratio of 12 to 60 ? 5. Are all ratios equal ? 6. How does the ratio of 12 to 4 compare with the ratio of 10 to 5 ? 7. Name two equal ratios. Name tAvo others. 8. Name two unequal ratios. Name two others ? 9. What two numbers have the same quotient as 16 divided by 4 ? As 28 divided by 4 ? As 60 divided by 12 ? 10. Express in both forms the ratio of two numbers which have the same ratio as 6 to 12. 11. How does the ratio of 7 to 14 compare with the ratio of 5 to 20 ? 12. How does the ratio of $15 to 15 compare with the ratio of 18 ft. to 6 ft. ? Ans. They are equal to each other. 672. Proportion is an equality of ratios. Thus, the ratio 8 : 4 = 6 : 3, is a proportion. That is, Four quantities are in propoi'tion, when the first is the same multiple or part of the second, that the third is of the fourth. 673. The Sign of Proportion is a double colon ( : : ), or the sign ( = ). Thus, the proportion above is expressed 8 : 4 : : 6 : 8. Or, 8 : 4 = 6 : 8 l^he first form is read " 8 is to 4 as G to 3." The second is read " the ratio of 8 to 4 equals the ratio of 6 to 3." Proportion, 275 674. The Terms of a proportion are the numbers compared. 675. The Antecedents of a proportion are ^(i first and third terms. 676. The Consequents are the second dmdi fourth terms. Thus, in the proportion 4:8:: 3:6, the 4 and 3 are the antecedents, and 8 and 6 the consequents. 677. In every proportion there must be at least four terms ; for the equaUty is between tivo or more ratios, and each ratio has two terms. 678. A proportion may, however, be formed from three numbers, for one of the numbers may be repeated, so as to form ttvo terms ; as, 2 : 4 : : 4 : 8. Note, — When a proportion is formed of three numbers, the middle number is called a mean proportional. 679. The Extremes of a proportion are the first and last terms. 680. The Means are the tivo middle terms. Thus, in the proportion 9 : 12 : : 18 : 24, 9 and 24 are the extremes, 12 and 18 the means. Eead the following : (Art. 673.) 1. 35 : 7 =r GO : 12. 7. 18 : 54 : : 21 : 63. 2. 42 : 14 = 75 : 25. 8. 23 : 92 : : 34 : 136. 3. 72 : 24 = 168 : 56. 9. 37 : 148 : : 41 : 164. 4. 144: 1 :: 1728:12. 10. 3.9 • . 8-14 • • 2 . 3 "8 • T- 5. 20 : 143y3g : : 2| : 17. 11. 816.05 : 85.35 : : 827.03 : 89.01. 6. 4^ : 54 : : 6 : 72. 12. 60 : 15 : : 80 : 20. 681. The relation of the four terms of a proportion to each other is such, that if any three of them are given, the other or missing term may be found. 276 Proportion. J)ErELOP3I ENT OF I* R I N C I P L E S . 682. 1. If the first three terms of a proportion are %, 4^ and 5, what is the fourth or missing term ? Analysis. — Representing the missing term by x, then tlie proportion is 2 6 2 : 4 : : 6 : a", and the ratio -a=-- These fractions reduced to a common 2x3? 4x6 denom. become - — ; = -r — - ; hence the numerators are equal. (Art. 667.) rt X t*/ rt X it/ But 2 X .1' is the product of the extremes and 4x6 the product of the means. CanceUing the factor 2, which is common to both, a; = 2 x 6, or 12, is the missing term required. 683. From the preceding example we derive the following Principles. i°. Bi every proportion the product of tlie extremes is equal to the jjrodud of the means. 2°. The 2^^oduct of the extremes divided ly either of the means, gives the other mean. 3°. The product of the means divided ly either extreme, gives the other extreme. 684. Find the missing term in the following : 2. 12 : 42 = 20 : x. 8. 400 rd. : 56 rd. = 195 : x. 3. 9 : 153 = 150 : x. 9. r. : 400 vests = $87.50 : IIOOO. 4. 175 : $900 = x:S5. 10. 130 lb. : x = $150 : 1850. 5. x:4:0 = 120 : 100. 11. 40 gal. : x = 180 : 60. 6. 24r:x = 12 : 144. 12. 16 yd. : 10 ft. = 72 : a:. 7. 187.5 : 7i = ^ : 15. 13. x : 75 = | : f. Simple Proportion. 685. Simple Proportion is an equality of two simple ratios. Note.— Of the three given numbers, two must always be of the same kind, and the third the same as the answer required. Simjple Pro])ortion, 277 Written Exercises. 686. To solve problems by Simple Proportion or by Analysis. 1. If 15 books cost $45, what will be the cost of 80 books ? BT ANALYSIS. OPERATION. Since 15 books cost $45, 1 book costs J^ of $45, ^ and 80 books will cost 80 times as much, or 4^ X o O __ ^^^ $240, Am. ^$ Ans. $240. BY PEOPOKTION. ) ^' As the answer is to be money, we operation. make $45 the third term ; and as 15 : 80 : : $45 : Ans. the cost of 80 books will be more 3 than that of 15 books, we make 80 ^^ y^ 80 the second term and 15 the first. And — ^ — = $240, Ans, We now have the two means and one extreme, to find the other extreme. Hence, we divide the product of the means by the given extreme, and the quotient is the other extreme or answer. (Art. 683.) Proof.— 80 x 45 = 240 x 15. (Art. 683, i°.) 687. From the preceding principles we have the following Rule. — I. Malce that number the third term, ivhich is the same hind as the answer. II. When the answer is to he larger than the hhird term, mahe the larger of the other two numbevs the second term; but when less, jjlace the smaller for the second term, and the other for the first. III. Multiply the second and third terms together, and divide the product by the first ; the quotient will be the fourth teJin or answer. Proof. — // the product of the first and fourth terms equals that of the second and third, the answer is right. Notes. — 1. The arrangement of the terms in the form of a proptyrtion is called " The Statement of the question." 2. The factors common to the first and second, or to the first and tMrd^ terms, should be cancelled. 278 Proportion. 3. The fij'st and secoiid terms must be reduced to the same denomina- tion. The third term, if a compound number, must be reduced to the lowest deno7Mnation it contains. 4. It is advisable for the pupil to solve the following examples both by- Proportion and by Analysis. 2. If 14 vests are worth 184, what are 23 vests worth ? By Analysis. — If 14 vests are worth $84, 1 vest is worth 1 fourteenth of $84, ot $6, and 23 vests are worth 23 times $6, or $138, Ans. By PfiOPORTiON.— 14 V. : 23 V. : : $84 : Ans. That is, 14 is the same part ot 23 as $84 are of the cost of 23 vests. Cancelling, etc., 6 1^ : 23 : : %U : 1138, Ans. 3. If 5 men can mow a meadow in 6 days, how long will it take 8 men to mow it ? 4. If 6 acres and 40 rods of land cost 1125, how much will 25 acres and 120 rods cost ? 5. If 15 meters of silk cost £4 10s., what will 75 meters cost ? 6. If a railroad car goes 35 mi. in 1 hr. 45 min., how far will it go in 3 days ? 7. If 84 lbs. of cheese cost $5|, what will 60 lbs. cost ? 8. If f of a ship is worth $6000, liow much is ^ of her worth ? 9. If a ship has sufficient water to last a crew of 25 men for 8 months, how long will it last 15 men ? 10. If the interest of $1500 for 12 mo. is $90, what will be the interest of the same sum for 8 mo. ? 11. If a tree 20 ft. high casts a shadow 30 ft. long, how long will be the shadow of a tree 50 ft. high ? 12. How long will it take a steamship to sail round the globe, allowing it to be 25000 miles in circumference, if she sails at the rate of 3000 miles in 12 days ? 13. How many hektars of land can a man buy for $840, if he pays at the rate of $56 for every 7 hektars? Cau8e and Effect 279 Cause and Effect. 688. The principles of proportion may also be explained by clie relations of the terms to each other, as causes and effects, 689. A Cause is that which produces something. An Effect is something which is produced. Thus, men at work, goods bought or sold, time, money lent, etc., are causes. Work done, provisions consumed, cost of goods, etc. , are effects. 690. In arithmetical operations it is assumed that like causes produce like effects, and the ratio between the effects is equal to the ratio between the causes which produce them. If 2 horses as a cause can move 3 tons as an effect, 6 horses as a cause will remove 9 tons as an effect ; that is 2 horses (1st C.) : 6 horses (2d C.) : : 3 T. (1st E.) : 9 T. (2d E.) Written Exercises. 691. 1. If 4 acres produce 60 bushels of wheat, how nauch will 9 acres produce ? Analysis. — In this example, the two causes are 4 acres and 9 acres; the first effect is 60 bu., the second effect is required. We make 60 bu. the given effect, the third term, and since the second effect must be greater than the first, we make 9 A., the greater cause, the second term, and 4 A. the first. Multiplying and dividing as before, the result is 135 bu., Ans. 2. If it requires 4 acres to produce 60 bushels of wheat, how many acres are required to produce 135 bushels ? Analysts. — In this example operation. two effects, 60 bii. and 185 bu., 1st. E. 2d E. 1st C. and one cause, 4 A., are given, 60 bu. : loO DU. : : 4 A. : Ans. the second cause is required. (135 X 4) -J- 60 = 9 Since the 2d effect is greater than Ans 9 acres the first, the 2d cause must also be greater than the given cause ; we therefore make 135 bu. the 2d term and 60 bu. the 1st term. The result is 9 acres. 1st C. 2d C. IstE. 4 A. : 9 A. : : 60 bu. : Ans. (60 X 9) . -^ 4 = 135 Ans. 135 bu. 280 Proj[>ortion. 692. When the terms of a proportion are considered in the relation of cause and effect^ the operations are the same as when considered in the relation of magnitude. (Arts. 687, 689.) Solve the following by either or both the preceding methods: 3. Bought 41 yd. of flannel for 116.40; how much would 8| yd. cost ? 4. Bought 18 kilos of ginger for 18.50; how much will 10| kilos cost ? 5. If a stage goes 84 kilometers in 12 hours, how far will it go in 15|- hours ? 6. If 26 horses eat 72 hektoliters of oats in a week, how many hektoliters will 25 horses eat in the same time ? 7. If a railroad car runs 125 kilometers in 5 hours, how far will it run in 12| hours ? 8. If 9 ounces of silver will make 4 tea spoons, how many spoons will 25 pounds of silver make ? 9. If 5J yd. of cloth are worth |;27J, what are 50^ yd. worth? 10. If 60 men can build a house in 90^ days, how long will it take 15 men to build it ? 11. What will 49 1\ yd. velvet cost, if 7f yd. 'cost £7 ISs. 4d. ? 12. At 7s. 6d. per ounce, what is the value of a silver pitcher weighing 9 oz. 13 pwt. 8 gr. ? 13. If 405 yd. linen cost £69 7s. 6d., what cost 243 yd ? 14. If A can saw a cord of wood in 6 hours, and B in 10 hours, how long will it take both together to saw a cord? 15. A cistern has 3 stop-cocks, the first of which will emj)ty it in 10 min. ; the second, in 15 min. ; and the third, in 30 min.; how long will it take all of them together to empty it? 16. A man and a boy together can mow an acre of grass in 4 hours ; the man can mow it alone in 6 hours ; how long will it take the boy to mow it ? 17. If the interest of $675.25 is 155.625 for 1 year, how much will be the interest of $2368.85 ? 18. What must be the length of a board which is 9f in. wide, to make a square foot ? 19. If 98J yds. carpeting \\ yard wide will cover a floor, how many yards f yd. wide will it take to cover it ? Compound Proportion, 281 Compound Proportion. 693. Compound Proportion is an equality between a com- pound ratio and a simple one. Thus, 8 : 4) h • : 13 : 3, is a compound proportion. That is, D : o ' 8x6:4x3:: 12 : 3 ; for, 8 x 6 x 3 = 4 x 3 x 12. It is read, '^ The ratio of 8 into 6 is to 4 into 3, as 12 to 3." Written Exercises. 694. 1. If 4 men earn $60 in 10 days, liow much can 6 men earn in 8 days ? Explanation. — Since the answer operation. is to be money, we make $60 the 4 m. : 6 m. third term. other numbers in pairs, two of a kind, placing them according as the answer would be greater or less than the third term, if it depended on each pair alone. Now, as 6 m. can earn more than 4 m., we place the larger for the second term and the smaller for the first. Agahi, as they will earn less in 8 d. than in 10 d., we place the smaller for the second term, and the large]' for the first. Reducing the compound ratio to a simple one, we have, 4 X 10 : 6 X 8 : : 60 : Ans. Dividing the prod, of the means by the extreme, cancelling, etc. ley, we make $60 the 4 m. : 6 m. ) _ We then arrange the 10 d * 8 d (" • ' '*^^'-' * ■^^^^' ^^^^ = 172, Ans. ^ X 10 695. From the preceding example we have the following Rule. — I. Make that niuiiber which is of the same kind as the ansiuei^ the thiixl teriiv. II. Tlien take the other ninnhers in pairs, or two of a kind, and arrange theiyv as in simple proportion. (Art. 687.) III. Maltiply the second and third terms together, and divide the product by the product of the first terms. TIxe quotient will l>e the answer. 282 Proportion, Proof. — If the product of the first and fourth terms equals that of the second and third terms, the luorh is right. Notes. — 1. The terms of eacli couplet in the compound ratio must be reduced to the same denomination, and the third term to the lowest denomination contained in it, as in Simple Proportion, 2. In Compound Proportion, all the terms are given in couplets ot pairs of the same kind, except one. This is called the odd term, or demand, and is always the same kind as the answer. 3. Problems in Compound Proportion may also be solved by Analysis and by Simple Proportion. Take the preceding example. By Analysis. — If 4 men can earn $60 in 10 d., 1 man can earn in the same time, \ of $60, which is $15, and 6 men can earn 6 times 15 or $90. Again, if 6 men earn $90 in 10 d., in 1 d. they can earn ^l of $90, which is $9 ; and in 8 d. they can earn 8 times 9, or $72, Ans. By Simple Propoktion.— 4 m. : 6 m. : : $60 : x, or $90. Again, 10 d. : 8 d. : : $90 : Ans., or $72. 2. If 8 men can clear 30 acres of land in C3 days, working 10 hours a day, how many acres can 10 men clear in 72 days, working 12 hours a day ? STATEMENT. 8 m. :10 m. | ^^^^^ , ^3 .2 br. ) OPEBATION. 63 d. : 72 d. V : : 30 : A7is. ^0 10 hr. : 12 br. ) 10 X 72 X 12 X 30 .,, , , 8 X 63 X 10 = '^^ ^'^ ^'^^• 10 12 30 360 = 51f A., Ans. Note. — When the vertical form of cancellation is used, the antecedents must be placed on the left of the line, and the consequents with the odd term on the right. 3. If a man can walk 192 miles in 4 days, traveling 12 hours a day, how far can he go in 24 days, traveling 8 hours a day ? 4. If 8 men can make 9 rods of wall in 12 days, how many men will it require to make 36 rods in 4 days ? Comjpoiind Proportion, 283 5. If 5 men make 240 pair of shoes in 24 days, how many men will it require to make 300 pair in 15 days ? 6. If 60 lbs. of meat will supply 8 men 15 days, how long will 72 lbs. last 24 men ? 7. If 12 men can reap 80 acres of wheat in 6 days, how long will it take 25 men to reap 200 acres ? 8. If 18 horses eat 128 bushels of oats in 32 days, how many bushels will 12 horses eat in 64 days ? 9. If 8 men can build a wall 20 ft, long, 6 ft. high, and 4 ft. thick, in 12 days, how long will it take 24 men to build one 200 ft. long, 8 ft. high, and 6 ft. thick ? 10. If 8 men reap 36 acres in 9 days, working 9 hours per day, how many men wdll it take to reap 48 acres in 12 days, working 12 hours per day? 11. If 1100 gain $6 in 12 months, how long will it take $400 to gain 118. Ans. 9 mo. 12. If $200 gain $12 in 12 mo., what will 1400 gain in 9 mo.? 13. If 6 men can dig a drain 20 rods long, 6 feet deejo, and 4 feet wide, in 16 days, working 9 hours each day, how many days will it take 24 men to dig a drain 200 rods long, 8 feet deep, and 6 feet wide, working 8 hours per day ? 14. If 3 lbs. of yarn will make 10 yards of cloth \\ yard wide, how many pounds will be required to make a piece 100 yards long, and 1^ yd. wide ? 15. A general wished to remove 80000 lbs. of provision from a fortress in 9 days, and it was found that in 6 days 18 men had carried away but 15 tons; how many men would be re- quired to carry the remainder in 3 days ? 16. If a man travels 130 miles in 3 days, when the days are 14 hours long, how long will it take him to travel 390 miles when the days are 7 hours long? 17. If the price of 10 oz. of bread is 5d., when corn is 4s. 2d. per bushel, what must be paid for 3 lbs. 10 oz. when corn is 5s. 5d. per bushel ? 18. If 6 journeymen make 132 pair of boots in 4^ weeks, working 5^ days a week, and 12f hours j^er day, how many pair will 18 men make in 13|^ weeks, working ^\ days per week, and 11 hours per day ? 284 Projportion, Partitive Proportion. 696. Partitive Proportion is dividing a number into two or more parts having a given ratio to each other. Oral Exercises. 697. 1. Cliarles and Robert divided 28 pears between them in the ratio of 3 to 4 ; how many had each ? Analysis. — Since Cliarles had 3 parts as often as Robert had 4, both had 3 + 4, or 7 equal parts. Hence, Charles had f and Robert | of 28. Now f of 28 are 12, and | are 16. Therefore, Charles had 12, and Robert 16 pears. Proof. — 12 pears +16 pears = 28 pears, 2. Divide 35 into 2 such parts that one shall be to the other as 3 to 2. 3. Divide 42 cts. into two such parts that one shall be to the other as 2 to 4. 4. A farmer had 56 acres of which he made 2 pastures in the ratio of 3 to 5 ; how many acres were in each ? 5. A man bought a cow and a calf for 150 ; the cow was worth 4 times as much as the calf ; what was the value of each ? 6. Divide |72 into two such parts that one shall be to the other as 4 to 8. Written Exercises. 698. To divide a number into two or more parts which shall have a given ratio to each other. 1. A and B divided $145 in the ratio of 2 to 3 ; how much had each ? Solution. — The sum of the pro- operation. portional parts is to each separate part 5:2 : : $145 : A's S. as the number to be divided is to each 5*3 : • 1^145 : B's S. man's share That is, 5 (2 + 3) is to 3 ^^^^ %)^r^ = *58,' A's s! as $145 to A's share. Affam, 5 is to ^ '^ / • * ' 3 as $145 to B's share. Hence, the (^1^^"^ x 3) -f-5 = $87, B's S. Partitive Pro])ortion, 285 EuLE. — I. Mahe the niunher to he divided the third term; each proportional part successively the second term ; and their sum the first. 11. T]ie product of the second and third terms of each proportion, divided by the first, will he the corresponding part required. 2. Divide 312 into three parts which shall be to each other as 3, 4, and 6. 3. A man having 198 sheep, wished to divide them into three flocks which should be to each other as 1, 3, and 5 ; how many will each flock contain ? 4. A farmer raised 500 bushels of grain, composed of oats, wheat, and corn in the proportion of 3, 4, and 5|-; how many bushels were there of each kind ? 5. A man paid 15.28 for pears, oranges, and melons, the prices of which were as 2, 4, and 6 ; how much did he pay for each kind ? 6. A father divided 13479 among his four sons in proportion to their ages, which were as 4, 6, 8, and 10 ; how many dollars did each receive ? Q U ESTI ON S. 654. What is ratio ? How found ? 655. What are the terms of a ratio ? 656. The first terra called? 657. The second? 658. The two terms? 659. Ratio denoted? 662. What numbers can be compared? 663. A simple ratio ? 664. Compound ? 666. Reciprocal ? 669. Name the prin- ciples of ratio ? 673. What is proportion ? 673. The sigfn ? 674. The terms ? 677. How many terms must there be in a proportion? 678. What is a mean proportional ? 679. The extremes ? 680. The means ? 683. Principles of proportion ? 685. What is a simple proportion ? 687. Which number is the third term ? How arrange the other terms ? How find the fourth term ? 688. How else may proportion be explained ? 689. What is a cause ? An effect ? 693. What is compound proportion ? 695. Which number is made the third term? How arrano-e the remaining numbers? How find the re- quired term ? 69o. What is partitive proportion ? 698. How arrange the terms ? How find the answer ? ARTNERSHiP 699. Partnership is the association of two or more persons for the transaction of business. 700. The association is called a Firm, Company, or House. 701. The persons associated are called Partners. 702. Tlie Capital is the money or property furnished by the Partners. 703. The Assets of a firm are the various kinds of property in its possession. 704. The Liabilities are its debts. 705. The Net Capital is the excess of its property above its liabilities. Written Exercises. 706. To find each Partner's Share of the Profit or Loss, when their capital is employed for the same time. 1. A and B formed" a partnership ; A put in 1400 and B $300 ; they make 1364. What was each man's share of the profit ? By Analysis.— $400 + $300 = $700, the whole capital. Hence, A had |g-a, or f of the gain ; and | of $364 = $208, A's. B had f ga, or f " " " f of $364 = $156, B's. Hence, EuLE. — I. Take such a part of the gain or loss, as each partner's stock is of the ivJiole capiial. By Percentage.— The gain $364 is fUJ = ^^^, or 53% of the whole capital. Therefore, X .53 = $308, A's share ; and $300 x .53 = $156, B's share. Hence, II. Find the % v;hich the profit or loss is of the whole capital, and multiply each man's capital hy it. (Art. 464.) Partnership. 287 Note. — The per cent method is preferable, when the partners or share- holders are numerous. 2. A and B formed a partnership ; A put in $648 and B $1080, agreeing to divide the profit in proportion to their capital ; what was each one's share of the gain ? 3. A, B, and C are partners ; A furnishes $600, B $800, and $1000 ; they lose $480 ; what is each man's share of the loss ? 4. A, B, 0, and D make up a purse for a stock speculation ; A puts in $300, B $400, C $600, and D $800 ; they make $2400 ; what is each man's share ? 707. To find each partner's share of the Profit or Loss, when their capita! is employed for unequal times. 5. A and B enter into partnership; A furnishes $400 for 8 months, and B $600 for 4 months ; they gain $280 ; what is each one's share of the profit ? Analysis. — In this case the profit of each partner depends on two ele- ments, viz. •. the amount of his capital and the time it is employed. But the use or interest of $400 for 8 months equals that of 8 times $400, or $3200, for 1 mo. ; and of $600 for 4 mo. equals 4 times that of $600, or $2400, for 1 month. The respective capitals, then, are equivalent to $2400 and $3200, each employed for 1 mo. Since A furnished $3200 and B $2400, the whole •apital = $5600. Now $280h-$5600 = .05 or 5;^. Therefore, $3200 X .05 = $160, A's share. $2400 X .05 =: $120, B's share. Hence, the Rule. — Multiply each -partner's capital hy the time it is employed. Consider these products as their respective capitals, and proceed as in the last article. 708. The Rule for Partnership is also applicable to problems in Bankruptcy, the General Average of losses at sea, and other distributions; the sum of the debts or property in question corresponds to the tvhole amount of capital, etc. 6. A and B formed a partnershi]) ; A put in $300 for 2 months, and B $200 for 6 months ; they gained $150 ; what was each man's just share of the gain ? 288 Partnership. 7. A, B^ and C enter into partnership ; A puts in 1500 for 4 mo., B $400 for 6 mo., and C $800 for 3 mo.; they gain $340 ; what is each man's share of the gain ? 8. A and B hire a pasture, together for $60 ; A put in 120 sheep foip 6 months, and B put in 180 sheep for 4 months; what should each pay ? 9. The firm A, B, and C lost $246 ; A had put in $85 for 8 mo., B $250 for 6 mo., and $500 for 4 mo.; what is each man's share of the loss ? 10. A man failing in business owed A $1200, B $1800, and C $2400 ; his assets were $2700 ; how much did each receiye ? Ans. A received $600, B $900, and C $1200. Pnoor.— $600 + $900 + $1200 = $2700, the assets. 11. A bankrupt owes A $1200, B $2300, $3400, and D $4500 ; his whole effects are worth $5600 ; how much will each creditor receive ? 12. A railroad company went into bankruptcy whose liabilities were $36300, and assets $12100 ; how much did the company pay on a dollar, and how much did a creditor receive who ]iad a claim of $15270 ? 13. A, B, and C freighted a vessel with flour from New York to New Orleans ; A had on board 1200 barrels, B 800, and 400. On her passage 400 barrels were thrown overboard in a gale ; what was the average loss ? 14. A Liverpool packet being in distress, the master threw goods overboard to the amount of $10000. The whole cargo was valued at $72000, and the ship at $28000 ; wliat per cent loss was the general average ; and how much was A's loss, whc had goods aboard to the amount of $15000 ? Q U ESTi ONS. 699. What is partnership ? 700. What is the association called ? 702. What is the capital ? 703. Assets? 704. Liabilities? 705. Net capital ? 706. How find each partner's share of profit or loss, when their capital is employed for the same time ? 707. When for unequal times ? Jl\ - ^ *-<—m. NVOLUTION. fe ■ : ■ ^^^ W ^^ Oral Exercises. 709. 1. What is the product of 3 multiplied by itself ? 2. What is the product of 3 taken 3 times as a factor ? 3. What is the product of 4 taken 3 times as a factor ? 4. What is the product of 5 taken twice as a factor ? 5. What is the product of 3 taken 4 times as a factor ? 6. What is the product of f multiplied by itself ? 7. What is the product of | taken twice as a factor ? 8. What is the product of .4 taken twice as a factor ? Of .3 taken three times ? Of .04 twice ? Definitions. 710. Involution is finding a power of a number 711. A Power of a number is the product of two or more equal factors. Thus, 2x2x2 = 8, and 3x3 = 9; 8 and 9 are powers of 2 and 3. 712. Powers are 7iamed according to the Clumber of times the factor is taken to produce the given power. 713. The First Power is the number itself. 714. The Second Power is the product of two equal factors, and is called a Square. 715. The Third Power is the product of three equal factors, and is called a Cube. Note. — The second power is called a square because the area of a square is found by multiplying one of its sides by itself. The thii'd power is called a cube because the contents of a cube are found by taking one of its sides three times as a factor. (Art. 429.) 390 Involution, 715. An Exponent is a small figure placed above a number on the right to denote the power. It shows that the number above which it is placed is to be raised to the power indicated by this figure. Thus, 2' = 2, the first power, or number itself. 2^ = 2 X 2, the second power, or square. 2^ = 2 X 2 X 2, the third power, or cube. 2^ — 2x2x2x2, ihe fourth power, etc. Notes. — 1. The term exponent is from the Latin exponere, to represent. 2. The exponent of the first power being 1, is commonly omitted. 717. The expression 2^ is read, *^ 2 raised to the fourth power, or the fourth power of 2." 9. Eead the following: 9^ ■1%\ 25^, 245^, SSl^o, 465i^ lOOO^^. 10. Read G^x^S 25^x48^ 1408— 75^, 256io^97^ 11. Express the 4th power of 85. 13. The 7th power of 340. 12. Express the 5th power of 348. 14. The 8th power of 561. 718. To find any required Power of a Number. 1. What is the 4th power of 8 ? Solution.— 8^ = 8x8x8x8 = 4096, Ans. Rule. — Tahe the number as many tunes as a factor as there are units in the exponent of the required power. Notes. — 1. A common fraction is raised to a power by involving each term. Thus, {If = j%. 2. A mixed number should be reduced to an improper fraction, or the fractional part to a decimal ; then proceed as above. Thus, (2i)'2 = (1)2 = 2^5 ; or 2i = 2.5 and (2.5)-^ = 6.25. 3. All powers of 1 are 1 ; for 1x1x1, etc. = 1. Raise the following numbers to the powers indicated: 2. 63. 5. 55. 8. 4.033. 11. (1)4. 3. 36. 6. 74. 9. 2.00033. 12. {if- 2322. 7. 353. 10. 300.053. 13. (3J)^. Formation of Squares, 291 Formation of Squares. 719. To find the Square of a Number in the Terms of its Parts. 1. Find the square of 5 in the terms of the parts 3 and 2. Illustration. — Let the shaded part of the diagram represent the square of 3 ; its contents are equal to 3 x 3, or 9 sq. ft. 1st. To preserve the form of the square, equal additions must be made to two adjacent sides ; for, if made on one side, or on opposite sides, the figure will no longer be a square. 2d. Since 5 is 2 more than 3, it follows that two rows of 3 squares each, must be added at the top, and 2 rows on one of the adjacent sides, to make its length and Irreadth each equal to 5. Now 2x3 plus 2x8 are 12 squares, or twice the product of the two parts 2 and 3. But the diagram wants 2 times 2 small squares, as represented by the dotted lines, to fill the upper corner on the right, and 2 times 2 or 4 is the square of the second part. We now have 9 (the sq. of the 1st part), 12 (twice the prod, of the two parts 8 and 2), and 4 (the square of the 2d part.) But 9 + 12 + 4—25, the square required. 2. Find the square of 7 in the terms of 5 and 2. Ans, 25 + 20 + 4. Proof.— 7 x 7 = 49. 3. Find the square of 25 in the terms of its tens and units. Analysis. — The product of 2 tens or 20 by 20 is 400 (the square of the tens) ; 20 x 5 plus 20 X 5 is 200 (twice the prod, of the tens by the units) ; and 5 by 5 is 25 (the square of the units). Now 400 + 200 + 25 zr 625, or 25^. Hence, the 25 = 20 + 5 25 20 + 5 125 400 + 100 50 + 100 + 25 625 = 400 + 200 + 25 Rule. — TJie square of any niunher consisting of tens and units is equal to the square of the tens, plus twice the product of the tens hy the units, plus the square of the units. 4. What is the square of 34 in the terms of its tens and units? i VOLUTION. Q> T^^ Oral Exercises. 720. 1. What are the two equal factors of 9 ? 16 ? 25 ? 2. Name the two equal factors of 36 ? 49 ? 64 ? 3. What are the three equal factors of 8 ? 27 ? 125 ? 4. Name the four equal factors of 16 ? Of 81 ? 5. Of what is 49 the square ? 6. Of what is 27 the third power? 7. Of what is 125 the cube? Definitions. 721. Evolution is finding a root of a number. 722. A Root is one of the equal factors of a number. Moots are named according to the number of equal factors they contain, 723. The Square Root is one of the tivo equal factoids of a number. Thus, 5 X 5 = 25; therefore, 5 is the square root of 25. 724. The Cube Root is one of the three equal factors of a number. Thus, B X 3 X 3 ~ 37 ; therefore, 3 is the cube root of 27, etc. 725. The character (y^) is called the Radical Sign. It is a corruption of the letter R, the initial of the Latin radix^ a root. 726. Roots are denoted in tivo tmys : 1st. By prefixing to the number the Radical Sign, witli a figure placed over it called tlie Index of the root ; as ^4, <^8. 2d. By ^fractional exponent placed above the number on the right. Thus, 9^, 27^ denote the square root of 9, and the cube root of 27. Sqiiare Root 293 Notes. — 1. The figure over the radical sign and the denominator of the exponent, denote the name of the root, 2. In expressing the square root, it is customary to use simply the radical sign (/y/), the 2 being understood. Thus, the expression /^%h — 5, is read, " the square root of 25 = 5. " 727. A Perfect Power is a number whose exact root can be found; as, 9, 16, 25, etc. 728. An Imperfect Power is a number whose exact root can not be found. This root is called a Surd. Thus, 5 is an imperfect power, and its square root 2.23 + is a surd. Note. — All roots as well aspowei's of 1, are 1. Eead the following expressions : 8. ^/40. 10. 119i 12. 1.5^ 14. ^256. 16. ^^ff. 9. ^15. 11. 243i 13. ^29. 15. i^45.7. 17. ^^|. 18. Express the cube root of 64 both ways ; the 4th root of 25 ; the 7th root of 81 ; the 10th root of 100. Square Root. 729. Extracting the Square Root is finding one of two equal factors of a number. 730. To find how many figures the Square of a Number contains. Illustration. — 1. Take 1 and 9, the least and greatest integer that can be expressed by one figure ; also 10 and 99, the least and greatest that can be expressed by two integral figures, etc. Squaring these numbers, 12 z= 1 ; 102 _ 100 ; 1002 — 10000. 92 = 81 ; 992 = 9801 ; 999^ = 998001. 2. Take .1 and .9, the least and greatest decimals that can be expressed by one figure ; also .01 and .99, the least and greatest that can be expressed by two decimal figures, etc. Squaring these, .12 = .01 ; .012 ^ .0001 ; OOP .-=: .000001. .92 = .83 ; .992 = .9801 ; .9992 = .998001, etc. 294 Evolution. 731. From these illustrations we discover the following Principles. i°. Tlie square of a number contains twice as many figures as the root, or twice as many less one. 2°. If any number is separated into periods of tiuo figures each beginniyig with units 'place, the number of figures in the square root loill be equal to the nimiber of periods. Note. — If tlie number of figures in the given number is odd, the left Jiand period will have but one figure. 732. 1. Required the length of one side of a square garden which contains 16 sq. rods. Illustkation. — Let the garden be repre- sented by the adjoining diagram. Now as the garden is square, its sides are equal, and the length of one side is one of the two equal factors, or the square root of 16. But 16 = 4 X 4. Hence, the length of a side is 4 rods. Proof.— 4 rd. x 4 rd. = 16 sq. rods, the given area. a a -ia a ° 4 X 4 = 16 pq. rods. 2. What is the length of one side of a square which contains 625 square feet ? OPERATION. 4 ro ds. o ■^ 625 ) 25 45 ) 225 225 Analysis. — Since 625 contains three figures, it must liave two i^eriods ; its square root tw'o figures, and first period on the left one figure. The greatest square of 6 (hundreds) the left hand period is 4 (hundreds) and its root is 2 (tens) which we place on the right for the first figure of the root. Sub- tracting the square of 2 from the period used, we annex to the remainder the next period for a dividend. Since the additions are to be made on two sides of the square, we place 4, the double of tlie root, on the left of the dividend for a trial divisor, and find it is contained in 22, 5 times, the right hand figure being omitted. Placing the o on the right of the root and of the trial divisor, we multiply the divisor thus increased by this figure, and subtracting the product there is no remainder. The square root or answer, is 25 feet. Square Root. 295 yO feet. 5 leet. Geometrical Illustration. 733. 1. Take any number as 625 sq. ft., the square root of which is to be found. Let the shaded part of the diagram represent the square of 2 tens, the first figure of the root ; then 20 x 20, or 400 sq. ft., will be its contents. Subtracting the contents from the given area, we have 625-400 = 225 sq. ft. to be added to this square. To preserve its form, the addition must be made equally to two adjacent sides. The question is, what is the width of the addition. Since the length of the square is 20 ft., adding a strip 1 foot wide to two sides will take 20 + 20 or 40 sq. ft. Now if 40 sq. ft. will add a strip 1 foot wide to the square, 225 sq. ft. will add a strip as many ft. wide as 40 is contained times in 225 ; and 40 is contained in 225, 5 times and 25 over. That is, since the addition is to be made on two sides, we double the root or length of the side found for a trial divisor, and find it is contained in 225, 5 times, which shows the width of the addition to be 5 feet. Now the length of each side addition being 20 ft., and the width 5 ft., the area of both equals 20 x 5 + 20 x 5, or 40 x 5 = 200 sq. feet. But there is a vacancy at the upper corner on the right, whose length and breadth are 5 ft. each ; hence its area = 5x5, or 25 sq. feet ; and 200 sq. ft. + 25 sq. ft. ■= 225 sq. ft. For the sake of finding the area of the two side additions and that of the corner at the same time, we place the quotient 5 on the right of the root already found, and also on the right of the trial divisor to complete it. Multiplying the divisor thiis completed by 5,. the figure last placed in the root, we have 45 x 5 = 225 sq. ft. Subtracting this product from the divi- dend, nothing remains. 2. What is the square root of .576 ? Solution.— y^ySTG = a/,5766 = .75 + , Ans, 3. Find the sq. root of 234.09. Solution.— V234.O9 = 15.3, Ans. Hence, 296 Evolution, 734. To extract the square root we have the following General Rule. 1. Separate the number into periods of two figures each, beginning at units, and count both ways. IT. Find the greatest square in the first period on the left, and' place its root on the right. Subtract this square from the period, and on the right of the remain- der place the next period for a dividend. III. Double the part of the root thus found for a trial divisor ; and finding how many times it is contained in the dividend, omitting the right hand figure, annex the quotient both to the root and to the divisor. IV. Multiply the divisor thus increased by the last figure placed in the root, subtract the product from the dividend, and place the next period on the right of the remainder. V. Proceed as before, till the root of all the periods is found. V^oo^.— Multiply the root by itself. (Art. 722.) Notes. — 1. If tliere is a remainder after the root of the last period is found, annex periods of ciphers, and proceed as before. The figures of the root thus obtained will be decimals. 2. If the trial divisor is not contained in the dividend, annex a cipher both to the root and to the divisor, and bring down the next period. 3. It sometimes happens that the remainder is larger than the divisor; but it does not necessarily follow that the figure in the root is too small. 4. The left ?innd period in ^chole numhcrs may have but one figure ; but in deciinals, each period must have two figures. Hence, if the number of decimals is odd, a cipher must be annexed to complete the period. Find the square root of the following numbers : 4. 576. 9. 538.245. 14. 287.65. 5. 1600. 10. 61.7646. 15. .776961. 6. 1225. 11. 8476.124. 16. 1073.741824, 7. 291.64. 12. 1232136. 17. .00053361. 8. 864.91. 13. 5314491. 18. 617230.2096. Sqiiare Root, 297 735. To find the Square Root of Fractioiw. 1. What is the square root of f|^ ? Solution. — \'\\ = VtV = f» ^^ns. Hence, the EuLE. — Reduce the fraction to its simplest form and find the square root of each term separately. Notes. — 1. If either term of the given fraction, when reduced, is an imperfect square, reduce tlie fraction to a decimal, and proceed as above. (Art. 249.) 2. Mixed numbers should be reduced to improper fractions, or the fractional part to a decimal. 2, What is the square root of ^^^ ? Ans. |. Find the square root of the following fractions : ^' TT6* °* 46T56- '• 65§^36' 4 2 5 6 6 1024 8 2 ft 1 6 9. What is the square root of 20 J ? Solution.— y'20 J = -^/^ = |» or 4^, Ans. Find the square root of the following : 10. 18-2. 12. 52^. 14. II of 144, 11. 40|f. 13. 113ff. 15. 16. What is the square root of -g^ of |f oij^^ of 4096 ? 17. Required the square root of 3 to 7 decimals. 18. Required the square root of 12 to eight decimals. A.PPI^ICATIONS, 735, a. 1. What is the side of a square whose area contains 2025 sq. yards ? 2. A general has 906304 soldiers; how many must he place in rank and file to form them into a square ? 3. A man bought a square tract of land containing 3840 acres ; how many rods square is the tract ? 298 Evolution. 4. What is the side of a square, whose area is equal to that of a triangle containing 5184 sq. ft. ? 5. What is the side of a square equal in area to a rectangu- lar field 32 rods long and 18 rods wide ? 6. A landholder divided a tract of 3802J A. into four equal and square farms ; what is the length of one of their sides ? 7. A man having a garden 465 yards square, wished to extend it so as to make it 9 times as large ; how many 3'ards square will it then be ? 736. A mean proportio7ial between tivo ^uwihers is found hy extracting the square root of their jyroduct. (Art. 678.) 8. What is the mean proportional between 9 and 16 ? Solution.— 16 x 9 = 144 ; and y^144 = 12, Ans. Hence, Note. — The product of any square number by another square number is always itself a square. Find the mean proportional between the following numbers: 9. 4 and 16. 14. 28 and 54. 19. i and -J-. 10. 9 and 25. 15. 45 and 96. 20. f and if • 11. 25 and 36. 16. .04 and .16. 21. If and %\. 12. 49 and 64. 17. .64 and 6.25. 22. If and ^V 13. 81 and 64. 18. .09 and .36. 23. 1 2 1 ^^^^ 14 4 • 737. When the length of a rectangular field equal to a given area, is double, triple, etc., its width, its dimensions are found by extracting the square root of \, -j, etc., of the area, as the case may be. This root will be the width, and being doubled, tripled, etc., wdll be the length. 24. The length of a rectangular field containing 80 acres, is twice its breadth ; what are its length and breadth ? 25. The breadth of a rectangular farm containing 160 acres, is J its length ; what are its length and breadth ? 738. A Triangle is a figure having three sides and three a^igles. 739. A Eight-angled Triangle is one which has a right angle ; as ABC. ^ Square Root. 299 740. The Hypothenuse of a right-angled triangle is the side AC, opposite the right angle B ; the base is AB, the perpen- dicular is BO. 741. The relation of the sides of a tri- angle to each other may be illustrated as follows : Take any right-angled triangle as ABC, the base of which is 4 ft., the perpendicu- lar 3 ft., and the hypothenuse 5 ft. It will be seen that the square of the base contains 16 sq. ft., that of the perpendicu- lar 9 sq. ft., and that of the hypothenuse 25 sq. ft. Now 25 = 16 + 9. Hence we derive the following Principles. i°. Tlie sum of the squares of the Base and Perpendicular is equal to the square of the Hypothenuse. 2°. The square of the Hypothenuse diminished hy the square of the Perpendicular, is equal to the square of the Base. 3°. The square of the Hypothenuse diminished ly the square of the Base, is equal to the square of the Perpendicular. That is, 742. The square descrihed on the hypothenuse of a right- angled triangle is equal to the sum of the squares of the base and peiyendicular. 743. To find the Hypothenuse, when the Base and Perpendicular are given. 26. What is the length of a ladder which will just reach to the top of a house 32 feet high, when its foot is placed 24 feet from the house ? Solution.— Perpendicular (33)2 - 32 x 32 = 1024 Base (24)2 - 04 x 24 = 576 The square root of tlieir sum, 1600 = 40 ft., Ans. 300 Evolution, Hence we have the following EuLE. — Add the square of the base to the square of the perpendicular, and the square root of the sum will he the hypothenuse. Formula. — HyiMlmiuse = VBase^ + Perpendicular^ 27. The side of a certain school-room having square corners, is 8 yards, and its width 6 yards ; wliat is the distance between two of its diagonal corners ? 28. Two men start from the same place and at the same time ; one goes exactly south 40 miles a day, the other goes exactly west 30 miles a day ; how far apart will they be at the close of the first day ? 29. How far apart will the same travelers be in 4 days ? 744. To find the Perpendicular, when the Base and Hypothenuse are given. 30. A line 10 yd. long fastened to the top of a tree, reaches the ground 6 yd. from the base ; what is the height of the tree ? Solution.— Hypothenuse (10 yd.)^ = 10 x 10 = 100 Base (6yd.)2 = 6x6 =: 36 Ttie square root of their difference, 64 =; 8 yd., Ans. Hence, the Rule. — From the square of the hypothenuse subtract the square of the base, and the square root of the remain- der will be the perpendicular. Formula. — Perpendicular = ^Hypothenuse^ — Bas^. 31. A line 75 feet long fastened to the top of a flag-staff reaches the ground 45 feet from its base; what is the height of the flag-staff ? 32. A house is 40 ft. wide and the length of the rafters is 32 ft. ; what is the perpendicular distance from the beam to the ridgepole ? 33. The distance between the diagonal corners of a croquet ground is 17 yards, and its length is 15 yards ; what is its width ? Square Moot. 301 745. To find the Base, when the Hypothenuse and Perpendicular are given. 34. A ladder 50 ft. long was placed against the top of a house 40 ft. high ; what distance was the foot of the ladder from the house ? Solution.— Hypothenuse (50 ft.)2 = 50 x 50 = 2500 Perpendicular (40 ft.)2 = 40 x 40 =: 1600 The square root of their difference, 900 = 30 ft., Ans. Hence, the Rule. — From the square of the hypothenuse subtract the square of the jjerpeiidieular, and the square root of the reviainder will he the base. Formula. — Base = ^/ Hy2)othenuse^ — Perpendicular'^. 35. The slant height of a square pyramid is 40 ft., and its perpendicular height 32 ft., what is the distance from the center of the base to its side ? 36. The height of a tree on the bank of a river is 100 ft, and a line stretching from its top to the opposite side is 144 ft. ; what is the width of tlie river ? 37. The side of a square field is 30 rods ; how far is it be- tween its diagonal corners ? 38. If a square field contains 10 acres, what is the length of its side, and how far apart are its diagonal corners ? 39. If a school room is 40 feet long, 30 feet wide, and 14 feet high, what is the length of a diagonal drawn upon the floor ; and what is the length of a diagonal drawn from the floor to the ceiling ? 40. A park 53 rods long and 39 rods wide has a straight walk running from its diagonal corners ; what is the length of the walk ? 41. The side of a square room is 40 feet ; what is the dis- tance between its diagonal corners on the 'floor? 42. A tree was broken 35 feet from its root, and struck the ground 21 ft. from its base ; what was the height of the tree ? 302 Mvolution. Similar Plane Figures. 746. Similar Plane Figures are tliose whicli have the same form, and their like dimeiisions proportional . Notes. — 1. All circles and all rectilinear figures are similar, when their several angles are equal each to each, and their UTce dimensions propor- tional. 2. The like dimensions of circles are their diameters, radii, and circum- ferences. 747. The Areas of similar surfaces are to each other as the squares of their like dimensions. Conyersely, The Like Dimensions of similar surfaces are to each other as the square roots of their areas. 1. If one side of a triangle is 12 yards, and its area 36 square yards, what is the area of a similar triangle, the corresponding side of which is 8 yards ? Solution.— (12)-^ : (S)^ : : 36 : Ans., or 16 sq. yards. 2. If one side of a triangle containing 36 sq. yards is 8 yards, what is the length of a corresponding side of a similar triangle which contains 81 sq. yards ? Solution. — /y/36 : y^Sl : : 8 : Ans., or 12 yards. 3. If a pipe 1 inch in diameter will fill a cistern in 60 min., in what time will a pipe 2 in. in diameter fill it ? 4. If a gate 9 inches in diameter will empty a mill-pond in 16 hours, how large must a gate be to empty it in 4 hours ? 5. If one side of the base of a triangular pyramid measuring 16 square feet, is 20 inches in length, what is the length of a side of a similar pyramid, which measures 36 square feet ? 6. A man owns a building lot containing 20 square rods in the shape of a right-angled triangle, the perpendicular of whicli is 20 yards in length ; what is the perpendicular of a similar lot, which contains 30 square rods ? Cube Root 303 Cube Root. Oral Exercises. 748. 1. What number taken three times as a factor pro- duces 8 ? 27 ? 2. What is one of the three equal factors of 64 ? 3. Name one of the three equal factors of 125 ? 4. Name one of the three equal factors of 1000. Of 1728. Written Exercises. 749. The Cube Root of a number is one of its three equal factors. 750. To find the number of figures In the Cube of a Number, also in the Cube Hoot of a Number. 1st. Take 1 and 9, also 10 and 99, 100 and 999, etc., the least and great- est integers tliat can be expressed by one, tiDo, three, etc., figures. 3d, In like manner take .1 and .9, also .01 and .99, etc., the least and greatest decimals that can be expressed by one, two, etc., decimal figures. Cubing these, we have Roots. Powers. Roots. Powers. 1 13 = 1, .1 .13 = .001 9 93 = 729, .9 .93 = .729 10 103 = 1000, .01 .013 ^ .000001 99 993 = 970299, .99 .993 = .970299 100 1003 = 1000000, .001 .0013 ^ .000000001 999 9993 = 997002999, .999 .9993 = .997002999 omparir ig these roots and their cubes, we discover the followi PrI N CI PLES. 1°. The cube of a number cannot have more thati three times as many figures as its root, nor but two less. 28. If a number is separated into periods of three figures each beginning at units place, the number of figures in the cube root ivill be the same as the number of ijeriods. 304 Evolution. Notes. — The left hand period in \diole numbers may be incompletCf having only one or two figures; but each period of decimals must always have three figures. Hence, if the decimal figures in a given number are less than three, annex ciphers to complete the period. How many figures in the cube root of the following : 1. 340566. 3. 576.453. 5. 32.7561o 2. 1467. 4. 5.7321. 6. .456785. 751. To find the Cube of a number consisting of two figures in the terms of its parts. 1. Find the cube of 35 in the terms of its tens and units. 35 = 35 = OPERATION. 30 + 5 30 + 5 175 = 105 = (30x5) +52 302+ (30x5) 1225 =: 35 .= 302 + 2(30x5) +52 30 + 5 6125 = 3675 := (302 30-^ + 2(302 X5) +2(30x52) + 53 X 5) + (30 X 52) 42875 = 303 + 3(302x5) + 3(30x52) + 53. Explanation.— The cube of the tens, (SO^) = 27000 3 times the square of tens by units, 3 (30^ x 5) = 13500 3 times the tens by square of units, 3 (80 x 5'^) = 2250 and the cube of the units 5^ =125 Now 27000 + 13500 + 2250 + 125 = 42875. Hence, 752. The cube of any numher consisting of tens and units is equal to the cuhe of the tens, plus 3 times the square of the tens dy the units, plus 3 times the tens hy the square of the units, plus the cube of the units. Note. — Since the cube of a number consisting of tens and imits is equal to the cube of the tens, plus 3 times the square of the tens by the units, etc., when a number has two jieriods, it follows that the left hand period must contain the cube of the tens, or first figure of the root. 2. Find the cube of 32 in the terms of its tens and units. Cube Root 305 753. To Extract the Cube Root of a number. 1. What is the side of a cube which contains 27 solid feet ? Illustration. — Let the cube be represented by the adjoming diagram, each side of which is divided into 9 square feet. Since the length of a side is 3 feet, if we multiply 3 into 3 into 3, the product 27, will be the solid contents of the cube. (Art. 429.) Now, if we reverse the pro- cess, dividing 37 into three equal factors, one of these factors will be the side of the cube. Ans. 3 ft. 3 feet. ,...: ^ 'I ■ i I, ! \ Ii ' "■■ i 3 X 3 X 3 = 27 ft. 2. What is the length of one side of a cubical mound con- taining 15625 solid feet of earth ? OPERATION. 15625 ( 25 1200 300 _25^ 1525 7625 7625 Explanation. — 1. We separate the given num- ber into periods of three figures each, placing a point over units, then over thousands. This shows that the root must have two figures. 3. Beginning with the first period on the left, we find the greatest cube in 15 is 8, the root of which is 3. Placing the 3 on the right, we sub- tract its cube from the period, and to the remain- der bring down the next period for a dividend. This shows that we have 7635 solid feet to be added. 3. We square the root already found, which in reality, as there is to be another figure in the root, is 30 ; then multiplying its square 400 by 3, we write the product on the left of the dividend for a trial divisor ; and find- ing it is contained in the dividend 5 times, place the 5 in the root. 4. We next multiply 30, the root already found by 5, the last root figure ; then multiply this product by 3 and write it under the divisor. We also write the square of 5, the last figure placed in the root, under the divisor. Adding these three results together, multiply their sum 1535 by 5, and subtract the product from the dividend. The answer is 35. Tllustratiox by Cubical I^locks* Let the adjoining diagram represent a set of cubical blocks. Let the cube of 30, the tens of the root, be represented by the large cube. The remainder 7635 is to be added equally to three adjacent sides of this cube. * Every gchool in which cube root is taught, should be furnished with a set of Cubiral Blocks, 306 Evolution. 20 ft. 5 It. To ascertain the thickness of these side 20 ft. additions, we form a trial divisor by squar- ing 2, the first figure of the root, with a cipher annexed, for the area of one side of this cube, and multiply this square by 3 for the three side additions. Now 20'^ = 20 X 20 = 400 ; and 400 x 3 = 1200, the trial divisor. Dividing 7625 by 1200, the quo- tient 5, shows that the side additions are to be 5 ft. thick, and is placed on the right for the units' figure of the root. To represent these additions, place the corresponding layers on the top, front, and right of the large cube. But we discover three vacancies along the edges of the large cube, each of which is 20 ft. long, 5 ft. wide, and 5 ft. thick. Filling these vacancies with the corresponding rectangular blocks, we discover another vacancy at the junction of the corners just filled, whose length, breadth, and thickness are each 5 ft. This is filled by the small cube. To complete the trial divisor, we add the area of one side of each of the corner additions, viz., 20 x 5 x 3, or 300 sq. ft., also the area of one side of the small cube = 5x5, or 25 sq. ft. Now 1200 + 300+25 = 1525. The divisor is now composed of the area of 3 sides of the large cube, plus the area of one side of each of the corner additions, plus the area of one side of the small cube, and is complete. To ascertain the contents of the several additions, we multiply the divisor thus completed by 5, the last figure of the root ; and 1525 x 5 — 7625. Subtracting the product from the dividend, nothing remains. Hence, 754. To extract the cube root we have the following General Rule. I. Separate the given numher into periods of three figures each; begin luith units and count hoth ways. II. Find the greatest cube in the first period on the left, andy place its root on the riglvt. Subtract this cube froirv the period, and to the right of the remainder bring down the next period, for a dividend. III. Multiply the square of the root thus found, con- sidered as tens, by three, for a trial divisor; and finding how many times it is contained in the divi- dend, write the quotient for the second figure of the root. Cube Root, 307 IV. To complete the trial divisor, add to it three times the product of the root previously found ivith a cipher annexed, by the second root figure, also add the square of this second figure. V. Multiply the divisor thus completed hy the last figure placed in the root. Subtract the product from the dividend; and to the right of the remainder bring down the next period for a new dividend. Find a neiv trial divisor as before, and thus proceed till the root of the last period is found. Notes. — 1. If there is Si remainder after tlie root of tlie last period is found, annex periods of ciphers, and proceed as before. The root figures thus obtained will be decimals. 2. If a trial divisor is not contained in the dividend, put a cipher in the root, two ciphers on the right of the divisor, and bring down the next period. 3. If the product of the di\'isor completed into the figure last placed in the root exceeds the dividend, the root figure is too large. Sometimes the remainder is larger than the divisor completed ; but it does not necessa- rily follow that the root figure is too small. 3. What is the cube root of 130241.7 ? Explanation. — Having completed the period of decimals by annexing two ciphers, we find the first figure of the root as above. We place the next period on the right of the remainder, and the dividend is 5241. The trial divisor 7500 is not contained in the dividend ; therefore, placing a cipher in the root and two ciphers on the right of the divisor, we bring down the next period , and proceed as before. Extract the cube root of the following numbers ; 136241.700(50.6 + 125 750000 5241.700 9000 36 759036 4554216 687484 Eem. 4. 13824. 8. 1092727. 12. 91.125. 5. 571787. 9. 2357947691. 13. .253395799. 6. 373248. 10. 27054036008. 14. 164.566592. 7. 1953125. 11. 12.167. 15. 122615.327232. 308 Evolution. 755. To find the cube root of a common fraction, reduce the fraction to its loioest terms, then extract the root of its 7i2imera- tor a?id denominator. Notes. — 1. When either the numerator or denominator is not 2i perfect cube, the fraction should be reduced to a decimal, and the root of the deci- mal be found as above. 2. A mixed number should be reduced to an improper fraction. 16. What is the cube root of ^y^ ? Solution.— ^^ = ^f| = f , Ans. Find the cube root of the following : 1 n 3 T6_ 10 1520 91 iq2 23. Find the cube root of 2 to 4 places of decimals. 24. Find the cube root of 3 to 5 places of decimals. Applic.4^tions. 756. 1. What is the length of a side of a cubical box, which contains 389017 solid inches? 2. Find the side of a cu. vat, which contains 48228544 cu. feet? 3. What is the side of a cubical mound, which contains 1259712 solid yards ? 4. What is the side of a cube equal to a stick of timber 2 feet square and 128 feet long ? 5. What is the side of a cubical bin, which contains 500 bushels, allowing 2150.4 cu. in. to a bushel ? 6. What is the side of a cubical cistern, which holds 100 wine hogsheads ? 7. What is the side of a cube equal to a pile of wood 2421 ft long, 12 ft. wide, and 7 feet high ? Similar Solids. 757. Similar Solids are those which have the same form, and their like dimensions proportional. Notes. — 1. The like dimensions of spheres are their diameters, radii, and circumferences ; those of cubes are their sides. Cube Root. 309 2. The like dimensions of cylinders and cones are their altitudes, and the diameters or the circumferences of their bases. 3. Pyramids are similar, when their bases are similar polygons, and their altitudes proportional. 4. Polyhedrons {i. e., solids included by any number of plane faces) are similar, when they are contained by the sa7ne nmnber of similar polygons, and all their solid angles are equal each to each. 758. The Contents of similar solids are to each other as the oubes of their like ditnensions. Conversely, The Like Dimensions of similar solids are as the cube roots of their contents. 1. If a globe 4 inches in diameter weighs 32 lbs., what is the weight of a globe whose diameter is 5 inches ? Solution. — 4^ : 5^ : : 32 lbs. : Ans. 125 X 32 lbs. = 4000 lbs., and 4000 lbs. -^64 = 62.5 lbs., Aris. 2. If a sphere 3 inches in diameter weighs 4 lbs., what is the diameter of a sphere which weighs 32 lbs. ? Solution. — 4 lbs. : 32 lbs. : : 3^ : cube of diameter required. Now 32 X 27 = 864; then 864-T-4 = 216, and ^^216 = 6 in., Ans. 3. If a cannon ball 6 inches in diameter weighs 58 lbs., what is the weight of a similar ball 8 inches in diameter ? 4. If a cube of gold whose side is 3 inches is worth $6400, what is the worth of a cube of gold whose side is 8 inches ? 5. If a pyramid 60 feet high contains 12500 en. ft., how many en. ft. are there in a similar pyramid 30 ft. high ? 6. If a conical stack of hay whose height is 12 feet contains 5 tons, what is the weight of a similar stack whose height is 20 feet ? 7. If a cubical block of marble whose side is 4 inches weighs 12 pounds, what will a cubic foot of marble weigh ? 8. If a cylindrical cistern 6 feet in diameter will contain 30 hogsheads of water, how much will a similar cistern contain, whose diameter is 20 feet ? 310 Evolution. 759. The side of a cube whose solidity is double, triple, etc., that of a cube whose side is given, is found by CuMng the given side, rrmltiplying it hy the given proportion, and extracting the cube root of the product, 9. What is the side of a cubical mound, which contains 8 times as many solid feet as one whose side is 3 ft. Ans. 6 ft. 10. Required the side of a cubical vat, which contains 3 times as many solid feet as one whose side is 5 ft. 11. If a cube of silver whose side is 4 inches, is worth $200, what is the side of a cube of silver, worth $1000? 12. I have a cubical box whose side is 6 ft. ; I want another which will contain \ j^art as much. What will be the length of its side ? 13. Required the side of a cubical vat which shall contain ■^ part as much as one whose side is 12 feet ? Questions. 710. What is involution? 711. What is a power? 713. The first power? 714. The second ? 715. The third ? 716. What is an exponent ? 718. How find a power of a number ? 731. What is evolution? 722. What is a root? 723. Square root? 724. Cube root ? 727. A perfect power ? 728. Imperfect ? 729. What is extracting the square root? 731. Name the principles respecting squares and root ? 734. How extract the square root. 735. How find the square root of fractions ? 736. How find a mean proportional between two numbers ? 739. What is a right-angled triangle ? 740. Which side is the hypothenuse ? What are the other two sides called ? 741. Name the principles respecting right angled triangles. 742. To what is the square of the hypothenuse equal ? 746. What are similar figures ? 747. How do similar surfaces compare with each other ? 749. What is the cube root of a number? 750. Name the principles respecting the number of periods and figures in the root ? 752. To what is the cube of a number consisting of tens and units equal? 754. How extract the cube root ? 755. How find the cube root of a fraction ? 757. What are similar solids? What are the like dimensions of spheres? Of cubes? Of cylinders and cones? Of pyramids? 758. How do the contents of similar solids compare with each other ? EOGKESSIOI^. Definitions. 760. A Progression is a series of numbers which regularly mcrease or decrease. 761. The Terms of a Progression are the numbers which form the series. T\\q first and last terms are the extremes ; the others, the means. 762. Progressions are of two kinds, aritlunetical and geo- metrical. Arithmetical Progression. 763. An Arithmetical Progression is a series w^hich increases or decreases by a common difference. 764. The Common Difference of a progression is the differ- ence between any two of its consecutive terms. 765. In an ascending series, each term is found hy adding the common difference to the preceding term. Thus, If the first term is 1 and the common difference 3, the series is 1, 4, 7, 10, 13, 16, 19, etc. 766. In a descending series, each term is found by subtract- ing the common difference from the preceding term. Thus, If 15 is the first term and 2 the common difference, the series is 15, 13, 11, 9, 7, 5, 3, 1. Notes. — 1. An Arithmetical Progression is sometimes called &n.Bqui- Mfferent Series. In every progression there may be an infinite number of ferms. 2. An Arithmetical Mean between two numbers is found by taking half their sum. 312 Progression, IQl. In Arithmetical Progression there are five elements or paints to be considered : the first term, the common difference, the last term, the numher of terms, and the sum of the terms. Let (I = the first term I — the last term. (I = the common difference. ti = the number of terms. s = the smn of the terms. The rehation of these five quantities to each other is such that if any three of them are given, the othe?' tivo can be found. 768. To find the Last Tei'm, when the First Term, the Common Difference, and Number of Terms are given. 1. Find the last term of an increasing series having 7 terms, its first term being 3, and its common difference 2. Analysis. — From the definition, each succeeding term is found by add- ing the common difference to the preceding. The series is : 3, 3 + 2, 3 + (2 + 3), 3 + (2 + 2 + 2), 3 + (2-f2 + 2 + 2), etc. Or, 3, 3 + 2. 3 + (2x2), 3 + (2x3), 3 + (2x4), etc. 2. Find the last term of a decreasing series having 5 terms, the first term being 24, the common difference 2. Analysis. — In a descending series, each succeeding term is found by subtracting the common difference from the preceding. Hence, the series is 24, 24-2, 24-(2 + 2), 24-(2 + 2 + 2), 24-(2 + 2 + 2 x2). etc. Or, 24, 24-2, 24-(2x2), 24-(2 x 3), 24-(2 x4), etc. That is, 769. The last term is equal to the first term, increased or diminished by the product of the common difference into the number of terms less 1. Hence, the Rule. — I. Multiply the mnnher of teTins less one by the common clifference. II. WJien the series is ascending, add this product to the first term; when descending, subtract it from the first term, FOKMULAS.-? = I « + (» - J) X d. Or, ( a — {n — 1) X a. Arithmetical Progression. 313 770. To find the First Term, when the Last Term, the Common Difference, and Number of Terms are given. 1. Find the first term of a decreasing series the last term of which is 2, the common difference 3, the number of terms 6. Analysis. — The first term of a decreasing series will be the last term increased by the product of the common difference by the number of terms less one. The series is 3 + 3x5, 2 + 8x4. 2 + 3x3, 2 + 3x2, 2 + 3, 2. 2. Find the first term of an increasing series, the last term of which is 45, the common difference 5, and the number of terms 7. Analysis. — The first term of an increasing series will be the last term diminished by the product of the common difference by the number of terms less one. The series is 45-5x6, 45-5x5, 45-5x4, 45-5x3, 45-5x2, 45-5x1, 45. Hence, the Rule. — I. Multiply the ninnher of terms less one hy the coimnon difference. II. When the series is ascending, subtract this product from the last term; when descending, add it to the last term. ^ \l — (n — 1) X d. Or, Formulas. — a — { j ) y, { t + {ii — 1) X d. Note. — Any term in the series may be found by the preceding rules. For, the series may be supposed to stop at any term, and that may be considered the last. 3. Find the last term of an ascending series, the first term of which is 5, the common difference 3, and the number of terms 12? 4. The first term of a descending series is 40, the common difference 3, and the number of terms 11 ; what is the last? 5. The last term of an ascending series is 87, the number of terms 16, and the common difference 4 ; what is the first term ? 6. What is the amount of ^250, at Q% simple interest, for 21 years ? 314 Progression, ITL. To find the Number of Terms, when the Extremes and the Common Difference are given. 1. The extremes of an arithmetical series are 4 and 37, and the common difference 3 ; what is the number of terms? Analysis. — The last term of a series is equal to tlie first term increased or diminished by the product of the common difference by the number of terms less 1. (Art. 769.) Now 37—4, or 83, is the product of the common difference 3, by the number of terms less 1. Consequently 33-J-3, or 11, must be the number of terms less 1 ; and 11 + 1, or 12, is the answer required. Hence, the Rule. — Divide the difference of the extremes by the coimnon difference, and add 1 to the quotient. Formula. — n = \ —^ h 1, 2. The age of the youngest child of a family is 1 year, the oldest 22, and the common difference of their ages 3 yr.; how many children in the family ? 3. The extremes of an arithmetical series are 8 and 96, the common difference 4 ; what is the number of terms ? 4. A laborer worked for 50 cts. the first day, 54 cts. the second, 58 cts. the third, and so on till his wages were 12 a day ; how many days did he work ? 772. To find the Common Difference, when the Extremes and the Number of Terms are given. 1. The extremes of a series are 3 and 21, and the number of terms is 10 ; what is the common difference ? Analysis. — The difference of the extremes 21 — 3 = 18, is the product of the number of terms less 1 by the common difference, and 10—1, or 9, is the number of terms less 1 ; therefore 18-5-9, or 2, is the common differ- ence required. (Art. 764.) Hence, the . EuLE. — Divide the difference of the extremes hy the nujnber of terms less 1. Formula. — d = ^ ' ~ - n — 1 Aritlimetical Progression, 315 2. The ages of 10 children form an arithmetical series ; the youngest is 3 yr. and the eldest 30 years ; what is the differ- ence of their ages ? 3. A military company appropriated 1108 for 8 target prizes, the highest of which was $24, and the lowest $3 ; what was the common difference in the prizes ? 4. The amount of $600 for 45 yr. at simple interest is $3120 ; what is the rate per cent ? 5. The amount of $1500 for 27 years is $1620; wdaat is the rate per cent ? 773. To find the Suin of all the terms, when the Extremes and the Number of Terms are given. 1. Eequired the sum of the series having 7 terms, the extremes being 3 and 15. Analysis.— (1.) The series is 3, 5, 7, 9, 11, 13, 15. (2.) Inverting the same, 15, 13, 11, 9, 7, 5, 3. (3.) Adding (1.) and (2.), 18 + 18 + 18 + 18 + 18 + 18 + 18=twice thesnm. (4.) Dividing (3.) by 2, 9+ 9+ 9+ 9+ 9+ 9+ 9=63. the sum. By inspecting these series, we discover that half the sum of the extremes is equal to the average value of the terras. Hence, the EuLE. — Multiply half the sinn of the eoctr ernes by the ninnher of terms. Formula.— s — ] — ^— x n. Note. — From the preceding illustration we see that, Th(} sum of the extremes is equal to the sum of any tim terms equidistant from them ; or, to twice the sum of the middle term, if the number of terms he odd. 2. How many strokes does a common clock strike in 12 hours? 3. Find the sum of all the terms, the extremes being and 300, and the number of terms 1200. 4. A father deposited $1 in the bank for his daughter on her first birthday, %^ the next, $7 the next, and so on; how much did she have when she was 21 years old ? 316 Progression, Geometrical Progression. Definitions. 774. A Geometrical Progression is a series of numbers which increase or decrease by a common ratio. 775. The Terms of a geometrical progression are the num- bers which form the series. Note. — The series is called Ascending or Descending, according as the terms increase or decrease. (Arts. 763, 764.) 776. In an ascending series the ratio is greater than one. Thus, 2, 4, 8, 16, 32, 64, etc., is an ascending progression. 777. In a descending series the ratio is tess than one. Thus, 1, 1, \, I, iV» O' 6*^-' is ^ descending progression. 778. In Geometrical Progression there are also five elements or parts to be considered, viz. : i\\Q first term, the last term, the numher of terms, the ratio, and the sum of all the terms. Let ct = the first term. I = the last term. V = the ratio. 11 = the number of terms. s = the sum of the terms. 779. To find the Last Term, when the First Term, the Ratio, and the Number of Terms are given. 1. Required the last term of an ascending series having 6 terms, the first term being 3, and the ratio 2. Analysis. — From the definition, the series is 3, 3x2, 3x(2x2), 3x(2x2x2), 3 x (2 x2 x2 x2), etc. Or, 3, 3 X 2, 3 X 22, 3 X 23, 3 x 2^, etc. Now, 3 X 25 = 3 X 32 = 96, A ns. That is, Each successive term = 1st term x ratio raised to a power whose ex- ponent is one less than the number of th.e term. Hence, the Geometrical Progression. 317 KuLE. — Multiply the first term hy that power of the ratio whose exponent is 1 less than the number of terms. Formula. — I = a x ^•»-^. Notes. — 1. Any term in a series may be found by tlie preceding rule. For, tlie series may be supposed to stop at that terra. 2. The preceding rule is applicable to Compound Interest ; the principal being the first term of the series ; the amount of $1 for 1 year the ratio ; and the number of year >< plus 1, the number of terms. 2. A father promised his son 1 ct. for the first example he solved, 2 cts. for the second, 4 cts. for the third, etc. ; what would the son receive for the tenth example ? 3. What is the ami of $375 for 4 yr., at b% compound int. ? 4. "What is the amount of 11200 for 5 years, at Q'-c comj^ound interest ? Of $2500 for 4 years, at 7^ ? 780. To find the First Term, when the Last Term, the Ratio, and the Number of Terms are given. 1. The last term of a progression is 96, the number of terms 6, and the ratio 2 ; what is the first term ? Analysis. — Reversing the steps of the preceding rule, we have 96-i-2^ = 96-V-33 = 3, Ans. Hence, the EuLE. — Divide the last term by that power of the ratio whose exponent is 1 less than the number of terms. Formula. — a = I -^ ?♦"-'. 2. The last term of a series is 192, the ratio 3, and the num- ber of terms 7 ; what is the first term ? 781. To find the Stiin of all fJie Terms, when the Extremes and Ratio are given. 1. Required the sum of the series whose first and last terms are 2 and 162, and the ratio 3. Analysis. — Since each succeeding term is found by multiplying the preceding term by the ratio, the series is 2, 6, 18, 54, 162. (1. ) The sum of the series, = 2 + 6 + 18 + 54 4- 162. (2.) 3 times the sum = 6+18 + 54+162 + 486. Subt. (1.) from (2.), we have 486—2 = 484, or twice the sum. Therefore, 484^2 = 242, the sum required. Hence, the 318 Progression, EuLE. — Multiply the last term dy the ratio, and sub- tracting the first terin from the product, divide the remainder by the ratio less 1. -r^^ {I X r) — a FORMVLA.—S = ~ ~ r — 1 2. The first term is 4, the ratio 3, and the last term is 972 ; what is the sum of the terms ? 3. What sum can be paid by 8 instalments ; the first being II, the second $2, etc., in a geometrical series ? 4. A man bought a dozen sheep, agreeing to pay 1 ct. for the first, 2 cts. for the second, 4 cts. for the third, etc.; what did he pay for the 12 sheep ? 5. A housekeeper bought 12 chairs, paying 2 cts. for the first, 6 cts. for the second, and so on ; what did they cost ? 782. To find the Sum of a Descending Infinite Series, when the First Term and Ratio are given. Note. — In a descending infinite series the last term being infinitely small, is regarded as 0. Hence, the Rule. — Divide the first term by the difference between the ratio and 1, and the quotient ivill be the sum required. 1. What is the sum of the series f, ^, -J-. ^, continued to infinity, the ratio being |^? Ans. 1^. Note. — The preceding problems in the Progressions embrace their ordi- nary applications. Others might be given, but they involve principles with which the pupil is not supposed to be acquainted. Questions. 760. Wliat is progression? 761. The terms? 763. An arithmetical progression ? 765. How is each term found in an ascending series ? 766. In a descending series ? 767. Name the parts. 768. How find the last term ? 770. The first term? 771. Number of terms. 773. The common difference ? 773. The sum of all the terms ? 774. What is geometrical progression? 778. Name the parts. 779. How find the last term ? 780. The first? 781. The sum of all the terms? 783. The sum of a descending infinite series ? ENSUFvATION. 1^ V ^ Definitions. 783. Mensuration is the process of measuring lines, sur- faces, and solids. 784. A Line is length without breadth or thickness. 785. A Straight Line is one that does not change its direction, and is the shortest dis- tance between two points in the same plane. 786. Parallel Lines are those which are equally distant from each other at every point. 787. Curved Lines are those which change their direction at every point. 788. A Horizontal Line is one that is parallel to the horizon or water level. 789. A Perpendicular Line is a straight line meeting another straight line, so as to make the two adjacent openings equal. As AB and CD. (Art. 792.) 790. A Perpendicular to a horizontal line is called a Vertical line. 791. A Plane Angle is the opening be- tween two straight lines drawn from the same point. Thus, the opening between AB and AC is an an- gle, the lines AB and AC are called the sides, and the point A the vertex of the angle. *.A ^:-\ ' s * t % « D 320 Mensuration, 792. A Right Angle is one of the two equal angles formed by the meeting of two straight lines perpendicular to each other. Thus, the adjacent angles ABC and ABD are liofht angles, and the lines AB and CD are perpendicular to each other. 793. An Acute Angle is one that is less than a right angle ; as AOB. A' 794. An Obtuse Angle is one that is ^ greater than a right angle ; as BCD. Note. — All angles except right angles are called oblique angles. 795. A Surface is that which has length and breadth, with- out thickness. Surfaces are either plane or curved. The surface of a table is plane, that of an orange is curved. 796. A Plane Figure is one which represents a surface all the parts of which are in the same plane. 797. A Polygon is a plane figure bounded by three or more straight lines. 798. The Perimeter of a polygon is the line by which it is bounded. 799. A Regular Polygon has all its sides and all its angles equal. 800. A polygon having three sides is called a triangle ; four sides, a quadrilateral ; five sides, ^ pentago7i ; six sides, a hex agon ; seven sides, a heptago7i ; eight sides, an octagon ; etc. 801. A Triangle is a polygon having three sides ^ and three angles. 802. The Base of a triangle is the side AB on which it is supposed to stand. A D B 803. A Vertical Angle is the angle opposite the base; as C. Area of Triangles, 321 804. The Altitude of a triangle is the perpendicular CD drawn from the vertical angle to the base. 805. An Equilateral Triangle is one having three equal sides. Isosceles. Scalene. Equilateral. 806. An Isosceles Triangle is one having only two equal sides. 807. A Scalene Triangle is one having all its sides unequal. Area of Triangles. 808. It is proved by Geometry that Tlie area of a triangle is equal to half the area of a parallelo- jram of equal base and altitude. Illustration. — Let ABCD be a paralle''ogram whose altitude is the perpendicular J^B. Connect the diagonal corners by the straight line BD, and the parallelogram will be divided into two equal triangles, the altitude of each a e being EB. The area of a parallelogram or rectangle is equal to the length multi- plied by the breadth . 809. To find the Area of a Triangle when the Base and Altitude are given. 1. What is the area of a triangle whose base is 30 ft. and its altitude 12 feet ? Let the base AD of the triangle ABD be 30 ft., and EB, its altitude, be 12 ft. Then 30 x 12 = 360 sq. ft., the area of the parallelogram. And 30 X 6 (1^ the altitude) = 180 sq. ft., area of triangle. Hence, the Rule. — Multiply the base hy half the altitude. 323 Mensuration, 2. What is tlie area of a triangle wliose base is 45 feet, and its altitude 20 feet ? 3. What is the area of a triangle whose base is 156 feet, and its altitude 63 feet ? 4. Find the number of acres in a triangular field whose base is 227 rods and altitude 65 rods. 5. What is the area of a triangle whose base is 135 yds., and its altitude is half its base ? 6. Find the number of sq. feet in the gable end of a build- ing 40 ft. wide, and 12 J^ ft. from the beam to the ridgepole. 810. To find the Area of a Triangle, when the Three SIcfes are given. From half the sum of the three sides subtract each side respec- tively ; then multiply half the sum and the three remainders together, and extract the square root of the product. 1. What is the area of a triangle whose sides are respectively 10 feet, 12 feet, and 16 feet ? Solution.— (10 + 12 + l6)-=-2 = 19 feet. 19-10 = 9 ; 19-12 = 7 ; 19 -16 = 3. Now 19x9x7x3 = 3591, and y'3591 = 59.92+ sq. ft. 2. What is the area of an equilateral triangle whose side is 12 yds. ? 3. What is the area of an isosceles triangle whose base is 30 feet and sides 20 feet ? 4. How many acres in a triangular field whose sides are 45, 53, and 64 rods ? 811. To find the Altitude, when the Area and Base are given. Rule. — Divide the area by half the base. 1. What is the altitude of a triangle whose area is 27-1 square yards and base 5 yards ? Ans. 11 yards. Quadrilaterals. 323 2. What is tlie altitude of a triangle whose area is 210 sq. yds. and its base 140 yards ? 3. What is the altitude of a triangle whose base is 150 rods and its area 11250 square rods ? 812. To find the Base, when the Area and Altitude are given. KuLE. — Divide the area hy half the altitude. 1. What is the base of a triangle whose area is 154 sq. ft. and its altitude 14 feet? Arts. 22 feet. 2. What is the base of a triangle whose area is 40 acres and its altitude 160 rods? 3. Find the base of a triangle whose area is 5260 sq. yd., and altitude 200 yards. Quadrilaterals. 813. A Quadrilateral is a polygon bounded by four straight lines. A quadrilateral is either a mTallelogram, a trapezoid, or a trapezium. 814. A Parallelogram is a quadrilateral haying its opposite sides equal and parallel. 815. The Altitude of a quadrilateral hav- ing two parallel sides is the perpendicular distance between these sides ; as, AL. 816. A Rectangle is a right-angled parallel- ogram. Note. — When the four sides of a rectangle are equal it is called a square. (Art. 345.) 817. A Rhomboid is an oblique-angled par- ielogram. 818. A Rhombus is an equilateral rhomboid. 324 Mensuration. 819. A Trapezoid is a quadrilateral which has two of its sides parallel. 820. A Trapezium is a quadrilateral having four unequal sides, no two of which are parallel.* Note. — The Diagonal of a plane figure is a a straight line connecting two of its angles not adjacent ; as AB. 821. To find the Area of a Parallelogram, when the Base and Altitude are given. 1. What is the area of a rectangle whose base is 88 feet and altitude 30 feet ? Solution.— 58 x 30 = 1740 sq. ft., Ans. 2. What is the area of a rhomboid whose base is 63 feet and its altitude 40 feet ? Solution.— 63 x 40 = 2520 sq. ft., Ans. Hence, the EuLE. — Multiply the base by the altitude. Note. — The area of a square, a rectangle, a rhomboid and rhombus is found in the same manner. 3. How many acres in a field 120 rods long, and 90 rods wide ? 4. How many acres in a field 800 rods long, and 128 rods wide ? 5. Find the area of a square field whose sides are 65 rods in length. 6. A man fenced off a rectangular field containing 3750 sq. rods, the length of which was 75 rods ; what was its breadth ? 7. One side of a rectangular field is 1 mile in length, and it contains 160 acres ; what is the length of the other side? * The majority of Authors define these terms as in the text. Others, among whom arc Le2;endre, Dr. Brewster, Younjr, and De Morgan, apply the definition here given of A Trapezium to the Trapezoid, and vice versa. Quadrilaterals, 325 8. The length of a rhombus is 17 ft., and its perpendicular height 16 ft.; what is its area? Ans. 272 sq. ft. 9. What is the area of a rhomboid whose altitude is 25 rods, and its length 28.6 rods ? 822. To find the Area of a Trapezoid, w^en its Parallel Sides and Altitude are given. 1. Find the area of a trapezoid whose parallel sides are 28 and 36 feet and its altitude 12 feet. Solution.— The sum of the parallel sides 28 + 36 = 64 ft., i of 64 = 32 ft., and 32 ft. x 13 (the altitude) = 384 sq. ft., Ans. Hence, the EuLE. — Multiply half the sum of the parallel sides hy the altitude. 2. The parallel sides of a trapezoid are 25 yd. and 21 yd., and its altitude 16 yd. ; what is its area ? 3. Find the area of a trapezoid whose parallel sides are 25 rods and 37 rods, and its altitude 18 rods. _ , 823. To find the Area of a Trapezhon, when the Diagonal and Perpendiculars are given. 1. A man bought a city lot in the form >/r^\ of a trapezium, the diagonal of which was y^ \ ^\^^ 84 ft. and perpendiculars from the opposite <^^ ^ -^ angles 12 ft. and 16 ft. ; what was its area? N. | ^^ Solution. — The sum of the perpendiculars is ^^580 at b%, on w^hich a payment of 185 had been made every 6 months ; using the U. S. rule. 187. A and B enter into partnership ; A furnished $240 for 8 mo., and B 1559 for 5 mo. They lost $118 ; how much did each man lose ? 188. In 25 kilogrammes how many pounds, Troy weight ? 18 -^ 1 189. Reduce — — '—^ to its simplest form. 190. What is the area in acres of a triangle whose base is 156 rods and its altitude 63 rods ? 191. Suppose a certain township is 6 miles long and 4|- miles wide, how many lots of land of 90 acres each does it contain ? 192. How^ many strokes would a clock wiiich goes to 24 o'clock, strike in a day ? 193. The extremes are 3 and 19, the number of terms 9; what is the com. dif.^ and the sum of the series ? 194. A man spent 13 the first holiday, l?45 the last, and each day $3 more than on the preceding ; how many holidays did he have, and how much did he spend ? 195. What is the area of a circle whose diameter is 120 rd. ? 352 Test Questiom. 196. How much should be discounted on a bill of $3725.87, due in 8 mo. 10 da., if paid immediately, money being worth 5 per cent ? 197. Bouglit bonds at 115 aiid sold at 110, losing $300. How many bonds of $1000 each did I buy? 198. What is the amount of 1225, at 6 per cent compound interest for 4 years ? 199. A steamer goes due north at the rate of 15 miles an hour, and another due west 18 miles an hour ; how far apart will they be in 24 hours ? 200. Find the cost, at 30 cts. per sq. yd., of plastering the bottom and sides of a cubical cistern that will hold 300 bbls. 201. Find the surface and the diagonal of a cube of granite containing 162144 cu. inches. 202. What is the area of a circle whose circumference is 160 yards ? 203. What is the solidity of a prism whose height is 25 ft., and its base an equilateral triangle whose side is 12 feet? 204. What is the solidity of a prism whose base is 6 ft. square and its height 15 feet? 205. What is the solidity of a triangular prism whose height is 20 feet, and the area of whose base is 460 square feet? 206. Required the solidity of a square pyramid, the side of whose base is 25 feet, and whose height is ijQ feet. 207. Required the solidity of a cone, the diameter of whose base is 30 feet, and whose height is 96 feet. 208. Required the solidity of a cylinder 20 feet in diameter and 65 feet long. 209. How many acres in a triangular field whose base is 325 yd. and its altitude 160 yd. ? 210. If a scholar receive 1 credit mark for the first example he solves, 2 for the second, 4 for the third, and so on, the num- ber being doubled for each example, how many marks will he receive for the twelfth ? 211. What rate of income will U. S. 34^ bonds yield, if bought at 102, and payable at par in 25 years? 212. What per cent income will Alabama 9's yield, bought at 85 and paid at par in 15 years ? Test Questions, 353 III. Problems FROM Entrance Examination Papers of Various Colleges. Harvard University, 1880, '81. 859. 1. Find the greatest common divisor of 315, 504, 441. 2. Find the square root of 2 to the nearest ten-thousandth. 3. A wall which was to be 36 feet high was raised 9 feet in 6 days by 16 men ; how many men will be needed to finish the w^ork in 4 days ? 4. A tradesman marks his goods at 25 per cent above cost, and deducts 12 per cent of the amount of any customer's bill, for cash. What per cent does he make ? 5. A tunnel is 2 miles 21 chains 13.2 yards long. Find its length in meters. [1 mile = 1.61 kilometres.] 1 7_5 q 3. _|_ 45. 6. Simplify lliV-^^^- 7. Find the value in cubic decimeters of {-|- of 87 cu. meters 62 cu. decimeters 300 cu. centimeters. 8. If 27 men, working 10 hours a day, do a piece of work in 14 days, how many hours a day must 12 men work, to do the same amount in 45 days ? 9. What sum of money, at 6 per cent annually compounded interest, will amount to $2703 in 1 yr. 4 months? 10. Arrange in order of magnitude, fl, ^, 0.89. Yale College, 1880. 11. Add (fxixl), tV f, and tV 12. Divide (p - ^) by A. 13. Find the fourth term of a proportion of which the first, second, and third terms are, respectively, 3.81, 0.056, 1.67. 14. Reduce 133 sq. rd. 8 sq. ft. to a decimal of an acre. 15. In a board 4 meters long and 0.4 meters wide, how many square decimeters ? 3A 16. Divide H oi -{-^ oi ^) bv ^-, and add the quotient to 4 iF' 354 Test Questions, 17. Find V-Ti:, to three decimal places. 18. Find, to three decimal places, the number which has to 0.649 the same ratio which 58 has to 634. 19. A man bought a piece of ground containing 0.316 A. at 53 cents a square foot ; what did he pay for the piece ? 20. A grocer buys sugar at 18 cents a kilo, and sells it at 1 cent per 50 grams ; how much per cent does he gain ? Columbia College, 1881. 21. Define a fraction. Give the rule for the addition of frac- tions, and the reason for each step of the operation. 22. Eeduce 126 grams to ounces. 63f yd. to meters. 23. From f of a gallon take If of a pint. What difference, if any, between the subtraction of compound numbers and that of simple ones ? Between the subtraction of fractions and that of integers ? 24. If 30 lb. of cotton will make 3 pieces of muslin 42 yd. long and f yd. wide, how many pounds will it take to make 50 pieces, each containing 35 yd., 1\ yd. wide? 25. A, B, and formed a partnership, and cleared 112000. A put in $8000 for 4 mo., and then added 12000 for 6 mo.; B put in $16000 for 3 mo., and then withdrawing half his capital, continued the remainder 5 mo. longer ; put in $13500 for 7 mo. How divide the profit ? 26. Find the sum of 3^, 6f, 8j^, 65f, reduce the fractional part to a decimal, and extract the cube root of the result. Dartmouth College, 1879, '80. 27. Find the I, C 7n. and the g, c, d, of 6, 8, 20, and 36. 28. How many metres in 25 feet ? 29. Find the square root of 3530641. 30. Gold was quoted at |1.12|-; what was a %1 greenback worth ? 31. 11200 includes a sum to be invested and a commission of h% of the sum invested; what is the sum invested? 32. Find the sum and product of -|, J, f . 33. Find the cube root of 3845672000. Test Questions, 355 34. Find the square root of 3534400.5. 35. A platl'orni bears a weight of 100 lb. per square foot ; what is the weight in kilograms per square meter? 36. A horse that cost 6^ per cent of $25000, was sold for $1000 ; what was the loss per cent ? College of City of New York, 1880, '81. 37. Eeduce f of ^ of -y- to a decimal, carrying out the opera- tion to four places. 38. If two men, working 8 hours, can carry 12000 bricks to the height of 50 fe^t, how many bricks can one man, working 10 hours, carry to the height of 30 feet ? 39. I buy goods to the amount of $4,978.70, payable in 4 mo., with interest at b^^, and give my note without interest. What must be the face of the note ? 40. A man lost ^, \, and f of his money, and then had $2600 left ; what sum had he originally, and how much per cent had he lost? 41. Sold a fire engine for $7050, and lost %% on its cost ; for how much ought I to have sold it to gain 12;;^^' ? 42. What sum of money put at interest 6 yr. 5 mo. 11 da., at 7^, will gain $3159.14? 43. For what sum must a note be drawn at 60 days to net $1200 when discounted at b% ? 44. Extract the square root of 3286.9835 to the fourth deci- mal place. 45. Extract the cube root of 30.625. Amherst College, 1881. 46. Find the greatest common divisor of 1263 and 1623. 47. Find the least common multii)le of 18, 24, 36, and 126. 48. A cable that weighs one ton per mile weighs how much per foot ? 49. When it is 10 o'clock in Boston what time is it in Am- herst, the longitude of Boston being 71° 7' 45" W. from Green- wich, that of Amherst being 72^ 31' 50" ? 50. Reduce 1 hr. 25 min. 30 sec. to the decimal of a day. 51. Of what number is f the J part? 356 Test Questions. 52. What must be the face of a note which discounted at a bank at Q% for 30 days and grace, would yield $200 ? 53. Sold a house for $5000 and thereby gained 20^. Should I have gained or lost, and how much per cent, if I had sold it for $4000 ? 54. Find the square root of 5.6169. 55. The meter is 39.37 inches. Find how many kilometers there are in a mile. Vassar College, 1880. 5e. Add ^^1 to 4^f . •5 + 1 ^ — ^ 57. Multiply 48 ten thousandths by two and one thousandth, and divide the result by one million. 58. Express 462 mm. in higher denominations. 59. What is 1% of 140 books? What per cent of 30 ft. is 25 inches ? 60. If I lose 10;^ by selling goods at 28 cents per yard, for what should they be sold to gain 20%'? 61. What principal will yield an interest of $339.20 in 5 yr. 4 mo. at Q% ? 62. What must be the length of a box, 1 meter wide and 1 meter deep, to contain 4500 liters ? 63. Cube .01. Square 1.001. 64. Extract the square root of 4.932841. 65. A can do a piece of work in 10 days; A and B can do the same work together in 7 days; in how many days can B working alone do the Avork ? New York Normal College, 1880. 66. What will it cost to floor a room 17^ ft. long and 16 ft. wide, at the rate of $1.10 per sq. yd. ? 67. A man has a capital of $12500 ; he puts 15;^ of it in stocks, 33|C^ in land, and 25^^ ni mortgages ; how many dollars has he left ? 68. A grocer bought 500 bags of coffee, each bag containing 49^ pounds, at 12 cents a pound, and sold at a profit of 16-|%' ; for what did he sell it ? Test Questions, 357 69. If I buy a house for $5620 and receive $1803 for rent in 2 yr. 3 mo. 15 da., what rate of int. do I get for my money? 70. Find the face of a note payable in 90 da. at 1i%, so that the proceeds shall be 12050 ? 71. A merchant owes 12400, of which $400 is payable in 6 mo., $800 in 10 mo., and $1200 in 16 mo. ; what is the equa- ted time ? 72. If it costs 17.20 to transport 18 J cwt. 5 J miles, what will it cost to transport 112 J tons 62|- miles ? 73. Extract the square root of 1051 to three places of deci- mals. 74. What is the cube root of 403583419 ? Cornell University, 1880, 75. What is the value of 50 lb. 8 oz. of gold, at $20.59;|- per ounce ? 76. Given the metre equal to 39.37 inches, reduce one mile to kilometers. Give the metric table of weights. 77. Divide f of 7| by f of 12^. Prove the result by reduc- ing the fractions to decimals and working the example anew. 78. How long must $125 be on interest at 7^ per cent to gain $15 ? 79. Eeceived 6 per cent dividend on stock bought at 25 j^er cent below par ; what rate of interest did the investment pay ? 80. How many liters in 20 bu. 3 pk. 4 qt., the bushel being 2150.42 cubic inches, and the metre 39.37 inches? 81. Simplify (l + ^^) - (l + ^). 82. If one kilometer equals five-eighths of a mile, how many turns will a wheel make in 20 miles, the circumference of the wheel being 4 meters 5 millimeters ? 83. What is the difference between the true and bank dis- count of $250, due 10 mo. hence, at 7f/ ? 84. If 8 men spend $32 in 13 weeks, what will 24 men spend in 52 weeks ? 358 Te8t Questions, Trinity College, 1880. 85. Subtract thirty million twenty-six thousand three from 45007021. Find what number must be added to the difference to make one hundred million, and write the answer in words. 86. The sum of | and -^j is diminished by ^. How many times does the difference contain y\ of the sum of \, ^, and yV-^ 87. Divide 375 by .75, and .75 by 375, and find the sum and the difference of the quotients. 88. A loaded wagon weighs 2 T. 3 cwt. 48 lb. ; the wagon itself weighs 18 cwt. 75 lb. The load consists of 215 packages, each of the same weight. Find the weight of each, and reduce it to grams and kilograms. 89. Define interest, and give and explain the rule for com- puting the interest on any sum of money, for any time, and at any rate per cent. 90. Extract the square root of 184.2 to 3 decimals. 91. How many hektoliters of oats can be put into a bin that is 2 meters long, 1.3 meters wide, and 1.5 meters deep ? Wesleyan University, 1881. 92.- Add t-^ and if±4i I + 4i 7i - 4| 93. If money is worth 3 per cent, what is the premium on government 3^ per cent bonds ? 94. How many liters in 6 gallons of water? 95. How many cords of stone will it take to build a Avail 2 ft. thick and 6 ft. high about a rectangular cellar whose inte- rior dimensions, when the wall is completed, shall be 20 ft. long and 16 ft. wide? 96. How long must a note of $243, at ^%, run that its in- terest may equal the int. on a note of $125, for 7 mo., at 5^? 97. Multiply \/2 by a/.123, and carry the result to 3 decimals. 98. Eeduce 5 mi. 3 fur. 10 rd. to kilometers. 99. If 5 horses will consume 8 bu. 1 pk. 6 qt. of oats in G da., w^hat quantity of oats will 7 horses consume in 11 da. ? PPENDIX Roman Notation. 860. The Roman Notation is the method of expressing num- bers by seven ccqntal letters, viz. : I, V, X, L, 0, r>, M. 1, 5, 10, 50, 100, 500, 1000. 861. To express other numbers, these letters are combined as in the following Table. I II III IV V VI VII VIII IX X 1 2 3 4 5 6 7 8 9 10 XI XII XIII XIV XV XVI XVII XVIII XIX XX 11 12 13 14 15 16 17 18 19 20 XXI XXV XXX XXXI XL L LX LXX XO 21 25 30 31 40 50 60 70 90 100 CI = 101 ex = 110 CL = 150 cc = 200 D = 500 DC rr 600 M = 1000 MC = 1100 MD = 1500 MM = 2000 MDCCLXXVI = 1776. MDCCCLXXXII = 1882. 862. The Roman Notation is based upon the following gen- eral principles : 1st. Repeating a letter repeats its value. Thus, I denotes one ; II, two ; III, three ; X, ten ; XX, twenty, etc. 2d. Placing a letter of less value before one of greater value, dhainislies the value of the greater by that of the less ; placing the less after the greater, increases the value of the greater by 360 Appendix. that of the less. Thus, V denotes five, but IV denotes only four, and VI six. 3d. Placing a horizontal line over a letter increases its value a thousand times. Thus, I denotes a thousand; X, ten thou- sand ; 0, a hundred thousand ; M, a million. Notes. — 1. The letters C and M are the initials of the Latin centum, a hundred, and mille, a thousand. 2. The radix of the system appears to be doubtful. Some have sup- posed that at first it was five (V), and %Yas subsequently changed to ten (X), forming a combination of the quinary and decimal systems, 3. Others maintain, more plausibly, that it proceeds according to the alternate scale of 5 and 2, thus uniting the Unary with the quinary scale. That is. Five times one (I) are five (V). Two times five (V) are ten (X), Five times ten (X) are fifty (L). Two times fifty (L) are one hundred (C). Five times one hundred (C) are five hundred (D). Two times five hundred (D) are one thousand (M). English Numeration. 863. By the English Numeration, numbers are divided into periods oi six fgures each, and then each period is subdivided into units, tens, hundreds, thousands, tens of thousands, and hundreds of tliousands, as in the following T ABLE. 5 S 5 ill 1 s 03 .2 32 CD o o Cm O 5M s o o o O i !<-( 3 00 O O s=l CS 3 X! o O ^ si o o '*-. ^ .c o Eh J=i rjj O s . ^ .£1 iE o ^ 5t Eh .a 00 , o Cm O rj-j Cm o Cm o O r/1 tM o o o '2 s 00 CO '6 Ol s 73 QQ "■«. 13 7J s 13 ■33 rS 00 s -a 00 *9A G n o a p -«j C o r^ c ••^ P c o «—■ "» S a> j3 ^ - 21 and set the first product figure in tens' place ; then ^^ adding this partial product to the multiplicand, we ' '^ have 7S65 for the answer. 7665 days, AllS. 362 Aijpendix, 867. To Multiply Two or More Numbers by the Same Multiplier. 1. A grocer sold 4 pounds of tea to one caslomer, 3 lb. to another, and 5 lb. to another; how much did it all come to, at 7 dimes a pound ? Solution. (4 + 3 + 5) x 7 = 84 dimes, Ans. Hence, tlie ^i]!^^. — Miiltiplij the siuih of the iiin7ibers by the Jiiidti- plier. 868. To Multiply a Mixed Number, whose Fractional Part is |, by itself. 1. What is the product of 3|^ into 3|^ ? Solution. — The integral part of tlie given number is 3, and 3 + 1 =4. Now 3 into 4 = 12, and 12 + ] = 12], Ans Hence, the Rule. — Multiply the ijite^ral part by one more than itself, and to this result annex ^. 869. To Multiply by 9, 99, 999, or any number of 9's. 1. How much will 99 carriages cost, at 235 dollars apiece ? Explanation. — Since 1 carriage opekation. costs $235, 100 carriages will cost 100 $23500 Price of 100 C. times as much, or $23500. But 99 is 1 qo;^ an -in less than 100 ; therefore, subtracting tlie price of 1 carriage from the price of 100 $23265 ^' ^^ 99 C. gives the price of 99 carriages. 870. To Divide by 5. 1. A merchant laid out $873 in flour, at $5 a barrel ; how many barrels did he get ? Explanation. — We first multiply the dividend " ' "^ by 2, and then divide tlie product by 10, by cutting 2 off the right-hand figure. The hgure cut oli' is written over the divisor, and the fraction, reouced. 110 ) 17416 to its lowest terms, is annexed to the quotient. ^^4|, AnS. Note. — This contraction depends upon the princij)le that any given di- visor is contained in any given dividend just as many times as twice that divisor is contained in twice that dividend, three times that divisor in three t'mes that dividend, etc. Contractions in Division, 363 871. To Divide by 25. 1. A farmer paid $150 for cows, at $25 apiece ; how many cows did he buy ? Explanation. — We first multiply the dividend by 4, and then divide the product by 100. (Art. 118.) OPERATION. 150 4 1100 ) 6100 Ans. 6 cows. 872. To Divide by 125. 1. A man bought land for $12150, at $125 an acre ; how many acres did he buy ? Explanation. — We multiply the dividend by 8, and divide the product by 1000. (Art. 118.) Placing the remainder over the divisor, we re- duce the fraction to lowest terms, and annex it to the quotient. The answer is 97^ acres. Note. — This contraction multiplies both the dividend and divisor by 8. (Art. 119, 3°.) OPERATION. 12150 11000 ) 971200 Ans. 97-iA. 873. Demonstration of Finding the Greatest Common Divisor by Continued Divisions. 1. Find the g. c, d. of 30 and 42. Ans. 6. Two points are to be proved ; 1st. That 6 is a common, divisor of the given numbers. 2d. That 6 is their greatest common divisor. That 6 is a common divisor of 30 and 42 is easily proved by trial. Next, we are to prove that 6 is the greatest common divisor of 30 and 42. If the greatest common divisor of these numbers is not 6, it must be either greater or less than 6. But we have shown that 6 is a common divisor of the given numbers ; therefore, no number less than 6 can be the greatest common divisor of them. The assumed number must therefore be greater than 6. By the supposition, this assumed number is a divisor of 30 and 42 ; hence, it must be a divisor of their difference, 42 — 30, or 12. And as it is a divisor of 12, it must also divide the product of 12 into 2, or 24. Again, since the assumed number is a divisor of 30 and 24, it must also be a di- visor of their difference, which is 6 ; that is, a greater number will divide a less without a remainder, which is impossible. Therefore, 6 must be the greatest common divisor of 30 and 42. 364 Appendix. 874. To find the Excess of 9's in a Number. 1. Let it be required to find the excess of 9's in 7548467. Beginning at the left hand, add the figures together, and as soon as the sum is 9 or more, reject 9 and add the remainder to the next figure, and so on. Adding 7 to 5, the sum is 12. Rejecting 9 from 12, leaves 3 ; and 3 added to 4 are 7, and 8 are 15. Rejecting 9 from 15, leaves 6 ; and 6 added to 4 are 10. Rejecting 9 from 10, leaves 1 ; and 1 added to 6 are 7, and 7 are 14. Finally, rejecting 9 from 14 leaves 5, the excess required. 875. Hence we derive this property of the number 9 : Any nurnher divided hy 9 will leave the same remainder as the sum of its digits divided hy 9. Notes. — 1. It will be observed that the excess of 9's in any tico digits is always equal to the sum, or the excess in tlie sum, of those digits. Thus, in 15 the excess is 6, and 1 + 5 = 6; so in 51 it is 6, and 5 + 1 = 6. 2. The operation of finding the excess of 9's in a number is called cast- ing out the 9's. 2. What is the product of 746 multiplied by 475 ? OPERATION. Proof by Excess of G's. Proof by Mult. 746 Excess of 9's in multiplic'd is 8 475 475 " " multiplier is 7 746 3730 Now 8x7 = 56 2850 5222 '^^16 excess of 9's in 56 is 2 iqqo 2984 The excess of 9's in product is 2 3325 Ans. 354350 ^^^ ^-^ Ans. 354350 876. To prove Multiplication by Excess of 9's. Fijul the excess of 9's in each factor separately ; then multiply these excesses together, and reject the 9's from the result ; if this excess agrees with the excess of 9's in the answer, the worh is right. Note. — The preceding is not a necessary but an incidental property of the number 9. It arises from the hno of increase in the decimal notation. If the rndix of the system were 8, it would belong to 7 ; if the radix were 12, it would belong to 11 ; and universally, it belongs to the number that is one less than the radix of the svpter.i of notation. Circulating Decimals. 365 Circulating Decimals. 877. A Circulating Decimal is one in which the same figure or set of figures is continually repeated in the same order. 878. In reducing common fractions to decimals, we find that in one class of examples the division is complete, and the quo- tient is an exact decimal. In another class, the same figure or set of figures is repeated again and again, and the division will not terminate, though continued indefinitely. The former are Terminate Decimals, the latter are Circulating Decimals, and the figure or figures repeated the Repetend. Thus, 4 = .5, f = .4, I = .75, | = .625, etc., are exact decimals. But i = .333333 + , if = .424242 + , i| ^ .297297 + , are interminate. 879. By inspection we see that the denominators of the first class are t\iQ prime numbers 2 or 5, or are composed of the fac- tors 2 or 5. In the second class, the denominators contain other prime factors than 2 and 5. 880. To find whether a common fraction will produce a ter- minate or a circulating decimal, Resolve the cleiioiiiiuator into its prime factors. If it contains no other factors than 2 and 5, the quotient will he a terminate decimal. If it contains any other prime factors than 2 and 5, the quotient will he a circulating decimal. 881. Circulating decimals are expressed by writing the rep- etend once, and placing a dot over the^r^^ and last figure. Thus, the repetends .33333+ and .297297+ are written .3 and .297. 882. When the repetend begins with tenths, the decimal is called a Pure Repetend ; as, .4 ; .297. 1. |. 5. -AV- 9. 2. i- 6. 1 3 16* 10. 3. A- 7. 7 16* 11. 4. e- 8. 4. 12. 36G Appendix. 883. When the repetend is preceded by one or more decimal figures, it is called a Mixed Eepetend; as .27; .42631. Note. — The decimal figures hefore the repetend are called the finite part of the decimal ; as 2 and 43 in the mixed repetends above. 884. Change the following fractions to terminate or circu- latimj decimals, and mark the repetends in each. (Art. 249.) JLj6 14 1_6 18 _3JL 21- •^*' 30' ■'■°* 180* _9 15 2.i iQ 2l1. -T- 16 2_2 on _3 885. To Reduce a Pure Repetend to a Common Fraction. 1. Reduce .24 to a common fraction. Analysis. — Since the repetend con. operation. sists of two figures, if we multiply it by q \\ ^ i qq 94 2424 100, the decimal part of the product mil be the same as the repetend. Now, if 0.24 X 1 = 0.2424 we subtract the repetend from this prod- ^ a ; 24 y 99 — 24 uct, the remainder will have no decimal y^ 'J^j '^■^ and will be 99 times the given repetend. ^24 :zr ^ = -S_ Ans, Therefore, once the given repetend is f f or 3^, Ans. Hence, the EuLE. — Take the repetend for the numerator, and Tnahe the denominator as many 9's as there are figures in the repetend. Reduce the fraction thus produced to its lowest terms. Reduce the following to common fractions: 2. .18. 5. .123. 8. .1007. 11. .25121. 3. .72. 6. .297. 9. .6435. 12. .142857. 4. .69, 7. .045. 10. .4158. 13. .076923. Circidating Dechnals, 367 886. To Reduce a Mixed Repetend to a Common Fraction. 14. Keduce 0.227 to a common fraction. OPERATION. SoLUTioN.-Subtracting the finite part ^27 Given decimal, from the given mixed repetend, both re- t?- • garded as integers, we have for the uumer- ^^^^^^^ paiT. ator 235, and for the denominator 990, ^^5 Numerator, that is, as many 9's as there are figures in 990 Denominator. the repetend with as many ciphers an- 22^5 4_5 .5 i^<. nexed as there are figures m the jimte part, and the result is'fff = jfg = A, Ans. Hence, the EuLE. — For the numerator, siibtractthe finite part from the given mijoecl repetend, both regarded as integers ; and for the denominator, talce as many 9's as there are figures in the repetend, with as many ciphers annexed as there are figures in the finite part. Explanation. — Since the repetend operation. has two figures, if we multiply the q 927 x 100 = 22.7272 given mixed repetend by 100, and from the product subtract once the given 0.227 X 1 = .2272 mixed repetend, the remainder (22.5) • ;, will be 99 times the given mixed repe- ^•'^^^ X J9 — 22.5 tend; and once the mixed repetend = 0.227 ^^ ^^"^ rr: ^^^ ^^^ = ||3. But 225 is the difierence between 327 the given mixed repetend, TWS '^^ TtE ^^^ 2T? -^ '^^* regarded as an integer, and 2 the finite part of it, regarded as an integer, and f |f = ^%, the same as before. Change the following to common fractions : 15. .6472. 17. .5925. 19. .5925. 21. .9285714. 16. .1004. 18. .0227. 20. .4745. 22. .008497133. 887. Circulating decimals, when reduced to common frac- tions, may be added, subtracted, multiplied, and di\dded, like other common fractions. 23. What is the sum of .5925 + .4745 + .0227 ? 24. What is the difference between .6435 and ,4158 ? 368 Appendix. Table of Prime Numbers from I to 1009. 1 59 139 233 337 439 557 653 769 883 2 61 149 239 347 443 563 659 773 887 3 67 151 241 349 449 569 661 787 907 5 71 157 251 353 457 571 673 797 911 7 73 163 257 359 461 577 677 809 919 11 79 167 263 367 463 587 683 811 929 13 83 173 269 373 467 593 691 821 937 17 89 179 271 379 479 599 701 823 941 19 97 181 277 383 487 601 709 827 947 23 101 191 281 389 491 607 719 829 953 29 103 193 283 397 499 613 727 839 967 31 107 197 293 401 503 617 733 ^ 853 971 37 109 199 307 409 509 619 739 857 977 41 113 211 311 419 521 631 743 859 983 43 127 223 313 421 523 641 751 863 991 47 131 227 317 431 541 643 757 877 997 53 137 229 331 433 547 647 761 881 1009 SURVEYOR'S Measure. 888. Surveyor's Measure is used in measuring land, in lay- ing ont roads, establishing boundaries, etc. 889. The Linear Unit usually employed by surveyors is Guntefs Chain, which is 4 rods or i!>Q feet long, and contains 100 links. It is subdivided as in the following Ta b l e. 7.92 iuclies {in.) = 1 link, I, 25 links = 1 rod or pole, . . r. 4 rods = 1 cJuiin, . . . ch. 80 cliains = 1 mile, m. Notes. — 1. Ountefs chain is so called from the name of its inventor. Measurements by it are usually given in chains iir\6. hundredths of a chain. 2. In measuring roads, etc., engineers use a chain, or measuring tape, 100 feet long, each foot being divided into tenths and hundredths. 3. The mile of the table is the common land mile, which contains 5380 feet. Government Lands. 369 890. The Measuring Unit of Land is the Acre. Table. 625 sq. links = 1 sq, rod or pole, . sq, rd. 16 sq. rods = 1 sq. chain, . . . sq. c. 10 sq. chains, or ) ^ . 160 sq. rods ) 640 acres = 1 sq. mile, . . . . sq. mi. Notes. — 1. The Rood of 40 square rods is no longer a unit of measure. 2. A Square, in Architecture, is 100 square feet. G-OVERNMENT LaNDS. 891. The i^iiblic lands of the United States are divided into Townships, which are subdivided into Sections, Half-Sections, Quarter-Sections, etc. A Towiifship is 6 miles square, and contains 36 sq. mUes. A Section is 1 mile square, and contains 640 acres. A Half-Section is 1 mile long by ^ mile wide, and contains 320 acres. A Quarter-Section is 160 rods square, and contains 160 acres. 892. The method adopted by the GoYernment in siirvejring a new territory is the following : First. — A line is run North and South, called the Principal Meridian. Second. — A line is run on a parallel of latitude E. and W. called the Base Line. TJiird. — Lines are run 6 miles apart parallel to the principal meridian. Fourth. — Other lines are run 6 miles apart parallel to the base line, forming townships, or squares^ each containing 36 sq. • miles. 893. Townships are designated by their number N. or S. of the base line. 894. A line of townships running N". and S. is called a Range, and is designated by its number E. or W. of the prin- cipal meridian. Thus, 370 Appendix. T. 39 N., R. 14 E. 3d P. M., describes the township ihat is in the 39th tier North of the base line, and in the 14th range E. of the 3d principal meridian. 895. A Township is divided into Sections each 1 mile square and contains 640 acres. Thus, A Section = 1 mi. X 1 mi. — 640 acres A Half Section = 1 " X i " = 320 " A Quarter Section = 1 " X i " = 160 " A Hall-quarter Section = 1 " X i " = 80 " A Quarter-quarter Section = 1 " X tV " = 40 " 6 5 4 3 2 1 7 8 9 10 11 13 18 17 16 15 14 13 19 20 21 23 23 24 30 29 28 27 26 25 31 32 33 34 35 36 The adjoining diagram represents a A Township. township divided into sections, which N are numbered commencing- at the N.E. corner, and running W. in the North tier, E. in the second, etc. Each section is divided into 4 quar- ter sections, called N.E., S.E., N.W., and S.W. quarters, each containing w 160 acres. Thus, S.E. \, sec. 16, T. 39 N., R. 14 E. 3d., P. M., is read, "Southeast quarter of section 16, tier 39 north, range 14 east of third principal meri- dian." 1. A colony of 224 persons took \\]) a township of land and divided it equally among them ; how many acres did each receive ? 2. What part of a section did each colonist receive, and what did it cost him, at 11.25 an acre ? 3. What will it cost to enclose a quarter section of land with a fence 5 rails high, at $2 for every 3 rods ? 4. If you pay $1.75 an acre for a half section of land, and sell a quarter section for 12.50, how much will your remaining quarter cost you ? 5. A company of speculators bought a township at 11.50 an acre ; they sold 10 sections at 12.25 an acre, 15 sections at $3.50, 8 sections at 14, and the balance at S5 an acre ; how much did they sell at 15, and what was the gain on the whole ? Explain by diagram. Pounds in a Bushel. 371 896. Table of Pounds Avoirdupois in a Bushel, as fixed by Law in the several States named. It is becoming common in some parts of this country and in England to sell grain and other produce by weight and not by measure, a much more equitable system than that which has long prevailed. w K • !S b-l COMMODITIES. 60 52 no o 56 60 56 1 60 52 -^ 60 56 60 50 5J> « 60 56 1 60 56 « 1 60 56 60 56 o 1 60 60 1 S 60 58 60 54 •i 60 56 1 60 56 "Si 60 56 o 60 56 s 1 60 56 i 60 56 Wheat Indian Com in ear 52 56 Oats 32 50 40 28 45 32 48 40 32 48 50 35 48 52 48 52 32 32 30 46 46 32 48 42 32 48 42 35 30 48 4R 32 48 48 48 50 32 48 34 46 42 32 47 48 32 46 46 36 45 42 32 48 42 Barlev .... Buckwheat 52 50 Rve 54 56 54 60 56 60 56 60 56 60 32 56 56 60 56 60 56 60 56 56 60 56 60 56 60 56 56 56 60 56 60 Clover Seed Timothy Seed — 45 45 45 45 45 45 46 Blue Grass Seed. . 14 14 14 14 14 56 Flax Seed 56 56 56 56 56 55 55 56 Hemp Seed 4i 44 44 44 44 Notes. — 1. Beans, peas, and potatoes are usually estimated at 60 lb. to the bii., but the laws of N. Y. make 63 lb. of beans to a bushel. In Illinois, 50 lb. of common salt, or 55 lb, fine, are 1 bu. In N. J., 56 lb. of salt are 1 bu. In Ind., Ky., and Iowa, 50 lb. are 1 bu. In Penn., 80 lb. coarse, 70 lb. ground, or 63 lb. fine salt are 1 bu. In Maine, 30 lb. oats, and 64 lb. of beets or of ruta-baga turnips are 1 bu. In New Hampshire, 30 lb. of oats are 1 bu. 3. Grain, seeds, and small fruit are sold by the bushel, stricken or level measure. Large fruit, potatoes, and all coarse vegetables by heaped measure. 897. Capacity Measures, estimated by Avoirdupois Weight. 63| pounds, or 1000 oz. 100 pounds 100 pounds 196 pounds 200 pounds 280 pounds §40 pounds 1 cubic foot of water. 1 keg of nails. 1 quintal of dry fish. 1 barrel of flour. 1 barrel offish, beef, or porko 1 barrel of salt. 1 cask of lime. 372 Appendix, Apothecaries' Fluid Measure. 898. Apothecaries' Fluid Measure is used in mixing liquid medicines. 60 minims, or drops (Tt|^ ot gtt.) = 1 fluid draclim, . . f 3 . 8 fluid draclims — 1 fluid ounce, . . . /s . 16 fluid ounces = 1 pint 0. 8 pints = 1 gallon, ..... Cong. Note. — Gtt. for guttce, Lsitm, signifying- drops; 0, for oaarius, Latin for one-eighth ; and (Jong.y congiariwm, Latin for gallon. 899. The following approximate measures, though not strictly accurate, are often useful in practical life: 45 drops of water, or a common teaspoonful = 1 fluid drachm. A common tablespoonful = | fluid ounce. A small teacupful, or 1 gill = 4 fluid ounces. A pint of pure water = 1 pound. 4 tablespoonfuls, or a wine-glass = \ gill. A common -sized tumbler = 4 pint. 4 teaspoonfuls = 1 tablespoonful. 900. The following linear units are often used : 11 statute miles = 1 geographic or nautical mile. 60 geographic, or ) a a *i * «^. . , - = 1 degree on the equator. 69| statute mi., nearly, ) 360 degrees = 1 circumference of the earth. A knot, used for measuring distances at sea, is equivalent to a nautica DiUe. 4 inches = 1 hand, for measuring the height of horses. 9 inches = 1 span. 18 inches = 1 cubit. 6 feet = 1 fathom, for measuring depths at sea, 120 fathoms = 1 cable's length. 3.3 feet = 1 pace, for measuring approximate distances. 5 paces = 1 rod, " " " Annual Interest 373 Leap Years. 901. A Solar Day is the time between the departure of the sun from a given meridian and his return to it. 902. A Mean Solar Day is the average lengtli of all the solar days in the year, and is divided into 24 hours, the first 12 being designated by a. m., the last by p. m. Note. — a.m. is au abbreviation of ante meridies, before midday ; p. m., of post meridies, after midday. 903. A Solar Year is the time in which the earth, starting from one of the tropics or equinoctial points, revolves around the sun, and returns to the same point. It is thence called the tropical year, aud is equal to 365 da. 5 hr. 48 min. 49.7 sec. Notes. — 1. The excess of tlie solar above the common year is 6 hours or i of a day, nearly ; hence, in 4 years it amounts to about 1 day. To provide for this excess, 1 day is added to the month of February every 4th year, which is called Leap year, because it leaps over the limit, or runs on 1 day more than a common year. 2. Every year that is exactly divisible by 4, except centennial years, is a leap year ; the others are common years. Thus, 1876, '80, etc., were leap years'; 1879, '81, were common. Every centennial year exactly di- visible by 400 is a leap> year; the other centennial years are common. Thus, 1600 and 2000 are leap years ; 1700, 1800, and 1900 are common. Annual Interest. 904. Annual Interest is interest that is payable every year. 905. To Compute Annual Interest ^ when the Principal, Rate, and Time are given. 1. What is the amount due on a note of $500, at 6^, in 3 yr. with interest payable annually ? SOLUTION. Principal $500.00 Interest for 1 year is $30 ; for 3 years it is $30 x 3, or 90.00 Interest on 1st annual interest for 2 yr. is 3.60 2d " " " l''- is 1.80 The amount is $595.40 374 Appendix. Rule. — Find the interest on the pTincipal for the given time and rate ; also find the simple legal int. on eaeh annual int. for the time it has remained unpaid. The sum of the principal and its int., with the int. on the unpaid annual interests, will he the amount. Note. — When cotes are made payable "with interest annually," sim- ple interest can be collected, in most of the States, on the annual interest after it becomes due. This is according to the contract, and is an act of justice to the creditor, to comjjensate him for the damage he suffers by not receiving his money when due. 2. What is the amount of a note of $1500, payable in 4 yr. 3 mo. 10 da., with int. annually at 5%? 906. Connecticut Rule for Partial Payments. I. When the first payment is a year or more from the time the interest commenced : Find the amount of the principal to that time. If the payment equals or exceeds the interest due, subtract it from the amount thus found, and considering the re- mainder a new principal, proceed as before. II. When a payment is made before a year's interest has accrued : Find the amount of the principal for 1 year ; also, if the payment equals or exceeds the interest due, find, its amount from the time it was made to the end of the year ; then subtract this a?nount from the amount of the principal, and treat the remainder as a nciu principal. III. If the payment be less than the interest : Subtract the payment only from the amount of the principal thus found, and proceed as before. $650. New Havei^, April 12, 1878. 1. On demand, I promise to pay to the order of George Sel- den, six hundred fifty dollars, with interest, value received. Thomas Sawyer. Indorsements:— May 1, 1879, rec'd $116.20. Feb. 10, 1880, rec'd $61.50. Dec. 12, 1880, rec'd $12.10. June 20, 1881, rec'd $110. What was due Oct. 21, 1881 ? Partial Paijments. 375 SOLUTION. Principal, dated April 12, 1878 $650.00 Interest to first payment, May 1, 1879 (1 yr. 19 da.) 41.06 Amoimt, May 1, '79 691 .06 First payment, May 1, 79 116.20 Remainder, or New Principal, May 1, '79 574.86 Interest to May 1, '80, or 1 yr. (2d payment being short of 1 yr.). . 34.49 Amount, May 1, '80 609.35 Amount of second payment to May 1, '80 (2 mo. 20 da.) 62.32 Remainder, or New Principal, May 1, '80 547.03 Amount, May 1, '81 (1 yr.) 579.86 Third payment (being less than interest due) draws no interest. . . 12.10 Remainder, or New Principal, May 1, '81 567.76 Amount, Oct. 21, '81 (5 mo. 20 da.), . 583.85 Amount of last payment to settlement (4 mo. 1 da.) 112.22 Balance due Oct. 21, '81 $471.63 Note. — For additional exercises in the Connecticut Rule, the student is referred to Art. 554. 907. Vermont Rule for Partial Payments on Notes bearing Annual Interest. I. When payments are made on notes bearing interest, such payments shall be applied, " First, to liquidate the interest that has accrued at the time of such payments ; and secondly, to the extinguish- ment of the principal. " II. When notes are made ^^ with interest annually," Tlie annual interests which remain unpaid shall he subject to simple interest froin the time they become due to the time of settlement. III. If payments have been made in any year, reckoning from the time such annual interest began to accrue, the amount of such payments at the end of such year, with interest thereon from the time of payment, shall be applied : 376 Appendix. " First, to liquidate the simple interest that has accrued from the unpaid annual interests. " Secondly, To liquidate the annual interests that have hecome due. " Thirdly, To the extinguishment of the principal. ^1500. BuRLiKGTOJS", Fel. 1, 1877. 1. On demand, I promise to pay to tlie order of Jared Sparks, fifteen hundred dollars, with interest annually at 6^, value re- ceived. Augustus Wakren. Indorsements :— Aug. 1, 1877, received 1160; Nov. 1, 1880, $250. Kequired the amount due Feb. 1, 1881. SOLUTION. Principal $1500.00 Annual interest to Feb. 1, 78 (1 yr. at 6;^ 90.00 Amount 1590.00 First payment, Aug. 1, 77 $160.00 Interest on same to Feb. 1, 78 (6 mos.) 4.80 104.80 Remainder, or New Principal 1425.20 Annual interest on same from Feb. 1, 78, to Feb. 1, '81 (3 yr.). . . 256.53 Interest on first annual interest from Feb. 1, 79 (2 yr.).. $10.26 Interest on second annual int. from Feb. 1, '80 (1 yr.) 5.13 15.39 Amount 1G97.13 Second payment, Nov. 1, '80 $250.00 Interest on same to Feb. 1, '81 (3 mo.) 3.75 253.75 Balance due Feb. 1, '81 $1443.37 908. New Hampshire Rule for Partial Payments. I. When on notes drawing annual interest. Find the interest upon the ])rincipal from date of note to the end of the year next after the first payment, also upon each annual interest to the same date. IT. If the first payt. be larger than the sum of interests due, Find the int. on such payt. from the time it was made to end of the year, and deduct the sum of payt. a,nd int, from the amount of principal and interests. Partial Payments. 377 III. If less than the annual interests accruing, Deduct the payment juitJiout interest froTn the sinn of annual and simple interest, and upon the balance of such interest cast the simple interest to the time of the next payment. IV. If less than the simple interest due, Deduct it from the simple interest, and add the bal- ance witJiout interest to the other interests due when the next payment is made. Proceed thus to the end of the year after the last pay- ment, being careful to carry forward cdl interest unpaid at the end of each year.* 1. A agrees to pay B $2000 in 6 yr. from Jan. 1, 1870, with interest annually. On July 1, 1872, a payment of 1500 was made; and Oct. 1, 1873, $50. What was due Jan. 1, 1876? SOLUTION. Principal $2000.00 First year's interest ... $120.00 3 yr. simple int. thereon 14.40 134.40 Second year's interest 120.00 1 yr. simple int. thereon 7.20 127.20 Third year's interest 120.00 $2381.60 First payment, July 1, 1872 $500.00 Int. thereon from July 1, '72, to Jan. 1, '73 15.00 515.0 Balance of principal $1866.60 Interest on same for fou.rth year 111.99 + Second payt. (less than the int. accruing during the year) 50.00 Balance of fourth year's interest unpaid $61.99 + Annual interest on balance of principal for fifth year 111.99 + " sixth " 111.99 + Simple int. on unpaid bal. of fourth year's int. for 2 yr 7.43 + Simple interest on fifth year's interest for one year 6.71 -f- Balance of principal .... 1866.60 Amount due January 1, 1876 $2166.71 * Abstract of N. H, Court Rule, Report of Hon. C. A. Downs, State Superintendent. 378 Appendix. 909. The Twelve Per Cent Method of Computing Interest. 1. Find the int. of 1275.20, for 3 yr. 4 mo. 10 cla., at 12^. SOLUTION. Int. of $275.20, 1 yr. at 1% = $275.20 x .01 = $2,752. 1 yr. at 12% = $2,752 x 12 = $33,024. 1 mo. at 12% - 12 mo. at 1% = $2,752. (Art. 578.) 3 yr. at 12% = $33.024x3 $99,072 4 mo. (1 of 1 yr.) = $33.024-=-3 11.008 10 da. (i of mo.) = 2.752h-3 .917 Hence, the Ans. $110,997 EuLE. — For 1 year: Find the interest on the principal at 1%, by moving the decimal point two places to the left, and multiplying the result hy 12. For 2 or more years : Multiply the interest for 1 year hy the nujnber of years. For months and days : Proceed as in Art. 537. AVERAQE OF MIXTURES. 910. To find the Average Value of a Mixture, when the Quan- tity and Price of each Article are given. 1. A man mixed 45 bii. oats worth 25 cts. a bushel with 38 bu. corn at 50 cts., and 56 bu. rye at 60 cts. ; what was the mixture worth a bushel ? OPERATION. Solution. — The whole number of bush- |0. 25 X 45 = 111. 25 els mixed is 45 + 38 + 56 = 139. The whole ^0 v '^K 1 Q 00 cost of the mixture is $11.25 + $19.00 ^-^^X^O — iJ.UU + $33.60 = $63.85. 0. 60 X 56 = 33. 60 Now, $63.85-139 = $0.46 nearly, the I39 ) $63.85 price of 1 bushel of the mixture. Hence, the Ans. $0.46. Rule. — Divide the value of the whole mixture hy the sum of the articles mixed. Notes. — 1. If an article costs nothinpf, as water, its value is ; but the quantity used must be added to the other articles. 2. The process of finding the average value of mixtures is often called Allio-ation. Average of Mixtures. 379 2. A grocer had three kinds of sugar, worth 6, 8, and 12 cents per pound; he mixed 112 lb. of the first, 150 lb. of the second, and 175 of the third together. What was the mixture worth per pound ? 5s. JL 4 8s. 1 lis. i 12s. \ 911. To find the Propovtiotial Parts of a Mixture, the Mean Price and the Price of each Article being given. 3. A grocer desired to mix 4 kinds of tea, worth 5s., 8s., lis., and 12s. a pound, so that the mixture should be worth 9s. a pound ; in what proportion must they be taken ? Analysis. — First find how much it takes operation. of each article to gain or lose a w;< /if of the QqJ 123 mean price. Since the mean price is 9s. a — pound, 1 lb. at 5s. gains 4s, ; hence, to gain Is. takes \ lb,, which we place in Col. 1. Again, 1 lb. at 12s, loses 3s. ; hence, to lose 9s. Is. takes | lb., which we place also in Col. 1, opposite the price compared. In like man- ner, 1 lb. at 8s. is required to gain Is., while 1 lb. at lis, loses 2s, ; hence, to lose Is, takes | lb. We place these results in Col. 2, opposite their prices. Reducing the fractions in Col. 1 and 2 to a common denominator separately, the numerators are the proportional parts required. Hence, the Rule. — I. Write the prices of the articles in a colinnn in their order, with the mean price on the left. II. Tahe them in pairs, one less and the other greater than the mean price, find how inuch is required of each article to gaix or lose a unit of the mean price, and set the results in Col. 1, opposite to its price. Com- pare the other couplet in like manner, setting the results in Col. 2. III. Finally, reduce the numbers in each column sep- arately to a common denominator ; the numerators will he the proportional parts required. Notes, — 1. If there are three articles, compare the price of the one which is greater or less than the mean price with each of the others, and take the sum of the two numbers opposite this price. 380 Appendix. 2. The reason for considering the articles in pairs, oiie above, and the other below the mean price, is that the loss on one may be counterbalanced by the gain on another. 3. When the given prices are integers, the same results are readily found by taking the diflference between the price of each article and the mean price, and placing it opposite the price with which it is compared, as 4. How much coffee at 9, 11, and 14 cents a pound, will form a mixture worth 12 cents a pound ? 5. How much g-inger at 15, 18, 21, and 22 cents a pound, will form a mixture worth 19 cents a pound? 912. To find the Other Quantities when the Mean Price of the Mixture and the Quantity of one of the Articles are given. 6. How many pounds of starch worth 11 and 15 cents a pound, must be mixed with 16 lb. at 10 cents, so that the mix- ture may be worth 13 cents a pound. Analysis. — If neither article were limited, opebation. the proportional parts would be 2, 2, and 5. (Art. 911, N. 1.) But the quantity at 10 cts. is limited to 16 lb., and its proportional part is 2 lb. Now, 16-i-2 — 8. Therefore, multiplying each of the proportional parts by 8 gives 16 ; 16 and 40 lb. the mixture required. Hence, the KuLE. — Fijid the proportional parts as if the quantity of neither article were limited. (Art. 911.) Divide the limited quantity hy its proportional part, and' multiply each part found hy this quotient; the product luill he the quantity required. Note. — When the quantities of two or more articles are given, find the amrage value of them, and considering their sum as one quantity, proceed as above. 7. How much corn at 45, 56, and 65 cents per bushel, must be mixed with 25 bu. of oats at 40 cents, so that the mixture may be worth 50 cants a bushel ? Col. 1. 2. 10 2 2 13 11 2 2 15 3+2 5 N S W E R S . Article 50. 19. 1418. Arts. 70, 71. 31. 2280. 12. 8879. 13. 7889. 14. 7979. 15. 9798. 16. .S6178. 17. 966 mi. 18. 987 Acres. 20. 1836. 21. ir83. 22. 2604. 23. 7512. 24. 21241. 25. 10562. 26. 2742. 2. 10292. 3. 10083. 4. 27886. 5. 34339. 8. 216.9. 9. 182.19. 10. $242.19. 32. 5583. 33. 7271. 34. 84841. 35. 5482. 36. 14935. 37. 985. 38. 7065. 27. 2355. 28. $627. 11. $3684.939. 39. $301.12. 40. $5072.35. Arts. 55, 5f), 29. 630 lb. 2. 12054 yd. 3. 14792 rods. 30. ij^3789. 31. $3582. 32. $1323. 33. $279075. 34. .■>595522. Art. 72. 1. 113 yd Art. 74. 1. 439.25. 4. 19747 wk. 2. $221. 2. $1291. 5. 28143 lb. 3. 189 gents. 3. 83412.7. 7. 415.034. 35. $2115306. 4. 1003 bu. 4. $72320. 8. 114.634. 36. $1606895. 5. 374 bu. 5. $985.25. 10. $8S.967. 37. 7448208453. 6. $1989. 6. 146 trees. 11. $104,721. 38. $.^068, farm. 7. $479. 7. $1090. 12. $100.84. $6136, all. 8. $1659. 8. 12520 bu. 13. $93,833. 39. .^16646. 9. $3023. 9. $1910.89. 40. $37650. 10. $1763. 10. $5491. Art. 60. 41. 1727. 11. $3747. 11. $5250. 42. $8475. 12. 1825. 12. 627067. 1. $6821. 43. 14,507,407. 13. 2600. 13. $21422. 2. $2324. 44. 7,597,197. 14. 3085. 14. 22225. 3. $4900. 45. 17,364,111. 15. 1306. 15. 16,014,400. 4. $1444. 46. 8,919,371. 16. 4098. 16. 184.815,000, 5. 503 trees. 47. 1,767,697. 17. 1108. 000. 6. 73 yr. 48. 50,155,783. 18. 4531. 17. 1486.75. 7. $1648. 19. 14520. 18. 5 times. 8. $34950. 20. 24622. 19. 61483.95. 9. $33700. Art. 69. 21. 125028. 20. 26973. 10. 565. 22. 64303. 21. 34059.5. 11. 742. 2. 3232. 23. 224066. 22. 1912, B's. 12. 1530. 3. 3244. 24. 103S75. 4482, C's. 13. 1779. 4. 3424. 25. 420486. 23. 5986. 14. 1597. 5. 3525. 26. $16014. 24. 33086330. 15. 1757. 6. 3213. 27. $1315. 16. 2379. 7. 501 Acres. 28. $5385. Arts. S4, 85. 17. 2619. 8. $1134. 29. 708. 18. 1020. 9. 412292. 30. 942. 2. 17501b. 382 Answers. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 24. 26. 3. 4410 sheep. 4. 2022 1b. 5. 3000 ft. 6. 4345 yd. 7. 12768 bu. 8. 20712 in. 1924.5.- 402.12. 434.98. 60.221. 787.14. $26116.02. $3381.19. -150981.28. $74241.84. $7264,854. $138.24. $1455.78. $7.68. $7,857. $60. .^3000. 27. $42120. 28. $6000. 29. $45.36. 30. $21150. 1. 2. 3. 4. 5. 6. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Art. 87. 1445. $23646. $173.34. 996.84 lb. 234588 yd. Art. 88. 16425d. 91350 lb. $8991. $16884. 68520 ft. 1564 sheep. $6000. $17920. 17945 bu. 48000 A. 20(i952. 98982. 204336. 368109. $1849.65. $3201.44. 438480. $1443.099. 31,968,868. 30. 48,053,208. 21. 34,628,175. 22. 65,404,110. 23. 639,756. 24. 1560.975. 25. 3,071,926. 26. 3,007,368. 27. 24,631,008. 28. 35,497,655. 29. 42,546,240. 30. 62,355,319. 31. 849,126,321. 32. 1.219,641,537. 33. 1,988,907,892. 34. 2,758,104,145. 35. $576. 36. $3744. 37. 6930 mo. 38. 58050 ds. 39. $632. 40. $149. 41. 1549.25. 42. $488.25. 43. $42255. 44. $1895. 45. 168750 1b. 46. $57649, cost. $3367, dif. 47. $534842. 48. $618. 49. $22. 50. $56.50. 51. 1624 trees. 52. 1584 pupils. 53. 62234. 54. 20,345,400. 55. 1776 bu. 56. 368,640. 57. 283,410. 58. 270,592. 59. 111,168. 60. 92,538. 61. 493,480. Art. 89. 2. $2295. 3. 4. $684. 5. 8610 miles. 6. $4950. 7. $312. 8. 25760 bu. 9. $16128. 10. $91080. Art. 91. 6. 468001b. 7. 809600 pp. 8. 476,000. 9. 534,860,000. 10. 1,204,670,800,000. 11. 26,900,785,000,000. 12. 890,634,570,000,000. 13. 946,030,506,800,000. 14. 3,840,000. 15. 10,940,000. 16. 2,075,994,000. 17. 390,677,500,000. 18. 372,000. 19. 11,840,000. 20. 373,520,000. 21. 3,603,200,000. 22. 55,447,000,000. 23. 37,800,000,000. 24. 25,800,000,000. 25. 4,059,360,000. 26. 14,760,000,000. 27. 6,204,000,000. 28. 1,672,650,000,000. 29. 1,075,635,900,000. 30. 450,230,874,000. 31. 6,980,161,370,000. 32. 834,271,780,000. 33. 779,984,000,000. Art. 93, 17. 103 cts. 18. 3600 soldiers. 19. $18200. 20. 452 miles. 21. $504. 22. $1492. 23. $4632. 24. $234. 25. 8288d. s. q. 26. 1460 mi. 27. 2,015,028. 28. 8,498,120. 29. 404,444,040. 30. 6,342,737,821. 31. 351,039,462,230. Arts. 107, 110. 3. 1275. 4. 1173. 5. 2468. 6. 1317. 7. 11449. Answers, 383 8. 11155. 10. 378. 11. 246. 12. 427. 13. 1234. 14. 546. 15. 1234. 16. 335. 17. 349. 19. 913|. 20. 661|. 21. 820|. 22. 4639 1. 23. 15290|. 24. 20588|. 25. 8731i. 26. 9124f. 27. 120421f. 28. 71410|. 29. 96043i. 30. 87105|. 31. 82401 . 32. 8369f. 33. 56234. 34. 1533. 37. 640.87. 38. 80.666. 39. 24631 yd. 40. $5.42. Art, 112, 2. 2862f|. 3. 12431^^. 4. 13967Jy. 5. 29493tV 6. IO9O3V 7. 1231|f. 8. 192.3011- 9. 24218|f. 10. $1.20||. 11. $1.27^. 12. $1.3541^. 13. $2.007i|. Art. 115, 1. 61 shares. 2. 31||yr. 3. 48|ilihd. 4. 5. 6. 7. 21. 22. 23. 24. 25. 26. 27. 28. 29. 43ff mi. 75 dresses. 51|| m. $50. 73.6 A. 45 If casks. 27 days. $31.14i§. $3.15i|. 37.4^V 9. 10. 11. 12. 13. 14. 15. 199.2 16. 17. 18. 19. $1.36|i. 20. 1.463V 340ff. 175ff. 65^^ $1.53i|. 84/^. 149f|. 837ff. 1607f|. 962||. 1006ff. 901b. 14911 A. 126 boxes. 30. 288 eggs. 32. 245iif. 33. 34. 35. 36. 37. 38. 39. 133HI. 9633«o^e- 720|ff. 3011fa|. 3938IM. 6671fti. 7318ff|. 40. 121.93f|-|. 41. 3.1885fi|- 42. 673.888|ff. 43. 456.607iW7- 44. 2680.52.^111. 45. 2.3631tt|f|. 46. 5.2558f4|||. 47. 5.109|fif|. Arts, 117, 118, 2. 3452 and 31 rem. 3. 672 and 487 rem. 4. 642 and 3544 rem. 5. 73 and 64159 rem. 8. 340 bar. 9. $456.50. 10. 80 bales. 11. $40. 12. 292^^ lb. 13. 4160 men. Art. 121. 1. 14. 2 12 4! 100 s. 150 g. 5. 118 B., 155 A. 6. 69 cts. 7. 82 vr. 8. $283. 9. 392 mi. 10. 1 mi. 11. $5243, B. $17176, C. $23684, all. 12. $621. 13. 730 sch. 14. $20 per A. $1868, gain. 15. $66.52|i|. 16. Cows; $6094. 17. 228 A., B.'s. 114 A., C.'s. 1710 A., all. 18. 18, sm. No. 882, gr. No. 19. 406 oxen. 20. 38,818,897, dif, 21. $3780. 22. 83. 23. 105374. 24. 14. 25. 1213. 26. 43. 27. 8117. 28. 723.8. 29. 49312 mem. 30. 3082 men. 31. $718. 32. 196 sofas. Art, 125o 6. 112. 7. 120. 8. 13f. 384 Answers. 9. 5h 10. 13. 11. 22i. 12. mh 18. 3. 14. Wt- 15. 15 tons. 17. Ifi bags. 18. 33f bar. 19. 126 bar. Art. 143. 2. 5x5x3x3. 3. 47 X 2 X 2 X 2. 4. 48 X 2 X 2 X 2. 5. 3x3x2x2x2x2 x2x2. 6. 7x3x2x2x2x2 x2. 7. 199 X 2 X 2. 8. 3x8x3x2x2x2 x2x2. 9. 7x5x3x8x3. 10. 19 x5x 3x3x2x2 11. 37x5x5x5x2x2 12. 67 X 43 X 2 X 2 X 2 x2. 18. 6029 x2x2x2x2. 14. 1297 X 2 X 2 X 2. 15. 5x5x2x2x2x2 x2x 2x2x2x2 x2. 16. 508x2x2x2x2 x2x2x2 17. 193 X 2 X 2 X 2 X 2 x2x2x2x2x2 ArU 14:4, 19. 2 and 2. 20. 2. 21. 2. 22. 2. 23. 2, 8, and 7. 24. 2, 2, 2, and 3. 25 2, 2, and 2. 26 5 and 5. 27 2 and 2. Art. 149. 2 21. 3 13. 4 . 19. 5. 15. 7 22B ' • 2 8 8* 6. 3. 8.. Iff. 7. 4. 8. 12. 9. iff. 9. 0. 10. 5. Art. 185^ Art. 150. 2. fi. 3 ifi 1. 6. A 16 *• 25- 2. 15. 5. \. 3. 12. 6. tV 4. 1. 7. iM. 5. 5. 6. 4. 8. ii. 7. 4. Q 22 8. 4. 10. f. 9. 8. 10. 4bu. 11. 4 A. 11 i|t 12. f^. 12. 21. 13. J^. 13. 60ft. wide; 10 1., 14. |. 2 1., and 15 1. 15. TT- 14. $252, price ; 5 li.. 1 1 16 1 9 h., and 11 b. J-U. 5. 17 431 Art. 157. 2. 240. ^?^ i«^ 8. 12600. 1. 37i 4. 504. 2 44-*- 5. 1134. /O. '±'±13. 6. 144. 3. 30. 7. 130645. 4 28i. 8. 533610. 5. 31i. 9. 156240. 6. 53V 10. 144. 7. 14l|. 11. 2520. 8. 28#8. 12. 262080. 13. 1921506000. 9. 22|ft. 14. 360. 10. 46tV 15. 1584 ft. 11. 21t%V 16. $60. 12 lOA^TT. 17. 840 gal 13. 40i|f. 18. 12 hr. 19. 24 hr. 14. 993^^. 20. 120 br. 15. 13^V 16. 110^. Art. 182. 17. 131iA|. 9 40 18. l|i q 65 19. SIMM- 4. m- 20. 12TV¥r- 5. m- 21. 60if rd. 6. m. 22. $153J3VV c \ ■x/ Answers. 385 Art, ISO, 9. 67 2. 3. 69 10. 9^. 214 11. 41. 4. 968 12. A5 8 9 ^sjo- 5. 18. 1119 l^T^- 6. 3 231 5^ * 4980 313 •\¥- 2543 14. 41tV. 7. 15. ^^h- 8. 16. ^\U- 9. 17. m.]io- 10. 18. 111 62 9 11. 19. 1389 12. 5 * 57 6 ill45_ 20. 651^0. 13. 21. 199if i lb. 15 22. 289|ii ra. Art. 195, 23. 553/,- yd. 2. 3. 4. 5. 88 30 25. Uf. 3 5 2 7 26. 16|. ¥5» T5- 3 5 6 27. 33i§. ^0' ^0- 4 8 6 3 28. 16H. ITT' TT4- 29 mi 6. 15 7500 113400 2S3oO(J» 2 8 3 5 0' 19845 30. 37H. ■J8¥50(T- 31. 222 7. 120528 208656 --"^s* 3^657^' 3^6o92^' 96768 32. oqi 1 3¥65¥^- 33. 5I5V 34. 16511 yd. 297f J m. Art, 196, 35. 9. 2 4 2 5 16 60' ^0' 60- 10. 264 385 234 ^T6' ^T^' 1?T^- Art, 204 11. 45 32 15 T(J^' TTJ^' TTTS"- 8. 4079 ssso- 12. 735 600 672 490 ¥TT> ?'T0' ^TTJ' 8TT5' 4. 1 6 T^I-- 13. 189 180 168 8 4 2TI7' ^T0» 2T(y> "JTo- 5. 4 9 1 T650- 02 5f. 14. 672 1240 94 5 T40 0' ITOI^' TIOO"- 6. 15. 210 395 1092 5^5» o^Z' F55 • 7. 16. 3 5 9 15 12 TT)F' TF5> TOI^- 8. 3^^- 17. 12 175 680 ■^80' 2B^0' 2?(J- 9. 8 77* 18. 154 2590 180 2T5' 245"' 2T5- 10. 1 •> • 19. 972 189 100 132' T32' 132* 11. 3H- 20. 117 85 63 11^9' T89' 189- 12. 13. If- Art, 201, 14. 391 3. 1JL9_ J^ 1 6 5 • 15. llfi- 4. 0653 ~ 7 2 8 • 16. n- 5. Q2 4 3 ^2 8 0- 17. mh 6. 1t¥77. 18. 317|i rd. 7. 51. 19. 357:j?T7 T. 8. 21. 20. 303 1 lb. 21. 131611 bu. 22. |15|. 23. mil A. 24. 38U yd. Art, 200, 8. If. 4. i^V 5. 4. 6. 5|. 7. 7. 8. 28|. 9. 4^r. 10. 5. 11. 5. 13. 702. 14. 1988. 15. 4941. 16. 3537. 17. 769^. 18. 27612. Alt, 208. 2. 6|. 3. A 4. 45. 5. 174. 6. 60." 7. 255. 8. 36. 9. 432. 10. 1191. 12. 1178. 13. 8450. 14. 1280. 15. 341111 16. 4496. 17. 8118. 18. 1042811. 19. 5611|i 20. 5086^%. 21. 43452. 22. 74269 i|. 23. 91806. Art, 211, 5. 1 3- 6. 3 7- 7. 21 8. 386 Answers. 5(?- _3_ 35- rc 5 9. 10. 11. 12. 11 If. 13. 2UV 14. 7ii. 15. $1031. 1. $12||. 2. 136 cts. 3. 112i cts. 4. 23r)~cts. 5. mi 6. 33 L cts. 7. 292.1 cts. 8. 344] cts. 9. $57if. 10. 630 cts. 11. 1161 12. $4|.' 14. $5i^i. 15. $5if. 16. 12371 cts. 17. 4061 "cts. 18. 300;^ cts. 19. 806i cts. 20. $7. 21. $10if. 22. 273f. 23. .f3«\. 24. $41|t. 25. 621-1% cts. 26. $831^3. 27. 391f cts. 2S. $1331 29. $65}f. 80. $615/^. 31. 743| m. 32. 2310. 33. 32|. 34. 197f. # .> 6 t> TUIT975T' 147 ¥67- 38. 39. 40. 41. 42. 43. 44. 867 o 7- 80. 1561^11 109A«s. J>-22_ 1325* 6-" 60- Art, 215. 2. H. t5. 3 1. 4 _7JL ^- 7 5 6' O. 7j. 6 109 "• 3T50' 7 __L7_1 '• 1850* Q 29 '^- 7 7 5- 10 -8 ' 9 ^^- 2394* ^** 5625* 1R 4 15. l\ T.; 1 part. 16. |if. 2. 294. 3. 432. 4. 576. 5. 1350. 6. 48. 7. 171. 8. 1344. 9. 804. 10. 11045. 11. 461yd. 12. 144 d. 14. 9i 15. 18. 16. 411f. 17. 81614. 18. 12^.^ 19. 22f. 20. 480 sh. 21. 56{|yd. 22. 34 1 cloaks. 23. 52 c. 3 rem. 24. 30x\ br. Art, 220, 4 4^ 5. 20. 6. 7. 1 _? J5 3_ ■^1096- 4 Art, 221. 1. 5i^lb. 2. 10^ c. 3. 8y\v bar 4. %^jh- 5. 7 cts. 6. Q 44 o 7. p^m- 8. 16. q (feO fi 3 3 10. llf^ T. 11. 87t^V sacks. 12. 157y%V bales. 13. i^%,. " 15 _38_ ^^' 1T05' 16. f. 1 7 -8J 18 7 '•-!- 19. 171. 30. tI r. 22 5^^ "•*• •-'3 4' 23. y|^. 24. ff|. 25. M. 26. 3V7. 27 1_7 7 28 19 "9 '^*-'- 1 9 8 4 • 1058 ¥"(>T5- 3. If. 4. tV 5. 1^. 6. 2^0- 7 1 J. Q 41 9. U. 10. 1^. 11. 163 20G* 12. 313|. 13. 805 8 3 7' 14. 33^^' 15. 2 58 16. 1V0-. 17. 167 iso- 18. ItV 19. 451 6T8' 20. 'iOfil4 3 Art. 226, 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 1 ^^• _7 496* 11 T¥3^- 19 TT5' 1 T40' 7 2600* J)_0 6 1 5 7 9" 5 3F- _2JL 477- 2 3- 10 IT- 113 280* 1 7 A ^3 ^• 30 3 7' 2^7 38* 65 m- 3 8* 3 515 53 7 3- __3_ 20- Art, 228, 2. 64. 3. 70. 4. 135. 5. 160. 6. 165. 7. 2891. 8. 844f. 9. 1249 vV 10. $14560. 11. $7080. 12. 315 s. 13. $455. Aiiswers. 387 Art, 230. -( 117 ^- ¥¥3- 2 729 • T2TT- 3. 2x3x3x3x3x8. 2x2x2x3x3x2 x3x3. 3x2x3x3x3x3 x8. 4. 200, 1. c. d. 5. 6^^ G. 4i|. 7. 1011 J. 8. iileft. 'f- so- lo. 411|m., both go. 401 ra., (jif. 11. $48}. 12. lU. 13. 13|. 14. 17tV. 15. 32. 16. 22|. 17. 39tV 18. 134^5^. 19. 249^^. 20. $20|. 21. $130|f. 23. $29056}. 23. 112||. O/l 3 5 '*"*• 333- 25. 53V 26. $12521^. 27. $18521^1. 28. S3212}f§. 29. $2681i. 30. 15| lots. 31. $32yVlost. 32. Hi books. 33. $82|. 34. $75^. 35. 305 eggs. 36. $12250. 37. $36752. 38. $1000. 39. 64 ft. 40. 75 ft. 41. 102 marbles. 42. $112i 43. 108 ds. C. 44. 15360 men. 45. 100. 46. 100. 47. 38. 48. 13|. ^O. 2 9- 50. 3H. 51. $3311. 53. 131 oz. 53. $3808. 54. $4750. Art, 245, 3. .3000,3.0004,7.0080. 3. .300, .060, .008. 4. .0300, .1350, .7000, .3363. 5. .3600, .3750, .0336, .3060. 6. .0450, .6100, .0035, .1080. Art, 24:7. 3 -'- 4. 14 ^* 2 5' 5. I 6. f. 7. Jo. Q 5 10 -5- nl 12. 13. 1 _3 5 07' 3 1_ 125- 4 100 7_ TOOOITOO' 14 15 16 17 -33 ^*- 4 OTIT- IS i^ 5000000' Art, 249, 3. .35. 3. .75. 4. .635. 5. .8. 6. .135. 7. .375. 8. .75. 9. .875. 10. .45. 11. .35. 13. .4. 13. .65. 14. .43. 15. .033. 16. .09375. 17. .33UV. 18. 75.6." 19. 89.75. 30. 39.635. 31. 65.135. 33. 8.075. 33. .83. 34. 27.8125. 25. 93.1875. 26. .66666|. 27. .33333.1. 28. .44444|. 29. .16379:t%. 30. .13666|. Art, 254, 2. $182,731. 3. 238.4313. 4. 1073.16387. 5. 386.5134. 6. 133.3387. 7. $411,135. 8. $738. Art, 25(i, 3. $3,635. 4. $1743.35. 5. $5.75. 6. $1305.90. 7. $170,675. 8. $533.65. 9. •806.41m. 116.09 m. Arts. 259, 2 GO. 3. .135918. 4. 16.141753. 5. 37.478884. 6. 441.3336144. 7. 813.475. 8. 5151. 9. 164.0635. 10. 1503.4375. 11. 47.0331027. 388 12. .1719375. 13. 3G75. 14. 17.91)53125. 15. .00187441562. 16. 2374.2125. 17. 14G47.5 cts. 18. 53200 cts. 19. $3803.625. 21. 46384.2. 2.>. 6423.02. 23. 25460. 24. 3004. Art. 263, 4. .9128 + . 5. .9408 + . 6. .008911 + . 7. .09. 8. 10. 9. .000102 + . 10. 3.4542 + . 11. 846.37105 + . 12. .116|. 13. .7. 14. 9.991.2844 + . 15. 8930. 5972 + . 16. 761409.375. 17. 73.79512 + . 18. 124.33^. 19. 48 coats. 20. 8 yd. 21. 15.15+ hr. 22. 56 bar. 23. 4.637+ d. Art. 264, 25. .2425. 26. .45631. 27. .032463. 28. .00008534. 29. .00642564. 30. .5634527. Art. 279, 1. $30.68. 2. $12.13. 3. $45,805. 4. $196.51. 5. $362.50, cost. $71.50, dif. 0. $16,865, dif. 7. $93,625. Anstvers. 8. $471.25. 9. $108.75, gain. 10. $1080. 11. $1601. 12. $1180. 13. $8429, gain. 14. $7540. 15. 5 liats. 16. 35. 65 If w. 17. $2,574 + . 18. 465.551 bii. 19. $10,282. 20. $136,986 + . 21. 9 yd. 22. 252 bar. 23. $.125 + . . 24. 600 T. Art, 284, 2. $438. $300. 4. $137.60. 5. $112.25. 6. $250. 8. $1061.33i 9. $364. 10. $300. Art, 285, 12. 1700 yd. 13. 450 lb. 14. 1950 bu. 15. 962 cans. 16. 128 yd. 17. 132 hoes. Art, 286, 20. $355.25. 21. $()91.08. 22. $315.75 23. $4733 82. 24. $111.65. 25. .1S44.10. 26. $100.49. 27. $1866.90. 28. $191.93. Art. 287, 31. $68.31. 32. $19,642. 33. $652 03125. Art, 299, 1. $17.77. o $158.256|. 3! $360,705. 4. $416.02. 5. $2659.275. 6. $1067.65. 7. $429.8825. 8. $372,755. 9. $1058.05. Art, 331. 2. 105.735 m. 4. 64 liters. 6. 3.990 Km. 7. $240.50. 8. $54. 10. 26| Km. 11. 174.52592 Km 12. $4.50. 13. 54 cts. 14. $819. 15. $1.15. 16. $23,125. 17. $105.60. 18. $2.40. 19. 21 kilos. Art. 332, 21. 3 sq. m. 22. 37.8 sq. m. 23. 85.2 sq. m. 24. 1044 centars. Art. 333. 26. 853.632 cu. m. 27. 768 cu. m. 28. $87. 29. $12,375. 30. $59.8598. Art. 396, 26612 ft. 3. 825264 gr. 4. 434035 lb. 5. 3393088 oz. 6. 121528.5 ft. 7. 140304 gr. 8. 1539776 oz. 9. 10524 gi. 10. 42960 pt. 11. 2936 pt. 12. 161856 sq. in. 13. 1395553.5 sq. ft. 14. 3335 cu. ft. 15. 364620 min. 16. 184584 Ixr. 17. 522028 sec. 18. 16362 d. 19. 197150 far. 20. 120475020 sec. 21. 119.84. 22. $10.92, gain. 23. $2597.82. Art. 400, 3. 33 bbl. 30 g. 3 qt. 4. 92hlid.ll»bl.27gal. 2qt. 5. 27231b.7oz. I4pu^. 6. 74 lb. 11 pwt. 4 gr. 7. 22 cwt. 26 lb. 8 oz. 8. 263 T. 2 cwt. 95 lb. 4 oz. 9. 150 rd. 2 ft. 4 in. 10. 9 m. 880 ft. 11. 313sq.rd.49.75sq.ft. 12. 437 A. 102 sq. rJ. 13. 129 C. 56 cu. ft. 14. 1350 bu. 28 qt. 15. 452 bu. 14 qt. 16. 649 com. vr. 20 da. 17. 122385 wk. 5 d. 18. £4878, 7s. 8d. 19. $6.90. 20. $750.75. 21. $1.89. Art, 402. 3. 9 hr. 20 min. 4. 5 d. 14 lir. 24 min, 5. 3 fur. 22 rd. 3 ft. Sin. 6. 1 pk. 5 qt. 12 pt. 7. 274 A. 45 sq. rd. 21-i| sq. yd. 8. I of a gill. 9- iH, pt. 10. 9s. 3d. 11. 3 qt. .048 pt. 12. 15 hr. 34.56 sec. 13. 85 lb. 9.6 oz. 14. 3 pk. .5248 pt. 15. 5 yd. 1 ft. 2.04 in. Ansivei's. Art. 403, 19. If bu. 20. f i gal. 21. ^¥o% wk. 22. ^oV(j lb. 23. 47 A 24. .828125 bu. 25. .75625 d. 26. .88^ yd. 27. .697911 lb. 38. .05 gal. 29. .000125 T. 30. 2.3 yr. Art, 404, 31. 12 ^3* qo 14 O/J. 12T- 83. 195 ¥8".r- 34. .12648 + . 35. .405 + . Art. 405, 3. 39.1482 mi. 4. 19.8131 gal. 5. 15.89 bu. 6. 4.2324 oz. 7. 303.68365 lb. 9. 148.87775 A. 10. 4287.92 cu. ft. Art, 40(L 12. 58.293+ m. 18. 6286.959+ Kg. 14. 236.58559+ li. 15. 72.497 + kl. 16. 143.228+ Kg. 17. 6000 sq. m. 18. 16.378+ hektars. 19. 410.748+ cu. m. 20. 27985.715+ cu. m Art, 407, 2. £29, 7s. Id. 3. 23 gal. 2 qt. 1 gi. 4. 16 wk. 6 da. 4 hr. 48 min. 6. 184 bu. 3 I'k. 7 qt. 7. 249 A. 157 sq. rd. 389 8. 6 hhd. 53 gal. 3 qt. 9. 200yr. llmo.Owk. 4 da. 10. 101 mi. 160 rd. 11. 109 sq. yd. 8 ft. 142 in. 12. 73 C. 69 ft. 177 in. 13. 177mi. 242rd. 4yd. 2 ft. 4 in. 15. lbu.2pk. lqt.|Jpt. 16. 9 hr. 37 min. 25? J 17. 8 oz. 3 pwt. 22.4 gr. 18. 5 d. 16 hr. 6 min. 51 f sec. 19. 1 gal. 20. 18s. 5d. 2! far. Art, 40 S, 2. 58 lihd. 6 gal. 2 (p. 3. 6 oz. 18 pwt. 2 gr. 4. 113^ yd. 5. 9 mi. fur. 18 rd. 7 ft. 10 in. 6. 54 A. 149 rd. 38 ft. 7. 128cu.ft.l652cu.in. 8. 48 C. 106 ft. 58 in. 10. 3 yr. 5 mo. 21 d. 11. 6 yr. 4 mo. 26 d. 12. 12 yr. 2 mo. 28 d. 13. 1 yr. 7 mo. 21 d. 14. 5 yr. 4 mo. 15 d. 16. 191 d. 17. 150 d. 18. 74 d. 19. 150 d. 20. 222 d. 21. 10= 54' 13'^ 22. 98'^ 0' 57". 23. 19^ r 54". 24. 79^ 20' 15". Art. 410, 2. 68 lb. 1 oz. 4 pwt. 3. 195 gal. 8 qt. pt. Igi- 4. 286mi. 150rd. 3 yd. 1 ft. 6 in. 5. 8 1b. 10 oz. 7 pwt. 6. 1538 gal. 1 qt. 7. 532mi. Ofur. lOrd. 390 Answei's, Art, 411, 2. 5 oz. 8 pwt. 8 ^Y. 3. 10 lb. lU oz. 4. 4 m. 177 r. 7 ft. 6 in. 5. 19 bu. pk. 2 qt. Alt. 415. 2. 2° 15' 30''. 3. 10° 53' 1.95". 4. 11° 59'. 5. 9° 17' 6". 6. 12° 20'. Art, 416. 2. 29 mill. 35.2 1 sec. 3. 41 min. 18/- sec. 4. 5 h. 45 m. 16.6 s. 5. 1 lir. 6 min. 30 sec. 6. 40 min. 28 sec. 7. 12 min. 13 sec. 8. Ihr.l7min. 20isec. 3 hr. 21 min. | sec. 9. N. Y., 9 A.M. Rich., Va., 8 h. 46 min. 29 sec, A.M. San Fr., 5 h. 46 min. 25 sec, A.M. Art, 424. 3. 36|yd. 8. 48f yd. 4. 18 A. 44 sq. rd. 5. 435.6 ft. 6. $10890. 7. ^1^1980. 8. 20358 sq. meters. 9. 3.25 meters. 10. $2. 11. 640 A. 12. 26 A. 65 sq. rd. 13. 50 rd. 14. 11.6973 Ha. 15. 5 A. 159 r. 260} sq.ft. 16. 2 sq. rods. 17. 20 A. 1120 sq. ft. 18. $57500 gain. 19. $76. 20. $47,782. 21. $9498 50. 22. 1330.56 tiles. 23. $11760. 24. 120 rd. wide. $2560. cost. 25. 41^ planks. Art. 429, 2. 4200 cu. ft. 3. 23328 cu. ft. 4. 377 1 loads. 5. $125.92 If. 6. $2823.331. 7. 198 cu. yd. 13 cu. ft. 648 cu. in. 8. 137 cu. yd. 24 cu. ft. 1512 cu. in. 9. 30 ft. Art, 431. 1. 6f*6 cords. 2. 14^6 cords. 3. $13.8125. 4. 48 cu. ft. ; 80 cu. ft. 5. 1536 ; 3072 cu. ft. 6. 128 ft. high. 7. 6 cord ft. 8. $2.95y\. 9. 5ift. Arts. 432, 436, 1. 63 perch 9 cu. ft. 2. $300.96. 3. 76545 bricks. 4. 146966.4 bricks. 5. $1138.9896. 3. 19 1 board ft. 4. 17^ board ft. 5. 110 board ft. 6. $66. 7. 20 board ft. 8. 120 board ft. 9. 42 1 b. ft. ; 3f cu. ft. 10. $5 50. 11. 12 ft. 12. 260 cu. ft. 13. 2 0. 4 ft. 14. 500 ft. 15. $11.25. 16. 65 1 ft. 17. $3>0125. 18. 273;^ cu. ft. $297.97,4, cost. 19. $7,875. 20. 3038 cu. ft. Arts, 438, 439. 3. 1795ff. 4. 136*11 hhd. 5. 21371 II hbd. 6. 6 ft. 11|1| in. 7. 267i| ft. 8. $162.72 + . 9. 373|ff cu. ft. 11. 208 bu. 12. 7 ft. 9| in. 13. 5 ft. 14. 10 ft. 15. $360. 16. 44.8 bu. 17. 16if T. 18. 12.6 T. Art. 441. 1. 110.4 A. 2. $453.03125. 3. $708 and 270, rem 4. 8100 cu. in. 5. 9288 cu. in. 6. 2160 cu. ft. 7. 3f cords. 8. 31 1 If cords. 9. $5.00. 10. $250. 11. $6. 12. 114yV sq. yd. 13. 156 sq. yd. 14. 72 yd. 15. 85 yd. 16. $33825. 17. $30750. 18. $49.095^f. 19. 880000 times. 20. 20 da. 20 hr. 21. 18849 ,V;1 wk. 22. 2700 bricks. 23. 369062 in. 24. 17400 shingles. 25. 144 farms. 26. 220320 bricks. 27. 30 da. 10 hr. 28. 76 yr. 37 da. 7 hr. 46 min. 40 sec. Answers, 391 Art. 452. 13. 8if%. 17. $4800, entire cost. 3. .40 or 40%. 14. 212|f%. $5.05 i%, cost per bar 4. .171 or 17|%. 15. 3|%. 18. $16000, whole cost 5. 1* y /*^ .30 or 30 /o. 16. 1%. $20 per bar. 6. .48 or 48^. 17. 18. 16|%. Art, 47 S. Art. 460, 19. 66|%. 1. $1347.80. o 20. $2863 for 3d. | 2. $5.79. 3. 4 5^1207. £2024.16. 6506 bu. 3 pk. 2 qt. 25342 5 9' 1st. 3. $537.50. 5. 45|Mt%. 2d. 4. $56.25. 6. 2240 lb. 299%%, 3d. 5. $252. 7. $62.50. 6. $448,121. 8. $588. Art, 466, 1 7. $380.70." 9. 10. 11. dE1218. $359.25. 194625 A. 3. 4. 288. 2340. 8. 9. $1485. $1787.50. 12. $2048.50. 5. £3428|. 10. $791,464. 13. 2725 ft. 3| in. 6. 1000. 11. $2843.75. 14. $188058.33i. 7. 8. 8(i00 yd. 312. 12. $312.0605. 15. $432, the first. 9. 250. 13. $1163.75. 16. $5580, the second. 10. $120. 14. $29250. 17. $7245. 11. $40000. 18. $3833.60. 12. $21150. Art, 474, 19. $911.25. 13. 62|. 15. m%. 14. 46.8. 16. 22% fo. Art. 462, 15. 43100. 17. 20%. 16. $7200. 18. 31A%. 30%. 23. 23. $6480. 3724 HI. 17. $184. 19. 24. $303000. .$1060. 20. 100%. 25. 10862 men. 18. $3675. 21. 25%. 26. 1780 sheep. 22. 4HMf%- 27. $10680. Art, 469, 23. 4f%. 3. 4856xV 24. 33i%. Art, 464, 4. 2281^3. 25. 23|i%. 3. 4. 29i%. 1%. 5. 6. 26000. 2200. Art. 475. 5. 8H%. rr 1 . $3100. 26. $5478.26^. 6. 6A%. 8. S7000. ■27. $17036.363^. 7. 11%. 9. 5200. 28. $24375. 8. 63i%. 10. 2234:^. 29. $4175.36. 9. 90 % , Henry. 11. 1363,V 30. $6970. 94%, sister. 12. 2705 bu. 3 pk. 4 qt. 31. $371. 16|. 10. m%. Apt. 32. $516.25. 11. 156| bu. sold. 13. $7000. 33. $980.20. 62^%. 14. 15000. 34. $1634.71|. 12. 25 9;, wife. 15. $7179.18 _'-". 35. $3435 20. $3125, each child. 16. $3085.71 36. $1696.2281^. 392 Answers. Art. 47 f 5ii7, 2. 1 yr. 10 mo. 28 d. 3. 16 yr. 8 mo. 4. 10% 10 yr. 2. $7142.857. 3. $11666.06|. 4. $12800. 5. $4237.288. 6. $446,428. 7. $1168.831. Art, 55f> 2. $102.04. 3. $139.50. 4. $209.02. 5. $348.21. 6. $1289.01. Art, 301, 2. $4690.34. 3. $560.36. 4. ^261.69. 5. $1524.46. 6. $1174.42. 7. $1194.05 8. $1520.12. 9. $1938.35. Ai't, 565, 2. $780,045, pr. worth. $70,205, true dis. 3. $1170.11. pr. vvortli. $102.39, true dis. 4. $2631.82, pr. worth. $263.18, true dis. 5. $4881.86, pr. worth. $768.89, true dis. 6. $9527.44. pr. worth. $472.56, true dis. 7. $41.60. 8. $3214|, pr. worth. $1607^33, true dis. 9. $12380.95. Art, 57 (K 2. $639,925. 3. $816.34. 4. $1258.84. 5. $18.61. 6. $821.76. Art, 571. 2. $518.45. 3. $4473.01. 4. $5342.81. Art, 574, 2. $418.50. 3. $511.65. 4. $657.90. 6. $2211.84. 7. $3147.54. 8. $3623.29. 9. $5606.25. 10. $8881.50. 11. $696.62. Art, 575, 2. $73.68. 3. $4.80. 4. $150. 5. $375. Art, 5 S3, 2. 4 mo. 3. 6 mo. 4. Hyr. 5. 3 mo. 6. 6| mo. 7. 2 yr. 3 mo. Art, 584. 9. Au^. 15th, 1879. 10. MaV 4th, 1880. 11. Aug. 19th. 12. Sept. 6th. 13. Sept. 21st. 14. Jan. 5th, 1881. Art, 5S(i, 2. Sept, 3. Nov. Bal. 4. Apr. Bal. 5. Dec. Bal. 0. July Bal. 7. Aug. Bal. , 2d. 28th. due, $150. 1st. due, $1730. 20th, 1879. due, $140. 28th. due, $100. 13th. due, $1275. Art. 601. 1. $280. 2. $750. 3. $510. 4. §420. Art, 602. 5. $2461. 0. $2604. 7. $10048.50. 8. $13725. 9. $12525. Art, 604, 2. 15%. 3. m%. 5. m-fc. Art. 605. 7. $9750. 8. $10890. 9. $21875. 10. $47062.50. Art. 606. 12. 125 shares. 13. 50 shares. 14. 60 shares. 15. 73 shares. 16. 40 shares. 394 Answers. Arfs.607,10. 18. 66|%. 19. G2L%. 20. 1331 fc. 22. $78750. 23. $14437.50. 24. $24040. 25. $28333.331. 2C. $6360. 29. 4:fjfo. 30. 5Hf%. 31. $666. 66|. 34. 311^. 35. lUf%. Arts. 624,25. 3. 4. 5. 7. 8. 9. 11. 12. 13. 14. 15. 16. $862.75. $970,125. $2035. $5285.29. .$3445.75. $5075. $2439.02. $3419.69. $454.78. $2308.64. $4160.16. $2971.77. Arts, 632, 34. 2. $1683.22. 3. .$1843.092. 5. $145.54. 6. $195.26. 8. $675.50. Art. 635. 10. £1337, 8s. 11.W. 11. 12360 fr. 12. 7740 fr. 13. 3200 marks. 14. 85331 marks. Art. 641. 2. 3. 4. $499.20. $47.50. $97.50. $24.50. 6. 7. 8. 9. 10. 11. 12. 13. 15. 16. 17. 18. 19. 20. 21. 23. 24. 25. 26. 27. Oft $35.28. $37.50. $7.44. $5.15. 756 m. 5624 bu. 5106 m. 84 II Km. £1." $53333.33. $427.50. $75.74. $23,125. 60 d. 50 d. 4:1 d. 1 ^\ mill. 2y\ lir. 8ds. 4i hr. Art. 643. 2. 99 lb. lOf oz. 3. 98 lb. 7/3 oz. 4. 69T. 1285 fib. 5. .07AV 6. 93bu.5.12qt. 7. $0,831. 8. 818.568 m. 9. 4858 55+ lb. 10. 9581b. lO.loz. 11. $6.75. 12. 189 yd. 13. 10|doz. 14. 325 lb. Art. 645. 2. $12. 3. $80. 4. $250. 5. $250. 6. $872.50. Art. 647. 2. 107 b. 44 q, A. 85 1). 22 f q., B. 87 b. 4iq.,('. 3. $600, A's. $375, B's. $525, C's. 4. 666|bar.,A's. 800 bar., B's. 1000 bar., C"s. 5331 bar.,D's. 5. $315, A's. $525, B's. $420. C's. 6. $1250, X's. $1750, Y's. $2000, Z's. 7. $0.66|. $200, 1st. $266.66|, 2d. $333,331, 3d. 8. $0.80. 9. $64.14, A's. $105.12, B's. $147.76, C's. 10. $0.10 on $1. $500, B rec'd. 11. $0.4223 + . 12. 100 bar., A's. 66| bar., B's. 33^ bar., C's. 14. $100, A. $120,BandC. 15. $30. 16. S40.02, A's. $88.28, B's. $117.70, C's. 17. $332.50. S's $525, Jones'. 18. $508.83, A's. $677.75, B's. $938.42, C's. Art. 649. 2. 8581. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 2554^. 2666f.' $288. $1.20. 875 pears. 425. 9S0. 270 mi. $56. 13. 385. 14. $2.10. 15. 22 T. 15001b. 16. 2 coats. 17. 21 tubs. 19. 240slieep. 20. $288. 21. 1440 men. 22. 48 ft. 23. $296. 24. $14400. 25. 72 yr. 26. 21i9i. 27. 72 pupils. 28. $15600. 29. 60 trees. Art. 650. 3. 4. 5. 6. 7. 8. 9. 10. 11. 40 shares, $62.30. $4932.64. $2187.50. $16841. 8%. $3869.72 + . $1619.31. $1010.50. 12. Um. 13. 14. 15. $900, A's. $1040, B's. $1060, C's. $770,625. $3894.4325. Art. 651. 2. 324 in one. 432 in other. 3. 312 in one. 936 in other. 4. 108|, one. 326 1, other. 5. 151 A., one. 604 A., other. 6. $1312.89,one. $1641.11,ot'r. 7. 180, first, 240, second. 300, third. 9. 378, 1st. 252. 2d. 315, 3d. 11. 205 Ansivers, 395 12. 13. 14. 15. 11. 1211 Kl. 40 peaches. 80 pears. 160 apples. 64, 1st. 32, 2d. 96, 3d. Art, 652, 17. 7i ds. 18. 2| mo. 19. 20 men. 20. 720 mi. 21. 224 bu. Art. 653, 23. 90 cts., 1 pt. 40 cts., 14 pt. 24. 30 cts., I'pt. 24 cts., 5 pt. Art, 668, 1. 3|. 2. 2/j. 3. 214. 4 2 3 K 13 »• 3^- 6. 7. 8. 9. 10. 11. 12. 1 1' 27 6?- 11 2F' 8 7- 4 9- _8_ 15- 49 73' 13: 16. 14. 11. 15. 1151. 16. 16|. 17. 31 18. ih- 19. 32. 20. 3\- J Art. 21. 7. 22. A- 670. 23. f 24. 112. 25. f|. 26. VW- 27. 192. 28. 432. 29. I183. 2. 70. 3. 2550. 4. 7xV. 5. 48. 6. 288. 7.' 375. 8. 27.3. 9. 35 vests. 10. 736.? lb. 11. 131 gal. 12. 45. 13. 75. Art, 687, 3. 3|d. 4. $515. 5. £22, 10s. 6. 1440 min, 7. $3.75. 8. $5000. 9. 13^ mo. 10. $60. 11. 75 ft. 12. lOOd. 13. 105 1iektars. Art, 692, 3. $3.44. 4. $5.01. 5. 108.5 Kil. 6. 69f«3Hl. 7. 318|Km. 8. 133 3^ spoons. 9. $251|. 10. 362 d. 11. £51, 3s. 2d. 12. £3, 12s. 6d. 13. £41, 12s. 14. 3f hr. 15. 5 min. 16. 12 br. $195.14. U\9, in. 164L yd. 6d. 17. 18. 19. Art. 605, 3. 4. 5. 6. 7. 8. 9. 10. 12. 13. 14. 15. 16. 17. 18. 768 m. 96 men. 10 men. 6d. 7.2 d. 170| bu. 80 d. 6 men. $18. 90d. 25 lb. 60 men. 18 d. 3s. 1.7d. 792 pr. Art. 608, 3. 72, 1st. 96. 2d. 144, 3d. 3. 22 sb., 1st. 66 sb., 2d. 110 sb., 3d. 4. 120 bu. oats. 160bu. wlie't. 220 bu. corn. 5. $0.88, pears. $1.76,or'ng's. $2.64,merns. 6. $497, 1st. $745.50, 2d. $994. 3d. $1242.50,4tli. Art. 706, 2. t, A's. h B's. 3. $120. A's. $160, B's. $200, C's. 4. $342.86, A's. $457.14, B's. $685.71, ("'.s. $914.29, D's. Art, 708. 6. $50, A's. $100, B's. 7. $100, A's. $120, B's. $120, C's. 8. $30. 9. $40.02, A's. f88.28, B's. $177.70, C's. 10. $600, A's. 11. $589.47, A's. $1129 82, B's. $1670.18, C's. $2210 53, D's. 12. $5090. 13. 16|%. 14. 10%, or $1500, A's loss. Art, 718, 2. 216. 3. 729. 4. 53824. 5. 3125. 6. 2401. 7. 42875. 8. 65.450827. 9. 8.003600540- 027. 10. 27013502.25- 0125. 11 **!- 13. 111.56640625. Art, 710, 4. 900 + 240 + 16. Art, 734, 4. 24. 5. 40. 6. 35. 396 Answers. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 17.07 + . 29.4 + . 23.20 + . 7.859 + . 92.06 + . 1110.016 + , 2305.317 + , 16.96 + . .881 + . 32.768. .0331. 785.64 Art, 735, 3. i. 4. I 5. ^. 6. .36288 + . 7. 1. 8. .4848 + . 10. ^. 11. 6|. 12. ri 13. 10|. 14. 9|. 15. .751 + . 16. 14A. 17. 1.7320508. 18. 3.46410161 + . 1. 45 yd. 2. 952. 3. 783.836 rd. 4. 72 ft. 5. 24 rds. 6. 390 rds. 7. 1395 yd. sq. ArU 736, 9. 8. 10. 15. 11. 30. 12. 56. 13. 72. 14. 38.8844442. 15. 65.72 + . 16. .08. 17. 2. 18. .18. 19. 1. 20. A. 21. M. 22. .3. OQ 3 1 24. 80 rd., width. 160 rd., length. 25. 80 rd., breadth. 320 rd., length. Art, 74,3. 27. 10 yd. 28. 50 m. 29. 200 m. Art. 744, 31. 60 ft. 32. 24.98 ft. 33. 8 yd. Art. 745. 35. 24 ft. 36. 103.614+ ft. 37. 42.4264 rd. 38. 40 rd., length, side. 56.5685 rd., diago- nal. 39. 50 ft., floor diago- nal. 51.92+ ft., other diagonal . 40. 65.802+ rd. 41. 56.5685 ft. 42. 75.816+ ft. Art. 747. 3. 15 min. 4. 18 in. 5. 30 in. 6. 24.49+ vd. 1. o 3. 4. 5. 6. Art. 750, 2 fig 2 fig 2 fig 3 fig 3 fig 2fiff Art, 752, 2. 303 + 3(30-^x2) + 3(30x22) + 2'. Art, 754. 4. 24. 5. 83. 6. 72. 7. 125. 8. 103. 9. 1331. 10. 3002. 11. 2.3. 12. 4.5. 13. .632 + . 14. 5.48. 15. 49.68. Art, 755, 17. .601 + 18. i\. 19. f. 20. If. 21. 2.39 + . 00, 31. 23. 1.2599 + . 24. 1.442249 + . Art, 756. 1. 73 in. 2. 364 ft. 3. 108 yd. 4. 8 ft. 5. 8 ft. 6.44 in. 6. 9 ft. 5.3 in. 7. 58.8+ ft. Art. 75S, 3. 137.48+ lb. 4. $121362.96. 5. 1562.5 cu. ft. 6. 23^V T. 7. 163840000 1b. 8. llll|hhd. Art. 759. 10. 7.2+ ft. 11. 8 in. 12. 3 ft. 13. 4 ft. > Answers. 397: Art, 76S. 1. 15. Art, 769, 2. 16. Art. 770. 3. 38. 4. 10. 5. 27. 6. $550. Art. 771. 2. 8 children. 3. 23. 4. 381 d- Art. 77*^. 2. 3 yr. 3. $3. 4. ^%. 5. ^V%. ^1 ^ 773. 2. 78 strokes. 3. 180000. 4. $651. 3. $455.81. 4. $1605.87, anit. $32,76.975, amt. Art. 7S0. 2 -^-^- '*• 243- Art. 7 2. 1456. 3. $255. 4. $40.95. 5. $5314.40. SI. Art. S09, 2. 450 sq. ft. 3. 4914 sq. ft. 4. 46 A. 17i sq. rds. 5. 4556] sq: yd. 6. 250 sq. ft.* Art. 810. 2. 50.91 + sq. yd. 3. 198.43+ sq. ft. 4. 7 A. 58.14 sq. rd. Art. 811. 2. 3 yd. 3. 150 rd. Art. 812. 2. 80 rd. 3. 52.6 yd. Art. 821. 3. 67iA. 4. 640 A. 5. 26 A. 65 sq. rd. 6. 50 rd. 7. 80 rd. 9. 4 A. 75 sq. rd. Art. 822. 2. 368sq. yd. 3. 558 sq. rd. Art. 823. 2. 50 A. 125 sq. rd. 3. 12480 sq. yd. Art. 829. 2. 141.372 yd. 3. 314.16 rd. Art. 830. 2. 30 rd. 3. 200 yd. 4. 8 rd. 5. 16 ft. 9. 10. Art. 831. 7854 sq. ft. 11309.76 sq. rd. 2037.178+ sq. yd. 15.91 ft. 10.472 ft. 706.86 sq. ft. 203.7178 A. 7.97 rd. Art. 841. 3. 126 sq. ft. 4. 54 sq. in. 5. 152 sq. ft. 6. 576 sq. ft. 7. 640 sq. ft. 8. 4084.08 sq. ft. Art. 842. 3. 375 cu. ft. 4. 9200 cu. ft. 5. 565.488 cu. ft. 6. 45945.9 cu. ft. Art. 846. 3. 7744 cu. ft. 4. 3817.044 cu. ft. 5. 441 cu. ft. Art. 850. 2. 4.91 sq. ft. 3. 201062400 sq. m Art. 831. 2. 523.6 cu. in. 3. 259,777,100,10S cu miles. 4. 381.7+ cu. in. Art. 852. 2. mf. ft. Art. 855. 2. 44.982 gal. 3. 1059.5286 liters. 4. 548.4375 gal. Art. 856. 1. 2526rVT. 2. 378if T. Art. 858. 1. 164 girls. 304 pupils. 2. 62f4f rd., breadth. 330f A. 3. .568431. 4. .881 A. 398 Answers. 5. 45f ft., height 6. $1,462. 7. 8157 da. 8. $28.95. 9. $4702.50. 10. 7. 11. 5 bbl. 152 lb. 12. 88704 steps. 13. $64G0.40. 14. 12 ft. 15. $34.57. 16. 39.163 yd. 17. 3949, one. 4705, other. 18. $462.50, one. $1037 50, other. 19. 469, less. 1407, greater. 20. $315, B's share. $1260, A's share. 21. 511?, one. m~ij, other. 22. 1 = g-. c. d. 261648 = 1. c. 111. 23. 23 p. of 36 yd. 24. 16 A. 25. $1258250. 26. 139180. 27. 132x7x2. 28. 4 days. 29. 15, j;-.c. . 85. $201.60. 86. $1546.40. 87. 20%. 88. f^, A's share. ^Yw^, B's share. ^Wt' C's share. 89. $1.76tV 90. 7%. 91. $545.73. 92. 14?, da. 93. $2000. 94. $5.00 95. 40 vd. 96. $185. 97. $9000. 98. $85. 99. 250 shares. 100. $18277.91. 101. 101%. 102. 16326 yr. 11 mo 4.44 da. 103. $3.41. 10-1. $1320, As share. $880, B's share. $440, O's share. 105. 103.9. 106. 80 rd. 107. SOg^ vd., or 52 yd. 108. 13.34 + , diagonal. 11250 lb. 109. $10.3125. 110. $53.90. 111. 260 bu. 11.424 qt. 112. $550.92. 113. 71f%. 114. $1525.55. 115. 16875. 116. 240 rd. 117. $408.96. 118. $175. 119. $362.30. 120. 80.6 ft. 121. 56" 8'. 122. 1 da. 6 hr. 25 min. 123. 4165 cu. ft. 124. 470 1 cu. yd. 125. 1190sq. rd. 126. 2101 mi. 127. 107fV lb., entire weight. 35f| lb., average weight 128. 129. ^T 130. $10. 131. 88° 45'. 132. 169UCU. ft. 133. $401.20. 1.34. 22 min. 40 sec. 135. .635. 136. $24634.63, inves't. $615 87, com. 137. $18.75. 8 6^ qt. Ansvjers. 399 138. $6. 139. 6|lir. 140. 34J4 T. 141. 14.5 ft. 142. 486 tiles. 143. 47J/j%. 144. 7()jU%- 145. $3.75. 146. 3432 ft. 147. .44A. 1 48 30 -•' **^- X'±0. .OW, 0- 149. 3|mo. 150. /«. 151. d^jdn. 152. .299980317. 153. 8.55 ft., nearly. 154. 5 h. 56 m. 32 s. 155. $99. 156. POI.72. 157. $12.50. 158. 38.47 sq. rd. 159. 42 in. 160. i\%, one. -j^j;, other. 161. $125000. 162. $7.22. 163. Iff. 164. 1500 lb., Nit. 250 lb., S. and C. 165. 1.1071. 166. $1200, IPt. $1600, 2d. $1800, 3d. 167. $190000. 168. $167.20. 169. 93|%. 170. $107.10. 171. $9.70. 172. 2 yr. 4^ mo. 173. $300. ' 174. 15fda. 175. $1000, or 10%. 176. 384 sq. ft. 177. .062. 178. $654.50. 179. 35 30' eastward. 180. Iff. 181. 525. 182. $413.44. 183. 59.92 sq.ft. 184. 249.41 sq. yd. 185. $55924.05. 186. $130.80. 187. $48.05, A's. .$69.95, B's. 188. 66.98 lb. Troy. 189. 40. 190. 30.7125 A. 191. 192 lots. 192. 300 strokes. 193. 2, com. difference. 99, sum. 194. 15 holidays. $360. 195. 70 A. 109.76 sq. rd. 196. $3600.84. 197. 6 bonds. 198. $284.06. 199. 562.34 mi., nearly. 200. $19.40. 201. 123.888 sq. ft. 7.87 ft. 202. 2037.178 sq. yd. 203. 1558.75 cu. ft. 204. 540 en. ft. 205. 9200 ca. ft. 206. 13750 cu. ft. 207. 22619.52 cu. ft. 208. 20420.40 cu. ft. 209. 210. 3048. 211 • 3tV%. 212 • llyf Art, 859. 1. 63, g-. c. d. 2. 1.4142. 3. 72 men. 4. 10%. 5. 3654.7 meters. 6. 3|- 7. 73677.6846 cu. dm 8. 71ir. 9. $2500. 10. M. .89,11- 11. m- 12. 825 1288- 13. .0245i|f. 14. .831114 A., or .831433+ A. 15. 160 sq. dm. ifi -; 1361 17. .218. 18. .059. 19. $7295.43. 20. lli%. 32. 4| oz., nearly. 58.293+ meters 23. 41 pt. 24. 7501b. 25. $2331.12//j, A's $4662.24i|f, B's $5006.61i|f, C's. 26. 4.38 + . 27. 360, I.e. 111. 2, j» . c. d. 28. 7.6199 meters. 29. 1879. 30. $.88|. 31. $1142.86. 32. 2.358^, sum. .291f , product. 33. 1566.712 + . 34. 1880.0001 + . 35. 488.2468 Kg. 36. 36|f%. 37. 1.178*. 38. 12500 bricks. 39. $5061.68. 40. $21000 at first. 87if % loss. 41. $8400. 42. $7000. 43. $1210.59. 44. 57.3332. 45. 3.128. 46. 3, g. c. d. 47. 504, I.e. 111. 48. 63^07. • 49. 9 o'clock 54 m. 23 3 & 50. .059375 day. 51. If 52. $'301.11. 53. 4% loss. 54. 2.37. 55. 1.60933 Km. 56. ly^. 400 Ansivers. 57. .0096048. Art, SS4, 58. 4 dm. 6 cm. 3 m. 1. .8. 59. 1.05 books. 60. 37| cts. 3. 3. .875. .75. 61. $1060. 4. .575. 63. 4.5 meters. 5. .088. 63. .000001. 6. .8125. 1.002001. 7. .4375. 64. 3.331. 8. .857143. 65. 2^ da. 9. .3. 66. $34.23. 10. .761904. 67. $3313.50. 11. .36. 68. $3447.50. 69. 14 f,. 13. .17073. 70. $3087.75. 13. .416. 71. 13i mo. 14. .53. 73. $9973.97. ; 15. .590. 73. 33.419. 16. .36. 74. 7:\0. 17. .313. 75. $13530 34. 76. 1.60933 Km. 18. .31. 77. .5. 19. .484375. 78. 1 yr. 7 mo. 6 da. 30. .13. 79. 8%. 80. 735.6 liters. Art, 885. 81. -iz- 3. 2 83. 8000 turns, nearly. o 11' 83. $.95, difference. o. TT- 84. $381. 4. 23 33- 85. 85018983. 5. _41_ 3 3 3* 86. 9tV. 6. il 37* 87. 500.003, sum. 7. 6 TTT* 499 998, difference. 8. 97 88. 90. 5 Kg. 317.4 cr. 13.573 + . 9. 10. 65 4 2 91. 3.9 hektoliters. TOT- 11. 13. 13. 35121 93. 93. O 851 \fo premium. 999¥¥' 1 94. 33.713+ liters. 1 13* 95. 7^ cords. 96. 5 mo. 4 da. Art. 886, 97. .495 + . 15. 178 2T5- 98. 8.65 Km. 16. 11_3 TT25- 99. 31 bu. 3 pk. 5 qt. 17. 2933 7¥Fff- 18. 19. 30. 21. 33. 1 .!¥» ^7' 361 ^so- ls T¥- 127457 1560I)OOTy* Art, 887, 33. 1.0897. 34. .3377. Art, 895, 1. 103f A. 3. $138.57 3. $533.33 4. $160. 5. 1930 A. $43530 gain. Art. 905, 3. $1836.66. Art, 910, 3. $0.09^^7. Art, 911. 4. 3 lb. at 9 cts. 2 lb. at 11 cts. 4 lb. at 14 cts. 5. 3 lb. at 15 cts. 3 lb. at 18 cts. 1 lb. at 31 cts. 4 lb. at 33 cts. Art, 912, 7. 35 bu. at 40 cts 10 bu. at 45 cts 8i bu. at 56 cts 161 bu. at 65 cts J- Thomson, J*Il Complete ^;raded arthinie- Dept» tio, oral and writt e n, upon the mau or instruct /ton, 3Ja$s. "It is an admirable work for students who desire to find in printed form the facts of classic mythology."— .ffcv. L. Clark Seelye, Fres. Smith College, Northamp- ton, Mass. " The subject is a difficult one from the nature and extent of the materials and the requirements of our schools, llie author avoids extrefaie theories and states clearly the facts with modest limits of interpretation. I think the book will take well and wear Avell."— C. F. F. Bancroft, Fh.D., Frin. Fhillips Academij, Andover, '^"^^' Price, by Mail, Post-paid, $1.00. Clark k Maynard, Publishers, New York. Yb Joa^^ Two-Book Series of Arithmetics. By James B. Thomson, LL.D., author of a Matliematical Course. 1. FIRST LESSONS IN ARITHMETIC, Oral and Written. Fully and handsomely illustrated. For Primary Schools. 144 pp. 16mo, cloth. 2. A COMPLETE GRADED ARITHMETIC, Oral and Writ- ten, upon the Inductive Method of Instruction. For Schools and Academies. 400 pp. 12mo, cloth. This entirely new series of Arithmetics by Dr. Thomson has been prepared to meet the demand for a complete course in two books. The following embrace some of the characteristic features of the books : First Lessons.— This volume is intended for Primary Classes. It is divided into Six Sections, and each Section into Twenty Lessons. These Sections cover the ground generally required in large cities for promotion from grade to grade. The book is handsomely illustrated. Oral and slate exercises are com- bined throughout. Addition and Subtraction are taught in connection, and also Multiplication and Division. This is believed to be in accordance with the best methods of teaching these subjects. Complete Graded.— This book unites in one volume Oral and Written Arithmetic upon the inductive method of instruction. Its aim is twofold : to develop the intellect of the pupil, and to prepare him for the actual business of life. In securing these objects, it takes the most direct road to a practical knowledge of Arithmetic. The pupil is led by a few simple, appropriate examples to infer for himself the general principles upon which the operations and rules depend, instead of taking them upon the authority of the author without explana- tion. He is thus taught to put the steps of particular solutions into a concise statement, or general formula. This method of developing prin- ciples is an important feature. It has been a cardinal point to make the explanations simple, the steps in the reasoning short and logical, and the definitions and rules brief, clear and comprehensive. The discussion of topics which belong exclusively to the higher depart- ments of the science is avoided ; while subjects deemed too diiiicult to be appreciated by beginners, but important for them when more advanced, are placed in the Appendix, to be used at the discretion of the teacher. Arithmetical puzzles and paradoxes, and problems relating to subjects having a demoralizing tendency, as gambling, etc., are excluded. All thjit Is obsolete in the former Tables of Weights and Measures is eliminated, and the part retained is cori-ected in accordance with present law and usage. Examples for Practice, Problems for Review, and Test Questions are abundant in number and variety, and all are different from those in the author's Practical Arithmetic. The arrangement of subjects is systematic; no principle is anticipated, or used in the explanation of another, until it has itself been explained. Subjects intimately connected are grouped together in the order of their dependence. Teachers and School OflBcers, who are dissatisfied with the Arith- metics they have in use, are invited to confer with the publishers. Clark & MAYNARD, Publishers, New York.