LIBRARY OF THE University of California. Gl FT OF WVVV--'?'^ Class Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/completearithmetOOfishrich Robinson's Sljorttr Course. THE COMPLETE AEITHMETIO. ORAL AND WRITTEN. By DANIEL W. FISH, A.M., ^„0B OF BOBI^SON'a BBKIE8 OF PROGIU«SSIVB ABITHMETICh. IVISON, BLAKEMAN, TAYLOR NEW YORK AND CHICAGO. & CO., EOBINSON'S Shorter Course FIRST BOOK IN ARITHME TIC. Primary. COMPLETE ARITHMETIC In One Volufne* COMPLETE ALGEBRA. ARITHMETICAL PROBLEMS. Oral and Written. ALGEBRAIC PROBLEMS. KEYS to Complete Arithmetic and Problems, and to Complete Algebra and Problems, in separate volumeSy for Teachers. Arithmetic, oiial and written, usually taught in THREW books, is now offered, complete and thorough, in ONE booh, " the complete arithmetic:'' * This Complete Arithmetic is also f>ublished in two volumes. TAUT if» nnd TA-HT IJ» are each bound separately ^ and in cloth. ^-^Ui- Copyright, 1874, by DANIEL W. FISH. rpHE design of the author, in the preparation of this work, has been to furnish a text- book on the subject of arithmetic, com- plete not only as a treatise, but as a comprehensive manual for the class-room, and, therefore, embodying every necessary form of illustration and exercise, both oral and written. Usually, this sub- ject has been treated in such a way as to form the contents of three or more graded text-books, the oral exercises being placed in a separate volume. In the present treatise, however, the whole subject is presented in all its different grades ; and the oral, or men- tal, arithmetic, so called, has been inserted, where it logically and properly belongs, either as introductory to the enunciation of prin- ciples or to the statement of practical rules — the treatment of every topic from the beginning to the end of the book being thoroughly inductive. In this way, and by carefully constructed analyses^ applied to all the various processes of mental arithmetic, the pupil's mind cannot fail to become thoroughly imbued with clear and accurate ideas in respect to each particular topic before he is required to learn, or apply to written examples, any set rule whatever. The intellect of the pupil is thus addressed at every step ; and every part of the instruction is made the means of effecting that mental development which constitutes the highest aim, as well as the most important result, of every branch of education. This mode of treatment has not only the advantage of logically training the pupil's mind, and cultivating his powers of calculation, but must also prove a source of economy, both of time and money, inasmuch as it is the means of substituting a single volume for an entire series of textbooks. 1 fif^925 IV PREFACE. As the time of many pupils will not permit them to pursue this study through all of its departments, the work is issued in two parts, as well as in a single volume. This will, it is thought, be also con- venient for graded schools, in supplying a separate book for classes of the higher and lower grades respectively, without requiring any unnecessary repetition or review. The author feels assured that, on examination, this work will commend itself to teachers and others, by the careful and progres- sive grading of its topics, the clearness and conciseness of its deji- nitions and rules, its improved methods of analysis and operation, and tho great number and variety of its progressively arranged examples, both oral and written, embodying and elucidating all the ordinary business transactions. The use of equations as a form of expression in these examples will be found to possess many advan- tages, not only as an arithmetical drill, but also in familiarizing the pupil with tho use of algebraic symbols. All obsolete terms and discarded usages have been studiously ignored, and many novel features introduced, favorable to clearness as well as brevity. The work has been carefully adapted, in other respects, to tlio present time, recognizing and explaining all the recent changes in Custom-house Business, Exchange, etc., and pre- senting, in connection with the examples for practice under each topic, information not only fresh but important. Attention is especially called to the manner in which United States Money is introduced in connection with the elementary rules ; to the comprehensive treatment of all tho various departments of Percentage, so essential at the present time in commercial transac- tions ; to the articles on Measurements and Mensuration, and the vast amount of valuable information given in connection with this part of the subject. In these respects this part of the work will be found to be particularly adapted to the wants of High Schools and Academies, as well as of Mercantile and Commercial Colleges. The Remews interspersed throughout the book will be found to be just what is needed by the student to make his progress sure at each step, and to give him comprehensive ideas of the subject as he advances. Carefully constructed Synopses have also been PREF A C E. V inserted, with the view to afford to both teacher and pupil a ready means of drill and examination, as well as to present, in a clear, concise, and logical manner, the relations of all the different depart- ments of the subject, with their respective sub-topics, definitions, principles, and rules. Great jDains have also been taken to make this work superior to all others in its typographical arrangement and finish, and in the general tastefulness of its mechanical execution. The author takes pleasure in acknowledging his indebtedness for many valuable suggestions received from teachers of experience and others Interested in the work of education ; and to Henry Kiddle, A. M., late Superintendent of Schools in the city of New York, for valuable assistance, especially in the higher departments of Percentage, and for important suggestions in relation to other parts of the work. How nearly the author has accomplished his purpose, to give to the public, in one volumCy a clear, scientific, and complete treatise on this subject, combining and systematizing many real improvements of practical value and importance to the business man and the student, the intelligent and experienced educator must decide. D. W. F. T N order to teach any subject with the best success, the instruc ^ tor should not only fully understand it, in all its principles and details, but should also clearly perceive what particular faculties of the mind are concerned in its acquisition and use. Arithmetic is pre-eminently a subject of practical value ; that is, it is one to be constantly applied to the practical affairs of life. But this is true only in a limited sense. Very few ever need to apply to any of the purposes of business more than a small part of the principles and rules of calculation taught in the text-books. Every branch of business has its own requirements in this respect, and these are all confined within very narrow limits. The teaching of arithmetic must, therefore, to a great extent, oe considered as disciplinary, — as training and developing certain faculties of the mind, and thus enabling it to perform its functions with accuracy and dispatch. The following suggestions, having reference to this twofold object of arithmetical instruction are pre- sented to the teacher, as a partial guide, not only in the use of this text book, but in the treatment of the subject as a branch of education. Seek to cultivate in the pupil the habit of self-reliance. Avoid doing for him anything which, either with or without assistance, he should be able to do for himself. Encourage and stimulate his exertions, but do not supersede them. Never permit him to accept any statement as true which he does ' not understand. Let him learn not by authority but by demonstra- tion addressed to his own intelligence. Encourage him to ask questions and to interpose objections. Thus he will acquire that most important of all mental habits, that of thinking for himself. SUGaESTIONS TO TEACHERSi Vll Carefully discriminate, in the instruction and exercises, as to which faculty is addressed, — whether that of analysis or reasoning, or that of calculation. Each of these requires peculiar culture, and each has its appropriate period of development. In the first stage of arithmetical instruction, calculation should be chiefly addressed, and analysis or reasoning employed only after some progress has been made, and then very slowly and progressively. A young 3hild will perform many operations in calculation which are far beyond its powers of analysis to explain thoroughly. In the exercise of the calculating faculty, the examples should be rapidly performed, without pause for explanation or analysis; and they should have very great variety, and be carefully arranged so as to advance from the simple and rudimental to the complicated and difficult. In the exercise of the analytic faculty, great care should be taken that the processes do not degenerate into the mere repetition of formulcB, These forms of expression should be as simple and con- cise as possible, and should be, as far as practicable, expressed in the pupil's own language. Certain necessary points being attended to, the precise form of expression is of no more consequence than any particular letters or diagrams in the demonstration of geomet- rical theorems. Of course, the teacher should carefully criticise the logic or reasoning, not so as to discourage, but still insisting upon perfect accuracy from the first. The oral or mental arithmetic should go hand in hand with the written. The pupil should be made to perceive that, except for the difficulty in retaining long processes in the mind, all arithmetic ought to be oral, and that the slate is only to be called into requi- sition to aid the mind in retaining intermediate processes and results. The arrangement of this text-book is particularly favora- ble for this purpose. Definitions and principles should be carefully committed to memory. No slovenliness in this respect should be permitted. A definition is a basis for thought and reasoning, and every word which it contains is necessary to its integrity. A child should not be expected to frame a good definition. Of course, the pupil should VUl SUOGESTIOKS TO TEACHERS. be required to examine and criticise the definitions given, since this will conduce to a better understanding of their full meaning. In conducting recitations, the teacher should use every means that will tend to awaken tJiouglit, Hence, there should be great variety in the examples, both as to their construction and phrase ology, so as to prevent all mechanical ciphering according to fixed IBethods and rules. The Bvles and FormulcB given in this book are to be regarded ae mrrimaries to enable the pupil to retain processes previously ana- lyzed and demonstrated. They need not be committed to memory, since the pupil will have acquired a suflScient knowledge of the principles involved to be able, at any time, to construct rules, if he has properly learned what precedes them. In the higher department of arithmetic, the chief difficulty con- sists in giving the pupil a clear idea of the nature of the business transactions involved. The teacher should, therefore, strive by careful elucidation, to impart clear ideas of these transactions before requiring any arithmetical examples involving them to be per- formed. When the exact nature of the transaction is understood, the pupil's knowledge of abstract arithmetic will often be sufficient to enable him to solve the problem without any special rule. The teacher should be careful not to advance too rapidly. The mind needs time to grasp and hold firmly every new case, and then additional time to bring its new acquisition into relation with those preceding it. Hence the need of frequent reviews, in order to give the pupil a comprehensive as well as an accurate and permanent knowledge of this subject. The Synopses for Betiew interspersed throughout this work are designed for this purpose. The whole or a part of a Synopsis, embracing one or more topics, may be placed upon the blackboard, and the pupil required to give briefly but accurately the siibdimsions, definitions, principles, etc., involved in each. By this means, if further tested by questions, a thorough and well classified know- ledge of the whole subject will be permanently impressed upon his mind. PAGE Pkeliminaby Definitions 1 Notation and Numeration 3 Arabic Notation 4 Synopsis 12 Addition 13 Synopsis 22 Subtraction 23 Synopsis 34 Multiplication 35 Synopsis 50 Division 51 Synopsis 76 Properties op Numbers 77 Divisibility op Numbers 79 Factoring 80 Common Divisors 83 Multiples 88 Cancellation 92 Synopsis 96 Fractions 97 Definitions 98 General Principles 100 Reduction 101 Addition 109 Subtraction 112 Multiplication 115 Division 124 Relation of Nujibers 132 Synopsis 141 Decimals 142 PAQl Notation and Numeration 144 Decimal Currency 150 Reduction 151 Addition 156 Subtraction 158 Multiplication 159 Division 162 Circulating Decimals 165 Short Methods 169 Ledger Accounts, 175 Accounts and Bells 176 Synopsis 183 Denominate Numbers 184 Measures OF Extension 186 Measures of Capacity 191 Measures of Weight 194 Synopsis 198 Measures of Time 199 Measures of Angles 200 Miscellaneous Measures 202 Measures of Value 204 Synopsis 208 Reduction of Denom. Integers. . . 209 Reduction of Denom. Fractions.. 216 Addition 225 Subtraction 227 Multiplication 230 Division 231 Longitude and Time 233 Duodecimals 239 COKTENTS. PAGE Synopsis 240 Measurements— Surfaces 241 Land 245 Rectangular Solids 249 Boards and Timber 254 Capacity of Bins, Cisterns, Etc. 257 Synopsis 264 Percentage 265 Profit and Loss 276 Commission 284 Synopsis 292 Interest. 293 Problems in Interest 304 Compound Interest. 309 Annual Interest 312 Partial Payments 314 Discount 318 Savings Banks 326 Synopsis 329 Stocks 330 Insurance 340 Life Insurance 344 Taxes 348 Synopsis 352 Exchange 353 Arbitration of Exchange 362 Custom-house Business 366 Equation of Payments 369 Averaging Accounts 374 Synopsis 382 Ratio . . 383 Proportion 387 Partnership. 401 Alligation 407 Synopsis 413 Involution 419 Evolution 425 Progressions 439 Annuities 449 Synopsis 453 Mensuration 454 TRLA.NGLES 455 Quadrilaterals 459 CmcLES 461 Similar Plane Figures 464 Solids 467 Prisms 467 Pyramids AND Cones 469 Spheres 472 Similar Solids 473 Gauging 474 Synopsis 476 Metric System 477 Vermont Partial Payments 491 Vermont Taxes 495 Tables 497 Answers 499 N. B. — Editions of this book are bound with, and without, the answers. The edition with answers will be supplied unless other- wise ordered. -»-^-^ -^ a F^^. -^f v^'^Sr^ PREBli i W«RV DEFIMTIONS ] ARTICLE 1. Arithmetic is the Science of Num- -^^ bers, and the Art of Computation. As a science, Arithmetic treats of the nature and properties of numbers. As an art, it teaclies how to apply a knowledge of num- bers to practical and business purposes. ' 3. A JJtiit is one, or a single thing ; as one, one boy, one year, one dozen. 3. A Numher is a unit, or a collection of units ; as 07ie, three, five boys ; it answers the question, How many? 4. An liitec/ral Nwinber or Integer is a num- ber which expresses whole things ; as seven, four days. 5. The Unit of a Number is one of the collection of units which constitute the number. Thus, the unit of twelve is one, of twenty dollars is o?ie dollar. 6. A Concrete Number is a number that is applied to a particular kind of object, or quantity; as three houses, four dollars, five minutes. 7. An Abstract Number is a number that is not applied to any object ; a>sfour, seven, eight. 2 DEFINITIOKS. 8. Like JS'uinbers are such as have the same kind of unit, or express the same kind of quantity. They may be either concrete or abstract ; as eight and nine, six days. and ten days, two rods &yq feet, and five rods three feet. 9. Unlike Niimhers are such as have different kinds of units, or express different kinds of quantity ; as ten months and eight miles j seven dollars and five iarrels. 10. A Scale in Arithmetic, is a succession of units, increasing and decreasing according to a certain law, or rule. Scales are uniform or varying, 11. A Tin! form Scale is one in which the law of increase and decrease is the same throughout the entire succession of units. 13. A^ Varyinr/ Scale is one in which the law of increase and decrease is not the same throughout the entire succession of units. 13. A Decimal Scale is one in which the laAV of increase and decrease is 2tnifor7nly ten.' exeh cise s. 14. 1. How many units in two ? In five cents ? In six dollars ? In seven acres ? 2. What is the unit of six cents ? Of nine books ? 3. Are two trees and five trees like or unlike numbers ? 4. Are they concrete or abstract? Why? 5. What kind of numbers are seven and nine? Are five acres and seven cords ? Are four coats and six coats ? 6. Name two numbers that are Ulce and abstract. 7. Name two numbers that are like and concrete. 8. Name three numbers that are unlilce and concrete. KOTATIOK AKD NUMERATION. One Thousand. One Hundred and Eleven. NOTATIOlsr A'ND ITUMEEATION. 15« In representing numbers, objects are regarded as arranged in groups of tens ; hence we have single things, or units; next, groups containing ten units, or te?ij next, groups containing ten tens, or one liu7idred; and again, groups containing ten hundreds, or one thousand, etc. 16. This method of grouping is called the Decimal Systetu, from the Latin word decern, which signifies ten. 17. dotation is a method of writing, or represent- ing numbers by characters. 18. JVumeration is a method of reading numbers represented by characters. 4 NOTATIOK AKD K U M E R AT I O 1^. 19. The number of objects may be represented by words, or by characters. 30. The characters may be gHXiqv figures or letters. 31. Figures are characters used to express numbers. 32. The Arabic Notation is the method of ex^ pressing numbers by figures. It is so called because it was invented by the Arabs. 33. This method employs ten different characters, or , figures, to represent numbers, viz. : \ fe^ d^^ Figures. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. JVaifieS. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. 34. The first character, or cipher, is called Naught, or Zero, and when standing alone, has no value. The other nine are called significant figures, because each has a value of its own. They are also called digits. These ten characters, when combined according to cer- tain principles, can be made to express any number. 35. The first nine numbers are each represented by a single figure, and are called miits of ihQ first order. 36. By grouping ten ones, or units of the first order into a larger collection, there is formed a unit of the second order, called ten, which is represented by writing the figure 1 with a cipher after it ;. thus, 10. 37. In the same manner are represented Two tens, or Twenty, by 20 Three tens, or Thirty, " 30 Four tens, or Forty, " 40 Five tens, or Fifty, " 50 Six tens, or Sixty, by 60 Seven tens, or Seventy, " 70 Eight tens, or Eighty, « 80 Nine tens, or Ninety, " 90 NOTATION AND NUMERATION, 38. The numbers between ten and twenty are repre- sented bV/ /riting 1 in the second place, and the units in the first place. Thus, Eleven 11 Fourteen 14 Twelve 12 Fifteen 15 Thirteen 13 ' Sixteeq, 16 29. In like manner, the numbers between 20 and 30 are represented, thus, H' i ^ Seventeen 17 Eighteen 18 Nineteen 19 Twenty-one 21 Twenty-two 22 Twenty-three 23 Twenty-four 24 Twenty-five 25 Twenty-six 2^ Twenty-seven 27 Twenty-eight 28 Twenty-nine 29 30. The greatest number that can be expressed by two figures is 99. 31. By grouping ten units of the second order, or ten tens, into a larger collection, there is formed a unit of the third order, called a hundred, represented by writing the figure 1 with two ciphers after it ; thus, 100. 33. In like manner are represented Two hundred by 200 Six hundred by 600 Three hundred " 300 Seven hundred '' 700 Four hundred " 400 Eight hundred " 800 Five hundred " 500 Nine hundred " 900 33. The numbers from one hundred, to nine hundred and ninety-nine, are represented by writing the hundreds m the third place, the tens in the second place, and the units in Wv^ first place. 34. The greatest number that can be expressed by three figures is 999, 6 DOTATION AKD KUMERATIOKJ 35. Orders of Units are denoted by the position of the figures used in expressing a number. Thus, 532 represents 2 units of the first orders 3 units of the second order, or 3 tens, and 5 units of the third order, or 5 hundred^, and is read five hundred and thirty -two. 36. Principles. — 1. Ten units of any order in number make one unit of the next higher order. 2. When any order of units in a number is vacant, the place is filled tvith a cipher. EXERCISES. 37. Express the following numbers by figures 1. One hundred twenty. 2. Four hundred eighty. 3. Seven hundred six. 4. Five hundred seven. 5. Seven hundred. 6. Three hundred eight. 7. Six hundred ninety. 8. Eight hundred five. 9. Seven hundred ten. 10. Six hundred eleven. 11. Nine hundred seven. 12. Two hundred sixty. 38. Copy and read the following, and name the num- ber of hundreds, tens, and units in each : (1-) (3.) (3.) (4) (5.) (6.) 67 321 190 840 592 219 85 406 761 269 904 807 77 289 345 793 531 395 98 672 402 503 762 608 39. By grouping ten units of the third order, or ten hundreds, into a larger collection, there is formed a unit of the fourth order, called a iliousand, represented by writing the figure 1 with three ciphers after it ; thus, 1000* NOTATION AND NUMERATIOK. 40. In like manner are represented Two thousand by 2000 Three thousand " 3000 Four thousand " 4000 Five thousanvl '' 5000 Six thousand by 6000 Seven thousand " 7000 Eight thousand '' 8000 Nine thousand '' 9000 41a The numbers from one thousand, to nine thousand nine hundred and ninety-nine, are represented by writing thousands in the foitrth place, hundreds in the third place, tens in the second place, and units in the^r^^ place. Thus, 5304 represents 4 units of the first order, units of the second order, or tens, 3 units of the third order, or hundreds, and 5 units of the fourth order, or thousands, and is read five thousand three hundred and four. 42. The greatest number that can b3 expressed by four figures is 9999. 43. In the same manner, other 7iew orders are formed to represent larger numbers, by grouping ten units of the fourth order to form Wiq fifth order, or tens of thousands ; and ten units of the fifth order, to form the sixth order, or hundreds of thousands, etc. Thus, 432076 represents 6 units of the firs:t order, 7 units of the second order, units of the third order, 2 units of the fourth order, 3 units of the fifth order, and 4 units of the sixth order, and is read four hundred thirtj-two thousand and seventy-six. From the preceding illustrations it is obvious, that 44. Moving a figure one place to the lefty increases it& representative value tenfold ; and, 45. Moving a figure one place to the right, diminishes its representative value tenfold. 8 KOTATIOK AKD KUMERATIOK, EXEItCISES. 46. Write in figures and read : 1. Two units of the third order^ four units of the second order, and three units of the first order. 3, Five units of the fourth order, six units of the third order, and two units of the second order. 3. Seven units of the fourth order, eight of the second order, and three of the first. 4. One unit of the third order and four of the second. 5. Three units of the fifth order, two of the third, and one of the first. 6. Eight units of the fourth order, and five of the second. 7. Two units of the sixth order, nine of the fifth, fou of the third, one of the second, and seven of the first. 47. Express the following numbers by figures ; 1. Thirty-seven thousand. 2. Sixteen thousand one hundred. 3. Twelve thousand five hundred fifty. 4. Forty-nine thousand five hundred twenty-seven. 5. Fifteen thousand two hundred six. 6. Seventeen thousand twenty-four. 7. Sixty thousand six hundred eight. 8. Seven hundred twenty thousand. 9. Two hundred forty thousand five hundred. 48. Copy, and read the following, naming the num ber of units of each order : (1.) (2.) (3.) (4.) (5.) 1542 1020 32507 76387 528031 3473 1256 53106 627324 600320 NOTATIOI^ AKD NUMERATION. 9 49. This method of numeration groups the successive orders into periods of three figures each. The periods are commonly separated by commas, each period taking the name of its lowest order, as shown in the following Periods. Name. Numeration Table. 6th. 5th. 4th. 3d. OQ g o ^ H o PQ O 2d. OQ O H 1st. Orders OF Units IN THE Periods. Number. CO "§ S.1:! 5 S-- H 5 KhP fflHt^ HHh^ WHr- n-...^ 30,291,040,027,30 0, ^ s « CO PS g*5 4 The number is read 30 quadrillion , 291 trillion, 40 billion, 27 mzY- ?iQm^, say, 2, 10, 13, 19 tens, equal to 1 hundred and 9 tens. Write the 9 tens in the tens' place, 4nd reserve the 1 hundred to add to the next column. Lastly, adding the 1 hundred reserved, say 1, 8, 13, 17 hundreds, equal to 1 thousand, and 7 hundreds, which write in hundreds' and thousands' places. Hence the sum is 1792. In like manner, copy, add, and prove. (2.) (3.) (4.) (5.) 276 miles. 876 feet. $20.30 $145.24 307 " 94 " 7.56 36.60 638 " 142 " 13.08 105.08 425 " 507 " 25. .75 6. Find the sum of $370.21, $2.49, $3.07, and $.94. 7. Find the sum of 2008, 1400, 706, 300, and 77. 8. If 4 loads of coal weigh respectively 1922, 1609, ilOO, and 1873 pounds, what is the entire weight ? EuLE. — I. Write the numbers so that figures of the same ae paid $146.30 ; how much did the other pay? SUBTRACTIOIir. 27 31. A house and lot sold for $7856, whicli was one thousand one hundred and ten dollars more than the cost. What was the cost ? 32. A certain city has a population of 246857, which is 25324 more than it had last year. What was its population last year ? OBAZ EXEItCISES. 93. 1. A man having $20, paid $7 for a hat, and $8 for a vest. How many dollars had he left ? Analysis. — The difference between $20, and the sum of $7 and $8, which is $5. 2. A boy had 25 cents, and gave 15 cents for a slate and 10 cents for some paper. How many cents had he left? 3. Ella having 16 cents, Jane gave her 9 more, and James gave her enough to make her number 36. How many did James give her ? 4. Subtract by 7's from 63 to 0. 5. By 7's from 80 to 3. 6. By 8's from 64 to 0. 7. By 8's from 85 to 5. 8. By 9's from 90 to 0. 9. By 9's from 86 to 5. 10. By lO's from 100 to 0. 11. By ll's from 119 to 9. 12. By ll's from 125 to 4. 13. By 12's from 129 to 9. 14. By 12's from 150 to 6. 15. Count by 7's from 2 to 86, and back from 86 to % 16. Count by 8's from 4 to 100, and back to 4. 17. Count by 9's from 7 to 115, and back to 7. la Count by lO's from 16 to 136, and back to 16. 19. Count by ll's from 9 to 119, and back to 9. 20. Count by 12's from 20 to 140, and baok to 20. 21. How many are 5 tens less 3 tens? 50—30 ? 28 SITBTRACTIOK. 22. How many are 6 tens less 4 tens ? 60— 40 ? 23. From 6 tens 5 units subtract 4 tens 3 units. 24. From 8 tens 7 units subtract 5 tens 6 units. 25. From a cask containing 52 gallons, 27 gallons were drawn out. How many gallons remained ? Analysis. — The difference between 52 gallons and 27 gallont 97 is 2 tens and 7 units. 2 tens or 20 from 52 leaves 32, and 7 from 82 leaves 25. Hence 25 gallons remained in the cask. 26. From a piece of cloth containing 46 yards, 24 yards were cut. How many yards were left ? 27. A man bought a watch for $40, and a chain for $15, and sold both for $63. How much did he gain ? 28. How many are 6 and 40, less 5 and 20 ? 29. How many are 7 and 30, taken from 5 and 50 ? 30. Eighteen plus 12 equals 40 minus how many ? 31. Twenty-two plus 15, equals how many plus 10? 32. William haying 75 cents, gave 25 cents for a book and 20 cents for a slate. How many cents had he left ? 33. A farmer sold a horse for $96, which was $23 more than the horse cost. What did he cost ? Find the omitted term in the following equations : 34. 12+ 8— 6=? 43. 54—12= ? +12 35. 46 + 13—14=? 44. 17 + 23=56—? 36. 57-13+ 8=? 45. 18 + 25 = 23+? 37. 60— (24 + 6)=? 46. 64—48=30— ? 38. 28+ 6=40—? 47. 75—30= ? +U 39. 42—12=18+ ? 48. 16 + 38=60—? 40. 30 + 25=? +40 49. 43+ ? =27 + 28 41. 27—11=19—? 50. 80- ? =100—40 42. 36 + 16=60—? 51. 22 + 54=64+? SUBTRACTION^. 29 WRITTEN EXERCISES. 93. When any figure of the subtrahend is greater than the corresponding figure of the minuend. 1. From 953 subtract 674. Analysis. — Write the numoers as before (90), and subtract each order of units sepa rately. Since 4 units cannot be subtracted from 3 units, increase the 3 units by a unit from the next higher order, or 10 units, making 13 units 4 units from 13 units leave 9 units, which write in the units' place. Since 1 of the tens was united with the units, there are 4 tens left. As 7 tens cannot be subtracted from 4 tens, increase the 4 tens by a unit from the next higher order, or 10 tens, making 14 tens. 7 tens from 14 tens leave 7 tens, which write in the tens' place. Since 1 of the hundreds was united with the tens, there are 8 hundreds left. 6 hundreds from 8 hundreds leave 2 hundreds, which write in the hundreds' place. Hence the remainder is 279. In like manner, solve and prove the following : OPERATION. 8 14 13 Minuend 953 Subtrahend 674 Remainder 379 (2.) (3.) (4.) (5.) From 3273 6345 6702 7465 Subtract 1425 2462 4384 3270 (6.) (7.) (8.) From 42670 miles. 51062 acres. 246700 feet. Take 14384 66 24300 18030 « When one of the given numbers contains centSy and the Dther does not, fill the vacant places with two ciphers. From (9-) $325.17 (10.) $279.00 (11.) $105.08 (12.) $7.00 Take 84.36 183.42 67.00 .84 30 SUBTRACTIO]S". EuLE. — I. Write the subtrahend under the minuend^ placing units of the same order in the same column. II. Begin at the right, and suMract the units of each order of the subtrahend from the units of the correspond* ing order of the minuend, and write the result beneath. III. If the units of any order of the subtrahend are fjreater than the units of the corresponding order of the minuend, increase the latter by 10, and subtract; then diminish by 1 the units of the next higher order in the minuend, and proceed as before. Pkoof. — Add the remainder to the subtrahend, and if the sum is equal to the minuend, the loork is correct. Instead of diminishing by 1 the units of the next higher order in the minuend, we may increase by one the units of the next higher order in the subtrahend. Subtract 13. 20762 from 53120. 14. $73.16 from $138. 15. $247 from $382.28. From 16. $430.09, take $272.46. 17. 15200 rods, take 6472 rods. 18. 120764 tons, take 75028 tons. How many years from the date of each of the following events to the present year ? 19. Figures were used by the Arabs in the year 890. 20. Decimal fractions were invented in 1464. 21. Printing was invented in 1441. 22. The telescope was invented by Galileo in 1610. 23. The electric telegraph was first used in the United States in 1844. 24. The first passage of the Atlantic Ocean by steam tvas in 1839. REVIEW. SI What is the difference between 28. 7620 and 13420? 29. $4027 and $703.41? 30. $1076 and $2340.50? 25. 34726 and 47062 ? 26. 57600 and 20012 ? 27. 70361 and 1005 ? 31. 2762 + 10341 and 45701 + 1200? 32. 3000 + 42301 and 720 + 1684 + 7342? 33. A merchant bought a quantity of goods for $1248.65, and sold them for $1540. How much did he gain? 34. Sold a horse for $250.75, which was $28 more thaji he cost. How much did he cost ? 35. A man having $15740.80, gave $5085 for a store, and $7640.75 for goods. How much money had he left? 36. If a piece of property bought for $7086.86 is sold at a loss of $1562.09, for how much is it sold ? Find the second member of the following equations : 37. 12346 + 840 + 1046—3846= ? 38. $210 + $809.76 — ($15.21 + $308.76)ir: ? 39. $600.09— $276.25 + $5682— $654= ? 40. $1032.07 + $68. 05 + $.98— $1000=? 41. 476281—12672—8720 + 20000=? REVIEW. ORAL EXAMrZES. 94. 1. The sum of two numbers is 46, and one r* them is 18 ; what is the other ? 2. The difference of two numbers is 16, and the greatei is 32 ; what' is the less? 3. The difference of two numbers is 24, and the less is 26 ; what is the greater ? 32 SUBTRACTION^. 4. A boy haying 28 peaches gave 8 to his brother, 7 to his sister, and lost 4 ; how many had he left ? 5. If a lady buy some thread for 10 cents, some needles for 5 cents, and some ribbon for 20 cents, and give the clerk 50 cents, how much change should he return? 6. In a garden are 47 fruit trees ; 15 of them are peach irees, 12 plum trees, and the remainder pear trees. How many pear trees are there ? 7. A lady having 3 ten-dollar bills and 1 five-dollar bill, bought a bonnet for $11, a pair of gaiters for $7, a.nd a scarf for $3. How much money had she left ? 8. A man died at the age of 64 years, having been married 36 years. What was his age when he married ? 9. In a public school there are 75 pupils, and 47 of them are girls ; how many of them are boys ? 10. A man sold 25 sheep, then bought 12, and then had 20. How many had he at first ? 11. A merchant gave $52 for a box of goods, and paid $5 freight; for how much must he sell them to gain $15 ? 12. A man gave his watch and $10 in money for a har- ness valued at $75. How much did he get for his watch ? 13. A man having received $45 for labor, paid $15 for a coat, $7 for a barrel of flour, and $6 for a ton of coal. How much had he left? 14. A man bought a vest for $7, a pair of pants for $12, and three shirts for $9, and gave in payment 3 ten* dollar bills. How much change should he receive ? Find the required term in the following equations : 15. 42— (10 + 12)=? 16. 9 + 16=:30— ? 17. 36-141=154- ? 18. 36—8 + 9 + 12=? 19. 7 + i6_8=:22— ? 20. 14 + 28—16 — 9=? REVIEW. 33 WJtITTEN EXAMPLES. 95, 1. The subtrahend is 260346, and the remainder 72304. What is the minuend ? 2. The difference is $310.62, and the minuend $1206.28. What is the subtrahend ? 3. What is the sum of 4062 and 12356 increased by the difference between 15000 and 975 ? 4. From the sum of 23462 and 9030, subtract the dif- ference between 34000 and 7640. 5. From the difference between 19876 and 6032, sub- tract the difference between 12000 and 673. 6. From what sum must $.62 be taken to leave a remainder of $14.60? 7. There were 67374 miles of railway in the United States in 1872, and 71564 miles in 1873. How much Was the gain in one year ? 8. A man has $10000. How much must he add to this, to be able to pay for a farm worth $13640 ? 9. California contains 158933 square miles, and Texas 237321 square miles. How much larger is Texas than California ? 10. Mt. Blanc is 15572 feet high, and Pike's Peak 12000 feet. What is the difference in their height ? 11. A man willed $125000 to his wife and two children. To his son he gave $44675, to his daughter $26380, and the remainder to his wife. What was his wife's share ? 12. A merchant of Nashville goes to New Orleans with $21600. He invests $7638.50 in groceries, $3210.65 in crockery, $1245.18 in wooden ware, and the remainder in hardwarg. How much does he invest in hardware ? 34 SUBTKACTIOK. 13. The population of London in 1870 was 3250000 5 of New York, 9M292 ; and of Brooklyn, 396099. How much greater was the population of London than of New York and Brooklyn ? 14. A man owns property valued at $75860, of which $45640 is invested in real estate, $25175.75 in personal property, and the remainder he has in bank. How much has he in bank ? x5. Three persons bought a hotel valued at $42075. The first agreed to pay $8375.50, the second agreed to pay twice as much, and the third the remainder. How much was the third to pay ? 16. A had $725.40, B had $180.36 more than A, and had as much as A and B together minus $214. How much had ? 17. 376 + 1684+573 — (931 + 1000)=? 18. $27.62 + $30.50--$14.00— $7.62=? 19. 17300 + 6840— (5800 + 1386) =25300— ? 20. (48036— 7690)— (3600 + 1873)=18321+ ? o H b w 96. SYNOPSIS FOR EEVIEW. " 1. Subtraction. 2. Difference, or Remainder. 3. Minuend. r 1. Definitions. i 4. Subtrahend. 5. Sign of Subtraction. 6. A Parenthesis, or Vinculum. 2. Principles, 1 and 2. 3. Subtraction of Dollars and Cents. 4 Rule, I, II, III 5. Proof. 1. How the numbers should be written. 2. If one number contains cents, and the other does not O HA L EXERCISES. 97. 1. If a man earns $3 a day, how many times $3 does he earn in 4 days ? $3 + $3 + $3 + $3 are how many ? 2. There are 7 days in 1 week. How many days are there in 3 weeks ? How many are three 7's, or 3 times 7 ? 3. There are 4 pecks in 1 bushel. How many pecks in 4 bushels. Four 4's, or 4 times 4 are how many ? 4. Is the result the same whether we say 4 times 6, or 6 times 4 ? 5. What is the difference between six 5's and five 6's ? 6. How many are three 8's ? Eight 3's ? 7. Add by 2's from to 24. 8. Multiply from times 2, to 12 times 2. Operation.— times 2 is 0, once 3 is 2, twice 2 are 4, 3 times 2 are 6, 4 times 2 are 8, 5 times 2 are 10, and so on. 9. Subtract by 2's back from 24 to 0. 10. Multiply back from 12 times 2 to times 2. Operation.— 12 times 2 are 24, 11 times 2 are 22, 10 times 2 ani ^\ 9 times 2 are 18, 8 times 2 are 16, and so on. 11. Multiply from times 3 to 12 times 3, and back. 12. Multiply from times 4 to 12 times 4, and reverse. 13. Multiply from times 5 to 12 times 5, and reverse. 14. Multiply from times 6 to 12 times 6, and reverse. MULTIPLICATION. MuLTiPLiCATioiir Table. ll 2 3 4! 5 6 7 1 8 9 lO 11 1 12 2 4| 6| 8|10|13|14|16 18 30 33 1 34 3 6 9 13 15|l8 31 34 37 30 33 36 4 8 13 16 30 34 38 33 36 40 44 1 48 5 10 15 30 35 30 35 40 45 60 55 60 6 8 13 18|34 30 36 43 48 54 60 66 73 14 31 38 1 35 i 43 49 56 63 70 77 84 16 34 33 40 48 56 64 73 80 88 96 9 18 37 36 45 54 63 73 81 90 99 1 108 10 30 30 40 50 60 70 80 90 100 110 130 11 33 33 1 44 1 55 66 77 88 99 110 131 i 133 12 34 36 1 48 1 60 1 73 i 84 1 96 108 130 133 i 144 DEFINITIONS. 98. Multiplication is the process of taking one of two numbers as many times as there are units in the other. Or, it is a short method of adding equal numbers. 99. The Multiplicand is the number to be mul- tiplied. 100. The Multiplier is the number by which to multiply. It shows how many times the multiplicand is to be taken. 101. The Product is the result obtained by the mul- tiplication. The multiplicand and multiplier are called the factors of the product. MULTIPLICATION. 37 103. The Sign of Multiplication is x . It is read times, or multiplied iy. When placed between two numbers, it shows that they are to be multiplied together. Thus 9 x 7 is read 9 multiplied 6y 7, or 7 times 9. Since changing the order of the factors does not change the re- sult, 9x7 may be read, 7 times 9, or 9 times 7. 103. Pkikciples. — 1. The multiplier is dlimys re- garded as an abstract number. 2. TJie 7mdtipUcand and product are like numbers, and may be either concrete or abstract. In examples containing concrete numbers, the concrete number is the true multiplicandy but when it is the smaller, it is often, for convenience, used abstractly as the multiplier. OBAL EXERCISES, 104. 8x4=? 10 X Oziz? 11 X 8=? 7x7=r? 9x 5 = ? 9x12=? 6x9 = ? 7x11==? 11x10=? 0x5 = ? 12x 6 = ? 10x12=? 10x8=? 9x 7=? 9x11 = ? 8x 8 + 10=? 10x10-14=? 15 X 2 + 15=? 9x 4—10=? 7x12 + 16=? 11x11— 9=? 12x 6 + 15=? 8x 0- 7=? 12x + 25=? 10x12—25=? 0xl2x 8=? 10x12—16=? 8x11 — 12=? 1x12+ 8=? 12x11 — 12=r 9x 9 + 19=? 12x10—30=? 12x12+ 6=? 1. At 7 cents each, what is the. cost of 5 pencils ? Analysis. — Since 1 pencil costs 7 cents, 5 pencils will cost 5 times 7 cents, or 35 cents. 38 MULTIPLICATIOK. 2. What is the cost of 4 tons of coal, at $8 a ton ? 3. What is the cost of 5 hats, at $5 a piece ? 4. At 9 cents each, what will 3 melons cost ? 5. What will 5 yards of gimp cost, at 11 cents a yard? 6. 12 inches make a foot. How many inches in 4 feet? 7. At $4 a cord, what will 9 cords of wood cost ? 8. Multiply from times 6 to 12 times 6, and reverse • 9. Multiply from times 7 to 12 times 7, and reverse. 10. Multiply from times 8 to 12 times 8, and reyerse. 11. Multiply from times 9 to 12 times 9, and reverse. 12. What cost 6 pairs of boots at $8 a pair? At $9 ? 13. At 8 cents each, what cost 9 books ? 10 books? 14. What cost 9 barrels of flour at $9 a barrel ? At $10 ? 15. 7 days make a week. How many days in 7 weeks ? 16. If a man earn $12 in 1 week, how much will he earn in 8 weeks ? In 9 weeks ? 17. Multiply from times 10 to 12 times 10, and reverse, 18. Multiply from times 11 to 12 times 11, and reverse. 19. Multiply from times 12 to 12 times 12, and reverse. 20. At 12 cents a yard, what cost 9 yards of calico ? 21. What cost 10 pounds of ginger,at 11 cents a pound ? 22. At $11 a hundred, what will 11 hundred posts cost? 23. How many bushels of grain can be put in 8 bins, each containing 12 bushels ? 24. How many are 8 times $4, minus $7 ? 25. How many are 7 times 9 pounds, plus 10 pounds ? 26. How many are 6 times 12 rods, less 20 rods ? 27. James gave 5 cents each for 6 oranges. How much change should he receive from 50 cents ? 28. How much more than $35 will 7 tons of coal cost, at $6 a ton ? MULTIPLICATIOI^. 39 WniTTEN EXEHCISBS , 105. When the multiplier consists of but one order of units. 1. How many are 4 times 73 ? l«rr OPERATION. 73 73 73 73 Analysis. — To obtain tlie result by Addition. First find the sum of four 3's, or 4 times 3 units, which is 12 units, equal to 1 ten and 2 units. Write the 2 units in the units' place, and reserve the 1 ten to be added to the sum of the tens. Next, the sum of four 7's, or 4 times 7 tens, is 28 tens, and 28 tens plus 1 ten reserved are 29 tens, or 2 hundreds and 9 tens, which write in the Sum 2 9 2 hundreds' and tens' place. Hence the sum is 292. 2d operation Multiplicand 7 3 Multiplier 4 Product 292 Analysis. — In this operation, the multi- plicand 73 is written but once ; and as it is to be taken 4 times, write the multiplier 4 under it, and commence at the right to multiply. 4 times 3 units are 12 units, or 1 ten and 2 units. Write the 2 units in units' place and reserve the 1 ten to add to the product of the tens. Next, 4 times 7 tens are 28 tens, and 28 tens plus 1 ten reserved are 29 tens, or 2 hundreds and 9 tens, which write in the hundreds' and tens' places. Hence the product is 292, equal to the sum in the first operation. Solve by both methods, 2. 3 times 84. 4. 5 times 234. 6. 4 times $204. 3. 4 times 135. 5. 6 times 352. 7. 5 times $425. 8. Multiply 4621 by 4 ; by 5 ; by 6 ; by 7. 9. Multiply 3062 by 6 ; by 7 ; by 8 ; by 9. What is the product 10. Of $5642 by 6? by 5? by 7 ? by 9? 11. Of 20372 feet by 7 ? by 9? by 5? by 6? 40 MULTIPLICATION. 12. What cost 527 barrels of flour, at $9 a barrel ? Although $9 is the true multiplicand, for convenience, we may use 9 for the multiplier, and 527 as the multiplicand (103, Note), but the product is dollars, since the true multiplicand is dollars. This is obvious, since 527 barrels at $1 a barrel, would cost $527, and at $9 a barrel, 9 times $527, etc. 13. What cost 326 tons of coal, at $6 a ton ? 14. What cost 1238 cords of wood, at $5 a cord? 15. What cost 752 pounds of nails, at 7 cents a pound? Operation.— 7 cents x 752 = 5264 cents = $52.64. When either factor contains cents, the product is cents, and may be changed to dollars and cents by putting the point ( . ) two places from the right, and prefixing the sign (8). (16.) (17.) (18.) (19.) Multiply $43.72 $136.04 87 cents. $2.06 By 8 7 9 6 Product $349.76 $952.28 $7.83 $12.36 20. At 6 cents a pound, what cost 675 pounds of rice ? 21. At $4.37 a yard, what is the cost of 7 yards of cloth? 22. At $124.50 an acre, what will 5 acres of land cost? 23. What is the cost of 8 building lots, at $2015 each I oitAJL EX euci s t:s. 106. 1. 9 times $12 are $108. Which number is the Multiphcand ? The Multipher ? The Product ? 2. If 6 men can build a wall in 7 days, in how many days can 1 man build it ? Analysis. — It will take 1 man 6 times as many days as it will 8 men, to build the wall; and 6 tim^s 7 days are 42 days. Eence it will take 1 man 42 days. MULTIPLICATION". 41 3. If 7 men can do a piece of work in 10 days, how many days will it take 1 man to do the same work ? 4. How many horses will consume as many bushels of oats in one day as 7 horses will consume in 5 days ? 5. If 3 barrels of flour last 9 persons 4 months, how long will the same quantity of flour last 1 person? 6. If a man earns $18 a week, and spends 19 for board and other expenses, how much will he save in 8 weeks ? 7. If Henry earn $5 a week, and James $4, how much will both earn in 7 weeks ? 8. What is the difference in the cost of 6 yards of rib- bon at 9 cents a yard, and of G yards at 11 cents a yard? 9. What will be the cost of 6 cows at $26 each ? Analysis. — Six cows will cost 6 times $26. 6 times 6 unita are 36 units, or 3 tens and 6 units, and 6 times 2 tens are 12 tens, which plus 3 tens and 6 units, are 15 tens and 6 units, or 156 Hence 6 cows will cost $156. 10. What cost 7 pounds of figs, at 23 cents a pound? 11. What cost 8 pounds of coffee, at 42 cents a pound ? 12. At $36 a ton, what will 6 tons of guano cost ? 13. At $18 a barrel, what will 9 barrels of pork cost ? 14. At $5 a barrel, what are 33 barrels of apples worth. 15. At 17 a week, what is the cost of 21 weeks board ? 16. What cost 20 pouuds of beef, at 12 cents a pound. 17. Two men start from the same place, and travel in opposite directions, one at the rate of 6 miles an hour, the other, of 8 miles an hour. How far apart will they bo at the end of 6 hours ? 8 hours ? 9 hours ? 18. A woman sold a grocer 5 pounds of butter at 30 cents a pound, and received in payment 12 pounds of sugar at 9 cents a pound. How much was still due her ? 4» MULTIPLICATION. Find the second member of the following equations : ^ 19. 20 + 12—3x6=1? 20. 16-^7 + 4x0==? 21. 7x12 — 6x11=? 22. 60-(0xl2) + 15=:? ^.3. 20x3 + (40-7x5)z=: 24. 3x0 + 4x7=:? 25. (55-7)— 20^1= ? 26. (7 + 5)-(6 + 4)=? 27. 14x0 + 45 — 15=1? 28. 100—12x7 + 20-4: WRITTEN JEXEBCISES. lOT. When the multiplier consists of two or more orders of units. 1. Multiply 678 by 46. OPERATION. 678 46 Multiplicand Multiplier 1st Partial Prod. 4 6 8 2d Partial Prod. 2 712 678 X 6 6 7 8x40 Entire Prod. 31188 = 678x46 Analysis.— Write the numbers as before. Since 46 is composed of 6 units and 4 tens, 46 times any number is equal to 6 times the num- ber, plus 4 tens, or 40 times the number. 6 times 678 is 4068, the first partial product. 4 tens times 8 units are 32 teTis, or 3 hun- dreds and 2 tens. Write the 3 tens in the tens' place, in the second partial product, and reserve the 3 hundreds to add to the product of hundreds. 4 tens times 7 tens are 28 hundreds, and 28 hundreds plus 3 hun- dreds reserved, are 31 hundreds, or 3 thousands and 1 hundred. Write the 1 hundred in the hundreds' place in the second partial product, and reserve the 3 thousands to add to the product of thousands. 4 tens times 6 hundreds are 24 thousands, and 24 thousands plus B thousands reserved, are 27 thousands, or 2 tens of thousands and 7 thousands, which write in the second partial product. The sum of the partial products is the entire prodnct 31188. * The operations of multiplication and division, indicated by signs, must be performed before those of addition and subtraction, unless otherwise indicated by a parenthesis or vinculum. MULTIPLICATIOlSr. ^ In like manner, multiply 2. 473 by 27. 5. $36.45 by 34; by 47. 3. 738 by 35. 6. $70.65 by 55; by 64. 4. 609 by 56. 7. $29.07 by 76 ; by 83. KuLE. — I. Write the multiplier under the multiplicand^ w that units of the same order stand in the same column. When the multiplier consists of one figure. n. Begin at the right and multiply the units of each order of the multiplicand hy the multiplier. Write in the product the units of each result^ and reserve the tens to add to the next result. When the multiplier consists of more than one figure. III. Multiply the multiplicand hy the units of each order of the multiplier successively, beginning at the right, and write the right-hand figure of each partial product under the order of the multiplier used. The sum of the partial products is the required product. Proof. — Review the worTc carefully, or multiply tht multiplier ly the multiplicand ; if the results are the same, the worJc is prohdbly correct. When there are ciphers in the multiplier, multiply by the sig. uificant figures only, since the product of any number by is 0. 8. Multiply 6432 by 75 ; by 67 ; by 136. 9. Multiply 23072 by 128 ; by 243 ; by 307. 10. Multiply $420.06 by 204; by m^\ by 408. What is the value 11. Of 67 hogsheads of sugar, at 137.75 a hogshead? 12. Of 2347 acres of land, at $136 an acre ? 13. Of 64 horses, at $219.75 each ? 44 MULTIPLICATION. 14. What will be the cost of building a line of telegraph 274 miles long, at $967 a mile? 15. If 1049 pounds of seed cotton be raised from an acre of land, how many pounds will 386 acres produce ? 16. If a cotton mill manufactures 628 yards of cloth 1 a day, how many yards can it make in 297 days ? What is the product 17. Of 2572 bushels by 94 ? 18. Of $403. 06 by 127? 19. Of 86072 pounds by 208? 20. Of 316 times $487.46? 21. Of 507 times 30975 days? 22. Of 325 times 6408 cents? 29. Find the cost of 386 railway coaches; at $7034.75 each. 30. What cost 802 tubs of butter, at $27.08 each? 23. Of 370607 by 4071 ? 24. Of 600326 by 2645 ? 25. Of 730096 by 5006 ? 26. Of 2407068 by 3406? 27. Of 408091 by 2407 ? 28. Of 73069 by 46035 ? 31. 236x63x28=:? 32. 439x0x142=? 33. 1927x613x802=? 34. 4605x2034x576=? 35. How many yards of shirting in 49 bales, each bale containing 26 pieces, and each piece 57 yards? 36. What is the cost of 128 barrels of beef, each con- taining 216 pounds, worth 13 cents a pound ? 37. Three schooners, ship 239 cords of wood each, and a fourth ships 248 cords. What is the value of the whole at $4.25 a cord? 38. If it require 108 tons of iron rail for 1 mile of .rack, how many tons will be required for 476 miles, and vf hat will be its value at $145 a ton ? 39. A crop of cotton was put up in 472 bales, the average weight of which was 588 pounds. What was the weight of the whole crop, ^nd its value at 18 cents a pound? MULTIPLICATION. 45 108. To multiply by the factors of a number. The Factors of a number are the numbers which multiplied together will produce it. Thus, 6 and 7 are factors of 42 ; 2, 4, and 5 are factors of 40. The pupil should carefully distinguish between the factors and le parts of a number. The factors are multiplied, but the parts are added, to produce a number. A factor is always a part, but a part is not always a factor. Thus, 2 and 9, 3 and 6, 2, 3, and 3, are factors of 18 ; but the parts of 18 are 9 and 9, 10 and 8, 6 and 12, 7 and 11, etc. 109. Principle. — The product of any mimber of fac- tors will le the same in whatever order they are multiplied. 1. Multiply 468 by 36. OPERATION. 3 6 = G X 6, or 9 X 4, or 1 2 X 3. 468 468 468 468 36 6 9 12 2808 2808 4212 5616 1404 6 4 3 16848 16848 16848 16848 It will be observed that the multiplicand, multiplied by the given multiplier, or by any set of factors into which it can be sepa- rated, produces the same result. In like manner, multiply 3. $73.04by48=z8x6. 3. 50076 by 72=6x4x3. 4. 46502by 84r=:7x4x3. 5. $206.14 by 96=4x4x6. 6. $780.91 by 108. 7. 140086 by 120. 8. 380509 by 144. 9. $457.52 by 240. 46 MULTIPLICATION^. EuLE. — I. Separate the multiplier into ttvo or more factors. II. Multiply the multiplicand by one of the factors, the resultiiig product iy another factor, and so continue until ill the factors have been used. The last product loill be the product required, 10. What will 56 acres of land cost, at $164.50 an acre? 11. At 28 cents a pound, what will be the cost of 24 sacks of coffee, each containing 64 pounds ? 12. What is the value of 107 pieces of cloth, each piece containing 42 yards, at $4.28 a yard ? 110. When either the multiplicand or multi- plier, or both, have ciphers on the right. 1. Multiply 286 by 100. OPERATION. Analysis. — Since removing a figure one 'place to 2 8 6 the left, increases its value ten times (44), annex- \{\^ ing a cipher to a number multipUes it by 10 ; an- nexing two ciphers multiplies it by 100, etc. Hence 2 8 6 286 X 100=28600, the product required. 2. Multiply 3240 by 600. OPEKATION. Analysis.— 3240=334 X 10, and 600=6x100. 3 2 4 First multiply together the two factors 324 and 6, /> Q Q and then multiply their product 1944, by 10 x 100, or by 1000, by annexing three ciphers, which 19 4 4 gives 1944000, the required product. What is the product 3. Of 372 by 10 ? By 100 ? By 1000? By 10000? 4. Of 860 by 50 ? By 400 ? By 1500 ? By 3000 P EuLE. — To the product of the significant figures, annex as many ciphers as there are ciphers on the right of either or of both of the factors. REVIEW. 47 What is the product Find 5. Of $4.72 by 100 ? 8. 120 times 5000. 6. Of $30.40 by 60 ? 9. 600 times 21000. 7. Of $1300 by 700 ? 10. 1000 times 104000. 11. 42030090x3020=? 12. 7000600x50040=? 13. There are 640 acres in 1 square mile. How many acres in 150 square miles ? In 200 ? In 420 ? 14. The salary of the president is $50000 a year. How much does he receive in 8 years ? REVIEW. O BAL BXAMVIjBS, 111. 1. The sum of 8 + 12 + 16 equals the product of 9 X what number ? 2. The sum of 40 — 14 and 12 + 4 equals 7 x what number ? 3. The difference between 35 + 15 and 24—10 is equal to the product of what two factors ? Three factors ? 4. The product of what two factors is equal to the sum of 9, 20, and IX ? 5. The product of 8 times 9 is equal to 6 times what number ? 6. The sum of 25, 13, 8, and 10 is equal to the pro- duct of what three factors ? 7. What is the sum of 3 times 3x4, and 5 times 4 x 3't What is the difference ? 8. What is the product of 15 + 24—14 by 16-12 ? 9. Which is greater, 9 x 13—6, or 12 times 8—20 ? 10. How much less --s 60—5 x 8 than 16 + 14—10 ? 48 MULTIPLICATION. 11. Charles is twice as old as George, and George is 12 years old. What is the sum of their ages ? 12. What is the cost of 4 brooms at 30 cents each, an(J 6 pounds of sugar at 11 cents a pound ? 13. Mary had 18 cents, and Belle had 3 times as many ]3S3 9 cents. How many had both ? 14. A young man earned 19 a week, and spent $5 a week for board. How much did he save in 12 weeks ? 15. A woman sold a grocer 4 dozen of eggs at 24 cents a dozen, and received in payment half a pound of tea worth 50 cents, and 2 pounds of sugar at 11 cents a pound. How much was still due her ? 16. A boy bought a book for 36 cents, a slate for 20 cents, and a pencil for 4 cents. How much change should he receive from a 1 dollar bill ? 17. A lady bought 9 yards of silk at $3 a yard, 3 pairs of kid gloves at $2 a pair, 4 pairs of hose at half a dollar a pair. She gave in payment 4 ten dollar bills. How much change should she receive ? Find the required term in the following equations : 24. 7x12-25x0=? 25. 3x0x5 + 16=2x? 18. 19 — 7 + 28—11 = ? 19. 8x9 — 16 = 7x? 20. 21 + 6x7=40+? 21. 10x12 — 9x11==? 22. 75—5x12=35—? 83. 44 + 19-(50— 23) = ? 26. 28 + 12-6 X? =16 27. 9x12 + 10 = 120—? 28. 42—20 + 14 = ? x9 29. 8 + 55 — (? x8) = 7 113. By a little practice, numbers containing three oi four figures may be multiplied mentally, by first multi- plying the highest order of units, and adding the pro- duct of each lower order as found. REVIEW. 49 1. Multiply 324 by 2, Operation.— 2 times 3 hundreds are 600 ; 2 times 2 tens are 4 tens, or 40, and 600 + 40 are 640 ; 2 times 4 are 8, and 640 + 8 are 648. Omitting all but results, tlie required product will be easily and promptly obtained by a strictly mental process. Thus, 600, 640, 648. In like manner, find the product of 2. 3 times 330. 5. 4 times 425. 8. 234 X 2. 3. 3 times 342. 6. 6 times 241. 9. 501 X 3. 4. 4 times 150. 7. 5 times 615. 10. 255 X 4. WMITT EN EXAMPLES. 113. 1. If I receive S1500 salary, and pay $370 foi board, $281.50 for clothing, $112.75 for books, and $196.65 for other expenses annually, what can I save in 3 years ? 2. A merchant bought 7 hogsheads of sugar at $46.45 a hogshead, and sold it for $53.62 a hogshead. How much did he gain ? 3. Paid $2709 for 388 barrels of flour, and sold the same at $9.12 a barrel. How much was the gain ? 4. If a man have an income of $5670 a year and his daily expenses average $7.25, how much can he save in a year of 365 days ? 5. What number must be added to 272 x 400 to make the amount 126720 ? 6. What is the difference between 40706 — 308 x 56, and 97x340— 12400? 7. Multiply 98 + 6 x (37 + 50) by 64—50 x 5— 10. 8. Multiply 675-(77 + 56) by (3 x 155) -(214-28). 9. A man owing $15760, gave in payment 5 lots of land, worth $730 each, 5 horses, valued at $236.50 each, an interest he had in a coal mine worth $2000, and $1728.75 in money. How much remained unpaid ? 60 MULTIPLICATIO]S^. 10. A farm-house is worth 13246, the farm is worth 3 times as much plus $1200, and the stock is worth twice as much as the house, less $1875. What is the value of the whole, and of the farm and stock ? 11. What is the difference in the cost of 48 horses at $184.50 each, and of 130 sheep at $4.80 a head ? 12. Bought 150 barrels of flour for $1150^ and finding 25 barrels of it worthless, sold the remainder at $9 a bar- rel. Did I gain or lose, and how much ? Complete the following equations : 13. (142 + 405) X (1000 — 850) — 5000 c^ ? 14. (97 X 1000) — (75 x 500 — 420) + 1500 = ? 15. $73.46 — ($.94 + $3.02) + $47 x 35 =:^ ? 16. $246.08 X 104 + ($2000 — $240.50) x iO ~ ? 114. SYNOPSIS FOR EEVIEW. f 1. Multiplication. 2. Mu.Hiplic»nd. 1. Definitions. ^ 3. Multiplier. 4 Product. ^5, Sign (^ of Multiplication. 2. Pbinciples, 1 and 2. O < I— I fin H 3. Rule— I, II, III. 4. Proof. 5. When either factor contains cents. f 1. Definition of factors. 6. By Factors. ^ 2. Principle. 7. Multiplier and Multiplicand. 3. Rule. 1, When one or botli have ciphers on the right. 3. Rule. ORAZ ^XJEHCISES, 115. 1. How many 4's are 12 ? Are 16 ? Are 24 ? 2. How many lots, of 5 acres each, in 20 acres ? 3. How many 5's in 15 ? In 30 ? In 35 ? In 50 ? 4. How many barrels, each holding 3 bushels, will be required for 18 bushels of apples? 21 bushels ? 5. How many times can 6 yards of cloth be taken from a piece containing 30 yards. 6. How many times can 6 cents be taken from 23 cents, so as to have 5 cents remaining ? 7. Distribute $28 equally among 7 men. How many dollars will each receiye ? Do you find how many times 7 men are contained in $28, or do you find one of 7 equal parts of $28 ? 8. How do you find one of 8 equal parts of a number ? Of 9 equal parts ? Of 6 equal parts ? 9. What is one of 4 equal parts of 40 ? Of 36 ? Of 48 ? 10. What is one of 6 equal parts of 30 ? Of 42 ? Of 48 ? 11. What is one of 7 equal parts of 56 pounds ? 12. How many times 8 cents are 48 cents? Is the result a concrete or an abstract number ? 13. What is one of 8 equal parts of 48 cents ? Is the result a concrete or an abstract number ? 52 DIVISION. DEFINITIONS. 116. Division is the process of finding how many times one number is contained in another of the same kind, or of finding one of the equal parts of a number. 11*7. The Dividend is the number to be divided. 118. The Divisor is the number by which to divide. 119. The Quotient is the result of the division, and shows hoio many times the dividend contains the divisor. The division is said to be exact when there is no remainder. The part of the dividend remaining when the division is not exact is called the Bemainder, and must always be lesa than the divisor. 130. The Sign of Division is -^. It is read divided hy. It shows that the number before it is to be divided by the one after it ; thus 54 -t- 9 is read 54 dimded ly 9. 131. Division is also indicated by placing the dividend above the divisor with a line between them ; thus, ^ is read 72 divided hy 8. 133. Principles. — In finding how many times one number is contained in another : 1. The divisor and dividend are like numbers, and the quotient an abstract number. In finding one of the equal parts of a number: 2. The dividend and quotient are liJce numbers, and the divisor an abstract number. 3. The dividend is equal to the product of the divisor by the quotient, plus the remainder. DIVISION. 53 V£3. 36-^9=? 43 -=-7=? 40-;-5=? O R AZ, EXEnv is Ea 63^9=? 56^8=? 45-j-5=? 64-^8=? 66-^6= ? 72-r-9= ? --^ = ? ^ = ? ff = ? e = ? ft = ? ¥ = ? 1. Divide by 3, from 3 in 3 to 3 in 24. 84-T- 7=? 73-T-13= ? 96-^ 8=? w = ? Operation. — 2 in 2, once ; 2 in 4, twice ; 2 in 6, 3 times ; 2 in 8, i times ; 2 in 10, 5 times, and so on to 2 in 24, 12 times. In the same manner, divide 2. By 3, from 3 in 3, to 3 in 36. 3. By 4, from 4 in 4, to 4 in 48. 4. By 5, from 5 in 5, to 5 in 60. 5. By 6, from 6 in 6, to 6 in 72. 6. By 7, from 7 in 7, to 7 in 84. 7. By 8, from 8 in 8, to 8 in 96. 8. By 9, from 9 in 9, to 9 in 108. 9. By 10, from 10 in 10, to 10 in 120. The pupil may reverse the above ; thus, 2 in 24, 12 times ; 2 in 22, 11 times ; 2 in 20, 10 times, and so on. Also combine the two ; thus, 3 in 3, once ; 3 in 6, twice, 2 in 6, 3 times ; 3 in 12, 4 times, 4 in 12, 3 times ; and so on to 3 in 36, "•2 times, 12 in 36, 3 times, 134. Division may also be regarded as a short method if performing several subtractions of a number. Thus, 24 - 6 = 18 ; 18 - 6 = 12 ; 12 - 6 = 6 ; 6 - 6 = 0. We have performed four subtractions of 6, hence there are four 6's in 24, or 6 is contained in 24, 4 times. 54 DivisiON^. 135. Since one number is contained in another as many times as it is a factor of the other, division may be regarded as the reverse of multiplication. In Multiplication, loth factors are given to find the product; in Division, one factor and the product (an-^ swering to the dividend) are given to find the other factor^ which answers to the quotient Thus, 6 X 4 = 24, the factor 6 being taken 4 times ; hence there faefour 6's in 24, or 6 is contained in 24, 4 times. 136. The Object of Division is tioofoJd. First. To find how many times one number is contained in another of the same hind. Ex. At 5 cents each, how many pencils can be bought for 20 cents. Since 5 cents taken 4 times equals 20 cents (5 x 4== 20), it follows that 5 cents is contained in 20 cents 4 times. Analysis. — As many pencils can be bought for 20 cents, as 5 cents are contained times in 20 cents, which are 4 times. Hence, etc. 137. Second. To separate a given number into as many equal parts as there are units in another. Ex. If 4 pencils cost 20 cents, what is the cost of 1 pencil ? Since 5 cents taken 4 times equals 20 cents, it follows that 5 cents is one of the four equal parts of 20 cents (5 + 5 + 5 + 5=20), and we say one-fourth of 20 cents is 5 cents. Analysis. — Since 4 pencils cost 20 cents, 1 pencil costs one-fourth of 20 cents, which are 5 cents. 138. The equal parts into which a unit or whole thing I divided are called fractions, 139. The names of these equal parts of a unit vary according to the number of these parts ; thus^ one-half is one of two equal parts, one-third is one of three equal parts into which the whole thing or number is divided. DI VI SI OK. 55 So in like manner we ]mNQ fourths, fifths, sixths, sevenths, eighths, tenths, twelfths, twentieths, etc. 130. These jmrts are expressed by writing the number denoting the name of the parts below a short horizontal line as a divisor, and the number of parts taken or used, above the line as a dividend. Thus, ^, signifies 1 divided by 2, and is read, one-half f , signifies 2 divided by d, and is read two-thirds, -^, signifies 7 divided by 12, and is read, seven-twelfths, etc. ORAL EXEHC IS ES. 131, 1. If a number is separated into tiuo equal parts, what is each part called ? Ans. One-half of the number, written ^, 2. If $18 are equally divided between two poor families, how much does each receive? What part of the whole? 3. What is one-half of 12 ? Of 16 ? Of 20 ? Of 24 ? 4. If a number is separated into three equal parts, what is each part called ? One'third of the number, ^. 5. If 15 peaches are equally distributed among 3 boys, what part of the whole will each receive ? 6. What is one-third of U5? Of 21 days? Of 30 rods? 7. Divide an acre of land into four equal parts. What is one of the parts called ? One-foui^th of an acre, ^. 8. What are 2 of the parts called ? Two-fourths^ |. Three of the parts ? Three- fourths^ f . 9. If 48 marbles are given to 4 boys, to each an equal number, what part of the whole does 1 boy receive? Two boys ? Three boys ? How many marbles ? 10. What is one-fourth of 24? Of 48 miles ? 56 DIVISION. 11. If a number is divided into Jive equal parts, what is each part called ? One- fifth of the number, \. Two parts ? Two-fifths, |. 12. If $20 are paid for 5 barrels of apples, what part of $20 is paid for 1 barrel ? . For 2 barrels ? For 3 barrels ? 13. What is ^ of 30 ? Of $40 ? Of 45 rods ? 14. If a number is divided into six equal parts, what is each part called ? One-sixth of the number, -J-. 15. If into seven equal parts ? One-seventh, ^. 16. If into eight equal parts ? One-eighth^ \. 17. If into nine equal parts? One-ninth, ^. 18. If into ten equal parts ? One-tenth, yV 19. If into twelve equal parts? One-twelfth, ^^, 20. Find one-half of 2, one-half of 4, one-half of 6, one-half of 8, and so on to one-half of 20. 21. Find one-third of 3, one-third of 6, one-third of 9, one-third of 12, and so on to one-third of 30. 22. Find i of 4, J of 8, J of 12, J of 16, to J of 40. 23. Find i of 5, ^ of 10, | of 15, | of 20, to | of 50. 24. Find i of 6, -i- of 12, ^ of 18, i of 24, to ^ of 60. 25. Find -ij- of 7, j of 14, \ of 21, { of 28, to | of 70. 26. Find i of 8, | of 16, i of 24, i of 32, to ^ of 80. 27. Find i of 9, ^ of 18, i of 27, i of 36, to | of 90. 28. Find -^ of 10, j\ of 20, ^ of 30, to ^^o of 100. 29. How do you find |, i, J, ^, i, etc., of any number? 30. How many yards of cloth, at $4 a yard, can be bought for $36 ? Analysis. — As many yards as $4 are contained times in $36, \;rliicli are 9 times. Hence 9 yards can be bought for $36. 31. At $6 a ton, how many tons of coal can be bought for $24? For $30? For $54? For $72? ..DIVERSITY B 32. If 7 cords of wood cost $42, what does 1 cord cost. Analysis. — Since 7 cords of wood cost $42, 1 cord costs 1 seventh of $42, or $6. Hence 1 cord costs $6. 33. A man sold 8 bushels of cranberries for $32. How much did he receive a bushel for them ? 34. A farmer gathered 108 bushels of apples from 9 treea What was the average number of bushels to each tree. 35. A merchant paid $96 for 8 pieces of dress goods. What was the cost of each piece ? 36. If a farm of 120 acres is divided into 12 equal lots, how many acres does each lot contain ? WBITTIjN jsxehcishjs. 133. When the divisor consists of but one order of units. 1. Divide 875 by 7. ,_ ^ ^ Analysis. — Write the divisor at the OPERATION. , ,. ^ , ,. ., _. . _. ., , ^ . , left of the dividend with a line between Divisor. Dividend. Quotient. 7)875(135 *''^™- , . . . OT, . 1 1 , ^ ^ 7 IS contained m 8 hundreds, 1 hun- i dred times, with a remainder. Write 1 7 the 1 hundred at the right of the divi- ■^ ^ dend, for the first figrure of the quotient. Multiply the divisor 7 by the 1 hundred ^ ^ of the quotient, and write the product, 3 ^ 7 hundreds, under the hundreds of the dividend. Subtract, and to the remain- der 1 hundred, annex the 7 tens of the dividend, making 17 tens. 7 is contained in 17 tens, 2 tens times, with a remainder. Write tie 2 tens in the quotient. Multiply the divisor 7 by the 2 tens, and subtract the product from the partial dividend, 17 tens. To the remainder. 3 tens, annex the 5 units of the dividend, making 35 units. 7 is cohtained in 35 units, 5 times, which write in the quotient. Multiplying and subtracting as before, nothing remains. Hence, etc. 58 DIVISION. The solution of the preceding example may be abbre- viated by what is termed Short Division, as follows : Analysis. — 7 is contained in 8, once, and OPERATION. \ remainder. 1 prefixed to 7 makes 17. 7 is 7)875 contained in 17, 2 times and 3 renfainder. 3 Ouotient 12 5 Prefixed to 5 makes 35, and 7 is contained iu 35, 5 times. Hence the quotient is 135. 133. In Short Division only the quotient is writ- ten, the operations being performed me7itally. It is generally used when the divisor does not exceed 12. In like manner, divide and analyze the following : (2.) (3.) (4.) (5.) 6 ) 7944 7 )9464 8 ) 8928 5 ) 6895 6. Divide 92352 by 8 ; by 6 ; by 4. 7. Divide 83762 by 7 ; 79880 by 6 ; 3263 by 8. Analysis. — Since 8 is not contained in 3 OPERATION. thousands, unite the 3 thousands and 3 hun- 8)3263 dreds, making 32 hundreds. 8 is contained Ouotient 4 7'*^ in 32 hundreds, 4 hundreds times, which write in the hundreds' place in the quotient. Next, 8 is not contained in 6 tens, so write a cipher in tens' place in the quotient, and unite the 6 tens and 3 units. 8 is contained in 63 units 7 times and 7 units remainder, which write over the divi- sor and add as a part of the quotient. Hence the quotient is 407|. Proof. — Multiply the quotient 407 by the divisor 8, and the product is 3356; 3256 plus the remainder 7, equals the dividend 3263. (Prin. 3.) 8. Divide 8135464 by 6 ; by 8 ; by 7 ; by 5 ; by 9. 9. Divide $48.56 by 8 cents. Eight cents may be written $ 08 (73). OPERATION. When the divisor and dividend are i^, 08) $4 8.56 like numbers, the quotient is an abstract 6 7 times number (Prin. 1). Hence 8 cents are contained in $48.56, 607 times. DIVISION. 59 10. Divide $48.56 by 8. OPERATION. When the divisor is an abstract number, the 8 ) $ 4 8 . 5 6 dividend and quotient are like numbers (Prin. 2). $6.0 7 Hence 1 eighth of $48.56 is $6.07. Solve and prove, (11.) (12.) (13.) (14.) 9 ) $217.62 7 ) $6.44 $7 ) $644 $.07 ) $6.44 $24.18 $.92 92 times. 92 times. How many times 15. Are $8 contained in $15096 ? In $58424 ? In $23064? 16. Is 7 contained in 330457 ? In 19278 ? In 918271? 17. Is 9 contained in 436281 ? In 605675 ? In 1039126? Find 18. 1^/tt of $863.25. 19. 1 sixth of 34807 tons. 20. 1 ei^A^A of 20673 days. 21. 1 ninth of $7384.50. What is 22. I of 500322 miles? 23. i of 32876 men? 24. ^ of 60349 acres ? 25. ^ of 760344 rods? 26. How many barrels of flour at $8 a barrel, can be bought for $12736 ? For $7068 ? 27. If 75000 bushels of grain are put into 8 bins of equal size, how many bushels does each bin contain ? 28. If 9 acres of land cost $976.50, what is the cost of 1 acre? 29. How many oranges can be bought for $3.72, at 4 cents a piece ? 30. At 8 cents a yard, how many yards of ribbon can be bought for $7. 28? 31. Paid $1792 for 7 horses. What did each cost ? 60 DIVISIOl^. ORAL EXEHC IS BS 134. 1. The quotient of two numbers is 15, and the divisor is 8. What is the dividend ? 2. The dividend is 96, and the quotient is 6. What is the divisor ? 3. The quotient is 12, the remainder is 9, and the divisor is 11. What is the dividend ? 4. If 12 yards of cloth cost $35, for how much a yard must it be sold to gain $13 ? 5. A man received $50 for 5 barrels of pears, and paid all but $14 for 4 chairs. What did each chair cost ? 6. If 4 weeks' board cost $28, what will 9 weeks' board cost ? Analysis. — One week's board will cost 1 fourth of $28, or |7 ; and 9 weeks' board will cost 9 times $7, or $63. 7. If 8 yards of silk cost $32, what will 12 yards cost ? 8. What will 15 sheep cost, if 5 sheep cost $35 ? 9. How many cords of wood at 4 dollars a cord, will pay for 6 barrels of flour at $8 a barrel ? Analysis.— Six barrels of flour will cost 6 times $8, or $48 ; and $4, the price of 1 cord of wood, are contained in $48,12 times. Hence,etc. 10. How many days' labor at $4 a day will pay for 3 tons of coal at $6 a ton, and 2 tons of hay at $15 a ton ? 11. How many pounds of meat at 12 cents a pound, will cost as much as 9 pounds of cheese at 8 cents a pound ? Complete the following equations : 12. 8x0 + 6x4—8=? 13. 10x12—0x6-^6=? 14. 9x11—54-^6 + 20=? 15. 63-^7xO + 12=? 16. (108^12) X 11— 25= ? 17. 90— 18-~(44-7x6)=? DIVISION. 61 WB ITT JEN EXERCISES. 135* When the divisor consists of more than one order of units. 1. Divide 5437 by 26. OPERATION. Analysis. — 26 is contained in 54 iH^isor. Dividend. Quotient, hundreds, 2 hundred times, with a re- 26)5437(209-^ mainder. Write the 2 hundreds in 5 2 the quotient, and multiply the divisor ^ 26 by this quotient figure, and subtract the product, 52 hundreds, from 54 ^ ^ "* hundreds, the first partial dividend, 3 Remainder, and there remains 2 hundreds. To this annex the 3 tens of the dividend, making 23 tens for the second partial dividend. 26 is not contained in 23, so write a cipher in the quotient and bring dowm the 7 units of the dividend, making 237 units for the third partial dividend. 26 is contained in 237 units 9 times, with a remainder. Write the 9 units in the quotient, and multiplying and subtracting as before, there remain 3 units, which write over the divisor, and annex as a part of the quotient. Hence the quotient is 209/^. 136. Ijong Division is the process of dividing when the subtractions are written, 2. Find how many times 204 is contained in 1041835. OPERATION. PROOF. Divisor. Dividend. Quotient. 204)1041835(5107^^ 5107 Qaotient. 10 2 2 4 Divisor. 218 20428 204 10214 -1435 1041828 14 2 8 7 Remainder. 7 Remamder. 10 418 3 5 Dividend. 62 DIVISION. 3. Divide 32762 by 14; by 16 ; by 23 ; by 28. 4 Divide 130426 by 58 ; by 63 ; by 81 ; by 74. Rule. — I. Write the divisor at the left of the dividend^ with a line between them. II. Find how many times the divisor is contained, in the least numier of the left hand orders of the dividend that will contain it, and write the result for the first figure of the quotient. III. Multiply the divisor ly this quotient figure^ sub- tract the product from the partial dividend used, and to the remainder annex the figure of the next lower order of the dividend for a new 'partial dividend, and divide as before. IV. Proceed in the same manner until all the orders of the dividend have been used. V. If any partial dividend does not contain the divisor j tvrite a cipher in the quotient, and annex the next order of the dividend, and proceed as before. VI. If there be at last a remainder, write it after the quotient ivith the divisor underneath. Proof. — Multiply the divisor by the quotient, and to the product add the remainder, if any. If the work is correct, the result toill be equal to the dividend. 1. If the product of the divisor and quotient be greater than the partial dividend, the quotient is too large^ and must be diminished. 2. If any remainder is equal to or greater than the divisor, the quotient is too small and must be increased, 137. When the divisor and dividend are both concrete numbers, they must be of the same narne. Hence, if one be dollars, and the other cents, or dollars and cents, before dividing, change so that both may be cents. DIVISIOIT. 63 138. Since 100 cents make 1 dollar, there are 100 times as many cents as dollars. Hence, To change a number representing dollars to a number represent- ing cents, annex two ciphers (110), omit tbe sign ($) and write the word cents after it. To change dollars and cents to the same form, omit the sign ($) and the point ( . ) and write the word cents at the right. 5. Divide $46.92 by 23. 6. Divide $46.92 by 23 cents. OPERATION. OPERATION. 23)$46. 92(12.0 4 2 3)4692(204 times. 46 46 92 92 92 92 7. Divide $46.92 by $23. 8. Divide $46 by 23 cents. OPERATION. 23)4600(200times, 46 OPERATION. 23 0) 46 9 2 (2^^ times. 4600 92 00 In like manner divide, and prove the following : 9. $325.72 by 34. 14. 10. $938.07 by 63. 15. 11. $3176.46 by 126. 16. 12. $49.56 by 14 cents. 17. 13. $87.36 by 21 cents. 18. How many times 19. Is 47 contained in 30176 ? In 27865? In 103474? 20. Is 185 contained in 200376 ? In 4701625 ? 21. The annual receipts of a company are $570685o What is the average a day, if there are 313 working days? $288.96 by $.43. $810.98 by $.46. $594 by 18 cents. $1385 by $105. J.48 by $7. 64 DIVISION. 22. If 867 shares of railroad stock are valued at $84099^ what is the value of each share ? 23. A plantation of 736 acres was sold for $55936 What was the price of an acre ? 24. Paid $17100 for a farm, at the rate of $36 an acrCo How many acres did it contain ? 25. How many horses, at $125 each, will $4735 buy, and how much money will be left ? Divide Di vide 26. 33490 by 85. 34. 863256 by 736. 27. 740070 by 135. 35. 1646301 by 381. 28. 1554768 by 216. 36. 5226412 by 2567. 29. 5497800 by 175. 37. 11214887 by 3076, 30. 3931476 by 556. 38. 75862500 by 10115. 31. 5120401 by 587. 39. 313194105 by 7153. 32. 1018090 by 1669. 40. 1246038849 by 269181. 33. 73484248 by 2624. 41. 2331883954 by 6739549 139. To divide by the factors of a number. 1. Divide 644 by 28, using the factors. OPERATION. 4)644 7 )161 23 Analysis. — Since 28 is equal to 4 times 7, divide either by 28, or by its factors 4 and 7. Now, 644 -f- 4=161 ; but this quotient is 7 times too great, and must therefore be divided by 7 ; hence, 161-^7=^23 the true quotient. Factors. 6228 by 36, or by 4, and 9. li. Divide 3. Divide 27360 by 96, or by 3, 4, and 8. 4. Divide 526050 by 126, or by 2, 7, and 9, 5. Divide 73416 by 168, or by 4, 6, and 7. DIVISION. 65 6. Divide 5831 by 84, using the factors, 3, 4, and 7. OPERATION. Analysis. — Since 84 is 3)5831 equal to 3x4x7, divide by A\IQ 4 3 2 ^^' ^^ ^^ ^^^ factors 3, i, I and 7. 7 )48 5 o . .3x3= 9 5831 -j- 3 = 1943, and a re- fiQ 2v4-y3 '2 ^ mainder of 2, which being a — part of the dividend, is also True Remainder. 3 5 a part of the true remainder. 6 9 M Quotient. 1^4^ ^^ = 4^^> ^^^ ^ ^^- mainder of 3. Since a unit of the first quotient 1943, equals 3 units of the dividend, this second remainder 3 being a part of 1943, equals 3 x 3, or 9 units of the dividend. 485 -f- 7 ==: 09, and a remainder of 2. Since a unit of the second quotient 485 equals 4 units of the first quotient 1943, this third re- mainder 2 being a part of 485, equals 2 x 4 x 3, or 24 units of the dividend. Hence the first partial remainder is 2, the second is 9, the third is 24, and the true remainder 35 ; and the quotient 69 |f. 7. Divide 139074 by 72/ using its factors 3, 4, and 6. 8. Divide 7360479 by 96, using its factors 2, 6, and 8. KuLE. — I. Separate the divisor into two or more factors, II. Divide the dividend hy one of these factors, and the quotient thus obtained hy another factor, and so on U7itil all the factors have teen used as divisors. III. If there he remainders, multiply each remainder hy all the divisors preceding the one that produced it. IV. Add the products and the remainder from the first iivision, if any, and the sum tuill he the true remainder. 9. Divide 1315125 by 315, or by 5, 7, and 9. 10. Divide 73522 by 135, or by 3. 5, and 9. 11. Divide 401976 by 245, or by 5, 7, and 7. 66 DIVISIO]S^, 140. When the divisor has ciphers on the right. 1. Divide 4067 by 10. OPERATION. 1|0 )4Q6|7 4 6 . . 7 Rem. 4 6 jV Quotient. Analysis. — Sioce removing any order of figures one place to tlie right, dimia islies its value ten times (45), by cutting off, or taking away, the right-hand fig- ure of a number, each of the remaining figures, being removed one place to the right, is diminished in value ten times, or divided hy 10. For similar reasons, cutting off two figures divides by 100, cutting off three figures, divides by 1000, and so on. The remaining figures are the quotient, and those cut off, the remainder. Divide 2. 37684 by 100. 3. 103076 by 1000. 6. Divide 2416700 by 6000. 4. 267104 by 10000. 5. 5023001 by 100000. Analysis.— Resolve 6000 into the factors 1000, and 6. First divide by 1000, by cut- ting off the three right-hand figures of the dividend. The quotient is 2416, and a re- mainder of 700. Next divide 2416 by 6 ; the quotient is 402 and a second remainder of 4 thousands, which prefixed to the first re- mainder 700 gives a true remainder of 4700. Hence the quotient is ^U-^sTFTFF- OPERATION. 6 [ )2416 | 7 402 . . 4700 Rem. 40 2|-^fg- Quotient. In like manner, divide 7. 307200 by 900. 8. 7820305 by 28000. 9. 5761321 by 2040. 10. 8073160 by 14800. EuLE. — I. Cut off the ciphers from the right of the divisor, and as many figures from the right of^ the dividend, II. Divide the re7naining part of the dividend hy the remaining part of the divisor. DIVISION. 67 III. Prefix the remainder, if any, to the figures cut off, and the result will be the true remainder. 11. If it require $34400 to pay a regiment of 800 men, how much does each man receive ? 12. At $3400, how many lots can be bought for $68000 ? 13. How many bales, each weighing 470 pounds, can m made of 39500 pounds of cotton ? GENERAL PRINCIPLES OF DIVISION. 141. The quotient depends upon the relative values of the dividend and divisor. Hence, any change in the value of either dividend or divisor, will produce a change in the value of the quotient, But some changes may be made upon both dividend and divisor, which will not affect their relative values, and consequently will not affect the quotient. To illustrate, let 54-^-9 = 6, be the fundamental equation, with which the following are to be compared : 1. (54x3)-^9=:lG2~9 = 18. Multiplying tlie dividend by 3 multiplies the quotient by 3. 2. 54-i-(9_i-3)— 54-^3:^=18. Dividing the divisor by 3 mul- tiplies the quotient by 3. 3. (54-r-3)-^9rzil8-^9=:2. Dividing the dividend by 3 di- vides the quotient by 3. 4. 54-^(9 x3)=:54-^27=:2. Multiplying the divisor by 3 di- > vides the quotient by 3. ►?• (54x3)-^(9x3)=162-T- Multiplying both dividend and 27 = 6. divisor by 3 does not change the quotient. 6. (54-^3)-r-(9-^3)=iil8-~ Dividing both dividend and di- 3 = 6. visor by 3 does not change the quotient. ^8 DIVISIOK. These six equations illustrate the following 143. GexVEral Pmiq^ciPLES of Division. 1. Multiplying the dividend, or ) Multiplies the quo- Dividing the divisor, ) tient. 2. Dividing the dividend, or ) Divides the quo- Multiplying the divisor, ) tient. 3. Multiplyiyiq or dividing both \ ^ , , .. ./ / 7 7 7 ^7 [Does not chanm dividend and divisor by they,^ . ^ , \ the quotient, same number, ) These three principles may be embraced in one GENERAL LAW. 143. A change in the dividend produces a like change in the quotient, but a change in the divisor produces an OPPOSITE change in the quotient, GENERAL REVIEW. OTtAL EXAMPLES, 144. 1. The sum of three numbers is 40. One of the numbers is 12, and another is 15. What is the third ? 2. The difference of two numbers is 16, and the smaller is 12. What is the larger ? 3. The difference of two numbers is 18, and the larger is 30. What is the smaller ? 4. The product of two numbers is 132, and one of the numbers is 11. What is the other ? 5. What five numbers less than 10 will divide 120 without a remainder ? 6. The sum of two numbers is 21, and the greater 12. What is the product of the two numbers ? REVIEW. 69 7. The quotient of two numbers is 45, and the divisor 8. What is the dividend ? 8. How many times can 8 bushels of grain be taken from a bin containing 52 bushels, and what will remain ? 9. A news-boy sold 24 papers at 4 cents each, and there- by gained 48 cents. At what rate did he buy the papers ? 10. The dividend is 240 and the quotient 12. What is the divisor ? 11. The quotient is 20, the remainder 8, and the divi- sor 9. What is the dividend ? 12. A drover bought 10 sheep at $8 a head, and sold them for $96. How much did he gain a head ? How many 13. In each of 5 equal parts of (9 x 12— 8 x 6) ? 14. In each of 9 equal parts of (56—0 x 7 + 16) ? 15. In each of 7 equal parts of (72—40 + 37—20) ? 16. If 5 men can build a wall in 9 days, in how many days can 3 men build it ? Analysis. — It will take 1 man 5 times 9 days, or 45 days ; and ^ men can build it in 1 third of 45 days, or 15 days. 17. How long will it take 7 men to do the same work that 14 men can perform in 3 days ? 18. If 9 days' work will pay for 6 tons of coal at $6 a ton, what is the price of a day's labor ? 19. How much pork can be bought for 96 cents, if 9 pounds cost 72 cents ? 20. If 5 men can build a wall in 8 days, how many men can build it in 4 days ? Analysis. — It will require 8 times 5 men, or 40 men, to build it in 1 day, and 1 fourth of 40 men, or 10 men, to build it in 4 daya 70 Divisioiq-. 21. How many men will be required to do the same work in 5 days that 4 men can do in 40 days ? 22. If 6 men can dig a ditch in 5 days, how many men would be required to dig it in 1 day ? In 2 days? In 3 days ? In 6 days ? In 10 days ? 23. At the rate of 24 miles in 8 hours, how many milep would a man walk in 12 hours ? 24. If a woman pay 60 cents for some lemons, at the rate of 10 cents for 6, and sell them at the rate of 9 for 20 cents, how many cents will she gain ? 25. If 5 barrels of flour are worth $60, how many cords of wood at $4 a cord will pay for 3 barrels ? 26. If 12 yards of cloth cost $40, for how much must it be sold a yard to gain $20 ? 27. What cost 9 quarts of milk, if 4 quarts cost 24 cents? 28. How many bags will be required to hold 108 bushels of wheat, if 4 bags hold 9 bushels ? 29. To 6 add 8, subtract 4, multiply by 5, add 6, divide by 8, and what is the result? 30. How much greater is 7 times 8 plus 4, than 72 divided by 9, multiplied by 7 ? 31. How much less is 10 times 10, diminished by 4 times 10, plus 12, than 100 divided by 10, plus 8 times 11? Find the required term in the following equations : 32. 25 + 9—32 + 4=::? 33. 4x12 + 3x9==? 34. 60— 12-T-6x?=56 35. 72-=-9x22— 10=? 36. 120-~20 + 48-v-?=9 37. 96-=-8x9=? xl2 38. (32 + 12-^11) X ? =80 39. (132-^-ll— 4)x9=:60+? 40. 42 + 24—15=? +10 41. 48 + 31-^-48-36 = 16- ? 42. (120-7xl2)-f-6=?-^ll 43. 49 + 14-^(28-19)=25— ? REVIEW. 71 WRITTEN EXAMPLES, 145. 1. Subtract 2520 from the sum of 3472, 450, 1254, and 56 ; divide the remainder by 113, and multiply the quotient by 205. What is the result ? 2. How many times can 236 be subtracted from 2124? 3. How many times 236 will produce 2124 ? 4. The factors of a number are 36 + 114, and 5640 -^3007. What is the number ? 5. The product of two numbers is 30128, and one of the numbers 4200-^75. What is the other? 6. Divide the product of 204 and 378 by their difference. 7. What must be added to the sum of $12.36 and $7.62, to amount to $30. 76 ? 8. What is the difference between 746 x 23 and 1S975 -^25? 9. A man owing a debt of $3000, paid $756.50 at one time, $1289.75 at another, and then made a third pay- ment large enough to reduce the debt to $925.60. What was the third payment ? 10. How many pounds of butter at 40 cents a pound are worth as much as 1600 bushels of oats at 75 cents a bushel ? 11. If a man gain $638.75 by selling 365 barrels of flour at $9.25 a barrel, at what price did he buy it ? 12. The multiplier is 36, and the product 170352 ; if the multiplier is 1 fourth as great, what is the product ? 13. The multipher is 204, and the multiplicand is 17605 ; if the multiplicand were one-fifth as great, what would be the product ? 14. If a mechanic receives $1500 a year for his labor, and his expenses are $968, in what time can he save enough to buy 28 acres of land at $133 an acre ? 72 DIVISION. 15. With the multiplier 48, the product is 166656 ; with a multiplicand 1 third as great, what would be the product? 16. The divisor is 16, the quotient 12624 ; with a divi- sor 1 fourth as great, what would be the quotient ? 17. The divisor is 24, and the quotient is 43950 ; if the divisor be made 6 times as large, what will be the quotient? 18. The quotient is 91864 ; with a divisor 1 ninth ai great, what would be the quotient ? 19. A grocer bought two kinds of syrup ; one for 54 cents a gallon, and the other for 62 cents. What was the average cost a gallon ? Operation.— (54 cents + 62 cents)-^2=58 cents. The amrafje of two numbers is one-half Wi^ay sum, the average of three numbers is one-third their sum, etc. 20. A merchant bought equal quantities of 3 kinds of tea, some at 60 cents, some at 78 cents, and some at 90 cents a pound. What was the average cost a pound? 21. A keeper of a toll bridge received $104 toll on Monday, $97 on Tuesday, $128 on Wednesday, and $99 on Thursday. What were the average daily receipts? 22. Sold 3 city lots for $1500, $2976, and $1895, respec- tively. What was the average price ? 23. If a young man receive a salary of $25 a week, and he pays $8.75 for his board, and $4.65 for other expenses, in how many weeks can he pay a debt of $487.20 ? 24. A man having $4578 paid out all but $1642 in 8 weeks. What was the average amount paid out each week? 25. Bought 140 acres of land for $7560, and sold 86 acres of it at $75 an acre, and the remainder at cost. How much was gained ? REVIEW. 73 26. A father gave his property to his 4 children. To the first he gave $6780, to the second $8200, to the third $1526 more than to the first, and to the fourth $1345 less than to the third. What was the value of his property? 27. The sum of two numbers is 184, and their differ- ence is 42. What are the numbers ? Analysis. — Since 184 is the sam of the numbers, if the differ^ ence 42 be subtracted from the sum 184, the remainder 142 will be twice the less number. 142 -;- 2 = 71 the less number ; and 71 + 4'^' = 113 the greater number. Or, if the difference 42 be added to the sum 184, the amount 226, will be twice the greater number. 226 -v- 2 = 113 the greater num- ber ; and 113 — 42 = 71 the less number. Proof.— 113 + 71 = 184 the sum. 28. The sum of two numbers is 5672, and their differ- ence is 1974. What are the numbers ? 29. A man paid $1250 for a horse and carriage, the horse being valued at $190 more than the carriage. What was the value of each ? 30. At a town election the whole number of votes cast for two candidates was 3789, and the majority for the successful candidate was 227. How many votes did each receive ? 31. Two men are worth $28475, and one is worth $4625 more than the other. How much is each man worth ? 32. A grocer wishes to put 240 pounds of tea into threo kinds of boxes, containing respectively 5, 10, and 15 pounds, using the same number of boxes of each kind. How many boxes will be required ? 33. Sold a quantity of wood for $2492, that cost $1424, thus gaining $3 a cord. How many cords were there, and what was the cost per cord ? 74 DIVISION. 34. What number divided by 36, the quotient increased by 48, the sum diminished by 37, the remainder multiplied by 14, and the product increased by 216-^72, is 269? Find the missing term in the following equations : 35. (15341 -f-29) x (8430-^-1405) = 1587 x? 36. [4500+ (12000 — 1375) -T-121 x 25] x 48= ? x24 37. 732 X 6~(15 x 24-^9"xl0)+ (42 x 234^26)= ? 38. 450 + (24-^12y~x5-^(90-^6) + (3~xIi — 18=? 146. The pupil should illustrate the following prob- lems by original examples : Pboblem 1. Given several numbers, to find their sum. 2. Given the sum of several numbers and all of them bait one, to find that one. 3. Given the parts, to find the whole. 4. Given the whole and all the parts but one, to find that one. 5. Given two numbers, to find their difference. 6. Given the greater of two numbers and their differ- ence, to find the less. 7. Given the less of two numbers and their difference, to find the greater. 8. Given the minuend and subtrahend, to find the re- mainder. 9. Given the minuend and remainder, to find the sub- trahend. 10. Given the subtrahend and remainder, to find the minuend. 11. Given two or more numbers, to find their product. REVIEW. 75 12. Given the product and one of two factors, to find the other factor. 13. Given the multiplicand and multiplier, to find the product. 14. Given the product and multiplicand, to find the Qiultiplier. 15. Given the product and multiplier, to find the mul- tiplicand. 16. Given two numbers, to find their quotient. 17. Given the divisor and dividend, to find the quotient 18. Given the divisor and quotient, to find the dividend. 19. Given the dividend and quotient, to find the divisor. 20. Given the divisor, quotient, and remainder, to find the dividend. 21. Given the dividend, quotient, and remainder, to find the divisor. 22. Given the final quotient of a continued division and the several divisors, to find the dividend. 23. Given the quotient of a continued division, the first dividend, and all the divisors but one, to find that divisor. 24. Given the dividend and several divisors of a con- tinued division, to find the quotient. 25. Given two or more sets of numbers, to find the difference of their sums. 26. Given two or more sets of factors, to find the sum of their products. 27. Given two or more sets of factors, to find the dif- ference of their products. 28. Given the sum and the difference of two numbers, to find the numbers. 76 DIVISION. 147. SYNOPSIS FOR EEVIEW. {1. Division. 2. Dividend. 3. Divi sor. 4e Quotient. 5. Remaindei 6. Sign of Division. Pbinciples, 1, 3, and 3. Relation op Division to Subtraction. Relation of Division to MuLTiPLiCATioif, Objects op Division. ] 1. Illustrate. 2. Equal Parts. Short Division. 1, 2. 1. Definition, Method 8. Long Division. Division op Dollars AND Cents. 10. Division by Factors. 11 ^ 1. Definition. 2. Method. 3. Rule, 1— VI. 4. Proof. 1. When divisor and dividen i are concrete, but unlike. 2. How to change dollars to cents. 3. How to change dollars and cents to cents. i 1. Method. 2, Rule, I, II, III, IV j When the Divisor has Ciphers j 1. Method. ON the Right. i 2. Rule, I, II, III 12. General Principles of Division, 1, 2, 3. . 13. General Law, 148, 1. What two numbers, besides the number itself dtod 1, will give a product of 8 ? 16 ? 25 ? 42 ? 64 ? 2. What numbers, other than the given number and 1, will exactly divide 9 ? 15 ? 36 ? 48 ? 55 ? 8. Of what sets of two numbers is 24 the product ? 4. Of what sets of three numbers is 36 the product ? 5. What are the smallest numbers, other than 1, that will exactly divide 18 ? 21 ? 49 ? 55 ? 6. What is the largest number, v>ther than the given number itself, that will exactly divide 22 ? 24 ? 30 ? 40 ? 7. Name the numbers between 12 and 30, that are the product of two factors greater than 1. Between 30 and 50. 8. Name the numbers between 5 and 20, that have no other factors than the numbers themselves and 1. 9. Of what number are 7 and 8 the factors ? 2, 5, and 7? 4, 5, and 3? 2, 3, 5, and 10? DEFINITIONS. 149. The Properties of Winnbers are those ![ualities or elements which necessarily belong to numbers; Numbers are either Integral, Fractional, or Mixed, 150. "^An Integral Niiniber or Integer is a number' representing whole things. ()^.) Thus, 8, 23, 30 men, 45 pounds are integral numbers. Integral numbers are either E^en or Odd^ Prime or Composite. 78 PEOPERTIES OF ]^UMBERS. 151. An Even Number is a number that is exactly divisible by 2. All numbers whose unit figure is 0, 3, 4, 6, or 8, are even. 153. An Odd Number is a number that is not 3xactly divisible by 2. All numbers whose unit figure is 1, 3, 5, 7, or 9, are odd, 153. A Prime Number is a number that has no integral factors except unity and itself. Thus, 3, 3, 5, 11, 33, etc., are prime numbers. 3 is the only even prime number. 154. A Comimsite Number is a number that has other i7itegral factors besides unity and itself. Thus, 21 is a composite number, since 31=7 x 3. 155. The Factors of a number, are the numbers which multiplied together will produce it. (108.) Thus, 7 and 8 are factors of 56 ; 3, 4, and 7, of 84. 156. A Prime Factor is a prime number used as a factor, (153.) The prime factors of a number are also the prime divisors of it. 157. An Exact Divisor of a number is one that will divide that number without a remainder. Thus, 6 is an exact divisor of 48, and 9 an exact divisor of 73. 1. The Ikact Divisors of a number are also the factors of that number. 3. An exact divisor of a number is sometimes called the mea^urt >f that number. 3. When a number is a factor, or divisor, of each of two or more numbers, it is called a common factor, or divisor, of those numbers 158. Numbers are prime to each other when they have no common integral factors, or divisors. Thus, 9 and 14, 16 and 35 are prime to each other. DIVISIBILITY OF NUMBERS. 79 DIYISIBILITY OF NUMBEES. 159. A number is said to be divisible by another, when there is no remainder after dividing. Any numbei is divisiile 1. By 2, if it is an even number. Thus, 20, 24, 36, and 44 are divisible by 2. 2. By 3, if the sum of its digits is divisible by 3. Thus, 135, 471, and 1134 are divisible by 3. 3. By 4, if its two right-hand figures are ciphers, or express a number divisible by 4. Thus, 300, 432, and 1548 are divisible by 4. 4. By 5, if it ends with a cipher or 5. Thus, 30, 45, and 235 are divisible by 5. 5. By 6, if it is an even number and di\dsible by 3. Thus, 168, 402, and 1314 are divisible by 6. 6. By 8, if its three right-hand figures are ciphers, or express a number divisible by 8. Thus, 3000, 2728, and 10576 are divisible by 8. 7. By 9, if the sum of its digits is divisible by 9. Thus, 217683 and 401301 are divisible by 9. 8. By 10, if it ends with one or more ciphers. Thus, 40, 500, 3000 are respectively divisible by 10, 100, and 1000. 9. By 7, 11, and 13, if it consists of but four places, the first and fourth being occupied by the same signifi- cant figures, and the second and third by ciphers. Thus, 2002, 3003, and 5005, are divisible by 7, 11, and 18. 80 PROPERTIES OF NUMBERS. 10. An odd number is not divisible by an even number. 11. If an even number is divisible by an odd number, the quotient will be an even number. Thus, the quotient of 36 divided by 9, is 4 ; of 42 by 7, is 6. 12. If an even number is divisible by an odd number^ t is also divisible by Uoice that number. Thus, 28 is divisible by 7, and also by twice 7. 13. Every odd number except 1, increased or else diminished by 1, is divisible by 4. Thus, 11 increased by 1, or 17 diminished by 1, is divisible by 4. 14. Every prime number except 2 and 3, increased or else diminished by 1, is divisible by 6. Thus, 23 increased by 1, or 31 diminished by 1, is divisible by 6. EXEItCIS ES . 160. Find by inspection some of the exact divisors of the following numbers : 1. 1536. 4. 6105. 7. 32472. 3. 1683. 5. 12936. 8. 71460. 3. 3348. 6. 43560. 9. 197200 FAOTOEIKG. OJRAZ EXERCISES. 161. 1. What are the even numbers from 12 to 36? 2. What are the odd numbers from 12 to 36 ? 3. What are the prime numbers from 12 to 36 ? 4. What are the composite numbers from 12 to 36 ? 5. Name all the prime factors of 36. 6. Name all the composite factors of 36. , 7. What are the prime factors of 35 ? 49 ? 60 ? 8. What are the composite factors of 32 ? 48 ? 72 ? FACTORIN^G. 81 9. What prime factors are common to 31 and 42 ? 10. What composite factors are common to 36 and 72 ? 11. What factors are common to 18 and 30 ? To their sum and difference ? 12. What factors are common to the sum and difference of 20 and 40 ? 13. What prime factors are common to 14 and 4 times 14 ? 14. What two composite factors are common to 24 and 3 times 24 ? 15. What is the largest, and what the smallest prime factor of IS, 30, and 45 ? DEFINITIONS AND PEINCIPLES. 1Q2. Factoring is the resolving of a composite number into its factors, and is performed by division. 163. An Exponent is a small figure written at the right of a number, and a little above, to show how many times the number is used as a factor. Thus, 2^ = 2 X 2 X 2, and denotes tliat 2 is used as a factor 3 times, 5^, denotes that 5 is used as a factor 4 times, 164. Pkinciples. — 1, The prime factors of a nuvfiber^ or the prod^ict of any tiuo or more of them, are the only exact divisors of that number. 2. A factor of a number is a factor also of any number !)/* times that number. 3. A factor common to two or more numbers is a factor of their sum, and also of the difference of any two of them. 4. Every composite nmnher is equal to the product of its prime factors. 82 PEOPERTIES OF JSTUMBERS. WMITTEN BXEB ClSm S . 165. To find all tlie prime factors of a composite number, 1. What are the prime factors of 2772 ? OPERATION. Analysis. — Since the given number is even, di vide it by 2, the least prime factor, and the result also hj 2, which gives an odd number for a quotient. Next divide by the prime factors 3, 3, and 7, suc- cessively, obtaining for the last quotient 11, which not being divisible, is a prime factor of the given number. Hence the divisors 2, 3, 3, 3, 7, and the last quotient 11, are all the prime factors, or divisors, of 2772, and may be written 2\ 3^ 7, 11. 2)2772 2 )1386 3 )693 3)231 7 )77 11 In like manner find the prime factors or divisors 2. Of 1050. 4 Of 2445. 6. Of 2205 3. Of 1140. 5. Of 2366. 7. Of 2310 EuLE. — Divide the given numher ly any prime factor of it, and the resulting quotient by another, and so continuf the division until the quotient is a prime number. The several divisors and the last quotie7it are the prime factors. Proof. — The product of all the prime factors is equal to the given number. (Prin. 4.) Eesolye the following numbers into their prime factors 8. 1155. 9. 2934. 10. 6300. 11. 2205. 12. 13981. 13. 32320. 14. 21504. 15. 29925. 16. 12673. 17. 10010. 18. 28665. 19. 31570e COMMON DIVISORS. 83 commo:n" diyisoes. ORAL EXERCISES. 166. 1. Name two exact divisors of 12. Of 15. Of 20. S. Name three exact divisors of 24. Of 48. Of 72. 8. What number is an exact divisor of 27 and of 56 ? 4. What are the prime divisors of 15 ? 55 ? 49 ? 77 T 5. What are the composite divisors of 72 ? 84 ? 120 ? 6. What prime divisor is common to 28, 35, and 42 ? 7. Name a common measure of 22, 44, and 66. 8. Name the greatest common measure of 16, 32, and 64. 9. Of what three numbers is 12 a common divisor? 10. What two numbers will exactly divide 15 and 30 ? Their sum and difference ? 11. What is the smallest exact divisor of the sum and difference of 10 and 15 ? Of 21 and 56 ? 12. What is the greatest exact divisor of the sum and difference of 16 and 24 ? Of 18 and 45 ? 13. Find the greatest common measure of 14, 42, and 56. 14. Find the greatest common divisor of 27, 36, aiid 45. ^ DEFINITIONS AND PEINCIPLES. 167. A Common Divisor of two or more numbers is a common factor of each of them. 168. The Greatest Common Divisor of two or more numbers is the greatest common factor, and is the product of all the common prime factors. 169. Principles. — 1. The only exact divisors of a number are its prime factors^ or the product of two or more of them> 84 PROPERTIES OF JSTUMBERS. 2, An exact divisor divides any number of times its dividend. 3. A commo7i divisor of two or more numbers will divide their sum, and also the difference of any two of them, 4 The greatest common divisor of tivo or more numbers IS the product of all their com^non prime factors. WniTTEN EXEItCISES. 170. When the numbers can be readily factored. 1. What is the greatest common divisor of 42, 63, and 126 ? 1st OPERATION. ANALYSIS. — By factoring the given nuiDL- j^2zz:7vSv2 ^^^^y *^^ prime factors common to all of 63 = 7x3x3 them are 7 and 3. Hence 7 x 3 = 21 is the greatest common divisor of 42, 63, and 126 = 7x3x6 126. (Prin. 4) 2d operation. Analysis. — Since the given numbers 3)42 63 126 ^^® exactly divisible by 3, and the result- ing quotients by 7, they are also divisible 7 )14 21 42 by 7x3, or 34 (Prin. 1.) 2 3 G If there were other factors of the great- est common divisor, then the quotients 2, 3, and 6 would be exactly divisible by them. Find the greatest common divisor 2. Of 42 and 112. 3. Of 96 and 544 4. Of 40, 75, and 100. 5. Of 72, 126, and ^16. Rule. — Separate the numbers into their prime factors and find the product of all that are common. Or, I. Write the numbers in a line, and divide by any prime factor common to all the numbers. II. Divide the quotients in like 7nanner, and so continue the division till all the quotients are prime to each other. III. The product of all the divisors will be the greatest common divisor. (Pri:n^. 4,) COMMOK DIVISORS. 85 537 459 68 68 15 3 136 What is the greatest common divisor 6. Of 144 and 720 ? I 8. Of 126, 210, and 252 ? 7. Of 308 and 506 ? I 9. Of 72, 96, 120, and 384 ? 171. When the numbers cannot be readily factored. 1. Find the greatest common divisor of 527 and 1207. OPERATioii. Analysis. — Draw two vertical lines, and 12 7 place the greater number on the right, and the 10 5 4 ^^^^ ^^ *^® leity one line lower down. Di- vide 1307 by 527, and write the quotient 2 between the vertical lines, the product, 1054, under the greater number, and the remainder 1 7 153, below. Next, divide 527 by this remainder 153, writing the quotient 3 between the verticals, the product 459, on the left, and the remainder 68, below. Again, divide the last divisov 153, by 68, and write the product, and remainder in the same order as before. Finally, dividing the last divisor 68, by the last remainder 17, there is no remainder. Hence 17, the last divisor, is the greatest common divisor of 537 and 1207. Proof. — Now, observing that the dividend is always the sum of the product and remainder, and that the remainder is always the difference of the dividend and product, trace the work in the reverse order, as indicated by the arrow line in the diagram below. 17 divides 68, as proved by the last division ; it will also divide 2 times 68, or 136 (Prin. 2). Since 17 divides both itself and 136, it will divide 153, their sum (Prin. 3). It will also divide 3 times 153, or 459 (Prin. 2) ; and, since it is a common divisor of 459 and 68, it must divide their sumy 527, which is one of the given numbers. It will also divide 2 times 527, or 1054 (Prin. 2) ; and, since it divides 1054 and 153, it must divide their sum, 1207, the greater number (Prin. 3). Hence, 17 is a comman divisor of the given numbers. ILLUSTRATION. A 1207 527 i59 68 68 1054 153 136 -^ 17 86 PEOPEKTI'SS OF NTJMBEBS. 11207 527 ^^ 459 68 Again, tracing the work in the direct order, as indicated in the following diagram,tlie greatest com- mon divisor, whatever it is, must divide 2 times 527, or 1054 (Pein.2). And since it will divide both 1054 and 1207, it must divide their dif- ference, 153 (Prin. 3). It will also divide 3 times 153, or 459 (Prin. 2) ; and as it will divide both 459 and 527, it must divide their difference, 68 (Prin. 3). It will also divide 2 times 68, or 136 (Prin. 2); and as it will divide both 136 and 153, ix must divide their difference, 17 (Prin. 3); hence, it cannot be greater than 17, 1054 153 136 y 17 Thus, it has been shown, 1st. That 1 7 is a common divisor of the given numbers. 2d. That their greatest common divisor, whatever it be, cannot be greater than 17. Hence it must be 17. In like manner, find the greatest common diyisor 2. Of 316 and 664. 3. Of 679 and 1869. 4. Of 1080 and 189. 5. Of 2192 and 458. 6. Of 825 and 1372. 7. Of 2041 and 8476. 8. Of 7241 and 10907. 9. Of 2373 and 6667. EuLE. — I. Draiu two vertical lines, and write the two numbers, one on each side, the greater Clumber one line above the less, II. Divide the greater number by the less, writing the fuotient between the verticals, the product under the divi- dend, and the remainder beloiv. III. Divide the less number by the remainder, the last divisor by the last remainder, and so on, till nothing re- mains. The last divisor is the greatest common divisor. COMMON DIVISORS. 87 IV. If more than two numbers are giveriy first find the greatest common divisor of tivo of them, and then of this divisor and one of the remaining mimiers, and so on to the last ; the last common divisor found is the greatest common divisor of all the given numbers. 10. What IS the greatest number that will divide 3281 and 10778 ? 10353 and 14877 ? 11. What is the greatest number that will divide 620, 1116, and 1488? 396, 5184, and 6914 ? 12. A man having a piece of land, the sides of which are 240 feet, 648 feet, and 420 feet, wishes to inclose it with a fence having panels of the greatest possible unU form length ; what will be the length of each panel ? 13. A farmer wishes to put 231 bushels of corn, 393 bushels of wheat, and 609 bushels of oats into the largest bags of equal size, that will exactly hold each kind. How many bushels must each bag hold ? 14. A forwarding merchant has 15292 bushels of wheat, 1520 bushels of corn, and 504 bushels of beans, which he wishes to ship, in the fewest bags of equal size that will exactly hold either kind of grain ; how many bags will it take ? 15. Three persons have respectively $630, $1134, and $1386, with which they agree to purchase horses, at the highest price per head, that will allow^ach man to invest all his money. How many horses can each man buy ? 16. How man.^ rails will inclose a field 5850 feet long by 1729 feet wide, the fence being straight, and 7 rails high, and the rails of equal length, and the longest that can be used ? 88 PBOPEETIES OF NUMBERS. MULTIPLES. ORAL EXEMC ISES, 173. 1. What numbers between 5 and 30 are exactly divisible by 4 ? By 6 ? 7 ? 8 ? 9 ? 2. What numbers less than 40 are exactly divisible by 7 ? 3. What prime factors are common to 6, and 5 times 6? 4. Name some numbers exactly divisible by 4 and 6 ; by 3 and 7 ; by 5 and 7 ; by 8 and 10. 5. By what three prime numbers can 42 be divided ? 6. Name some numbers of which 3 and 4 are factors. 7. Find the least number exactly divisible by 3, 4, and 5. DEFINITIONS AND PEINCIPLES. 173. A 31ultiple of a number is a number exactly divisible by the given number; or, it is any product or dividend of which a given number is a factor, 1. A number may have an unlimited number of multiples. 2. A number is a divisor of all its multiples, and a multiple of all Its divisors. 174. A Common Multiple of two or more given numbers is a number exactly divisible by each of them. 175. The Least Common Multiple of two or more given numbers is the least number exactly divisible by each of them. Two or more numbers can have but one least common multiple. 176. Peinciples. — 1. A multiple of a number contains each of the prime factors of that number. 2. A commo7i multiple of two or more numbers contains each of the prime factors of those numbers. Hence, MULTIPLES. 89 3. The least common multiple of two or more numbers is the least number that contains each of the prime factors of those numbers. 4. A common multiple of two or more numbers may be found by multiplying the given numbers together. WRITTEN EXEHCISES. 1-7*7. To find the least cominoii multiple. FIRST METHOD. 1. Find the least common multiple of 30, 42, and ^^. OPERATION. Analysis. — The least common QQ 2ySv5 multiple cannot be less than the ,^ ^ o 7 largest number 66, since it must contain 66 ; hence it must con- 66=:2x3xll tain all the prime factors of 66, 3x3xllx7x5=:2310 ^^^^^^ ^^^ ^' ^' ^^^ ^l- ^^^^n. 1.) But the least common multiple of 66 must also contain all the prime factors of each of the other num- bers, and since the prime factors 2 and 3 of 66 are common also to 42 and 30 omit them, and annex the factors 7 and 5 to those of 66, and the series 2, 3> 11, 7, and 5 are all the prime factors of the given numbers, and their product 2x3xllx7x 5=2310, is the least common multiple of the given numbers. (Prin. 3.) 2. Find the least common multiple of 24, 42, and 17. 3. Find the least common multiple of 8, 12, 20, and 30. 4. Find the least common multiple of 10, 45, 75, and 90. Rule. — I. Resolve each of the given numbers into its prime factors. 11. Multiply together all the prime factors of the largest number, and such prime factors of the other numbers as are not found in the largest number, and their product will be the least common multiple. 90 PROPERTIES OF NUMBERS. Find the least common multiple 5. Of 30, 66, 78, and 42. I 7. Of 16, 60, 140, and 2ia 6. Of 21, 30, 44, and 126. I 8. Of 16, 48, 80, 32, and 66. SECOND METHOD. 178. 1. Find the least common multiple of 18, 24, and 54. OPERATION. Analysis. — Write the numbers in a hori zontal line, with a vertical line at the left Since 2 is a prime factor of one or more '^ of the given numbers, it must also be a 18 34 54 9 13 27 3 4 9 4 3 3 3 4 9 factor of the least common multiple of those numbers. (Prin. 3.) Hence, divide by 2 and write the quotients underneath. For a like reason divide again successively by 3 and 3, writing the quotients and undivided numbers in a line below, omitting to write any quotient when it is 1. Since there is no factor common to 4 and 8, they are prime to each other, and hence the divisors 2, 3, and 3, with the numbers 4 and 3 in the last line, are all the prime factors of the given numbers, and their product 216 is the least common multiple. (Prin. 3.) If in any example, any of the smaller numbers are exactly con- tained in the larger, they may be omitted in finding the least com- mon multiple, inasmuch as a number that will contain a given number, will contain any factor of that number. Thus, if required to find the least common multiple of 8, 12, 24, 72, and 120, omit all the numbers except 72 and 120, since the others are factors of these, and the least common multiple of 72 and 120, will be the least common multiple of all the numbers. 2. Find the least common multiple of 32, 34, and 36. 3. Find the least common multiple of 84, 100, and 224. EuLE. — I. TVrite the mimiers in a horizontal line, 07nit* ting such of the smaller numhers as are factors of tht larger, and draw a vertical line at the left. II. Divide hy any prime factor that 2vill exactly divide two or more of the given numhers, and write the quotients and undivided members in a line underneath. MULTIPLES. 91 III. In like manner divide the quotients and undivided numbers until they are prime to each other. IV. The product of the divisors ayid the final quotients and undivided numbers, is the least common multiple. What is the least common multiple 4. Of 4.QQ% and 5698 ? 5. Of 312, 260, and 390 ? 6. Of 24,10, 32, 45 and 25? 7. Of 153,204,102,andl020? 8. Find the least common multiple of the first eight even numbers. 9. Find the least common multiple of the first five odd numbers, 10. What is the least number of oranges that can be equally distributed among 16, 20, 24, or 30 boys ? 11. What is the shortest piece of rope that can be cut exactly into pieces either 15, 18, or 20 feet long? 12. What is the smallest sum of money which can be exactly expended for books at $5, or $3, or $4, or $6 each ? 13. What is the product of the least common multiple of 12, 16, 24, and 32, multiplied by their greatest common divisor ? 14. Divide the least common multiple of 7, 42, 6, 9, 10, and 630, by the greatest common divisor of 110, 140, and 680. 15. What is the smallest sum of money which can b€ exactly expended for sheep at $8, or cows at $28, or oxeij at $54, or horses at $162 each ? 16. What is the smallest quantity of grain that will fill an exact number of bins, whether they hold 36, 48, 80, ot 144 bushels? M FBOPEBTIES OF KUKBEBS. OANOELLATIOE". OltAIi JSXEBCISMS . 179. 1. Diyide 72 by 24. One-half of 72 by one-half of 24. One-third of 72 by one-third of 24. 2, Divide one-fourth of 72 by one-eighth of 24. 8. Diyide 36 by 9. One-third of 36 by one-third of 9o i. What factors are common to 72 and 24 ? 6. What is the quotient of 12 x 6 divided by 12 x 2 ? 6. Divide 3x3x3 by 3x3. 4x5 x2 by 2x2 x5. 7. Divide7x6x2by2x6x7. 5 x6 x4 by 3 x4x6. DEFINITIONS AND PEINCIPLES. 180. Cancellation is the process of abridging operations in division by rejecting equal factors from both dividend and divisor. 181. Prii^ciples. — 1. Rejecting a factor from any numler divides the number ly that factor. 2. Rejecting equal factors from both dividend and divp 8or does not change the quotient. WJtITTBN BXEJtCISES, 183. 1. Divide 56 x 24, by 48 x 7. 1st operation. Analysis. — Indicate the 56x24__0 X/?X^X4__ operation to be performed * — — 4 in t}ie example, by writing 48x7 0X0X;J the numbers that constitute the dividend, above a line, and those that constitute the divisor below it. Resolve these numbers into their factors, and the dividend will consist of 8 X 7 X 6 X 4, and the divisor of 8 x 6 x 7. Rejecting equal factors from both dividend and divisor, there remains the factor 4 in the dividend. Henoe the quotient is 4. CAKCELLATI02^, 93 2d operation. t 4 00 X ti = 4 Analysis.— Since it is evident that 8 will divide both 56 and 48, reject 8 as a factor of 56, retaining the factor 7, and also of 48, re- taining the factor 6. Again, since 6 will divide both 24 in the dividend and 6 in the divisor, reject 6 as a fac- tor from both, retaining the factor 4 in the dividend. Finally, jrejecting the factor 7, common both to the dividend and to the divisor, there remains only the factor 4 in the dividend, which is the required quotient. 2. Diyide the product of 44, 30, 7, and 6, by- product of 33, 18, and 14; or, diyide 55440 by 8316. the OPERATION. 2 10 55440 __ Mx$0x1lx$ 8316^ "" $$xl$X^4 ^ 3 $ t Or, $0 10 20 By many it is thought more convenient to write the factors of the dividend on the right of a vertical line, and the factors of the divisor on the left 3. Divide 13x7x5x3by3x5x7. 4. Diyide 42 x 18 x 6 x 4 by 36 x 21 x 6. EuLE. — I. Cancel all the factors common to loth divi- dend and divisor. II. Divide the product of the remaining factors of the dividend hy the product of the remaining factors of the divisor^ and the result will be the quotient. When a factor equal to the number itself is canceled, the unit 1 remains, since a number divided by itself gives a quotient of 1. If the 1 occur in the dividend, it must be retained; if in the divisor, it need not be regarded. 94 PROPERTIES OF NUMBERS. 5. What is the quotient of 35 x 33 x 28, divided by 15 x 14x11? 6. What is the quotient of 140 x 39 x 13 x 7, divided bj 7x26x21? 7. Multiply 11 times 21 by 28, and divide the product by 14 times 13. 8. How many times is the continued product of 14, 9, 3, 20, 5, and 6 contained in the continued product of 183, 18, 70, 12, and 5 ? 9. If 213 X 190 X 84 X 264 is the dividend, and 56 times 36 multiplied by 30 is the divisor, what is the quotient ? 10. (240x56xl8)-r-(60x28x9) = ? 11. (72 X 48 X 28 X 5) -^ (84 X 15 X 7 X 6) = ? 12. (66 Xl8x27x25)-r-(84x45x 7x30) = ? 13. (80 X 60 X 50 X 16 X 14)-^(70 x 50 x 24 x 20) = ? 14. Multiply 64 by 7 times 31, divide the product by 8 times 56, multiply this quotient by 15 times 88, divide the product by 55, multiply this quotient by 13, and di- vide the product by 4 times 6. What is the quotient ? -.^ ^' A^u 4-- 4. « 12x60x27x35 15. Fmd the quotient of zr — — — j^ — -— . ^ 7 X 15 X 42 X 108 \n -c.- A 4.1. \' 4. .77x100x18x64 16. Fmd the quotient of -- — -- — — — —7-. ^ 25x11x49x16 17. How many tons of hay at $18, must be given for 45 cords of wood at $4 a cord ? 18. How many flour barrels at $.80 each, will pay for 112 bushels of corn at $.70 a bushel ? 19. How many tubs of butter, each containing 56 pounds, at 30 cents a pound, must be given for 7 barrels of sugar, each containing 195 pounds, at 10 cents a pound ? CANCELLATION. 95 20. A laborer gave 12 days' work for 48 bushels of potatoes, worth 50 cents a bushel. What were his daily earnings ? 21. A grocer sold 24 boxes of soap, each containing 55 pounds, at 10 cents a pound, and received as pay 88 bar rels of apples, each containing 3 bushels. How much were the apples worth a bushel ? 22. Sold 20 pounds of butter at 27 cents a pound, which exactly paid for 15 pounds of coffee. What was the price of the coffee a pound ? 23. A farmer exchanged 240 bushels of corn, worth $.75 A bushel, for an equal number of bushels of barley, worth $1 a bushel, and oats, worth $.50 a bushel. How many bushels of each did he receive ? 24. A farmer bought two kinds of cloth, one kind at $.75 a yard, and the other at $.90, buying twice as many yards of the first kind as of the second. He paid for the cloth, 132 pounds of butter at 40 cents a pound. How many yards of each kind of cloth did he buy ? 25. A merchant bought 6 loads of oats, each load con- taining 22 bags, and each bag 2 bushels, worth 56 cents a bushel. He gave in payment 8 boxes of tea, each con- taining 24 pounds. What was the tea worth a pound ? 26. How many bushels of oats at $.60 a bushel, will pay for 12 tons of coal at $7.20 a ton ? 27. How many chests of tea, each containing 63 pounds worth 87^ cents a pound, must be given for 21 bags oi coffee, each weighing 28 pounds, worth 37^ cents a pound ? 28. How many days' work, at $1.25 a day, will pay for 75 bushels of corn, at $.80 a bushel ? n PROPEKTIES OF NUMBERS, 183. SYJSrOPSIS FOR EEVIEW. 1. Definitions. 2. Divisibility of Numbers. 3. Factoring. 4 Common Divisors. - 5. MuiiTIPLES. 6. Cancellation. ' 1. Properties of Numbers. 2. Ini^ gral Number, or Integer. 3. Even Number. 4. Odd Number. 5. Prime Number. 6. Composite Number. 7. Factors. 8. Prima Factor. 9. Exact Divisor. 1. How to find whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10; also by 7, 11, or 13. 2. Other properties of even and of prime numbers. 1. Definitions. / ^- ^<'<'t<»^riff, L 2. Exponent, 2. Principles, 1, 2, 3, 4. 3. Rule. 4. Proof. {1. Divisor, 2. Common Divisor- 3. G, G, Divisor. 2. Principles, 1, 2, 3, 4. 3. Rule (1st), I, II, III. 4. Rule (2d), I, II, III, IV. fl. Multiple. 2. G. Multiple, 3. L. G Multiple 2. Principles, 1, 2, 3. 3. Rule (1st), I, II. 4. Rule (2d), I, II, III, IV. 1. Definition. 2. Principles, 1, 2. 3. Rule, I, IL ORA.L EXERCISES. 184, 1. If any unit, as an apple, or a yard, be divided into 2 equal parts, what is each part named ? One-half. 2. If the unit be divided into 3 equal parts, what name is given to 1 of the parts ? To 2 of the parts ? 3. If the unit be divided into 5 equal parts, what is each part named ? What name is given to 3 of the parts ? 4. How many halves are there in a unit ? How many thirds? Fourths? Fifths? Sixths? Sevenths? 5. If a mile be divided into 4 equal parts, what part of the whole mile is 1 of the parts ? 3 of the parts ? 6. What is 1 of 5 equal parts of a unit called ? What are 2 of 6 equal parts called ? 4 of 10 equal parts ? 7. What is meant by 1 sixth of a unit ? By 3 fourths ? 8. What are 3 of the 7 equal parts of a week called ? 9. Which is the smaller, one-third or one-fourth ? One- fifth or one-third ? 10. Which is the greater, one-fourth or one-sixth ? 185. PKii^CTPLES. — 1. The less the number of equal parts into which a unit is divided, the greater is the VALUE of each part. 2. The GREATER the number of equal parts into which a unit is divided, the less is the value of each part, 5 98 FRACTIONS. DEFINITIONS. 186. A Fraction is one or more of the equal parts of a unit. Thus, 1 half and S thirds are fractions. 187. A Fractional Unit is 07ie of the equal parts into which any unit is divided. Thus, 1 fourth and ' 1 fifth are fractional units of fourths and fifths. Fractional units take their name and their "calue from the number of parts into which the integral unit is divided. 188. A fraction is usually expressed by two numbers, called the Numerator and the Denominator , one written over the other with a line between them. A fraction written in this form is sometimes called a Common Frac- tion. Thus, One-third is written \ Three-fourths " | Five-sixths ^^ f Seven-eighths ^^ ^ Nine-tenths is written -^ Seven-twentieths " ^^5- Twelve-thirty-fifths " H Thirty-six forty-ninths ^^ J| 189. The Denominator of a fraction shows the number of equal parts into which the unit is divided, and also indicates the name of these parts. It is written ieloio the line. Thus, in the fraction |, 8 is the denominator and shows that the unit is divided into eight equal parts, named eighths, 190. The Numerator of a fraction shows the num- ier of equal parts taken to form the fraction. It is written above the line. Thus, in |, 7 is the numerator, and shows that 7 of the 8 equal parts are taken, or expressed by the fraction 191. The Terms of a fraction are its numerator and denominator. Thus, 6 and 7 are the terms of the fraction fy/ FRACTIONS. 99 Express by figures, 1. Five-ninths. 2. Seven twenty-fifths, 3. Nine-eighteenths. 4. Twelve twentieths. 6. Eight thirty-sixths. 6. Twenty-six forty-eighths. 7. Twenty-seven two-hundredths* 8. Forty-three ninety-ninths. 9. Sixteen one-hundred-eighths. 10. Fifty-five eighty-ninths. Copy and read, 7. t\; A; i+; H; -A\; I'h; ^. 8.^; ^;m;m; m; ^-^; m- \Q2i.^Fractions are Proper or Improper. 193. A Proper Fraction is a fraction whose nu- merator is less than its denominator. Its value is less than a unit. Thus, f , f, and \^ are proper fractions. 194. An Improper Fraction is a fraction whose numerator equals or exceeds its denominator. Its value is equal to, or greater than a unit. Thus, f, ^, and ^^ are improper fractions. 195. A Mixed JSFiiinber is an integer and a fraction united. Thus, 12| is equivalent to 12 + f. 196. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 9 is l-=-9=| ; of 16, itisl-7-16:=3ig-, etc. 197. The JRecipj^ocal of a Fraction is 1 divided by that fraction, or it is the fraction inverted. Thus, the reciprocal of f is 1-f-f i=:-| ; of -^, it is ^. 198. The Value of a fraction is the quotient of its numerator divided by its denominator. Thus. ^=A. 100 FRACTIONS 1. Analyze the fraction -g. j Analysis. — | is a fraction ; 8 is the denominator, and shows that the unit is divided into 8 equal parts ; ^ is the fractional unit, since it is one of the eight equal parts into which the unit is divided ; 7 is the numerator, and shows that seven of these equal 'parts are taken ; 7 and 8 are the terms of the fraction. It is a proper frac- ion, since the numerator is less than the denominator ; its 'oalue is 'ms than 1 ; and it is read semn-eighths. In like manner, analyze 2. |. I 3. k^. I 4. |. I 5. fi. I 6. V. 199. Since fractions indicate division, all changes in the terms of a fraction will affect the value of the fraction according to the laws of division ; hence if we substitute the General Principles of Division (143), we shall have the following 300. GENERAL PRmCIPLES OF FRACTIONS. 1. Multiply inq the numerator y or) ,^ ,,. ,. ,, ^ ,. •n- '7' J J • ^ ( Multiplies the fradioru Dividing the denommator, ) '■ -' 2. Dividinq the numerator, or ) _ . . , ,. ^ Ti€ 1.' 1 ' ±11 ' i r Divides the fraction. Multiplying the denominator ^ ) -^ 3. Multiplmnq or dividinq loth ) _ ■ _ , 7-7 * . , ( Does not chanqe the numerator and denominator > , ^,t .. ,. , ,, , \ value of the fraction. / oy the same numoer, ) j j 301. These three /principles may be embraced in one GEKEKAL LAW. A change in tlie numerator produces a like change in the value of the fraction ; but a change in the denominator produces an opposite change in the value of the fraction. BE DUCT I ON. 101 REDUCTION. • 202. To reduce fractions to higher or lower terms* ORAL, EXERCISES. 1. One-half is equal to how many fourths? Analysis. — Since 1 is equal to 4 fourths, ^ is equal to 1 half of 4 fourths or 2 fourths. 2. One-third of a mile is how many sixths of a mile ? 3. One-half of a dollar is how msmj fourths of a dollar ? 4. Name some equivalent fractions for halves. Thirds. 5. Express | in^terms 3 times as great. 4 times as great. 6. The denominators four, six, eight, and ten, are mul- tiples of what number ? 7. Multiply both terms of f by 3, and show that the value of the fraction is not changed. Analysis. -:-If both terms of | are multiplied by 3, the resulting fraction is y%, which is equivalent to f , since the fractional unit is J as great, while the number taken is 3 times as great. 8. .Name three equivalent fractions for f ; for ^; for f . 9. Change f to twelfths. To eighteenths. 10. 8 twelfths are how many thirds ? Analysis. — Since 1 third is equal to 4 twelfths, 8 twelfths are equal to as many thirds as 4 twelfths are contained times in 8 twelfths, which is 2 times. Hence there are f in ^. 11. How msinj fourths of a rod are 9 twelfths of a rod? 12. Divide both terms of ^ by 5, -and show that the value of the fraction is not changed. Analysis.— If both terms of ^f are divided by 5, the resalting fraction is f , which is equivalent to ^f , since the fractional unit is 5 times as great, while the number taken is ^ as great. 102 FBACTIONS. 13. Change Jf to an equivalent fraction having a de- nominator 1 half as great. 1 third as great. 14. Change ^ to a fraction having lower terms. Jf. |^. 15. In what lower terms can |f be expressed ? 16. Change -^ to its lowest terms. ^. If. ^. |-g. / 17. Name two common divisors of -Jf . -||-. |^. ||^. 18. Express -^- in terms 4 times as great. 19. Express W in terms 6 times as great. DEFINITIONS. 203^ Meduction of Fractions is the process of changing their form without altering their value. 304. A fraction is reduced to Higher Terms when the numerator and denominator are expressed in larger numbers. Thus, |=|, or ■^. 305. A fraction is reduced to Lower Ter?ns when the numerator and denominator are expressed in smaller numbers. Thus, ^=f , or f . 306. A fraction is reduced to its Lowest Terms when its numerator and denominator are prime to each other. Thus, ^=f ; i|=|. 307. Fractions are changed to higher terms by Multi- plication, and to lower terms by Division. All higher terms of a fraction are multiples of its lowest terms. 308. Principle. — Multiplying or dividing loth terms of a fraction iy the same number does not change the value iif the fraction. (300, 3.y REDUCTION. 103 WRITTEN JSXEB CISISS . 209. 1. Chauge |- to a fraction whose denominator is 30. OPERATION. Analysis. — First, divide 30, the required de- Q Q _^ g __ g nominator, by 6, tlie denominator of the given fraction. The quotient 5 is the factor employed j- ^ ^ z=z |-§^ to produce the required denominator. Hence, multiply both terms of f by 5 (200, 3), and |§ 18 the required fraction. 2. Change ^ to a fraction whose denominator is 96. 3. Change |^ to a fraction whose denominator is 105. 4 Eeduce -^^ to its lowest terms. OPERATION. Analysis.— Dividing both terms JtX t 8 =: -A ; -A^ i = 4 ^^ ^^^ ^^^^^ fraction j\%, by 8, Or, tVit J it — I dividing both terms of -^^ by 3, the result is f . Since the terms of I are prime to each other, the lowest terms of -ff^ are f . The same result is obtained more directly, by dividing both terms by their greatest common divisor, 24. 5. Eeduce ff to its lowest terms. 6. Eeduce ^^ to its lowest terms. 7. Eeduce -^ff to its lowest terms. EuLES. — 1. To reduce a fraction to MgJier terms. Divide the required denominator hy the denominator of the given fraction, and multiply the terms of the given fraction hy the quotient. 2. — To reduce a fraction to its loivest terms. Reject all factors common to the terms of the given frac- tion. Or, Divide the terms of the given fraction iy their greatest common divisor. 104 FRACIIONS. 8. Change ■j'^ to a fraction whose denominator is 180. 9. Keduce ^ and -/y each to sixty-thirds. 10. Reduce |, J-, and ^, each to 130ths. 11. Eeduce ^, ^, J|, and ||, each to 132ds. 13. Change 168-4-253 to the form of a fraction in its lowest terms. 81 -^ 63. 160 -j- 400. 324 -^ 613. Eeduce to their lowest terms, 13. Iff- 14. Mi- ls, m- 16. m- 17. ^^. 18. tV%. 30. ilM. mi ylfSJA ortnr. 31. 33. 33. 24. Tii^h- 310. To reduce an integer or a mixed number to an improper fraction. ORAZ EXERCISES. 1. In 3 units, how many, fourths ? Analysis. — Since in 1 unit there are 4 fourths, in 3 units there are 3 times 4 fourths, or 12 fourths. Hence 3 = ^-. 2. In 4 bushels, how many eighths of a bushel ? 3. How many sevenths of a week in 6 weeks ? 4. How many 9ths in 5 ? 6 ? 8 ? 10 ? 12 ? 5. How many tenths of a dollar in $7 ? In $9 ? 6. How many half dollars will p\iy for a ton of coal that cost $7 ? For a barrel of flour that cost $10 ? 7. How may an integer be changed to thirds? To sixths ? To eighths ? To tenths ? 8. In 5| how many eighths ? Analysis. — Since 1 is equal to 8 eighths, 5 equals 5 times 8 eighths, or 40 eighths, and f added make 43 eighths. Hence 5| = ^i-. 9. In 6 J cords of wood^ how m^^w'^ fourths of a cord? BEDUCTIOK. 105 10. How many 6ths in $8f ? In 12| rods? 11. Among how many boys can you distribute 5} quarts of chestnuts, if you give ^ of a quart to each ? 12. Among how many poor famihes can 4-| tons of coal be distributed, if each family receive ^ of a ton ? WRITTEN EXEnClS ES. 311. 1. Change 75 to the form of a fraction having 27 for its denominator. OPERATION. Analysis. — Smce 1 is equal to 27 twenty 75x27 = 2025 sevenths, 75 is equal to 75 times 27 twenty- sevenths, or 2025 twenty-sevenths. Hence 7 5zz:^ff^ 75 = Hf^. 2. Change 49i^ to hoelfths. OPERATION. Analysis. — Since 1 is equal to 12 twelfths^ 4 9 7 49 is equal to 49 times 12 twelfths, or 588 -| 2 twelfths; ' ^' ' ^ '' " " " ' * 595 twelfths. Hence 49yV = %K 5 8 8 twelfths An integer is reduced to a fractional form ^„„ „ ^^^ by writing 1 under it for a denominator. W + tV=W Thus,93=f;23 = V. 3. Change 81 to a fraction having 24 for its denominator. 4. In 78 pounds, how many sixteenths of a pound ? 5. In 42f weeks, how many sevenths of a week ? 6. How many 20ths of a ton in 16^^ tons ? In 21f| tons ? EuLE. — Multiply the integer by the required denomina- tor, and to the product, add the numerator of the fraction^ and under the result twite the required denominator. Eeduce 7. 207 to fifteenths. 10. 543^ to fortieths. 8. 13611 to eighteenths. 11. 184|f to ninety-fifths. 9. 472^ to twenty-sixths. 12. 2014||- to eighty-fourths. 106 FEACTIOKS. 13. Eeduce 204|J days to twenty-fourths of a day. 14. Change 312 to a fraction whose denominator is 126. 15. Reduce 2146^^ to an improper fraction. 16. Change 1006^^ to an improper fraction, t 31 3. To reduce an improper fraction to an inte- ger, or a mixed number. OliAL EXEJtCISBS, 1. How many units are ^ ? Analysis. — Since 4t fourths equal 1, 18 fourths are as many times 1 as 4 fourths are contained times in 18 fourths, which is 4f times. 2. How many times 1 are -^ ? ^ ? |f ? ^ ? f^ ? 3. How many yards are -^ of a yard ? ^^ ? -^^ ? 4. How many dollars are $^ ? $^ ? Iff ? $f ^ ? 5. In If of a foot, how many feet ? In ^^ of an acre, how many acres ? In ^^ of a ton, how many tons ? WRITTEN EXEHCIS ES. 313. 1. Eeduce ^^ to a mixed number. OPERATION Analysis. — Since 9 ninths equal 1, 21 8 _ 9 1 Q . Q L o ^ 2 ^^^ "^^*^® ^^^ ^^^ *^"^®^ ^ ninths. 2. Change -^^ to a mixed number. 3. In ^^ of a dollar, how many dollars ? 4. How many rods in -^f of a rod ? Rule. — Divide the numerator hy the denominator. Eeduce to integers or mixed numbers, 11. ^|i. 5. w- 8. 6. ^iF- 9. 7. -w- 10. 12. mw"- 13. ^^1j!W^. EEDUCTIOK. 107 314. To reduce fractions to equivalent fractions having a common denominator. ORAL EXERCISES, 1. How ms^uy fourths in 1 ? In |^? 2. How many ninths in 1 ? In J ? In f ? 3. Express f , ^, and |, each as twelfths. 4. Change f and | to fractions of the same denominator. 5. What is a multiple of 4 ? Of 6 ? Of 8 ? Of 9 ? 6. What is a common multiple of 3 and 4 ? Of 4 and 5 ? 7. What is the least common multiple of 3, 4, and 6 ? 8. What is the least common multiple of the denomi- nators of J, f , and f ? Of f , f , and f ? 9. Keduce f and J to eighteenths. To twenty-sevenths. 10. Name some fractions that can be changed to 16ths. 11. Name four fractions that can be changed to 24ths. DEFINITIONS AND PEINCIPLES. 315. A Common Deno^ninator is a denomina- tor common to two or more fractions. 316. The Least Common Denominator of two or more fractions is the least denominator to which they can all be reduced. Since all higher terms of a fraction are multiples of its corresponding lowest terms (307, Note), hence the fol- lowing 317. Principles. — 1. A common denominator of two or more fractio7is is a common multiple of their denominators. 2. The least common denominator of two or more frac- tions is the least common multiple of their denominators. 108 FKACTIOKS. WMITTEN BXEMCISES. 218. 1. Keduce I, |, and f to equivalent fractions having a common denominator. OPERATION. Analysis. — Multiply eacli denominator by the OyQwK 30 o*^^^ *^o, and the product, 30, is a common de nominator of the three. (Pkin. 1.) I X 3 X 5 3 But since the value of the fractions is not tc 3x|x6=-5^ be changed, each numerator must be multiplied -|J|x3=-jf hy the same multiplier as its denominator. Hence, multiplying the terms of ^ by 3 and 5, the result is Jf ; of f , by 2 and 5, the result is fg ; and of | by 2 and 3, the result is Jf . Or, To find the numerators, take such part of the common denomina- tor 30, as the given fraction is part of 1. Thus, i of 30 is 15, etc. Keduce to fractions having a common denominator 2. -f- and f . 3. T^ and f . 4. f, I, and |. 5. VV. i andl. 6. A, \, and i. 7. h h and A. 8. Change |, f , and ^ to equivalent fractions having the least common denominator. OPERATION. Analysis.— First find the least 7)3 7 14 |z=|-| common multiple of the given de- "~^ ^j ^ g 3g nominators, which is 42. This must * ■^^ be the least common denominator 2x3x7 = 42 -^ — ^ of the given fractions. (Prin. 2.) 9. Change |, ^, and |^ to equivalent fractions having ■Uie least common denominator. Rule. — 1. To reduce two or more fractions to equiva« lent fractions having a common denominator. Multiply the terms of each fraction iy the denominators of all the other fractions. BBDUCTIOK. 109 3. To reduce them to their least common denominator. I. Find the least common multiple of the denominators of the given fractions for their least common denominator. II. Divide this common denominator hy the denomina- tor of each of the given fractions, and multiply its numer- ator by the quotient. The products are the new numerators. Mixed numbers must first be reduced to improper fractions. Reduce to fractions having the least common denomi- nator. 10. \, H. and if. 11- -^j. ii and ^\. 1^. M. A. and li. 13. 1H> h H. and H. 14. I, 24, I, andl,-V 15. ^h -h, 7, and IJ. 16. H. -A'o. ii> and ^, 17. li, -3%, H. and ^. additio:n^. ORAL BXBItCISE S. 219. 1. What is the sum of | and | ? Of f and |? 2. How many times 1 is the sum of f , f , and ^ ? 3. Sold ^g^ of an acre of land to one man, -^ to another, and ^ to a third. How much was sold to all ? How are fractions added that liave a common denominator ? 4. Mary paid $| for some ribbon, and $| for a pair of gloves. How much did she pay for both ? Analysis. — She paid the sum of $f and $f . f is equal to 3^, and f is equal to \^ ; y% and \^ are ||, or 1^. Hence she paid flyV 5. A man having | of a ton of coal, bought f of a ton more. How much had he then ? How are fractions added that have different denominators ? 110 FRACTIONS. 6. Henry gave $f for a book, $J for a slate, and IJ for a bottle of ink. What did he pay for all ? 7. What is the sum of |, |, and | ? Of f , J, and ^ ? 8. Find the sum of f, ^, and ^g^. Of f , f, and -^^g-. 9. Find the sum of ^, |, and J. Of |, J, and 3^. 10. A farmer sold 31^ tons of hay to one man, and 5 J to another. How much did he sell to both ? Analysis. — The sum of 3J tons and 5| tons. 5 and 3 are 8 ; and J and I are f , which added to 8 makes 8| tons. 11. A man bought 6^ cords of wood at one time, and 7^^ at another. How much did he buy in all ? How are mixed numbers added ? 12. A man paid S25| for a watch, and sold it for $6^ more than he gave for it. What did he sell it for ? Find the sum 13. Of f and 3f 14. Of 5^ and i. 15. Of 1| and f . 16. Of 2^and6|. 17. Of 8^ and -5^. 18. Of 15i and |. 19. Of 2i and If. 20. Of 5i and f . 21. Of l^V and 12|. 220. Pkinciple. — Fractions can be added only when they have a common denominator, and when they express parts of like units. WMITTEN EXERCISES. 221. 1. Find the sum of |, VV^ and ^. OPERATION. Analysis. — Reduce the given ivKC- I _|- JL 4- JL nz ^ ^^'il^^^ tions to equivalent fractions having 24 + 3 fi ^16 75 -j 1 the least common denominator, which ^^ — TIT — t is 60 (217, 2). Then add their nu- merators, and write the sum, 75, over the common denominator 60, and Jf = 1 J is the required result. ADDITION. in 2. What is the sum of 14f , 25|, and 7| ? OPERATION. 14f = 14if 25| = 25|i 7|- 7H 46 + lt = 48i Find the sum 3. Of A. il. and A. 4. Of I, J, and f . Analysis.— The sum of the frac- tions is f f = 2 J, which added to the sum of the integers 46, gives 48J the required sum. 5. Of 42, 31^^, and 9^\. 6. Of 204^, 50if , and 7^. Rule. — I. Reduce the given fractions to equivalent frac- tions having the least common denominator, and write the sum of the numerators over the common denominator. II. When there are mixed numbers or integers, add the fractions and integers separately, then add the results. 9. 18A + 24 + 1H-? 10. | + 6tV + 21|- + 77=? 12. 124f + 325^^ + 40|f=? 13. Bought 3 pieces of cloth containing 105f, 86f, and 58f yards respectively ; how many yards in all ? 14. If it takes 5^ yards of cloth for a coat, 3^ yards for a pair of pantaloons, and |^ of a yard for a vest, how many yards does it take for all ? 15. Four cheeses weighed respectively 46-|, 48f, 49y\. and 57} pounds. What was their entire weight ? 16. What number is that from which if 244- is taken, the remainder is 63f f ? 17. A farm is divided into 4 iBelds : the first contains 29^^ acres, the second 50ff- acres, the third 41^ acres, and the fourth 69| acres. How many acres in the farm ? 113 PBACTIONS. SUBTRAOTIOF. ORAIi EXEItC IS ES, 332. 1. What is the difference between f and -| ? 2. What is the difference between y^ and j\ ? How is one fraction subtracted from another, each having thd same denominator ? 3. A gentleman who owned a sail-boat sold /^ of it. What part did he still own ? 4. A boy having %l, gave $^ for a neck-tie. What had he left? Analysis. — He had left the difference between $f and $|^. f is equal to y^, and J equals y\ ; -{^ less j% are f^. 5. A man owning f of an acre of ground, sold -J- of an acre. What part remained ? How is one fraction subtracted from another having a different denominator ? 6. Subtract ^ from | ; | from J ; f from -^, 7. Find the difference between ^ and | ; f and |. 8. From a piece of cloth containing 12-|^ yards, 5^ yards were cut. How many yards remained ? Analysis. — The difference between 12 J yards and 5 J- yards. \ from ^ leaves f , and 5 from 12 leaves 7. Hence 7f yards remained. 9. If a ton of coal costs $7f, and a cord of wood $4^, what is the difference in their cost ? How is one mixed number subtracted from another ? 10. What is the value of 3^—2^ ? 8^—2^ ? ^\—\ ? 333. Principle. — Fractions can he subtracted only when they have a common denominator, and when they express parts of like units. SUBTRACTIOl^. 113 WRITTEN EXJEHCISISS 334. 1. Froji J subtract ^j. OPERATION. Analysis. — Reduce the J — ^ = II — if = 3g-^g — II given fractions to equiva- lent fractions having the least common denominator. Hence f | — ^^f = f f . 2. From 134^ take 76 1. OPERATION. Analysis. — Reduce J and f to equivalent 134i^::::134^ fractions having the least common denomi- )v Q5 —. 7511 nator. As Jf cannot be taken from ^, take — 1 or If from 134, leaving 133, and add it to ^\, 5 7|^ making!}. Then ^| from f| leaves i|> and 76 from 133, leaves 57. Hence 57Jf is the result. 3. From ^j take -i^f . 4. From -^ take -^. 5. From || take ^. 6. From 36f take lOf 7. From 112y|^ take 56. 8. From 204^^^ take 39yV EuLE.— I. Reduce the given fractions to equivalent frac- tions having the least common denominator, and lorite the difference of the numerators over the common denominator. II. When there are mixed nmniersy siibtract the frac- tional and integral parts separately, and add the results. If the mixed numbers are small, thej may be reduced to improper fractions and subtracted according to the usual method. Find the difiEerence between 9. I and ^. 11. f| and 2|. 13. 63| and 71^. 10. If and f. 12. 16 and 33^. 14. 106 and 95f|. 15. From -^V take yi^. 17. From 410^ take 226|. 16. From 1,6^35^ take -^. 18. From 428^ take 180|f 19J[ A farmer having 208 acres of land, sold 92-^ acres. How many acres had he left ? 114 FRACTIONS. ADDITION AND SUBTRACTION. ORAL MXJERCIS ES, 335. 1. How much less than 2, is i+| ? 2. How much greater than 2, is f + i + l|- ? 3. What is the difference between 4 and 2| ? 5^ and 7^ ! 4. Mr. Smith sold ^ of his farm to one man, ^ to another, and -J to a third. What part had he left ? 5.! Paid $6^ for a ton of coal, and $3|- for a load of wood. What change must be returned for a ten-dollar bill ? 6. What is the difference between If +f and 5-^^? 7. What is the difference between ■^+i and i+f ? Find the second member of the following equations 10. 3-(|-i) = ? 11. 6i + i-l|=? 13. 13— (8— 2|) + li='-'\ 13. (36— 14) — (| + 3|=? 14. (l| + 3f)-(i+|) = ? 15. (5-34H-_(9A-6) = ? 16. 8TV- 3i+| + 10=? 17. 133sV+9i-li-3i=? WMITTEN EXAMPLMS. 336. 1. The sum of two numbers is 134J, and the less is 36-^. What is the greater ' 3. What number added to 147^ will make 316f ? 3. What number added to 3074- +310f will make 700|? 4. What number must be added to the difference ol 1861- and 314f to make 1043f|? 5. What fraction added to the sum of \, -^^, and ^, will make m ? 6. What must be added to -J, that the sum may be ^ ? MULTIPLICATIOlir. 115 7. Bought a quantity of barrel staves for $160-|, and of lumber for $1136|. Sold the staves for $205| and the lumber for $1240^. What was the whole gain ? 8. A man bought a ton of hay for $15|, a barrel of jBiour for $9^^, and a barrel of apples for $3y\. What change ihould be given to him for 3 ten-dollar bills ? Complete the following equations : 9. * + f-FR=? 1^. 41i + 56- 24^V-4H::=? 13. 120— 51f + 90^— f=? 14. 342-(21A+^-9)=? 10. 8| + 2|-5^=? 11. 48— (164— 3^)=? 15. 176^ + 132| - 26B - ^ = ? 16. $1000 — $500 + $107^% + ^91^ = ? MIJLTIPLIOATIOIS". 237. When one factor is a fractional number, ORAL EXMBC I8E8. 1. What part of a mile is 3 times |^ of a mile ? 2. What part of a dollar is 4 times ^ of a dollar ? 3. How many times 1 is 3 times | ? 4 times J ? 4. At $4 ^ pound, what will 4 pounds of tea cost ? Analysis. — ^Fonr pounds will cost 4 times $f , or %^^y equal to |2i. 5. At $1 a bushel, what is the cost of 6 bushels of oats ? Of 7 bushels ? Of 8 bushels ? Of 9 bushels ? 6. If a horse eat -f- of a bushel of grain in a day, how much will 4 horses eat ? 6 horses ? 8 horses ? 10 horses ? 7. What cost 12 baskets of pears, at $f a basket ? With each class of oral questions in Art. 227, the pupil may Bolve the corresponding written examples on pages 118 and 119. 116 FEACTIONS. 8. What cost 9 pounds of butter at $| a pound ? 9. Show that multiplying the numerator of ^ by 4 multiplies the fraction by 4. 10. Show that dividing the denominator of ^ by 4 multiplies the fraction by 4. How many ways to multiply a fraction by an integer ? 11. Multiply A by 5; ^ by 6; ^ by 5 ; jV by 8. 12. lit $4f a box, what will 5 boxes of raisins cost ? Analysis. — They wiU cost 5 times $4f. 5 times $| are $3f , and 5 times $4 are $20. $20 + $3f = $23f . Hence, etc. 13. At 7^ cents a pound, what will 9 pounds of rice cost ? 14. At $9| a barrel, what is the cost of 6 barrels of flour ? Of 8 barrels ? Of 9 barrels ? Of 10 barrels ? 15. What will 8 yards of cloth cost, at $5^^ a yard ? 16. Multiply 7i by 9 ; 9f by 6 ; 10^ by 7 ; 12^ by 8. 17. What is ^ of 12 yards ? ^ of 24 men ? i of $30 ? 18. Multiplying by ^, ^, J, |-, etc., is the same as dividing by what integers ? When a fractional part of an integer, or of a fraction, is to be taken, the word of, and not times, should be used. 19. At $7 a ton, what will f of a ton of coal cost ? ANALYSis.~It wiU cost f of $7, or 3 times J of $7. :J of $7 is |lf , and 3 times $lf are $5J. Hence, etc. 20. At $12 a gross, what will J of a gross of butts cost ? 21. What will ^ of a ton of hay cost, at $15 a ton ? 22. What is I of 6 ? fof2? fof8? T^^of9? 23. At $5 a yard, what will f of a yard of cloth cost ? 24. If a man can build a wall in 28 days, in what time can he build I of it? f of it ? fofit? 25. If an acre of land produce 45 bushels of corn, how much will f of an acre produce ? |? f? f? ^? MULTIPLICATION. 117 36. What is I of $5? Of $7? Of 116? Of 125? 37. Multiply 50 by I ; 49 by 4 ; 63 by ^ ; 81 by ^. 38. In $1 are 100 cents ; how many cents in J of a dollar? Tni? i? i? ^? ^a^? ^? 29. How many cents in -^-^ of a dollar ? In^?|?|? 30. Which is greater^ f of 15, or 15 xf ? 31. Show that a fraction of an integer equals the pro dnct of the integer by the fraction. How is an integer multiplied by a fraction ? 32. At $12 a ton, what will 5f tons of cheese cost ? Analysis.— It will cost 5| times $12. 5 times $12 are $60, and f of $12 are $4|, which added to $60, make $64J, 33. At 15 cents each, what will 4| melons cost ? d^" What will 7f weeks' board cost, at $9 a week ? 35. How much is 6| times 12 ? 5f times 20 ? 36. Multiply 4 by 81 ; 6 by 7f ; 8 by 9^^^. What 37. Is yV of legations ? 38. Is I of 47 pounds ? 39. Is I of 90 rods? 40. Is -5 of 56 days ? Find the value How many 41. jAre 10 times 6f tons t 42. Are 9 times ll^s^ miles ? 43. Are 7f times 12 men ? 44. Are 12 J^ times 9 minutes ? 45. Of |xl5. 49 46. Of 38xf 50. 47. Of 56xf 51. i8. Of /t X 7. 53. 53. OfT^+|x5. 54. Ofi^-3xi. 55. Of4T^ + 3x5| 56. Of9x9i^ + 30i Of|-x7+i. Of|x9-3i. Of 37 X 1 + 3^. Of 6f x7-lf 338. Principles. — 1. Multiplying the numerator or dividing the denominator multiplies the fraction. (300, 1.) 3. The product of an integer by a fraction is equal to sttch part of tlie integer as the fraction is of a unit. 118 PBACTIONS. WJTITT E 839. 1. lOJtiply WV N JEXJSMCIS JES Itiply ^ by 9. OPERATION. Or, Or, 2- Analysis.— Multiply tho numerator 7 by 9, or divide the denominator 37 by 9 ; either operation will give 2^, the required result. (Pbin. 1.) By uaing the vertical line and cancellation, both operations are combined and shortened. In the first operation, the number of parts or of fractional units is increased, while their size or value remains the same ; in the second operation, the size of the parts is increased, while their nunUier remains unchanged. In like manner multiply 3. A by 12. 3. H ^J 9. 4. ^ by 13. ^ by 15. ^ by 36. -1% by 21. 8. 9. 10. Th by 17. m by 22. U by 44. 11. Multiply 72 by i. OPERATION. 72xi=72-f-9x4=33 Or, 72xi=-^^|^=32 Find the product 12. Of 75 by ^Sj.. 13. Of 7by^V 14. Of 56 by T^. Or, Analysis. — To multi- ^. 8 ply 73 by i, is to find | '"^ of 73. f of 73 is 4 times ^ i of 73, which is 33. 33 (Prin. 3.) 15. Of 168 by If. 16. Of 200 by A. 17. Of 315 by i|. 18. Of 19byi4. 19. Of 448 by /f. 20. Of 572 by ^. A fraction is multiplied by a number equal to its denominator by cancelling the denominator. Thus, J x 8=7. Cancelling a. factor of the denominator multipliea the fraction by that factor. Thus, j\ x 4=|. MULTIPLICATION. 110 21. Multiply 17| by 6. OPERATION. Or, 17| = H^ 17| 6 103i Multiply 22. 127f by 12. 33. 85i by 15. S0 155 08 3 310 103i ANALyBis.—To multiply 17f by 6, multiply the fraction |, and the in- teger 17 separately and add theif products, which gives 103 J, the re' quired product. Or, Reduce the mixed number to an improper fraction, and multiply as in Ex. 1, which gives the same result. 24. 128^T by 42. 25. 246f by 16. 26. 314iV by 48. 27. 750^ by 17. 28. Multiply 140 by 9|. 140 H 1260 1353^ OPERATION. Analysis.— To multiply 140 by 9|, multiply by the fraction f , and by the integer 9, separately, and add their products, which gives 1353|^, the re- quired product. Or, Reduce the mixed number to an improper fraction, and multiply as in 1 3 5 3 -J- Ex. 1, which gives the same result. Or, 9f = -^ 140 29 4060 Multiply 29. 96byl2|. 31. 304 by 24^. 33. 560 by 23^^ 30. 216 by 16f 32. 198 by 18f 34. 715 by 14,^, 35. Multiply 327yV by 72 ; 2466 by 84| ; 759 by J|. 36yWhat will 120 dozen of hose cost at $4f a dozen ^ 37. At $20 a ton, what will If of a ton of hay cost ? a 38. if a city lot is worth $3145, what is -^ of it worth? / 39. What will 142 yards of curbing cost at $6-| a yard? 40. At $f| a yard, what is the cost of 8 yards of cloth? Of 24 yards ? Of 64 yards ? Of 120 yards ? 120 FRACTIOKS. 330. When both factors are fractional nvimbers. ORAL JEXJEHCISES. 1. A boy having ^ ot a melon, gave ^ of it to his sister. What part of the melon did she receive ? 2. What part of 1 is i of i? Is J of ^ ? Is ^ of ^ ?\ 3. Vhat part of lis I of I? -|off? ioff? iof^? 4. Which is greater, | of |, or |^ of | ? ^ of |, or J- of |? lofi,oriof|? 5. If I own -f of an acre of land, and sell J- of it, what part of an acre do I sell ? What part do I retain ? 6. If a yard of silk is worth $f , what is ^ of a yard worth ? 7. A boy having $| gave f of it for a knife. What part of a dollar did he pay for the knife ? Analysis.— He paid f of $|, or 5 times J of $|. J of $f is $j\, and 5 times $^ are ^f , or $f . 8. At $^ a gallon, what will f of a gallon of syrup cost ? Fractions with the word of between them are sometimes called Compound Fractions. The word of is equivalent to the sign ( x ) of multiplication. Thus, f or of f x H -^ A x 5^. 45. Find the value of J-^, or of f^ -^TT|. 46. If a man spend $4| a month for tobacco, in what time will he spend $2^ ? Find the value 47. Of ^±'A 48. Of if-f 49^0f|x^--6|-5^. 50.,Of (16f -T-18})xl7. 51.iOf 9|x8xTV-^-^4• 53./Of (7^-5-^)-(4i+6i). EELATIOIT OF NUMBEES. 339. Numbers to be compared with each other, must be so far of the same nature, that one may properly be eaid to be a part of the other. Thus, wc may compare a dap with a week, since the one is thci (Seventh part of the other ; bat we cannot say, that a day is any part of a mile, therefore a day cannot be compared with a mile. 240. Principle. — Only like numbers are so related as to be compared with each other. RELATION OF NUMBERS. 133 341. To find what part one number is of another. DUAL EXERCISES. 1. What part of 5 is 3 ? ANAI.YSIS. — Since 1 is ^ of 5, 3 is 3 times J or f of 5 ; or it ig 9 i nded by 5. Hence 3 is | of 5. 2. What part of 9 is 5 ? Of 12 is 7 ? Of 24 is 18 ? 3. 10 yards are what part of 25 yards? 8 pounds, of 20 pounds ? 9 eggs, of a dozen ? 10 ounces, of a pound ?a 4. ($15 are what part of $50? v $60, of $72? /9 days, of 90 days?x(6 days, of a week?^ 7 months, of a year? 5. If an acre of land can be bought for $48, what part of 9n acre can be bought for $8 ? For $12 ? $16 ? $24 ? f,. What part of 3 is | ? /ANALYSIS. — 1 is i of 3, and | of 1 is f of J of 3, or i x f =^\. Or, 3 = V" ^ ^^^ relation of %*- to f is the same as that of their numerators 34 and 5, or -f^. Hence f is /^ of 3. 7. What part of 9 is i? Of 8 is -A-? Of 20 is |? 8. What part of 15 is 1| ? Of 18 is 2^ ? Of 25 is 6^? 9. f of a month is what part of 8 months? 10. What part of f isf? Analysis.— 1 fifth is J of 4 fifths, and 1, or 5 fifths, is 5 times J or } ; hence f is f of f , or ^, equal to f . Or, 1=11, and |=|t, and the relation of | to f is the same as ^.hat of 10 to 12, or it=f. Hence | is f of |. 11. What part of | is I ? Of A is | ? Of | is ^ ? 12. What part of ^ is |? Of If is i^? Of -^ is ^P 13. What part of 3^ is f ? Of 4^ is 3 ? Of ^ is If ? 14. What part of 7i is 1^^ ? Of If is \\ ? Of 3| is 2^ ? 15. What part of 9 miles are | of 8 miles ? i of 10 miles? 184 FRACTIONS. WRITTEN £:XMBCI8 B8, 243. What part of 1. 96 is 72? 2. 56 is -J? 3. 120 is 90? 4. ilisff? 9. 150isl2i? 10. 24|isi|? 11. 160is26|? 12. 212^ is 424? 5. If is A? 6. 6isf|? 7. 80is5i? 8. 13|is2|? 13. A man having $150, gave $25 for a robe, and | of the remainder for a harness. What part of $150 had he left ? 14. Bought a horse for $275, and sold him for $160. For what part of the cost was he sold ? 15. If from 18| yards of cloth 2-| yards are cut, what part of the whole is taken ? 16. If 15 tons of coal cost $112^, what part of $112| will f of a ton cost ? 343. To find a number when a fractional part of it is given. OnAJL JEXEBCI8ES. 1. 7 is -J of what number ? Analysis. — 7 is J of 5 times 7, which is 35. Hence 7 is J of 35. 2. 12 is \ of what number? \ of what number? 3« 9^ is \ of what number ? \ oi what number? 4. 7| is -^ of what number ? -^ ot what number ? 5. ^3Q,is I of what number ? Analysis. — Since 36 is | of a certain number, \ of the number Is \ of 36, or 12 ; and the number is 4 times 12, or 48. 6. 42 is f of what number? ^ of what number? 7. 75 is 4 of what number ? f of what number? 8. 84 is -If of what number ? ^^ of what number ? 9. 15 J is f of whafc number ? -^ of what number ? RELATION OF NUMBERS. UffiS 10. -| is I of what number ? f of what number? 11. If is ^ of what number ? | of what number? 12. 3f is -^ of what number? f of what number ? 13. 36 is I of how many times 4? Analysis. — 36 is f of 8 times l of 36 which is 32, and 4 is con* tained in 32, 8 times. Hence 86 is | of 8 times 4. 14. 28 is yV of how many times 8 ? 12 ? 9 ? 16 ? 15. 35 is I of how many times | of 28 ? | of 30 ? 16. 16| is J of how many times | of 56 ? t^ of 48 ? 17. I is 4 of how many times ^ of ^ ? ^ of | ? 18. I of 56 is yV of what number? Analysis. — | of 56 is 3 times J of 56, which is 21 ; and 21 is yV of 10 times ^ of 21, which is 30. Hence f of 56 is j\ of 30. 19. f of 27 is f of what number? | of what number ? 20. I of f of 64 is I of what number ? 21. |- of J of 72 is ^ of f of what number? 22. i of 54 is I of how many times 5 ? 7 ? 8 ? 9 ? 23. f of ^ of 63 is i of I of how many times 10 ? 9 ? 24. 4 of 56 is f of 3 times what number? Analysis.— 4 of 56 is 32, and 32 is f of 36, and 36 is 3 times i of 36, which is 12. Hence f of 56 is 5 of 3 times 12. 25. I of 64 is f of 9 times what number ? 26. f of 21 is J of 8 times what number ? 27. Paid $60 for a sideboard, which was | of the cost; of a bookcase. What was the cost of the bookcase ? 28. A scarf cost $lf, which was | of the cost of a v^rst What was the cost of the vest? 29. Paid $100 for a sleigh, which was | of 3 times wnat I paid for a harness. What did I pay for the harness ? Written Exercises of this kind are included in the review example* 13i FRACTIOJS^S. REVIEW OF FRACTIONS. ORAL EXAMPLES, 344. 1. What fraction added to 4 will make | ? 2. What number taken from 25f will leave 7f ? 3. If the sum of two fractions is -|f and one of them ii J, what is the other ? 4. From what number must 3f be taken to leave 5^ ? 5. A boy spends | of his earnings for board, and \ for clothing. What part has he left ? 6. The less of two numbers is 5^^ ^^^ their difference J. What is the greater ? 7. What number divided by | will give a quotient of 1| ? 8. The product of two numbers is 4, and one of them is 18 ? What is the other ? 9. If 2 be added to both terms of the fraction |, will its value be increased, or diminished, and how much ? 10. If 2 be added to both terms of the fraction |, will its value be increased, or diminished, and how much ? 11. If a box of tea cost $21|, what will f of a box cost? 12. A man owning f of a steam-mill sold ^ of his share. What part of the whole mill does he still own ? 13. A farmer sold 40 acres of land, which was /^ of his whole farm. How many acres were there in his farm ? 14. A man sold f of his farm, and had 100 acres left. How many acres had he at first ? 15. Bought a watch and chain for $120, the chain cost- ing f as much as the watch. What did each cost ? 16. A, B, and C together own a yacht. A owns f of it, and B. f of it. What part does own ? REVIEW. 137 17. A farmer put all his grain into 4 bins : in the first he put f of it, in the second ^, in the third ^, and in the fourth 40 bushels. How many bushels of grain had he ? 18. Bought 6 mats at $| each, and had $5 left. Ho^ much money had I at first ? 19. How many bushels of grain can be put into 15 bags, if they hold 2| bushels each ? 20. If 5 men can do a piece of work in 10| days, how many days will it take one man to do the same ? 21. If a man can build 6 rods of wall in 1 day, how many rods can he build in 7| days ? 22. How much less than $10 will 7 pounds of tea cost, at $f a pound ? 23^" George having $1|, gave | of it for a knife. What part of a dollar did he give for his knife ? 24. At $12|^ a ton, what will f of a ton of hay cost ? 25. Bought a cow for $45^, and sold her for j\ of what she cost. What did I lose ? 26. If a man has 22| bushels of clover-seed, and he sells f of it, how much has he left ? 27. What will 4^ days' wages come to at ^2^ a day? 28. A man spent | of his money, and then found that $15 was f of what he had left. What had he at first ? 29. A man paid $30 for a cow, f of the cost of which was f of the cost of a horse. What did the horse cost ? 30. How many pounds of tea worth $-^ a pound, must be given for 9 bushels of apples worth $| a bushel ? 3r:f How many building lots of yV ^^ ^^ ^^re each are contained in 1^ acres of land?X 32/At $1 each, how many books can be bought for $3^?><^ 33. If I of a box of figs cost $1J, what will 1 box cost? V 138 FRACTIONS. 34;; If 5^ dozens of eggs cost $lf , what is the cost of 1 dozel3^^ Of 2^ dozens ? Of 3 J- dozens ? 3ot /|f f of a barrel of flour cost $8, what cost 9 barrels ? 36. If 3 yards of flannel cost $|- , what will 8 yards cost ? 37yHow much tea can be bought for |4|^, at $f a pound? 38. If I of a bushel of quinces cost $|, what will 1 bushel cost ? 2 J bushels ? 3| bushels? 39. If a gallon of syrup cost $|, how many gallons can be bought for $^^ ? For $1| ? For || ? 40. A man having $24, gave f of his money for clover- seed at $5 J a bushel. How many bushels did he buy ? 41. What number taken from 2^ times 12f leaves 20f ? 42. A coal dealer sold f of what coal he had on hand for $90, at the rate of $6 a ton. How many tons had he ? WRITTEN EXAMTLlSa, 345. 1. Change \ of f , |, |, and f , to equivalent frac- tions whose denominator shall be 72. 2. Find the least common denominator of f , f , J, and If. 3.. The less of two numbers is 1206| and their differ- ence 470f . Find the greater number. 4. Find the value of (3 x | x | x 4|) — (3| x | x 4 x |). 5. What number multiplied by | will produce 1825| ? 6. What number diminished by f and f of itself leaves a remainder of 144 ? "X 7. If I of a farm is valued at $1729J, what is the value of the whole ? 8. A man gave ^, |, and \ of his money for different objects and had $1500 left. How much had he*at first ? REVIEW. 139 9. K the dividend is |, and the quotient -^, what is the divisor? 10. \A man owning | of a cotton mill, sold f of his share for $4560|-. What was the value of the mill? 11. f A stone mason worked 23^ days, and after paying ^ of his earnings for board and other expenses, had $53-| left. (What did he receive a day ? 12. Gave 6f pounds of butter at 36 cents a pound, for 3^ gallons of oil. What was the oil worth a gallon ? 13.1 A person having 271^ acres of land, sold ^ of it to one man, and f of it to another. What was the value of the remainder at $57| an acre ? 14. A lyian's family expenses are $2465^^ a year, which is f of his income. What does he save ? J 15. If 7i tons of hay cost $120, how many tons can be "^ bought for $78 ? 16. If a man travel 240 miles in 5f days, how far would he travel in 3|- days ? 17. A can do a certain piece of work in 8 days, and B can do it in 6 days : in what time can both together do it ? 18. A, B, and C can do a piece of work in 5 days ; B and C can do it in 8 days : in what time can A do it alone ? 19. If I of 4 acres of land cost $205|, what will | of 2 icres cost ? 20. Bought I of 25^ yards of cloth for \ of $177f What was the cost per yard ? 21. If I of a farm is worth $9000, what is ^ of it worth ? 22. If 8 be added to both terms of the fraction ||, will its value be increased, or diminished, and how much ? 23. If 8 be added to both terms of the fraction ^, will its value be increased, or diminished, and how much ? 140 FEACTIONS. 24. How many bushels of oats at $f a bushel, will pay for I of a barrel of flour at $d^ a barrel ? 25. 'A man at his death left his wife $12500, which was ■J- of f of his estate. At her death she left -| of her share to her daughter. What part of the father^s estate did the iaughter receive from her mother ? 26. Paid $1837^ for 3675 bushels of oats. What was the cost a bushel ? 27. A merchant bought a cargo of flour for $2173^^, and sold it for 11 of the cost, thereby losing $.25 on a barrel. How many barrels of flour did he purchase ? 28. A man owning | of 156|- acres of land, sold ^ of | of his share. How many acres did he sell, and what was the value of the remainder of his share, at $42^ an acre ? 29. A horse and wagon cost $360 ; the horse cost 2-|^ times as much as the wagon. Find the cost of the wagon. 30. If $7^ will buy 3^ cords of wood, how many cords can be bought for $31|^ ? 31.\ A dealer sold 7 barrels of apples for $32|^, which was f as much as he received for all he had left, at $4 a barrel. How many barrels in all did he sell? 32. If f of 9 bushels of wheat cost $13|, what will | of a bushel cost ? 33. \A man engaging in trade lost | of the money he in- vepted, after which he gained $740, and then, had $3500. What was his loss ? 34. A boy having lost ^ of his kite-string, added 45| feet ; the string was then f of its original length. What was its original length ? 35. There are two numbers the sum of which is 4^, and their difference |. What are the numbers ? REVIEW. 141 36. A man invests ^ of his money in cotton, J in sugar, ■^ m molasses, and the remainder, which is $2543, in dried fruits. What is the amount of each investment, and the total amount ? 37. If ^ of 3^ times 1, be multiplied by -g, the pro- duct divided by f, the quotient increased by 4J-, and the 3um diminished by f of itself, what is the remainder ? Keduce to their simplest form : I off 38. 39. _ii±i_ x3. 40. 41. iofj |ofi" I of 15 "^ 11 X If + Ion I off 4|xA Complete the following equations : /2i-| of 1| 42, 43. n/ • 8f~* 7 45. 1 i-. 1 + 1 44. (7i+lH-8f +3tt)-(3i+||--3| + 143V)= ? 246. SYNOPSIS FOR EEVIEW. f 1. EquAL Pabts. 3. Peinciples, 1 and 3. tfRAPTTHMQ J 3- DEFiNmoNS. 5 1- A Fraction. FRACTIONS. < ] 8. A Fractional Unit. 4. Expression op Fractions. 5. Teems. \i Denominator. Numerator. 143 FRACTIONS. SYNOPSIS FOR REVIEW.— CoNTii^UED. 1. Classification. 2. Definitions. 1. Proper Fractions. 2. Improper Fractions. 1. Mixed Numbers. 2. Reciprocal of a Number. 3. " Fraction. Value of a Fraction. General Principles, 1, 2, 3. General Law. r r 1. Reduction. .2. HigJier Terms, 1. Defs. 1 3 ^^^^^ ^^^^^ 202. t 4. Lowest Terms, 2. Principle. . 3. Rules, 1, 2. 210. Rule. REr>UCTION. - 212. Rule. f 1. Common DeTiomU ' 1 nator. 1. Defs. 1 2 ^^^ Common De^ 214. I nominator. . 2. Principles, 1, 2. ,3. Rule (1); I, II. (2.) "™- u:Brn.. «— uir:™ MULTIPLICAT] ""■{^ 7. Principles, 1, 2. ) Bute I, 77(1). ►0. Principle. f i^w^e /, // (2). 8. 9. 10. Division. 11. \ 233. Principle. ) ^^^^ ^ ^ ^^ I 236. Principle, f Relation of Numbers. Principle. \ ^m^ ORJLJO JEXEHCISJES, 247. 1. If a unit be divided into 10 equal parts, what is each part called ? What are 2 parts ? 3 parts ? 4 parts ? 2. What is the fractional unit ? 3. If 1 tenth of a unit be divided into 10 equal parts^ what is each part called ? What are 2 parts? 4 parts? 5 parts ? 7 parts ? 12 parts ? 25 parts ? 4. Whatis^VofiV? T^of^? T^of^? 3^of^? 5. If a unit be divided into 100 equal parts, or each tenth into 10 equal parts, what are the parts called ? 144 DECIMALS. 6. Wliat part of 1 tenth is 1 hundredth ? How many hundredths in 1 tenth ? 7. Tf 1 hundredth of a unit be divided into 10 equal parts, what is each part called ? What are 3 parts ? What are 8 parts? 9 parts ? 15 parts ? 8. Whatis^^oof^oofi^cr? -foOfTio? Aofyio? 9. If a unit be divided into 1000 equal parts, or each hundredth into 10 equal parts, what are the parts called? What are 12 parts ? 26 parts ? 42 parts ? 10. AVhat part of 1 hundredth is 1 thousandth ? How many 1 thousandths is 1 hundredth ? EOTATIOI^ AND NUMEEATIOI^. 348. A Decimal Fraction is one or more of the decimal divisions of a unit. Thus, -^, -^, y|^, t|^, etc., are decimal fractions. Decimal Fractions are commonly called Decimals,^ (16.) 349. Decimals are like other Fractions, except that their denominators increase and decrease by the uniform scale of 10. The fractional units are, therefore, always tenths, hundredths, thousandths, etc. 350i The Decimal Sign ( . ), called the decimal point, is used to distinguish a decimal from an integer, and must always be placed before the numerator of the decimal. - * The terms fraction and decimal will hereafter be used to dis- tinguish the common from the decimal form of expression. Thus, ■^^^y and .75, are two forms of expressing the same thing* For convenience we shall call the first form a fraction, and the other a decimal. KOTATIOK AND NUMERATION. 145 251. The position of the decimal sign indicates the denominator, and determines the value of the decimal expression. Thus, -^ is expressed .7. -^^^ is expressed .126. t¥o" " .36. T^oWo" " .1425. 353.^The Denominator of a decimal fraction is always 10, 100, 1000, etc., or 1 with as many ciphers annexed as there are figures in the given decimal. Thus, .4 z= ^ ; .09 =z t! ; .007 = ^\^, etc. 353. The Numerator of a decimal fraction when expressed alone, must have as many decimal places as there are ciphers in the denominator. Thus, -j^ = .8; ,3^:=. 12; -,1^^^=.. 125, etc. If the numerator does not contain as many figures as there are ciphers in the denominator, prefix ciphers until the number of places is equal to the number of ciphers in the denominator, and prefix the decimal point. Thus, -^ = .07 ; YoV?y = -^^^^ ^^^' 2i54c. Decimal fractions may be written in two ways ; either as other fractions, the denominator being expressed, or, in decimal notation, the denominator being omitted. Thus, ■^, or .5 is read 5 tenths and is ^^ of 5 units, yf 0. « .05 '' 5 hundredths, " -^^^5 tenths. T()Vo>"-005 " 5 thousandths, " ^ ^^ 5 hundredths. 355. The value of any decimal figure is always ^ of the value of the same figure in the next place to the left. 356. When an integer and decimal are written to- gether, the expression is a Mixed Numler (195). Thus, 7.12 and 26.134 are mixed numbers. \ 357. The relation of decimals and integers to each other is clearly shown by the following 7 146 decimals. Table. • og 1 S 1 CQ OQ ^ -§ OQ -4^ 1 •i-H % § 1:5 o 1 QQ CD o5 g •T3 C3 QQ o -1-3 5 .2 r— t -d a nd ^ 1 g n3 tg ^ 1 1 1 is 1 CQ M "A 4-» 1 ;« •^ ^ 9 8 7 6 6 4 3 2 1 .2 3 4 5 6 7 8 9 Integers. Decimals. The number is read 987 million 654 thousand 321, ««d 23 million 456^ thousand 789 hundred-millionilis, A decimal takes the name of its right-hand order. 358.' In decimals^ as in integers, make the order of tinits the starting-point of notation and of numeration, extending the scale to the left of the units' place in writing integers, and to the right of the units' place in writing decimals. The first order to the left of units is tens, and the first order to the right of units is tenths ; the second order to the left of units is JiundredSy and the second order to the right is handredth% ; the third order to the left is thousands, and the third order to the right is thousandths, and so on, the integers on the left, and the decimals on the right, equally distant from the units' place, corresponding in name. 359. Hence, both in integers and in decimals, the value of any jBgure is determined by the position of that figure, and is always ten times the value of the same figure in the next lower order, or 1 tenth the yalue of the same figure in the next higher order. Hence, 360. In writing decimals, vacant orders must be filled with ciphers. (36, 2.) KOTATIOI^ AKD NUMERATION. 147 Dictation exercises, both oral and toritten, should be given, until the pupil can lorite and read decimals with rapidity and correctness. OrcUy thus, Ques., "The denominator of a fraction is 100, the nu- merator 7 ; what will express the decimal ? '' The prompt response should be, ''Pointy naught, seven, read, seven-hundredths" (.07). Ques, '* The denominator is 1000, the numerator 35." Ans, ''Point, naught, three, jive, read thirty-five thousandths *' (.035), etc. Also the converse ; thus, Qves. '* Point, naught, eight ; what will express the fraction?" Ans. " The numerator is eighty the denom- inator one hundred, and the fraction is eight-hundredths" (xf^j). Ques, " Point, naught, one, five?" Ans. " The numerator is fifteen, the denominator is one thousand, and the fraction fifteen-thou- sandths " (xif ^), etc. WMITTEN EXEItCISJES, 261. Express in the form of a fraction. 1. 2. .12. .16. 3. 4. .138. .003. .2162. .0056. Express in the form of a decimal, 9. 10. 363 206 iinnr- 11. 12. toVtt* 13. 14. 3 27 10 0- 15. 16. .14036. .00035. 42 10 000' 10000 0* 10660- Prefixing a cipher to a decimal multiplies the denominator by 10, and hence divides the decimal by 10 (300, 2). Thus, .5 = -j§^ ; .05 = ^f^ ; .005 zz: .^^ ; or, .5 -^ 10 = .05 ; .05 -^ 10 = .005, etc. 363. Kejecting a cipher from the left of a decimal divides the denominator by 10, and hence multiplies the decimal by 10 (300, 1). Thus, .007 = y^Vir; -07 =t^; .7 ~ -^^^ ; or, .007 x 10 = .07 ; .07 x 10 = .7. 364. Annexing a cipher to a decimal multiplies both numerator and denominator by 10, and hence reduces the fraction to higher terms (300, 3). Thus, .3 = -^; .30 = ^; .300 = tW^ ./ 148 DECIMALS. 365. Eejecting a cipher from the right of a decimal divides both numerator and denominator by 10, and hence reduces to lower terms (300, 3). Thus, tVoV = -6^^; ^:zz.60; -i^^.6. From the foregoing explanations are deduced the fol- lowing 366. 'Principles. — 1. Decimals are governed hy the same latvs of notation as integers. Hence, 2. The value of any decimal figure depends upon the place it occupies at the right of the decimal sign, (358.) 3. Every removal of a decimal figure one place to the right diminishes its value tenfold. (363.) 4. Every removal of a decimal figure one place to the left increases its value tenfold. (363.) 5. Ciphers may he annexed or rejected at the right of any decimal^ without changing its value. (364, 365.f WniTTEN EXERCISES. 367. Express in figures and decimally ; 1. Seventy-five thousandths, yf^ = •^''^• 2.'^Fifteen hundredths. 3. Seven thousandths. 4. Fifty-three thousandths. 5. Nine ten-thousandths. 6. 22 ten-thousandths. 7. 245 ten-thousandths. 8. 1042 hundred-thousandths. 9. 14605 millionth^V 10. au 12. tW^v- 13. m^i^. 1^* 1000 00' 15. 84^1^(7. 16. i^iA/WV 17. 60y,jl^. I EuLE. — I. Write the numerator of the decimal as if an integer, toriting ciphers in the place of vacant orders to give each significant figure its proper value, and place the decimal point before tenths, y NOTATIOX AKD NUMERATIOl^. 149 II. (Read the decimal as if an integer, and give it the name of its right-hand order. V In like manner express decimally the following frac- tions and mixed numbers : 18. 596 thousandths. 19. 625 ten-thousandths. 20., 12 ten-thousandths. 21. 74 millionths. 22. 105 ten-millionths. 23. 9.9010 billionths. 24. Four hundred thirty-seven thousand five hundred 49 millionths. 25. Three million forty thousand 12 ten-millionths. 26. Six hundred and 24 hundred-millionths. 27. Four hundred ninety-five million seven hundred five thousand and 43075 ten-millionths. 28. Four million seven hundred thirty-five thousand and 903624 hundred-millionths. 29. 1 7 1 (5"- 33 205AV 30. 10^0 0- 34 68iooLo- 31. iWVVA. 35 705^5^^- 32. 1 SIO* 36 300^«Wi^o-. 1000 0,0 00- Copy and read the following decimals and mixed num- bers : 37. .705. 45. 18.0031. 53. .00078. 38. .0023. 46. 6.306. 54. .305004a 39. .3607. 47. 49.0703. 55. .0003006. 40. .00705. 48. 10.0064. 56. 42.0637. 41. .400564. 49. 22.09042. 57. 108.0094. 42. .000256. 50. 1.10106. 58. 230.40685. 43. .0010275. 51. 14.00370. 59. 30.26002015. 44. .0000407. 52. 70. 00063. 60. 8.040103463 150 DECIMALS. DECIMAL CUERENOY. \ 268. Cu7*rency is coin, bank-bills, treasury notes, etc., employed in trade and commerce. 369. A Decimal Currency is a currency whose denominations increase and decrease by the decimal scale 370. The Legal Currency of the United States is a decimal currency ; it is sometimes called Federal Money ^ because issued by the Federal Government. Table. 10 mills {m.) make 1 cent, c, or ct. 10 cents ''1 dime. d. 10 dimes or 100 cents ^^ 1 dollar. $. 10 dollars '^ 1 eagle. E. 371. Since the dollar is the unit of United States Money, dimes, cents, and mills are respectively tenths, hundredths, and thousandths of the unit. 373. Dollars should be written as integers, with the sign ( I ), prefixed ; and dimes, cents, and mills, as deci- mals, with the decimal point at their left, or before tenths. Thus, 7 dollars 3 dimes 4 cents 5 mills, are written $7,345. 373. The denominations eagles and dimes are not re- garded in business operations, eagles being tens of dollars, and dimes tens of cents. Thus, $34.27 is read 34 dollar? ^7 cents, instead of 3 eagles 4 dollars 2 dimes 7 cents. 374. Since the two places of dimes and cents, or of tenths and hundredths are appropriated to cents^ when the number of cents is less than 10, write a cipher in the place of tenths. Thus, 9 cents are written $.09. (73.) REDUCTION. 151 275. The half-cent may be written, either as a frac- tion {^), or as 5 mills. Thus, thirty-seven and a half cents are written $.37i, or $.375. 376. Cents are often written as fractions of a dollar. Thus, $9.28 may be also written $9^^^. 377. In business transactions, if the mills in the final result are 5 or more than 5, they are considered a cent, if less than 5, they are not regarded. Thus, $5,197, would be called $5.20, and $5,194 would be called $5.19. 278. Principles. — 1. Decimal currency is expressed according to the decimal system of notation. 2. All the operations in Decimal Currency are the same as the corresponding operations in Decimals. V/ REDUOTIOK OF DECIMALS. 279. To reduce decimals to units of lower or higher orders. ORAIj EXETtC is ES, 1. How many tenths in 2 units ? In 5 units ? 2. How many tenths in 20 hundredths ? In .40 ? 3. How many hundredths in 2 units ? In 4 units ? 4. How many hundredths in 200 thousandths ? 5. How many hundredths in 5 tenths ? In .6 ? .7 ? .8 ? 6. How many thousandths in .06 ? In .25 ? .48 ? .75 ? 7. How many hundredths in .150 ? In .260 ? In .2500 ? 8. In 400 thousandths how many hundredths ? Tenths ? 9. How many tenths of a dollar in $6 ? Hundredths ? 10. Change 4 dollars 50 cents to cents. To mills. 11. How many dollars are 300 cents ? 540 cents ? 12. How many cents are 2600 mills ? Dollars ? / IM DECIMALS. 13. What is the decimal expression for 5 cents ? Ans, Sign, point, naughty five; read five hundredths ($.05). 14. Express decimally 7 cents ; 9 cents ; 15 cents. 15. 'Express decimally 7 mills ; 5 cents 6 mills. 16.1 Express decimally 2 dollars 45 cents and 6 mills. Ans. Sign, two, point, four, five, six; read, two and four hundred fifty -six thousandths dollars ($2,456). 17. What is the decimal expression for 84 cents 5 mills ? 18. 'Change .3 to hundredths; to thousandths. 19.VChange .4 and .05 to thousandths ; .07 and .01 20i Change .5, .08, and .023 to equivalent decimals, having a common denominator of 1000. Also, .14, .009, and .6. .7, .007, and .091. 21. (Eeduce .7, .150 and .600, to equivalent decimals, having the least common denominator. Also, .50, .250, and .1700. .43, .006, and .0214. 380. From the foregoing it appears, 1. That dollars may be reduced to cents by annexing two ciphers ; and to mills, by annexing three ciphers. Omit the sign $ and write cts. or m. after the result. 2. That cents may be reduced to mills by annexing one cipher. 3. That cents may be reduced to dollars by pointing off tivo figures from the right ; and mills to dollars, by pointing off three figures from the right, and prefixing the sign ( $ ). 4. That mills may be reduced to cents by pointing oC jne figure from the right. 5. That two or more decimals are reduced to a common denominator by annexing or rejecting ciphers at the right until the decimal places of all are equal. REDUCTIOK. W3 5. $57 to mills. 6. 86 cents to mills. (380, 2.) 7. $.763 to mills. 8. $.471 to mills. 12. 846 mills to cents. 13. 50000 mills to dollars. 14. 61040 cents to dollars. wit ITT EN EXERCISES. 281, Reduce 1. $85 to cents. (380, 1.) 2. $615 to cents. 3. $24.06 to cents. 4. $9,206 to mills. Change 9. 486 cts. to dollars. (280, 3.) 10. 32462 cents to dollars. 11. 40327 mills to dollars. 15. Seduce .7, .05, and .304, each to hundred-thou- sandths. (280, 5.) 16. Eeduce 2.5, .107, and .0008, each to ten-thousandths. 17. Change 4, 2.17, .136, and .0408 to equivalent deci- mals having a common denominator. 18. Reduce 9 tenths, 24 thousandths, 109 hundred- thousandths, and 47 millionths to equivalent decimals having the least common denominator. Also, 19. 100.03, 41.0034, .475, .0753, and 6.00044. 20. .84003, 120.4, 5.00031, and 15.240007. 282. To reduce a decimal to a fraction. DUAL EXERCISES. 1. How many halves in -^ ? In -/^ ? In -f^ ? 2. How many fifths in ^ ? In ^ ? ^Vb"? -^ ? 3. How many fourths in ^§4? In .50 ? In .75 ? 4. How many twentieths in -^ ? In -^-^ ? In .20 F 5. In .50 how many halves ? Fourths ? Tenths ? tu DECIMALS WMITTJEN BXEBCISES. 383. 1. Change .375 to an equivalent fraction. OPERATION. Analysis. —The numerator is 375, the de- .3 75 = ^Wtr = i nominator 1000, and the decimal expressed as a fraction is -f^^^ = f . Hence .375 = |. Change to equivalent fractions, 2. .16. 3. .125. 4. .625. 5. $.75. 6. $.375. 7. $.655. 8. .024. 9. .5625. 10. .3125. 11. $.875. 12. .0008. 13. .9375. Rule. — Omit the decimal point, supply the proper de* nominator^ and then redtice the fraction to its lowest terms. 14. Eeduce .13J^ to an equivalent fraction. Operation.— 13^ = ^ = i?- = -?_. ^ 100 300 15 Reduce to fractions in their lowest terms, 15. $.37i. 16. e.62|-. 17. $.08^. 18. .06i. 19. .58^. 20. .93f. 21. $.33 J. 22. $.66|. 23. ('$.16f. 24. .1M^. 25. ,i444|. 26. .0008}. Express b^ 27. $15.4. f an integer and a fraction, 29. $9,625. 1 31. 24.26|. 33. 38.41|. 28. $36.75. 30. $27,375. 1 32. 84.05|. 34. 104.00|. 384, To reduce a fraction to a decimal. OnAIj EXJEItC IS ES , 1. How many tenths in |^ ? How many hundredths <, Bow many thousandths? 2. How many tenths in | ? Hundredths in f ? In f ? 3. How many hundredths in -^ ? In -^ ? In -^^ ? BEDUCTIOlff. ISd WBITT EN EXERCISES. 385. 1. Eeduce |- to an equivalent decimal. oPERATioiff. Analysis. — Annex the same ^ io.ojL 626 a<^ K number of ciphers to both terms 8 - 8-000 - 1000 --^^^ Of the fra^etion and divide the re- suiting terms by 8, the significant figure of the denominator, to cotaai the decimal denominator 1000, Then change to the decimal form, (253.) 2. Reduce yIt ^^ ^^ equivalent decimal. OPERATION, Analysis. — Since yf ^^ = 12 5)2.000 Tk of ^ units, and 2 units equal 2000 thousandths, -^ .016 Or, of 2000 thousandths is 16 JLJ._ _ Q 1 (3 thousandths, or .016. tIt — tI^ o"oir — To ©"^ Eeduce to equivalent decimals 3. 4. 5. 6. 8. ^^V 9. 10. tIt- Rule. — I. Annex ciphers to the numerator and divide by the denominator. 11. Point off as many decimal places in the result as there are ciphers annexed. T lie sign + is sometimes placed after the result to indicate that there is still a remainder. Thus, f = .666 + , or . Reduce to five decimal places : 11. f I 12. ^. 1 13. Eeduce to equivalent decimals : «• 14. m- 15. ^^. 16. ^. 17. Tk- 18. 9 164 4- 19. ^ of |. ! 31. f of $2f 20. ^offf. I 22. $2f Xyf?. Change to the decimal form : 23. lOlf . I 25. Hi. 24. $225|. 1 26. 8.6f. 27. $.93^. 28. $40^. 39 ?^ of At ^^- 8 ^'3tV- 156 DECIMALS. ADDITIOK. ohjlij :ex:e h cis:e s . 286. 1. What is the sum of fV and ^ ? .6 and .4 ? 2. What is the sum of yf^ and ^ ? .11 and .15 Y 3. What is the sum of .12 and .20 ? .15 and .25 ?,^ 4. Find the sum of 6 mills and 9 mills. .008 and .021. 5. What is the sum of 4 and .09 ? Of .04 and .009 ? How many decimal figures in the sum of tenths and tentJis f Of tenths and hundredths ? Of hundredths and thousandths f Of tenths and thousandths ? In adding several decimals, each having a dif- ferent number of decimal places, how many places will there be in the sum ? 387. Since decimals and integers increase and decrease uniformly by the scale of ten, decimals expressing like parts of a unit may be added y subtracted , mnltipliedy and divided in the same manner as integers. The pupil should obtain and express all results in decimal form. WJtITTJSN EXEJRCISES, 388. 1. Find the sum of 12.07, 326.2086, .768, and 1.9. OPEBATiON. Analysis. — Write the numbers so that units 12.0700 of the same order stand in the same column. Q 2 6 2 8 6 After reducing the decimals to a common de- 7 A ft n nominator by annexing ciphers (280, 5), or supposiug them to be annexed, add as in inte^ 1.9 gers, placing the decimal point before tenths \t 3 40.94 6 6 the sum. In like manner find the sum 2. Of .375, .24, .536, .0437, .50039, and .008236. 3. Of 405.327, 64.03, .84673, 121.8, and 7.00327. 4. Of $18.19, $142,095, $.964, $5,125, and $40.50. ADDITION. 157 EuLE. — I. Write the numbers so that units of the same order stand in the same column and the decimal points in the same vertical line. 11. Add as in addition of integers, and place the deci- mal point before the order of tenths in the sum, 5. What is the sum of 37 thousandths, 54 ten-thou- sandths, 407 hundred-thousandths, and 12345 millionths? 6. Find the sum of 45 units, 25 tenths, 360 hundredths, 75 thousandths, 52 ten-thousandths, and 406 millionths. Find the sumX 7. Of $25f, $81.09, $16^, tS^, $150^, and $|. 8. Of 103.60-1, 6.0|, .37012, and 40.0034|. 9. Of 24.6^, 47.32-^, 5.3784|, and 2.64878f. 10. Of 61.843 acres, 8^ acres, 21.04 acres, 15-|4 acres, and 3f acres. " 11. Bought a ton of coal for $7f, a barrel of sugar for $28^, a chest of tea for $23.08, and a barrel of flour for $10.87i. ^Vhat was the cost of all ? In the reduction of each fractif)n, carry the decimal to at least Jive places, to Insure accuracy in the fourth. 12. Find the sum of -^, f, ff, ^^, and -^, in deci- mals, correct to the fourth place. 13. A man bought a farm for $6736.75, which was $325| less than he sold it for. What did he sell it for? 14. How many rods of fence will enclose a field, the sides of which are respectively 34.72 rods, 48|^ rods, 152.17 rods, 95| rods, and 56-| rods ? 15. Paid for building a house $3450.75, for painting the same $518|, for furniture $1204.37^, and for carpets $810f . What was the cost of the whole ? WB DEOIHALS. SUBTEAOTIOE". OMAL EXERCISES. 389. l./From -^ take ■^. From .9 take .7. 2. From ^ take yV^. From .36 take .12. 3. From y|^ take y^j^-. From .028 take .010. 4. From -^ take ^. From 45 cents take 20 cents. 5. Find the difference between f and -^. J and .25. 6. Find the value of ^—,^ ; of .5~i ; .65— .5. 7. Find the value of $i— 30 cents ; 80 cente--$.6-^ AN How many decimal places in the remainder, if there 2Lte three in the minuend and one in the subtrahend ? If two in the minuend and four in the subtrahend? If none in the minuend and three in the subtrahend ? WniTTEN EXERCISES. 390. 1. From 3.16 subtract .2453. OPERATION. Analysis. — Write the given numbers as in Addi- 3.1600 *ion, the subtrahend under the minuend, reducing 2 4 5 3 *^^^ decimals to a common denominator, by annexing ciphers (280, 5), or supposing them to be annexed, 2.9147 and then subtract as in integers. 2. From 324.07 take 70.20681. 3. From $1034 take $500.94. ! EuLE. — I. Write the sultraliend under the minuend, so that units of the same order stand in the same column. II. Suitract as in subtraction of integers, and place the decimal point before the orders of tenths in the remainder^ Find the difference, decimally, between ^. %l^ and 143^. 5. 1.0066 and .630482. 6. $143| and $304.96. 7. 2 and .00345. MULTIPLICATIOI^. 159 8. 10.0402 and 26 millionths. 9. 115 and 115 tenths. 10. 5 and 125 ten-millionth s. 11. $.875 and $|. K 12. $Hand$.75. 13. 7.005 ^nd .7005. 14. .93^andl.l69f. 15. 1| and 1875 million ths. 16. $200 and $70f|. 17. .4 and .04 J. 18. U and T^. 19. .1^\ and .01|. 20. A speculator having 7346 acres of land, sold at dif* ferent times 364^ acres, 1235.125 acres, 2700^^ acres, and 850. 65 acres. How much had he left ? 21.( A man bought an overcoat for $36f , a sack for $18|, and pants for $8.12|^, and gave in payment one fifty, and two ten-dollar bills. What change should he receive ? Find the decimal value + 23>Of2|-H+(.9-T^). 34. Of .37i + ^ + 4.2 — (2 — .68|). 25. Of $250 - ($170^f - $Ui) + $^. 26. Of $48^ + $.97 - m + $.62i + MULTIPLICATIOIT. OXAI, EXEUCISES. 391. 1. What is 5 times ^? 6 times .3 ? 4 times .5 ? 2. What is 7 times yfj-? 5 times .08 ? 6 times .09 ? 3. What is t3^ X 3 ? 3x.7? 4x.6? .5x7? 4. What is if ^ X 5 ? 5x.04? .05x7? 8x.06? 5. Whatis^x^? .4x.3? .8x.7? .6x.9? 6. WhatisT^xA? .5X.05? .12x.6? .7x.llP 7. What is y^ X 1^7 ? .03x.07? .15x.06? 8. What is 8 times $.6 ? 7 times -^-^ of a dollar ? 9. What is 8 X. 5? 8x.05? 8x.005? 8 x. 0005? 160 DECIMALS. How many decimal places in the product of units multiplied by tenths f Tenths by tenths? Tenths by hundredths? Hundredths by hundredtJis ? If there are two decimal figures in the multiplicand, and two in the multiplier, how many are there in the product? If three in the multiplicand and one in the multiplier? How many decimal places ^ are there always in the product ? X. 392. PRiisrciPLE. — The number of decimal places tn any product is equal to the decidual places in both factors. V/ WniTTEN JEXJSMCISJSS. 393?L Multiply .04 by .8. OPERATION. Analysis. — Multiply as in fractions. (232.) . 6 4 Thus, .64 X .S = j%\ X j\ = 3-VA = .512. Or, o Multiply as in integers, and since hundredths mul- — '- — tiplied by tenths produces thousandths, the product • 5 1 2 must contain three decimal places. (Prin.) Multiply Multiply 2. 1.245 by .27. 3. .4056 by 35.05. 4. 7.25 by .00012. 5. $506^ by .048f EuLE. — Multiply as in multiplication of integers, and from the right of the product point off as many figures for decimals as there are decimal jjlaces in both factors. 1. If there are not as many figures in the product as there are decimals in both factors, supply the deficiency by prefixing ciphers. 2. To multiply by 10, 100, 1000, etc., remove the decimal point in the multiplicand as many places toward the right as there are ciphers in the multiplier. (266, 4.) Multiply and express the product decimally : 6. I324| by .324. 7. $175.64 by .205. 8. 5.728 by 100. 9. .6207 by 1000. 10. 5^ hundredths by 25. 11. 26000 by 26 thousandths. 12. 84 tenths by 244 hundredths. 13. 7| tenths by .06^. MULTIPLICATION. 161 Find the value 14. Of 3.126 X. 046 X. 3. 15. Of 9f X .07^ X 10. 16. Of 18.75 X 1.001 x-|. 17. Of .25 of T^ X .04f 18. Of 327ix.9x4J. 19. Of $8.56 X. 06^x100. 20. Of 18^^ X. 0062^x1000. 21. Of .01 of f xl00x.08f 22. Bought 156 pounds of cheese at $.12^ a pound, 327 pounds of coffee at $.26| a pound, and 17 barrels of apples at $2.87|^ a barrel. What was the cost of the whole ? 23. If an acre of land produce 127.25 bushels of pota- toes, how many bushels will 4.375 acres produce? What is the value 24. Of 170 barrels of apples, at $2f a barrel? 25. Of 100 cords of wood, at 14.38 a cord ? 26. Of 204yV acres of land, at $72f an acre ? 27. Of 580^ pounds of sugar, at 9 J cents a pound ? 28. Of 126 mules, at |97| each? 29.1 What is the cost of 3f bales of cloth, each bale con- taining 36.75 yards, at $.85 a yard? 30.^^ A farmer sold 300 bushels of oats at $.45 a bushel, 16| cords of wood at $3| a cord. He received in payment 125 pounds of sugar at $.12J a pound, 36 pounds of tea at $1 a pound, 6 barrels of flour at $8.3 7|^ a barrel, and the remainder in cash. How much cash did he receive ? Complete the following equations : 31. $450.75 — $241^ x 3.24 + $18/^ = ? 32. ($200 - $1251^) X (f + 2.5) =: ? 33. 3.0065 X .304 + 40^^ x 10 = ? 34. .00493 X 1000 x (1 — ^ + .025) = ? 35. (4 — .00036 + .316) - (.75 + 3^ — 1|) = ? 36. (^Vo X .08^ + .03685 X f ) X 100 .= ? M8 - DECIMALS. DITISIOIT. ORAL EXERCISES. 394. 1. What is i of ^V? iof,%? iof^Hir? 2. What is I of .8 ? \ of .42 ? \ of .072 ? 3. Divide .8 by 4 ; .56 by 7 ; .120 by 10 ; .0048 by 12. 4. Divide H^Jt^C). m^l^' 5. Divide 4.8 by 6. Analysis. — 4.8 equals 48 tenths, and ^ of 48 tentTis is 8 tenths^ or .8. 6. Divide .48 by 6 ; .48 by .06 ; .048 by .006. 7. Divide ^ by ^^ (.6 -r- .12) ; 7.5 by 2.5. 8. Multiply -^ by ^^ (.8 X .9). Divide .72 by .9. 9. MiiltiplyxfobyTlo(-08x.09). Divide .0072 by .09. 10. The product of two factors is .096, one of which is .8 ; what is the other ? How many decimal places in the quotient when tenths are divided hy units? Tenths hy tenths? Hundredths hy tenths? Thousandths by hundredths ? If there are two decimal figures in the divisor and three in the dividend, how many are there in the quotient? If three in the di- visor and three in the dividend? If none in the divisor and three in the dividend? If two in the divisor and none in the dividend? 395. Pki^stciples. — 1. The dividend mu^t contain at hast as many decimal places as the divisor^before division is possible, 2. Since tits dividend is the product of the divisor and quotient, it contains as many decimal places as both divisor and quotient. Hence, 3. The quotient must contain as many decimal 'places as the number of decimal places in the dividend exceeds those in the divisor,\ DIVISION. 163 WBITTEN MXJEBCISB8, 296. 1. Divide .952 by .7. OPERATION. Analysis. — Divide as in fractions. (238.) Thus, ,7) .952 .952^.7-TWiJ^A-TmxY=«l=1.36. Or, Divide as in integers, and since the dividend con- tains three decimal places, and the divisor one, the quotient must have two decimal places. (Prin. 3.) Divide 2. 81.6 by 3.6. 3. 675 by .15. 4. .952 by 4.76. Divide 5. $41.25 by 33. 6. $518.70 by $14.25. 7. 345.15 by .075. Exile. — Divide as in division of integers, mid from the right of the quotient point off as many figures as the deci- mal places in the dividend exceed those in the divisor. 1. If the number of figures in the quotient be less than the excess of the decimal places in the dividend over those in the divisor, the deficiency must be supplied by prefixing ciphers. 2. If there be a remainder after dividing the dividend, annex ciphers, and continue the division : the ciphers annexed are deci- mals of the dividend. 3. In most business transactions, the division is considered suf- ficiently exact when the quotient is carried to 4 decimal places, unless great accuracy is required. 4. To divide by 10, 100, 1000, etc., remove the decimal point in the dividend as many places to the left as there are ciphers in the divisor. (266, 3.) 8. Divide 88.476 by 1.2 ; by 3.6 ; by m\ ; by 1.04. 9. Divide $56.05 by .59; $408,371 by 27. 10. Divide $6.45 by $.45 ; $52 by $.65 ; 293.75 by 45f . 11. Divide .0026 by .003 ; 3 by .450 ; 75 by 1000. 12. What is the quotient of 75.15208 divided by 24? by .24? by .024? by .0024? by .00024 ? 13. Divide $3875 by 10; by 100 ; by 1000; by 10000. 164 DECIMALS. What is the value of 14.1645.5 — 1000. 18. 3-t-18|. 22. $27-~37i. 15. $1000-^$.02. 19. 4.2-^31i. 23. .001 — 100. 16. $56-^.007. 20. 17^-r-lOOO. 24. 100-^.001. 17. 1.904-r-4.76. 21. .73|-v-100. 25. $48|^$f. 26. Divide .24 by 72 ; | of .24 by -^ of .042. 27. If 64 tons of iron cost $4816, how many tons can )e bought for $1730.75? 28. How many coats can be made from 32.4 yards of cloth, allowing 2.7 yards for each coat? 29. At $287f each, how many horses can be bought for $4885.80 ? 30./ If 125 bushels of potatoes cost $82^, how many bar- rels, each containing 2^ bushels, can be bought for $224.40 ? 31. If 3^ cords of wood cost $11.37|, what will 20| cords cost ? 32. How much sugar can be bought for $46.75, if | of a hundred pounds cost $6-| ? 33. Gave lOf cords of wood, worth $4| a cord, for 7.74 barrels of flour. What was the flour worth a barrel ? 34. A man sold a horse for $125, and received in pay- ment 12^ yards of cloth at $3^ a yard, and the balance in tea at $.62|^. How many pounds of tea did he receive ? Find the second member in each of the following equa- tions : 35. /Of (1.008^18 + 63-^4000 xlOO)-f==? 36. lot 714-.714-^(.34— .034 x .25 of 6) = ? 37. Of (.48-^800xl0000 + 6.4-^-.08)-^.125z=? 38. Of (34x.l93 + 2.7x.4|)-r-(4.81— fof 1.662)=? 39. Of ($262.90-^$.56)x.0084+T02|xl00= ? 40. Of ($1260 X 3.49)-r-$10.47-$8504-$6.80= ? CIRCULATING DECIMALS. 165 OIECULATING DECIMALS. O UAL EXERCISES, 397. 1. What are the prime factors of 10 ? Of 100? 2. Change to the decimal form i ; J ; f ; | ; i^« (385.) What are the prime factors of each of the denominators of these factions ? Are they the same as the prime factors of 10 ? Can these fractions be reduced to perfect decimals? 3. Change to the decimal form, extending to four places, i ; f ; ^ ; A- Can these fractions be reduced to perfect decimals ? What are the prime factors of their denominators ? 4. Change to the decimal form, extending to three places, i ; tV ; 1^- Can these fractions be reduced to perfect decimals ? What are the prime factors of their denominators ? How do the decimals produced by these fractions differ from the decimals produced by the fractions in examples 2 and 3 ? What kind of decimals are all fractions equivalent to, that in their lowest terms have denominators containing the factors 3 or 5 ? 5. What figure is constantly repeated in reducing to a decimal i? f? f? A? e. If a decimal consists of 3 repeated indefinitely, what fraction is it equal to ? 7. Is there any difference between | and f ? f and | ? f and If? Ilandffff? 8. Is there any difference between -^ and || ? || «id mi ? 9. If the numerator is 4444, what must be its denom- inator so that the fraction may equal % ? To change a repeating decimal number to an exact fraction, what figures must be used in the denominator? 166 DECIMALS. DEFINITIONS AND PEINCIPLES. 398. A Finite Decimal is a perfect decimal, oi one that terminates with the figures written ; as, .25, .375. 399. A Circulating Decimal is a decimal in which a figure, or set of figures, is constantly repeated in the same order; as, .333+, .727272 + . 300, A Hepetend is the figure or set of figures, con- tinually repeated. The repetend is written but once, and when it consists of a single figure a point is placed over it ; when it consists of more than one figure, points are placed over the first, and over the last figure. Thus, the circulating decimal .666 -f , and .297297 -f , are written .6, and 297. 301. A JPure Circulating Decimal is a deci- mal which commences with a repetend ; as .7, or .279. 303, A Mixed Circulating Decimal is a deci- mal in which the repetend is preceded by one or more deci- mal places called the finite part of the decimal ; as, .27, or .04648, in which .2 or .04 is called the finite part. 303. The law for the formation of repetends will be apparent from the following : 1. i =:.1111+ =.1. 2. ^ =.01010+ =.6i. 3. rh =.001001 + =.00i. 4..^^— .00010001 + =.6ooi 5. i =.4444+ =.4. 6. If =.2323+ =.23. 7. ill =.135135+ =.i35. S.Hff =.17281728 + =.i728 304. Pkikciples. — 1. Hvery fraction in its lowest termsy whose denominator contains no other prime factors than 2 or 5 is equivalent to a finite decimal. 2. Every fraction in its lowest terms, whose denomina- tor contains other prime factors than 2 or b is equivalent to a circulating decimal. CIRCULATING DECIMALS. 167 3. Every fraction in its loivest terms, whose denomina- tor contains 2 or 6 with other prime factors is equivalent to a mixed circulating decimal. 4. Every pure circulating decimal is equal to a fraction whose numerator is the repetend, and tuhose denominator consists of as many ^^s as there are places in the repefend. WRITTEN JEXJEJiCISBS. 305. To change a fraction to a finite or to a circulating decimal. 1. Change -^V to a finite decimal. (385.) 2. Change to finite decimals, i ; i ; A 5 if ; If ; tV? ; and ^. (Peik. 1.) 3. Change to a pure circulating decimal ^\. Operation.— ^V = 7.000000 ^ 27 = .259259 + = .259. (Prin. 2.) 4. Change to pure circulating decimals, -f 5 1 r A ; if ; H ; and A. 5. Change to a mixed circulating decimal f. Operation.—! = 5.0000 -h 6 = .8333 + = .83. (Prin. 3.) 6. Change to mixed circulating decimals ^ ; -5^; ^f ; H ; and TtVff- 7. Change to finite, or to circulating decimals the fol- lowing fractions: ^; i; A; fi; if; ^; t¥t ; A; and iff. 306. To change a pure circulating decimal to a fraction. 1. Change .216 to a fraction. OPERATION. Analysis. — Since .001 = ^^^ (303), 216 6 1 « 2 1 « ft ^^ equal to f ^f , which reduced to its lowest 316 = fif =z -^ ^gj.j^g equals ^. Hence .216 == sV Change to fractions. 2. 3. .45. .66. DECIKAIiS. 4. .297. 6. .334. 5. .675. 7. .4158. Rule. — Write the figures of the repetend for the numero' tor of a fraction, and as many 9's as there are places in the repetend for the denominator, and reduce to its lowest terrm In like manner change to fractions. 8. 9. .279. .32i. 10. 11. .6435. .i067. 12. 13. .95121. .923076. 14. Eeduce 2.297 to an improper fraction. 15. Reduce 12.081 to an improper fraction. 307. To change a mixed circulating decimal to a fraction. 1. Change .227 to a fraction. OPEKATION. 1st. .227 = ^ + ^-=^ Or 2d. Or 3d. Analysis.— Sincfi the repetend is not fl-, but U of tV = 5 -f^-^y write the finite 22 P^^^ ^^^^ t^® repe- tend each as frac- tions and add them, the reasons for which will appear more clearly in the second solution. Or, by an abbreviated method of reducing the fractions to a coiii» mon denominator, 2 x 99 — 2 x 100 — 2 ; hence, 2 x 100 + 27 —3 = 225 is the numerator of the equivalent common fraction. 2. Change to fractions, .57; .048; .1004; .6472. 3. Change to mixed numbers, 7.543 ; 2.564 ; 7.0126. ■^"=-^H = 1 = 1-0 2 2 7 given decimal. 2 finite part. .225 m^h SHORT METHODS. 189 EuLE. — Reduce the finite part and the repetend of the given decimal each to the form of a fraction, Tlten add them, and reduce to lotvest terms. Or, From the given decimal subtract the finite part for a numerator, and for a denominator write as many d's as there are figures in the repetend, with as many ciphers annexed as there are figures in the finite part. Change to fractions, 4. .04648. 6. .9385714. 8. .0126 5. .7852. 7. .35i35. 9. 5.27. To addy suUract, multiply^ or dimde circulating decimals, reduce them to fractions, and then perform the required operation. For a fuller development of *' Circulating Decimals " and " Con- tinued Fractions," see " Robinson's Higher Arithmetic." SHORT METHODS. obatj exercises , 308. 1. What part of $1 are 8| cents ? 16| cents ? l^ cents ? 25 cents ? 50 cents ? 2. At 25 cents a pound, what cost 22 pounds of coffee ? Analysis, — Since 25 cents are $J, 22 pounds will cost 22 times %h or $y» equal to $5^, or $5.50. Or, At $1 a pound, 22 pounds will cost $22, and at %\ a pound, \ of $22, which is $5|, or |5.50. 3. What is the cost of 80 pounds of beef at 12| cents a pound ? At 16| cents ? At 20 cents ? At 25 cents ? 4. At 33-1^ cents a can, what will be the cost of 25 cans of sweet com ? Of 37 cans ? Of 54 cans ? Of 60 cans ? 5. What is the cost of 160 pounds of sugar at 6^ cents a pound ? At 8-J^ cents ? 10 cents ? 12i^ cents ? 8 170 DECIMALS. 6. How mauy pounds of raisins, at 16| cents a pound, can be bought for $5 ? Analysis. — Since 16 1 cents are $^, |5 will buy as many pounds of raisins as $J is contained times in $5, which are 30 times. Hence, etc. 7. At $.50 a bushel, how many bushels of oats can be bought for $15 ? For $16^ ? For $25 ? 8. At 12^ cents a yard, how many yards of calico can I buy for 27 pounds of butter, at 33^ cents a pound ? 9. What is the cost of 40 pairs of shoes, at $1.25 a pair? Analysis. — At $1 a pair, the cost would be $40 ; but since the price is $1 + $|, the whole cost is $40 + J of $40, or $50. 10. At $1.50 each, what is the cost of 48 chairs ? 11. What is the cost of 60 yards of cloth, at $1.12| a yard ? At $1. 16| ? At $1.25 ? At $1.33^ ? At $2.50 ? 12. At $2.25 a pair, what is the cost of 12 pairs of shoes ? Of 16 pairs ? Of 18 pairs ? 20 pairs ? 25 pairs ? DEFINITIONS. 3091 Quantity^ in commercial transactions, is the amount of anything bought or sold, and is estimated by the number of times it contains the measuring unit. 310. Price is the value in money of each measuring unit of any commodity. 311. Cost is the value of the entire quantity. 313. An Aliquot Part or Even Part^ of a number is such a part as will exactly divide that number. Thus, 2, 2^, 3^, and 5, are aliquot parts of 10. An aliquot part may be either an integer or a mixed number, while a component factor must be an integer. short methods. 171 Aliquot Parts of One Dollar. 5 cents = ^ of $1. 10 cents = iV of $1. 20 cents = -J- of $1. 25 cents = ^^ of $1. 60 cents = i of $1. 6 J cents = ^oi $1 8^ cents = i^ of $1, 12^ cents = i of $1 16| cents = i of $1 33^ cents = i of $1 V WniTTEN EXEMCISES. 313. To find the cost of a quantity when the price is an aliquot part of one dollar. 1. What cost 951 bushels of oats, at $.33^ a bushel? operation. Analysis. —At $1 a bushel, the cost would be $951 ; 3)951 but since the price is ^ of $1 a bushel, the cost is ^ of $951, which is $317. Or, the cost is ^ as many dollars as there are bushels, and ^=317. Hence, etc. 317 2. What cost 750 slates, at 33 J cents each ? At 25 cents ? 3. At $.50 each, what cost 631 shad? 1250 ? 1605? EuLE. — Tahe such a fractional part of the given num" ler or quantity as the price is of one dollar. 4. What is the cost of 12 sacks of coffee, each sack con- taining 43 pounds, at 33^ cents a pound ? 5. A merchant sold 5 pieces of prints, each containing 28 yards, at 16f cents per yard, 6 pieces of sheeting, each containing 34 yards, at 8^ cents per yard, and received in payment 41 bushels of oats at $.50 a bushel, and the bal- ance in money. How much money did he receive ? 172 DECIMALS. 6. At $1.12^ a foot, what cost 334 feet of wire fence? OPERATION. 8)324 Analysis.— At $1 a foot, the cost would be $324 ; 4 5 ^^* since the cost is $1 + $J, the entire cost is '— $324 + 1 of $324, which is $364.50. $364.5 7. At $1.33| each, what will 642 steel shovels cost ? 8. What cost 320 cloth caps, at $1.20 each ? 314. To find the quantity when the cost is ^iven, and the price is an aliquot part of one dollar. 1. How many barrels, at $.50 each, can be bought for $213 ? OPEBATioN. Analysis. — Since $J wiU pay for ^0iQ_i_^i 4-98 ^ barrel, $213 will pay for as many barrels as $J is contained times in Or, 213x2=426 $213, or 426 barrels. Or, since $1 will pay for 2 barrels, $213 will pay for 213 times 2, or 426 barrels. 2. How many baskets of pears can be bought for $318, at $.33| each ? At $.50 each ? 3. How many pine-apples can be bought for $240, at 16| cents each ? At 20 cents ? At 25 cents ? EuLE. — Divide the cost by such a fraction as will ex press the price as an aliquot part of one dollar. 4. How many pounds of cheese can be bought for $350, at 6i cents a pound ? At 8^ ? 10 cents ? 12| cents ? 5. How many cocoa-nuts, at $.25, can be bought foi $150.75? SHORT METHODS 173 315. To lind the cost when the quantity and the price of 100, or lOOO are given. 1. What cost 564 cedar posts, at 112.25 for 100 posts ? 1st operation. Anai.ysis.— -At $12.25 a post, the cost $12.25 would be $12.25 x 564 = $6909. But n QA since $12.25 is the price of 100 posts, — $6909 is 100 times the cost. Hence di- 100) 6909.00 vide by 100 (296, Note 4), and the $69.09 result is $69.09. Or, Since 564 is 5 and 64 hundredths 2d operation. (5 g4)^ if 1 hundred cost $12.25, 5.64 will $12.25 X 5.64=$69.09 cost 5.64 times $12.25, or $69.09. If the price is by the thousand, divide the product by 1000, or re- duce the quantity to thousands and decimals of a thousand before multiplying. 2. What is the cost of 1684 pounds of beef, at $9.37i a hundred pounds ? 3. What cost 22840 raikoad ties, at $174.55 a thousand ? 4. How much is the freight on 4575 pounds of merchan- dise from New York to Baltimore, at $.98 for 100 pounds ? EuLE. — Multiply the price iy tJie quantity reduced to hundreds and decimals of a hundred, or to thousands and decimals of a thousand, and point off in the product as in multiplication of decimals. In business transactions, the letter C is sometimes used for hun- dreds, and M for thousands, when the price is by the 100, or 1000. What is the cost, 5. Of 536720 bricks, at $8.75 per M.? 6. Of 2108 feet of pine boards, at $3.12|- per 0. ? 7. Of 2700 pine-apples, at $16^ per 100 ? ^ 8/ Of 875 feet of scantling, at $10^ per M. ? V- 9. Of 2160 oysters, at $1.86 per 100 ? 174 DECIMALS. 10. Of 3080 fence pickets, at $5| per 1000 ? 11. Of 28643 feet of timber, at $11| per M. ? 12. Of 1480 pounds of maple sugar, at $12. 37^ per 100 I 13. What is the value of 3700 cedar rails, at $5f per C? 14. What is the value of 12500 shingles, at $6| per M.? 15. Find the cost of 527 feet of boards at $15^ per M. and of 972 feet of siding at $1.62|^ per 0, 31 6. To find the cost, when the quantity and the price of a ton of 2000 pounds are given, 1. What is the cost of a load of hay, weighing 2280 pounds, at $18.50 a ton? OPEBATION. 2 ) 1 8. 5 Analysis.— Since $18.50 is the cost of 2000, $9.25 i^^ $18.50, or $9.25 is the cost of 1000 pounds ; ' and 2280 pounds will cost 2.280 times $9.25, or "^'^^ $21.09. $21.09 2. At $4.75 a ton, what will a load of plaster weighing 2806 pounds cost ? 3. What is the freight on 21672 pounds of iron, at $2.80 a ton? EuLE. — Multiply one-half the price of a ton hy the mim' ber of thousands and decimals of a thousand in the given quantity J as in 315. 4. What is the value of 150 sacks of guano, each sack ^ntaining 162} pounds, at $51| a ton ? 5. Find the value of 6340 pounds of Lehigh coal, at $7} a ton, and 5080 pounds of soft coal at $6J a ton. 6. At $26.44 a ton, what will be the cost of 1526 pounds of bone dust? LEDGER ACCOUNTS. 175 LEDGER ACCOUNTS. 317. A Ledger is the principal book of accounts kept by business men. Into it are transferred, in a con- densed form, all the items of the Journal, or Day Book, for convenient reference and preservation. 318. The deMfs (marked Dr.) are placed on the left, and the credits (marked Cr.) are placed on the right, 319. The Balance of an Account is the differ- ence between the debit and credit sides. When this is settled, or paid, the account is said to be balanced. 320. Find the balance of the following Ledger Ac- counts : (1-) (2.) Dr. Cr. Dr. Cr. 1506.76 $42.17 $2371.67 $4763.84 194.32 36.24 571.84 7061.39 173.26 8.42 90.50 8242.76 71.32 10.71 2037.69 364.96 39.46 94.30 94.46 410.31 152.60 347.16 876.54 5724.27 71.78 40.00 679.81 6317.66 320.00 12.94 4930.71 2431.27 48.50 271.19 104.13 163.55 63.41 500.50 1987.67 7063.21 56.00 11.44 143.84 451.09 410.10 81.92 522.71 200.00 72.22 10.10 3114.60 1807.36 137.89 107.09 152.91 768.72 276.44 207.16 9328.42 3024.27 176 DECIMALS. AOOOU]^TS AE"D BILLS. 321. An Account^ in commercial transactions, is a record of debits and credits. 332. A Debtor is a person who owes another money, goods, or services. 323. A Creditor is a person to whom money, goods, or services are due from another. 324. A JBill is a written statement of money paid, of goods sold or delivered, or of services rendered. It is sometimes called an Invoice. An account or bill should always state tbe place and the time of each transaction, the names of both the parties, the price or value of each item, and the entire cost. 325. A JBill is receipted when the words " Re- ceived Payment" are written at the bottom, and the creditor's name is signed either by himself, or by some authorized person. 326. The following abbreviations are in general use : @ At. Disc't Discount. Net Without disc't. % or Acc't Account. Do. The same. No. Number. Am't Amount. Doz. Dozen. Pay't Payment. Bal. Balance. Dr. Debtor. Pd. Paid. Bbl. Barrel. Exch. Exchange. Per By. Bo't Bought. Fol. Folio. Prem. Premium. B. L. Bill of Lading. Fwd. Forward. Prox. Next month. f^ Per cent. Fr't Freight. Rec'd Received. Co. Company. Ins. Insurance. Sund*fi i SondrieSo Cr. Creditor. Inst. This month. Ult. Last month. Com. Commission. Int. Interest. Yd. Yard. Dft. - Draft. Mdse. Merchandise. Yr. Year. The character @ is always followed by the price of a unit. Thus, 5 yd. of cloth @ $3.25, signifies, 5 yards of cloth at $3.25 a yard ; J lb. of tea @ $.90, signifies J a pound of tea at $.90 per pound. ACCOUNTS AND BILLS. 177 3211, Required the foofengs and balances of the fol- lowing bills and accounts : New York, May 10, 1875. A. S. Mann & Co., Bought of Halsted, Haynes & Co. 886 yd. Muslin, @ 26^ . .$J^?.c/^ 98J " Canton Flannel, . . '' W • - / 7 7^ 162 '* Victoria Gingham, . ** 16^^ . . J? C* . '/ 110 '* Cassimere, . . . . " $2.87^ . .^ ;^:> ■ ^ Find the footing of this bill. (2.) Boston, June 20, 1876. Messrs. C. P. Mead & Son, BoH of Belknap, Bro. 216 pairs Boys' Kip Boots, . . . . @ $2.25 160 '' '* Brogans, . . . . " 1.12^ 75 " Women's Fox'd Gaiters, . " 1.25 110 " " Enameled Boots, " 1.87| 6 cases Men's Calf Boots, . ..." 75.50 lease Drill, 648 yd., " .14^ 36 gross Silk Buttons, " .87^ Belknap, Bro. Mr. Chas. Elliott, Received Payment, (3.) Charleston, S. C, Oct. 4, 1874. BoH of Wm. J. AiKiN. 8 bales, ea. 485 lb., Ordinary Tex. Cotton, @ ISJ)^ 6 " '' 506 " Upland, Middlings, . ** 21|^ 3 hhd., 215 gal., N. O. Molasses (N. Crop)," 60^ Bec*d Payment ly draft on N. T., Wm. J. Aikin. 178 DECIMALS. (4.) Messrs. Cook & Cheney, Chicago, Sept. 10, 1876. BoH of Baker & Ellis. 275 bbL Flour, State Superfine, . . @ $7.10 146 " " Minnesota Ex., . . " 7.87i 94 '* " Wisconsin XX, . . *' 8.12J 650 bu. Wheat, No. 1, Red Winter, '' 1.75 # W 400 " ♦* Illinois, No. 1, . . '' 1.82 t> 368 " Corn, Southern White, . ** .87i $ Bodd Paym't by note at 4 mo.. Baker & Ellis. Mr. James Wilde, (5.) San Francisco, Jan. 1, 1875. To Hodge akd Son, Dr. 1874 Sept. Oct. Nov. Dec. 10 16 15 20 26 To 75 lb. Sugar, " 1 caddy Japan Tea, 22 lb., " 1 sack Rio Coffee, 116 lb., " 1 *' Rice, 751b., . . . " 1 box P. & G. Soap, 60 lb., *' 25 lb. Mackerel, .... " 9 gal. Molasses, .... " 18 lb. Soda Crackers, . . " 12 *' Dried Beef, . . . " IboxS. G. Starch, 281b., 12ij^ 9^ lOJ^ Bec'd Paym% Hodge & Son, Per Henry Scott. ACCOUiq^TS AJfcTD BILLS. (6.) Mr. Jacob E. Kekt, 179 Detroit, May 28, 1877. To George W. Parker, Dr. Jan. 6 << (( Mch. 20 if (( « a April 16 it (( Fop BuildiBg Out house as per contract, " Extra Labor, " 15 days' work of self, @ $3J " 7 " " of son, *• 1.50 - " 784 ft. Boards, . . . . 2^ per C. - " 2 days' work, . . " 3.50- '* Nails, Hinges, and Sundries, . , , $150 X^4 crv r/V 50 75 (7.) Statement of Account. St. Louis, Nov. 6, 1875. Messrs. Wood & Cole, To Phelps & Dodge, Dr. April 15 << t( June 21 Aug. 10 Oct. 3 (t (( May 25 July 14 (( <( Sept 5 «< 12 To 30 tons Eng. Iron, . . . @ $34.30 '* 12 cwt. Eng. Blister Steel, '' 15.25 " 6 doz. Hoes (Trowel Steel), " 9.78 *• 30 Buckeye Plows, ..." 10.45 " 12 Cross-cut Saws, ..." 12.12| " 37 cwt. Bar Lead, ..." 6.90 Cr. By 22 M. feet of Boards, . . @ $27.60 « 36 M. " Plank, . . " 13.37J " 45 M. Shingles, . . . . " 3.62^ " Draft on New York, . " 46 C. feet Scantling, . 1.3 BcU. due Phelps & Dodge, $500 180 DECIMALS. (8.) Account Current; Balanced by Note, Geo. B. Damon & Co., In % with Gkay & Banks. Br. Or, 1876 1876 Aug. 2 To 796 lb Butter@$.28 Nov. 3 By 27 bbl. Pears® $9.25 Sept. 17 " 972 " Cheese" .09 " 24 " 56 "Apples'- 1.67 (t 24 " 4811/2" Lard " .12 Dec. 1 " 70Mbu. Corn" .70 Oct. 4 18 " 81 doz. Tallow" .16 Eggs " .26 1877 22 " ^VA " Peas" 1.95 '* 31 '' 15bbl. Salt "2.40 Jau. 2 " Note at a mo. to Bal. Dec. 15 " 963 lb. Hams " ,14 - - ^^~^^-^_^ - Gray & Banks. PHrLADBLTHiA, Jan. 2, 1877. REVIEW. WRITTEN EXAMPZE& 328. What is the cost, 1. Of 7^ barrels of flour, if 4| barrels cost $38 ? 2. Of 9^ tons of coal, if .875 of a ton cost $5,635 ? 3. Of 14.25 yards of cloth, if 36.48 yards cost $54.72 4. Of 100 pounds of pork, if .93 cwt. cost $6,975 ? 5. Of 25.42 acres of land, if .125 of an acre cost $15| 6. Of 1 ton of plaster, if 1680 pounds cost $2,856 ? 7. Of .8 of a pound of tea, if 1 pound cost $.62|? 8. Of 18640 feet of timber, at $6^ per 0. ? 9. Of 1375 pounds of potash, at $121| a ton? 10. Of 19600 bricks, at $9| per M. ? 11. Of .625 of a ton of coal, at $7| a ton ? 12. Of 35 yards of cloth, if 29 yards cost $101^? 13. Of 1 bushel of potatoes, if 28.8 bushels cost $9.60 O*" ^ 101 IFOR^ii^;^ REVIEW. iol 14/ If 36 boxes of raisins, each containing 36 pounds, cost $194.40, what is the price per pound? 15. What will be the freight on 10860 pounds of mer- chandise from New York to St. Louis, at $1.62|^ per 0. ? 16. How much must be paid for 1220 feet of boards, at $25^ per M.; 1866 feet of scantling at $2.12 J^ per C. ; and 9525 feet of lath at $3^ per M. ? 17. If I pay $1.37 a bushel for wheat, $.95 for rye, and $.73 a bushel for corn, how much, of each an equal num- ber of bushels, can I purchase for $70.15 ? 18/ Bought 27i barrels of sugar for $453.75, and sold it at a profit of |4.62| a barrel. At what price was it sold ? 19. Three persons bought 645 tons of coal, and divided bO that ttie first had .375 of it, the second ^, and the third the remainder. How much did the third receiye ? 20. What is 814^ X 26^ correct to 5 decimal places ? /^21. A person haying $55.92, wished to purchase ?' equal number of pounds of tea, coffee, and sugar • tea at $.87i, the coffee at $.18f, and the sugar r' How many pounds of each could he buy ? / ^2. A dealer bought 240000 feet of lu' per M., and retailed it out at $2J per ^ whole gain ? // 23. Three hundred seventy-fi'^ dry goods, valued at $8000, ^ would a man lose who ov . 24. Bought 150 bar of wheat @ $1.44, @ $8^, and ail rel must th on the ^ 182 DECIMALS. /^25. Sold 20900 feet of timber for $339.62 1, and gained thereby $78.37J. What did it cost per C. ? / 4)26. Eeduce (^ -^ ||) x i of | to a decimal. //27. A farmer exchanged 28^ bushels of oats worth $.75 per bushel, and 453 pounds of middlings worth $1^ per hundred, for 12520 pounds of plaster. What was the plaster worth per ton ? / , 28. A merchant tailor bought 27 pieces of broadcloth, each piece containing 19| yards, at $4.31:} a yard ; and sold it so as to gain $381.87^, after deducting $9.62^ for freight. For what was the cloth sold per yard ? /^29/ If 10| cords of wood cost $34.12|,what cost 60f cords? ^30. If 1^ hundred pounds of sugar cost $l2|, how many pounds can be bought for 193 1, at the same rate ? 31. Paid $108 for grain, f^ of it being barley at $.62J per bushel, and | of it wheat at $1.87^ per bushel; the rest of the money was paid for oats at $.37^ per bushel. "^ow many bushels of grain were bought ? -^^at is the yalue of (-^ + S "^ ^-^^ ? ^r sold to a merchant 3 loads of hay weigh- 1826, 1478, and 1921 pounds, at $17.60 pounds of pork at $5.25 per C. He — ^^ vards of sheeting @ $.18, 11^ balance in money. How ^ purchase of rye at ^d com at $.73 h kind; how BEVIE W. 183 35. A farmer had 150 acres of land, which he could have sold at one time for $100 an acre, and thereby have gained $3900; but after keeping it for a time he was obliged to sell it at a loss of $2250. What did the land cost him an acre, and for how much an acre did he sell it ? 36. Bought 2500 bushels of wheat @ $1.40, and 735 bushels of oats @ $.54 ; I had 1470 bushels of the wheat floured, and sold it at a profit of $435. 87^, and I sold 528 bushels of the oats at a loss of $30. Afterward I sold the remainder of the wheat at $1.25 per bushel, and of the oats at $.45 per bushel. Did I gain or lose, and how much ? 339. SYNOPSIS FOE KEVIEW, 1. Notation and Numeration. 2. Decimal cctrbency. . 3. Reduction. . _ _ } 1. Decimal Fractiona. 1. Defs. ^3 .. ^g^ 2. DenomiDator — how composed. 8. Numerator— Kiecimal places in. 4. Two ways of writing decimals. 5. Value of decimal figures — how determined. 6. Starting point in notation and numeration. 7. Principles, 1, 2, 3, 4, 5. ^ 8. Rule, I, II. C 1. Currency. 1. Defs. \ 2. Decimal Currency, I 3. Federal Money. 2. 271, 272, 274, 275, 277. , 3. Principles, 1, 2. 1. Art. 279. 2. 280, 1, 2, 3, 4, 5. 3. 282. 283. 4. 284. 285, Rule, I, H. r-\ DECIMALS. SYNOPSIS FOE EEVIEW— Continued. f 4. Addition. 5. Subtraction. 6. Multiplication. 7. DiTISION. 8. Circulating Decimals. Short Methods. 10. Ledger Accounts. 11. Accounts and Bills. Eule, I, II. Rule, I, II. j 1. Principle. ( 2. Rule. \i Principles, 1, 2, 3. Rule. 1. Fini:e BecinKjU. 2, Girc, Decimal. " 1. Definitions, -j 3. Repetend. 4. Pure Girc. Dec, 5. Mixed Girc. Dec 2. Principled, 1, 2, 3, 4. 3. 306. Rule. L 4. 307. Rule, 1, 2. ' 1. Definitions. 2. 313. Rule. 3. 314. Rule. 4. 315. Rule. 5. 316. RuU. 1. Quantity. 2. Price. 3. Go8t. ^ 4 Aliquot Part, \ 1- D^fi'iiti""^- \ 3.' bT^' Account ( 2. Position of Debits and Credits. 1. Definitions. 1. Account 2. Debtor. 3. Greditor. , 4. Bill. 2. Receipt of a Bill. 3. Mercantile Abbreviations. DEFINITIONS. 330. A Denominate Number is a concrete number, and may be either simple or compound ; as, 8 quarts, 5 feet 10 inches, etc. 331. A Simple Denominate Number consists of a unit or units of but one denomination ; as, 16 cents, 24 hours, 30 barrels, etc. 333. A Compound Denominate Number consists of units of two or more denominations of the same nature ; as, 10 pounds G ounces, 5 yards 2 feet 8 inches, etc. 333. In integral numbers, and in decimals, the law of increase and decrease is by the uniform scale of 10 ^ but in Compound Numbers, the scale varies. . MEASUEES. 334. A Measure is a standard unit established by law or custom, by which quantity, such as extent, dimen- sion, capacity, amount, or value, is measured or estimated. Thus, the standard unit of Measures of Extension is the yard ; of Liquid Measure, the wine gallon ; of Dry Measure, the Win- chester bushel ; of Weight, the Troy pound, etc. Hence the length of a piece of cloth is ascertained hy applying the yard measure ; the capacity of a cask, by the use of the gallon measure ; of a bin, by the use of the bushel measure ; the weight of a body, by the pound weight, etc. 186 DEKOMIiq^ATE N UMB EES. 335. Measures may be classified into six kinds : 1. Extension. 2. Capacity. a Weight. 4. Time. 5. Angles or Arcs. 6. Money or Value. MEASUEES OF EXTENSION. 336* Extension is that which has one or more of the dimensions length, breadth, and thickness. It may be a line, a surface, or a solid. 337. The Standard Unit of measures of extension, whether linear, surface, or solid, is the yard/\ LINEAR MEASUER 338. Linear or Long Measure is used in meas- uring lines and distances. 339. A Line has only one dimension — length. 1 Inch. 2 Inches. 1 12 Inches (m.) = 1 Foot 3 Feet = 1 Yard 5| Yards, or ) 16i Feet f = 1 Rod 320 Rods = 1 Mile 3 Inches. Table. , .ft . . yd, . . Td. . . mi. mi. 1 = rd, ft in. 320 = 5280 = 633G0 1 = 16i= 198 1 = 12 1. The Inc7i is generally divided into halves, quarters, eighths^ iixteenths, and sometimes into tenths or twelfths. 2. Civil and mechanical engineers, and others, use decimal divi- sions of the foot and inch. extensiois^. 187 Other Deii^omikatiois'S. 3 Barley-corns, or sizes =1 Inch. Used by shoemakers. jto measure the height of 4 Inches =1 Hand. " \ horses at the shoulder. 9 Inches = 1 Span. Among sailors, 8 spans = 1 fieithom. A-^1.888 Inches =1 Sacred Cubit. 6 Feet ==1 Fathom. Used to measure deptha at sea, 120 Fathoma =1 Cable's Length- 3 Feet =1 Pace. 1. 152f Common Miles =1 Geog. Mi. Used to meas. distances at sea. 3 Geographic Miles =1 League. 60 Geographic, or ) _^ Decree \ ^^ Latitude on a Meridian, or 69.16 Statute Miles ) ~~ ^ of Longitude on the Equator. 360 Degrees =the Circumference of the Earth. 1, A Knot is 1 geographical or nautical mUe, used to measure the speed of vessels. 2. The geographic mile is ^ of ri^,or ^j^-^fj of the circumference of the earth. It is a little more than 1.15 common miles. 340. Cloth Measure is practically out of use. In measuring goods sold by the yard, the yard is divided into halves, fourths, eighths, and sixteenths. At custom houses, in estimating duties, the yard is divided into tentlis and hundredths. 341. Surveyors^ Linear Measure is used by land surveyors in measuring roads and boundaries of land. Table. mi. ch. rd. L in. 7 92 Inches = 1 Link . 25 Links = 1 Rod . 4 Rods = 1 Chain 80 Chains = 1 Mile . 1. A Ounter's Chain is the unit of measure, and is 4 rods, or 66 feet long, and consists of 100 links. 2. Engineers commonly use a chain, or measuring tape, 100 feet long. 8. Measurements are recorded in chains and huridredthi. l. 1 = 80 = 320 = 8000 = 63360 rd. 1 = 4 = 100 = 792 ch. 1 = 25 = 198 mi 1 = 7.92 188 DEKOMIKATE KUMBERS. SURFACE OE SQUAEE MEASUEE. 343. Surface or Square Measure is used in computing areas or surfaces. 34:3. A Surface has two dimensions — length and ireadth. 344. The Area of a surface is expressed by the product of the numbers that represent these two dimensions. 345. A Square IB Si -plane ^gare bounded by four equal sides, and 1 inch. having four right angles. A Square Inch is a square each side of whicli is 1 inch in length. Table. 144 Square Inches {sq. in,) = 1 Square Foot . . . .sq.ft. 9 Square Feet = 1 Square Yard . . . . sq, yd. 80 J Square Yards = 1 Square Rod or Perch . sq rd,y P. 160 Square Rods , = 1 Acre A, sq.mi. A, sq. rd, sq. yd. sq.ft. sq, in. 1 = 640 = 102400 = 3097600 = 27878400 = 4014489600 346. Surveyors^ Square Measure is used by suryeyors in computing the area or contents of land. Table. ^25 Square Links {sq. I) = 1 Pole P. 16 Poles = 1 Square Chain . . sq, ch. 10 Square Chains = 1 Acre A, 640 Acres = 1 Square Mile . . sq. mi 36 Square Miles (6 miles square) = 1 Township . . . Tp. Tp, sq. mi. A. sq. ch. P. sq. I 1 = 36 = 23040 = 230400 = 3686400 = 2304000000 EXTENSION. 189 1. The Acre is the unit of land measure. 2. Measurements of land are commonly recorded in square miles, acres, and hundredths of an acre. For Notes and Applications, see ** Measurements " (467, 468). CUBIC OR SOLID MEASURE. 347. Cubic or Solid Measure is used in com- puting the contents or volume of solids. 348. A Solid or Sody has three dimensions— lengthy breadth^ and thickness. 349. The Toltime of a body is "expressed by the product of the numbers that represent these dimensions. 350. A Cube is a body bounded by six equal squares, csilled faces. The sides of these squares are called the edges of the cube. A Cubic Inch is a cube each side of which is 1 inch in length. ^qr ^llllllllllli 1 inch. Table. 1728 Cubic In. {cu. in.) = 1 Cubic Ft. cu.ft. I cu. yd. cu.ft. cu. in. 27 Cubic Ft. = 1 Cubic Yd. cu. yd A 1 = 27 == 46656 351. Wood 3Ieasure is used to measure wood and rough stone. Table. 16 Cubic Feet = 1 Cx)rd Foot cd.ft 8 Cord Feet, or ) 128 CubicFeet \ = ^ ^^^^ ^^• \ Perch of Stone, ) ^ , 24| Cubic Feet - 1 ] or of Masonry \ ■ ' P'^' For Notes and Applications, see " Measurements " (474-477). 190 DENOMINATE NUMBERS. ORAZ EXERCISES, 353. 1. How many inches in 3 feet ? In 2 ft. 6 in. ? 2. How many feet in 48 in. ? In 67 in. ? In 75 in. ? 3. In 5 yd., how many feet ? In 6| yd.? In ^ yd. ? 4. How many quarters in 3 yd. 2 qr. ? Eighths in 5 qr. ? 5. At 6 cents a quarter, what cost 3 yd. 3 qr. of cord? 6.. Hovr many yards in 96 in. ? In 25 ft. ? In 108 in. ? 7. In 22 yd., how many rods ? In 3 rd., how many ft. ? 8. If a vessel sail 4 leagues an hour, how many hours will she be in sailing 75 miles ? 9. How high is a horse that measures 16 hands? 10. How many fathoms deep is a body of w^ater that requires 45 ft. of line to measure it ? 11. A vessel sunk in 9^ fathoms of water: what was the depth of the water in feet? 12. What part of a foot are 9 in. ? Of a yard are 12 in. ? 13. How many rods is ^ of a mile ? ^ ? ^ ? | ? 14. What part of a mile are 80 rods ? 32 rd. ? 64 rd. ? 15. At $1^ a foot, what wiU 6 yd. 1 ft. of lead pipe cost ? 16. What part of a mile are 20 ch. ? Are 60 ch. ? 17. At $1 a rod, what will it cost to dig a trench J of a mile long ? 18. How many square yards in 54 sq. ft. ? In 84 sq. ft. ? 19. In a piece of zinc 12 in. long and 9 in. wide, how many square inches ? 20. Find the difference of 6 ft. square, and 6 sq. ft. ? 21. In a lot 12 rd. long and 10 rd. wide, how many square rods ? What part of an acre ? 22. How many yards of carpeting a yard wide, will cover a fioor 15 ft. long and 12 ft. wide ? EXTENSION. 191 2|. What will it cost to pave a court 10 ft. by 15 ft., at $.50 a square foot ? 24. At 20 cents a square yard, what will it cost to paint a ceiling 18 ft. by 10 ft. ? 25. How many cubic feet in 2 cu. yd. ? In 3 cu. yd.? 26. How many cubic inches in 1 cu. ft. 20 cu. in. ? 27. What part of a cubic yard are 9 cu. ft. ? Are 12 cu. ft. ? 28. How many cubic feet in 3 cd. ft. ? In 4 cd. ft. ? 29. In J of a cord, how many cord feet ? Cubic feet ? 30. In 2 perch of stone, how many cubic feet ? 31.1 How many cubic inches in a 10 inch cube ? 32. What is the difference between 4 cubic inches, and a 4 inch cube ? 33. How many blocks, each containing 1 cu. ft., are equal to a block 6 ft. long, 5 ft. wide, and 3 ft. thick ? MEASUKES OF CAPACITY. 853. Capacity signifies extent of room or space. 354. Measures of capacity are divided into two classes ; Measures of Liquids and Measures of Dry Substances. 355. The Units of Capacity are the Gallon for Liquid, and the Bushel for Dry Measure. LIQUID MEASURE. 356. lAquid Measure is used in measuring liquids. Table. 4 GiUs {gi.) — 1 Pint . . . pt, 2 Pints = 1 Quart . . qt. 4 Quarts — 1 GaUon . . gal. gal, qt. pt. gi. 1 = 4 = 8 = 32 1=2=8 1= 4 192 DEKOMIKATE NUMBERS. In estimating the capacity of cisterns, reservoirs, etc. hhd. bbl. gal, qt. pt. 31|^ Gal. = 1 Barrel . . . bbl. I 1 = 2 == 63 = 252 = 504 63 Gal. = 1 Hogshead . . hhd. I 1 = 31^= 126 = 252 1. The barrel and hogshead are not fixed measures, but vary when used for commercial purposes. 2. The tierce, hogshead, pipe, butt, and tun are the names of casks, and do not express any fixed measures. They are usually gauged, and have their capacities in gallons marked on them. 357. Ajyothecaries^ Fluid Measure is used in prescribing and in compounding liquid medicines. Table. 60 Minims, or drops {y(\) = 1 Fluidrachm . . /3 . 8 Fluidrachms = 1 Fluidounce . . /§_. 16 Fluidounces = 1 Pint ^• 8 Pints = 1 Gallon .... Cong. Cong. 1 = 0.8 =fl 128 =/3 1024 = 1^ 61440. / ^ , 1. Cong.y for congius, is the Latin for gallon; 0., for octaHus^ la the Latin for one-eighth. The minim is equivalent to a drop of water. A pint of watet weighs a pound. Drops are indicated in a physician's prescription by gtt. The symbols, as in Apothecaries' Weight, pr^c^cZe the numbers to which they refer ; thus, 0. 3 /| 6, is 3 pints 6 fluid ounces. DEY MEASUEE. 358. Dry Measure is used in measuring dry arti- cles, such as grain, fruit, roots, salt, etc. Table. bu, pk. qt. pt. 2 Pints {pt) = 1 Quart . . qt. 8 Quarts = 1 Peck . . pk. 4 Pecks = 1 Bushel . . bu. 1 = 4 = 32 = 64 1 = 8 = 16 1=2 For Notes and Applications, see "Measurements " (482). CAPACITY. 193 OMAI, EXERCISES, * 359. 1. How many gills in 3 pints ? In 2 qt. 1 pt. ? 2. How many pints in 1 gal. ? In 1 gal. 2 qt. 1 pt. ? 3. In 36 pints, how many quarts ? How many gallons ? 4. What part of a quart are 6 gi. ? What part of a gallon ? 5. What part of 2 gal. are 4 pints ? Are 8 pt. ? 2 qt. ? 6. How many gills in |- of a quart ? In ^ of a gallon ? 7. How many pints in 64 gills ? How many quarts ? Gallons ? 8; How many fluidrachms in 5 fluidounces ? 9. How many pint bottles will be required to hold 3 gal. 1 qt.iof sjrrup ? 2 gal. 3 qt. ? 10. (At 5 cents a pint, what will 2 gal. of milk cost ? 11.. If 10 gal. 2 qt. are drawn from a barrel of vinegar, how many gallons remain ? 12. If a gallon of wine cost $6, what will 3 pt. cost ? 13. How many barrels can be filled from 20 hogsheads ? 14. At 20 cents a quart, how many gallons of molasses win $4 buy? $6? $5.60? 15. How many pints in 6 quarts ? In 2 pk. 1 qt. ? Ip. How many quarts in 3 pk. 6 qt. ? In 1 bu. 2 pk. ? 17. In 96 qt., how many pecks ? How many bushels? 18. What part of 5 bu. are 5 pk. ? Of 1 bu. are 12 qt. ? 19. How many quart boxes will 1 bu. 2 pk. 6 qt. fill ? 20. At 20 cents a quart, what will ^ bu. of plums cost? 21. At 5 cts. a pt., what is a bushel of chestnuts worth ? 22. At $3.20 a bushel, how many quarts of peanuts can be bought for $2 ? .23. Bought \ bu. of chestnuts for $1^, and sold them for 8 cents a pint ? What was the gain ? 194 DEKOMIKATE NUMBERS. MEASUKES OF WEIGHT. 360. Weighty on the earth, is the measure of gravity, and varies according to the quantity of matter a body contains. 361. The Standard Unit of weight is the Troy pound of the Mint, and contains 5760 grains. TEOY WEIGHT. 363. Troy Weight is used in weighing gold, silver, and jewels, and in philosophical experiments. Table. 24 Grains (gr.) — 1 Pennyweight, 'pwt. 30 Pennyweights = 1 Ounce . . . oz. 12 Ounces = 1 Pound , , . Ih. A Carat is a weight of about 3.2 Troy grains, and is used to weigh Jiamonds and precious stones. The term carat is also used to express the fineness of gold, and means a ticenty -fourth part. Thus, gold is said to be 18 carats fine, when it contains 18 parts of pure gold, and 6 parts of alloy, or baser metal. APOTHECAEIES' WEIGHT. 363. Ax>othecaries^ Weight is used by physicians and apothecaries in prescribing and mixing dry medicines. Table. 20 Grains {gr, xx) = 1 Scruple 5C., or 3, 3 Scruples Oiij) =: 1 Dram dr., or 3. 8 Drams ( 3 viij) = 1 Ounce .... . (?s., or | . 12 Ounces (Ixij) = 1 Pound ?6., or Bb. Ibl = i 12 = 3 96 = B288 = gr. 5760. 1. Medicines are bought and sold by Avoirdupois Weight. 2. The pound, ounce, and grain are the same as those of Troy Weight, the ounce being differently divided. lb. oz. pwt. gr. 1 = 12 =r 240 = 5760 1=. 20= 480 1 = 24 WEIGHT. 195 8. Physicians write prescriptions according to the Roman nota- tion, using small letters, preceded by the symbols, writing j for i, when it terminates a number. Thus, 6 ounces is written, § vj ; 8 dr., 3 viij ; 14 sc, 3xiv., etc. 4. J^ is an abbreviation for recipe, or take ; a, aa., for equal quan- tities ; ij. for 2 ; ss. lor semi, or half; gr. for grain ; P. for pnrticula, or little part ; P. seq. for equal parts ; q. p., as much as you please. AVOIEDUPOIS WEIGHT. 364. Avoirdiij)ois Weight is used for weighing all coarse and heavy articles. Table. T, cwt. lb. oz. 16 Ounces (^.) =1 Pound . . . , lb. 1=20=2000=32000 100 Pounds =1 Hundred- weight cwj^. 1= 100= 1600 20 Cwt., or 2000 lb. =1 Ton . , . . . T. 1= 16 1. The Ounce is often divided into halves, quarterSy etc. 2. The long J or gross ton, hundred-weighty and quarter were for- merly in common use ; but they are now seldom used, except in estimating duties at the U. S. Custom Houses, and in weighing a few of the coarser articles, such as coal at the mines, etc. Long Ton Table. T. cwt. qr. lb. oz. 16 Ounces = 1 Pound . lb. 28 Pounds = 1 Quarter . qr. 4 Quarters = 1 Hund. . cwt. 20 Cwt., or 2240 lb. = 1 Ton . . T. =20=80=2240=35840 1= 4= 112= 1792 1= 28= 448 1= 16 3. Both custom and the law of most of the States make 100 pounds a hundred-weight. 365. The following denominations are also in use : 100 Pounds of Grain or Flour make 1 Cental. 100 Dry Fish (( 1 Quintal. 100 Nails " IKeg. 196 Flour tt 1 Barrel. 200 Pork or Beef (f 1 Barrel. 280 Salt at N. Y. S. works u 1 Barrel. 240 Lime « 1 Cask. 196 DEKOMINATE NUMBERS. 366. The weight of the bushel of certain grains and roots has been fixed by statute in many of the States; and these statute weights must govern in buying and sell- ing, unless specific agreements to the contrary are made. Table of Avoirdupois Pounds in a Bushel, As prescribed hy statute in t7ie several States named. COMMODITIES. Barley Beans Blue Grass Seed.. Buckwheat Castor Beans Clover Seed Dried Apples Dried Peaches — Flax Seed Hemp Seed Indian Corn .. Indian Corn in ear Indian Corn Meal. Mineral Coal Oats. Oniorrs Potatoes Rye Rye Meal Salt Timothy Seed — Wheat ;> 50 40 45 52 56 54 56 60 56 56 40 50 46 46 60 ^i^ 48 48 60 60 1414 52 52 46 60 60 25|24 33 33 56,56 44J44 56 50 50 TOjSO 32 35 48 57 oo'eo 56156 50 50 45,45 6060 56 50 50 46 46 48 42 56 32 56 56 60 60 48 50 48 64 46 60 56 34 60 47 48 56 32 56 60 46 46 50 50 56 60 45 42 48 42 28 28 56 56 56 50| 60 60 5656 60 1. In Pennsylvania 80 lb. coarse, 70 lb. ground, or 62 lb. fine salt make 1 bushel ; and in Illinois, 50 lb. common, or 55 lb. fine salt make 1 bushel. 2. In Maine 64 lb. of ruta-baga turnips, or of beets make 1 busliel WEIGHT. 197 ORAIi EXERCISES, 367. 1. How many grains in 3 pwt. ? In 35 ? 2. How many ounces in 60 pwt. ? In 100 pwt. ? 1 20 pwt. ? 3. How many ounces in 5 lb.? In 3 lb. 10 oz.? ^ lb.? 4. How many ounces in 40 drams ? In 64 dr.? 120 dr.? 5. How many pounds in 36 oz.? In 70 oz.? 110 oz.? 6. How many scruples in 10 drams ? In 80 grains ? 7. What will a gold chain, weighing 1 oz. 12 pwt., cost at $1 a pennyweight ? 8. What part of a pound Troy are 4 oz. ? 6 oz. ? 8 oz.? 9. How many parts of pure gold in a ring 16 carats fine ? 10. How many powders of 8 grains each, can be made from half an ounce of medicine ? 11. How many tablespoons, each weighing 2 oz., can be made from 2 lb. 10 oz. of silver ? 12. How many pills of gr. 5 each can be made from 3 1 32 of calomel? 13. What is the Yalue of a gold bracelet weighing 3 oz. 15 pwt., at $20 an ounce ? 14. How many ounces in 4 lb. Avoir. ? In 5 lb. 6 oz.? 15. How many pounds in 7 cwt. ? In 8|^ cwt. ? 16. How many cwt. in 600 lb. ? In 350 lb.? In 875 lb.? 17. In 3 T., how many hundred- weight ? How many lb. ? 18. What part of a cwt. are 25 lb. ? 50 lb. ? 75 lb. ? 19. How many cwt. in |^ of a ton ? In ^ of a ton ? 20. How many tons are 50 cwt. ? 80 cwt. ? 95 cwt. ? 21. What will a ton of hay cost, at 1 cent a pound ? 22. At 8 cents an ounce, what will 2|^ lb. of licorice cost ? 23. What will f lb. of candy cost, at 3 cents an oz. ? 24. At $2 a bushel, what must be paid for 3 bags of wrheat, each containing 120 lb. ? 198 DEKOMII^ATE LUMBERS, 368. SYNOPSIS FOE EEVIEW. Defikitioks. j * GQ Denominate Number. 2. Simple Number. 3. Comp. Denom. Number Definition op Mbastire. a. Classification. o M O m < o o 1. LiNEAB MBAS- UKE. 2. Square Meas- ure. 3. Cubic Meas- ure. 4. Wood Meas- ure. For what used. Table. Other Denominations. Surveyors' Linear Meas. For what used. ( 1. Surface. Defs. I 2. Area, ' 3. Square, Table. Surveyors' Square Meas. For what used. _ , 5 1. Solid, Defs. { Table. For what used. Table. 2. va 3. Cube. 1. Definition of Capacity. 2. Units of Capacity. 3. Liquid Meas- ure. (3. 4. Dry Measure. ] * 1. Definition of Weight. 2. Standard Unit. For what used. Table. Apoth. Fluid Measure For what used. Table. 3. Troy Weight. 4. Apothecaries Weight. 5. Avoirdupois Weight. For what used. Table. For what used. Table. For what used. Table. TIME. 19i MEASUEES OF TIME. 369. Time is a measured portion of duration. 370. The Unit of measure is the meari solar day. Table. = 1 Minute mirio 60 Seconds {sec.) 60 Minutes 24 Hours 7 Days 365 Days, or \ 12 Calendar Months ) 866 Days 100 Years da. \ 865 Hour Tit. Day da. Week wk. mo. yr. 1 = 12 = 1 Common Year . . yr. = 1 Leap Year . . . yr. = 1 Century .... (7. hr. min. sec, 8760 = 525600 = 31536000 \ 366 = 8784 = 527040 = 31622400 In most business transactions 30 days are considered a months and 12 months a year. Four weeks are sometimes called a lunar Tnonth, The calendar year is divided as shown in the diagram : 1. 1l\\q Solar Day \s the interval of time between two succes- sive passages of the sun across the meri- dian of any place. 2. The Mean Solar Day is the mean or average length of all the solar days in the year. 3. The Civil Day, used for business pur- poses and which cor- responds with the mean solar day,begins and ends at 12 o'clock, midnight. A.M. de- 365 oF366 days, notes the time before noon ; M., at noon ; and P.M., afternoon. 200 DElfOMIKATE NUMBERS. 4. The Solar Tear is exactly 365 da. 5 lir. 48 min. 49.7 sec. 5. The Common Tear consists of 365 da. for 3 successive years, every fourth year containing 366 da., one day being added for the excess of the solar year over 365 da. This day is added to the month of February, which then has 29 da., and the year is called Leap-year, 371. The following rule for leap year will render the calendar correct to within 1 day, for a period of 4000 years I. Every year exactly divisible ly 4 is a leap year, the centennial years excepted ; the other years are common years. Thus, 1876 is a leap year, but 1877 is a common year. II. Every centennial year exactly divisiUe by 400 is a leap year; the other centeiinial years are common years. Thus, the year 2000 is a L. year, but 1800 and 1900 are com. years. CIRCULAR MEASURE. 373. Circular or Angular Measure is used in measuring angles and arcs of circles, in determining lati- tude and longitude, the location of places and vessels, etc. 373. The TJnit is the Degree, which is -^-^ part of the circumference of any circle. 374. A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the Center, 375. The Circumfer" ence of a circle is the line that bounds it. 376. An Arc is any part of the circumference; as A D, D E. MISCELLANEOUS. 201 SHI. An Anffle is the difference in the direction of two lines proceeding from a common point called the ver^ tex. Thus, A D and D C B Fig. 2. are angles ^ and C is their vertex. "^ 378. A Right Angle is formed by drawing one line per- pendicular to another. Thus, ACE and E C B are right angles, a" 379. A Degree is one of the 360 equal parts ink) which the circumference of a circle is supposed to be divided. Thus, E and B (Pig. 1) are at the distance of 90°, or a right angle from each other, the vertex being at the center of the circle. 380. The Measure of an Angle is the arc of the circle included between its sides. Thus, the arc D B (Pig. 3) is the measure of the angle D C B. Fig. 3. 60 Seconds (") = 1 Minute 60 Minutes = 1 Degree 30 Degrees =: 1 Sign . 12 Signs, or 860° = 1 Circle Table. Gir, 8. "^ ' " 1=12=360=21600=1296000 1= 30= 1800= 108000 1= 60= 360 1= 60 8. Cir. 1. A Semi-Ch'ciim. is one-Ttalf of a. circumference, or 180°. 2. A Quadrant is one-fourth of a circumference, or 90°. 3. A Sextant is one-sixth of a circumference, or 60°. 4. A Sign is one-twelfth of a circumference, or 30°. 5. A degree varies with the size of the circle ; thus, a degree of long, at the Equator is 69.16 statute miles, at 30° of latitude it is 59.81 mi., at 60° of latitude it is 34.53 mi., and at 90°, it is nothing. 6. A minute of the earth's circumference is called a geographic, or nautical mile, and is a small fraction less than 1.16 common miles. 202 DEKOMINATE NUMBERS. COUNTING. 381. The following table is used in counting certaio. classes of articles : 12 Units or things = 1 Dozen . . . doz. 12 Dozen = 1 Gross . . . gro, 12 Gross = 1 Great Gross . G. gro. 20 Units or tilings = 1 Score . . , Sc. 1. Two things of a kind are often called a paiVy and six things •» »et; as a pair of horses, a set of chairs, spoons, etc. PAPEE. 383. The denominations of the following table are used in the paper trade: 24 Sheets = 1 Quire. 20 Quires = 1 Ream. 2 Reams = 1 Bundle. 5 Bundles = 1 Bale. 1 Bale = 5 Bundles. 1 Bundle = 2 Reams. 1 Ream = 20 Quires. 1 Quire = 24 Sheets. BOOKS. 383. The terms folio, quarto, octavo, etc., indicate the number of leaves into which a sheet of paper is folded. hen a sheet is folded into The book is called And 1 Bheet of paper makes 2 leaves a Folio, 4 pp. (pages). 4 " a Quarto or 4to, 8 " 8 " an Octavo or 8vo, 16 " 12 " a Duodecimo or 12mo, 24 " 16 " a 16mo, 82 " 18 " an ISmo, 36 " Clerks and copyists are usually paid by the folio for making copies of legal papers, records, and documents. 72 words make 1 folio, or sheet of common law. 90 '* 1 " chancery. MISCELLANEOUS. 203 OJRA.L EXERCISES. 384. 1. How many seconds in ^ min. ? In f min. ? 2. How many minutes in 120 sec. ? In 180 sec. ? 3. How many hours in 90 min. ? In 200 min. ? 4. How many hours in 2 da. ? In 3| da. ? In 5^ da.? 5. How many hours from 6 A. m. to 5 p. m . ? 6. In 4 wk. 3 da., how many days ? In 5 wk. 4 da. ? 7. How many minutes from 10 min. past 9 o'clock to 25 min. past 10 A. M. ? 8. How much time from 20 minutes before 11 A. m. to half past 10 o'clock p. m. ? 9. Which of the months have 30 da. each ? 31 da. each ? 10. How many days from Jan. 1 to March 5, inclusive ? 11. How many days from May 10 to July 16, inclusive ? 12. Which of the following are leap years, and which are common years : 1874 ? 1876 ? 1880 ? 1886 ? 1900 ? 13. How many centuries and years since the birth of Christ ? 14. How many leap years in every century ? 15. How many degrees in |^ a circle ? In | ? In J ? In ^ ? 16. How many geographic miles in 2° ^ In 3° ? In 4° ? 17. How many common miles in 6 geographic miles? 18. How many degrees in 360 nautical miles ? 19. How many degrees in ^ a quadrant ? In ^ a sextant? 20. How many degrees in ;^ of a circumference ? 21. What part of a circumference are 60°? 90°? 180°? 22. How many dozens in 2| gross ? In 3f gro. ? 23. How many dozens in |^ of a great gross ? In | ? 24. How many score in 100 ? Pairs in 50 ? Sets in 75 ? 25. In 1 B. of paper, how many reams ? How many quires ? 204 DENOMIITATE NUMBERS. 26. How many eggs in 5| dozen ? In 12 doz. and 7 ? 27. How many quires of paper in |^ a ream? In 3 J rm.? 28. How many years in 4 score years and 10 ? 29. How many sheets of paper will be required to make a 12mo book of 320 pages ? Of 480 pages ? 30. How many sheets will be required to make a quartc .6ook of 144 pages ? Of 240 pp. ? Of 360 pp. ? 31. How many 16mo books wiU the paper for 1 quariw book make ? MEASUEES OF YALUE. 385. Money is the measure of the value of things, and is used as a medium of exchange in trade. 386. Specie or Coin is metal struck, stamped, or pressed with a die, to give it a fixed legal value, and authorized by Government to be used as money. 387. Paper Money consists of bills and notes duly authorized by Government to circulate as substitutes for, or representatives of, money. 388. Currency is a term applied to all kinds of money employed in trade and commerce, both of coin and paper. 389. A Mint is a place in which the coin of a coun- try or government is manufactured. 390. An Alloy is a metal compounded with another .'>f greater value. In coinage, the less valuable or baser metal is not reckoned of any value. Gold and silver, in a pure state, are too soft for coinage ; hence they are hardened by compounding them with an alloy of baser metal, while their color and other valuable qualities are not im- paired. VALUE. 20^ UNITED STATES MONEY. 391. United States Money is the legal currency of the United States, and is sometimes called Federal Money. 393. The Unit of U. S. Money is the Gold Dollar. Table. E. $ d. ct. m. 10 Mills im.) - 1 Cent , . ct. 10 Cents = 1 Dime . . d. 10 Dimes = 1 Dollar. . $. 10 Dollars = 1 Eagle . . E. 1 = 10 = 100 = 1000 = 10000 1 = 10 = 100 =r 1000 1 z= 10 = 100 1 = 10 Federal money was adopted by Congress in 1786. Previous to this, pounds, shillings, and pence were in use. There is no coin for the mill. 393. The Coin of the United States consists oigold, silver, nickel, and ironze, and is as follows : 394. Gold. The double-eagle, eagle, half-eagle, quarter-eagle, three-dollar and one-dollar pieces. 395. Silver. The Tra^^-doUar, one-dollar, half-dol- lar, quarter-dollar, twenty-cent, and the ten-cent pieces. 396. Nickel. The five-cent, and three-cent pieces. 397. JBronze. The one-cent piece. 1. The half -dime and three-cent pieces, the bronze two-cent, and the nickel one-cent pieces are no longer coined. 2. The Trade-dollar weighs 420 grains, and is designed solely for purposes of commerce and not for currency. The legal-tender doWsiV weighs 412 1 grains. 3. The Standard purity of the gold and silver coins is, 9 parts (.9) pure metal, and 1 part (.1) alloy. The alloy for gold coins is silver and copper, the silver, by law, not to exceed j\j of the whole alloy. The alloy of silver coins is pure copper. 4. The five and tJiree cent pieces consist of 3 parts (.75) copper, and 1 part (.25) nickel. 5. The one cent piece consists of .95 copper, and .05 of zinc and tin. 206 DENOMIKATE NUMBERS. CANADA MONEY. 398. Canada Money is the legal currency of the Dominion of Canada, and is a decimal currency. The denominations are dollars, cents, and mills, and have the samo nominal value as the corresponding denominations of U. S. Money. 399. The Cohi of the Dominion of Canada is silver and bronze 400. The Silver Coins are the fifty-cent, twenty-five-cent, ten-cent, and five-cent pieces. 401. The Bronze Coin is the one-cent piece. 1. The gold coins in use are the Sovereign and the Half- Sovereign. 2. The intrinsic value of the 50-cent piece in United States coin is about iQl cents, of the 25-cent piece 23yV cents. In ordinary business transactions, they pass the same as United States coin. ENGLISH MONEY. 402. JEnglish or Sterling Money is the legaJ currency of Great Britain. 403. The Unit of Eng. Money is the Pound Sterling. Table. 4 Farthings {far.) = 1 Penny . , . . d. 12 Pence = 1 Shilling ...... 20 Shillings ^U Sovereign, or ^ i 1 Pound ....£. 1= 4 The value of a Sovereign in United States money is $4.8665. The character for pound (£) is written before integers ; 5 pounds ^£5. Other Denomin^atioks. 2 Shillings (s.) = 1 Florin . . . . fl. 5 Shillings = 1 Crown . . . . cr. 404. The Coin of Great Britain in general use consists of gold,, silver, and copper, as follows : 405. Gold. The sovereign and half-sovereign. 406. Silver. The crown, half-crown, florin, shilling, siX' penny, and three-penny piece. 407. Copper, The penny, half-penny, and farthing. sov. £. s. d. far. 1=20=240^960 1= 12= 48 1 = 10 = 100 = 1000 1 = 10 = 100 1 = 10 VALUE. 207 FEENCH MONEY. 408. French Money is the legal money of France and is a deciinal currency. 409. The Unit of French Money is the Silver Franc. Table. fr, dc. ct. m. 10 Milliines (m.) — 1 Centime . . . ct. 10 Centimes = 1 Decime . . , dc, 10 Decimes = 1 Franc . . . fr. The value of a franc in U. S. money of account is $.193. 410. The Coin of France consists oi gold, silver, and bronze. 411. Gold, The 100, 40, 20, 10, and 5 franc pieces. 412. Silver. The 5, 3, and 1 franc, the 50 and the 25 centime pieces. 413. Bronze. The 10, 5, 2, and 1 centime pieces. GERMAN MONEY. 414. The Empire of Germany has adopted a new and uniform system of coinage. 415. The Unit is the ''Mark^^ {Reichsmarh), equal to 23.85 cents, U. S. Money. A pound of gold .900 fine is divided into 139 1^ pieces, and the ^ part of this gold coin is called a " Mark," and this is subdivided into 100 pennies (Pfennige). 416. The Coin of the New Empire consists of gold, silver, and nickel, and is as follows : 417. Gold. The 20, the 10, and the 5-mark pieces. 418. Silver. The 2, and the 1-mark, and the 20-penny pieces 419. Niclcel. The 10, and the o-penny, and pieces of lest valuation. The 10-mark piece (gold) is equal to 3|^ P. Thalers (old). The 1-mark (silver) is equal to 10 S. Groschen, or 100 pennies. The 20-penny (silver) is equal to 2 S. Groschen, or ^ of a mark. The 10-penny (nickel") is equal to 1 S. Groschen, or ^ of a mark. 208 DENOMIKATE NUMBERS. 430. SYNOPSIS FOE EEVIEW. 00 . P tri S «' i ^ m O QQ P o m xn 1. Definition op Time. 2. Standard Unit. 3. Table. 4. Rule for Leap Year, I, II. 1. For what Used. 2. Standard Unit. r 1. Circle. 2. Circumference. 3. Defs. \ 3. Arc. 4 Right Angle. ( 5. Degree. 4. Measure of an Angle. 5. Table. 1. Counting. 2. Paper. 3. Books. 1. Defs. Table. Table. Table. 2. U. S. Money. 3. Canada Money. 4. English Money. 5. French Money. 6. German Money. r 1. Money. 2. Specie. 3. Pa. \ per Money. 4. Currency ( 5. Mint. 6. Alloy. r 1. Definition. \ 2. Unit. 8. Table. ( 4. Coin. ( 1. Definition. -? 2. Denominations, ( 3. Coin. 1. Definition. ' 2. Unit. 3. Table. I 4. Coin. 1. Definition. > 2. Unit. 3. Tabla [4. Coin. 1. Definition. [ 2. Unit. U. Coin. REDUCTION. 209 REDUCTION. 4l2YJ JReduction of Denominate Numbers is the process of changing their denomination without altering their value. 433. Denominate numbers may be changed from nigher to lower denominations, or from lower to higher denominations. 433, To reduce denominate numbers from high- er to lower denominations. ORAL EX ERCIS BS, 1. How many inches in 3 ft.? In 5 ft.? In 4 ft. 10 in.? 2. How many feet in 5 yd. 2 ft. ? In 1 rd. 10 ft. ? 3. Eeduce 12 fath. 4 ft. to feet. 15^ hands to inches. 4. How many quarters in 3|^ yd. ? How many eighths? 5. How many chains in 1 J mi. ? How many rods ? 6. In 3^ sq. yd., how many sq. ft.? In 7 sq. yd. 5 sq. ft.? 7. In 10 A., how many sq. ch. ? How many sq. rd. ? 8. Change 3 cu. yd. to cu. ft. 3 cd. ft. to cubic feet. 9. Change 4 cords to cord feet. 2 perch to cubic feet. 10. How many quarts in 2 gal. 3 qt. ? In b\ gal. ? 11. In 4 bu. 1 pk., how many pecks? Quarts? Pints? 12. In 2 pints, how many fluidounces? Fluidrachms? 13. How many pints in 3 pk. ? In 2 pk. 6 qt. ? ^ 14. How many half-pecks in 1^ bu. ? In 3^ bu. ? 15. In 5 lb. Troy, how many ounces ? In 5 lb. Ayoir. ? 16. How many pounds in 5 cwt. 20 lb. ? In 4| cwt. ? 17. In 5 dr., how many scruples ? How many grains ? 18. In 2 bu. 20 lb. of wheat, how many pounds ? 210 DENOMINATE NUMBERS. 19. How many minutes in 5 hr. 40 min. ? In 4^ hr. ? 20. In ^ a sign, how many degrees ? Geographic miles ? 21. In 10 gross 9 doz., how many dozen ? In 7^ gro. ? 22. In 2 reams of paper how many quires? Sheets? 23. In £5 10s., how many shillings ? In 3J soy. ? 24. In 5 francs how many centimes ? In 10^ francs ? 25. How many pence in 1^ crowns? In 12 florins ? 26. How many crowns in £5 ? How many florins? 27. How many pennies in 3 marks ? In 5^ marks ? 434. (principle. — Denominate Nuynlers are changed to lower denominations by Multiplication. WJtITTEN EXEKCISES , 435. 1. Eeduce 28 rd. 4 yd. 2 ft. 10 in. to inches. OPERATION. Analysis. — Since 1 rod equals 2 8 rd. 4 yd. 2 ft. 1 in. ^ yards, 28 rd. 4 yd. equal g 1 28 times 5J yd., ] us 4 yd.; ^ ^ yd. X 28 + 4 yd. = 158 yd., 1 5 8 yd. the number of yards in 28 rd. 3 4 yd. Since 1 yard equals 3 feet, ^ ' ^ **• 158 yd. 2 ft. equal 158 times 1 2 3 ft., plus 2 ft ; 3 ft. X 158 + 2 ft. 5 7 2 2 in ~ ^^^ ^**' ^^^ number of feet in 28 rd. 4 yd. 2 ft. Since 1 foot equals 12 inches, 476 ft. 10 in. equal 476 times 12 in., plus 10 in. ; 12 in. x 476 + 10 in. = 5722 in., the number of inches in 28 rd. 4 yd. 2 ft. 10 in. 2. Reduce 7 lb. 10 oz. 16 pwt. 11 gr. to grains. 3. In 3 T. 6 cwt. 21 lb. 12 oz., how many ou|ices? 4. How many inches in 12 fathoms 3 ft. 10 in. ? 6. Change 6 wk. 5 da. 9 hr. 25 min. to minutes. BEDUCTIOK. 211 EuLE. — I. Multiply the units of the highest denomina- tion of the given number, hj that numier of the scale that will reduce it to the next loiver denomination, and to the product add the number of that de7iomination given. II. Proceed in like manner with this and each sitccessive denomination obtained, until the number is reduced to the required denomination. Keduce 6. 12 mi. 36 rd. 10 ft. to ft. 7. 10 rd. 5^ ft. to inches. 8. 27| yd. to eighths. 9. 1 A. 15 sq. yd. to sq. ft. 10. 2 sq. mi. 125 A. to P. 11. 14 sq. mi. to acres 12. 3 mi. 51 ch. 6 Lto links. 13. 75 Cd. 6 cd. ft. to en. ft. 14 12 hhd. 21 gal. to pt. 15. 24 bu. 3 pk. to quarts. 16. Oong.4,0.5,f58tof3. 17. 31^ gal. to gills. Change 18. 7 T. 9 cwt. 18 lb. to lb. 19. 22 lb. 10 oz. to pwt. 20. ft) 16, 17, 3 3, to 3. 21. 1 common year to min. 22. The summer mos. to sec. 23. 1 leap year to hours. 24. 10 S. 22° 5' to min. 25. 5 bundles to quires. 26. 6 G. gro. to dozens. 27. 326^ soy. to pence. 28. 26^ fr, to centimes. 29. £341 to pence. 30. How much is 5 lb. 9 oz. 14 pwt. of gold dust worth, at $.75 a pwt. ? 31. How many rods of fence will enclose a farm f of a mile square? 32. If 1 barrel will hold 2 bu. 3 pk., how many barrels will be required to hold 1548 bu. 1 pk. ? 33. How many boxes, each containing 12 lb., can be filled from ^ hogshead containing 9 cwt. 60 lb. of sugar ? 34. If I buy 9 bu. of chestnuts at $4| a bushel, and retail them at 12^ cents a pint, what is my whole gain ? 21^ DEKOMIKATE KUMBERS. 35 J How many times will a wheel 16^ ft. in circumfer- ence revolve in running 42 miles ? 36. How many minutes less in every autumn of a com- mon year than in either spring or summer ? 37. If it require 4 reams 10 quires of paper to print a book, how many sheets are required ? 38. At 12| cents each, what will be the cost of 2 great gross of writing books ? 39. If a clock tick seconds, how many times will it tick during February, 1877? 40. If your age is 21 yr. 26 da., how many minutes old are you, if 5 leap years have occurred in that time ? 41. If a vessel sail 120 leagues in a day, how many statute miles does she sail? 42.^ How many pint, quart, and 2-quart bottles, of each an equarhumber, can be filled from a barrel of 31| gallons ? 43. In the eighteenth century, how many hours ? 44. How large an edition of a 12mo book can be printed from 2 bales, 2 bundles, 15 quires of paper, allowing 8 sheets to the volume ? 45. How many pages, 2 pages to each leaf, will there be in an 8vo book, containing 16 fully printed sheets ? How many pounds 46.^ In 36;^ centals of grain ? 47. In424bbl. of flour? 48. In 29.5 quintals of fish ? 49. Inll6|bbl. of salt? 60. In 63.25 kegs of nails ? 51. In .75 of 75 bu. of salt ? 52. In 125f bu. of wheat? 53. In I of 21648 bu. oats ? 54. In .7 of 40 bu. corn meal? ^ 55. In 7.5 casks of lime ? X What is the value in U. S. Money 66. Of 28 sovereigns ? 67. Of £25 10s. ^ 58. Of 25 francs ? 59. Of 42i marks ? BEDUCTION^. 213 436. ^To reduce denomijttate numbers from lower to higher denominations. ' ORAL, EXERCISES. 1. How many feet in 108 in. ? How many yards ? 2. How many square yards in 63 sq. ft. ? In 85 sq. ft.? 3. How many chains in 200 L? In 425 1. ? In 674 1.? 4. In 81 cu. ft., how many cu. yd. ? How many cd. ft.? 5. How many cords in 100 cd. ft. ? In 256 cu. feet ? 6. Change 120 sq. ch. to A. 80 P. to square chains. 7. In 162 in., how many hands ? Spans? Feet ? 8. In 112 pt., how many quarts ? Pecks ? Bushels ? 9. How many gallons in 46 qt. ? 96 pt.? 128 gi. ? 10. Eeduce 0. 160 to Cong. ; f 3 90 to f 3 . 11. Change 96 oz. to Troy pounds; to Avoir. 12. Reduce 345 to oz.; 3 75 to pounds. 13. In 400 pwt., how many oz. ? How many lb. ? 14. In 508 lb., how many cwt. ? In 1276 lb. ? 15. In 630 lb. of wheat, how many bushels ? 16. In 140 da. how many wk. ? Months, of 30 da. each ? 17. Change 1200 min. to hours. 84 hr. to days. 18. How many doz. are 240 eggs ? How many gross ? 19. How many degrees in 180'? Minutes, are 240"? 20. In 90 units, how many score ? Sets ? Pairs ? 21. In 120 quires of paper, how many reams ? Bundles? 22. In 120d. how many shillings? Crowns? Florins? 23. In 500 pennies, how many marks ? 437/ Prikciple. — Denominate numiers are changed to higher denominations hy Division/ 214 DENOMINATE NtJMBEBS. WMITTEN JEXJEItCISES. 438. 1. Change 5722 inches to rods. OPERATION. / Analysis.— Since 13 ia 1 2 ) 5 7 2 2 in. ■ make 1 ft, in 5722 in. ~~""~~" 1 A • there are as many feet as 3 )476 ft. + 1 in. ^^ ^^ ^^^ contained times 5 ^ ) 1 5 8 yd. + 2 ft. in 5722 in., or 476 ft. and rt 2 ^^ ^n. more. Since 3 ft. make 1 yd., 1 1 )3 1 6 half -yd. in 476 ft. there are 158 9 R rd 4- 4 vd ^^* ^^^ ^ ^** ™^^^- •^ * And since 5| yd. make 5732 in. =28 rd. 4 yd. 2 ft. 10 in. i ^d., in 158 yd. there are 28 rd. and 4 yd. more. In order to divide by 5J, both dividend and divisor may be re- duced to halves before dividing. In this case the remainder, if an;j, is halves, which may be reduced to integers. 2. Reduce 157540 minutes to weeks. 3. Reduce 80820 links to miles. 4. Change 487630 pwt. to pounds. Rule. — I. Divide the units of the given denomination hy that numher of the scale which is equal to a unit of th^ next higher denomination, and write the remainder as a part of the answer. II. In like manner, divide this and each successive quo- tient until reduced to the denomination required. TJie last quotient, loith the remainders annexed, is the required result. How many 5. Miles are 3168000 in.? 6. Acres are 256800 P.? 7. Sq. mi. are 27878400 sq. ft.? 8. Cu. ft. are 216840 cu. in.? 9. Cords are 38042 cu. ft.? Reduce 10. 30876 gills to hhd. 11. 27072 qt. to bushels. 12./ 66742 pt. to barrels. 13.^ 103720 pt. to gallons. 14. fz 8106 to Cong. REDUCTIOIS'. 215 How many- is. /Pounds Troy are 85894 gr.? 16. (Tons are 51570 pounds? 17^ Cwt. are 40607 ounces? 18/ Pounds are 3000 pwt. ? 19/ Bu. are 12060 lb. of wheat? 20.f Bbl. are 3038 lb. of flour? 21. /Bu. are 6496 lb. of oats ? 22.1 Quin. are 3172 lb. of fish? 23. Weeks are 3114061 sec. 24. Months are 8263420 min. ? 25. Degrees are 2007200'? 26. Deg. are 5270 Naut. mi. ? Eeduce 27. 120400 pens to gro. 28. 2734 eggs to dozens. 29. 5020 balls to scores. 30. 10738 sheets to rm, 31. 6048 quires to bun. 32. 24684d. to crowns. 33. 4076s. to florins. 34.; $194.66 to half-soY. 35. 42346 far. to £. 36. $86.85 to francs. 37. $225.40 to sov. 38. $47.70 to marks. 39. If the Atlantic Cable is 3200 mi. in length, and cost 10 cents a foot, what was its entire cost ? 40. If a cubic foot of gray limestone weigh 175 lb., what is the weight of a cubic yard ? 41. What is the cost of a load of oats weighing 1860 lb., at $.56 a bushel ? 42. In a storm at sea, a ship changed her longitude 423 geographic mi. How many degrees and minutes ? 43. How much time will a person gain in 40 yr., by rising 25 min. earlier and retiring 20 min. later every day, counting 9 leap years in the time ? 44. What will a peck of clover-seed cost, at $.12^ a lb. f 45. What will a ton of corn-meal cost, at $1.20 a bu. ? 46. An Illinois farmer sold a load of corn weighing 2496 lb., and a load of oats weighing 1920 lb. ; for the corn he received $.62 a bushel, and for the oats $.44 a bushel. What did he receive for both loads? 216 DENOMIl^^ATE NUMBERS. EEDUOTION OF DENOMHSTATE FRACTIONS. 439. A JDenoniinate Fif action is a fraction fv^hose integral unit is a denominate number. Thus, f of \ week, .7 of an acre, are denominate fractions. The Principles, Operations, and Analyses of the reduction of de- nominate fractions are essentially the same as those of denominate integers. 430. To reduce denominate fractions from higher to fractions of lower denominations. ORAL JS X EB CI S ES, 1. Eeduce -^ of a gallon to the fraction of a pint. Analysis. — Since in 1 gal. there are 4 qt., in ^^ gal. there are ^^ of 4 qt., or i qt. ; and since in 1 qt. there are 2 pt., in J qt. there are J of 2 pt., or ^ pt. Hence ^ gal. equals J pt. 2. Eeduce ^ lb. Troy to the fraction of an oz. 3. What part of a pint is yV ^^ a qt. ? ^ of a pk. ? 4. What decimal part of a day is .12 of a week ? Analysis.— Since in 1 wk. there are 7 days, in .12 wk. there are 12 of 7 da., or .84 da. 5. What part of a peck is .02 of a bu.? .07 bu.? .25 bu.? 6. Eeduce .5 gal. to the fraction of a quart ? Of a pint? 7. What part of an inch is -^\ of a foot ? ^ota yard ? 8. Change .04 of a pound to the decimal of an ounce. WRITTEN EXERCISES , 431. 1. Eeduce y|^ of a bushel to the fraction of a pint OPERATION. Analysis. — Same as for ora\ U^ bu. X 4 == -^ pk. questions. (430.) ^ , Q o i Multiply successively by 4, 8, -g^ pK. X « — f qt. ^^^ 2^ ^^^ numbers in the descend- ■| qt. X 2 = 4" P^* ing scale required to reduce bush* Or, ^ X f X f X f = f pt. els to pints. (425.) EEDUCTIOK. 2iT 2. Eeduce j^ of a rod to the fraction of a foot. 3. Change -^ of an ounce to the fraction of a grain. 4. What part of a pint is -^ if-^ of a hogshead ? 5. What part of a shilling is .012 of a £ ? EuLE. — Multiply the fraction of the higher denomina^ tion ly the numbers as factors in the descending scale suc- cessively ietween the given and the required denominatiouo (435.) 6. What part of an ounce is xAir ^^ ^ pound Avoir. ? 7. Eeduce y^t^ ^^ ^^ ^^^^ ^^ ^^^ fraction of a sq. rd. 8. Eeduce .005 of a bushel to the decimal of a pint. 9. How many yards is f of -^ of a rod ? 10. Change .0000625 mi. to the decimal of a foot. 11. What part of an ounce Troy is f of -|- of 2 pounds? 12. What part of a yard is -^^ of a mile ? 33. What fraction of a link is ^^^ of a rod? 14. What part of a minute is .000175 of a day? 15. What part of a sq. rd. is y-^ of 4^ times A -^- ? 433. To reduce denominate fractions to integers of lower denominations. ORAL EXERCISES. 1. How many hours in f of a day ? Analysis. — Since in 1 da. there are 24 hr., in f of a day tlierearc I of 24 hr., or 16 hr. Hence | da. equals 16 hr. 2. How many minutes in -^ hr.? In ^ hr. ? In f hr. ? 3. How many quarts in | pk. ? In |^ bu. ? In -^ bu. ? 4. How many ounces in .5 of a pound ? Analysis. — Since in 1 lb. there are 16 oz., in .5 lb. there are .6 o\ 16 oz., or 8 oz. 10 "ZIS DEKOMIKATE LUMBERS. 5. How much is .7 hr. ? .25 hr. ? .15 hr. ? .8 hr. ? 6. How many yards in y\ of a rod ? In f of a rod ? 7. How many cwt. in f of a ton ? How many pounds? 8. Change to pints ^ gal. J pk. ^ bu. f of 2 pk. 9. Change ^ of an acre to sq. rd. f sq. yd. to sq. ft. 10. How many pecks in .75 bu. ? Quarts in 1.25 pk. ? 11. Change J lb. to oz. .45 oz. to pwt. .53 cwt. to lb„ WUITTEN BXEJRCISES, 433. 1. Reduce f bu. and .645 da., each to integers of lower denominations. 1st operation. 4pk.x|i=-%Q-pk. = 3^pk. 8qt. Xi = f qt. =2|qt. 2pt. x|=i pt. = lipt. I- bu. = 3 pk. 2 qt. 1| pt. 1st operation. 24 hr. X.645 = 15.48 hr. 60 min. X.48 = 28.8 min. 60 sec. X .8 =48 sec. .645 da*=15 hr. 28 min. 48 sec. 1. The analyses of the above are the same as in (425) and (430). 2. The following methods may be regarded as most convenient in practice, since the operations are performed without rewriting the fractional part of each product, 2d operation. 2d operation. 5 .6 4 5 da. A 24 6)20(3pko2qt. IJpt. 2580 4 1_8 1290 2 2 1 5.4 8 hr. 6)8 pt. 8 |bu,=3pk.2qt. IJpt. 6^ 5 6 ) 1 6 qt. 2 8.8 min.. 2 1^ 60 4 4 8.0 sec. .6 4 5 da. — 1 5 hr. 2 8 min. 4 8 sec. BEDU CTIOiq". 219 Reduce to integers of lower denominations, 2. iiotsi £. 3. .35 lb. Apoth. 4. T^ of a mi. 5. .75 lb. Troy. .625 of a fath. .55 lb. Ayoir. EuLE. — I. Multiply the given fraction or decimal hy that numler in the scale that will reduce it to the next loiver denomination, (4^5.) II. Proceed in like manner with the fractional part of each successive product, until it is reduced to the denomi- nation required. III. The integral parts of the several products, arranged in their proper order, is the required result Find the value in integers of lower 8. Of 4 mo. 9. Of .555 £. 10. Of ^^ A. 11. Of I of I lb. 12. Of -^ cu. yd. 13. Of .1934 S. 14. Of f 1 .7. 15. Offf^T. 16. Of .875 hhd. 17. Of I sq. rd. 18. Of ft) if. 19. Of f 0. gro. 20. Of .67 lea. 21. Of .125 bbl. 29. At 8^ cents a pound, what will denominations, 22. Of .578125 bu. 23. Of .6625 mi. 24. Of i of 5^ T. 25. Of I of 3| A. 26. Of f of 3| Cd. 27. Of i of .225 ml 28. Of ,3125 ream. ^ T. of cheese cost ? 434, To reduce denominate fractions from lower to fractions of higher denominations. oBjLL ex ercisbs, 1. Reduce f of a peck to the fraction of a bushel. AnaIiYsis. — Since 4 pk. make 1 bu., there are J as many bushels as pecks ; J of f pk. = ^^ or ^ bu. 2. Reduce f of a pint to the fraction of a gallon. 3. What part of a pound Avoir, is f oz. ? 4. What part of a week is | da. ? i da. ? | da. ? | da. ? ago DEKOMIXATE KUMBEKS. 5. What decimal part of a gallon is .28 of a quart ? Analysis. — Since 4 qt. make 1 gal., there are J as many gallon^ as quarts ; J of .28 qt. is .07 gal. 6. Change .32 of a pint to the decimal of a quart. 7. What decimal of a pound Troy is .48 oz. ? .84 oz, ? 8. What decimal of a week is .35 da. ? .63 da. ? 0. Change .72 in. to the decimal of a foot. Of a yd. WRITTEN EXERCISES. 435* 1. What fraction of a bushel is ^ of a pint ? OPERATION. . T^. ., . . Analysis. — Divide successively f pt. -V- 2 = f qt. ^^ 2, 8, and 4, the numbers in the •| qt. -f- 8 = -^ pk. ascending scale, required to reduce -Xr pk. -T- 4 zz: -J^ bu. P^^*^ *^ bushels. (428.) Hence 2. Eeduce ^ of a gill to the fraction of a gallon. 3. Change f of a shilling to the fraction of a £, 4. Eeduce .64 of a pint to the decimal of a bushel. 5. What part of a pound Troy is .576 of a grain ? KuLE. — Divide the fraction of the lower denoynination iy the numbers as factors in the ascending scale successively hetiveen the given and the required denomination. (438.) 6. What decimal of a ton is .8 lb. ? .36 of a cwt. ? 7. Eeduce ^f of a cord foot to the fraction of a cord ? 8. Eeduce .216 gr. to the decimal of an ounce Trojo 9. What part of a ton is f of a pound ? 10. What part of a day is f of a minute? .12 hr. ? 11. What decimal of a rod are 3.96 in. ? 12. How much less is | of a pint than -^-^ of a hhd. ? 13. What part of an acre is ^ of a square rod ? REDUCTIOIS". 221 436. To reduce a compound denominate num- ber to a traction of a higher denomination. ORAL JEXER C IS ES. 1. What part of a pound are 4 oz. ? 8 oz. ? 10 oz. ? 2. What part of a foot are 9 in. ? What part of a yard ? 3. What part of a bushel are 2 pk. 4 qt. ? Analysis.— 1 bu.r=32 qt, and 2 pk. 4 qt.-20 qt. ; 20 qt.r^ff bu. >^f bu., or .625 bu. Hence 2 pk. 4 qt.=:| bu., or .625 bu. 4. What part of a yard are 2 ft. 6 in.? Are 18 in. ? 5. What part of 3 lb. Troy are 1 lb. 6 oz. ? Are 9 oz. ? 6. What part of 5 gal. are 2 gal. 2 qt. ? 3 gal. 1 qt. ? 7. Eeduce 12 oz. to the decimal of 3 lb. Avoir. 8. What fraction of 3 Cd. 6 cd. ft. are 2 Cd. 4 cd. ft. ? 9. What part of 3 pk. are 1 pk. 4 qt. ? Are ^ pk. ? WRITTEN EXERCISES, 437. 1. What decimal of a pound Troy are 2 oz. 14 pwt. ? 1st operation. Analysis. — Since 20 pwt. make 2 0)14 pwt. 1 oz., there are ^ as many ounces 1 2)2 7 OZ ^^ pennyweights ; and -^^ as many - — '- pounds as ounces (428). Hence .225 Ib.m^ lb. 2 oz. 14 pwt.=.225 lb., or changed to 2i fraction, by 283, is ^% lb. 2d operation. Analysis. — In order to iBnd 2 oz. 1 4 pwt. =1 5 4 pwt. '^^^ 'P^^'rt one compound number I ]i) __ 2 4 nwt ^^ ^^ another, hoth must be like * o o K 11. * numbers, and must be reduced to 2lV = A l"^* ^= * '^ ^ ^ !"• the lowest denomination in either. Thus, 2 oz. 14 pwt. are equal to 54 pwt., and 1 lb. is equal to 240 pwt. Hence 2 oz. 14 pwt.=/^% lb = -^ lb., or, reduced to a decimal, by 285, .225 lb. 2. Eeduce 3 gal. 3 qt. 1^ pt. to the fraction of a bbl. 3. Reduce 3 cd. ft. 8 cu. ft. to the decimal of a cord. 222 DEKOMIKATE KUMBEES. EuLE. — I. Divide the units of the lo7vest denomination given hy that number in the scale which is equal to a unit of the next higher denomination, and annex the quotient as a decimal to the number given of that denomination. II. Proceed in like manner until the whole is reduced to the denomination required. Or, Reduce the given number to its lowest denomination for the numerator of the required fraction, and a unit of the required denomination to the same denomination for the denominator, and reduce the fraction to its lowest terms, or to a decimal. 1. If the given number contain a fraction, tlie denominator of this fraction must be regarded as the lowest denomination. 2. The pupil may be required to give the answers either in the form of a fraction, or of a decimal, or both. 4. Eeduce 13 gal. 3 qt. 3.62 gi. to the decimal of a hhd. 5. What part of a pound Troy are 10 oz. 13 pwt. 9 gr.? 6. What fraction of 2 T. 7 cwt. 28 lb. are 5 cwt. 91 lb.? 7. What part of 3 A. 80 P. are 51.52 P. ? 8. What part of a f ^ are f 3 5 m36 ? 9. Change 126 A. 4 sq. ch. 12 P. to the decimal of a Tp. 10. What decimal part of 25° 42' 40" are T 42' 48"? 11. From a hhd. of molasses 28 gal. 2 qt. were drawn, ^hat part of the whole remained ? 12. What decimal of a league are 2 mi. 3 rd. 1 yd. 3| in.? 13. What part of 3 bbl. of flour are 110 lb. 4 oz.? 14. What decimal part of a ton is \ of 22f lb. ? 15. Reduce .45 pk. to the decimal of 1^^ bu. 16. What part of 54 cords of wood are 4800 cu. ft. ? 17. Change 18s. 5d. 2^ far. to the fraction of a £. REVIEW. 223 REVIEW. WRITTEN EXAMPLES. 438. 1. How many steps of 30 in. each must a pers'on take in walking 21 miles ? 2. How long will it take one of the heavenly bodies to move through a sextant, at the rate of 3' 12" a minute? 3. Eeduce £10 18s. Gd. to United States Money. 4. Paid $425.75 for 2|^ tons of cheese, and retailed it at 12-1^ cents a pound. What was the whole gain ? 5. Eeduce 580 francs to United States Money. 6. Change $291.99 to Sterling Money. 7. What cost 30 bu. 2 pk. 1 qt. of beans, at $4.20 a bu. ? 8. Bought 15 cwt. 22 lb. of rice at $4.25 a cwt., and 6 cwt. 36 lb. of pearl-barley at $5.60 a cwt. What would be gained by selling the whole at Q^ cents a pound ? 9. How many bushels of corn in 36824 lb., Illinois standard ? Louisiana ? New York ? 10. 5000 bu. of oats in Ohio are equal to how many bushels in Connecticut, by weight ? In New Jersey ? 11. If I buy 16 T. 3 cwt. 3 qr. 24 lb. of Eng. iron, by long ton weight, at 3d. a lb., and sell the same at $140, by the short ton, what do I gain by the transaction ? 12. How many carats fine is a piece of gold | pure ? 13. How many acres in a piece of land 105 ch. 85 L long, and 40 ch. 15 1. wide ? 14. If 10 lb. of milk make 1 lb. of cheese, what will \\ jost at 1 cent a pound to manufacture the cheese thai Jan be made from 90000 lb. of milk ? 15. At $75| an acre, what is the value of a farm 189.5 rd. long and 150 rd. wide ? 324 DEKOMIKATE KUMBEES. 16. What cost 2 bu. 3 pk. 6 qt. of green peas, at $.30 a peck ? 17. What cost 3 T. 17 cwt. 20 lb. of hay, at $22f a ton ? 18. If a grocer's scales give ^ oz. short of true weight on every pound, of how much money does he defraud his Customers, in the sale of 3 bbl. of sugar, each weighing 2 mt. 10 lb., at 12| cents a pound? 19. If 37 A. 128 P. are sold from a farm containing 170 A. 16 P., what part of the whole remains ? 20. Paid $526.05 for 3^ tons of cheese, and retailed it at 9| cents a pound. How much was the whole gain ?. 21. How many bushels of oats in Connecticut are equivalent in weight to 2500 bushels in Iowa ? 22. How many centals of barley in California are equiva- lent to 1500 bushels in Missouri ? 23. A man sold 12 bu. 3 pk. 6 qt. of cranberries at $3^ a bushel, and took his pay in flour at 4 cents a pound. How many barrels did he receive ? 24. If 3 T. 12 cwt. 20 lb. of ground plaster cost $15.75, what will be the cost of 5 T. 80 lb. at the same rate ? 25. Bought 37 Cd. 48 cu. ft. of wood for $129.81, and there was but 13 Cd. 59 cu. ft. delivered. What part ol the money should be paid ? 26. If a grocer's gallon measure is too small by 1 gi., what does he make dishonestly in selling 2 hhd. of mo- lasses, averaging 58 gal. 2 qt. 1 pt. each, worth $.80 a gal.? 27. How many reams of paper are required to supply 4500 subscribers with a weekly newspaper for 1 year ? 28. A publisher printed an edition of 10000 copies of a 12mo book of 336 pp. How much paper did he use, allowing 1 quire to each ream for waste ? ADDITIOiq-, ADDITIOIsr. 439. Denominate numbers are added, subtracted^ mul- tiplied, and divided by the same general methods as are smployed for like operations in Simple Numbers. The corresponding processes are based upon the same principles. The only modification of the rules needed is that which is required by a varying scale instead of a uniform scale of 10. The principles will be made sufficiently plain In the operations and analyses to render special rules unnecessary. WMITT EN EXERCISES. 440. 1. Find the sum of 4 cwt. 46 lb. 12 oz., 12 cwt. ^\h., 2J cwt, and 21 1 lb. Analysis. — Write the numbers so that units of the same denomination stand in the same column, and begin at the right to add. The sum of the ounces is 30, equal to 1 lb, 14 oz. Write the 14 oz. under the column of ounces, and add the 1 lb. to the pounds of the next column. T Q 2 14 '^^^ ^^^ ^^ *^^ pounds is 102 lb., equal to 1 cwt. and 2 lb. Write the 2 lb. under the column of pounds, and add the 1 cwt. to the cwt. of the next column. The sum of the cwt. is 19 cwt., which write under the column of cwt. Hence the entire sum is 19 cwt. 2 lb. 14 oz. 2. What is the sum of -^ wk., | da., and | hr. ? 1st operation. Analysis. — First find the da. hr. min. sec. value of each denominate ^ wk. := ^ 21 36 00 fraction in integers of lower 3(jg^--_ ][4 24 00 denominations (433), and I hr. = 2 2 3 OPERATION. cwt. lb. OZ. 4 46 12 12 9 8 2 25 21 10 12 22 30 then add the resulting com pound numbers. Or, 836 DENOMIIJfATE NUMBERS. 2d operation. Reduce the given 3 ^^ __. 3 ^]^^ fractions to fractions of 3 1 , ^j 1 the same denomination f nr. — -^f^ WK. (434), then add the re. yV Wk. + ^ + Th = fit wk. suits and find the value Jl^ wk. — 5 da. 12 hr. 22 min. 30 sec. of their sum in integers of lower denominations. If denominate fractions occur in the given numbers, they should be reduced to integers of lower denominations (433) before adding. 3. Add 7 yd. 2 ft., 5 yd. IJ ft., 2 ft. 9^ in., 3 yd. 1 ft. 6i in., 2| ft., ^ yd. 4. Add 5 Cd. 7 cd. ft., 2 Cd. 2 cd. ft. 12 cu. ft., 6 cd. ft. 15 cu. ft., 7| Cd., and 3 Cd. 2 cu. ft. 5. What is the sum of If hhd., 36 gal 3 qt. IJ pt., I gal., 2 qt. I pt., and 1.75 pt. ? 6. What is the sum of | of a day added to ^ of an hour ? 7. To 4 of a hhd. add | of 10 gal. 8. What is the sum of 22^ cwt., 26 J lb., and 14 oz. ? 9. Add 5^ Pch., 18 cu. ft., 86.6 cu. ft., and f Pch. 10. Find the sum of lb 4 | 6 3 5, and ft) 6 m 3 9}. 11. A Missouri farmer received $.75 a bushel for 4 loads of corn; the first contained 48.4 bu., the second 2626 lb., the third 36f bu., and the fourth 41 bu. 52 lb. What did he receive for the whole ? 12. Bought three loads of hay at $15 a ton. The first weighed 1.125 T., the second 1| T., and the third 2750 lb What did the whole cost ? 13. When B was born, A's age was 3 yr. 9 mo. 24 da. ^ when C was bom, B's age was 12 yr. 19 da. ; when D was born, C's age was 5 yr. 11 mo., and when E was born, D's age was 10 yr. 1 mo. 20 da. What was A's age when E was born ? OPEKATION, rd. yd. ft. in. 25 2 2 6.3 12 4 11.6 12 3(1)1 6.7 i = 1 6 SUBTEACTION. 227 SUBTEAOTIOE". WJtITTEN BXEItCISES. 441. 1. From 25 rd. 2 yd. 2 ft. 6.3 in., subtract 12 rde 4 yd. 11.6 in. Analysis. — Write the numbers so that units of the same denomina- tion stand in the same column, and begin at the right to subtract. Since 11.6 in. cannot be subtracted from 6.3, take 1 ft., equal to 12 in., from the 2 ft., leaving 1 ft. and add it to the 6.3 in., making 18.3 in. •^ ^ ^ ^ • * Subtract 11.6 in., and write the re- mainder, 6.7 in., under the inches. Since 1 ft. has been taken from 2 ft., subtract ft. from 1 ft., and write the remainder 1 ft. under the feet. Since 4 yd. cannot be taken from 2 yd., take 1 rd., equal to 5} yd., from 25 rd., leaving 24 rd., and add it to the 2 yd., making 7^ yd. Subtract 4 yd. from 7 J yd., and write the remainder, ^ yd., under the yards. Since 1 rod has been taken from 25 rd., subtract 12 rd. from 24 rd., and write the remainder, 12 rd., under the rods. The J yd., reduced to feet and inches, and added to 1 ft. 6.7 in. of the remainder, gives 12 rd. 4 yd. .7 in. 2. From 1| bu. subtract f bu. OPERATION. Analysis. — First 1|^ bu. z=z 1 bu. 2 pk. 4 qt. pt, find the value of each 4 ^^^ __ 3 1 14- denominate fraction in integers of lower de* 3 2^ nominations (433), Or, and subtract the less value from the greater. 1| DU.=J^bu.; J^bu.-|bu.=i|bu. Or, reduce the given 1^ bu. = 3 pk. 2 qt. ^ pt. fractions to fractions of the same denomi- nation (434), then subtract the less from the greater, and find the value of their difference in integers of lower denominations. 228 DEKOMINATE iq^UMBERS. 3. From a pile of wood containing 42 Cd. 5 cd. ft., take 16 Cd. 6 cd. ft. 12 cu. ft., and how much remains? 4. From the sum of f of 3f mi. and 174- ^^-^ ^^^e 120^ rd. Find the difference between 10. f lea. and -^ mi. 11. f gross and f doz. 12. 3 2 31 and 14. 13. .9 da. and ^ wk. 14. fl A. and 84.56 P. 5. S^^ cwt. and 48f lb. 6. £f and f of f s. 7. /^ lb. and 5 lb. 4 oz. 8 p wt. 8. .659 wk. and 2 wk. 3f da. 9. m hhd. and 3.625 gal. 15. If from a hhd. of molasses 14 gal. 1 qt. 1 pt. be drawn at one time, 10 gal. 3 qt. at another, and 29 gal. 1 pt. at another, how much will remain ? 16. Of a farm containing 250 A., two lots were re- served, one containing 75 A. 136.4 P., and the other 56 A. 123.3 P. ; the remainder was sold at $62^ an acre. What did it sell for? 17. From 1 T. 11 cwt. 30 lb., take I of a long ton. 18. From a pile of wood containing 125f Cd., was sold at one time 26 Cd. 7 cd. ft. ; at another, 30 Cd. 4| cd. ft. ; at another, 37^^ Cd. How much remained ? 443. To find the interval of time between two dates. 1. How many yr., mo., da., and hr., from 3 o'clock p. m. of May 16, 1864, to 9 o'clock A. m. of Sept. 25, 1875 ? OPERATION. Analysis. — Since the later date yr. mo. da. hr. expresses the greater period of time, 1875 9 25 9 write it as the minuend, and the ear- 15^fi4. ^ Ifi 1^ ^^®^ ^^*^ ^^ *^^ subtrahend, writing • the denominations in the order of the 11 4 8 18 scale, then subtract. SUBTRACTIOK. 22S 1. When hours are to be obtained, reckon from 12 at night, and d minutes and seconds, write them still at the right of hours. 2. In finding the difference of time between two dates, 12 mo. are usually considered a year, and 30 days a month. 3. When the time is less than a year, the true number of days Ib each month and parts of a month is added, 4. The day on which a note, draft, or contract is dated, and that yn which they mature^ are not hoth included. The former is gen^ arally omitted. 2. The war between England and America was com- menced April 19, 1775, and peace was restored Jan. 20, 1783. What length of time did the war continue ? 3. The American Civil War began April 11, 1861, and closed April 9, 1865. What time did it continue ? 4. How long has a note to run that is dated Jan. 16, 1873, and made payable July 10, 1875 ? 5. A note dated May 28, 1875, was paid Feb. 10, 1876. What length of time did it run ? 6. A person started on a tour of the world at 9 o'clock A.M., Sept. 3, 1874, and returned to the same depot at 3 P.M., July 15, 1876. What time was he absent ? 7. How many years, months, and days from your birth- day to this date ; or, what is your age ? 8. How many days from June 20th to the 10th of Jan, following? 9. What length of time elapsed from 12 o'clock m., Jan. 10. 1876, to June 16, 9 o'clock a.m. ? 10. What length of time elapsed from 16 min. past IG o^clock A.M., July 4, 1873, to 22 min. before 8 o'clock P.M., Dec. 12, 1875 ? 11. What length of time will elapse from 40 min. 25 sec. past 12 o'clock m., April 21, 1875, to 4 min, 36 sec. before 5 o'clock p.m., Jan. 1, 1878 ? DENOMIJTATE NUMBEKS. MULTIPLIOATIOK. WltlTTEN EXEItCIS JE8. 443. 1. Multiply 28 rd. 2 yd. 2 ft. by 7. OPERATION. Analysis. — Write the inulti« % 8 rd. 2 yd. 2 ft. plier under the lowest denomina. w tion of the multiplicand, and multiply. 19 9 1 (I) 2 7 times 2 ft. are 14 ft., equal to ^ = 1 6 in. 4 yd. 2 ft. Write the 2 ft. under - q Q '^ Z TT the feet, and reserve the 4 yd. to add to the product of yards. 7 times 2 yd. are 14 yd., and 4 yd. added make 18 yd., equal to 3 rd. IJ yd. Write the 1| yd. under the yards, and reserve the 3 rd. to add to the product of rods. 7 times 28 rd. are 196 rd., and 3 rd. added make 199 rd., which write under the same denomination. The ^ yd. is equal to 1 ft. 6 in., which added to the product, gives 199 rd. 2 yd. 6 in. for the entire product. 1. The multiplier must be an abstract number. (103.) 2. When the multiplier is large and is a composite number, the work may be shortened by multiplying successively by its factors. (109.) 2. In 9 bbL of walnuts, each containing 2 bn. 3 pk. 6 qt., how many bushels ? 3. If a man cut 3 Cd. 36 cu. ft. of wood in 1 da., how many cords can he cut in 12 days ? 4. Multiply 8 gal. 3 qt. 1 pt. 3.25 gi. by 96. 5. If 1 A. produce 42 bu. 1 pk. 5 qt. 1 pt. of com, how many bushels will 64 A. produce ? 6o Multiply 0. 8 f 1 9 f 3 6 m 34 by 24. 7. What will 84 yd. of cloth cost, at £1 8s. 9id. a yd. ? 8. If $80 will buy 3 A. 24 P. 20 sq. yd. 4 sq. ft. of land, how much will $4800 buy ? 9 24 bu. Opk. 6 5 120 bu. 5 3pk. 1 6 4 (C MULTIPLICATION. 231 9. How many bushels of grain in 47 bags, each con- taining 2 bu. 2 pk. 6 qt. ? OPERATION. Analysis.— Multiplying At^ /Q y^ K\ fO *^® contents of 1 bag by 9, ^ ' and the resulting product 2 bu. 2 pk. 6 qt. by 5, gives the contents of 45 bags, which is the com- ~ , . rt 1 posite number next les$ 6 qt. m 9 baffs. ^. ^. • ^ ^ than the given pnme num- ber, 47. Next find tho 45 (i contents of 2 bags, which, ^ added to the contents of 45 bags, gives the contents of 126 bu. 1 pk. 2 qt. '' 47 '' 45 + 2, or 47 bags. 10. If a load of coal by the long ton weigh 1 T. 6 cwt 2 qr. 26 lb. 10 oz., what will be the weight of 67 loads ? 11. Multiply 4 yd. 1 ft. 4.7 in. by 125. 12. Multiply 7 T. 15 cwt. 10.5 lb. by 1.7. 13. At $1.37|- a gallon, what will be the cost of 5 casks of wine, each containing 28 gal. 2 qt. 1 pt. ? 14. A farmer sold 4 loads of oats, averaging 41 bu. 3 pk. each, at $.75 a bushel. What did he receive for the whole ? DIYISIOI^. 444, 1. Divide 56 lb. 9 oz. 12 pwt. by 6. OPERATION. Analysis.— Write the divisor at the left lb. oz. pwt. of the dividend. The object is to find i 6)56 9 12 8ixt7i of a compound number. 7 ~ ~~ J of 56 lb. is 9 lb. and a remainder of 2 lb. Write the 9 lb. in the quotient, and reduce ihe 2 lb. to ounces, which, added to 9 oz., make 33 oz. ^ of 83 oz. is 5 oz. and a remainder of 3 oz. Write the 5 oz. in the quotient, and reduce the 3 oz. to pwt., which added to 12 pwt., make 72 pwt. J of 72 pwt. is 12 pwt., which write in the quotient 232 DENOMINATE NUMBERS. 2. Divide 358 A. 57 P. 6 sq. yd. 2 sq. ft. by 7. 3. Divide £35 9s. 7d. by 5 ; by 7; by 8. 4. Divide 282 bu. 3 pk. 1 qt. 1 pt. by 9 ; by 10 ; by 12, When the divisor is large, and is a composite number, the work may be shortened by dividing successively by its factors. 5. Divide 254 yd. 4 ft. 3^ in. by 21 ; by 42. 6. Divide 196 Cd. 4 cd. ft. 12 cu. ft. by 72. 7. How many iron rails, each 16 ft. long, are required to jay a railroad track 26 mi. long ? 8. Divide 24 sq. mi. 140 P., by 22f 9. Divide 202 yd. 1 ft. 6f in. by |. 10. Divide 336 bu. 3 pk. 4 qt. by 4 bu. 3 pk. 2 qt. Reduce both dividend and divisor to the same denomination, and divide as in simple numbers. 11. How many boxes, each holding 1 bu. 1 pk. 7 qt., can be filled from 356 bu. 3 pk. 5 qt. of cranberries? 12. Divide 311 gal. 1 qt. 1 pt. by 53. OPERATION. 5 3 ) 3 1 1 gal. 1 qt. 1 pt. ( 5 gal. 3 qt. I pt. 265 4 6 gal. rem. 2 6 qt. rem. 4 2 1 8 5 qt. in 46 gal. 1 qt. 5 3 pt. in 26 qt. 1 pt 159 53 2 6 qt. rem. 13. The aggregate weight of 41 hhd. of sugar is 19 T. *5 cwt. 22 lb. What is the average weight ? 14. If a town 4 mi. square be equally divided into 63 farms, how much land will each farm contain ? LONGITUDE AND TIME. 233 LONGITUDE AI^D TIME. 445. The Longitude of a place is its distance east or west from a given meridian, measured on the equator. The meridian from which longitude is reckoned is called the first meridian, and is marked 0°. All places east of this, within 180°, are in east longitude, and all places west, within 180°, are in west longitude. The English and Americans usually reckon longitude from the meridian of Greenwich, England ; tlie French, from Paris. 446. Since the earth revolves on its axis once in 24 hours, the sun appears to pass from east to west around the earth, or through 360° of longitude once in 24 hours of time. Hence in 1 hour the sun appears to pass through ^ of 360% or 15° ; in 1 minute, through -g-V of 15°, or J 5' ; and in 1 second, through -^^ of 15', or 15''. CoMPAKisoi!^ OF Longitude and Time. ^or a difference of There is a difference of 15Mn Long. 1 lir. in Time. 15' " *t 1 min. " ** 15" " i( 1 sec. " ** r « <« 4 min. '* '* V " <« 4 sec. ** *' -^fi «< u yVsec. " " 1. Since the sun appears to move from east to west, when it is 12 o'clock at one place, it will be past 12 o'clock at all places east; and hefore 12 at all places west. Hence, knowing the difference of time between two places, and the exact time at one of them, the exact time at the other is found by adding their difference tc the given time, if it is east, and by subtracting, if it is west. 2. If one place is in east and the other in west longitude, the difference of longitude is found by adding them, and if the sum i& greater than 180', by subtracting it from 360°. 234 DEI^OMIKATE NUMBEBS, OJIA.Z BXERCISES. 4:4:11. 1. The earth revolves on its axis once in every 34 hr. What part of a revolution does it make in 12 hr. ? 2. How many degrees of the earth's surface pass under ihe sun's rays in 24 hr. ? In 12 hr. ? In 4 hr. ? In 1 hr. ? 3. How many degrees of longitude cause a differenco of 1 hr. in time ? 2 hr. ? 3 hr. ? 4. When it is 6 o'clock at Chicago, what is the hour 15° east of Chicago ? 15° west of Chicago ? 5. When it is noon in New York, what is the hour 15° east of New York ? 30° west of N. Y. ? 6. When it is 3 o'clock at Washington, what is the time 15° 15' east of Washington? 30° 30' west ? 7. What difference of longitude causes a difference of 1 hr, in time ? Of 1 minute ? Of 1 second ? 8. If the difference in the time of Boston and of St. Louis is 1 hr. 15 min., what is the difference in their longitude ? 9. A man left New Orleans and traveled until his watch was 1 hr. 2 min. too fast. How far had he trav- eled, and in what direction ? 10. Two persons, at different points, observe an eclipse of the moon; one seeing it at 9^ p. m., and the other at midnight. What is the difference in their longitude ? 11. A tourist leaves home at 12 m. on Monday, and on Saturday finds his watch 1 hr. 15 min. slow. In whai direction has he been traveling ? How far ? 12. A and B start from opposite points and travel towards each other. When they meet, A's watch is 40 min. slow and B's 1 hr. fast. How far apart are the two points of starting, and in what direction did each travel? LONGITUDE AND TIME. 235 WMITTBN EXERCISES. 448. To find the difference of longitude between two places, wlien the difference of time is known. 1. When it is 9 o'clock at Washington, it is 7 min. 4 secc past 8 o'clock at St. Louis. Find the diff. of longitude. Analysis. — Since every OPERATION. liour of time corresponds 9 lir. min. sec. to 15° of long., and every % ^ 4 minute of time to 15' of '' -T^./« . m. long., and every second of 5 2 5 6 Diff. in Time. time to 15" of long. (446), 1 5 there are 15 times as many 1 3^ 1 4' 0^' Diff. in Long. ^^^- "^^•' ^^^ «^^- ^^ *^® difference of longitude, as Or, 4 ) 5 2 min. 5 6 sec. tliere are hr., min., and 13° 14' ^^^* ^^ *^® difference of time. Or, Since 4 min. of time make a difference of 1° of long., and 4 sec. of time a difference of 1' of long., tliere will be \ as many degrees of long, as tliere are minutes of time, and \ as many minutes of long, as there are seconds of time. 2. The difference in the time of Washington and of St. Petersburg!! is 7 hr. 9 min. 19J^ sec. What is the differ- ence in their longitudes ? 3. When it is 12 o'clock m. at Rochester, N. Y., it is 9 hr. 1 min. 37 sec. A. m. at San Francisco. The long, of Eochester being 77° 51' W., what is the long, of the latter? EuLE. — Multiply the difference of time expressed in hours, minutes, and seconds by 15 ; the product will be the difference of longitude in degrees, minutes, and seconds. Or, Reduce the difference of time to minutes and seconds, then divide by 4; the quotient taill be the difference of lon- gitude in degrees and mimdes. 236 DEi^OMINATE NUMBERS. 4. Noon comes 1 hr. 5 min. 42 sec. sooner at Quebec than at Chicago, whose longitude is 87° 37' 4.6". What is the longitude of Quebec ? 5. When the days and nights are of equal length, and it is noon on the first meridian, on what meridian is it t len sunrise ? Sunset ? Midnight ? 449. TJie followi7ig table of the Lo7igitude of places is compiled fro7n the Records of the U, S. Coast Survey, Albany 73° 44 50" W. Ann Arbor 83° 43' W. Astoria, Or 124° W. Boston 71° 3'30"W. Berlin 13° 23' 45 " E. Bombay 72° 54 E. Cincinnati 84° 29' 31" W. Chicago 87° 37' 45" W. Cambridge, Mass. 71" 7' 40" W. Jefferson City, Mo. 92° 8' W. Mexico 99° 5' W. New York 74° 3' w. New Orleans 90° 2' 30' w. Paris 2° 20' E. Rome 12° 27' E. Richmond, Va. . . 77° 25' 45' w San Francisco 122° 26' 45' W St. Paul, Minn. . . 95° 4' 55' W. St. Louis, Mo 90° 15' 15' W. Univ. of Virginia. 78° 31' 30' W. West Point, N.Y 73° 57' W. Washington, D.C 77° (yi5" W. 450. To find the difference of time between two places, when their longitudes are given. 1. Find the di£E. in the time of Cinn. and of St. Paul. 9 5° 84 4' 29 OPERATION. 5 5" Long, of St. P. 31 '' Cinn. Analysis.— Since 15° of Ion gitude make i difference of \ hr. of time and 15', a difference of Imin.of time, and 15", a difference of 1 sec. of time (446), there are j^ as many hours, minutes, and seconds of time as there are degrees, minutes^ and seconds of longitude. I 5 )10° 3 5' 2 4" Diff. of Long. 4 3 min. 2 1 1 sec. Diff. of Time. LOl^GITUDE AKD TIME. 237 10° 35' 241' ^^y Z. min. sec. 4 2 min. 2 1 sec. |f == 42 21| Since 1° of long, makes a diff. of 4 min. of time, and V makes i diff. of 4 sec. of time (446), there is a diff. of 4 times as many min- utes and seconds of time as there are deg., min., and sec. of long. 2. Find the difference in the time of Ann Arbor, Mich., and of Cambridge, Mass. ? * 3. When it is half-past 3 o'clock p.m. at West Point, N. Y., what time is it at Bombay ? EuLE. — Divide the differe7ice of longitude expressed in degrees, minutes, and seconds, by 15 ; the quotient luill he the difference of time in hours, minutes, and seconds. Or, Multiply the difference of longitude hy 4, and the 'product will he the difference of time in minutes and seconds, which may he reduced to hours. Find the difference in time of 4. Washington, and Rome. 5. Chicago, and Paris. 6. N. Orleans, and N. York. 7. Albany, and Jefferson C'y. 8. Eichm'd, and St. Louis. 9. New York, and Mexico. 10. Ann Arbor, and Berlin. 11. Mexico, and San Fran. 12. When it is 6 a.m. at Boston, what time is it at Cin- cinnati ? At Chicago ? At St. Louis ?. 13. When it is 6 p.m. at the University of Ya., What time is it at Berlin ? At St. Paul ? At Astoria, Or. ? 14. How much later does the sun rise in New York than in Eome? Than in Paris? 15. In sailing from San Francisco to Bombay, will a chronometer gain or lose time, and how much ? * Take from the Table the required Longitude of the different places, 238 DENOMINATE NUMBERS. DUODECIMALS. 451. Duodecimals are fractions of a foot formed by successively dividing by 12 ; as, y^^ tJt^ tiVf^ ^^c. 453. The Unit of measure is 1 foot, which may be a linear, a square, or a cubic foot The 5c«fo is uniformly 12. 453. In the duodecimal divisions of a foot, the differ- ent orders of units are related as follows : 1' (inch or prime) = ^ij of a foot, or 1 in. Linear Meas. 1" (second) or -^ of ^^^ = yj^ of a foot, or 1 ** Square " 1'" (third) or jV of xV of yV = ttV^ of a foot, or 1 " Cubic '' Table. 12 Fourths C^'0=1 Third . . V" 12 Thirds =1 Second . . 1" 12 Seconds =1 Prime . . 1' 12 Primes =1 Foot . , . ft. l/if.z=12'=144"=1728"'=20736"" 1'= 12"=- 144'"-- 1728"" 1"= 13'"= 144"" The marks ', '^ '^^ ^'^^ are called Indices. Duodecimals are used by artificers in measuring surfaces and 5 and yV oi 140 is 14 Hence lOfe of 140 is 14. What is 2. 5^ of $80? 3. 1% of 200 lb. ? 4. 6^ of 150 men? 5. 25^ of 120 mi.? Find the amount 10. Of 100 A. +27^. 11. Of $75 + 5^. 12. Of 32doz. + 12i^. How much is 6. 12^^ of 72 gal. ? 7. 40^ of 60 sheep? 8. S% of 50 bu. ? 9. 50^ of $240 ? Find the difference 13. Of 90hhd.— 10^. 14. Of 63 Cd.— 33i^. 15. Of $200-2^^. 16. A farmer had 150 sheep, and sold 20^ of them. How many had he left ? 17. A mechanic who received $20 a week had his sal- ary increased S%. What were his daily wages then ? 18. From a hhd. of molasses 33-|-^ was drawn. How many gallons remained ? 19. A grocer bought 150 dozen eggs, and found 16-|^ ^f them bad or broken. How many were salable ? 20. A train of cars running 25 miles an hour increases its speed 12|^^. How far does it then run in an hour ? 511. Principle. — The percentage of any number is the same part of that number as the given rate is of 100%, PERCENTAGE. 269 n HITTEIf BXERCISES 612. 1. What is 11% of $4957? OPERATION. $4957 .17 $842.69 ANALYSiS-^Since 17% is .17, the required percentage is .17 of $4957, or $4957 x .17, which is $842.69. What is 2. db% of 695 lb. ? 3. 75^ of $8428? 4. 12^% of £2105 ? EuLE. — Multiply the base iy the rate, part of the hase as the rate is of 100^. This rule may be briefly expressed by the following FoEMULA. — Percentage = Base x Rate. Find 5. 33^% of 8736 bu. 6. i% of $35000. 7. 120^ of $171.24. Or, take such a What Find 8. Is 4:^% of 312.8 rd. ? 13. 84^ of 254 bu. 9, Is 105^ of $5728 ? 14. 25^ of f of a ton. 10. Is $3140.75 + 11^? 15. 1% of 16400 men. 11. Is2f mi. + 7i^? 16. i% of 1- of a year. 12. Is400ft.-3i^? 17. \% of if of a hhd 18. The bread made from a barrel of flour weighs 35^ more than the flour. What is the weight of the bread ? 19. A man having a yearly income of $4550 spends 20^ of it the first year, 25^ of it the second year, and 37|^ of it the third year. How much does he save in 3 years ? 20. A man receives a salary of $1600 a year. He pays 18^ of it for board, ^\% for clothing, and 16^ for inci- dentals. What are his yearly expenses, and what does he save? 270 PERCENTAGE. 21. A man owning | of a cotton-mill, sold 35^ of his share for $24640. What part of the whole mill did he still own, and what was its value ? 22. Smith had $5420 in bank. He drew out 15% of it, ihen 20^ of the remainder, and afterward deposited 12|^^ of what he had drawn. How much had he then in bank? 513. The base and percentage being given to find the rate. ORAL EXERC ISE8 , 1. What per cent, of 25 is 3 ? Analysis. — Since 3 is /^ of 25, it is ^\ of 100%, or 12%. Hence, 3 is 12% of 25. What per cent. 2. Of 24 is 18 ? 3. Of $16 are $4 ? 4. Of 200 figs are 20 figs ? 5. Of 40 lb. are 15 lb. ? 6. Of 124bu. are 2ibu.? 7. Of 2 A. areSOsq. rd.? 8. Of 1 da. are 16 hr.? 16. f of an acre is what per cent, of it? 17. f of a cargo is what per cent, of it ? 18. 2J times a number is what per cent, of it ? 19. If $6 are paid for the use of $30 for a year, what is the rate per cent. ? 20. If a milkman adds 1 pint of water to every gallon of milk he sells, what per cent, does he add ? 514. Principle. — The rate is the number of hundredths which the mrcentage is of the base. What per cent. 9. Are 6:^ mi. of 12i mi.? 10. Arel8qt. of 30qt.? 11. Are 16f cents of $1 ? 12. Is $J of $25? 13. Is f off? 14. Is^ of 2i? 15. Isf of3f? PERCENTAGE, 271 WRITTEN EXERCISES. 515. 1. What per cent, of 72 is 48 ? OPERATION. Analysis.— Since the per- 48 -i_ 72 661- m 66-i-^ centage is the product of the base and rate, the rate Or, 4i == I ; 100^ X I = 66|^ is the quotient found by di» viding the percentage by the base ; and 48 divided by 72 is f | = | = .66| ; hence the rate is 66|%. Or, Since 48, the percentage, is | of the base, the rate is f of 100%, or66|fc. What per cent. 2. Of 300 is 75 ? 3. Of 66 is 16^? 4. Of $20 are $21.60? What per cent. 5. Of $18 are 90 cents ? 6. Of 560 lb. are 80 lb. ? 7. Of 980 mi. are 49 mi. ? EuLE. — Divide the percentage hy the base. Or, take such a part of 100% as the percentage is of the base. Formula. — Bate = Percentage -^ Base, What per cent. 8. Of $480 are $26.40. 9. Of 192 A. are 120 A. ? 10. Of 15 mi. are 10.99 mi.? 11. Of 46 gal. are 5 gal. 3 qt.? 12. Of $4 are 30 cents ? 13. Of 6 bu 1 pk. are 4 bu, 2 pk. 6 qt. ? What per cent. 14. Are 448 da. of 5600 da.? 15. Are 5 lb. 10 oz. of 15 lb. Avoir. ? 16. Is 13.5 of 225? 17. Isfiof^%? 18. Is 3| of 18^ ? 19. Is22f of 182.4? 20. A grocer sold from a hogshead containing 600 lb. of sugar, J of it at one time, and ^ of the remainder at another time. What per cent, of the whole remained ? 21. A merchant owes $15120, and his assets are $9828. What per cent, of his debts can he pay ? Z7i PERCENTAGE. 516. The rate and percentage being given to find the base. ORAL BXE n C I S ES. 1. 18 is 3^ of what number ? Analysis. — Since 3%, or yf^, of a certain number is 18, yJtf ^ \ of .18, or 6, and \%% is 600. Hence 18 is 3% of 600. Of what number Of what are 3. Is 15 25^? 6. 30 1b. 20%? 25%.? 3. Is 24 75% ? 7. $84 12% ? 21% ? 4. Is 48 8% ? 8. 15bn. 30%? 50%? 5. Is 1.2 6%? 9. 16 doz. \^% ? 8i% ? 10. 12|^ of 96 is 33|^ of what number ? 517. Pri2!^ciple. — Tlie iase is as many times the iier- centage as 100% is times the rate, WniTTEN EXERCISES. 518. 1. 144 is 75^ of what number? OPERATION. Analysis.— Since the percent- 144 -i- . 75 z=: 192 *o^ ^s *^^ product of the base by A 1 AA . »7K 1 4 *^^ ^^^^' *^^ ^^^^ ^® equal to the Or, 100 -^ 75 = -t/^^- = f percentage divided by the rate ; 144 X I = 192 and 144-^.75 is 192. Or, Since the rate is .75, the per- centage is -^^^y or I of the base ; hence the base is J of the percent- age, and f of 144 is 192. 2. $54 are 15^ of what? 3. $18.75 are 2^^ of what? 4. 4.56 A. are h% of what? 5. 39.6 lb. are 7i^ of what? Rule. — Divide the percentage iy the rate. Or, take as, many times the percentage as 100^ is times the rate. Formula. — Base =z Percentage -=- Rate. PERCENTAGE. 273 Of what number 6. Is 828 120^ ? 7. Is 6199 105^^ ? 8. Is .43 71|^ ? 9. Is31i 311^? Of what 10. Are $281.25 37^^? 11. Are 14578 84^? 12. Are37ibu. 6^%? 13. Are 1260 bbl. 12^^? 14. 25^ of 800 bu. is 2^% of how many bushels ? 15. A farmer sold 3150 bushels of grain and had 30% of his entire crop left. What was his entire crop ? 16. A man drew 25^ of his bank deposits, and expended 33^% of the money thus drawn in the purchase of a horse worth $250. How much money had he in bank at first ? 17. If a man owning 45% of a steamboat sells 16f % of his share for $5860, what is the value of the whole boat ? 18. If $295.12 are 13|% of A's money, and 4|% of A's money is 8% of B's, how much more money has A than B ? 519, The amount, or the difference, and the rate being given to find the base, OItA.L, EXERCISES. 1. What number increased by 25^ of itself amounts to 60? Analysis. — Since 60 is the number increased by 25% of itself, it is ^f |, or I of the number ; and if f of the number is 60, the number itself is 4 times | of 60, or 48. 2. What number increased by S^% of itself is 130 ? 3. $70 are 40^ more than what sum ? 4. A man sold a saddle for $18, which was 12|^^ more than it cost him. What did it cost him ? 5. A grocer sold flour for $8.40 a barrel, which was 16f % more than he paid for it. What did he pay for it ? 274 PERCENTAGE. 6. What number diminished by 20^ of itself is 40 ? Analysis.— Since 40 is the number diminished by 20 % of itself, it is y^(f(y, or I of the number ; and if | of the number is 40, tho number itself is 5 times \ of 40, or 50. 7. What number diminished by h% of itself is 38 ? 8. What sum diminished by 50^ of itself is $20.50 ? 9. 68 yd. are \h% less than what number? 10. A tailor, after using 75^ of a piece of cloth, had 9| yards left. How many yards in the whole piece? 11. A sells tea at $.90 a pound, which is 10^ less than he paid for it. What did he pay for it ? WMITTEN EXEItCISES. 520. 1. What sum increased by Zl% of itself is $2055 ? OPERATION. Analysis.— Since 1-|-.37 = 1.37 t^^6 number is in- $2055-^1.37=:$1500 ^"^^'^^ ^^^^' ^^ ^^ .37 of itself, $2055 0^> is 137%, or 1.37 the \^ of $2055 = $2055-^-137 X 100i=$1500 number. Hence $2055 divided by 1.37, is the base or required number. Or, Since $2055, the amount, is \H of the base, 100 times yjy of $2055, or $1500, is the base. 2. What number increased by 18^ of itself equals 2950 ? 3. What sum increased by 15^ of itself is $6900 ? 4. What number diminished by 12^ of itself is 2640 ? OPERATION. Analysis. — Since ine number 1 _.12 r= .88 is diminished 12%, or by .12 of 2640 -^88 3000 itself, 2640 is 88 % , or .88 of the . ' number. Hence 2640 divided by Or, 2640-7-22x25 = 3000 .88 is the base or required num. ber. Or, Since 2640, the difference, is ^-^-^ or || of the base, 25 times i-^ of 2640, or 3000, is the base. What number increased 7. By 12% of itself is 3800 ? 8. By 10% is 39600 ? 9. By 15% is $2616.25? 10. By 22% is 1098 bu.? PERCENTAGE. 275 5. If the difference is $1000 and the rate 20%, what is the base ? 6. What sum diminished by 35% of itself equals $4810 ? EuLE. — Divide Uie amount ly 1 plus the rate; or, divide the difference ly 1 minus the rate, p 7? __ i ^^ount -T- (1 + Rate). I Difference —■ (1 — Rate). What number diminished 11. By ^% of itself is 740 ? 12. By 4% is 312 acres ? 13. By 8% is $2281.60? 14. By 37i% is $234,625? 15. A man sold 160 acres of land for $4563.20, which was 8% less than it cost. What did it cost an acre ? 16. A speculator bought 48 bales of cottop, and after- ward sold the whole for $2008.80, losing 7%. What was the cost of each bale ? 17. A dealer bought a quantity of grain by measure and sold it by weight, thereby gaining 1|^% in the number of bushels. He sold at 10% above the purchase price, and received $4910.976 for the grain. Eequired the cost. 18. A merchant, after paying 60% of his debts, found that $3500 would discharge the remainder. What was his whole indebtedness ? 19. The net profits of a mill in two years were $6970, and the profits the second year were 5% greater than the profits the first year. What were the profits each year ? 20. A man sold two houses at $2500 each ; for one he received 20% more than its value and for the other 20% less. Eequired Kis loss. 276 PERCENTAGE. APPLICATIONS OF PERCENTAGE. 531. The applications of percentage are those which are independent of time, as, Profit and Loss, Commission, Stocks, etc. ; and those in which time is considered, as. Interest, Discount, Exchange, etc. Since some one of the four formulas of percentage already considered will apply to any of these applications, the following will serve as a general EuLE. — Note what elements of Percentage are given in the proUem, and what element is required, and then apply the special rule or formula for the corresponding case. PROFIT AI^D LOSS, 533. Profit and Loss are terms used to express the gain or loss in business transactions. 533. Gains and losses are usually estimated at a rate per cent, on the cost, or the money or capital invested. 534. The operations involve the same principles as those of Percentage. 535. The corresponding terms are the following : 1. The Base is the Cost, or capital invested. 2. The Kate is the per cent, of profit or loss. 3. The Percentage is profit or loss. 4. The Amount is the cost plus the profit, or the Selling Price, 5. The Difference^ is the cost minus the loss, or the Selling Price. PROFIT AND LOSS, 277 ORAL HJXERCISES. 536. 1. A horse that cost $200 was sold at a gain of 12^. What was the gain, and the selling price ? Analysis. — Since the gain was 13%, it was yV% of $200,whicli is $24; andthese]lingpricewas$200 + $24r=:$224. Hence, etc. (510.) 2. A saddle that cost $25 sold at a loss of 10^. What was the loss, and the selling price ? 3. A tailor bought cloth at $6 a yard, and wished to sell it at a gain of 26%. At what price mnst he sell it ? 4. For how much must a grocer sell tea that cost 1.60 a pound, to gain 30% ? 5. A merchant buys gloves at $.75 a pair, and sells them at a profit of 33 J-^. For how much does he sell them ? 6. Bought a carriage for $160, and, after paying 10^ for repairs, sold it at 12|^^ profit. What was the gain, and the selling price ? 7. If butter bought at 36 cents a pound is sold at a loss of 16|^, what is the selling price ? 8. What must be the selling price of coffee that cost 25 cents a pound, in order to gain 20^ ? 9. At what price must an article that cost $5 be sold, to gain 100^? 120^? 150^? 200^? 537. 1. A merchant bought cloth at $5 a yard, and sold it at $6 a yard. What was the gain per cent.? Analysis. — The whole gain is the difference between $6 and $5, which is $1. Since $5 gain $1, or J of itself, the gain per cent is J of 100 % or 20 % . Hence, etc. (5 1 3.) 2. What is gained per cent, by selling coal at $7 a ton, that cost $6 a ton ? 3. Sold a piano for $300, which was | of what it cost. What was the loss per cent.? 278 PERCENTAGE. 4. Sold melons for $.75 that cost $.50. What was the gain per cent.? 5. What is gained per cent, by selling pine-apples at 30 cents each, that cost $15 a hundred ? 6. Sold a sewing machine at a loss of -^ of what it cost. What was the loss per cent.? 7. What % is gained on goods sold at double the cost ? 8. What % is lost on goods sold at one-half the cost ? 9. What per cent, profit does a grocer make who buys sugar at 10 cents and sells it at 12 cents ? 10. What per cent, is gained on an article bought at $3 and sold at $5 ? 538. 1. A dealer sold flour at a profit of $2 a barrel, and gained 25^. What was the cost? Analysis.— Since the gain was 25^ = y^/^, or J, $2 is \ of the cost ; $2 is J of 4 times $2, or $8. Hence, etc. (516.) 2. Sold hats for $1 less than cost, and lost 16f ^. What did they cost ? 3. A merchant sells silk at a profit of %\\ a yard, which is 40^ gain. What did it cost, and what is the selling price ? 4. If com selling for 21 cents a bushel more than cost gives a profit of 30^, what did it cost ? 5. Sold sheep at $2^ more than cost, which was a profit of 50^. What did they cost, and what is the selling price ? 6. Shoes sold at $.50 above cost give a profit of V^\%. tVhat did they cost ? 7. A farmer, by selling a cow for $12 less than she cost, lost 33^^. What did she cost ? 8. A grocer sells a certain kind of tea for 6 cents a pound more than cost and gains 5^. What did it cose ? PBOFIT AND LOSS. 279 539. 1. A watch was sold for $120, at a gain of 20^. What was the cost ? Analysis. —Since the gain was 20%, or i, of the cost, $120, the Belling price, is f of the cost. ^ of $120, or $20, is f of the cost, and f , or the cost itself, is 5 times $20, or $100. Hence, etc. (518.) 2. Sold tea at $.90 a pound, and gained 26%. What did it cost ? 3. A newsboy, by selling his papers at 4 cents each, gains 33^^. What do they cost him ? 4. A man sold a horse and harness for $330, which was 10^ more than they cost. What was their cost ? 5. If 20^ is lost by selling wheat at $1.60 a bushel, what would be gained if sold at 20^ above cost ? 6. John Rice lost 40^ on a reaper, by selling it for $60. For what should he have sold it to gain 40^ ? 7. If, by selling books at $2 a volume, there is a gain of 25^, at what price must they be sold to lose 16% ? 8. Two pictures were sold for $99 each ; on one there was a gain of 10^, on the other a loss of 10^. Was there a gain or loss on the sale of both, and how much ? WRITTEN EXEJtCISES, 530. 1. A hogshead of sugar bought for $108.80 was sold at a profit of 12|^^. What was the gain ? Operation.— $108.80 x ^^ = $13.60. (5 12.) Formula. — Profit or Loss = Cost x Rate %. Find the Profit or Loss, 2. On land that cost $1745, and was sold at again of 20^ 3. On goods that cost $3120, and were sold at 27^ gain. 4. On a boat bought for $2545|, and sold at 26% loa«. 280 PEBCEKTAGE. 5. On goods bought for $2560.75, and sold at 8^ loss. 6. On 25 tons of iron rails bought at $58 a ton, and sold at an advance of 17^%, 7. A merchant pays $6840 for a stock of spring goods, and sells them at an advance of 26^% on the purchase price. After deducting $375 for expenses, what is his gain? 8. A miller bought 1000 bushels of wheat at $1.84 a bushel, and sold the flour at 16f ^ advance on the cost of the wheat. What was his profit ? 9. Bought 128 tons of coal at $5.12|^ a ton, and sold it at a gain of 22^. What was the entire profit ? 10. A ship, loaded with 3840 bbl. of flour, being over- taken by a storm, found it necessary to throw 37^^ of her cargo overboard. What was the loss at $7.65 a bbl. ? 11. A man bought a pair of horses for $450, which was 25^ less than their real value, and sold them for 25^ more than their real value ; what was his gain ? 531. 1. Bought a house for $4380. For what must it be sold to gain 14:^% ? Operation.— $4380 x (1 + .14^) or 1.145=$5015.10. (512.) 2. At what price must pork, bought at $18.40 a barrel, be sold, to lose 15^? Operation.--$18.40 x (1-.15), or .85=$15.64. (512.) ^ a IT Ti • ( Cost X (1 + Rate ^ of Gain). Formula.— Selhng Prices i ^ , )r i^ ^ Z ^t ( ^ I Cost X (1 —Rate % of Loss). Find the Selling Price, 3. Of goods bought at $187.50, and sold at 11^ gain. 4. Of beef bought at $12| a barrel, and sold at 9^% loss. 5. Of cotton bought at $.14, and sold at a gain of 21|^. 6. Of cloth that cost $5} a yard, and was sold at a profit of 18|^ ? PROFIT AKD LOSS. 281 7. At what price must goods that cost $3|^ a yard be marked,, to gain 25^ ? To lose 20^ ? 8. Sold a lot of damaged goods at a loss of 16%. What was the selling price of those that cost $,62^ ? $1.25 ? 9. Bought a hogshead of sugar containing 9 cwt. 56 Ibo for $86.04, and paid $4.78 freiglit and cartage. At whaf price per pound must it be sold to gain 20^. 53^. 1. Bought wool at $.48 a pound, and sold it at $.60 a pound. What per cent, was gained ? Operation.— $.60 — $.48 = $.12 ; and $.12 -f- $.48 = .25 = 25^. (515.) 2. Sold for $10.02 an article that cost $12. What was the loss per cent. ? Operation.— $12— $10 02=$1.98 ; and $1.98-^$12=.16^=16i%. Formula. — Eaie % = Profit or Loss -7- Cost. Find the rate per cent, of profit or loss, 3. On sugar bought at 8 cents and sold at 9|- cents. 4. On tea bought at $1, and sold at $.87f 5. On goods that cost $275, and were sold for $330. 6. On grain bought for $1.25 a bushel, and sold for $1.60 a bushel. 7. On a sewing-machine sold for $72.96, at a gain of |R9.12. 8. On goods sold for $425.98, at a loss of $134.52. 9. Bought paper at $3 a ream, and sold it at 25 centg H quire. What was the gain per cent. ? 10. A dealer bought 108 bbl. of apples at $4.62|-, an(^ sold them so as to gain $114.88|. What was his gain %' 11. If \ of an acre of land is sold for J the cost of at acre, what is the gain per cent. ? 282 PERCENTAGE. 12. If ^ of an acre of land is sold for the cost of ^ of an acre^ what is the loss per cent. ? 13. If -I of a chest of tea is sold for what the whole chest cost, what is the gain per cent, on the part sold ? 533. 1. A speculator sold grain at a profit of 33^%, by ivhich he made 25 cents on a bushel. What did it cost? OPEiiATiON.-$.25--.33i=$.75. Or, $.25^i=$.75. (518.) 2. Lost $45.75 on the sale of a horse, which was 20^ nf the cost. What was the cost ? Operation.— $45.75^.20=$228.75. Or $45. 75 -f- J =r $228.75. Formula. — Cost = Profit or Loss ~ Rate %, Find fche Cost, 3. Of goods sold at $1500 profit, or a gain of 16%. 4. Of flour sold at a loss of $.88, or 10^, on a barrel. 5. Of wheat sold at a loss of 6 cents, or 4^, on a bu. 6. Of lumber sold at an advance of $4.95 per M., or 35^ gain. 7. If a grocer sells his stock at a profit of 15^, what amount must he sell to clear $2500 ? 8. A and B engage in speculation. A gains $2000, which is 12^^ of his capital, and B loses $500, which is 6% of his capital. What sum did each invest ? 534. 1. A furniture dealer sold two parlor sets for $450 each ; on one he made 15^, on the other he lost 15^o What did each cost him ? $450-^(1 + .15)=r$391.30 + , cost of one. Opebation- '^ $450^(1 -.15) =$529.41 + , cost of the other. (520.) Formula. — Cost = Selling Price -7- \ )^ t^ \ t j, n x ^ ( {1— Rate % of loss.) PROFIT AKD LOSS. 283 Find the Cost, 2. Of coal sold at $6, being at a loss of 12|-^. 3. Of grain sold at $.96 a bushel, at a gain of 28^. 4. Of silk sold for $5.40 a yard, at a profit of 10^. 5. Of hops sold at 16 cents a pound, at a loss of 20^. 6. Of fruit sold for $207.48, at a loss of 15^. 7. Having used a carriage 1 year, I sold it for $125 which was 25^ below cost. What should I have received had I sold it for 10^ above cost ? 8. B sold a span of horses to C and gained 12^^ ; C sold them to D for $550, and lost 16f^. What did the horses cost B ? 9. If a piece of property increases in value each year at the rate of 25^ on the value of the previous year, for 4 years, and then is worth $16000, what did it cost ? 535. 1. Bought cloth at $3.60 a yard. At what price must it be marked that 12^% may be abated from the asking price, and still a profit made of 16|^ ? __ { Selling Price =$3. 60 x (1 + .16|) = $420. OPERATION. I Marking Price=$4:.20M^-.12i)=$i.S0. (519.) 2. At what price must shovels that cost $1.12 each be marked in order to abate 6%, and yet make 25^ profit ? 3. How must a watch be marked, that cost $120, so that 4:% may be deducted and a profit of 20^ be made ? 4. A merchant, on opening a case of goods that cost $.80 a yard, finds them slightly damaged. How must he mark them, to fall 25^ in his asking price, and sell at cost? 5. Bought land at $60 an acre ; how much must I ask an acre, that I may deduct 25^ from my asking price, and still make 20^ on the purchase price ? j884 percektage. OOMMISSIOK 536. An Agent or Commission Merchant is a person who buys or sells merchandise, or transacts other business for another, called the Principal 537. Commission is the fee, or compensatiouj allowed an agent or commission merchant for transacting business, and is usually computed at a certain rate pet cent, of the money involved in the transaction. 538. A Consignment is a quantity of goods sent to a commission merchant to be sold. 539. The Consignor is the person who sends the goods for sale. A consignor is sometimes called a Shipper. 540. The Consignee is the person to whom the goods are sent. He is sometimes called a Correspondent. 541. The ^et Proceeds of a sale or other transac- tion is the sum of money that remains after all expenses of commission, etc., are paid. 542. A Guaranty is security given by a commis- sion merchant to his principal for the payment of goods sold by him on credit. 543. An Account Sales is a written statement made by a commission merchant to his principal, contain- ing an account of goods sold, their price, the expenses^ and the net proceeds. 544. A JBroker is a person who buys or sells stocks^ bills of exchange, real estate, etc., for a commission, which is called Brokerage. COMMISSION. 285 545. The principles and operations of Percentage in- volved in Commission and Brokerage are the same as those already treated. 546. The following are the corresponding terms : 1. The Sase is the amount of sales, money invested, or collected. 2. The Hate is the per cent, allowed for seryices. 3. The Percentage is the Commission or Broker- age. 4. The Amount or Difference is the amount of sales, ylus or minus the commission. WRITTEN JEXBRCISES. 647. Find the Commission or Broherage, 1. On a sale of flour for $2575, at 2|^. Opekation.— $2575 x .025 = $64.37J. (5 1 2.) Formula. — Amount of Sales x Rate %-=. Commissioru 2. On the purchase of a farm for $13750, at 2|^. 3. On the sale of a mill for $9384, at 1%. 4. On the sale of $21680 worth of wool, at 1|^. 5. On the sale of 250 bales of cotton, ayeraging 520 lb., at 1 4| cents a pound ; commission 1^%. 6. On the sale of 175 shares of stock, at $92} a share ; brokerage, \%, 7. On the sale at auction of a house and the furniture for $9346.80, at 6}^. 8. A commission merchant sells 225 bbl. of potatoes at $3.25 per bbl., and 316 bbl. of apples at $4| per bbl. What is his commission at 4^^ ? 286. PERCENTAGE. 54:8. Find the rate of commission or brokerage, 1. When $89 commission is paid for selling goods foi $3560. Operation. —89 -^ 8560 = .02 J = 2^ % . (515.) Formula. — Commission -r- Amount of Sales =Rate%. 2. When $165 com. is paid for selling goods for $4950* 3. When $63 is paid for collecting a debt of $1260. 4. When $117.75 is paid for selling a house for $7850. 5. When $235.40 is paid for buying 26750 lb. of wool at 32 cents a pound. 6. When $125 is paid for the guaranty and sale of goods for $2500. , 7. Paid my K 0. agent $74.25 for buying 26400 lb. of rice, at 4^ ct. a lb. What was the rate of his commission ? 549. Find the Amount of Sales, 1. When a commission of $147 is charged at 3|^. Operation.— $147 -^ .035 = $4200. (517.) Formula. — Commission -v- Rate % = Amount of Sales. 2. When $92.80 commission is paid at 3^%. 3. When $210 commission is charged at 6%. 4. When $24 brokerage is paid at ^%. 5. When $135 commission is charged at 1^%. 6. Paid an attorney $72.03 for collecting a note, which was a commission of 7^^. What was the face of the note ? 550. Find the Amount of Sales, 1. When the net proceeds are $4875, commission 2^%. Operation.— $4875 -i- .975 = $5000. (5 19.) Formula. — Mt Proceeds-^(l—Rate %) =Amt, of Sales. 2, When the net proceeds are $3281.25, commission 12|-^ COMMISSION. 387 3. When the net proceeds are $560, and the com. 4^. 4. After deducting 6^% commission and $132 for storage, my correspondent sends me $33654.25 as the net proceeds of a consignment of pork and flour. What was the gross amount of the sale ? 551. Find the amount to he invested, 1. If $9500 is remitted to a correspondent to be invest- fid in woolen goods, after deducting b% commission. Operation.— $9500 -h 1.05 = $9047.62. (519.) Formula. — Amount Remitted -~ (1 + Rate %) = Sum Invested. 2. If $4908 be remitted, deducting 4|-^ commission. 3. If $3246.20 be remitted, deducting 2% commission. 4. If $1511.25 be remitted, deducting ^% commission. 5. If $10701.24 be remitted, deducting ^% brokerage. 6. A dealer sends his agent in Havana $6720.80, with which to purchase oranges and other fruits, after deduct- ing his commission of 6%. What sum did the agent invest, and what was the amount of his commission ? 7. What amount of sugar can be bought at 8 cents a pound, for $2523.40, after deducting a commission of If ^. 8. Remitted to a stockbroker $10650, to be invested in stocks, after deducting ^% brokerage. What amount of stock did he purchase ? 9. A broker received $45337.50 to invest in bond and! mortgage, after deducting a commission of 2^%, What amount did he invest, and what was his commission ? 10. Sent $250.92 to my agent in Boston, to be invested in prints at 15 cents a yard, after taking out his commis- sion of 2%, How many yards ought I to receive ? 288 PEECEIS^TAGE. REVIEW. oraij exehc isbs , 553. 1. If stoves bought at $36 each are sold at f profit of 8-J-^, what is the gain ? 2. What will be the expense of collecting a tax of $1000, allowing 5^ ? 3. What will a broker receive for selling $600 worth of stock, at 1^ brokerage ? 4. A man having $250 spent $80. What per cent of his money had he left ? 5. If a man sells a building lot that cost $300, at an advance of 166f ^, what is his gain ? 6. I of 30^ is what per cent, of 72^ ? Of 144^ ? Of 180^? 240^? 7. Bought a horse for 20^ less than $200, and sold him for 10% more than $200. What per cent, was gained? 8. How many bushels of wheat at $2 a bushel can an agent buy for $2040, and retain 2% on what he expends as his commission ? 9. If by selling land at $150 an acre I lose 25^, how must I sell it to gain 40^ ? 10. A boy bought bananas for $3 a hundred, and sold them for 5 cents each. What per cent, did he gain ^ 11. Bought cannel coal at $19 a ton, which was b% less than the market price. What was the market price ? 12. Paid an agent $150, or a commission of 1^%, foi selling my house. For what sum was the house sold ? 13. If an article is sold so as to gain f as much as it cost, what per cent, is gained ? REVIEW. 289 14. A merchant tailor sold some linen coats at $1.80 each, wnich was 33^% below the marked price. What was the marked price ? 15. A grocer bought 40 gal. of maple syrup at the rate of 4 gal. for $6, and sold it at the rate of 5 gal. for $8. What was his whole gain, and his gain per cent. ? 16. How much Avheat must a farmer take to mill that he may bring away the jlour of 4^ bushels, after the miller takes his toll of 10^ ? WMITTEN EXERCISES. 553. 1. After taking out lh% of the grain in a bin, there remained 40 bu. 3^ pk. How many bushels were there at first ? 2. The net profits of a farm in 2 years were $3485, and the profits the second year were b% greater than the profits the first year. What were the profits each year ? 3. A has 32^ more money than B ; what per cent, less is B's money than A^s ? 4. Bought 450 bushels of wheat at $1.25 per bushel, and sold it at $1.40 per bushel. What was the whole gain, and the gain per cent. ? 5. A man drew out of the bank | of his money, and ex- pended 30^ of 50^ of this for 728 bu. of wheat, at $1.12| a bushel. What sum had he left in bank ? 6. Sold goods to the amount of $47649, at a profit of 16f ^. Kequired the cost and the total gain. 7. A broker received $37.50 for selling some uncurrent money, charging \% brokerage. How much did he sell ? 8. If f of a farm is sold for what f of it cost, what is the gain per cent.? 13 290 PERCE Is^TAGE. 9. An architect charged 1% for plans and specifications, and 1|% for superintending a building that cost $25000. What was the amount of his fee ? 10. If a stationer marks his goods 60% above cost, and then deducts 50^, what per cent, does he make or lose ? 11. Sold a farm for $14700, and lost 12^. What per cent, should I have gained by selling it for $21000 ? 12. If an article bought at 20^ below the asking price is sold at 16% below that price, what is the rate of gain ? 13. A commission merchant sold a consignment of goods for $5250, and charged 3^% commission, and 2^% for a guaranty. Find the net proceeds. 14. Smith & Jones bought a stock of groceries for $13680. They sold } of the entire stock at 15^ profit, J at 18|^, -J- at 20^, and the remainder at 33|^ profit. What was the whole gain, and the average gain per cent. ? 15. Give the marking prices at 25^ advance, of the following bill of goods, and the amount when sold at a reduction of 10^ from those prices : 1 Case of Prints, 450 yd., @ $.12 3 Pieces Cassimeres, 65 '^ @ 3.25 1 Bale Ticking, 244 ^^ @ .20 25 Dress Shawls, @ 7.36 1 Gr. gross Clark's Thread, 144 doz., @ .70 50 Gross Buttons, @ 1.00 16. How much would the above bill of goods amount to if sold at 6^% below a marking price of 15^ above cost ? 17. What would be the net proceeds of a sale of 18 cwt. 75 lb. of sugar, at $9f per cwt., allowing 2^% commission, and $16f for other charges ? COMMISSlOiT. 29 J 18. A broker receives 17125 to invest in cotton, at 11| cents a pound. If his commission is 2^%y how many pounds of cotton can he buy ? 19. If the sale of potatoes at $.75 a barrel above cost gives a profit of 18|^, how much must be added to this price to reahze a profit of 31^% ? 20. An agent in Chicago purchases 1000 bbl. of flour at $6.80, and pays 5 cents a barrel storage for 30 days ; also, 3000 bu. of wheat at $1.20. He charges a commis- sion of 1^% on the flour, and 1 cent a bushel on the wheat. What sum of money will balance the account, and what is the amount of his commission ? 21. An agent in Boston received 28000 lb. of Texas cotton, which he sold at $.12^ a pound. He paid $45.86 freight and cartage, and after retaining his commission, he remits his principal $3252.89 as the net proceeds of the sale. What was the rate of his commission ? 22. The following bill of goods was sold at auction : 1| bbl. A Sugar, 312 lb., l^'' PuIy/' 96 '' 1 Chest Y. H. Tea, 84 '' 1 Box Soap, 60 '' 1^ Sacks Java Coffee, 110 '' 184 lb. Codfish, Allowing a commission of 4^% for selling, find the entire profit or loss, and the gain or loss per cent, on the whole. 23. A merchant in New York imported 2400 yd. of English cloth, for which he paid in London 10s. sterling a yard, and the total expenses were $255. He sold the cloth for $3.81 a yard, TJ. S. money. What was his whole gain, and his gain per cent. ? $.12i that cost $.11^ •14i " " .14 1.10 " " 1.12|. .13 " " .10| .22i " " .24^ .07^ " " .08f 292 PERCENTAGE. 554. SYNOPSIS FOE EEVIEW. 1. Definitions. 2. Elements. 1. Percentage. 2. Per Cent. 8. Sign of Per Cent. 4. Rate, or Rate % . 5. Base. 6. Percentage. 7. Amount. 8. Difference. 3. 510. 1. Principle. 2. 4. 513. 1. Principle. 2. 5. 516. 1. Principle. 2. 6. 519. 1. Principle. 2. 7. Applications of Percentage. 8. Profit and Loss. 9. Commission. How many considered. How many must be given. Rule. 3. Formula. Rale. 3. Formula. Rule. 3. Formula. Rule. 3. Formula. 1. 2. ri. 2. 3. Diff't kinds. U: Without Time. With Time, General Rule. Definition. To estimate gains and losses. 1. Base. f 1. Definitions. Correspond- J 2. Bate. ing terms. 3. Percentage. L 4. Am't and Diff. 1. Agent y or Qorru mission Merchant. 2. Commission. 3. Consignment. 4. Consignor. 5. Consignee. 6. Net Proceeds. 7. Guaranty. 8. Account Sales 9. Broker. 2. Prin. and Operations Involved. r 1. Base. 3. Correspond- J 2. Bate. ing terms. ] 3. Percentage. 4. AmH and Diff. ORAL EXERCISES . 555. 1. When 5^ is charged for the use of money^ how many dollars should be paid for the use of $100 ? For the use of $200 ? Of $500 ? Of $50 ? 2. At 1% a year, what should be paid for the use of $100 for 2 years ? Of $200 for 3 years ? 3. If $500 is loaned for 3 years, what should be paid for its use, at h% a year ? At 6^ a year ? 4. If I borrow $250, and agree to pay 4:% a year for its use, how much will be due the lender in 5 years ? 5. If $7 is paid for the use of $100 for 1 year, what is the per cent. ? 6. If $50 is paid for the use of $100 for 5 years, what is the per cent. ? 7. If $14 is paid for the use of $200 for 1 year, what is the per cent. ? 8. At Q%, what decimal part of the money borrowed is equal to the money paid for its use ? At 7^ ? S% ? 9% ? DEFINITIONS. 556* Interest is a sum paid for the use of money. 557. The Principal is the sum for the use of which interest is paid. 558. The Mate of Interest is the per cent., or number of hundredths, of the principal, paid for its use for one year. 294 PERCEKTA (^E, 559. The Atnount is the sum of the principal and the interest. 560. Legal Interest is the interest according to the rate per c^nt. fixed by Jaw. 561. Usury is a higher rate of interest than is al- lowed by law. 563. The legal rates of interest in the different States are as follows : Name of State. Alabama Arkansas* Arizona California*. . . . Canada and Ireland Connecticut . . . Colorado* Dakota . Delaware Dist. Columbia. Engl and and France Florida* Georgia Idaho Illinois Indiana.. Iowa Kansas Kentucky Louisiana Maine '^ Maryland Massachusetts*. . Michigan Rate. 8% 6f^ Any. 10% Any. 10% Any. 6% 7% 10% Any. 7% Any. 6% 6% 10% 5% 8% Any. 7% 10% 10% 8% 6% 6% 10% 6% 10% 7% 13% 6% 10% 5% 8% 6% Any. 6% Any. 7% 10% Name of State. Minnesota Mississippi.. . . . Missouri Montana New Hampshire. New Jersey . . . . New York North Carolina. . Nebraska Nevada* Ohio Oregon -Pennsylvania... . Rhode Island*. . . South Carolina*. Tennessee Texas Utah* Vermont Virpfinia West Virginia. . . Washington T.*. Wisconsin Wyoming Rate. 7% 6% 6% 10% 6% 6% 6% 6% 10% 10% H 10% 6% 6% 8% 10% 6% 10% 7% 12% 12% 10% 10% »7tf 15% Any. 8% 12% Any. Any. 10% 12% Any. *i2%' Any. 10% 1. When the rate per cent, is not specified in accounts, notes, mortgages, contracts, etc., the legal rate is always understood. 3. Where two rates are specified, any rate above the lower, and not exceeding the higher, is allowed, if stipulated in writing. 3. In the States marked thus (*) the rate per cent, is unlimited U agreed upon by the parties in writing. INTERESTo 295 563. In the operations of interest there are jive parts, or elements, namely : The Principal ; the Rate per Cent, per Annum (for one year) ; the Interest ; the Time for which the principal is lent ; and the Amount^ or sum of the Prin. and Int. 564. These terms correspond respectively to Base^ Rate, Percentage, and Amount in Percentage, excluding Time, which is an additional element in Interest. OBAJL MXJEnCIS ES. 565. 1. At 3^ for 1 yr., what decimal part of the prin- cipal equals the interest ? At 5^ ? At S% ? At l^% ? 2. What is the interest of $20 for 1 year at 5^? Analysis. — Since the interest of any sum at 5 % for 1 jr, is .05 of the principal, the interest of $20 for 1 yr. at 5% is .05 of $20, or $1. 3. What is the interest of $50 for 1 yr. at 5^ ? 6^ ? 7^? 4. What is the interest of $80 for 1 yr. at 7^? 8^? 10^? 5. At 7^ for 5 yr., what decimal part of the principal equals the interest ? ANALYsr-". — Since the interest at 7% for 1 yr. is .07 of the prin- cipal, the interest for 5 yr. is 5 times .07, or .35 of the principal. Or, it is 5 times the interest for 1 year. 6. At 6^ for 3 yr., what decimal or fractional part of the principal equals the interest ? At 7^ for 6 yr.? At 5^ for 5 yr. ? At ^% for 2 yr.? At 10^ for 4 yr. ? 7. Find the interest of $30 for 3 yr. at 5^. Analysis. — Since the interest of any sum at 5% for 1 yr. is .05 of the principal, for 3 yr. it is .15, and .15 of $30 is $4.50. Or, the interest for 1 yr. is .05 of $30, or $1.50, and for 3 yr. it is 3 times as much, or $4.50. 8. Find the int. at 6^ of $20 for 2 yr. Of $40 for 3 yr. 9. Find the int. at 8^ of $5 for 5 yr. Of $10 for 10 yr. 296 PEECEKTAGE. 10. At Q% for 2 yr. 6 mo., what decimal part of the principal equals the interest ? Analysis. — Since the interest of any sum for 1 yr. at 8% is .08 of the principal, the interest on the same for 2 yr. 6 mo. is 2\ times .08, or .20 of the principal. Or, it is 2 J times the interest for 1 yr. 11. At ^% for 3 yr. 3 mo., what decimal part of the principal equals the interest ? At 9^ for 3 yr. 3 mo. ? 12. Find the int. of $9 for 2 yr. 4 mo. at 1%. At 8^. 13. What is the int. of $1000 for 2 yr. 3 mo. at 10^ ? For 4 yr. 6 mo. ? For 5 yr. 3 mo. ? For 8 mo. ? 566. Pkikciple. — The interest is the product of three factors J namely, the principal, rate per annum, and time {expressed in years or parts of a year). WniTTEN JSX JER CIS ES 567. To find the interest or amount of any sum at any rate per cent., for years and months. 1. Find the amount of $97.50, at 7%, for 2 yr. 6 mo. OPERATION. Analysis. — Since the interest of $97.50 any sum at 7% for 1 yr. is .07 of rvrt> the principal, the interest of $97.50 ' — at 7% fjr 1 yr. is .07 of $97.50, or $6.8250 Int.forlyr. $6,825; and the interest for 2 yr. 2-|- 6 mo. is 2^ times the interest for 1 T^«"^ T . * o « yj-M or $17.06i, and $17.06^4 $97.50 17.0625 Int. for 2 yr. 6 mo. ^ I^wi\/.i .i a * =$114,561:, the Amount. 97.50 Principal. $^14.5625 Amount. Find the interest and the amount, 2. Of $450 for 3 yr. 9 mo. at 6^. For 8 mo. at 7%. 6. Of $247 for 5 yr. 3 mo. at 6^%. For 10 mo. at 8^, 4. Of $500 for 4 yr. 2 mo. at 10^. For 11 mo. at 5^ INTEREST. 29? EuLE. — I. Multiply the principal hy the rate, and the product is the interest for 1 year. II. Multiply the interest for 1 year hy the time in years, and the fraction of a year ; the product is the required interest. III. Add the principal to the interest for the amounts Formula. — Interest = Principal x Rate x Time. Find the interest, 5. Of $36.40 for 1 yr. 7 mo. at 6^. At 7^. At ^% 6. Of $750.50 for 3 yr. 1 mo. at b%. At 8^. At 9^ 7. Of $1346.84 for 2 yr. 4 mo. at ^1%. At 7^^. 8. Of $138.75 for 4 yr. 3 mo. at 10^. At 12^^. 9. Find the amount of $640 for 5 yr. 6 mo. at 7^. 10. Find the amount of $56.64 at 8^ for 3 yi\ 3 mo. 11. Made a loan of $1040 for 1 yr. 9 mo. at ^%. How much is due at the end of the time ? 12. If a note for $375, on interest at 8^, dated June 10, 1874, be paid Sept. 10, 1876, what amount will be due ? 568. To find the interest on any sum of money, for any time, at any rate per cent. Obvious Eelations between Time and Interest. I. The interest on any sum for 1 year at 1% is .01 of the principal. It is therefore equal to the principal with the decimal point ra moved two places to the left. II. The interest for 1 mo. is -^ of the interest for 1 yr. III. The interest for 3 days is ^, or ^, of the interest for 1 month ; hence any number of days may readily be reduced to tenths of a month by dividing by 3. 298 PERCENTAGE. TV. The interest on any sum for 1 month, multiplied by the number of mmiths and tenths of a month in the given time, and the product by the number expressing the rate, will be the required interest. 569. 1. Find the int. of $361.20 for 1 yr. 3 mo. 24 da. at 7^. OPERATION. 13.612 (.01 of the Prin.) Int. for 1 yr. at 1% (568, I). .301 Int. for 1 mo. at 1% (568, II). 15.8 Number of months and tenths (568, III). $4.7558 Int. for 1 yr. 3 mo. 24 da. at 1%, 7 133.2906 Int. for 1 yr. 3 mo. 24 da. at 7% (568, IV). AVhat is the interest, 2. Of $137. 25 fori yr. 6 mo. 10 da. at 6^? At 4^? 3. $510.50 for 3 yr. 7 mo. 15 da. at 0% ? At 8^ ? 4. Of $1297.60 for 2 yr. 11 mo. 18 da. at 7% ? At 7^%? EuLE. — I. To find the interest for 1 yr. at 1%. Remove the decimal point in the given principal two places to the left. II. To find the interest for 1 mo. at 1%. Divide the interest for 1 year ly 12. III. To find the interest for any time at 1%. Multiply the interest for 1 month hy the niimher of months and tenths of a month in the given time, IV. To find the interest at any rate %. Miiltiply tJie interest at 1% for the given time hy the num- ber expressing the given rate. 5. Find the int. of $781.90 for 1 yr. 1 mo. 12 da. at 7^. 6. Find the int. of $3000 for 11 mo. 21 da. at 10^. INTEREST. 299 7. What is the ami of $1049 for 2 yr. 3 mo. 9 da. at ^% ? 8. What is the amt. of $216.75 for 3 yr. 5 mo. 11 da. at S%? 9. Required the int. of $250 from Jan. 1, 1873, to Maj 10, 1875, at 7^. 10. Eequired the amount of $408.60 from Aug. 20 to Dec. 18, 1876, at 10^. 11. What is the interest on a note for $515.62, dated March 1, 1873, and payable July 16, 1875, at 7^? 12. A man sold his house and lot for $12500 ; the terms were, $4000 in cash on delivery, $3500 in 9 mo., $2600 in 1 yr. 6 mo., and the balance in 2 yr. 4 mo., with 6% interest. What was the whole amount paid ? 570. SIX PER CENT. METHOD. At 6% per annum, the interest of $1 For 12 mo. . . . . is 6 cents, or .06 of the principal, '' 2 ^^ or ^ of 12 mo., ^^1 cent, ^^.01 '' '' " 1 " ''-^'' 12 '' " \ " ^^005 '' '' " 6da.^'^^^ 1^^ "^" " m\ " " " \ " " \ " 6 da. " .000| " " 571. Principles. — 1. The interest of any sum at 6% is ONE-HALF as many hu7idredths of the principal as there are months in the given time. 2. The interest of any sum at 6% is one-sixth as many thousandths of the principal as there are da ys in *,he given time. Thus, the interest on any sum Sit 6 fo for 1 yr. 3 mo., or 15 mo., is J of .15, or .075, of the principal ; and for 18 da. it is ^ of .018, or .003, of the principal. Hence, for 1 yr. 3 mo. 18 da., it is .075 + .003 = .078 of the principal. It is evident that an odd month is J of .01, or .005; and that any number of days less than 6 is such a fractional part of .001 as the days are of 6 days. 300 PEE GENT AGE. ORAJL BXEItCISBS, 573. What is the interest, 1. Of $1 at 6^ for 1 year? 2 years? 3 years? 5 years! 8 years? 13 years? 2. Of U at 6^ for 1 month ? 2 mo. ? 3 mo. ? 4 mOc f 5 mo.? 7n)o.? 9 mo.? 10 mo.? 15 mo.? 18 mo? At (j%, what is the interest, 3. Of $1 for 1 yr. 4 mo. ? 1 yr. 7 mo. ? 2 yr. 2 mo. ? 4. Of $1 for 1 day ? 6 da. ? 12 da.? 19 da.? 24 da.? 33 da. ? 36 da. ? 45 da. ? 63 da. ? 5. Of $1 for 1 mo. 12 da.? For 3 mo. 15 da. ? For 6 mo. 25 da. ? For 7 mo. 11 da.? For 11 mo. 18 da.? Find the interest, 6. Of $1, at 6^, for 1 yr. 3 mo. 6 da. For 1 yr. 9 mo. 18 da. For 1 yr. 5 mo. 19 da. 7. Of $1 at 6^ for 2 yr. 1 mo. 9 da. For 3 yr. 24 da. 8. Of $1 at 6^ for 5 yr. 5 mo. 5 da. For 4 yr. 7 mo. 10 da' At 6;^, find the interest, 9. Of $1 for 2 yr. 6 mo. Of $2. Of $3. Of $5. 10. Of $1 for 4 yr. 2 mo. Of $10. Of $20. Of $30. 11. Of $5 for 1 yr. 4 mo. For 2 yr. For 2 yr. 8 mo. 12. Of $1 for 33 da. For 63 da. For 93 da. For 123 da. 13. Of $6 for 33 da. Of $4 for 63 da. Of $2 for 93 da. 14. If the interest of a certain principal at %% is $1& what would the interest be at b% ? 7^ ? 8^ ? 9^ ? 5% is J less than 6%; 7% is J more than 6%; 8% is J more, etc. 15. If the interest of a certain principal is $16, what would the int. be at 3^? 4^^? 5^? 7^^? 8^? 12^? 16. If the interest of a certain principal is $30, what would the int. be at 2^? 4^? 7^? 8^? 10^? 14^? INTEREST. 301 WBITTJEJSr EXJERCISE8, 573. 1. What is the int. of $427.20 at 6^ for 2 yr. 5 mo. 27 da. ? OPERATION. Analysis. — Since the in- 2 yr. 5 mo. = 29 mo. $427.20 terest of $1 for 2 yr. 5 mo. iof.29 =.145 .1494 27 da. is $.149i or of any snm is .149i of the princi- nf.027 =^004i $63.8664 pal (57.1), $427.20 x.l49i Int. = .1491^ of the Prin. = $63,866+ is the required interest. Find the interest at Q% of 2. $597.25 for 7 mo. 18 da. 3. $418.75 for 1 mo. 25 da. 4. $309.18 for 2 yr. 24 da. 5. $1298 for 3 yr. 1 mo. 13 da. 6. $2000 for 2 yr. 7 mo. 24 da. 7. $4010forlyr. lmo.l3da. EuLE. — Multiply the given principal iy the decimal ex- pressing the interest of $1 ; or ty the decimal expressing one-half as many hundredths as there are months, and one- sixth as many thousandths as there are days, in the given time, and the product loill he the required interest. To find the interest at any other per cent, by this method, increase or diminish the interest at 6% by such part of itself as the given rate is greater or less than 6% . 574. To compute Accurate Interest^ that is, reckoning 365 da. to the year, use the following EuLE. — Find the interest for years and aliquot parts of t year iy the common method, and for days take such part of 1 yearns interest as the numher of days is of 365. Or, When the time is in days and less than 1 year, find the interest by the common method, and then suhtract -^ pan of itself for the common year, or ^, if it be a leap year. 302 PERCEIl^TAGE. 1. Find the accurate interest of $1560 for 45 da. at 7^. The exact int. of $1560 for 45 da. at 1% = I12^^ili5^ 13.46+- > It is $13.65 - ^^^'^l'^^ = $13.46 + . To 2. Find the exact int. of $1600 for 1 yr. 3 mo. at 6%. 2. What is the difference between the exact interest c£ I64S.40 at S% for 1 yr. 3 mo. 20 da. and the interest reckoned by the 6% method ? 4. Find the exact interest of $875.60 at 7% for 63 da. 5. Eequired the exact interest on three XJ. S. Bonds of $1000 each, at 6%, from May 1 to Oct. 15. 6. What is the exact interest on a $500 U. S. Bond, at 5%, from NoY. 1 to April 10 following ? 575. Find the interest, by any of the ordinary methods, 1. Of $721.56 for 1 yr. 4 mo. 10 da. at 6%. 2. Of $54.75 for 3 yr. 24 da. at 6%. 3. Of $1000 for 11 mo. 18 da. at 7%- 4. Of $3046 for 7 mo. 26 da. at 8%- 5. Of $1821.50 from April 1 to Nov. 12 at 6%. 6. Of $700 from Jan. 15 to Aug. 1 at 10^. 7. Of $316.84 from Oct. 20 to March 10 at 7%. What is the amount 8. Of $3146 for 2 yr. 3 mo. 10 da. at 7%? 9. Of $96.85 for 3 yr. 1 mo. 27 da. at 6% ? 10. Of $1008.80 for 10 mo. 16 da. at 6^% ? 11. Of $2000 for 15 da. at 12^^? 12. Of $137.60 for 127 da. at 10^? 13. If $1671.64 be placed at interest June 1, 1874;, what amount will be due April 1, 1876, at 7^ ? INTEREST. 303 14. How much is the interest on a note for $600, dated i^-^eb. 1, 1872, and payable Sept. 25, 1875, at 6%? 15. If a man borrow $9700 in New York, and loan it in Colorado, what will it gain at legal int. in a year ? 16. Eequired the mterest of $127.36 from Dec. 12, 1873, to July 3, 1875, at 4.^%, 17. A note of $250, dated June 5, 1874, was paid Feb. 14, 1875, with interest at S%, What was the amount ? 18. A note for $710.50, with interest after 3 mo., at 7^, was given Jan. 1, 1874, and paid Aug. 12, 1876. What was the amount due ? 19. A man engaged in business was making 12 1^ an- nually on his capital ot $16840. He quit his business and loaned his money at 7^%. What did he lose in 2 yr. 3 mo. 18 da. by the change ? 20. A man borrows $2876.75, which belongs to a minor who is 16 yr. 5 mo. 10 da. old, and he is to retain it until the owner is 21 years old. What will then be due at 8^ simple interest ? 21. A speculator borrowed $9675, at 6^, April 15, 1874, with which he purchased flour at $6.25. May 10, 1875, he sold the flour at $7f a barrel, cash. What did he gain by the transaction ? 22. A man borrows $10000 in Boston at 6%, reckoning 360 da. to the year, and lends it m Ohio at S%, reckoning 365 da. to the year. What will be his gain m 146 days? 23. A tract of land containing 450 acres was bought at $36 an acre, the money paid for it being loaned at b^%. At the end of 3 yr. 8 mo. 24 da., f of the land was sold at $40 an acre, and the remainder at $38|^ an acre. What was gained or lost by the transaction ? 304 PERCEJ^TAGE. PEOBLEMS m INTEKEST. 576, Interest, time, and rate given, to find the principal. ORAL EX EMCIS ES. 1. What sum of money will gain $10 in 1 yr. at 5% ? Analysis. — The interest of $1 for 1 yr. at 5 % is .05 of the prin- cipal, and therefore $10 -J- .05, or $200, is the required sum. Or, Since $.05 is the interest of $1, $10 is the interest of as many dollars as $.05 is contained times in $10, or 200 times. Hence, etc. What sum of money will gain, 2. $20 int. in 2 yr. at 6% ? 3. $25int. in5yr. at 5^? 4. $60 int. in 2 yr. at 6%? 9 5. $84 int. in 2 yr. at 7^ 6. $50 int. in 6 mo. at 10^? 7. $30 int. in 3 mo. at S% ? WBITTEN EXERCISES. 57T. 1. What sum of money, put at interest 3|^ yr. at 6%, will gain $346.50 ? OPERATION. Int. of $1 for 3 J yr. at 6% = $.21. Analysis.— Same as in $346.50 -$.21 ==1650 times; oral exercises. (576.) $1 X 1650 = $1650. What principal 2. Will gain $49.50 in 1 yr. 3 mo. at 6% ? At 5^? 3. Will gain $153.75 in 3 mo. 24 da. at 7%? At S%? EuLE. — Divide the given interest ly the interest of %X tor the given time, at the given rate. Formula. — Principal = Interest -t- {Rate x Time). What sum of money 4. Will gain $213 in 5 yr. 10 mo. 20 da. at 7fo ? 6. Will gain $173.97 in 4 yr. 4 mo. at 6% ? At 12^ ? INTEREST. 305 6. A man receives semi-annually $350 int. on a mort- gage at 7^. What is the amount of the mortgage ? 578. Amount, rate, and time given, to Und the principal. oraij exercises, 1. What sum of money will amount to $107 in 1 yr. at 7^? Analysis. — Since the interest is .07 of the principal, the amount is 1.07, or \^ly of it. If $107 is |^J of the principal, y J^ of the prin- cipal is ^^^ of $107, or $1 ; and Ifg, or the principal itself, is $100. Or, Since $1.07 is the amount of $1, $107 is the amount of as many dollars as $1.07 is contained times in $107, or $100. What sum of money will amount to 2. $130 in 5 yr. at 6^? 3. $228 in 2 yr. at 7%? 4. $412 in 6 mo. at 6^? 5. $250 in 10 yr. at 10^? 6. $350 in 15 yr. at 5^? 7. $260in3yr. 9mo. at8^? WRITTEN EXERCISES. 579. 1. What sum will amount to $337.50 in 5 yr. at 7^? OPEKATION. Am't of $1 for 5 yr. at 7^ = $1.35. Analysis. — Same as $337.50 -. $1.35 = 250 times ; in oral exercises. (578.) $1 X 250 = $250. What principal 2. Will amount to $1028 in 4 mo. 24 da. at 7%? 3. Will amount to $1596 in 2 yr. 6 mo. at 6^% ? 4. Will amount to $1531.50 in 3 mo. 18 da. at 7%? EuLE. — Divide the given amount by the amount of $1 for the given time, at the given rate. Formula. — Prin.=z Amt.-^ (1 + Rate x Time). 306 PEECENTAGE. 5. What is the principal which in 217 days, at 5^^, amounts to $918.73? 6. What principal in 3 yr. 4 mo. 24 da. will amount to $761.44 at 5^? 580, Principal, interest, and time given, to find the rate. ORAIj jexbmc isbs. 1. At what rate will $100 gain $14 in 2 years ? Analysis.— Since the interest of $100 is $14 for 2 yr., it is $7 foi 1 yr., and $7 is .07 of $100, the principal. Hence the rate is 7% . Or, Since the interest of $100 for 2 yr. at 1 % is $2, $14 is as many per cent, as $2 is contained times in $14, or 1%, At what rate will 2. $300 gain $60 in 4 yr. ? 3. $500 gain $100 in 5 yr. ? 4. $400 gain $84 in 3 yr. ? 5. $5 gain $1 in 3 yr. ? 6. $120 gain $60 in 10 yr. ? 7. $150 double itself in 10 yr. ? W MITT EN EXERCISES. 581, 1. At what rate per cent, will $1600 gain $280 interest in 2^ years ? OPERATION. Int. of $1600 at 1% for ^ yr. = $40. Analysis. - Same ^^r.^ ^Ar. ^ i' -. ^ »v w,./ as in oral exercises. $280 -T- $40 = 7 times ; 1^ x 7=7^. (5^q_) At what rate per cent 2. Will $2085 gain $68.11 in 5 mo. 18 da. ? 3. Will $1500 gain $252 in 2 yr. 4 mo. 24 da. ? jSuLE. — Divide the given interest iy the interest of the given principal, for the given time, at 1%. Formula. — Rate = Int, -~ iPrin. xl%x Time). INTEREST. 307 4. A house that cost $14500 rents for $1189. What per cent, does it pay on the investment ? 5. At what rate will $1500 amount to $1684.50 in 2 yr. 18 da. ? 6. At what rate per month will $2000 gain $120 in 90 da.? 7. A man invests $15600, which gives him an annual income of $1620. What rate of interest does he receive? 8. At what rate per annum will any sum double itseK in 4, 6, 8, and 10 years, respectively? AX, 1%, any sum will double itself in 100 yr.; hence, to double itself in 4 yr., the rate will be as many times 1 % as 4yr. are con- tained times in 100 yr„ or 25%, etc. 9. At what rate per annum will any sum triple itself in 2, 5, 7, 12, and 20 years, respectively ? 10. I invest $49500 in a business that pays me $297 a month. What annual rate of interest do I receive ? 11. Which is the better investment, and how much, one of $4200, yielding $168 semi-annually, or one of $7500, producing $712^ annually? 583. Principal, interest, and rate given, to find the time. oraij jexehcises. 1. In what time will $200 gain $56 at 7^ ? ANAiiYSis. — The given interest, $56, is ^y^^, or .28, of the princi- pal, $200 ; therefore, the time is as many years as .07, the given rate, is contained times in ,28, or 4 times. Hence, etc. Or, the interest of $200 at 7% for 1 yr. is $14 ; therefore, the time is as many years as $14 are contained times in the given inter- est, $56, or 4 years. Hence, etc 308 PBRCEi^TAGE, In what time will 2. $40 gain $10 at 6% ? 3. $500 gain $100 at 4^? 4. $25 gain $20 at 6^ ? 5. $1000 gain $250 at 5^ ? 6. $5 gain 90 cents at 6^ ? 7. $50 gain $12^ at 10^? WRITTEN EXERCISES. 683. 1. In what time will $840 gain $78.12 at 6^ ? OPERATION. $840 X. 06 =$50.40 Int. for 1 yr. ANALYsis.—Same as in I78.12~$50.40 = 1.55. the oral exercises. (582,) 1 yr. X 1.55 = 1 yr. 6 mo. 18 da. In what time 2. Will $175.12 gain $6.43 at 6^? 3. Will $1000 amount to $1500 at 7^^? EuLE. — Divide the given interest ly the interest of the given principal, at the given rate for 1 year. Formula. — Time = Interest -r- {Prin. x Rate). 4. In what time will $8750 gain $1260 at 2% a month? 5. How long must $1301.64 be on interest to amount to $1522.92 at 5^? 6. How long will it take any sum of money to double itself at d%, 6%, 6%, 7^%, and 10^, respectively ? At 100%, any snm of money will double itself in 1 year ; hence \o double itself at 10%, it will require as many years as 10% is tontained times in 100%, or 10 yr. 7. How long will it take any sum to triple itself at i^5 5^, 7^, S%, and 12|-^, respectively ? 8. In what time will the interest of $120, at 8%, equal the principal ? Equal half the principal ? Equal twice the principal ? IKTEREST. 309 OOMPOU:C^D II^TEEEST. 584. Compotmd Interest is interest not only on the principal, but on the interest added to the principal when it becomes due. on AT. EXERCISES. 585. 1. What is the comp. int. of $500 in 2 yr. at 6% f Analysis. —The simple interest of $500 for 2 yr. is $60 ; the in- terest of the first year's interest, $30, for the second year is $1.80, which, added to $60, gives $61.80, the compound interest. Or, The interest of $500 for 1 yr. at 6% is $30, and the amount is $530, which is the principal for the second year ; the interest of $530 for 1 yr. at 6% is $31.80, which added to $530 gives $561.80, the final amount ; and deducting $500, the original principal, gives $61.80, the compound interest. What is the compound interest 2. Of $600 for 2 yr. at b% ? 3. Of $100 for 2 yr. at 7^? 4. Of $300 for 2 yr. at 10% ? 5. Of $1000 for 2 yr. at b% ? What is the amount at compound interest, 6. Of $800 for 2 yr. at b% ? 7. Of $2000 for 2 yr. at 10^? 8. Of $400 for 2 yr. at 4^? 9. Of $500 for 2 yr. at 8^? WRITTEN EXAM:PLES. 586. 1. What is the comp. int. of $750 for2 yr. at 6^? OPERATION. ANALYSis.—Since the amount is 1.06 $750 Prin. for 1st yr. of the principal, the amount at the end ]^^Q5 of the first year is $795, which is the ~T principal for the 2d year, and the amount $795 Prin. for 2d yr. ^t the end of the 2d year is $842.70. 1.06 Hence, by subtracting the given princi-. $842T70 Total amount. P^^' *^^^' *^^ ^^^^^^ ^« *^® compound ^^^ interest, $92.70. 750. . 7 Compound int. 310 PERCEl^TAGE. 2. What will $350 amt. to in 3 yr. at 1%, comp. int.? 3. What is the compound int. of $1200 for 3 yr. at b%} KuLE. — I. Find the amount of the given principal for the first period of time at the end of luhich interest is due, and make it the princi2Ml for the second period, II. Find the amount of this principal for the next period; and so continue till the end of the given time, III. Subtract the given principal from the last amount, and the remainder loill te the compound interest. When the time contains months and days, less than a single period, find the amount up to the end ot the last period, and com- pute the simple interest upon that amount for the remaining months and days, which add to find the total amount. 4. What will $864.50 amount to in 4 yr. at 8^, com- pound interest ? 5. What is the compound interest of $680 for 2 yr. at ^%, interest being payable semi-annually ? 6. What is the compound interest of $460 for 1 yr. 5 mo. 18 da. at 6^, interest payable quarterly ? 7. What will be the amount of $1250 in 3 yr. 7 mo. 18 da. at 5^, interest being semi-annual ? 8. Find the compound interest of $790 for 9 mo. 27 da. at 8^, payable quarterly. The computation of compound interest may be abridged by usin^ the following table. To use the table, multiply the given principal by the number In the table corresponding to the given number of years and the given rate. If the interest is not annual, reduce the time to periods, and the rate proportionally. Thus, 2 yr. 6 mo., by semi-annual payments, at 7%, is the same as 5 yr. at 3|%; and 1 yr. 9 mo., quarterly payments, at 8%, the same as 7 yr. at 2%. IKTEREST. 311 587. Table shoioing the amt, of^l, at 2^, 3, 3^, 4, 5, 6, 7, 8^ 9, 10, 11, and 12%, co7npound int., from 1 to 20 years. Yrs. 2J per cent. 3 per cent. 3^ per cent. 4 per cent. 5 per cent. 6 per cent. 1 1.025000 1.030000 1.035000 1.040000 1.050000 1.060000 2 1.050625 1.060900 1.071225 1.081600 1.102500 1 123600 3 1.076891 1.092727 1.108718 1.124864 1.157625 1.191016 4 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477 5 1.131408 1.159274 1.187686 1.216653 1.276282 1.338226 6 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519 7 1.188686 1.229874 1.272279 1.315932 1.407100 1.503680 8 1.218403 1.266770 1.316809 1.368569 1.477455 1.593848 9 1.248863 1.304773 1.362897 1.423312 1.551328 1.689479 10 1.280085 1.343916 1.410599 1.480244 1.628895 1.790848 11 1.312087 1.384234 1.459970 1.539454 1.710389 1.898299 12 1.344889 1.425761 1.511069 1.601032 1 795856 2.012197 13 1.378511 1.468534 1.563956 1.665074 1.885649 2.132928 14 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904 15 1.448298 1.557967 1.675349 1.800944 2.078928 2 396558 16 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 17 1.521618 1.652848 1.794676 1.947901 2.292018 2.692773 18 1.559659 1.702433 1.857489 2.025817 2.406619 2.854339 19 1.598650 1.753506 1.922501 2.1C6849 2 526950 3.025600 20 1.638616 1.806111 1.989789 2.191123 2.653298 3.207136 Yrs. 7 per cent. 8 per cent. 9 per cent. 10 per cent. 11 per cent. 12 per cent. 1 1.070000 1.080000 1.090000 I.IOOCOO 1.110000 1.120000 2 1.144900 1.166400 1.188100 1.210000 1.282100 1.254400 3 1.225043 1.259712 1.295029 1.331000 1.867631 1.404908 4 1.310796 1.360489 1.411582 1.464100 1.518070 1.573519 5 1.402552 1.469328 1.538624 1.610510 1.685058 1.762342 6 1.500730 1.586874 1.677100 1.771561 1.870414 1.973822 7 1.605781 1.713824 1.828039 1.948717 2.076160 2.210681 8 1.718186 1.850930 1.992563 2.143589 2.304537 2.475963 9 1.838459 1.999005 2.171893 2.357948 2.558036 2.773078 10 1.967151 2.158925 2.867364 2.593742 2.839420 3.105848 11 2.104852 2.331639 2.580426 2.853117 3.151757 3.478549 12 2.252192 2.518170 2.812665 3.138428 3.498450 3 895975 13 2.409845 2.719624 3.065805 3.452271 3.883279 4.363492 14 2.578534 2.937194 3.341727 3.797498 4.310440 4 887111 15 2.759031 3.172169 3.642482 4.177248 4.784588 5.473565 16 2.952164 3.425943 3.970306 4.594973 5.310893 6.130392 17 3.158815 3.700018 4327633 5.054470 5.895091 6.86C040 18 3379932 3.996019 4717120 5.559917 6.543551 7.689964 19 3.616527 4315701 5.141661 6.115909 7.263342 8.612760 20 3869684 i 4.660957 5,604411 6 727500 6.063309 9.646291 312 PERCENTAGE. 9. Find by the table the compound interest of $950 for 1 jr. 5 mo. 24 da., at 10^, interest payable quarterly. OPERATION. 1 yr. 5 mo. 24 da. = 5 quarters of a year +2 mo. 24 da. 10% per annum = 2|% per quarter. Amount for 5 yr. at 2^% = 1.131408 of principal. $950 X 1.181408 rn $1074837, amount for 1 yr. 3 mo. Interest of $1074.837 for 2 mo. 24 da. at 10% = $25,079. $1074837 + $25,079 = $1099.916, total amount. $1099.916 - $950 = $149,916, compound interest. 10. Find the amount, at compound interest, of $749.25 for 10 yr. 4 mo., at 7^, interest payable semi-annually. 11. What sum placed at simple interest for 3 yr. 10 mo. 18 da., at 7^, will amount to the same as $1500 placed at compound interest for the same time, and at the same rate, payable semi-annually ? 12. At S%, interest compounded quarterly, how much will $850 amount to in 1 yr. 10 mo. 20 da. ? 13. What will $500 amount to in 20 yr. at 1%, comp. int.? 14 A father at his death left $12500 for the benefit of his only son, 14 yr. 8 mo. 12 da. old, the money to be paid him when he should be 21 years of age, with Q% interest compounded semi-annually. What did he receive ? A]^]^UAL INTEEEST. 588. Annual Interest is interest on the principal and on each year's interest remaining unpaid, but so computed as not to increase the original principal. It is allowed in tlie case of promissory notes and other contracts which contain the words, '' with interest payable annually,'* or with " compound interest." In such cases, the interest is not compounded beyond the second year. IlSrTEREST. 313 WRITTEN EXEBCISES , 589. 1. Find the annual interest and amount of $8000 for 5 yr., at Q% per annum. OPERATION. Analysis.— The in Int. of $8000 for 5 yr. at 6^ =$2400. terest on $8000 for 1 '' " $480 for 10 yr. at 6^ == $288. y^. at 6% is $480, and $2400 + $288=:$2688, Annual int. %V/in"el'^^^^^^^^ $8000 + $2688 = $10688, Amount. first year, remaining unpaid, draws interest for 4 yr. ; that for the second year, for 3 yr. ; that for the third year, for 2 yr.; and that for the fourth year, for 1 yr., the sum of which is equal to the interest of $480 for 4 yr. + 3 yr. + 2 yr. + 1 yr.— 10 yr. ; and the interest of $480 at 6% for 10 yr. is $288. Hence the total amount of interest is $2400 + $288, or $3688, and the amt. is $10688. 2. What is the annual interest of $1500 for 4 yr. at 1% ? Rule. — Compute the interest on the principal for the given time and rate, to which add the interest on each yearns interest for the time it has remained unpaid. To oUain the latter, when the interest has remained unpaid for a number of years, multiply the intered for one year by the product of the number of years and half that number diminished by one. Thus, if the time is 9 yr., the interest for 1 yr. should be multi- plied by 9 X (9 — 1) -i- 2, or 9x4 = 36. Since the interest for ihe first year draws 8 years' interest, that for the second year 7 years' interest, etc., and the sum of the series 8 + 7 + 6 + 5 + 4 + 3 + 2 + lis36. 3. What will $3500 amt. to in 10 yr., annual int., at 8^ ? 4. What is the difference between the annual interest and the compound interest of $2500 for 6 yr. at 6^ ? 5. Find the amt. of $575, at 8% annual int., for 9| yr. 14 314 PEBCEKTAGE. 6. $800. Macon, June 15, 1872. Four years after date, for value received, I promise to pay Robert E, Parh, or order, eight hundred dollars, with in- terest at seven per cent., payable annually. J. W. Burke. What amount is due on this note at maturity, no in- terest having been paid ? PARTIAL PAYMENTS. 590. Partial Payments are payments in part of the amount of a note, bond, or other obligation. 591. Indorsements are the acknowledgment of such payments, written on the back of the note, bond, etc., stating the time and amount of the same. 593. A Promissory Note is a written promise to pay a certain sum of money, on demand or at a specified time. 593. The Maher or Drawer of the note is the person who signs it. 594. The Payee is the person to whom, or to whose order, the money is paid. 595. An Indorser is a person who, by signing his name on the back of the note, makes himself respon- sible for its payment. 596. The Face of a note is the sum of money made payable by the note. 597. A I^egotiahle Note is one made payable to bearer, or to any person's order. When so made, it can be sold or transferred. PARTIAL PAYMENTS. 315 WniTTEN EXERCISES. 1. $800. New York, Jan. 1st, 1874. One year after date, I promise to pay Caleb Barlow^ or order, eiglit hundred dollars, for value received, with in* ierest. James Dunlap. Indorsed as follows : April 1, 1874, $10 ; July 1, 1874, J35 ; Nov. 1, 1874, $100. What was there due Jan. 1, 1875? Analysis. — The interest of $800 for 3 mo., from Jan. 1 to April 1, at 7%,* is $14; am*t, $814. Since the payment is less than the in- terest, it cannot be deducted for a new principal without com- pounding the int., which is illegal ; hence, find the int. of $800 to the time of the next payment, 3 mo., which is $14, and the amt. to that time, $828, from which deduct the sum of the two payments, or $45, leaving $783, a new principal. The int. of $783 for 4 mo., to Nov. 1, is $18.27; amt., $801.27; from which deduct the third payment, $100, leaving $701.27, the next principal, the amt. of which for 2 mo., to Jan. 1, 1874, is $709.45, sum due at that time. Piiri;rciPLE. — The principal must not he increased hy the addition of interest due at the time of any payment, so as to compound the interest. Upon this principle is based the rule prescribed by the Supreme Court of the United States : U. S. EtJLE. — I. Find the amount of the given princi" pal to the time of the first payment, and if this payment equals or exceeds tlie interest then due, subtract it from the amt, obtained, and treat the remainder as a neto principal, II. If the interest exceed the payment, find the amount of the same principal to a time whe^i the sum of the pay' ments equals or exceeds the interest then due, and subtract the sum of the payments from that amount. III. Proceed in the same manner with the remaining payments, * At the date of these payments the legal rate of interest in N. T. was 7;?. 316 PERCEKTAGE. $500. Philadelphia, Feb. 1, 1875. 2. Three months after date, I prmnise to pay to J. B Lippincott & Oo,^ or order, five hundred dollar Sy toith interest, tvithout defalcatiorio Value received. James Moij^^rob. Indorsed as follows: May 1, 1875, $40 ; Nov. 14, 1875, 18; April 1, 1876, $18; May 1, 1876, $30. What was due Sept. 16, 1876? OPERATION. Face of note, or principal $500.00 Interest to May 1, 1875, 3 mo., at 6% 7.50 Amount . 507.50 Payment, to be subtracted 40.00 2a^5 ^/ 13. Partial Paymhnts. 14. Discount. 1. Defs. i:: 2. 1. Defs. 15, Bank Dis- count. 1. Defs. Grace. 5. Maturity of Note. 6. Term of Discount. 7. Bank Check. 8. Proceeds or Avails 9. Pr^^f5^. 2. Corresponding Terms. 3. 614. Rule, I, II. 4. 616. Rule. 16. Savings-Bank Accounts — Role. mmmmi ^^^^^^^ 62 !• A Corpor*ation is an association of indi- viduals authorized by law to transact business as a single person. 623. A Charter is the legal act of incorporation defining the jpowers and obligations of the body incor- porated. 623. The Caintal StocU of a corporation is the capital or money contributed, or subscribed to carry on the business of the company. 624. Certificates of Stock or Scrip are the papers or documents issued by a corporation, specifying the number of shares of the joint capital which the holders own. 625. A Share is one of the equal parts into which capital stock is diyided. Tlie value of a share in the original contribution of capital varies in different companies. In bank, insurance, and railroad compa- nies, it is usually $100. 626. Stocks is a general term applied to shares of gtock of various kinds, Government and State bonds, etc. Stockholders are the owners of stock, either by original title or by subsequent purchase. The stockholders constitute the company. 62*7. The Far Value of stock is the sum for which the scrip or certificate was issued. 628. The Market Value of stock is the sum foi which it can be sold. STOCKS. 331 Stock is at par when it can be sold for its original or face value, or 100^ ; it is above par, or at a premium^ when it will bring more than its face value ; and it is below par, or at a discount, when it sells for less than its face value. Thus, when stock is at par, it is quoted at 100 ; when it is 5% above par, at 105 ; and when it is 5% below par, at 95. 639. JPrefniunij Discount^ and JBroIcerar/e are each a percentage computed upon the par value of the stock as the base. 630. A Stock Broker is a person who buys and sells stocks, either for himself, or as the agent of another. 631. Stock-johbing is the buying and selling of stocks with the view to realize gain from their rise and fall in the market. 633. An Instalhnent is a portion of the capital stock required of the stockholders as a payment on their subscription. 633. An Assessment is a sum required of stock- holders, to meet the losses, or to pay the business expenses of the company. 634. A Dividend is a sum paid to the stockholders from the profits of the business. Dividends and assessments are a percentage computed upon the par value of the stock as the base. 635. JSfet Earnings are the moneys left from the profits of a business after paying expenses, losses, and the interest upon the bonds. 636. A Bond is a written instrument securing the payment of a sum of money at or before a specified time. The principal bonds dealt in by brokers are Government, State, City, and Railroad bonds. 332 PERCEKTAGE. 637. Tf, S. Sands are of two kinds ; viz., those which are payable at a fixed date, and those which, while payable at a fixed date, may be paid at an earlier specified time, as the Goyernment may elect. 1. The former are quoted in commercial transactions by the rate of interest which they bear; thus. United States bonds bearing 6% iaterest are quoted JJ. 8. 6's. The latter are quoted in commercial iransactions by a combination of the two dates ; thus, U. 8. 5-20' 8, or TJ. 8. 6' 8 5-20 y means bonds of U. S. bearing 6 % interest, and pay- able at any time from 5 to 20 years, as the Government may choose. 2. When it is necessary to distinguish different issues bearing the same rate of interest, the year at which they become due is also mentioned ; thus, U. 8, 5*s of '11; TJ, 8. 5*8 of '74; U, 8. 6% 5-20, of 84 ; V. 8. 4's of 1907, 3. The 5-20*8 were issued in 1862, '64, '65, '67, and '70. They bore interest a,t Qfo, paid semi-annually in gold, but have nearly all been refunded at a lower rate of interest. 4. Bonds issued by States, cities, etc., are quoted in a similar manner. Thus, 8, G. 6's are bonds bearing Qfo interest, issued by the State of South Carolina. 638. A Coupon is a certificate of interest attached to a bond, to be cut off and presented for payment when the interest is due. 639. Currency is a term used to denote the circu- lating medium employed as a substitute for gold and silver. It consists, at present, in the United States, of U. S. Legal-tender Notes, or ^' Greenbacks,'^ and the Bills issued by the Nat. Banks, and secured by U. S. Bonds. If from any cause the paper medium depreciates in value, gold becomes an object of investment, the same as stocks. Gold being of fixed standard value, its fluctuations in price indicate changes in the value of the currency. Hence, when gold is said to be at a premium, currency is virtually below par, or at a discount. STOCKS. 33b OBATj BXEItCISES , 640. 1. Find the cost of 100 shares of Chicago and Eock Island Eailroad stock at 90 ; brokerage \%. Analysis. — Since the cost of one share is 90^ of $100, or $90, the cost of 100 shares is 100 times $90, or $9000, to which add the brokerage, J% of $10000, or $12J, and the sum $9012^, is the entire cost of the stock. 2. What cost 50 shares of N. Y. Central E. E. Stock, at par ; brokerage, \% ? 3. Find the cost of 10 shares of Bank Stock at 104 ; brokerage \%. 4. What is the cost of $2000 IT. S. 4's, at 112; broker- 641. !• A broker has $5010 to invest in bank stock at 25^ premium ; how many shares can he buy, charging ^% for brokerage ? Analysis. — Since the stock sells at 25% premium, each share with brokerage will cost $125 J ; hence he can buy as many shares as $125|- are contained times in $5010, or 40 shares. 2. A speculator invested $52000 in Ohio and Missis- sippi E. E. stock at 25f, allowing \% brokerage; how many shares did he buy? 3. If I invest $2350 in U. S. 4's, at 11 7|, brokerage \%, how many $1000 bonds do I receive ? 642, 1. A man bought a number of shares of mining e'ock at 60, and sold the same at 68, and gained $800 by the transaction. How many shares did he buy ? Analysis.— Since he bought at 60% and sold at 68%, he gained 8% of the par value; hence $800 is 8% of $10000, the par value, and the number of shares at $100 each is 100. 334 PEBCEi^^TAGEa 2. Bought E. E. stock at 90, and sold at par, gaining $1000. Eequired the number of shares. 3. I purchased stock at 110 and sold at 98, losing $1200. How many shares did I buy ? 4. A broker bought some stock at par, and sold it at 95, losing $2000. How many shares did he buy ? 643. 1. What sum must be invested in California 7's, at 110, to obtain therefrom an annual income of $1400 ? Analysis. — Since the annual income is $7 on each share, the number of shares must be equal to $1400 -j- $7, or 200 shares ; and 200 shares at $110 amount to $22000, the required investment. 2. What sum must I invest in stock at 115, paying 10^ yearly dividends, to realize an income of $2000 ? 3. What sum must be invested in N. Y. 7's at 103^^, in order to receive therefrom an annual income of $2100? 644. 1. What per cent, does money yield which is invested in S% stock at 120 ? Analysis. — Since each share costs $120, and pays $8 income, the per cent, will be jf^, or -^^ of 100^, equal to 6f %. 2. What per cent, does stock yield when bought at 90, paying Q% dividends ? When bought at 75 ? At 120 ? 3. What per cent, of interest does stock yield, which pays b% semi-annual dividends, if bought at 150? At 140? At 120? 645. 1. What should be paid for stock yielding 6^ dividends, in order to realize an annual interest on the investment of S% ? Analysis. — Since the annual dividend on each share is $6, this must be 8% of the sum required; and if 8% is $6, 1% is $f, and 100% is $75. Hence the stock must be bought for 75. STOCKS. 335 S5. For what must stock that pays ^% dividends be bought to realize 10% interest ? 9^ ? 8^ ? 3. For what should Missouri 6's be bought to pay 6% interest? 5^^ ? ^%? 8^? 646. 1. How much currency can be bought for 1500 In gold, when the latter is at a premium of 10% ? Analysis. — Since $1 in gold is worth $1.10 in currency, $500 in gold is worth 500 times $1.10, or $550. Hence, etc. 2. How much currency can be bought for |200 in gold, when the latter is at a premium of 9^ ? 3. What is $1000 in gold worth in currency, when the former is at a premium of 1%% ? Of 0^% ? Of 10|% ? 647. 1. How much gold can be bought for $440 in currency, when the former is at a premium of 10^ ? Analysis. — Since $1 in gold is worth $1.10 in currency, $440 'will buy as many dollars in gold as $1.10 is contained times in $440, or $400 in gold. Hence, etc. 2. How much gold selling at 9^ premium will $1090 in currency buy ? $218 ? $654 ? 3. How much gold at 11^ premium will $444 buy? WniTT EK MXJEBCIS ES, 648. Find the cost |1. Of 220 shares of bank stock, the market value of which is 103|, brokerage 1%. Operation.-(103|% + J%) of $100 = $104, cost of 1 share. $104 X 220 = $32880, cost of 220 shares. (640.) Formula. — Entire Cost — {Market Value of 1 Shan + Brokerage) x iVb. of Shares. 1 2. Find the cost of 350 shares of Western Union Tele- graph stock, market value 97f, brokerage ^%. 336 PERCENTAGE. \d. A broker bought for me 15 one-thonsand-dollar U. S. bonds at 112;^, brokerage ^%. What was their cost ? ( 4. My broker sells for me 125 shares of stock at 127^. What should I receive, the brokerage being 1% ? 649. Find the number of shares { 1. Of bank stock at 105, that can be bought for $25260, mcluding brokerage at ^% ? Operation.— (105% +if^) of $100 = $105}, cost of 1 share. $25260 -f- $105i = 240, No. of shares. (641.) Formula. — iVo. of Shares = Investmejit -— Cost of 1 Share. \2. How many shares of N. J. Central R. R. stock at 107f , brokerage 1%, can be bought for $27000 ? \3. How many shares of Mo. 6^s at 97f, brokerage \%y will $21560 purchase? \ 4. Bought Pacific Mail at 29|, and sold at 31 J, paying \% brokerage each way. How many shares will gain $330 ? Operation.— (31^% - 39J%) - i% = H%, gain. $330 -^ $1.50 = 220, No. of shaim (642.) Formula. — No. of Shares = Wliole Gain or Loss -j- Gain or Loss per Share, \ 5. How many shares of stock bought at 97|- and sold at 102J, brokerage \% each way, will gain $990 ? \ 6. Lost $1680 by selling K Y. Central at 101 that cost 104. Brokerage being ^% each way, how many shares did I sell? \^7. How many shares of the Bank of Commerce bought at 110-|- and sold at 116f, brokerage \% on the purchase and the sale, will gain $1200? STOCKS. 337 650. Find the amount of investment l.jln 5 per cent, bonds, at 111, so as to realize there- from an annual income of $2500. Operation.— $2500 -I- $5, income on 1 share = 500, No. of shares. $111, price of 1 share x 500 = $55500, mvestment. (643.) Formula. — Investment = Price of 1 Share x No. of Shares, 2J What sum must be invested in Tennessee 6's at 85, to yield an annual income of $1800 ? 3.1 How much money must be invested in any stock at 105|-, which pays b% semi-annual dividends, to realize an annual income of $2000 ? 4. ^What sum invested in stock at $63 per share, will yield an income of $550, the par value of each share being }, and the stock paying 10^ annual dividends? 651. Find the rate per cent, of income, realized 1 1. From bonds paying 8^ interest, bought at 110. Operation. — $8, interest per sliare -i- $110, cost per share = .07x\, or7y\f^. (B4:4:.) Formula. — Rate % of Income =z Interest per Share -r- Cost per Share. [ 2. If stock paying 10^ dividends is at a premium of 12^^, what per cent, of income will be realized on an in- vestment in it ? 1 3. Which will yield the better income, 8% bonds at 110„ or 5's at 75 ? 1^. Which is the more profitable, and how much, to buy New York 7's at 105, or 6 per cent, bonds at 84 ? 15 338 PERCEKTAGE. \ 5. What per cent, of income does stock paying 10^ dividends yield, if bought at 106 ? ) 6. What per cent, will stock which pays 6% diyidends yield, if bought at a discount of 15% ? j 7. What rate per cent, of income sh-all I receive, if I buy XJ. S. 5's at a premium of 10^, and receive payment at par in 15 years ? 653. Find at what price stocTc ynust he bought l.\That pays 6% dividends, so as to realize an income of 7^% on the investment. Operation.— .06 ^ .075 =: .80 or 80% , price of stock. (645.) Formula. — Price of Stock = Dividend — Bate of In- come. \ 2. What must be paid for 6% bonds, that the invest- ment may yield S% ? \3. How much premium may be paid on stock that pays 10^ dividends, so as to realize 11^% on the investment ? y4. What must I pay for 5 per cent, bonds, that my investment may yield 7% ? \6. At what price must stock, of the par value of $50 a share, and that pays 6% dividends, be bought, to yield an income of 7^% ? \ 6. At what price must 6% stock be bought, to pay as good an income as S% stock bought at par ? As 9% stock ? 653, Find the value in currency, U. Of $3750 in gold, quoted art: 110|. Operation.— $1.10J x 3750= $4148. 75, value in currency. (646.) Formula. — Total Value in Currency =z Value of $1 in Currency x No. of Dollars in Gold. STOCKS. 339 ^2. Find the yalue of $4975 in gold, at a premium of j 3. What is the semi-annual interest of $8000 6% gold- bearing bonds worth in currency, when gold is at lllf ? 1 4. A merchant bought a bill of goods, for which he was to pay $7000 in currency, or $6625 in gold. Gold being at 109f, which is the better proposition, and how much in currency ? 654. Find the value in gold, \ 1. Of $2150 in currency, when gold is at a premium of 1^%. Operation.— $2150 ^ 1.105 = $1945.70, value in gold. (647.) FoEMULA. — Total Value in Gold = Amt of Currency -T- (1 + Premium). /2. What is $4500 in currency worth in gold, when the latter is at a premium of l^% ? At 11^^ ? At ^% ? 3. How much money must be inyested in XJ. S. 6's at 111, when gold is quoted at 110^, in order to obtain a semi-annual income of $2210 in currency ? 4. The Mechanics Bank of New York having $109737.50 to distribute to the stockholders, declares a dividend of 6}^ ; what is the amount of its capital ? 5. A man owns a house which rents for $1450, and the tax on which is 2f ^ on a valuation of $8500. He sells for $15300, and invests in stock at 90, that pays 1% divi- dends. Is his yearly income increased or diminished, and how much ? 6. If I have $36500 to invest, and can buy N. Y. Cen» tral 6's at 85, or N. Y. Central 7's at 95, how much more profitable will the latter be than the former ? 34:0 PERCENTAGE. 7. Which is the better investment, a mortgage for 3 yr, of $5000, paying 7^ interest, and purchased at a discount of 6%, and paid in full, without cost, at maturity, or 50 shares of stock at 95, paying S% dividends, and sold at the expiration of three years at 98 ? 8. Henry Ivison, through his. broker, invested a certain sum of money in New York State 6's at 107|^, and twice as much in U. S. 5's, at 98|^, brokerage in each case ^%. The annual income from both investments was $3348. How much did he invest in each kind of stock ? 9. A gentleman invested $12480 current funds in TJ. S. 6 per cent, bonds, at 104. What was his annual income in currency when gold was 110 ? 655. Insurance is a contract of indemnity against loss or damage. It is of two kinds : insurance on prop- erty, and insurance on life. 656. The Insurer or Underwriter is the party who takes the risk or makes the contract. 657. The Policy is the written contract between the parties. 658. The Fremitim is the sum paid for insurance, and is a certain per cent, of the sum insured. 659. Insurance business is generally conducted by Companiee^ whicli are either Joint-stock Companies, or Mutual Gompan/es. A Stock Insurance Company is one in wbicli the capita Vs owned by individuals called stockholders. They alone share the profits, and are liable for the losses. A Mutual Insurance Company is one in which the profits and losses are divided among" those who are insured. Some companies are conducted upon the Stock and Mutual plans wmbined, and are called Mix^yd G'^m'^r'nie^ IKSURANCE. 341 Insurance on property is principally of two kinds : Fire Insurance^ and Marine and Inland Insurance. 660. Fire Insurance is indemnity for loss of property by fire. 661, Marine and Inland Insurance is in- demnity for loss of vessel or cargo, by casualties of navi- gation on the ocean, or on inland waters. Transit Insurance refers to risks of transportation by land onlj, or partly by land and partly by water. The same policy may cover both Marine and Transit Insurance. Stock Insurance is indemnity for the loss of cattle, horses, etc. Most insurance companies will not take risks to exceed two-thirds or three-foarths the appraised value of the property insured. When only a part of the property insured is destroyed or dam* aged, the insurers are required to pay only the estimated loss ; and sometimes the claim is adjusted by repairing or replacing the property, instead of paying the amount claimed. 663. The operations are based on the principles ol Percentage, the corresponding terms being as follows : 1. The Base is the amount of insurance. 2. The Rate is the per cent, of premium, 3. The Percentage is the premium, ORAL EXERCISES, 663. 1. How much must be paid for insuring a house sjid furniture for $4000, at \\% premium ? Analysis. — Since the premium is 1 J % , or -^^ equal to ^*^ of ihe sum insured, the premium on $4000 will be ^V ^^ |4000, or $50. Hence, etc. (510.) 2. What will be the annual premium of insurance, at \%, on a building valued at $8000 ? 843 PBEOEIfTAGE. 3. What will be the cost of insuring a quantity of flour, valued at $1500, at ^% ? 4. What must be paid for insuring a case of merchan- dise, worth $640, at 2^% ? 5. A man owns f of a boat-load of corn valued a1 $1800, and insures his interest at 1|^. What premiuna does he pay ? 6. Paid $6 for insuring $300 ; what was the rate ? Analysis. — Since the premium on $300 is $6, the premium oil $1 is shi of $6, or $.03, equal to 2%. Hence, etc. (513.) 7. Paid $12 for an insurance of $800 ; find the rate. 8. Paid $24 for an insurance of $1000 ; find the rate. 9. At 2%, what amount of insurance can be obtained for $30 premium ? Analysis. — Since 2 % is yf ^ or ^ of the amount insured, $30, the given premium, is -^jj of the amount insured ; and $30 is ^^ ^^ ^^ times $30, or $1500. Hence, etc. (516.) What amount of insurance can be obtained, 10. On a house, for $75, at d% premium ? 11. On a boat load of flour, for $150, at |^ ? 12. On a car load of horses, for $90, at 4:^% ? 13. On a store and its contents, for $105, at If ^ ? WJtITTEN EXERCIS JE8, 664. Find the Premium . 1. For insuring a building for $14500, at 1\%. Operation.— $14500 x .015 = $217.50. (5 1 2.) Formula. — Preynium = Amount Insured x Rati Find the premium for insuring \2. A house valued at $5700, at i%. \3. Merchandise for $2750, at 1^. INSUEAKOE. 343 > 4. A fishing craft, for $15000, at 1^%. 5. If I take a risk of $25000, at If^, and re-insure | of it at 2^%, what is my balance of the premium ? 665. Find the Bate of Insurance^ 1. If $36 is paid for an insurance of $2400. Operation.— -$36 -^ $2400 = .015, or 1^%. (515.) Formula. — Bate of Insurance = Premium ~- Sum Insured. What is the rate of insurance, 2. If $280 is paid for an insurance of $16000 ? 3. If $4.30 is paid for an insurance of $860 ? 4. A tea merchant gets his vessel insured for $20000 in the Eoyal Company, at f ^, and for $30000 in the Globe Company, at ^%. What rate of premium does he pay on the whole insurance ? 666. To find the Amourit of Insurance, .1. A speculator paid $262.50 for the insurance of a cargo of corn, at 1^%. For what amount was the corn insured ? Operation.— $263.50 ^.015 =$17500, the sum insured. (518.) Formula. — Sum Insured == Premium -7- Bate. 2. If it cost $93.50 to insure a store for one-half of its value, at 1|^, what is the store worth ? 3. Paid $245 insurance at 4f ^ on a shipment of pork, to coyer | of its value. What was its total value ? 4. A merchant shipped a cargo of fiour worth $3597, from New York to Liverpool. For what must he insure it at 3^%, to cover the value of the flour and premium ? Opekation.— $3597 -s- (1 - .03J) or .9675 = $3717.829. (520.) 344 PERCENTAGE. 5. An underwriter agrees to insure some property for enough more than its value to cover the premium, at the rate of 26 cents per $100. If the property is worth $22163, what should be the amount of the policy? 6. For what sum must a policy be issued to insure a dwelling-house, valued at $35000, at ^%, a carriage-house rorth $9500, at |^, and furniture worth $4500, at |^, 10^ being deducted from the premium, which is to be covered by the policy ? 7. A person insured his house for f of its value at 40 cents per $100, paying a premium of $73.50. What was the value of the house ? 8. A dealer shipped a cargo of lumber from Portland to New York ; the amount of insurance, including the value of the lumber and the premium paid, at If ^, was $25200. What was the value of the lumber ? 9. A merchant had 500 bbl. of flour insured for 80^ of their cost, at 3^%, paying $107.25 premium. At what price per barrel must he sell the flour to gain 20^. LIFE INSUEANCE.* 667. Life Insurance is a contract by which a company agrees to pay a certain sum, in case of the death of the insured during the continuance of the policy. 668. A Term Life Policy is an assurance for one or more years specified. 669. A Wliole Life Policy continues during the life of the insured. * See note at bottom of page 346. IlfSURAKCB. 345 Premiums may be paid annually for life, or in 5, 10, or more Installments (called 5 -payment, 10-payment policies, etc.), or the entire premium may be paid in one sum in advance. The premium is computed at a certain sum or rate per $1000 insured, the rate varying with the age of the insured at the time the policy is issued. A policy of endowment is not in all respects an insurance policy, but is rather a covenant to pay a stipulated sum at the end of a Oertain period to the person named, if living. Most companies issue a form of policy that combines the princi- ples of Term Life Assurance and Simple Endowment, called for brevity Endowment Policy. Hence, 6T0. An Endowment Policy is one in which the assurance is payable to the person rnsured afc tne end of a certain number of years named, or to his heirs if he die sooner. An endowment policy is really two policies in one, and the assured pays the premiums of both. 671. A Dividend is a share of the premiums or profits returned to a policy-holder in a mutual life in- surance company. 673. A Table of Mortality shows how many per- sons per 1000 at each age are expected to die per annum. 673. A Table of Rates shows the premium to be charged for $1000 assurance at the different ages. Such a table is based upon the table of mortality, and the proba- ble rates of interest for money invested, with a margin or loading for expenses. 674. The following condensed table gives data from the American Experience Table of mortality, and the annual premium on the kinds of policies most in use. 340 PERCENTAGE. American Experience Table— Mortality and Premiums. 1 ANNUAL PREMIUM PER $1000. Life Table i Endow MENT (AND Term .1.GK. One Year Term (Net), Whole Life, Payments during life. Payment for 10 yr. only. Payment for 5 yr. only. Single Payment. Life). 10 years. 25 8.1 7.75 $19.89 $42.56 $73.87 $326.58 $103.91 26 8.1 7.82 20.40 43.37 75.25 332.58 104.03 27 8.2 7.88 20.93 44.22 76.69 a38.a3 104.16 28 8.3 7.95 21 48 45.10 78.18 34 .31 104.29 29 8.3 8.02 22.07 46.02 79.74 352.05 104.43 30 8.4 8.10 22.70 46.97 81.36 359.05 104.58 31 8.5 8.18 23.35 47.98 83.05 366.33 104.75 82 8.6 8.28 24.05 49.02 84.80 373 89 104.92 S3 8.7 8.38 24.78 50.10 86.62 381.73 105.11 34 8.8 8.49 25.56 51 22 88 52 389.88 105.31 35 8.9 8.60 20.38 52.40 90.49 398.34 105.53 40 9.8 9.42 31.30 59.09 101 58 445.55 106.90 45 11.2 10.73 37.97 67.37 115.02 501.69 109.07 50 13.8 13.25 47.18 77.77 131.21 567 13 112.68 The actual net cost of insurance for a single year at eacli age given in tlie table, on the mortality assumed, is as many dollars and tenths of a dollar as there are deaths, but discounted for 1 year. Thus, at age 25, deaths 8.1 per 1000, net cost, which is $8.10, dis- counted at ^fo by the insurance law, $7.75. If this sum, $7.75, is loaded for expenses at, say 25%, the total premium for 1 year is $9.69, if at 40%, then it would be $10.85. In a Term Life Policy the premium may vary, increasing slightly each year of the term, according to the assumed increasing liability to decease, or it may be averaged for the term so as to be the same each year. Note. — As there is no uniformity in the Tables and Methods used by different Life Insurance Companies, the pupil may very properly omit this subject. IKSUBAKCB. 347 WRITTEN EXEnCISESm 675. To find the amount of premium 1. For a life policy of $5000 issued to a person 30 7ears old. Operation.— $22.70 x 5 = $113.50. 2. For a life policy of $7500, age being 45. Rule. — Multiply the premium for $1000 assurance by ' the number of thousands. Formula — Premium = Bate per $1000 x iVb. of thou- sands. 3. Find the annual premium for an endowment policy of $10000, payable in 10 years, age 35. 4. What premium must a man aged 30 pay annually for life, for a life policy of $5000 ? What premium annually for 10 years ? What premium annually for 5 years ? What premium in a single payment ? OPERATION. Analysis. — Multiply the rate $22.70x5.000= $113.50 P®^ thousand doUars, found in $46,97 X 5.000 = $234.85 *^^ J^'^^ ™' T^^^^ ^^^ ^^' by the number of thousands, ex- $81.36x5.000= $406.80 pressing the hundreds, tens/and $359.05 X 5.000 =$1795.25 units decimaUy. 5. What annual premium will a man aged 35 years pay to secure an endowment policy for $5000, payable to him- self in 10 years, or to his heirs, if death occurs before? 6. If he dies at the beginning of the ninth year, how much will the assurance cost, reckoning simple interest at 6% ? 7. How much less would he have paid in the whole life (annual payment) plan, interest included ? 348 PERCEKTAGE. 8. A man aged 45 insures his life for $7500 on the sin- gle-payment plan, and dies 3 yr. 5 mo. afterward. How much less would his insurance have cost him had he in- sured on the annual payment plan, reckoning int. at 6% ? 9. A person aged 27 takes out a 10-year endowment policy for $5000 ; the dividends reduce his annual pre miums 16% on the average. Computing annual interest at 7% on his premiums, does he gain or lose, and how much ? 10. A man aged 35 years took out a life policy for $12000, on the 5-payment plan, and died 3 yr. 6 mo. afterward. What was gained to his estate by insuring, computing compound interest on his payments at 7^, also adding two dividends of $95 each ? TAXES. 676. A Tax is a sum of money assessed on the per- son, property, or income of an individual, for any public purpose. 677. A Poll Tax or Capitation Tax is a cer- tain sum assessed on every male citizen liable to taxation. Each person so taxed is called Q,poll 678. A Property Tax is a tax assessed on prop- erty, according to its estimated, or assessed, value. Property is of two kinds : Beal Property^ or Real Es^ tate, and Personal Property. 679. Real Estate is fixed property; such as houses and lands. 680. Personal Property is of a movable nature ; such as furniture, merchandise, ships, cash, notes, mort- gages, stock, etc. TAXES. 349 681. An Assessor is an oflBlcer appointed to deter- mine the taxable value of property, prepare the assess- ment rolls, and apportion the taxes. 683. A Collector is an officer appointed to receive the taxes. 683. An Assessment Roll is a schedule, or list^ containing the names of all the persons liable to taxation in the district or company to be assessed, and the valua- tion of each person's taxable property. 684. The Rate of Property Tax is the rate per cent, on the valuation of the property of a city, town, or district, required to raise a specific tax. WHITTEN BXBRCISES* ^685. 1. What sum must be assessed to raise $836000 net, after deducting the cost of collection at 5^ ? / Operation.— $836000 -^ .95 = $880000. (510.) Formula. — Sum to le raised -^(1 — Rate of Collection) = Sum to le Assessed. I 2. What sum must be assessed to raise a net amount of $11123, and pay the cost of collecting at 2^? \3o In a certain district, a school-house is to be built at a cost of $18500. What amount must be assessed to cover this and the collector's fees atd%? 4, The expense of building a public bridge was $1260.52^, which was defrayed by a tax upon the property of the town. The rate of taxation was 3|- mills on a dollar, and the collector's commission was 3^%. What was the valuation of the property ? 350 PERCENTAGE. 5. In a certain town a tax of $5000 is to be assessed* There are 500 polls, each assessed 75 cents, and the valuation of the taxable property is $370000. What will be the rate of property tax, and how much will be A's tax, whose property is valued at $7500, and who pays for 2 polls ? Operation.— $.75 x 500 = $375, amt. on polls. $5000 -$375= « « property. $4625 -5- $370000 = .0125, rate of taxation. $7500 X .0125 = $93.75, A's property tax. $93.75 + $1.50 = $95.25, A's whole tax. Rule. — L Find the amount of poll tax, if any, and suUract it from the whole amount to ie assessed; the remainder is the property tax. IL Divide the property tax iy the whole amount of faxaile property ; the quotient is the rate of taxation. III. Multiply each marHs taxable property hy the rate of taxation, and to the product add his poll tax, if any ; the result is the whole amount of his tax. A table sucli as tlie following is a great aid in calculating the amount of each person's tax, according to the ascertained rate. Assessor's Table. (Rate .0087.) Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. $1 $.0087 $ 9 $.0783 $ 80 $ .696 $ 700 $ 6.09 2 .0174 10 .087 90 .783 800 6.96 3 .0261 20 • .174 100 .87 900 7.83 4 .0348 30 .261 200 1.74 1000 8.70 5 .0435 40 .348 300 2.61 2000 17.40 6 .0522 50 .435 400 3.48 3000 26.10 7 .0609 60 .522 500 4.35 4000 34.80 8 .0696 70 .609 600 5.22 5000 43.50 TAXES. 351 6. Find by the table the tax of a person whose property is valued at $3475, the rate being .0087. Operation.— Tax on $3000 = $26.10 " " 400 = 3.48 '* *' 70 = .609 " " 5 = .0485 '* '* $3475 = $30.2325, or $30.23. Find by the table the tax of a person whose property ''/. Is $2596 and who pays for 5 polls at $.50. 8. Is $9785, polls 3 at $.75. 9. Is $12356, polls 4 at $1.25. 10. Is $25489, polls 5 at $.95. 11. A tax of $11384, besides cost of collection at ^\%, is to be raised in a certain town. There are 760 polls assessed at $1.25 each, and the personal property is valued at $124000, and the real estate at $350000. Find the tax rate, make an assessor's table for that rate, and find a person's tax, whose real estate is valued at $6750, personal property at $2500, and who pays for 3 polls. 12. In the above town, how much is B's tax on $15000 real estate, $2750 personal property, and 5 polls? 13. What is C's tax on $9786 and 1 poll? 14. How much tax will a person pay whose property is assessed at $7500, if he pays 1|^ village tax, ^% State tax, Rnd 1 J mills on a dollar school tax ? 15. The expense of constructing a bridge was $916.65, ^hich was defrayed by a tax upon the property of the town. The rate of taxation was 2J mills on a dollar, and the commission for collecting 3^; what was the assessed valuation of the property of the town ? Note. — Amt. to be raised -f- by rate = valuation. 352 PERCEKTA GE. 686. SYNOPSIS FOR EEVIEW. o m ^ ^ O 1. Defs. 2. 648. 3. 649. 4. 650. 5. 651. 6. 652. 7. 653. 8. 654. 1. Defs. 1. Corporation, 2. Charter. 3. Capital Stock 4. Certificate of Stock, or Scrip, 5. Share, 6. Stocks. 7. Stockholders. 8. Par Fa^?i7ici 18. D^/. iTwcZ^ (?/ IT". /^. J5(?7ic?5. 19. (7(??^ p(?7i. 20. Currency. i\r Formula. Amt. of Insurance. J 1. Defs. t 2. 675. 1. Defs. Life Insurance, 2. Term Life Policy. 3. Whole Life Policy. 4. Endowment Policy^ 5. Dividend. 6. Table of Mortality . 7. TaJ^/* ^/ Bates. Rule. Formula. ' 1. Tax. 2. Po^? r^aj. 3. Property Tax. 4. i^ea^ Estate. 5. Personal Property. 6. Assessor. 7. Collector. 8. Assessment Boll 9. Paif^ (?/ Property Tax, 2. 685. ) < Sum to be raised. Formula. ^ 3. 686. 1 '^^ ^^^ ( Amt. of Tax. Rale, I, II. IIL _^^^^ "^^^^ 687. Exchange is the giving or receiving of any mm. in one currency for its value in another. 15y means of exchange, payments are made to persons at a dis- tance by written orders, called BUls of Exchange, 688. Exchange is of two kinds, Domestic, or In^ land, and Foreign. 689. Domestic or Inland Exchange relates to remittances made between different places in the same country. 690. Foreign Exchange relates to remittances made between different countries. 691. A Bill of Exchange is a written request, or order, upon one person to pay a certain sum to another person, or to his order, at a specified time. An inland bill of exchange is usually called a Draft. 693. A Set of Exchange is a bill drawn in dupli- cate or triplicate, each copy being valid, until the amount of the bill is paid. These copies are sent by different conveyances, to provide against miscarriage. 693. A Sight Draft or Bill is one which requires payment to be made ^^at sight," that is, at the time it is presented to the person who is to pay it. 354 PERCENTAGE. 694. A Time Draft or Bill is one that requires payment to be made at a certain specified time after date, or after sight, 695. The Buyer or Remitter of a bill is the person who purchases it. The buyer and payee may be the same person. 696. The Acceptance of a bill or draft is the agree- ment by the drawee to pay it at maturity. The drawee thus becomes the acceptor, and the bill or draft, an acceptayice. 1. The drawee accepts by writing the word " accepted '' across the face of the bill, and signing it. 2. Three days of grace are usually allowed on bills of exchange, as well as on notes. When a bill Is protested for non-acceptance, ♦he drawer is bound to pay it immediately. 697. The Par of Exchange is the estimated value of the coins of one country as compared with those of another. It is either intrinsic or commercial, 1. The Intrinsic Par of Exchange is the comparative value of the coins of different countries, according to their weight and purity. 2. The Commercial Par of Exchange is the comparative value of the coins of different countries, according to their market price. 698. The Course or Bate of Exchange is the current price paid in one place for bills of exchange on another place. This price varies according to the relative conditions of trade and commercial credit at the two places between which the exchange is made. Thus, if New York is lart^ely indebted to London, bills of exchange on London will bear a high price in New York. EXCHANGE. 355 699. FORMS OF DRAFTS AND BILLS. A SIGHT DRAFT. $500. New York, July 1, 1874. At sights pay to the order of William Thompson^ five lundred dollars, value received, and charge to the acct, of Hekry J. Carpenter. To Harris, Jokes & Co., Cincinnati, 0. Other drafts have the same form as the above, except that in- stead of the words "at sight," " days after sight," or " days after date," are used. When the time is after sight, it means after acceptance. SET OF EXCHANGE. £700. New York, August 1, 1874. At sight of this First of Exchange (Second and Third of the same tenor and date unpaid), pay to the order of Samuel Monmouth, Seven Hundred Founds Sterling, for value received, and charge the same to the account of MoRTO]^, Bliss & Co. MoRTOK, KosE & Co., London. The above is the form of the first Mil ; the second requires only the change of " First " into " Second," and instead of " Second and Tliird of the same tenor," etc., '* First and Third." The Third Bill varies similarly. DOMESTIC OR INLAND EXCHANGE. The course of exchange for inland bills, or drafts, is always ex pressed by the rate of premium or discount. Time drafts, however, are subject to bank discount, like promissory notes, for the term of credit given. Hence, their cost is affected by both the course of exchange and the rate of discount for the time. 356 PERCEKTAGE. WRITTEN EXERCISES, 700. What is the cost 1. Of a sight draft on New Orleans for $1750, at 1^% premium ? Operation.— $1750 x l.Oli = $1771.87^. (512.) ^ ^ , ri ( 1 + Rcii^ of Premium. Formula.— C(?5^ = Face x i ' ^ ^ \. ^. ( 1 — Mate of Viscount, 2. Of a sight draft on Troy for $1590, at 1^% discount ? 3. Of a draft on Boston for $1650, payable in 60 days after sight, exchange being at a premium of 1^% ? Operation.— $1.0175 = Course of Exchange. $.0105 = Bank Dis. on $1, for 63 da. $1,007 = Cost of Exchange, for $1. $1,007 X 1650 = $1661.55, value of Draft 4. Of a draft on New York at 30 da. for $4720, at 1^% premium ? 5. Of a draft on New Orleans, at 90 da., for $5275, int being 7^, and exchange ^% discount? 6. Find the cost in Philadelphia of a draft on Denver, at 90 da., for $6400, the course of exchange being 101| ? 7. What must be paid in New York for a draft on San Francisco, at 90 da., for $5600, the course of ex- change being 102-^^? 701. Find the Face 1. Of a draft on St. Louis, at 90 da., purchased foi $4500, exchange being at 101|^ ? Operation. — $1,015 = Course of Exchange. $.0155 = Bank Dis. of $1, for 93 da., at 6%. $.9995 = Cost of Exchange of $1. $4500 -4- .9995 = $4502.25. (520.) EXCHANGE. 357 2. Of a draft on Eiclimond at 60 da. sight, purchased for ^797.50, interest 1%, premium 2\% ? a Of a sight draft bought for $711.90, discount 1\%. 4. A commission merchant sold 2780 lb. of cotton at 11 \ cents a pound. If his commission is 2^^, and the course of exchange 98^^, how large a draft can he buy to remit to his consignor ? 5. The Broadway Bank of New York having declared a dividend of h%y a stockholder in Chicago drew on the bank for the sum due him, and sold the draft at a pre- mium of 1\%, thus realizing $2283.18| from his dividend. How many shares did he own ? 6. A man in Eochester purchased a draft on Louisville, Ky., for $5320, drawn at 60 days, paying $5151.10. What was the course of exchange ? 7. Eeceived from Savannah 250 bales of cotton, each weighing 520 pounds, and invoiced at 12J cents a pound. Sold it at an advance of 25^, commission 1^%, and remitted the proceeds by draft. What was the face of the draft, exchange being ^% discount ? FOREIGN EXCHANGE. 703. Money of Account consists of the denomi- nations or divisions of money of any particular country, in which accounts are kept. The Act of March 3, 1873, provides that *' the value of foreign coin, as expressed in the money of account of the United States, ehaU be that of the pure metal of such coin of standard value ; and the values of the standard coins in circulation, of the various na- tions of the world, shall be estimated annuaUy by the Director of the Mint, and be proclaimed on the first day of January by the Secretary of the Treasury." 358 PEBCEKTAGE. •fa o I '^do 3^5 OS »-. o "^ s -as 2 a o OS .^CO Sfii f- ■ i 1. ^ t. • t. OcE 'oo'o3o':H3'o'os'o'oo'o'o:-5'ors o:;2:3'o.'s'2'o.':h'oo o'o'oo J5 !i a> o S S o ©j3 a <» <^ III III :bxchange. 359 704. Sterling Bills or Sterling Exchange are bills on England, Ireland, or Scotland. Such bills are negotiated at a rate fixed without reference to the par of exchange. Formerly such bills were quoted at a certain rate % above tbe old par value of a pound sterling, which was $4.44 J. As this was entirely a fictitious value, and always about 9% below the real value, the course of exchange always appeared to be heavily against this country, and thus tended to impair its credit. By the Act of March, 1873, "all contracts made after the first day of January, 1874, based on an assumed par of exchange with Great Britain of fifty- four pence to the dollar, or $4.44|- to the sovereign or pound sterling," are declared null and void. The par of exchange between Great Britain and the United States is fixed at $4.8665. 705. Exchanges with Europe are effected chiefly through the following prominent financial circles: London, Paris, Antwerp, Amsterdam, Hamburg, Frank- fort, Bremen, and Berlin. In exchange on Paris, Antwerp, and Switzerland, the unit is the franc^ and the quotation shows the number of francs and centimes to the dollar, Federal Money. In exchange on Amster- dam, the unit is the guilder, quoted at its value in cents ; on Ham- burg, Frankfort, Bremen, and Berlin, the quotation shows the value oifour reichsmarks (marks) in cents. Wit ITT EN JEXAMPIjES. 706. Find the cost 1. Of a bill of exchange on London at 3 days' sight, for £393 15s. 6d., exchange being quoted at 4.89^, and goldatl.lO|. OPERATION. £393 15s. 6d. = £393.775. $4,895 X 393 775 = $1927.529, gold value of bill. $1927.529 X I.IOJ = $2122.69, value in currency. 360 PERCEKTAGE. 2. Of a bill of exchange on Liverpool, for £473 5s. 9d. par yalue, in gold. 3. Of a bill of £625 4s. 3d. sterling, at 4.83|, gold 1.09|. 4. Of a bill on Paris for 495 francs, at 5.15 francs to the dollar, in gold. Operation.— -495 -^ 5.15 = $96.12, gold value of the bill. Analysis. — Since 5.15 francs cost $1, 495 francs will cost as many dollars as 5.15 francs are contained times in 495 francs, or $96.12. 5. Of a bill on Antwerp for 697J francs, at 5.17| francs to the dollar, in gold. 6. Of a bill on Geneva, Switzerland, for 1655 francs, at 5.15|, in cnrrency, gold being 1.09|. 7. Of a bill on Franhfort for 650 marks, at 94|, in gold. Opeiiation.~$.94375 -t- 4 x 650 = $153.36. Analysis.— Since $.94| is the value of 4 marks, 650 marks are worth 650 times J of $.94|, or $153.36. 8. Of a bill on Berlin for 1750 marks, quoted at 96^, in gold. 9. Of a bill on Hamburg for 2155 marks, at 95|, io currency, gold being 1. 10;^. 10. Of a bill on Amsterdam for 2500 guilders, quoted at 41f, brokerage ^%. Operation.— $.41625 x 2500 = 1040.625. $1040.625 X ,00} = $2.60, brokerage. $1040.625 + $2.60 = $1043.225, cost of bill. 11. Of a bill on Amsterdam for 1950 guilders, at 41^. 12. Bought exchange on Amsterdam, at 41 J, for 3750 guilders; on Hamburg, at 96^ for 1000 marks; and on London for £500, at $4.85. What was the cost of the whole in currency, gold selling at 109|- ? EXCHAKGB. 361 13. What will ifc cost to remit directly from Boston to Amsterdam, 12560 guilders, at 41|? 14. What will be the cost of remitting 13550 marks from New York to Frankfort, exchange selling at 94^^ and gold at 109|^ ; brokerage, ^% ? 707. What will be the face 1. Of a bill of exchange on London that can be bought for $5500, in currency, exchange selling at 4.86, and gold at 1.10 ? Operation.— $5500 currency -f- 1.10 = $5000, gold. (519.) $5000 -4- $4.86 = 1028.806 +. £1028.806 = £1028 16s. l|d. 2. Of a bill on Manchester^ England, that can be bought for $7500, gold ; rate of exchange, 4.86 ? 3. Of a bill on Berlin that cost $4000 in gold, eX' change 93f ? Operation.— ($4000 -4- $.9375) x 4 rz: 17066| marks. Analysis.— Since $.93J wiU buy 4 marks, $4000 wiU buy 4 times as many marks as $.93f is contained times in $4000, or 17066| marks- 4. Of a bill on Hamburg that cost $550 in gold, ex- change 94 J ? 5. Of a bill on Frankfort that cost $395.75 in gold, exchange 95^ ? 6. Of a bill on Geneva, Switzerland, that cost $325 in gold, exchange at 5.17 ? Operation.— 5.17 f r. x 325 = 1 680.25 francs. Analysis.— If $1 win buy 5.17 francs, $325 will buy 325 timea 5.17 francs, or 1680.25 francs.*^ 7. A merchant in New Orleans gave $6186, currency, for a bill on Paris, at 5.15^. What was its face ? 8. What is the face of a bill on Antwerp, that may be purchased in New York for $2500, exchange at 5.16f ? 16 S62 PERCENTAGE. AEBITEATIOlSr OF EXCHANGE. 708. Arbitration of JExchanr/e is the process of computing the cost of exchange between two places by means of one or more intermediate exchanges. Such ex- change is said to be indirect or circuitous. By this computation the relative cost of direct and indirect ex- change is ascertained. Sometimes, owing to the course of exchange between different places, it is more advantageous to remit by the latter than by the former. Arbitration is either simple or compound. 709. Simple Arbitration is that in which there is but one intermediate place. 710. Compound Arbitration is that in which there are several intermediate places. WJtITTEN EXERCISES. 711. 1. I owe 1500 marks to a merchant in Frankfort. Should I remit directly from ISTew York, or through Lon- don, exchange on Frankfort being 94, on London, 4.87^, and in the latter place on Frankfort 20.75 marks to the pound, and the London brokerage \% ? Operation.— $.94 x 1 500 -^4 3= $352.50, cost of direct exchange. 1500 marks h- 20.75 marks = £72.29. £72.29 + i % = £72.38. $4.87i X 72.38 = $352.85. $352.85 — $352.50 = $.35, loss by ind. exchange. 2. What will it cost to remit from Boston to Berlin 750 marks, by indirect exchange, through Paris, exchange in New York on Paris being at 5.15, and 4 marks at Paris being worth 4.91 francs, the brokerage being ati^? EXCHANGE. 363 3. What will it cost to remit 2500 guilders from New York to Amsterdam, through London and Paris, the rates of exchange being as follows : at New York on London 4.83, at London on Paris 34.75 francs to the pound, and at Paris on Amsterdam 2.09 francs to the guilder, broker- Thomas Browi^, Dr. March 10, 1874. To mdse. 1835. " 18, '' a (( 320. " 26, " i( 66 475. April 5, '' it 66 600. 12, ^' i( (6 250. Allowing 30 days' credit on each of the bills, what is the equated time of payment ? 6. Purchased goods as follows : Sept. 15, 1875, a bill of $275, on 3 mos. Oct. 10, '' " 351.50, " 60 da. " 28, " '' 415.75, '' 30 da. Nov. 3, " '' 500, " 4 mos. Dec. 15, '' '' 710, '' 3 mos. What was due on this account Aug. 10, 1876, com- puting interest at 7^ ? 7. I have four notes, as follows : the first for $425, due April 1, 1875 ; the second for 1615, due May 10, 1875 ; the third for $1500, due May 28, 1875 ; and the fourth for $750, due June 10, 1875. At what date should a single note be made payable, to be given in exchange for the four notes ? AYERAGING AOOOUI^TS. 735« An Account is a written statement of debit and credit transactions, with their respective dates. Debit means wliat is owed by the person with whom the account is kept ; credit^ what is due to him from the person keeping the account. 736. To Average an Account is to find, either AVERAGING ACCOUNTS. 375 the equated time of paying the balance, or the cash balance at any given time. Each item of a book account shoiild draw interest from tlie timi it becomes due. WniTTEN EXERCISES, 737. 1. Find the equated time of paying the balance of the following account. Dr. William Sampson. Cr. 1875. 1875. Jan. 11 To mdse. . . . $750 Feb. 10 By draft at 60 da. $500 Feb. 1 " at 3 mo. 600 Mar. 3 " cash . . . 700 Mar. 15 " " at 6 mo. 1500 Apr. 15 it (i 300 May 3 " at 4 mo. 900 Operation I. {MethocC by P roduets) Due. Amt, Days. Product. Paid, Amt. Jan. 11. 750 X 10 = 7500 Apr. 14. 500 X 103 = 51500 May 1. 600 X 120 = 72000 Mar. 3. 700 X 61 = 42700 Sept. 15. 1500 X 257 = 385500 Apr. 15. 300 X 104 = 31200 " 3. 900 X 245 = 220500 1500 1254(l}0 3750 685500 1500 125400 2250 )5e 0100 24811, or 249 da. Balance due 249 da. from Jan. 1, or Sept 7, Analysis. — Assuming for convenience Jan. 1 as the standard date, we find as in 734: the term of credit of each debit amount ; and, reckoning from the same date, the time to each credit amount. Multiplying each amount by its time in days, and adding the debit and credit products, we find the number of days' interest of $1 due to the debtor, and the number of days' interest of $1 he has already received. The difference, 560100, shows the number of days' inter- est of $1 still due, and as the balance is $2250, the time must be r^^ of 560100 da., or 249 da. Hence, the equated time is 249 da. from Jan. 1, or Sept 7. 376 PERCENTAGE. Dr. Operation II. (Method hy Interest) $750 to Jan. 11 (from Jan. 1)= 600 " Feb. 1 + 3 mo. 1500" Mar. 15 + 6 mo. 900 " May 3 + 4 mo =4 mo. =8 mo. 14 da., =8 mo. 2 da.. 10 da., int. at 1 % per mo. $2.50 24.00 127.00 72.60 $8750 $226.10 Cr, $500 to Feb. 10 + 63 da. = 3 mo. 13 da., int . at 1% per mo $17.17 700 * ' Mar. 3 = 2 mo. 2 da., it n 14.47 300 ' * Apr. 15 = 3 mo. 14 da., << it 10.40 $1500 $42.04 $226.10 - $42.04 = $184.06, int. at 1% per mo. due. Int. of balance, $2250, for 1 mo., at 1^= $22.50. Hence, $184.06 -^ $22.50 = 8.18 + mo., or 8 mo. 6 da. 8 mo. 6 da. from Jan. 1, or Sept. 7, Equated Time, In this operation, 12 % per annum or 1 % per mo. is assumed for convenience ; since the int. at 1 % per mo. is as many hundredths as there are months, and one-third as many thousandths as there are days. Thus, the int. of $249 for 2 mo. 9 da. is $4.98 + §.747 = $5,727(571). 2. Find the equated time of the following : Dr, William Simpson^. Cr. 1874. 1874. Aug. 5 To mdse. at 3 mo. $720 Oct. 10 By cash . . . $500 Sept. 10 .. - " 2 •♦ 850 Dec. 15 '' draft at 60 da. 450 Nov. 3 (t <( 1200 ** 25 " cash . . . 900 1875. 1875. Jan. 20 " sundr's at 5 mo. 620 Jan. 3 {( tt 2m EuLE 1. — I. Find the date at wliich each debit item is due, and each credit item is paid or due. II. Take the first day of the month in the earliest date on either side of the account as a standard date, and muU AVERAGI]NiG ACCOUKTS. 377 tiply each sum due or paid hy the numher of days between its time and the standard date. III. Add the products, and their difference divided hy the balance due will give the number of days between the standard date and the eqtiated time. Or, Rule 2. — Find the time of each item from the standard date as before, and com^pute the interest on each at 1% ^ month. The difference betiveen the amount of interest on each side divided by the interest of the bala7ice at 1% for one month zvill be the equated time. When the terms of credit are long, Rule 2 gives the shorter method. 3. Find the equated time of the following, allowing 60 da. credit on each debit item : Dr. John Driscoll. Cr. 1877. 1877. June 1 To mdse. . . $950 Aug. 1 By cash $700 July 6 « u 300 Sept. 20 " " . . 1000 Sept. 8 (( n 1900 Nov. 1 it tt 1200 Oct. 20 It it 2600 4. What is the equated time for the payment of the balance of the following account, allowing 4 months' credit on all the debit items ? Dr. DoDD, Browk & Co. Cr. 1878. 1878. Jan. 20 To mdse. . . $570 Feb. 14 By mdse. . . $490 - 28 300 Mar. 1 " cash . . 1000 Feb. 11 720 Apr. 2 (C (( 1800 " 26 835 Mar. 10 1150 '^ 28 960 Apr. 15 475 • 378 PERCENTAGE. 738. 1. Find the cash lalance of the following account on the 22d of August, allowing interest at Q% : Dr. George Hammokd. Cr. 1875. 1875. Mar. 15 To mdse., at 3 mo. $600 May 10 By cash . . $300 Apr. 3 " " " 4 mo. 700 July 1 <' <* 40) May 10 " « " 6 mo. 1000 Aug. 15 i< n 500 Operation. — By averaging the account, the equated time for paying the balance, $1100, is found to be Nov. 4. (734.) True present worth of $1100 for 74 da. (from Aug. 23 to Nov. 4) is $1086.60, or cash balance Aug. 22. Or, by Interest Method, as follows : Dr, Int. of $600, from June 15 to Aug. 22, 68 da., $6.71 (574.) ** *' 700, ** Aug. 3 " ** 19 da., 2.19 $8.90 Cr. [000, from Aug. 22 to Nov. 10, 80 da., $13.15 300, '' May 10 *' Aug.22, 104 da., 5.13 400, '' July 1 '' " 52 da.. 3.42 500, *' Aug. 15 ♦* " 7 da., .58 $22.28 8.90 Balance of interest due Hammond, $13.38 $1100 - $13.38 = $1086.62, Cash Balance, Aug. 22 Analysis. — Charge Hammond with interest on each debit item from the time it is due to date of settlement, and credit him with interest on each sum paid from the date of payment to date of set- tlement, also on each debit item which becomes due after the date of settlement. Hence, he is entitled to interest on $1000 from Aug. 22 to Nov. 10. As the balance of interest is in favor of Ham- mond, it must be deducted from the balance of the account, to ob- tain the cash balance. There is a slight difference in the results, but the interest method is the more accurate. By the use of Inter rjst Tables, it is also the shorter of the two methods. AVERAGIiq^G ACCOUNTS. 379 EuLE 1. — I. Average the account, and find the equated time of payment of the balance. II. If the date of settlement is prior to the equated time, find the present worth of the balance of account for the cash balance ; if later y find the interest of the balance of account for the intervening time, and add it to find the cash balance. Or, KuLE 2. — Find the interest on each debit and credit item, from the time it is due or paid to the date of settle- ment, placing on the same side of the account the interest on each item due prior to the date of settleme^it, and on the opposite side the interest on each item due after the date of settlement. If the balance of iiiterest is on the same side as the balance of the account, add it, if on the other side subtract it ; and the result will be the cash balance at the date of settlement. 2. I owe $1500 due May 1, and $750 due Aug. 15. If I give my note at 30 da. for $450, June 1, and pay $370 in cash July 15, what is the equated time for paying the balance ; and what would be due in cash Dec. 10, allow- ing interest at 7^? 3. When is the balance of the following account due per average ? Dr. 0. B. TiMPSOiq-. Cr, 1875. 1875. Aug. 10 To mdse. (^ 60 da. . $751.35 Oct. 3 By cash .... $300.00 Sept. 5 " " @30da. . 425.00 Nov. 15 '♦ note @ 90 da. . 450.00 Nov. 1 " " @90da. . 927.83 Dec. 20 " cash .... 5oo.oa Dec. 5 *' " @80da. . 1200.00 380 PEBCEKTAGE. 4. What is the cash balance of the above account Jan. 1, 1876, allowing interest at 10^ ? 5. Find the equated time, and cash balance July 1, of the following, allowing 7^ interest : Dr. Thomas Smith. Cr. Jan. 4 To mdse. @ 4 mo. $1600 Feb. 1 By mdse. @ 4 mo. $500 '* 6 " " @3mo. 1500 Mar. 2 " cash . . . 2000 Apr. 10 " " @60da. 3000 " 25 t( if 8150 « 28 " " @30da. 2500 Apr. 16 <( li 800 6. Average the following account, and find for what amount a note at 60 days should be given Aug. 1, to pay the balance, interest at 6^. Dr. ORSOiq^ Hinma:^. Cr. 1875. Apr. 2 May 15 To charges $87.25 35.75 1875. Feb. 25 Mar. 3 Apr. 1 By mdse. @ 8 mo. a i. @g u 300 500 739, 1. Average the following Account Sales ^ and find when the net proceeds are due. (543.) Account Sales of 1200 ibis, of flour received from Smith, Tyler <& Co.^ Cincinnati. Date. Buyer. Quantity. Price. Amount. 1876. May 1 June 5 ** 15 July 1 J. Brooke W. Long A. Bruce W. Case 300 bbl. 450 ** 250 ** 200 *' @ $5.50, 8 mo. @ 6.20, 4 mo. @ 6.50, 6 mo. @ 5.75, 2 mo. $1650.00 2790.00 1625.00 1150.00 $7215.00 AYERAGIKG ACCOUKTSo 381 Charges. Apr. 28. Freight $674.50 " " Cartage 37.50 May 1. Storage 191.00 Commission on $7215, @ 2J% . . . 162.34 Total charges . . . '. . $1065.34 Net proceeds due per average $6149.66 OPERATION. Average of sales, found by the method of Equation of Payments, Oct. i, which is the date at which the commission is due. Average of charges, including commission (Oct. 1), May 22, Equated time of $7215 due Oct. 1, and $1065.34 due May 22, Oct 24, date when the net proceeds are due. KuLE. — I. Average tlie sales alone, and the result will ie the date to ie given to the commission and guaranty. II. Make the sales the credits and the charges the deiits, and find the equated time for paying the balance. 2, Make an account sales, and find the net proceeds and the time the balance is due : Wm. Brown, of N. York, sold on acct. of J. Berry, of Chi- cago, June 1, 350 bu. Winter Wheat, @ $1.35, at 60 da.; June 15, 275 bu. Spring Wheat, @ $1.75, at 90 da. ; July 3, 1260 bu. Indian Corn, @ $.79, at 6 mo. ; and July 10, 375 bu. Eye, @ $1.02, at 3 mo. Paid freight. May 28, $567.50; cartage. May 30, $22.50; insurance, June 5, $56.25 ; and charged com. at 3 J^, and 1^% for guaranty. 3. Sold on account of Brown, Sampson & Co., at 6 mo. : Oct. 1, 1874, 13 hhd. sugar, averaging 1520 lb., @ $.12| ; Oct. 5, 15 chests Hyson Tea, each 95 lb., @ $1.05^ Paid charges : Oct. 3, Insurance, $85 ; Oct. 10, Cooper- age, etc., $24.50 ; Oct. 20, Cai'tage, $125. Charged com- mission and guaranty, 4^%. Make an account sales, and find the equated time for paying the net proceeds. 382 PERCENTAGE. 740. SYNOPSIS FOR REVIEW. w M 1. Defs. 3. Forms. 3. Inland Exch. 22. Custom- house Business. Exchange. 2, BomesticExchange. S. For- eign Exchange. 4. Bill of Exchange. 5. Set of Exchange. 6. Sight Draft or Bill. 7. Time Braft or BUI. 8. Buyer or Re- mitter. ^.Acceptance. 10. Par of Ex- change. 11. Course or Bate of Exchange, A Sight Braft. 2. Set of Exchange. 700.|^ ^ ,{ Costof Braft Formula. 701.r''^nFaceofBrafL 4. Foreign Exch'ge. 5. Arbitra- tion of Exch'ge. jl. Money of Account. 23. Equation op Payments. 24. Averaging Accounts. Defs. 1 rt Sterling Bills, or Exchange. Exchange with Europe — ^how effected, 706. [^ . .{Cost of Bill. 707.r''^''^\Fa^eofBill. rl. Arbitration of Exchange. Defs. \ 2. Simple Arbitration. 13. Compound Arbitration. Rule, I, II, III, IV. 1. Custom House. 2. Port of Entry. 3. Collector. 4. Clearance. 5. Manifest. 0. 1. Defs. \ Duties or Customs. 7. Tariff. 8. Specific Duty. 9. Ad VaL orem Duty. 10. Cross Wght, 11. Net Weight. 2. 726. To find the Duty. 1. Equation of Payments. 2. EquMed Time. 3. Term of 1. Defs. I Credit. 4. Average 2'erm of Credit. 2. Principle. 3. 733. Rule, I, 11. 4. 734. Rule, I, II, III. Proof. " 1. Defs. 1. Account. 2. To Average an Aca. 2. 737. Rule 1, 1, II, III. Rule 2. 3. 738. Rulel, I, n. Rule 2, 4. 739. Rule, I, n. OBAZ EXERCISES. 741. 1. A father is 30 years old, and his son 6 ; how many times as old as the son is the father? 2. 30 are how many times 6 ? 30 -^ 6 = ? 3. What part of- $30 are $6 ? Of 20 cents are 5 cents ? 4 What is the relation of 8 to 2 ? Of 40 rd. to 4 rd. ? 5. What relation has 12 to 3 ? 60 lb. to 20 lb. ? Compare the following, and give their relative values : 6. 75 with 5. 7. 25 with 6i. 8. 1 with 7. 9. 1 with 7. 10. 2^ with 3^. 11. .9 with .3. 12. 1.6 with $.2, 13. .42 with .3. 14. I with |. DEFUSriTIONS. 743. Matio is the relation between two numbers of the same unit value, expressed by the quotient of the first divided by the second. Thus the ratio of 12 to 4 is 12 ~ 4 ziz 3. 743. The Sign of ratio is the colon ( : ), or the sign of division with the line omitted. Thus, the ratio of 9 to 3 is expressed 9:3, or 9-^3, or in the form of a fraction |, and is read, the ratio of 9 to 3, or 9 divided by 3. 744. The Terms of a ratio are the two numbers eompared. 745. The Antecedent is the first term, or dividend. 146. The Consequent is the second term, or divisor. 384 RATIO. 747. The Value of a ratio is the quotient of the antece- dent divided by the consequent, and is an abstract number. Thus, in the ratio $18 : $6, $18 and $6 are the terms of the ratio , $18 is the antecedent ; $6 is the consequent ; and 3, the quotient oi $18 -7- $6, is the Tialue of the ratio. 748. A Simple Ratio is the ratio of two numbers ; as 10 : 5. 749. A Compound Ratio is the ratio of the products of the corresponding terms of two or more simple ratios. Thus the ratio compounded of the simple ratios, l\^^\ may be expressed \ ^^^ ^^ = f "^f^ '^p | = 72 : 48 : Or, f X A = I = 3 : 3. When the multiplication is performed the result is a simple ratio. 750. The Meciprocal of a ratio is 1 divided by the ratio (196), or it is the consequent divided by the ante- cedent. Thus the ratio of 8 to 9 is 8 : 9, or f, and its reciprocal is ■§-. The ratio of two fractions is obtained by reducing them to a common denominator, when they are to each other as their nume- rators (241). If the terms of a ratio are denominate' numbers, they must be reduced to the same unit value. 751 . Prom the preceding definitions and illustrations are deduced the following FoBMULAS. — 1. The Ratio = Antecedent -=- Consequent. 2. The Consequent = Antecedent -r- Ratio. 3. The Antecedent = Consequent x Ratio, PATIO. 385 753. Since the antecedent is a diyidend, and the con- sequent a divisor, any change in either or both of the terms of a ratio will affect its value according to the laws of division or of fractions (300)^ which laws become the General Principles of Eatio. 1. Multiplying the antecedent, ov ) ,^ ,, . ,. ^, ,, ^. . ,. ,: , \ Multiplies the ratio. Dividing the consequent, ) 2. Dividinq the antecedent, ox ) x^. .-r ^r -,.' ^ ' ,1 , r Divides the ratio. Multiplying the consequent, ) 3. Multiplying or dividing both ^ ^^^^ ^^^ ^^^^^ ^^^ antecedent and consequent r , • hy the same nuniber, ■ 753. These principles may be embraced in one general law. A change in the antecedent produces a like change in the ratio ; but a change in the consequent produces an OPPOSITE change in the ratio. EXERCISES, 754. 1. Express the ratio of 11 to 4 ; of 16 to 2 ; of 20 to ^ ; of $36 to $12 ; of 9 lb. to 27 lb. ; of 4^ bu. to 9 bu. 2. Can you express the ratio between $15 and 5 lb.? The reason ? 3. Indicate the ratio of 18 to 20 in tivo forms. What are the terms of the ratio ? The antecedeiit ? The con- sequent ? The hind of ratio ? The value of the ratio ? In like manner express, analyze, and give the value, 4. Of 80 to 120 ; of 12| to 37^ ; of 16^ to |. 5. Of 5.3 to 1.3; of f to A ; of ^^. 17 386 RATIO. 6. The antecedents of a ratio are 7 and 10, and the consequents, 5 and 4. What is the value of the ratio ? 7. The first terms of a ratio are 18, 12, and 30, the second, 54, 6, and 15. What is the kind of ratio ? Ex- press in three forms. Find its value in the lowest terms. Solve, and state ihQ formula applied to the following : 8. The consequent is 3 J, the antecedent ^ ; what is the ratio ? 9. The antecedent is 60, the ratio 7 ; what is the con- sequent ? 10. The consequent is t6.12|^, the ratio |^; what is the antecedent ? 11. The ratio is 2f , the antecedent ^ of | ; what is the consequent ? 12. The ratio is 6, the consequent 1 wk. 3 da. 12 hr. ; what is the antecedent ? 13. Express the ratio of 120 to 80, and give its value in the lowest terms. 14. Make such changes in the last example as will illustrate Prin^. 1. 15. With the same example, illustrate Prik. 2. 16. Illustrate by the same example Prik. 3. 17. Find the reciprocal of the ratio of 75 to 15. 18. Find the reciprocal of the ratio of 2 qt. 1 pt. to 4 gal. 1 qt. 1 pt. What is the ratio 19. Of 40 bu. 4.5 pk. to 25 bu. 2 pk. 1 qt. 20. Of 6 A. 110 P. to 10 A. 60 P. 21. Of 25 lb. 11 oz. 4 pwt. to 19 lb. 5 oz. 8 pwt. 33. Ofl?itoi^. OBAZ BXEBCISBS. 755. 1. What is the ratio of 4 to 2 ? Of 6 to 1 ? Of 14 to 7 ? Of 21 to 3 ? 2. Find two mimbers that have the same quotient as 8-^2. As 27 ~ 3. As 16 ~ 4. As 30 -^ 6. As 4 -=- ^. 3. Express in the form of a fraction the ratio of 26 to 13. Of 32 to 8. 4. Express in both forms the ratio of two other num- bers equal to the ratio of 10 to 2. Of 15 to 5. Of 12 to 3. 5. If 4 stamps cost 12 cents, what will 20 stamps cost at the same rate ? 6. What number diyided by 12, gives the same quo- tient as 20 -^ 4 ? 7. What number has the same ratio to 12, that 20 has to 4? 8. To what number has 48 the same ratio that 80 has to 5 ? That 24 has to 3 ? 9. The ratio of 20 to 5 is the same as the ratio of what number to 4 ? To 6 ? To 5|- ? To 6 J ? 10. The ratio of 45 to 9 is the same as the ratio of 15 to what number ? Of 30 to what number ? 11. 28 is to 7 as 8 is to what number ? 12. 56 is to 8 as what number is to 5 ? 13. 63 -T- what number equals the ratio of 36 to 4 ? 388 PROPORTION. DEFINITIONS. •JSG. A Proportion is an equation in which each member is a ratio ; or it is an equality of ratios. 75*7. The equality of the two ratios may be indicated by the sign = or by the double colon : : Thus, we may indicate that the ratio of 8 to 4 is eqaal to that of 6 to 3, in any of the following ways : 8:4 = 6:3, 8:4:: 6:3, \ = % 8-1-4 = 6-5-3. This proportion, in any of its forms, is read. The ratio of 8 to 4 i* equal to the ratio of 6 to 3, or, 8 is to 4- as 6 is to 3. 758. Since each ratio consists of two terms, every pro- portion must consist of at least four terms. Each ratio is called a Couplet, and each term is called a Proportional 759. The Antecedents of a proportion are the first and third terms^ that is, the antecedents of its ratios. 760. The Consequents are the second and fourth terms, or the consequents of its ratios. 761. The Extremes are the first and fourth terms. 763. The Means are the second and third terms. In the proportion 8 : 4 : : 10 : 5, 8, 4, 10, and 5 are the propor- tionals ; 8 : 4 is the first couplet, 10 : 5 the second couplet ; 8 and 10 are the antecedents^ 4 and 5 are the consequents ; 8 and 5 are the ex- tremes, 4 and 10 are the means. Three numbers are proportional, when the ratio of the first to the eecond is equal to the ratio of the second to the third. Thus the numbers 4, 10, and 25 are proportional, since 4 : 10 = 10 : 25; the ratio of each couplet being |. When three numbers are proportional, the second term is called a Mean Proportional between the other two. PROPORTION. 389 The proportion 8 : 4 : : 10 : 5 raay be expressed thus, | = ^ (757). Reducing these fractions to equivalent ones having a com- . , 8x5 10x4 mon denominator, -^r^ = -7^-« Since these fractions are equal, and have a common denominator, their numerators are equal, or 8 x 5 = 10 x 4. 763. Principles. — 1. The product of the extremes of u proportion is equal to the product of the means. 2. The product of the extremes divided iy either mean will give the other mean. 3. Tlie product of the means divided hy either extreme will give the other extrerne, EXEnCISES, 764. 1. The ratio of 4 to 10 is equal to the ratio of 6 to 15. Express the proportion in all its forms (757). Drill Exercise. — How many terms has a proportion ? What are they called ? How many ratios ? What are they called ? Name the proportionals in example 1 ; the couplets ; the ante- cedents ; the consequents ; the extremes ; the means. What is the product of the extremes ? Of the means ? What is the dividend of the first ratio ? The divisor of the second ratio ? The divisor of the first ratio ? The dividend of the second ratio ? In the frac- tional form what are the numerators ? The denominators ? 2. The ratio of 6 to 15 equals the ratio of 8 to 20. 3. The ratio of 4^ to 18 equals the ratio of 6 to 24, Change to the form of equations by Prik. 1 : 4. 12 : 1728 : : 1 : 144. 5. 2|:17::20:143t^. 6. 27.03: 9.01 :: 16.05 : 5.35. 7. f:f::|:A- 8. The extremes are 15 and 48, and one of the means is 10. Find the other mean. 9. TAe means are 25 and 75, and one of the extremes \s 12^. Find the other extreme. 390 PROPOBTIOK. The required or omitted term in a proportion, or in an operation, will hereafter be represented by a?. Find the term omitted in each of the following pro- portions : 17. ^Jd. : xjd. :: $9| : 127.25. 18. x: 9.01:: 16.05: 5.35. 19. I yd. : a; yd. : : $ J : $59.0625c 20. ^:|::a::|. 21. 2;:38i::8i:76f 22. 7.5:18 :: a; oz. : 7^ oz. 11. 8: 52:: 20: a:. 12. 12:^2::: 1:144. 13. a;: 20:: 120:50. 14. $80 : $4 : : a; : 4. 15. 2.5:62.5::5:ir. 16. $1 75.35 :$a;::i:f SIMPLE PKOPOETIOK 765. A Simple Proportion is an expression of equality between two simple ratios. It is used to solve problems of which three terms are given, and the fourth is required. Of the three given numbers, two mnst always be of the same kind ; and the third, of the same kind as the required term, 766. A Statement is the arrangement of these terms in the form of a proportion. WRITTEN EXEJtClSBS, 767. 1. If 4 tons of coal cost $24, what will be the cost of 12 tons at the same rate ? Analysis. — Since 4 tons and 12 tons have the same unit value, they can be compared, and will form one couplet of the proportion. For the same reason $24 the cost of 4 tons, and %x the cost of 12 tons, will form the other couplet. Then by Prin. 3, $03 = 24 x 12 -^ 4 = $72. 4T. STATEMENT. 12T. ::$24:fa; OPERATION. 12 x24-^4 = $72 Or By Cancellation 12 X t^^ 4 $«: = $72 PE0P0^TI02S-. 891 Proop.~4 X 72=12 X 24. (763, Prin. 1.) In practice, that number which is of the same unit value as the required term, is generally made the antecedent of the second couplet or third term of the pro- portion, and the required term, Xy the fourth term. The terms of thtj first couplet are so arranged as to have the same ratio to each other, as the terms of the second couplet have to each other, which is easily determined by inspection. The product of the means 12 And 24, divided by the given extreme 4, gives the other extreme, oi required term, $72. (763, Prin. 3.) Drill exercises Tike the following, will soon make the pupil familiar with the principles and operations of proportion. 2. If 4 horses eat 12 bushels of oats in a given time, how many bushels will 20 horses eat in the same time ? In this example, what two numbers have the same unit value ? What do they form ? What is the denomination of the third term ? Of the required term ? What is the antecedent of the second couplet? From the conditions of the question, is the consequent of the second couplet or required term, greater or less than the antecedent ? If greater, how must the antecedent and consequent of the first couplet compare with each other ? If less, how com- pare ? What is the ratio of the first couplet ? Why not 20 to 4 ? Make the statement. How is the required term found ? 3. If 96 cords of wood cost $240, what will 40 cords cost ? 4. If 20 lb. of sugar cost $1.80, find the cost of 45 lb. 5. If 18 bu. of wheat make 4 barrels of flour, how many barrels will 200 bu. make ? KuLE. — I. Make the statement so that two of the given numiers tuhich are of the same unit value, shall form the first couplet of the proportion, and have a ratio equal to the ratio of the third given term to the required term, II. Divide the product of the means iy the given extreme^ and the quotient will he the number required. 392 PROPOKTIOK. CAUSE AND EFFECT. 768. The terms of a proportion have not only the relations of magnitude^ but also the relations of cause and effect. Every problem in proportion may be considered as \ comparison of two causes and two effects. Thus, if 4 tons as a cause will bring when sold, $24 as an effecty 12 tons as a caiise will bring $72 as an effect. Or, if 6 horses as a cause draw 10 tons as an effect y 9 horses as a cause will draw 15 tons as an effect. 769. Since like causes produce like effects, the ratio of two like causes equals the ratio of two like effects pro- duced by these causes. Hence, 1st cause : 2d cause : : 1st effect : 2d effect. WRITTEN EXERCISES. 770. 1. If 8 men earn $32 in one week, how much will 15 men earn at the same rate, in the same time ? STATEMENT. ANALYSIS. — In this ex- ist cause. 2(1 cause. 1st effect. 2d effect, ample an effect is required. 8 men : 15 men : : $32 : $x The first cause is 8 men, OPERATION. ^^'^ ^^^^^^ ^^'^ ^^ ^^^' and since they are like ^X =: 1 D X o /i -— o = ^bi) causes they can be com- pared. The effect of the first cause is $32 earned, the effect of the second cause is $x earned, or the required term. Since like effects have the same ratio as their causes (769), the causes may form the first couplet, and the effects the second couplet of the proportion. The required term is readily obtained by (763, 3). 2. If 20 bushels of wheat produce 6 barrels of flour, ih >w m^iny bushels will be required to produce 24 barrels ? PBOPORTION. 393 STATEMENT. ANALYSIS. — In this ex- ist cause. 2d cause. 1st effect. 2d effect, ample a cause is required. 20 bu. : a;bu. :: 6 bbl. : 24bbl. The first cause is 20 bu., the second cause is x bu. OPERAT ION. ^^ the required term, ii; bu. = 2 X 2 4 -^ 6 = 8 bu. The effect of the first cause is 6 bbl. of flour, the effect of the second cause is 24 bbl. of flour. Since like causes have the same ratio as their effects (7B9), the statement is made as in Ex. 1, and the required term found by (763, 2). 3. If 5 horses consume 10 tons of hay iu 8 mo., bow many horses will consume 18 tons in the same time ? Drill Exercise, — In this example, what is the first cause ? The second cause? The first effect? The second effect? Is the re- quired term a cause or an effect ? A mean or an extreme ? What is the first couplet? What, the second? Make the statement. How is the required term found ? 4. If 8 yards of cloth cost $6, how many yards can be bought for $75 ? 5. How many men will be required to build 32 rods of wall in the same time that 5 men can build 10 rods ? EuLE. — I. Arrange the terms in the statement so that the ratio of the causes which form the first couplet^ shall equal the ratio of the effects which form the second cou2)let, putting X in the place of the required term, II. If the required term le an extreme, divide the loro- duct of the means hy the given extreme ; if the required term he a mean, divide the product of the extremes iy the given mean. To shorten the operation, equal factors in the first and second, or in the first and third terms may be canceled. Solve the following by either of the foregoing methods : 6. If 5 sheep can be bought for $20.75, how many sheep can be bought for $398.40 ? 394 PROPORTION. 7. When 10 barrels of flour cost $112.50, what will be the cost of 476 barrels of flour? 8. If a railroad train run 30 miles in 50 min., in what time will it run 260 miles ? 9. How many bushels of peaches can be purchased for $454.40, if 8 bushels cost $10.24? 10. K a horse travel 12 miles in 1 hr. 36 min., how far, at the same rate, will he travel in 15 hours ? 11. How many days will 12 men require to do a piece of work, that 95 men can do in 7^ days ? 12. If f of an acre of land cost $60, what will 45| acres cost? 13. At the rate of 72 yards for £44 16s., how many yards of cloth can be bought for £5 12& ? 14 If J of a barrel of cider cost $1^^, what is the cost of f of a barrel ? 15. H the annual rent of 35 A. 90 P. is $28450, how much land can be rented for $374.70? 16. What will 87.5 yd. of cloth cost, if 1 J yd. cost $1.26 ? 17. If by selling $5000 worth of dry goods, a merchant gains $456.25, what amount must he sell to gain $1000 ? 18. Bought coal at $4.48 per long ton, and sold it at $7.25 per short ton. What was the gain per ton ? 19. What will be the cost of a pile of wood 80 ft. long, 4 ft. wide, 4 ft. high, if a pile 18 ft. long, 4 ft. wide, 6 ft. high cost $30.24? 20. If 36 bu. of wheat are bought for $44.50, and sold for $53.50, what is gained on 480 bu. at the same rate ? 21. If a business yield $700 net profits in 1 yr. 8 mo., in what time will the same business yield $1050 profits ? PEOPORTIOK. 39S COMPOUND PROPORTION. 771. A Compound JProportion is an expression of equality between two ratios, one or both of which are compound. All the terms of every problem in compound proportion appear In couplets, except one, and this is always of the same unit value as the required term. The order of the ratios, and of the terms composing the ratios, is the same as in simple proportion. WJRITTEN EXMUC ISES, 773. 1. If 18 men build 126 rd. of wall in 60 da., working 10 lir. a day, how many rods will 6 men build in 110 da., working 12 hr. a day ? 11 STATEBfENT. 18 men : 6 men \ rods, rods. 60 days : 110 days )- : 126 : a; 10 hours : 12 hours ) OPERATION. 11 42 rods, ^xu^^n^m i_62. GO 2 ^ 1^X00x10 5^— y/jf $ 5 ^1$ n0 ^00 n u m' 5 463 934 42 92|=a:rd. Analysis. — All the terms in this example appear in couplets, ex- cept 126 rods, which is of the same unit value as the required term, And is made the third term of the proportion, and x rods, the fourth. The required number of rods depends upon three conditions : 1st, the number of men employed ; 2d, the number of days they work ; and 3d, the number of hours they work each day. Consider each condition separately, and arrange the terms of the same unit value in couplets, and make the statement as in simple proportion (767). Then find the required term by (763, S)« 396 PROPOBTIOK. 2. If 20 horses consume 36 tons of hay in 9 mo., ho'W many tons will 12 horses consume in 18 months ? Drill Exercise, — In this example, what is the denomination of the required term ? What given number has the same unit value T What will be the third term of the proportion? The fourth? How many couplets are there ? Name them. What kind of a ratio do they form ? How is the antecedent and consequent of each couplet determined ? How is a compound ratio reduced to a simple one? Make the statement. Is the required term a mean or an extreme ? How is it found ? (763, 3.) 3. If $320 will pay the board of 4 persons for 8 weeks, for how many weeks will $800 pay the board of 15 persons ? 4. If a man walk 192 miles in 6 days, walking 8 hr. a day, how far can he walk in 18 days, walking 6 hr. a day ? 5. If 6 laborers can dig a ditch 34 yards long in 10 days, how many days will 20 laborers require to dig a ditch 170 yards long? EuLE. — I. Form each couplet of the compound ratio from the numbers given, hy comparing those which are of the same unit value, arranging the terms of each in respect to the third term of the proportion, as if it were the first couplet of a simple proportion. (767.) II. Divide the product of the second and third terms by the product of the first terins, the quotient will be the num hr required. The same preparation of the terms by reduction is to be observed as in simple proportion. When possible, shorten the operation by cancellation. When the vertical line is used, write the factors of the dividend on the right, and the factors of the divisor with x on the left. PROPORTIOl^. 397 CAUSE AND EFFECT. 773. If we regard the conditions of each problem as the comparison of two causes and two effects^ the com- pound proportion will consist of two ratios, one or both of which may be compound, and the required term will be either a simple cause, or eflfect, or a single element of a compound cause, or effect. STATEMENT. 1st cause. 2d cause. 8 men : 8 days : 4 days ) OPERATION. %x = 13x4x3^01^^^^ %x 12 $ 4 $ $$20^ $x $240 ^X^ WRITTEN JEXEHCISJES. 774. 1. If 8 men earn $320 in 8 days, how much will 12 men earn in 4 days ? Or, Analysis. — In this example the first cause is 8 men at work 8 days, the second cause is 12 men at work 4 days ; the two form a compound ratio. The effect of the first cause is $320 earned, the effect of the sec- ond cause is $x earned, and is the required term ; the two effects form a simple ratio. The value of the required term depends upon two conditions : 1st, the number of men at work ; 2d, the number of days they work. Consider each condition separately, and arrange the terms of the same unit value in couplets, and make a statement in the same man- ner as in simple proportion. The required term being an extreme^ is found by (703, 3). 2. If it cost $41.25 to pave a sidewalk 5 ft. wide and 75 ft. long, what will it cost to paye a similar walk 8 ft. wide and 566 ft. long ? 398 PROPORTIONS". 3. How many days will 21 men require to dig a ditch 80 ft. long, 3 ft. wide, and 8 ft. deep, if 7 men can dig a ditch 60 ft. long, 8 ft. wide, and 6 ft. deep, in 12 days? 7:21 12: X STATEMENT. ( 60 : 80 ] 8: 3 '6:8 OPERATION. Or, X n 3 00 tr = 8 ^1 X 00 X ^ X 3 =z= ^ = 2f da. n i8 a; = 2f da. Analysis. — In this example the causes and the effects each form a compound ratio. The required term is an element of the second cause and a mean. Hence divide the product of the extremes by the product of the given means, and the quotient is the required factor or term, 2| da. (763, 2). 4. If 4 horses consume 48 bushels of oats in 12 days, how many bushels will 20 horses consume in 8 weeks ? KuLE. — I. Of the given numbers^ select those which con- stitute the causes, and those which constitute the effects, and arrange them in couplets as in simple cause and effect, putting X in the place of the required term. II. If the required term, x, be an extreme, divide the product of the means by the prodiict of the given extremes; if X be a mean, divide the product of the extremes by the product of the given means; the quotient loill be the re- quired term. Solve the following by either of the foregoing methods : 5. What sum of money will produce $300 in 8 mo., if $800 produce $70 in 15 months ? PROPORTIOK. 399 6. If 20 reams of paper are required to print 800 copies of a book containing 230 pages each, 40 lines on a page, how many reams are required to print 3000 copies of 400 pages each, 35 lines on a page ? 7. If 10 men can cut 46 cords of wood in 18 da., work- ing 10 hr. a day, how many cords can 40 men cut in 24 da., working 9 hr. a day ? 8. What is the cost of 36J yards of cloth IJ yd. wide, if 2J yards If yd. wide, cost $3.37i ? 9. A contractor employs 45 men to complete a work in 3 months ; what additional number of men must he employ, to complete the work in 2^ months ? 10. If a vat 16 ft. long, 7 ft. wide, and 15 ft. deep holds 384 barrels, how many barrels will a yat 17|- ft. long, 10|^ ft. wide, and 13 ft. deep hold ? 11. What is the weight of a block of granite 8 ft. long, 4 ft. wide, and 10 in. thick, if a similar block 10 ft. long, 5 ft. wide, and 16 in. thick, weigh 5200 pounds ? 12. If it cost $15 to carry 20 tons 1^ miles, what will it cost to carry 400 tons |- of a mile ? 13. If it take 13500 bricks to build a wall 200 ft. long, 20 ft. high, and 16 in. thick, each brick being 8 in. long, 4 in. wide, and 2 in. thick, how many bricks 10 in. long, 5 in. wide, 3| in. thick, will be required to build a wall 600 ft. long, 24 ft. high, and 20 ft. thick ? 14. What will 15 hogsheads of molasses cost, if 28| gallons cost $7 J? 15. At 6^6.. for If yards of cotton cloth, how many yards can be bought for £10 6s. 8d. ? 16. If $750 gain $202.50 in 4 yr. 6 mo., what sum will gain $155.52 in 1 vr. 6 mo. ? 400 PROPORTIOK. 17. In what time can 60 men do a piece of work that 15 men can do in 20 days ? 18. If 2|- yd. of cloth 6 quarters wide can be made from 1 lb. 12 oz. of wool, how many yards of cloth 4 quarters wide can be made from 70 lb. of wool ? 19. If the use of $300 for 1 yr. 8 mo. is worth $30, how long, at the same rate, may $210.25 be retained to be worth $42,891? 20. A farmer has hay worth $18 a ton, and a merchant has flour worth $10 a barrel. If the farmer ask $21 for his hay, what should the merchant ask for his flour? 21. How many men will be required to dig a cellar 45 ft. long, 34.6 ft. wide, and 12.3 ft. deep, in 12 da. of 8.2 hr. each, if 6 men can dig a similar one 22.5 ft. long, 17.3 ft. wide, and 10.25 ft. deep, in 3 da. of 10.25 hr. each ? 22. If a bin 8 ft. long, 4^ ft. wide, and 2^ ft. deep, hold 67^ bu., how deep must another bin be made, that is 18 ft. long and 3f ft. wide, to hold 450 bu. ? 23. What will 120 lb. of coffee cost, if 10 lb. of sugar cost $1.25, and 16 lb. of sugar are worth 5 lb. of coffee? 24. Two men have each a farm. A's farm is worth $48.75, and B's $43^; but in trading A values his at $60 an acre. What value should B put upon his ? 25. If 6 men in 4 mo., working 26 da. for a month, and 12 hr. a day, can set the type for 24 books of 300 pp. each, 60 lines to the page, 12 words to the line, and an average of 6 letters to the word, in how many months of 24 da. each, and 10 hr. a day, can 8 men and 4 boys set the type for 10 books of 240 pp. each, 52 lines to the page, 16 words to the line, and 8 letters to the word, 2 boys doing as much as 1 man ? OliAL EXJSItCISES, 775. 1. If John has 10 marbles, William 15 marbles, and Charles 25 marbles, what part of the whole has each ? 2. Two men bought a barrel of flour for $9, the first paying $4 and the second $5. What part of the flour belongs to each ? 3. Three men bought 108 sheep, and as often as the first paid $3, the second paid $4, and the third $5. How many sheep should each receive ? 4. If $45 be divided between two persons, so that of every $5, one receives $2, and the other $3, how many dollars does each receive ? 5. Two men hired a pasture for $36 ; one put in 2 horses for 3 weeks, the other 3 horses for 4 weeks. What should each pay? DEFINITIONS. 776. Partnership is the association of two or more persons under a certain name, for the transaction of busi- ness with an agreement to share the gains and losses. 777. A Flrm^ Company or House is any par- ticular partnership association. 778. The Capital is the money or property invested by the partners, called also Investment y or Joint- Stock. 402 PARTNERSHIP. 7*79. The JResources of a firm are the amounts due it, together with the property of all kinds belonging to it; called also Assets^ or Effects, 780. The Liabilities of a firm are its debts. *781. The Net Capital is the excess of resources oyer liabilities. WJRITTEN EXERCISES. 783. To apportion gains or losses according to capital invested. 1. A and B engage in trade ; A furnishes $400 capital, B $600. They gain $250 ; what is the profit of each? 1st operation^. {By Fractions) $400, A/s investment = -^^^^^ = f of the whole capital. 600, B/s - = T%V^ = f " $1000, whole " $250 X f = $100, A/s share of the gain. $250 X I = $150, B/s " 2d operation. {By Proportion) $1000 (whole cap.) : $400 (A.'s inv.) : : $250 (whole gam) : A.*s share. $1000 (whole cap.) : $600 (B.'s inv.) : : $250 (whole gain) : B/s share. 3d operation. {By Percentage.) $250 gain is y^^^iF = 35% of the whole capital. $400 X ,25 = $100, A.'s gain ; $600 x .25 = $150, B.'s gain. Analysis.— (7^^ Method) Since $400, A.'s investment, is xVA* or I, of the whole capital, he is entitled to | of the gain, or $100 ; and B is entitled to f of the gain, or $150. M Method. The ratio of $1000, the whole capital, to $400, A.'s investment, is equal to the ratio of $250, the whole gain, to A.'s share of the gain. Hence the proportions, etc. Sd Method. Since the gain is 25% of the whole capital, each partner is entitled to 25 % of his investment as his share of the gain. The third method (by dividend) is that generally adopted by joint* stock companies having numerous shareholders. PARTNEBSHIP. 403 2. At the end of the year, Norton, Smith & Co. take an account of stock, and find the amount of merchandise, as per inventory, to be $8400 ; cash on hand, $4850 ; due from sundry persons, $5273. Their debts are found to amount to $4223. S. Norton's inyestment in the busi- ness is $5000 ; E. Smith's, $4000 ; and 0. Woodward's, $2000. Make a statement showing the resources, lia- bilities, net capital, and net gain : and find each part- ner's share of the gain. OPERATION. Itesources. Mdse. as per inventory, $8400 Cash on hand, 4850 Debts due the firm, ...... 5273 $18523 Ziiabilities. Debts due to sundry persons, 4223 Net capital, .... $14300 Investments. S. Norton, $5000 R. Smith, 4000 C. Woodward, 2000 Total investments, $11000 Net gain, $3300 8. Norton's fractional part, yVinnr = rr of $3300 = $1500 R. Smith's " « ^oooj = 3^ of $3300 = $1200 C. Woodward's " " y\oop^ = ^a. of $3300 = $ 600 Proof.— $1500 + $1200 + $600 ~ $3300, total gain. 404 PARTNERSHIP. KuLE 1. Fmd loliat fractional part each partner^s in- vestment is of the whole capital, and take such part of the whole gain or loss for his share of the gain or loss. Or, 2. State by proportion, as the whole capital is to each partners investment, respectively, so is the whole gain or loss to each partner^s share of the gain or loss. Or, 3. Find what per cent, the gain or loss is of the tvhole capital, and take that per cent, of each partner's invest- mentfor his share of the gain or loss, respectively. 3. A furnishes $4000, B, $2700, and 0, $2300, to pur- chase a house, which they rent for $720. What is each one's share of the rent ? 4. Four persons rent a farm of 230 A. 64 P. at $7J an acre. A puts in 288 sheep, B, 320 sheep, C, 384 sheep, and D, 648 sheep ; what rent ought each to pay? 5. Prime & Co. fail in business ; their liabilities amount to $22000 ; their available resources to $8800. They ow^ A $4275, and B $2175.50 : what wiU each of these creditors receive? 6. Pour persons engage in manufacturing, and invest jointly $22500. At the expiration of a certain time, A's share of the gain is $2000, B's $2800.75, C's $1685.25, and D's $1014. How much capital did each put in ? 7. An estate worth $10927.60 is divided between two heirs so that one receives \ more than the other. What does each receive ? 8. Three persons engage in the lumber trade with a joint capital of $37680. A puts in $6 as often as B puts in $10, and as often as C puts in $14. Their annual gain is equal to C's stock. What is each partner's gain? PARTNERSHIP. 405 9. Ames, Lyon & Co. close business in the following condition : notes due the firm to the amount of $24843.75, cash in hand, $42375.80, due on account, $26500, mer- chandise per inventory, $175840. Notes against the firm, $14058.75, due from the firm on account, $12375.80. Ames invested $60000, Lyon, $40000, and Clark $25000c Make a statement showing the total amount of resources, liabilities, investments, net capital, net gain, and each partner's share of the gain. 783. To apportion ^ains or losses according to amount of capital invested, and time it is employed. 1. Three partners, A, B, and C, furnish capital as fol- lows : A, $500 for 2 mo. ; B, $400 for 3 mo. ; C, $200 for 4 mo. They gain $600 ; what is each partner's share ? OPERATION. 500 X 2 = 1000 = UU = J X ) ( $200, A^s share. 400 X 8 = 1300 = iiU = I X /• $600 = •< $240, B's '^ 200 X 4 = ^ = ^^00^ = Ax ) ( $160, Cb " 3000 Analysis. — The use of $500 for 2 mo. is the same as the use of $1000 for 1 mo. ; the use of $400 for 3 mo. is the same as that of $1200 for 1 mo. ; and the use of $200 for 4 mo. is the same as that of $800 for 1 mo. Therefore the whole capital is the use of $3000 for 1 mo. ; and as A's investment is $1000 for 1 mo., it is J of the capital, and hence he should receive i of the gain, or $200. For the same reason, B should receive f, and C y*^ of the gain, or $240 and $160, respectively. The other methods of operation may be applied by considering the products of investment and time as shares of the capital. Thus, $600 is 20% of $3000; and 20% of $1000, $1200, and $800 will give $200, $240, and $160, respectively, the sharee of gain required. 406 PABTI^EESHIP. 2. Barr, Banks & Co. gain in trade $8000. Barr fur- nished $12000 for 6 mo., Banks, $10000 for 8 mo., and Butts $8000 for 11 mo. Apportion the gain. EuLE 1. Multiply each partners capital ly the time it is invested, and divide the tvhole gain or loss among the partners in the ratio of these products. Or, 2. State by proportion : The sum of the products is to each product, as the tvhole gain or loss is to each partners gain or loss. 3. Jan. 1, 1876, three persons began business with $1300 capital furnished by A ; March 1, B put in $1000 ; Aug. 1, C put in $900. The profits at the end of the year were $750. Apportion it. 4. In a partnership for 2 years, A furnished at first $2000, and 10 mo. after withdrew $400 for 4 mo., and then returned it ; B at first put in $3000, and at tlie end of 4 mo. $500 more, but drew out $1500 at the end of 16 mo. The whole gain was $3372, Find the share of each. 5. The joint capital of a company was $5400, which was doubled at the end of the year. A put in | for 9 mo., B I for 6 mo., and C the remainder for 1 year. What is each one's share of the stock at the end of the year? 6. Crane, Child & Coe, forming a partnership Jan. 1, 1875, invested and drew out as follows : Crane invested $2000, 4 mo. after $1000 more, and at the end of 9 mOo irew out $600, Child invested $5000, 6 mo. after $1200 more, and at the end of 11 mo. put in $2000 more. Coe put in $6000, 4 mo. after drew out $4000, and at the end of 8 mo. drew out $1000 more. The net profits foi the year were $7570. Find the share of each» 784. Alliffation treats of mixing or compounding two or more ingredients of different values or qualities. 785. Alligation Medial is the process of finding the mean or average value or quality of several ingredients. 786. Alligatioit Alter^nate is the process of find- ing the proportional quantities to be used in any required mixture. WRITTEN EXAMPLES, 787. 1. If a grocer mix 8 lb. of tea worth $.60 a pound with 6 lb. at $.70, 2 lb. at $1.10, and 4 lb. at $1.20, what is 1 lb. of the mixture worth ? OPERATION. Analysis. — Since 8 lb. of tea at $.60 is $.60 X 8 =i: $4.80 worth $4.80, and 6 lb. at $.70 is worth 70 X 6 = 4 20 ^'^•^^' ^^^ ^ ^^* ^* ^^'^^ ^^ ^^^*^ $2.20, and 4 lb. at $1.20 is worth $4.80, the mix- 1.10 X 2 =z 2.20 ^^^g Qf 20 lb. is worth $16. Hence 1 lb. is 1.20 X^ = 4.8 worth ^^ of $16, or $16 -j- 20 = $.80. 20 ) $16. 00 2. If 20 lb. of sugar at 8 cents be mixed with 24 lb. at 9 cents, and 32 lb. at 11 cents, and the mixture is sold at 10 cents a pound, what is the gain or loss on the whole ? EuLE. — Find the entire cost or value of the ingredients^ and divide it ly the sum of the simples. 408 ALLIGATION. 3. A miller mixes 18 bu. of wheat at $1.44 with 6 bu. at $1.32, 6 bu. at $1.08, and 12 bu. at $.84. What will be his gain per bushel if he sell the mixture at $1.50 ? 4. Bought 24 cheeses, each weighing 25 lb., at 7)^ a pound; 10, weighing 40 lb. each, at 10^; and 4, weigh- ing 50 lb. each, at ISf; sold the whole at an average price of 9|^' a pound. What was the whole gain ? 5. A droyer bought 84 sheep at $5 a head ; 86 at $4.75 ; and 130 at $5|-. At what avera^^e price per head must he sell them to gain 20^? 788. To find the proportional parts to be used, when the mean price of a mixture and the prices of the simples are given. 1. What relative quantities of timothy seed worth $2 a bushel, and clover seed worth $7 a bushel, must be used fco form a mixture worth $5 a bushel ? OPERATION. Analysis. — Since on every ingredient used ^2 ^12) whose price or quality is less than the mean i ly 1 I Q i ^^^* rate there win be a gain, and on every ingre- ^21 ; dient whose price or quality is greater than the mean rate there will be a loss, and since the gains and losses must be exactly equal, the relative quantities used of each should be such as represent the unit of value. By selling one bushel of timothy seed worth $2, for $5, there is a gain of $3 ; and to gain $1 would require J of a bushel, which is placed opposite the 2. By selling one bushel of clover seed worth $7, for $5, there is a loss of $2 ; and to lose $1 would require I of a bushel, which is placed opposite the 7. In every case, to find the unit of value, divide $1 by the gain or loss per bushel or pound, etc. Hence, if every time J of a bushel of timothy seed is taken, J of a bushel of clover seed is taken, the gain and loss will be exactly equal, and ^ and J will be the propor- tional quantities required. ALLIGATION. 409 1 3 3 4 5 3 i 4 4 ■ 4 i 1 1 7 1 2 2 .10 i 3 3 To express the proportional numbers in integers, reduce these fractions to a common denominator, and use their numerators, since fractions having a common denominator are to each other as their numerators (241); thus, ^ and | are equal to | and f, and the pro- portional quantities are 2 bu. of timothy seed to 3 bu. of clover seed, 2. What proportions of teas worth respectively 3, 4, 7. and 10 shillings a pound, must be taken to form a mix- ture worth 6 shillings a pound ? OPERATION. Analysis. — To preserve the equality of gains and losses, always compare two prices or simples, one greater and one less than the mean rate, and treat each pair or couplet as a separate ex- ample. In the given example form two couplets, and compare either 3 and 10, 4 and 7, or 3 and 7, 4 and 10. We find that | of a lb. at 3s. must be taken to gala 1 shilling, and :|^ of a lb. at lOs. to lose 1 shilling ; also ^ of a lb. at 4s. to gain 1 shilling, and 1 lb. at 7s. to lose 1 shilling. These proportional numbers, obtained by comparing the two couplets, are placed in columns 1 and 2. If, now, the numbers in columns 1 and 2 are reduced to a common de- nominator, and their numerators used, the integral numbers in columns 3 and 4 are obtained, which, being arranged in column 5, give the proportional quantities to be taken of each. It will be seen that in comparing the simples of any couplet, one of which is greater, and the other less than the mean rate, the pro- portional number finally obtained for either term is the difference between the mean rate and the other term. Thus, in comparing 3 and 10, the proportional number of the former is 4, which is the difference between 10 and the mean rate 6 ; and the proportional number of the latter is 3, which is the difference between 3 and the mean rate. The same is true of every other couplet. Hence, when the simples and the mean rate are integers, the intermediate steps taken to obtain the final proportional numbers as in columns 1, 2, 3, and 4, may be omitted, and the same results readily found by taking the difference between each simple and the mean rate, and placing it opposite the one with which it is compared. 410 A L LI GAT I 01^. 3. In what proportions must sugars worth 10 cents, 11 cents, and 14 cents a pound be used, to form a mix- ture worth 12 cents a pound ? 4. A farmer has sheep worth $4, $5, $6, and $8 per bead. What number may he sell of each and realize an average price of $6^ per head ? EuLE. — I. Write the several prices or qualities in a column, and the mean price or quality of the mixture at the left, II. Form couplets ly comparing any price or quality less, with one that is greater than the mean rate, placing the part lohich must be used to gain 1 of the mean rate opposite the less simple, and the part that must he used to lose 1 opjjosite the greater simple, and do the same for each irimple in every couplet, III. If the propiortional numbers are fractional, they may be reduced to integers, and if two or more stand in ilie same horizontal line, they must be added; the final re- sults will be the proportional quantities required* 1. If the numbers in any couplet or column have a common fac- tor, it may be rejected. 2. We may also multiply the numbers in any couplet or column by any multiplier we choose, without affecting the equality of the gains and losses, and thus obtain an indefinite number of results, ^ny one of which being taken will give a correct final result. 5. What amount of flour worth $6^, $6, and S7|- per barrel, must be sold to realize an average price of %6^ pel barrel ? 6. In what proportions can wine worth $1.20, $lo80, and $2.30 per gallon be mixed with water so as to form a mixture worth $1.50 per gallon ? ALLIGATION. 411 789. When the quantity of one of the simples is limited. 1. A farmer has oats worth $.30, corn worth $.45, and barley worth $.84 a bushel. To make a mixture worth $.60 a bushel, and which shall contain 48 bu. of corn, how many bushels of oats and barley must he use ? 60^ '30 ^ 4 4 34^ 45 ^ 8 8 48 - .84 ^ A 5 5 10 60^ Ans. OPERATION. Analysis. — By the same process as in (788),the proportional quantities of each are found to be 4 bu. of oats, 8 of corn, and 10 of barley. But since 48 bu. of corn is 6 times the proportional num- ber 8, to preserve the equality of gain and loss take 6 times the proportional quantity of each of the other simples, or 6 x 4 = 24 bu. of oats, and 6 x 10 = 60 bu. of barley. Hence, etc. 2. A dairyman bought 10 cows at |40 a head. How many must he buy at $32, $36, and $48 a head, so that the whole may average $44 a head ? EuLE. — Find the proportional quantities as in (788) • Divide the giveti quantity hy the proportional quantity of the same ingredient, and multiply each of the other propor- tional quantities hy the quotient thus obtained. 3. A grocer having teas worth $.80, $1.20, $1.50, and $1.80 per pound, wishes to form a mixture worth $1.60 a pound, and use 20 lb. of that worth $1.50 a pound. 4. Bought 12 yd. of cloth for $30. How many yards must I buy at %Z\ and $1|^ a yard, that the average price of the whole may be $2| a yard ? 5. How many acres of land worth $70 an acre must be added to a farm of 75 A., worth $100 an acre, that the average value may be $80 an acre ? 413 ALLIGATION. OPJilKATlOH. r ^ i 3 3 27 10 « 7 13 i 2 3 2 3 18 27 .13 i 4 4 12 36 108 790. When the quantity of the whole compound is limited* 1. A grocer has sugars wortti 6 cents, 7 cents, 12 cents, and 13 cents per pound. He wishes to make a mixture of 108 pounds, worth 10 cents a pound; how many pounds of each kind must he use ? Analysis. — The proportion- al quantities of each simple found hy (788), are 3 lb. at 6 cts., 2 lb. at 7 cts., 3 lb. at 12 cts., and 4 lb. at 13 cts. Add- ing the proportional quantities, the mixture is but 12 lb., while the required mixture is 108, or 9 times 12. If the whole mixture is to be 9 times as much as the sum of the propor- tional quantities, then the quantity of each simple used must be 9 times as much as its respective proportional, or 27 lb. at 6 cts., 18 lb. at 7 cts., 27 lb. at 12 cts., and 36 lb. at 13 cts. 2. A man paid $330 per week to 55 laborers, consisting of men, women, and boys; to the men he paid $10 a week, to the women $2 a week, and to the boys $1 a week ; how many were there of each ? KuLE. — Find the proportional numlers as in (788). Divide the given quantity iy the sum of the proportional quantities, and multiply each of the proportional quanti- ties hy the quotient thus obtained. 3. How much water must be mijed with wine wortli $.90 a gallon, to make 100 gal. worth $.60 a gallon ? 4. One man and 3 boys received $84 for 56 days' labor ; the man receiyed $3 per day, ana the boys $-|, $J, and $1| respectively ; how many days did each labor ? BBVIEW. 413 •791. SYNOPSIS FOR REVIEW. RATIO. 1. Ratio. 2. Sign of Ratio. 3. Terms. 4. Antecedent. 5. Consequent. 6, 1. Deps. -j Value of a Ratio. 7. Simple Ratio. 8. Compound Ratio. 9. Reciprocal of a Ratio. 2. Formulas, 1, 2, 3. 3. General Principles, 1, 2, 3. 4. General Law. 1. Defs. - 1. Proportion. 2. Sign. 3. Couplet. 4. Proportional. 5. Antecedents. 6. Consequents. 7. Extremes. 8. Means. 9. Mean Proportional. 2. Principles, 1, 2, 3, 4 PROPOR. TION. 3. Simple Pro- portion. 1 Defs Jl- ^^^P^^-P^^i>^^'' what is the sum of all the terms ? 5. What is the sum of all the terms of the infinite pro gression 8, 4, 2, 1, J, },.... ? The last term of this progression may be conceived as 0. 6. What is the sum of all the terms of the infinite pro- gression 1, i, \, j\, ij, . . , .? 7. What is the sum of 1 + i + i + h ^^^-^ ^^ infinity? 8. The first is 7, the ratio 3, and the number of terms 4 ; what is the sum of all the terms ? First find the last term by Art. 840. 9. A drover bought 10 cows, agreeing to pay $1 for the first, $2 for the second, $4 for the third, and so on; wlrit did he pay for the 10 cows ? 10. If a man were to buy 12 horses, paying 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him ? 844. An Annuity is a sum of money payable an- nually. The term is also applied to a sum of money payable at any equal intervals of time. 845. A Certain Annuity is one which continues for a definite period of time. 846. A Perpetual Annuity or Perpetuity is one which continues forever. 847. A Contingent Annuity is one which begins or ends, or both begins and ends, on the occurrence of some specified future event or events. 848. An Annuity Forborne or in Arrears is one the payments of which were not made when due. 849. The Amount or Final Value of an an- nuity is the sum of all the payments increased by the interest of each payment from the time it becomes due until the annuity ceases. 850. The Present Worth of an annuity is such a sum of money as will, in the given time, and at the given rate per cent., amount to the final value. 851. An annuity is said to be deferred when it does not begin until after a certain period of time ; it is said to be reversionary when it does not begin until after the occurrence of some specified future event, as the death of a certain person; and it is said to be iv, possession when it has begun, or begins immediately. 450 AKKUITIES. ANNUITIES AT SIMPLE INTEREST. 853. All problems in annuities at simple interest may be solved by combining the rules in Arithmetical Pro- gression with those in Simple Interest. WniTTEN JEXJEJJtC IS ES, 853. 1. What is the amount of an annuity of $300 for 5 years, at 6 per cent, simple interest ? OPERATION. 300 + 372 „ -innrx ANALYSIS. — At the end of the 5th ^ ^ x5 = 1680 ^1 i. n • A 2 year the following sums were due : The 5th year's payment = $300, The 4th year's payment = $300 + $18 = $318, Tlie 3d year's payment = $300 + $36 = $336, The 2d year's payment = $300 + $54 = $354, The 1st year's payment = $300 + $72 = $372. These sums form an arithmetical progression, in which the first term is the annuity, $300, the common difference is the interest of the annuity for 1 year, and the number of terms is the number of years. The sum of all the terms of this progression is $1680 (832), which is the amount of the annuity. 2. A father deposits annually for ihe benefit of his son, beginning with his tenth birthday, such a sum that on his 21st birthday the first deposit, at simple int., amounts to $210, and the sum due his son is $1860. Find the annual deposit, and at what rate per cent, it is deposited. OPERATION. G X (1st term + 210) = 1860. (833.) Hence, 1st term = 310 — 210 = 100 = a. (210 - 100) ~ (12 - 1) =: -Vf =2lO = d. (830.) Akalysis.— Here $210, the first deposit, is the last term ; 12, the number of deposits, is the number of terms ; AlTl^trlTIEg. 451 and $1860, the final value of the annuity, is the sum of all the terms. Using the principle of 83^, we find the first term to be $100, which is the annual deposit. By 830, the common dif- ference is found to be $10 ; hence 10 per cent, is the required rate. 3. What is the amount of an annuity of $150 for 5| ysars, payable quarterly^ at 1^ per cent, per quarter ? 4. What is the present worth of an annuity of $300 for 5 years, at 6 per cent. ? 5. What is the present worth of an annuity of $500 for 10 years, at 10 per cent. ? 6. In what time will an annual pension of 8500 amount to $3450, at 6 per cent, simple interest ? 7. Find the rate per cent, at which an annuity of $6000 will amount to $59760 in 8 years, at simple interest. 8. A man works for a farmer 1 yr. 6 mo., at $20 per month, payable monthly ; and these wages remain unpaid until the expiration of the whole term of service. What is due the workman, allowing simple interest at 6 per cent, per annum ? ANNUITIES AT COMPOUND INTEREST. 854. All problems in annuities at compound interest may be solved by combining the rules in Geometrical Progression with those in Compound Interest. WniTTEN MXERC 1 SES. 1. What is the amount of an annuity of $300 for 5 years, at 6 per cent, compound interest ? OPERATION. Analysis.— At the end of the 300 X 1.06^— -300 _ 1 ^^1 . o 5th year the following sums .06 * ai^ ^^ $1263.71. 1 .OOO^/^O 3. Find the annuity whose amount for 25 years, at 6 per cent, compound interest, is $16459.35. 4. What is the present worth of an annuity of $700 for 7 years, at 6 per cent, compound interest ? 5. An annuity of $200 for 12 years is in reversion 6 years. What is its present worth, compound interest at 6^? 6. A man bought a tract of land for $4800, which was to be paid in installments of $600 a year ; how much money, at 6 per cent, compound interest, would discharge the debt at the time of the purchase ? 7. What is the present value of a reversionary lease of 1100, commencing 14 years hence, and to continue 20 years, compound interest at 5 per cent. ? E E V I E w . 453 855. SYNOPSIS FOE REVIEW. j 1. A Power. 2. Involution. 3. Base, or Root. 4. Ex- ^ 1. Defs. I ponent. 5. Square. 6. Cube. 7. Perfect Power. 2. Principle. 3. 802. Rule. 1. For Integers. 2. For Fractions. 4. 803. 1. Principle. 2. Geometrical Illustration. 5. 804:. 1. Principle. 2. Geometrical Illustration. j 1. Square Root. 2. Cube Root, etc. 3. Evolution ' 1. I>EFS. I ^ Radical Sign. 5. Index. 2. 810. Rule. 3. 812. Principles, 1, 2, 3, 4. 4. 813. Rule, I, II, III. For Fractions. 5. 814. Geometrical Illustration. 6. 818. Principles, 1, 2, 3, 4. 7. 819. Rule, I, II, III, IV, V, VI. For Fractions. 8. 820. Geometrical Illustration. 9. 822. Roots of a Higher Degree. Rule. r 1. Arithmetical Progression. 2. Terms. 3. Common ^ 1. Defs. ■< Difference. 4. Increasing Arithmetical Progression. ( 5. Decreasing Arithmetical Progression. 2. Quantities considered. 3. 829. Rule, I, II. Formulm. 4. 830. Rule. Formula. 5. 831. Rule. Formula. 6. 832. Rule. Formula. ( 1. Geometrical Progression. 2. Terms. 3. Ratto, 1. Defs. ■< 4. Increasing Oeom. Prog. 5. Decreasing Geom, ( Pr(?^. 6. Infinite Decrea>dng Geom. Prog. 2. Quantities considered. 3. 840. Rule, I, II. FormvlcB. 4. 841. Rule. Formula. 5. 842. Rule. Formula. 6. 843. Rule. Formula. 1. Annuity. 2. Certain Annuity. 3. Perpetuity. 4. Contingent Annuity. 5. Annuity in Arrears. 1. Defs. \ 6. Amount. 7. Present Worth of an Annuity 8. Deferred Annuity. 9. Reversionary Annuity. 10. Annuity in Possession. 2. Annuities at Simple Interest. ) 3. Annuities at Comp. Interest. \ P^^^^ms, how solved. 856. Mensuration is the process of findiDg the numher oi units in extension. LINES. 857. A Straight Line is a line that ■ does not change its direction. It is the short- est distance between two points. 858. A Curved lAne changes its direc- tion at every point. 859* Parallel lAnes have the same direction ; and being in the same plane and equally distant from each other, they can never meet. 860* A Horizontal Line is a line par- allel either to the horizon or water level. 861. A Perpendicular Line is a :g straight line drawn to meet another straight > line, so as to incline no more to the one side HorizontaL than to the other. A perpendicular to a horizontal line is called a tiertir co/line. ANGLES. 862. An Angle is the difference in the direction of two lines proceeding from a com- mon point, called the wrtex. Angles are measured by degrees. (301.) 863. A Itight Angle is an angle formed by two lines perpendicular to each other. 864. An Obttise Angle is greater than a right angle. 865. An Acute Angle is less than a right angle. All angles except right angles are called oUiqiie angles r V. TBI AKGLES. 455 PLANE FIGURES. 866. A Plane Figure is a portion of a plane surface bouifted by straight or curved lines. 867. A JFolygon is a plane figure bounded by straight lines. 868. The Perimeter of a polygon is the sum of its sides. 869. The Area of a plane figure is the surface included within the lines which bound it. (460.) A regular polygon has all its sides and all its angles equal. The attitude of a polygon is the perpendicular distance between its base and n side or angle opposite. A polygon of three sides is called a tngon^ or triangle ; of four sides, a tetra- gon^ or quadrilateral ; of five sides, 2l pentagon^ etc. Pentagon. Hexagon. Heptagon. Octagon. Nonagon. Decagon. TRIANGLES. 870. A Triangle is a plane figure bounded by three sides, and having three angles. 871. A KighUAngled Triangle is a triangle having one right angle. 872. The JT//pof/*enie.S6 of a right- angled triangle is the side opposite the right angle. 873. The JBase of a triangle, or of any plane figure, is the side on which it is supposed to stand. 874. The Perpendicular of a right-angled triangle is the side which forms a right angle with the base*. 875. The Altitude of a triangle is aline drawn from the angle opposite perpendicular to the base. 1. The dotted lines in the following figures represent the altitvde, 2. Triangles are named from the relation both of their eides and angles. Base. 456 MBKSURATION, 8 7 6, An Equilateral Triangle has its three sides equal. 877. An Isosceles Triangle has only two of its sides equal, ^8. A Scalene Triangle has all of its sides unequal Fig. 1. Fig. 2. Fig. 3. Equilateral. Isosceles. Scalene. 879. An Equiangular Triangle has three equal angles. (Fig. 1.) 880. An Acute-angled Triangle has three acute angles. (Fig. 2.) 881. An Obtuse-angled Triangle has one obtuse angle. (Fig. 3.) mOBZJEMS. 882. The base and altitude of a triangle being given to find its area. 1. Find the area of a triangle whose base is 26 ft. and altitude 14.5 feet. 14 5 Operation.— 14.5 x 26-^2=188^ sq. ft. Or, 26 x ---=188^ square feet, area, 2. What is the area of a triangle whose altitude is 10 yards and base 40 feet ? Rule. — 1. Divide the product of the base and altitude hy 2. Or, 2. Multiply the base hy one-half the altitude. Find the area of a triangle 3. Whose base is 12 ft. 6 in and altitude 6 ft. 9 in. 4. Whose base is 25.01 chains and altitude 18.14 chains. 5. What is the cost of a triangular piece of land whose base is 15.48 ch. and altitude 9.6? ch., at $60 an acre ? 6. At $.40 a square yard, find the cost of paving a triangular court, its base being 105 feet and its altitude 21 yards ? 7. Find the area of the gable end of a house that is 28 ft. wide, and the ridge of the roof 15 ft. higher than the foot of the rafters. TRIAl^GLES. 457 883. The area and one dimension being g^iven to find the other dimension. 1. What is the base of a triangle whose area is 189 square feet and altitude 14 feet ? Operation.— (189 sq. ft. x 2)-^ 14 r= 27 ft., lose. 2. Find the altitude of a triangle whose area is 20 J square feet and base 3 yards. Rule. — Double the area, then divide by the given dimension. Find the other dimension of the triangle 3. When the area is 65 sq. in. and the altitude 10 inches. 4. When the base is 42 rods and the area 588 sq. rods. 5. When the area is 6J acres and the altitude 17 yards. 6. When the base is 12.25 chains and the area 5 A. 33 P. 7. Paid $1050 for a piece of land in the form of a triangle, at the rate of $5J^ per square rod. If the base is 8 rd., what is itg altitude ? 884. The three sides of a triangle being given to find its area. 1. Find tbe area of a triangle whose sides are 30, 40, and 50 ft. Operation.— (30 -}- 40 + 50)-j-2 = 60 ; 60-30 = 30 ; 60-40 = 20 ; 60-50 = 10. ^60 X 30 X 20 X 10 =^ 600 ft., area. 2. What is the area of an isosceles triangle whose base is 20 ft., and each of its equal sides 15 feet? Rule. — From half the sum of the three side^, subtract each side separately ; multiply the half -sum and the three remainders together ; the square root of the product is the area. 3. Find the area of a triangle whose sides are 25, 36, and 49 in. 4. How many acres in a field in the form of an equilateral tri> angle whose sides each measure 70 rods ? 5. The roof of a house 30 ft. wide has the rafters on one sid^ 20 ft. long, and on the other 18 ft. long. How many square feet oi boards will be required to board up both gable ends ? 458 MEKSUKATIOK. 885. The following principles relating to rigfit-angled triangles have been established by Geometry : Principles.— 1. The square of the hypothenase of a right-angled triangle is equal to the sum of the squares of the other two sides, 2, TJie square of the base, or of the perpendicular, of a right-angled tri- angle is equal to the square of the hypothenuse diminished by the square of the other side, 886. To find the hypothenuse. 1. The base of a right-angled triangle is 12, and the perpendicular 16. What is the length of the hypothenuse ? Operation.— 12^ + 162=r400 (Prin. 1). ^^400=20, hypothenuse. 2. The foot of a ladder is 15 feet from the base of a building, and the top reaches a window 36 feet above the base. What is the length of the ladder ? Rule. — Extract the square root of the sum of the squares of the base and the perpendicular ; the result is the hypothenuse. 3. If the gable end of a house 40 ft. wide is 16 ft. high, what isi the length of the rafters ? 4. A park 25 chains long and 23 chains wide has a walk running through it from opposite corners in a straight line. What is the length of the walk ? 5. A room is 20 ft. long, 16 ft. wide, and 12 ft. high. What is the distance from one of the lower corners to the opposite upper corner ? 887. To find the base or perpendicular. 1. The hypothenuse of a right-angled triangle is 35 feet, and the perpendicular 28 fend to similar curved surfaces, as of cylinders, cones, and spheres. Hence, 4. The surfaces of all similar figures are to each other as the squares of their like dimensions. And conversely, 5. TJieir dimensions are as the square roots of their surfaces, monzjsMs, 1. A triangular field whose base is 13 ch. contains 2 A. 80 P. Find the area of a field of similar form whose base is 48 chains. OPBIIATION.--122 : 48^ : : 2 A. 80 P. : jc P.=6400 P.=40 A., area. (Prin. 2.) 2. The side of a square field containing 18 acres is 60 rods long. Find the side of a similar field that contains i as many acres. Operation.— 18 A. : 6 A. : : 60^: a;-=1200; ^/l2^=MMTd. + , side, (Prin. 3.) 3. Two circles are to each other as 9 to IG ; the diameter of the less being 112 feet, what is the diameter of the greater? Operation.— 9 : 13 : : 112^ : aj*^ = 3 : 4 : : 112 : ic = 149 ft. 4 in., diameter. (Prin. 2.) 4. A peach orchard contains 720 square rods, and its length is to its breadth as 5 to 4 ; what are its dimensions? Operation. — The area of a rectangle 5 by 4 equals 20 (898). 20 : 720 : : 52 : ic^ = 900 ; ^900 = 30 rd., length, 20 : 720 : : P : x' = 576 ; ^576 = 24 rd., toidth. 5. It is required to lay out 283 A. 107 P. of land in the form of a rectangle, so that the length shall be 3 times the width. Find the dimensions. 6. A pipe 1.5 in. in diameter fills a cistern in 5 hours ; find the diameter of a pipe that will fill the same cistern in 55 min. 6 sec. 7. The area of a triangle is 24276 sq. ft., and its sides in proportion to the numbers 13, 14, and 15, Find the length of its sides An feet. 466 MENSURATION. 8. If it cost $167.70 to enclose a circular pond containing 17 A. 110 P., what will it cost to enclose another ^ as large ? 9. If 63.39 rods of fence will enclose a circular field containing 2 acres, what length will enclose 8 acres in circular form ? REVIEW OF PLANE FIGURES. 91 !• 1. How much less will the fencing of 20 acres cost in the square form than in the form of a rectangle whose breadth is J the length, the price being $2.40 per rod ? 2. A house that is 50 feet long and 40 feet wide has a square or pyramidal roof, whose height is 15 ft. Find the length of a rafter reaching from a corner of the building to the vertex of the roof. 3. Find the diameter of a circular island containing 1} sq. miles. 4. What is the value of a farm, at $75 an acre, its form being a quadrilateral, with two of its opposite sides parallel, one 40 ch. and the other 22 ch. long, and the perpendicular distance between them 25 chains ? 5. Find the cost, at 18 cents a square foot, of paving a space in the form of a rhombus, the sides of which are 15 feet, and a per- pendicular drawn from one oblique angle will meet the opposite side 9 feet from the adjacent angle. 6. A goat is fastened to the top of a post 4 ft. high by a rope 50 ft, long. Find the area of the greatest circle over which he can graze. 7. How much larger is a square circumscribing a circle 40 rods in diameter, than a square inscribed in the same circle ? 8. What is the value of a piece of land in the form of a triangle, whose sides are 40, 48, and 54 rods, respectively, at the rate of $125 an acre ? 9. The radius of a circle is 5 feet ; find the diameter of another circle containing 4 times the area of the first. 10. How many acres in a semi-circular farm, whose radius is 100 rods ? 11. What must be the width of a walk extending around a gar- den 100 feet square, to occupy one-half the ground ? 12. An irregular piece of land, containing 540 A. 36 P. is ex- changed for a square piece of the same area ; find the length of one of its sides ? If divided into 42 equal squares, what is the length of the side of each 1 SOLIDS. 467 13. A field containing 15 A. is 30 rd. wide, and is a plane inclining in the direction of its length, one end being 120 ft. higher than the other. Find how many acres of horizontal surface it contains. 14. If a pipe 8 inches in diameter discharges 12 hogsheads of water in a certain time, what must be the diameter of a pipe which will discharge 48 hogsheads in the same time ? SOLIDS. 912. A Solid or Body has three dimensions, length, breadth, and thickness. The planes which bound it are called its faces^ and their intersections, its edges. 913. A Prism is a solid whose ends are equal and parallel, similar polygons, and its sides parallelograms. Prisms take their names from the form of their bases, as triangular^ quad' rangtUar, pentagonal, etc. 914. The Altitude of a prism is the perpendicular distance between its bases. 915. A Parallelopipedoii is a prism bounded by six parallelograms, the opposite ones being parallel. 916. A Cube is a parallel opipedon whose faces are all equal squares. 917. A Cylinder is a body bounded by a uniformly curved surface, its ends being equal and parallel circles. 1. A cylinder is conceived to he generated by the reyolution of » rectangle about one of its sides as an axis. 2. The line joining the centers of the bases, or ends, of the cylinder is its alti- tude^ or axis. Cube. Triangular Prism. Quadrangular Prism. Pentagonal Prism. Cylinder. 468 MEiiTSURATIO]^. ritO BJLEMS, 918. To find the convex surface of a prism or cylinder. 1. Find the area of the convex sur- face of a prism whose altitude is 7 ft., and its base a pentagon, each side of which is 4 feet. Operation. — 4 ft. x 5 = 20 ft., peri- meter, 20 ft. X 7=140 sq. ft., convex surface, 2. Find the area of the convex sur- face of a triangular prism, whose alti- tude is 8 1 feet, and the sides of its base 4, 5, and 6 feet, respectively. Operation. — 4 ft. + 5 ft. + 6 f t = 15 ft, perimeter, 15 ft. X 8i— 127j sq. ft., convex surf ace. 8. Find the area of the convex surface of a cylinder whose altitude is 2 ft. 5 in. and the circumference of its base 4 ft. 9 in. Operation.— 2 ft. 5 in. =29 in. ; 4 ft. 9 in. = 57 in. 57 in. X 29 = 1653 sq. in. = 11 sq. ft. 69 sq. inches, convex surface. Rule. — Multiply the perimeter of the base by the altitude. To find the entire surface, add the area of the bases or ends. 4. If a gate 8 ft. high and 6 ft. wide revolves upon a point in its center, what is the entire surface of the cylinder described by it ? 5. Find the superficial contents, or entire surface of a parallelo- pipedon 8 ft. 9 in. long, 4 ft. 8 in. wide, and 3 ft. 3 in. high. 6. What is the entire surface of a cylinder formed by the revo- lution about one of its sides of a rectangle that is 6 ft. 6 in. long and 4 ft. wide ? 7. Find the entire surface of a prism whose base is an equilateral triangle, the perimeter being 18 ft., and the altitude 15 ft. PYRAMIDS AKD C 1^ E S < 469 919, To find the volume of any prism or cylinder. 1. Find the volume of a triangular prism, whose altitude is 20 ft., and each side of the base 4 feet. Operation.— The area of the base is 6.928 sq. ft. (882). 6.928 sqc ft. x 20 = 138.56 cu. ft, wlume. 2. Find the volume of a cylinder whose altitude is 8 ft. 6 in., and the diameter of its base 3 feet. Opekation.— 32 X .7854=7.0686 square feet, area of base (905). 7.0686 sq. ft. x 8.5 = 60.083 cubic feet, wlume. Rule. — Multiply the areoi of the base by the altitude. 3. Find the solid contents of a cube whose edges are 6 ft. 6 in. 4. Find the cost of a piece of timber 18 in. square and 40 ft. long, at $.30 a cubic foot. 5. Required the solid contents of a cylinder whose altitude is 15 ft. and its radius 1 ft. 3 in. 6. What is the value of a log 24 ft. long, of the average circum- ference of 7.9 ft., at $.45 a cubic foot ? PYRAMIDS AND CONES. 920. A Pyramid is a body having for its base a polygon, and for its other faces three or more triangles, which terminate in a common point called the 'vertex. Pyramids, like prisms, take their names from their bases, and are called tri- angular, square^ or quadrangular^ pentagonal, etc. Pyramid. Frustum. Cone. Frustum. 921. A Cone is a body having a circular base, and whose con- vex surface tapers uniformly to the vertex. It is a body conceived to be formed by the revolution of a right-angled triangle about one of its sides containing the right angle, as an immovable axis. 922. The Altitude of a pyramid or of a cone is the perpendic- ular distance from its vertex to the piano of its base. 470 MENSURATIOK. 923. The Slant Height of a pyramid is the perpendicular distance from its vertex to one of the sides of the base ; of a cone, is a straight line from the vertex to the cii-curaference of the base. 924. The Jb^rustuni of a pyramid or of a cone is that part which remains after cutting off the top by a plane parallel to the base. riCO B JLBMS , 925. To find the convex surface of a pyramid or of a cone, 1. Find the convex surface of a triangular pyramid, the slant height being 16 ft. , and each side of the base 5 feet. Operation.— (5 ft. + 5 ft. + 5 ft.) x 16-5-2 = 120 sq. ft., conv. mrf. 2. Find the convex surface of a cone whose diameter is 17 ft. 6 in., and the slant height 30 feet. UtjIjE.^— Multiply the perimeter or circumference of the base by one-half the slant height. To find the entire surface, add to this product the area of the base. 3. Find the entire surface of a square pyramid whose base is 8 ft. 6 in. square, and its slant height 21 feet. 4. Find the entire surface of a cone the diameter of whose base is 6 ft. 9 in. and the slant height 45 ft. G. Find the cost of painting a church spire, at $.25 a sq. yd., whose base is a hexagon 5 ft. on each side, and the slant height 60 feet. 926. To find the volume of a pyramid or of a cone. 1. What is the volume, or solid contents, of a square pyramid whose base is 6 feet on each side, and its altitude 12 feet ? Operation. — x6x 12-^3 — 144 cu. ft., xolume. 2. Find the volume of a cone, the diameter of whose base is 5 ft. and its altitude 10^ feet ? Operation.— (52 ft. x .7854) x ioFTs = 68.72J cu. ft., wlume. Rule. — Multiply the area of the base by one-third the altitude. 3. Find the solid contents of a cone whose altitude is 24 ft., and ^he diameter of its base 30 inches. 4. What is the cost of a triang^ular pyramid of marble, whose altitude is 9 ft., each side of the base being 3 ft., at $2 J per cu. foot ? 5. Find the volume and the entire surface of a pyramid whose base is a rectangle 80 feet by 60 feet, and the edges which meet at the vertex are 130 feet. PYRAMIDS AKD COKES. 471 927. To find the convex surface of a frustum of a pyramid or of a cone. 1. What is the convex surface of a frustum of a square pyramid, whose slant height is 7 feet, each side of the greater base 4 feet, and of the less base 18 inches ? Operation.' — The perimeter of the greater base is 16 ft, of th€ less 6 feet. 16 ft, + 6 ft. X 7 -H 2 = 77 sq. ft., convex surface. 2. Find the convex surface of a frustum of a cone whose slant height is 15 feet, the circumference of the lower base 30 feet, and of the upper base 16 feet. Rule. — Multiply the sum of the perimeters ^ or of the circumfer- ences, ly one-half the slant height. To find the entire surface, add to this product the area of both ends, or bases. 8. How many square yards in the convex surface of a frustum of a pyramid, whose bases are heptagons, each side of the lower base being 8 feet, and of the upper base 4 feet, and the slant height 55 feet? 928. To find the volume of a frustum of a pyramid or of a cone. 1. Find the volume of the frustum of a square pyramid whose altitude is 10 feet, each side of the lower base 12 feet, and of the upper base 9 feet. OPEKATION.—122 + 9^ = 225 ; (225+ /y/144 x 8l) x ioT3=1110 cu. feet, volume. 2. How many cubic feet in the frustum of a cone whose altitude is 6 feet, and the diameters of its bases 4 feet and 3 feet ? EuLE. — To the sum of the areas of loth bases add the square root of the product y and multiply this sum hy one-third of the altitude. 3. How many cubic feet in a piece of timber 30 ft. long, the greater end being 15 inches square, and that of the less 12 inches? 4. How many cubic feet in the mast of a ship, its height being 50 ft., the circumference at one end 5 feet and at the other 3 feet ? 4tS MENSURATIOJSr. THE SPHERE. 929. A Sphere is a body bounded by a uniformly curved sup face, all the points of which are equally distant ^^^^^^^ from a point within, called the center. yj^^^^^^k 930. The DUnncfer of a sphere is a B'^J'^PmI straight line passing through the center of the ^^^^^^m ...^^ sphere, and terminated at both ends by its ^^^^Pr'' "' surface. 931. The Radius of a sphere is a straight line drawn from the center to any point in the surface. 932. To find the surface of a sphere. 1. Find the surface of a sphere whose diameter is 9 in. Operation.— 9 in. x 3.1416 = 28.2744 in., circvmference. 28.2744 in. X 9 = 254.4696 sq. in., surface. Rule. — Multiply the diameter hy the circumference of a great ^rcle of the ^here, 2. What is the surface of a globe 3 feet in diameter? 8. Find the surface of a globe whose radius is 1 foot. 933. To find the volume of a sphere. 1. Find the volume of a sphere whose diameter is 18 inches. Operation;.— 18 in. x 3.1416 — 56.5488 in., circumference. 56.5488 in. x 18 - 1017.8784 sq. in., mrface. 1017.8784 sq. in. x 18^=3053.6352 cu. in., volume. 'RjJLE..— Multiply the surface hy \ of the diameter^ or \ of the radius, 2. Find the volume of a globe whose diameter is 30 in. 3. Find the solid contents of a globe whose radius is 5 yards. 934. To find the three dimensions of a rectangu- lar solid, the volume and the ratio of the dimensions being- given. 1. What are the dimensions of a rectangular solid, whose volume is 4480 cu. ft,, and its dimensions are to each other as 2, 5, and 7 ? Operation.— ^4480 -^ (2 x 5 x 7) = 4 ; 4 ft. x 2 = 8 ft., height; 4 ft. x 5 = 20 ft., width; 4 ft. x 7 ;= 28 ft., length. REVIEW OF SOLIDS. 473 Rule. — I. Dimde the volume by theproduct of the terms proportional to the three dimensions, and extract the cube root of the quotient. II. Multiply the root thus obtained by each proportional term ; the products will be the corresponding sides, 2. What are the dimensions of a rectangular box whose voluniG is 3000 cu. ft., and its dimensions are to each other as 2, 3, and 4 ? 3. A pile of bricks in the form of a parallelepiped contains 30720 cu. feet, and the length, breadth, and height are to each other as 3, 4, and 5. What are the dimensions of the pile ? SIMILAR SOLIDS. 935. Similar Solids are such as have the same form, and differ from each other only in volume. Principles. — 1. The volumes of similar solids are to each other as ihe cubes of their like dimensions, 1. If the volume of a cube 3 inches on each side is 27 cu. in., what is the volume of one 7 inches on each side ? Operation.— 3^ : 7^ : : 27 cu. in. : cc = 343 cu. in., volume, 2. The like dimensions of similar solids are to each other as the cube roots of their volumes. 3. If the diameter of a ball whose volume is 27 cu. in. is 3 in., what is the volume of one 7 inches on each side ? Operation.— 'V^27 ; /^343 : : 3 : a? = 7 in., diameter. REVIEW OP SOLIDS. ritOliTjEMS. 936. 1. What IS the edge of a cube whose entire surface is 1050 sq. feet, and what is its volume ? 2. What must be the inner edge of a cubical bin to hold 1250 bn of wheat ? 3. How many gallons will a cistern hold, whose depth is 7 ft., the bottom bein^ a circle 7 feet in diameter and the top 5 feet in diameter ? 4. What is the value of a stick of timber 24 ft. long, the larger end being 15 in. square, and the less 6 in., at 28 cents a cubic foot ? 474 MEKSUEATIOK. 5. If a cubic foot of iron were formed into a bar ^ an inch square, without waste, what would be its length ? G. If a marble column 10 in. in diameter contains 27 cu. ft,, what is the diameter of a column of equal length that contains 81 cu. ft. 1 7. How many board feet in a post 11 ft. long, 9 in. square at the bottom, and 4 in. square at the top ? _. 8. The surface of a sphere is the same as that of a cube, the edge of which is 12 in. Find the volume of each. 9. A ball 4.5 in. in diameter weighs 18 oz. Avoir. ; what is the weight of another ball of the same density, that is 9 in. in diameter? 10. In what time will a pipe supplying 6 gal. of water a minute, fill a tank in the form of a hemisphere, that is 10 ft. in diameter? 11. The diameter of a cistern is 8 feet ; what must be its depth to contain 75 hhd. of water? 12. How many bushels in a heap of grain in the form of a cone, whose base is 8 ft. in diameter and altitude 4 feet ? GAUGING. 937. Gauging is the process of finding the capacity or volume of casks and other vessels. A cask is equivalent to a cylinder having the same length and a diameter equal to the meati diameter of the cask. To find the mean diameter of a cask {nearly). Add to the head diameter §, or, if the staves are but little curved, .6, of the difference between the head and bung diameters. To find the volume of a cask in gallons, Multiply the square of the mean diameter by the length {both in inches) and this product by .0034. 1. How many gallons in a cask whose head diameter is 24 inches, bung diameter 30 in., and its lengt,h 34 inches? Operatiox. — 24 + (30 — 24 x |) = 28 in., mean diameter. 282 X 34 X .0034 = 90.63 gal., capacity. 2. What is the volume of a cask whose length ig 40 inches, the diameters 21 and 30 in., respectively? 3. How many gallons in a cask of slight curvature, 3 ft. 6 in. long, the head diameter being 26 in., the bung diameter 31 in. ? FORMULAS 475 938. r X 3.1416 'r- .3183 L The Diameter ^ 2. The Circuin- ference 3. The Area 939. 1. The Surfa^ce 2. The Volume 3. The Diameter CIRCLES. ~ the circumference. = the side of an equal 8qua/re, = the side of an inscribed equi lateral triangle. — tlie side of an inscribed square = the diameter, = the side of an equal square, = the side of an inscribed equi- lateral triangle. = th^ side of an inscribed squa/re, = the radius. = the square of the radius. = the square of the diameter, = the sq*retf the circumference* SPHERES. Circumference x its diam, Radius^ x 12.5864. Diameter^ x 3.1416. Circumference' x .3183. Surface x J its diameter^ Radius X 4.1888. Diameter^ x .5336. Circumference^ x .0169. ^J'OfsuTface x .5642. X .8862 ^ 1.1284 X .8660 -H 1.1547 X .7070 -f- 1.4142 X .3183 -5- 3.1416 X .2821 -^ 3.5450 X .2756 -^ 3.6276 X .2251 ■^ 4.4428 X .15915 -^ 6.28318 -f- 3.1416 X 1.2732 -^ .7854 X 12.5663 -T- .07958 4. Ti Circumference 5. The Radius 6. The Side of Inscribed Cube ^Ofmlume x 1.2407. ^/ Of surface x 1.77255. ^ Of volume x 3.8978. _ i ^ Of surface x .2821. "" \ \/ Of volume x .6204 j Radius x 1.1547. i Diameter x .5774 476 BEVIBW. 940. SYNOPSIS FOE EEVIEW. ' 1. Definition. 2. Lines. 3. Angles. 4. Plane Fioubbs. f 1. Defs. - 6. Tri- ANOLES. 2. Prob- lems. ' To find 6. QirAi>- RILAT-- XRALS. 7. CmcLB. " r 1. Defs. 2. Prob- lems, 1. Triangle. 2. Bighi-angled Tri. 3. Hypothenuse 4. Base, 5. Ferpendicular. 6. Altitude. 7. Equi lateral Triangle. 8. Isosceles Triangle. 9. Scaleni Triangle. 10. EguiangiUar Triangle. 11. Acute angled Triangle. 12. OUuse-angled Triangle. Area of Triangle. Either Dimension. Area qf a Triangle. \ Rule. The Hypothenuse. The Base or Perp. 2. Parallelogram. 3. Bectangle. 4. Square. 5. Rhomboid. 6. Rhombus. 7. Trape- zoid. 8. Trapezium. 9. Altitude. 898. ) m fi ^ ( Parallelogram. \ 899. > ^^^°° J Trapezmd. J- Rule. 900.) *^^^^ (jVaixf^iwm. ) 882. 883. 884. 886. 1887. J 1. Quadrilateral. ' 1. Defs. 1. Circle, 2. Di r 904. ^ 905. 2. Prob- lems. " 906. 907. To find - 908. I 909. J imeter. 8. Radius. Diam. or Circum. Rule, 1, 2. ^r«a. Rule, 1, 2. Z)ia77i. or Circ. Rule, 1, 2, 3. Side of Ins. Square. Rule, 1, 2. Area of Circular Ring. Rule. Mecm Proportional. Rule. 8. SncTLAB Plane Figures. 1. Defs. 2. Prin. 1, 2, 3, 4, 5. 9. Solids. ^ 1. Defs. 918. 919. 925. 926. 927. 928. 932. 933. 934. a Similar Solids. 2. Prob- lems. ' 1. Solid or Body. 2. Prism. 3. Attitude. 4. Par- allelopipedon. 6. Cube. 6. Cylinder. 7. i^ro- mid. 8. (7o«e. 9. Altitude of Pyrwnid or Cone. 10. Slant Height. 11. Frustum. 12. >^«r«. 13. Diameter. 14. Radius. Conv. Surf. ofPrismor Cyl. Rule. Foi?w»w " " Rule. Coni?. SuTf. cfPyr. or Cone. Rule. Volume " " Rule To find -I Cbnt?. /Smj/. of Frustum, Rule Fo^ww^ " '* Rule. Surf ace (f Sphere. Rule. 7<3^w»ie " " Rule. Dim. ofRectang. Solid, Rula 1. Dtfs, 2. Pri»apfe«, 1, 2. L 10. GAxraiNG. 1. Definitions. 2. Rulea. -^t^ The edges of this cube are each 1 Me'ter^ or 10 Dec'i-rm'ters^ or 100 CeWti" me'ters. in length. ScAUB, ^Q of the Exact Size* 94:1. The ]i£etric System of weights and measures is based upon the decimal notation^ and is so called because its primary unit is the Me'ter. 942, The Me'ter (m.) is the lase of the system, and is th^ one ten-millionth part of the distance on the earth's surface from the equator to either pole, or 39.37079 inches. Meier means measure ; and the three principal units are units of length, capacity or volume^ and weight. 478 METKIC SYSTEM. 943. The Multiple Units^ or higher denominations, are named by prefixing to the name of the primary units the Greek numerals, Dek'a (10), Hek'to (100), KU'o (1000), and Myr'ia (10000). Thus, 1 dek'a-me'ter {Dm.) denotes 10 me'ters (m.) ; 1 hek'to-me'ter (Hm.), 100 meters ; 1 kil'o-me'ter {Km.)^ 1000 me'ters ; and 1 myr'ia-me''ter (ifm.), 10000 meters. 944. The Sub^niultiple Vnits^ or lower denominations, ure named by prefi.xing to the names of the primary units the Latin )rdinals, Dec'i (yV), Cen'ti (yj^), Mil'li (yirW- Thus, 1 dec'i-me'ter {dm.) denotes t'si or .1 of a me'ter ; 1 cen'ti-me'ter (cw.), ^o^ or .01 of a me'ter ; 1 mMi-me'ter (mm.), Tt?ooi or -001 of a me^er. Eence, it is apparent from the name of a unit whether it is greater ot less than the standard unit, and also how many times. 945. The Metric System being based upon the decimal scale , the denominations correspond to the orders of the Arabic Notation ; and hence are written like United States Money, the lowest denomina- tion at the right. Thus, S '5 ,2 « S 6 The number is read, 67015.638 me'ters. It may be expressed in other denominations by placing the decimal point at the right of the required denomination, and writing the name or abbreviation after the figures. Thus, the above may be read, 670.15638Hm. ; or 67.015638 Km. ; or 670156.38 dm. ; or 6701563.8 cm. ; or it may be read, 6 Mm. 7 Km. Hm. 1 Dm. 5 m. 6 dm. 3 cm. 8 mm. Write 3672.045 me'ters, and read it in the several orders ; read it in kil'o-me'ters ; in hek'to-me'ters ; in dek^a-me^ters ; in dec'i- me'ters ; in cen'ti-me'ters. The names mill, cent^ dime^ used in United States Money, correspond to mil'li^ cent% dec'i^ in the Metric System. Hence the eagle might be called the dek'a-dollar, since it is 10 dollars ; the dime, a deC'i-doUart since It Is -^^ of a dollar, etc. METRIC SYSTEM. 479 MEASURES OF LENGTH. 940. The Me'ter is tlie unit of length, and is equal to 89. or, 1.0986 yd. +. 4 in. Metric Denominations. U. S. Value. 1 Mirii-me'ter = .03937 in. 10 Mil'li-me'ters, mm. = 1 Cen'ti-me'ter = .3937 in. 10 Cen'ti-me'ters, cm. = 1 Decl-meHer = 3.937 in. 10 Dec'i-me'ters, dm, = 1 Me'ter z= 39.37 in. 10 Me'ters, m. =1 Dek'a-me'ter = 32.809 ft. 10 Dek'a-me'ters, Dm. = 1 Hek'to-me'ter=19.8842 rd. 10 Hek'to-me'ters,Sm. = 1 Kil'o-me'ter = .6213 mi. 10 Kiro-me'ters, Km. = 1 Myr'ia-me'ter= 6.2138 mi. Units of long measure form a scale of tens; hence, in writing numbers expressing length, one decimal place must be allowed for each denomina- tion. Thus, 9652 mm. may be written 965.2 cm., or 96.52 dm., or 9.652 m., or .9652 Dm. 1. The Metier is used in measuring cloths and short dis- tances. 2. The KiVo-me'ter is commonly used for measuring long distances, and is about | of a common mile. 3. The Cent'i-me'ter and MWli-me'ter are used by mechanics and others for minute lengths. 4. In business, i?€. «oS ""& «» - J» ^ ® 480 METEIC SYSTEM. Change the following to mdters : 327 Dm. 947 cm. 0.72 Km. 30674 mm. 28 Hm. 236 dm. 1.73 Hm. 83.062 cm. 16.8 Km. 43.5 cm. 85.4 Dm. 4000.5 dm. 1. Write 6 kilometers 6 dekameters 6 meters 6 decimeters 6 centio meters. Ans, 6.06666 Km., or 60.6666 Hm., or 600.666 Dm., etc. Write the following, expressing each in three denominations]: 2. 24:379 dm.; 15032036 cm.; 2475064 mm.; 30471 Dm. 3. 6704 Hm. ; 85 Km. ; 120000 m. ; 780109 cm. ; 75 m. Similar examples should be given, until the pupil is familiar with the reduc- tion of higher to lower, and of lower to higher denominations, by changing the place of the decimal point and using the proper abbreviations. 947. To add, subtract, multiply, and divide Metric Denomiuations. 1. What is the sum of 314.217 m., 53.062 Hm., and 225 cm. ? Operation. 314.217 m. + 5306.2 m. + 2.25 m. = 5622.667 m., Ans. 2. Find the difference between 4.37 Km. and 1246 m. Operation. 4.37 Km. — 1.242 Km. = 3.128 Km., Ans. 8. How much cloth in 8 J- pieces, each containing 43.65 m. ? Operation. 43.65 m. x 8.25 = 384.8625 m., Ans. 4. How many garments, each containing 3.5 m., can be made from a piece of cloth containing 43.75 Dm. ? Operation. 437.5 m. -s- 3.5 m. = 125 times; hence, 125 garments, Ans. Rule. — Beduce the given numbers to the same denominations , when necessary ; then proceed as in the corresponding operations with whole numbers and decimals, bxehcisbs, 1. Add 7.6 m., 36.07 m., 125.8 m., and 9.127 m. 2. Express as meters and add 475 dm., 3241 cm., and 725 mm. 3. Add 56.07 m., 1058.2 dm., 430765 cm., 6034.58 m., and express the result in kilometers. 4. From 8.125 Km. take 3276.4 m. Ans. 4.8486 Km. METRIC SYSTEM. 481 5. The distance round a certain square is 3.15 Km. How many meters will a man travel who walks around it 4 times? 6. How many meters of ribbon will be required to make 32 badges, each containing 40 centimeters ? Ans. 12.8 m. 7. What will be its cost, at 15 cents a meter ? 8. Find the difference between 25.3 Km. and 425.25 m. 9. If an engine runs 36.8 Km. in an hour, how far does it run between 8 o'clock and 12 o'clock ? 10. In what time will a train run from Boston to Albany, at the rate of 4G.55 Km. per hour, the distance being about 325.85 Km. ? 11. From a piece of cloth containing 45.75 m., a tailor cut 5 suits, each containing 7.5 m. How much remained ? 12. A wheel is 3.6 m. around. How many times will it revolve in rolling a distance of 1.08 Km. ? Ans, 300. MEASURES OF SURFACE. 94:8. The units of square measure are squares, the sides of which are equal to a unit of long measure. 1 sq. cm^xact Size. 100 Sq. Mil' \i-me'teTs(s2. mm.) = 1 sq. cm, = 0.155 sq. in. 100 Sq. Cen'ti-me'ters = 1 sq, dm. = 15.5 sq. in. 100 Sq. Dec'i-me'tera = \ ] T '"• I =\ ^^^if^' ^'■ ^ }l Centar (ea.) j 1 1.196 sq. yd. 100 Sq. Me'ters = i } ^ J ^f • i =i ^"^^^ «!• "^ ^ (1 Ar. (a.) f ( .C247 acre. 100 Sq. Dek'a-me^ters = \] ^S f^'"", tj ^i = 2.471 acres. ( 1 Hektar (Ea,) j 100 Sq. Hek'to-me'ters = 1 sq. Km. = .3861 sq. ml Units of square measure form a scale of hundreds; hence, in writing numbers expressing surface^ two decimal places must be allowed for each denomination. Thus, 36 sq. m. 4 sq. dm. 27 sq. cm. are written 36.0427 sq. m. ; tnd 6 Ha. 5 a. 3 ca. are written 6.0503 Ha., or 605.03 a., etc. 1. The Square Me'ter is the unit for measuring ordinary surfaces of small •stent, as floors, ceilings, etc. 2. The Ar, or Square Dek'a-m^fter, is the unit of land measure^ and is equal to 119.6 sq. yd., or 3.»54 sq. rd., or .0247 acre. 482 METBIC SYSTEM. EXERCISES, 1. Read 36145 sq. m., naming each denomination. Ans. 3 sq. Hm. 61 sq. Dm. 45 sq. m. 2. Write in one number 4 of each denomination from sq. Hm. to sq. mm., expressed in sq. Hm. Ans. 4.0404040404 sq. Hm. 3. Express the following, each in three denominations : 6 sq. Km. 6 sq. Hm. 24 sq. Dm. 5 sq. m. ; 16 sq. Dm. 8 sq. m. 4 sq. dm. 15 sq. cm. 4. In 15 sq. Hm. how many square meters ? 5. What is tlie surface of a floor 12 m. long and 7 m. wide? 6. Add 8 times 4 Ha., 7 times 9 a., and 12 times 14 ca. 7. What is the area of a piece of land 42 Dm. long and 36 Dm. wide? Ans. 1512 sq. Dm., or 15.12 Ha. 8. Divide 125000 ca. into 8 equal parts. 9. How many times is 2.50 sq. m. contained in 5 Ha. ? 10. How many meters of carpeting 0.6 m. wide will cover a floor 8 m. longjind 5.7 m. wide? Ans. 76 m. 11. At 15 cents a sq. m., what is the cost of painting a surface 20.5 m. long and 68 m. wide? Ans. $20.91. 12. A man having 5 Ha 8 a. 7 ca. of land, sold .3 of it, at $25 an ar. What did he receive for what he sold ? MEASURES OF VOLUME. 949. The units of cubic measure are cubes, the edges of which are equal to a unit of long measure. 1 cu. cm.y Exact Size, 1000 Cu. Dec'i-me'ters = 1 ^ o^ / x r =-^ ( 1 Ster (s.) ) I 1000 Cu. Mirii-me'ters {cu. mm.) = 1 cu. cm. = .061 cu. in. iAAA r>. r. w u S 1 ^^' ^^' \ ( -0353 cu. ft. 1000 Cu. Cen'ti-me'ters = in T-r^ n\r =-{1 ^^nr, t ^ ( 1 Li'ter {l.) ) 1 1.0567 li. qt. ( 35.3165 cu. ft. .2759 cord. Units of cubic measure form a scale of thousands; hence, in writing numbers expressing volume, three decimal places must be allowed for each denomination. Thus, 42 cu. m. 31 cu. dm. 5 cu. cm. are written 42.031005 cu. m. The ctUdc dec'i-me'ter, wheu used as a unit of liquid or dry measuie, is called a Wter. HETRIC SYSTEM. 483 WOOD MEASURE. 1000 Cu. Decl-me^ters {cu, dm) ) _ {\cu.m. ) j .2759 cord. 10 Decl-sters {ds.) ) (1 Ster, 8.y\ 35.8165 cu. ft 10 Sters = 1 Dek'a-Bter, Ds, = 2.759 cord. Units of wood measure form a scale of tens ; hence, but one deci mal is required for each denomination. Thus, 9 Ds. 4 s. 7 ds. are written 94.7 s. ; or 9.47 Ds. 1. The Cubic Me'ter i8 the unit for measuring ordinary solids ; as excavations, embankments, etc. 2. CuUc Cen'ti-mefters and Mil'li-^me'ters are used for measuring minute bodies. 3. The CulAc Me'ter when used as a unit of measure for wood or stone is called a Ster, 4 The common Cord is about the same as 3.6 sters ^ or 36 de&i-sters, JEXE R C IS JES, 1. Write 30 Ds. 6 s. 8 ds. Ans. 30.68 Ds. 2. Express in cu. m., 3 cu. m. 3 cu. dm. 3 cu. cm. 3 cu. mm. Ans, 3.003003003 cu. m. 3. Write and read the following, each in cu. dm., in cu. cm., and in cu. mm. : 16 cu. m. 275 cu. dm. ; 204 cu. m. .016 cu. dm. .024 cu. cm. ; 10 cu. m. 324 cu. dm. .016 cu. cm. 3244 cu. cm. 4. Express in cu. meters and add : 7 cu. m., 55 cu. dm., 12 cu. m., 6 cu. dm., 15 cu. cm., 10532 cu. cm. Ans. 19.071547 m. 5. From 36 cu. m. subtract 8 times 42 cu. dm. Ans. 35.664 m. 6. How many cubic meters of brick in a wall 16 m. long, 3 m. high, and 8 dm. thick ? Ans. 38.4 cu. m. 7. How many cu. meters of earth must be removed in digging a cellar 16.5 m. long, 8.2 m. wide, and 3.2 m. deep? 8- In a pile of wood 9.3 m. long, 2.8 m. high, and 1.5 m. wide, f how many sters? .^^5. 39.06 s. 9. At $2.25 a ster, what would be the cost of a pile of wood 5.6 m. long, 3.4 m. wide, and 2.5 m. high ? 10. If a cu. centimeter of silver is worth $.75, what is the value of a brick of silver 12.4 cm. long, 3.6 cm. wide, and 2.5 cm. thick? 484 METRIC SYSTEM. MEASURES OF CAPACITY. 950. The Lii'ter is the unit of ca- pacity, both of Liquid and of Dry Measures, and is equal in volume to one cu. deci-me'teVy equal to 1.0567 qt.Liquid Measure, or .908 qt. Dry Measure. lOMirii-li'ters, m?.=l Cen'ti-li'ter 10 CenHi-li'ters, cl. =1 Dec'i-li'ter 10 Dec'i-li'ters, dl—\Li'ter IOLi'ters, i. =1 Dek'a-li^er 10 Dek'a-li'ters, Dl =1 Hek'to-li'ter 10Hek^to-li'ters,J3?.= lORiro-li^ters, ^^.= rlKil'o-li'terorSter: :1 Myr'ia-li'ter {Ml.) .61 cu. in. 6,10 " " .908 qt. 9.081 *' : 2.837 bu. : (28.37bu.; ..308cu.; r283.72 bu. Liquid M, =.338fl'doz. = .845 gi. =1.0567 qt. =2.64175 gal. =26.4175 " _J28.37bu.| "~(1.308cu.ydj 264.175 " 2641.75 « 1. The lA'ter is used iu measuring liquids in moderate quantities. 2. The Hek'to-Wtei' is used for measuring grain, fruit, roots, etc., in large quantities, also wine in casks. 3. Instead of the KiVo-li'ter and MiVli-me'tery the CvJbic Mefter and CvUg Cm'ti-me'teTy which are their equals, may bo used. BXEJtCISES, 1. Write 5 kiloliters 5 liters 5 deciliters 5 centiliters. Ans. 5.00555 Kl., or 5005.55 1. 2. Read, naming each denomination, the following : 45624 cl. ; 306721 ml. ; 76031 dl. ; 89764 i. 3. In 3846 1. how many cl. ? How many Dl. ? Kl. ? dl. ? mL ? 4. Find the sum of 175 L, 25 HI., 42 cl., and 16 dL 5. From 6 times 25 HI. take 15 times 36 1. 6. Divide 5 HI. of com equally among 25 persons. Ana. 20 L 7. From a cask of wine containing 2 HI. of wine, 125 1. were drawn out. How much remained ? 8. How many HI. of wheat can be put into a bin 3 m. long, 2 m. wide, and 1.5 m. deep ? Ana. 90 HI. 9. What must be the length of a bin 1.5 m. wide, 1 m. deep, to contain 7500 liters of grain ? Ana. 5 j METRIC SYSTEM. 48S MEASURES OF WEIGHT. 951. The Gram is the unit of weight, and is equal to the vreight of a cu. cen'ti me' ter of distilled water. A Oram is equal to 15.432 gr. Troy, or .03527 oz. Avoir. 10 Mirii-grams, rag. = 1 Cen'ti-gram = .1543 + gr. Tr. 10 Cen'ti-grams, eg. = 1 Dec^i-gram = 1.5432 -f- " " lODec^i-grams, dg. = 1 Gram =^ .03527 + oz. A v. 10 Grams, g. = 1 Dek'a-gram - .3527+" " 10 Dek'a-grams, Bg. = 1 Hek'to-gram = 3.5274 -f " ** ^ A XT 1 /^ rr h\ Kir 0-gram, J i 2.6792 lb. Tr. 10 Hek'to-grams, Eg. =U _°.,, ')■= ]c^cv^Aa ii. a ^ ^ I or Ktl'o f (2.2046+ lb. Av. IC Kil'os, Kg, = 1 Myr'ia-gram = 22.046 4. « '* ,J^^?;f'^"^'^^''^^"^'i = lQuin'tal ^220.46 +« « 100 Kil'o-grams j 10 Quin'tals, Q., or j _ jTonneau, ) _ j 2204.62+ " " 1000 Kil'os, K. ) J or Ton\~ 1 1.1023 + tons. 1. The Gram is used for weighing letters, gold, silver, medicines, and ^ small, or costly articles. 2. The KU'o-gram or KWo is the weight of a cu, dm. of water, and is the unit of common weight in trade, being a trifle less than 2| lb. Avoir. 3. The Ton is the weight of a cu. m, of water, and is used for weighing very heavy articles, being about 204^ lb. more than a common ton. 4. The Aimr, oz. is about 28 g. ; the pound is a little less than I a kilo. £]X EltC I 8 ES, 1. Read 340642 eg. in grams ; in hectograms; in kilograms. 2. Change 16.5 T. to kilos ; to grams ; to decigrams. 3. If coffee is $.80 a kilo, what will 5 quintals cost? 4. How many boxes containing 1 gram each, will be required tc hold 1 kilo of quinine ? Arts. 1000. 5. If a letter weighs 3.5 g., how many such letters will weigh 1.015 Kg.? ^715.290. 6. A car weighing 6.577 T. contains 125 barrels of salt, each weighing 102.15 K. What is the weight of the car and contents ? 7. Find the difference in the weight of the car and its contents ? 486 METRIC SYSTEM. 952. To change the Metric to the Common Sys-* tern. 1, In 3.6 Km., how many feet? OPERATION. Analysis.— The meter i& 3.6 Km. X 1000 = 3600 m. *^« P^^^^^P^^ ^^j^^ ''\ *^® ^^^^ ' ^^«.x ^ ^w«,o^ . hence, reduce the kilometertj 39.37 m. X 3600 = 141732 m. ,^ ^.^ers. Since there are 141732 in. -^12 = 11811 ft., Am, 39.37 inches in 1 meter, in, 3600 m. there are 3600 times 39.37 in., or 141732 in. = 11811 ft. Therefore, 3.6 Km. are equal to 11811 ft. Rule. — Reduce the metric number to the denomination of the principal unit of the tcMe ; then multiply hy the equivalent , and reduce the product to the required denomination. EXEItCISES. 2. How many feet in 472 centimeters ? Ans, 15.485 ft 3. How many cubic feet in 2000 sters ? 4. How many gallons, liquid measure, in 325 deciliters ? 6. How many gallons in 108.24 liters? Ans. 28 gal. 2.77 qt 6. How many bushels in 3262 kiloliters? 7. How many acres in 436 ars ? An8. 10.774 A. 8. In 942325 centiliters, how many bushels? 9. In 456 kilograms, how many pounds ? Ana, 1005.024 lb. 10. In 42 ars, how many square rods ? 11. Change 75.5 hektars to acres. Ans, 186.56 A. 12. How many gallons in 24J liters of wine ? 13. How many pounds of butter in 124 kilos ! 14. In 28 sters, how many cords? Ans. 7.725 C. 15. In 72 kilometers, how many miles ? 16. Change 148 grams to ounces Avoirdupois. Ans, 5.22 oz- 17. Change 150.75 kilos to pounds. 18. How many sq. rods in 5 a. 85 ca. ? Ar^s. 23.13 sq. rd. 19. What is the weight of 24 cu. dm. 148 cu. cm. of silver, if cu. centimeter weighs 11.4 g. ? Ans. 737.556 lb. Tr. METRIC SYSTEM. 487 953. To change the Common to the Metric Sys- tem. In 10 lb. 4 oz. Troy, how many kilograms ? OPERATION. Analysis.— The gram^ 10 lb. 4 oz. — 10.25 lb. the principal unit of the 10.25 lb. X 5760 = 59040 gr. *^^!®' ^ expressed in crains : hence, reduce the 59040 gr.-^15.432gr. = 3825.75 g. f,„„^' ,,d ounces to 3825,75 g. -H 1000 = 3.82575 Kg., Ans, grains. Since 15.432 gr. make 1 gram, there are as many grams in 59040 gr. as 15.432 gr. is contained times in 59040 gr., or 3825.75 g. And since there are 1000 grams in a kilogram, dividing 3825.75 g. by 1000 g., the quotient is 3.82575. Therefore, there are 3.82575 Kg. in 10 lb. 4 oz. Rule. — Reduce the gwen quantity to the denomination in which the equivalent of the principal unit of the metnc table is expressed ; divide ly thi» equivalent, and reduce the quotient to the required denomination. exehcisjes. 2. In 6172.9 lb av., how many kilograms ? ^ns. 2800.009 Kg. 3. How many ars in a square mile ? 4. How many cu. decimeters in 1892 cu. feet ? 5. In 892 gr., how many grams? Ans, 57.8 g. 6. In 2 mi. 272 rd. 5 yd., how many kilometers? Ans. 4.59 Km. 7. How many sters in 264.4 cu. feet ? 8. How many liters in 3 bu. 1 pk. ? Ans. 1145 1. 9. How many grams in 6 lb. Troy ? In 6 lb. Avoir. ? 10. How many meters in 3 mi. 272 rd. ? 11. In 1828 cu. yd. how many cu. meters? Ans. 1397.52 cu. m. 12. In 3588 sq. yards, how many sq. meters? 13. Bought 454 bu. of wheat, at $3 a bushel, and sold the same at $8.75 per hektoliter ; how many hektoliters did I sell ? Did \ gain or lose, and how much ? Ans. 160 HI. ; gain, $38. 14. In 13 gal. 3 qt. 2 pt. 3 gi., how many liters? Ans. 53.351.+. 15. How many sq. meters of plastering in a room 1 8 ft. 6 in. long, 14 ft. wide, and 9 ft. 6 in. high? Ans. 81.427 sq. m. + . 488 METRIC SYSTEM. TEST PEOBLEMS. 954. 1. Find the weight of a barrel of flour (196 lb.) in Kg. ? 2. What is the cost of a carpet for a room 10.5 m. long, and 8.4 m wide, if the carpet is 84 cm. wide and costg $2.75 a meter? Ans. $288.75. 8. A farmer sold 540 HI. of wheat, at $2 a bushel, and invested ihe proceeds in coal at $7 per ton. How many tons did he buy ? Ans. 437.785 T. + . 4 What is the cost of a building lot 75 m. long and 62 m. wide, at $40 an ar ? Ans, $1860. 5. A bushel of wheat weighs 60 lb. What is the weight of 5 HI. of wheat, in kilograms ? A^is. 386.05 Kg. 6. What will be the cost of a pile of wood 15.7 m. long, 3 m. high, and 7.52 m. wide, at $1.50 a ster? 7. The new silver dollar weighs 412J gr. Troy. How many grams does it weigh ? Ans. 26.73 g. 8. How many acres of land in -24 6 Km. of a highway, which is 20 m. wide ? Ans. 121.573 A. 9. A bin is 4.2 m. long, 2.8 m. wide, and 1.5 m. deep. What will be the cost of filling it with charcoal, at 25 cts. a hektoliter ? 10. A merchant bought 800 m. of silk in Lyons, at 12.5 francs a meter ; he paid 75 cents a yard for duty and freight, and sold it in New York at $5 a yard. What was his gain ? Ans, $670.61. 11. What price per pound is equivalent to $2.50 per Hg. ? 12. If a man buys 5000 g. of jewels, at 35 francs a gram, and sella them at $15 a pennyweight, what was his gain or loss ? 13. If a field produces 40 HI. of oats to the hektar, how many bushels is that to the acre ? Ans, 45.93 bu. 14. What price per peck is equivalent to 80 cts. a dekaliter ? 15. What will be the cost of excavating a cellar 18.3 m. long, 10.73 m. wide, and 3.4 m. deep, at 20 cents per ster? 16. How many pounds Avoir, are there in 96.4 kilos of salt ? 17. How many liters will a cistern hold that measures on the inside 5.5 ft. long, 4 ft. 6 in. wide, and 4 ft. deep ? Ans. 2803.383 L METRIC SYSTEM, 489 18. How many meters of lining that is 60 cm. wide will line 15 m. of silk that is 75 cm. wide ? Ans. 18,75 cm. 19. A lady bought 40.5 m. of silk in Paris. What would be its value in Boston, at $475 per yard ? 20. A bin is 4 m. long, 2.3 m. wide. How deep must it he to contain 40 HI. of graia? Ans. 4.347 -f dnx. 21. How many sters of wood can be piled in a shed 8.5 m. longy 5.8 m. wide, and 4.2 m. high? What would be its value at $3.25 a cord? Ans. 207.03 s.; $185,665. 22. A dray is loaded with 60 bags of grain, each bag holding 8 Dl. ; allowing 75 K. of grain to the hectoliter, what is the weight of the load in metric tons ? Ans. 8.6 T. 23. How many meters of shirting, at $.18 per meter, must be given in exchange for 250 HI. of oats, at $1.20 per hectoliter? 24. A merchant shipped to France 50 barrels of sugar, each con- taining 250 lb., paying $2 per cwt. for transportation. He sold the sugar at $.34 per kilogram, and invested the proceeds in broadcloth, at $4 per meter. How many yards did he purchase ? 25. A cu. decimeter of copper weighs 8.8 Kg. What is the value of a bar of the same metal 15 dm. long, 9.6 cm. broad, and 6.4 cm. thick, at $1.30 a kilogram? Ans. $105.43. 26. How many bricks, each 20 cm. long and 10 cm. wide, will pave a walk 95.4 m. long and 2.1 m. wide; and what will they cost, at $1.75 per hundred? Ans. 10017 bricks ; $175,297. 27. What is the value of a pile of wood 40 ft. 6 in. lonj?^, 4 ft. broad, and 6 ft. 6 in. high, at $6.50 per dekastere ? 28. What will be the cost of building a wall 96 Dm. 6 m. 8 dm. long, 1 m. 6 dm. thick, and 2 m. 4 cm. high, at $6.75 a cu. meter? 29. A wine merchant imported to Boston 1000 dekaliters of wine, at a cost of $.75 a liter, delivered. At what price per gallon must he sell the same to clear $2000 on the shipment ? Ans. $3,596. 30. How many gallons of water will a cistern contain that is 3 m, deep, 2 m. long, and 1,5 m, wide; and what will be its weight in metric tons ? Ans, 2377.575 gals. ; 9 T. 490 METEIC SYSTEM. TABLE OP EQUIVALENTS. 955. The equivalents here given agree with those that have been established by Act of Congress for use in legal proceedings and in the interpretation of contracts. • 1 inch = 2.540 centimeters. 1 foot = 3.048 decimeters. 1 yard = 0.9144 meters. 1 rod — 0.5029 dekameters. 1 mile — 1.6093 kilometers. 1 sq. in. = 6.452 sq. centimeters. 1 sq. ft. = 9.2903 sq. decimeters. 1 sq. yard = 0.8361 sq. meter. 1 sq. rd. — 25.293 sq. meters. 1 acre = 0.4047 hektar. 1 sq. mile = 2.590 sq. kilometers. 1 cu. in. = 16.387 cu. centimeters. 1 cu. ft. = 28.317 cu. decimeters. 1 cu. yard = 0.7645 cu. meter. 1 cord = 3.624 sters. 1 liquid quart = 0.9463 liter. 1 gallon =: 0.3785 dekaliters. 1 dry quart — 1.101 liters. 1 peck = 0.881 dekaliter. 1 bushel = 3.524 dekaliters. 1 ounce av. = 28.35 grams. 1 pound av. = 0.4536 kilogram. 1 T. (2000 lbs.) = 0.9072 met. ton. 1 grain Troy = 0.064S gram. 1 ounce Troy = 31.1035 grams. 1 pound Troy = 0.3732 kilogram. 1 centimeter = 0.3937 inch. 1 decimeter = 0.328 foot. 1 meter = 1.0936 yds. = 39.37 in. 1 dekameter = 1.9884 rods. 1 kilometer = 0.62137 mile. 1 sq. centimeter = 0. 1550 sq. in. 1 sq. decimeter = 0.1076 sq. ffc. 1 sq. meter =: 1.196 sq. yards. 1 ar = 3.954 sq. rods. 1 hektar = 2.471 acres. 1 sq. kilometer = 0.3861 sq. mi. 1 cu. centimeter = 0.0610 cu. in. 1 cu. decimeter = 0.0353 cu. ft. 1 cu. meter = 1.308 cu. yards. 1 ster = 0.2759 cord. 1 liter = 1.0567 liquid quarts. 1 dekaliter = 2.6417 gallons. 1 liter - 0.908 dry quart. 1 dekaliter = 1.135 pecks. 1 hectoliter =: 2.8375 bushels. 1 gram = 0,03527 ounce A v. 1 kilogram = 2.2046 pounds Av. 1 metric ton — 1 .1023 tons. 1 gram = 15.432 grains Troy. 1 gram = 0.03215 ounce Troy. 1 kilogram = 2.679 pounds Tro^ PARTIAL PAYME2q-TS. 491 VERMONT EULE FOR PARTIAL PAYMENTS. 956. The General Statutes of Vermont provide the following Rule for computing interest on notes, when partial payments have been made : ** On all notes, hzlls^ or other similar obligations, whether made payable on demand or at a specified timey with interest, when payments are made, such payments shall be applied : fi/rst, to liqui date the interest that has accrued at the time of such payments; and, secondly, to the extinguishment of the principal. " On all notes, bills, or other similar obligations, whether made payable on demand or at a specified time, with interest annu- ally, the annual interests that remain unpaid sJiall be suhject to simple interest, from the time they become due to the time of final settlement ; but if in any year, reckoning from the time such annual interest began to accrue, payments ham been made, the amount of such payments at the end of such year, with interest thereon from the date of payment, shall be applied : first, to liquidate the simple inter- est that has accrued upon the unpaid annual interests ; secondly, to liquidate the annual interests that ham become due ; and thirdly, to the extinguishment of the principal." MXEB, C IS E8, $3458. Bradford, Vt., Sept. 13, 1869. 1, For 'oalu^ received, I promise to pay E. W, Colby or order three thousand four hundred and fifty-eight dollars, on or before ths first day of January, 1878, with interest. Samuel S. Green. Indorsed as follows: Dec. 16, 1870, $100; May 1, 1871, $1000; Jan. 13, 1874, $85 ; AprU 13, 1876, $450.75. What was due Jan. 1, 1878 ? Ans. $3239.90. $872. St. Johnsbury, Vt., Nov. 22, 1868. 2. For value received, I promise to pay James Ferguson or order eight hundred and seventy-two dollars, on demand, with interest annually, Sylvanus E. Boyle. Indorsed as follows: April 4, 1869, $28; July 10, 1872, $94.40; Dec. 10, 1874, $6.72 ; Jan. 14, 1877, i What was due Dec. 28, 1878 ? 4c92 PAETIAL PAYMENTS. OPERATION. Int. on Yearly Int. Int. Prifbo Int. of prin. to Nov. 22, 1869 . , . . . $52.82 $872 Am't of 1st payment 29.06 Bal. of unpaid yearly int. ...... 23.26 Int. of prin. to Nov. 22, 1872 156,96 Int. on 1 year's int. 3 years $9.42 Int. on bal. of unpaid yearly int. 3 years . 4.19 13.61 193783 Am't of 2d payment 96.48 Bal. of unpaid yearly int. 97.35 Int. of prin. to Nov. 22, 1875 156.96 Int. on 1 year's int. 3 years 9.42 Int. on bal. of unpaid yearly int. 3 years . 17.52 26.94 254.31 Am't of 3d payment .....*■,.. 7.10 Bal. of int. on int 19.84 Int of prin. to Nov. 22, 1877 104.64 Int. on 1 year's int, 1 year 3.14 Int. on bal. of unpaid yearly int. 2 years . 30.52 53.50 412.45 1284.45 Am't of 4tli payment 416.33 New principal 868.12 Int. of new prin. to Dec. 28, 1878 57.30 Int. on 1 year's int. 1 mo. 6 d. ,31 Due, Dec. 28, 1878 $925.73 Explanation.— We compute the interest for one year from the date of the note, as a payment is made within that year, and deduct the amount of the pay Tnent at the end of the year from the interest due. The balance of interest bears: mterest till Nov, 22, 1872. The amount of the payment at the end of this year exceeds the interest on interest due. We therefore deduct the amount of the payment from the total interest due, and have a balance of unpaid yearly inter- est, $97.35, which bears simple interest till Nov. 22, 1875. At this date the amount of the payment is less than the interest on interest due. We there- fore deduct the amount of the payment from the amount of interest on interest, and have a remainder of $19.84, which is without interest. The amount of un- paid yearly interest at this date bears simple interest till the next balance. PARTIAL PAYHEKTS. 493 The amount of the fourth payment, Nov. 22, 1877, exceeds the total interest due. We therefore deduct it from the sum of the interest and principal. The remainder forms a new principal, which bears simple interest to the settlement of the note, Dec. 28, 1878, and one year's interest on the same bears interest from Nov. 22, 1878, to Dec. 28, 1878, which interest, added to the new principal, gives the amount due Dec. 28, 1878— $925.73. In cases of annual interest with partial payments, like the above example, observe the following notes ; 1. To avoid compounding interest, keep the principal, unpaid yearly inter^ ests, and interest on yearly interest, in separate columns. 2. Deduct the amount of the payment or payments at the end of the year from the interest on the unpaid yearly interest, when it does not exceed this interest. The remainder never draws interest, but is liquidated by the first pay- ment that equals or exceeds it. 3. Deduct the amount of the payment or payments at the end of the year from the sum of the unpaid yearly interests and the interest on the unpaid yearly interests, when this amount exceeds the interest on the interest, but is less than such sum. The remainder is a balance of unpaid yearly interest which draws simple interest until canceled by a payment. 4. Deduct the amount of the payment or payments at the end of the year from the sum of the total interest due and the principal, when it exceeds the total interest due. The remainder forms a new principal, with which proceed as with the original principal. $5000. Newport, Yt., Oct. 19, 1862. 3. For value received, we jointly and severally promise to pay John Smith or hearer five thousand dollars^ sixteen years after date, with interest annually, Geo. S. Leazer. E. D. Crawford. Indorsed as follows : Jan. 13, 1866, $393 ; Sept. 24, 1866, $48 ; July 10, 1869, $493.47 ; Oct. 14,1873, $100; Dec. 12, 1877, $3200; April 15, 1878, $65. What was due Oct. 19, 1878? Ans. $7056.17. $420. Burlington, Vt., March 23, 1872. 4. For value received, I promise to pay Jas. B. Vinton or order four hundred and twenty dollars, six years from date, with interest annually. Geo. A. Bancroft. Indorsed as follows; Oct. 3, 1873, $40.23; March 1, 1874, $8; Sept. 13, 1875, $33.38. What was due March 23, 1878 ? Ans. $494.62. 494 PARTIAL PAYMEKTS. Barton, Vt. Aug. 20, 1872. 5. For value received, I promise to pay E. J. Baxter or order dx hundred and thirty-nine dollars, on demand, with interest annually, Samuel Macomber. Indorsed as follows : Oct. 14, 1877, $10 ; Dec. 24, 1878, $20. What was due March 30, 1879 ? Ans, $904.58. TABLE. Showing amount cf $1.00 from 1 to 20 years, at 4, 5, 6, 7 and 8 per cent., Anmujd Interest. Years. 4 per cent. 5 per cent. 6 per cent, 7 per cent. 8 per cent. Years. 1 . $1.0400 $1.0500 $i.oeit)o $1.0700 $1.0800 . 1 2 . 1.0816 1.1025 1.1236 1.1449 1.1664 . 2 3 . 1.1248 1.1575 1.1908 1.2247 1.2592 . 3 4 . 1.1696 1.2150 1.2616 1.3094 1,3584 . 4 5 . 1.2160 1.2750 1.3360 1.3990 1.4640 . 5 6 . 1.2640 1.3375 14140 1,4935 1.5760 . 6 7 . 1.3136 1.4025 1.4956 1.5929 1.6944 . 7 8 . 1.3648 1.4700 1.5808 1.6972 1.8192 . 8 9 . 1.4176 1.5400 1.6696 1.8064 1.9504 . 9 10 . 1.4720 1.6125 1.7620 1.9205 2.0880 . 10 11 . 1.5280 1.6875 1.8580 2.0395 2.2320 . 11 12 . 1.5856 1.7650 1.9576 2.1634 2.3824 . 12 13 . 1.6448 1.8450 2.0608 2.2922 2.6392 . 13 14 . 1.7056 1.9275 2.1676 2.4259 2.7024 . 14 15 . 1.7680 1.0125 2.2780 2.5645 2.8720 . 15 16 . 1.8320 2.1000 2.3920 2.7080 3.0480 . 16 17 . 1.8976 2.1900 2.5096 2.8564 3.2304 . 17 18 . 1.9648 2.2825 26308 3.0097 3.4192 . 18 19 . 2.0336 2.3775 2.7556 3.1679 3.6144 . 19 20 . 2.1040 2.4750 2.8840 3.3100 3.8160 20 ASSESSMENT OF TAXES. 495 VEEMONT METHOD OF ASSESSING TAXES. 957. The Or and List is the base on which all taxes are assessed ; it is 1^ of the appraised value of the real estate and personal property, together with the poll list. The Poll List is $2.00 for every male inhabitant, from 21 to 7C years of age, except such as are specially exempt by law. The General Statutes of Vermont provide that the listers in each town shall make a list of all the real estate and personal property, and the number of taxable polls in such town, and that the said list shall contain the following particulars : *' First. The name of each taxable person. *' Second. The number of polls and the amount at which the same are set m the list. *' Third. The quantity of real estate owned or occupied by such person. " Fourth. The value of such real estate. " Fifth. In the fifth column the full value of all taxable personal estate owned by such person. " Sixth. In the sixth column shall be set the one per centum on the value of all personal and real estate, together with the amount of the polls, which sum shall be the amount on which all taxes shall be made or assessed. The State and County Taxes are assessed by the Legislature. The minimum of the State School and Highway Taxes is fixed by law, and a higher rate left optional with the town. A Tovm Tax is assessed by vote of the town, a Village Tax by vote of the village, and a School District Tax by vote of the district. JEXJEBCISJES, 1. The town of Montpelier voted a town tax of $2.60 on each dollar of the grand list. The appraised value of the real estate was $702727, and of the personal property $309987, and there were 740 taxable polls. What was the grand list of the town ? How much money was raised by this vote ? What was John Hammond's town tax, who was 30 years of age, and w^hose property was ap- praised at $8927.75? 496 ASSESSMENT OF TAXES. OPERATION. 1702727 + $309987= $1012714, assessed value of the property. $1012714 X .01 = $10127.14, 1% of the assessed value. $2.00 X 740=$1480, the poll list. $10127.14-1- $1480=$11607.14, the grand list. $2.60 X 11607.14=130178.56, amount of money raised. $8927.75 X. 01 = $89.28,1% of the assessed value of John Ham mond's property. $89.28 + $2.00, his poll list = $91.28, John Hammond^s grand lisl $2.60 X 91 .28 =$237. 33, John Hammond's town tax. 2. The appraised value of property, both real and personal, in the town of Rutland, for the year 1878, was $3415264. The num- ber of taxable polls was 2066. The town voted to raise a tax of $28713.48. What was the tax on a dollar of the grand list ? Ans. $0.75. 3. The appraised value of the real estate in the city of Burling- ton was $2542373; of the personal property, $399937. There were 2040 taxable polls. The city voted to raise $60305.58 city tax. What was the amount of Henry Cook's tax, a resident, who was 73 years of age, and whose real estate was appraised at $750, and his personal property at $475.50 ? Ans. $22.06. 4. The grand list in the town of Chelsea was $4403.74. The ap- praised value of all the property was $368774. How many taxable' polls were there in that town ? Ans. 358. 5. The estimated cost of schools in school district No. 8, in the town of Cabot, for one year, was $765. The amount of public money received from the town was $71.50. The appraised value of the real estate in the district was $48545 ; of the personal estate $15428.75 ; the number of taxable polls in the district 103. How much tax on a dollar of the grand list must the district vote, to pay its expenses ? Ans. $0.82. 6. James Bell resides in Hardwick ; he is 44 years of age ; his property, both real estate and personal, is appraised at $8975.50. Hardwick voted a town tax of $1.60 on a dollar of the grand list. The highway tax is $0.40 : the state tax is $0.45 ; tlie state school tax is $0.09 ; the school tax is $0.86 ; and the county tax $0.04, on the dollar. What is the amount of his taxes ? Ans, $315.64. MEASURES. 497 FEENCH AND SPANISH MEASURES. 958. The old French Linear^ and Lafid Meas- ure^ is still used to some extent in Louisiana, and in other French settlements in the United States. Table. 12 Lines = 1 Inch. 6 Feet = 1 Toise. 12 Inches = 1 Foot. 32 Toises = 1 Arpent. 900 Square Toises = 1 Square Arpent. The French Foot equals 12.8 inches, American, nearly. The Arpent is the old French name for Acre, and contains nearly f of an English acre. In Texas, New Mexico, and in other Spanish settle- ments of the United States, the following denominations are still used: Table. 1000000 Square Varas =r 1 Labor = 177.136 Acres (American). 25 Labors = 1 League = 4428.4 Acres '* The Spanish Foot — 11.11 + in. (Am.) ; 1 Vara = 33^ in. (Am.) ; 108 Varas = 100 Yards, and 1900.8 Varas = 1 MUe. Other Denominations in Use. 5000 Varas Square — 1 Square League. 1000 Varas Square = 1 Labor, or -^ League. 6645.376 Square Varas = 4840 Square Yards = 1 Acre. 23.76 Square Varas = 1 Square Chain = -^ Acre. 1900.8 Varas Square = 1 Section = 640 Acres. TABLE FOR INVESTORS. 959. The following Table shows the rate per cent, of Annual Income from Bonds hearing 5, 6, 7, or 8 per cent, interesty and costing from 40 to 125. Purchase Price. 5%. 12.50 6%. 7%. 8%. Purchase Price. 5f.. 6%. 7f.. Sfc 40 15.00 17.50 20.00 83 6.02 7.22 8.43 9.6^ 41 12,20 14.64 17.08 19.52 84 5.95 7.14 833 9.5S 42 11.90 14.28 16.66 19.04 85 5 88 7.05 8.23 9.41 43 11.63 13.95 16.28 18.61 86 5.81 6.97 8.13 9.3( 44 11.36 13.63 15.90 18.18 87 5.74 6.89 8.04 9.U 45 11.11 13.32 15.56 17.78 88 5.68 6.81 7.94 9.0^ 46 10.86 13.0i 15.21 17.39 89 5.61 6.74 7.86 8.9^ 47 10.63 12.77 14.90 17.02 90 5.55 6.66 7.77 8.8^ 48 10.41 12.50 14.53 16.66 91 5.49 6.59 7.69 8.7S 49 10.20 12.25 14.29 16.33 92 5.43 6.52 7.60 8.6S 50 10.00 12.00 14.00 16.00 93 5.37 6.45 7.52 8.6C 51 9.80 11.76 13.72 15.68 94 5.31 6.38 7.44 8.51 52 9.61 11.53 13.46 15.38 95 5.26 6 31 7.36 8.42 53 9.43 11.32 13.20 15.09 96 5.20 6.25 7.29 8.39 54 9.25 11.11 12.96 14.81 97 5.15 6.18 7.21 8.24 55 9.09 10.90 12.72 14.54 98 5.10 6.12 7.14 SM 5G 8.92 10.70 ; 12.50 14.28 99 5.05 6.06 7.07 8.08 57 8.77 10.52 ' 12.27 14.03 100 5.00 6.00 7.00 8.0C 58 8.62 10.34 12.06 13.79 101 4.95 5.94 6.93 7.92 59 8.47 10.16 11.86 13.55 102 4.90 5.88 6.86 7.84 60 8.33 10.00 11.66 13.33 103 4.85 5.82 6.79 7.7G 61 8.19 9.83 11.47 13.11 104 4.80 5.76 6.72 7.69 62 8.06 9.67 11.29 12.90 105 4.76 5.71 6.66 7.61 63 7.93 9.52 11.11 12.69 106 4.71 5.66 6.60 7.54 64 7.81 9.37 10.93 12.50 107 4.67 5.60 6.54 7.47 65 7.69 9.23 10.76 12.30 108 4.62 5.55 6.48 7.40 66 7.57 9.09 10.60 12.12 109 4.58 5.50 6.42 7.33 67 7.46 8.95 10.44 11.94 110 4.54 5.45 6.36 7.27 68 7.35 8.82 10.29 11.76 111 4.50 5.40 6.30 7.20 69 7.24 8.69 10.14 11.59 112 446 5.35 6.25 7.14 70 7.14 8.57 10.00 11.43 113 4.42 5.30 6.19 7.07 71 7.04 8.45 9.85 11.26 114 4.38 5.26 6.14 7.01 72 6.94 8.33 9.72 11.11 115 4.35 5.21 6.08 6.95 73 6.84 8.21 9.58 10.95 116 4.31 5.17 6.03 6.89 74 6.75 8.10 9.45 10.80 117 4.27 5.12 5.98 6.83 75 6.66 8.00 9.33 10.66 118 4.23 5.08 5.93 6.7? 76 6.57 7.89 9.21 10.52 119 4.20 5.04 5.88 6.72 77 6.49 7.79 9.00 10.38 120 4.16 5 00 5.83 6.m 7a 6.41 7.69 8.97 10.25 121 4.13 4.95 5.78 6.61 79 6.32 7.59 8.88 10.12 122 4.09 4.91 5.73 6.55 80 6.25 7.50 8.75 10.00 123 4.08 4.87 5.69 6.5^^ 81 6.17 7.40 8.64 9.87 124 4.03 4.83 5.65 6.4 82 6.09 7.31 8.53 9.75 125 4.00 4.80 15.60 Q.4 ANSWERS , The answers to the introductory and more simple examples of many of the articles have been omitted. Art. 77. 1. $5.78. s. $89.18. 3. $137.87. 4^ $247.78. 6, $38.58. 6. $27.78. 7. $189.75. 8, $17.67. Art. 79. 2. 1646. 3. 1619. 4. $65.94. 5, $287.67. 6, $376.71. 7. 4491. S. 7504 lb. 9, 75686. 10, 72447. 11, $696.87. 12, $18.13. 13, $80.87. U^ $149.18. 15, 105233. 16. $220.34. 17. 181776. 18. 11965. 19. $944.66. m $7193.28. el. $3554.05. S2. 1547164. 23. $6692.23. 2U, 15873478. 25. $104560. Art. 91. 10, $14.11. 11. 3231. 12, $51.34 21. 22. 25. 13, 2123 tons. U. 2324 ft. 15. 2324 days. 16, $41.23. 17, $230.43. 18. $202.12. 19, 224113. 20. 721220. 210532. 4175. 151. 5113. $15.21. 26. $22.10. 27. $25.26. 28. 2710. 29. 34213. 30. $212.20. 31. $6746. 32. 221533. Art. 93. 2. 1848. 3. 3883. 4. 1318. 5. 4195. 6. 28286 miles. 7. 26762 acres. 8. 228670 ft. 9. $240.81. 10. $95.58. 11. $38.08. 12. $6.16. 13. 32358. U. $64.84 15. $135.28. 16, $157.63. 17. 8728 rd. 18, 45736 tons. 25. 12336. 26. 87588. 27. 69356. 28. 4800. 29. $3328.59. 30. $1264.50. 31. 33798. 32. 35555. 33. $291.35. 34^ $222.75. 35. $3015.05. 36. $552477. 37. 10386. 38. $695.79. 39. $5351.84. 40. $101.10. 41. 474889. Art. 95. 1. 332650. 2. $895.66. 3. 30448. 4. 6132. 5. 2517. 6. $15.22. 7. 4190 miles. 8. $8640. 9. 78388 sq. mi. 10. 3572 ft. 11. $53945. i^. $9505.67. i^. 1909609. U. $5044.25. 15. $16948.50. 16. $1417.16. 17. 702. 18. $36.50. 19. 8346. -^^. 16552. Art. 105. 12. $4743. i^. $1956. U^ $6190. ^(9. $40.50. 21. $30.59. ^^. $622.50. <^^. $16120. Art. 107o 2. 12771. 3. 25880. .4. 34104 J. $1239.30 ; $1713.15. 6. $3885.75; $4521.60. 7. $2209.82; $2383.74 8. 482400 ; 430944; 874752. 9. 2958216; 5606496 ; 7088104 10. $85692.24; $279759.96; $17188448. 11. $2529.25. 12, $319192. 13, $14064 U. $264958. 15. 404914. 16. 186516. 17. 241768. 18. $51188.62. 19. 17902976. 20. $154037.36. 21. 15704325 da, 22. 2082600 cts. 23. 1508741097. 24. 1587862270. 25. 8654860576. 26. 819847360a 500 AI^S WERS, ^7, 982275037. 28, 3363731415. 29. $2715413.50. SO, $21718.16. 31. 416304. 32. 0. 33. 947363302. 3J^. 5395144320. 35. 72618. 36. $3594.24. 37. $4101.25. 38. 51408 ; $7454160. S9. 277536; $49956.48. Art. 109. 2. $3505.92. 8. 3605472. ^. 3906168. 5. $19789.44. 6. 84338.28. 7. 16810320. 8. 54793296. 9. $109804.80. 10. $9212. 11. $430.08. 12. $19234.32. Art. 110. 5. $472. G, $1824. 7. $840000. 8. 600000. 9. 12600000. 10. 104000000. 11. 126930871- 800. 12. 350310024- 000. 13. 96000. 128000. 268800. U' $400000. Art. 113. i. $1617.30. 2, $50.19. S. $829.56. 4. $3023.75. 5. 17920. 6. 2878. 7. 37200. 8. 151218. 9. $7198.75. 10. $18801, Whole. Farm. $4617, Stock 11. $8232. 12. $25 loss. 13. 77050. U, 92500. 15. $1714.50. 16, $43187.32. Art. 133. 15. 1887; 7303; 2883. 16. 47208^ ; 2754; 131181|. 17. 48475-1; 672971 ; 115458J. 18. $172.65. 19. 5801i lb. 20. 2584| days. ;^i. $820.50. 22. 714741 mi. 8219 men. 20116J A. 63362 rd. 26. 1592 bbl.; 883f " ^7. 9375 bu. 28. $108 50. 29. 93 oranges. 30. 91 yd. .?i. $256. Art. 136. 3. 2340^; 204711 ; 1424it; ^4 1170/,. 4. 2248^1; 2070i|; 1610|f; 1762f|. Art. 138. 9. $9.58. m $14.89. 11. $25.21. i^. 354 times. 13. 416 " i4. 672 " 15. 1763 " 16. 3300 ** i7. 13A«^ " 18. ^%% - 2^. 642^V; 592H ; 2201f^. 20. 1083 AV; 25414tW 21. $1823/A. 22. $97. 23. $76. 2J^, 475 acres. 25. 37 horses ; $110 left. 26, 394. -^7. 5482. 28, 7198. ;^5/. 31416. 30, 7071. ^i. 8723. ,?^. 610. 33, 28004Jf If . 31^, 1172fff 35. 4321. .?^. 2036. .^7. 3645||fJ. . 2008; 178L ANSWERS. 501 SL $16550. $11925. S2. 24 boxes. ^3. 358 cords. $4 cost. ^I^. 288. 35, 2. 36, 10. 37, 1476. $8, 469. Art. 165. 2, 2. 3. 5^ 7. 3, 22, 3, 5. 19. Jf., 3, 5, 163. J. 2, 7, 13^ ^. 3^ 5, T. 7, 2, 3, 5, 7, 11. ^. 3, 5, 7, 11. 9, 2, 3^ 163. ;a 22, 3^ 5-', 7. 11. 3', 5, 7^. 12. 11, 31, 41. 13. 2«, 5, 101. i^. 2'^ 3, 7. i5. 3^5^7,l9. IQ. 19, 23, 29. i/. 2.5,7,11,13. 18. 3^ 5, 72, 13. 19, 2,5,7,11,41. Art. 170. 2. 14. 3. 32. .4. 5. 5. 18. g. 144. 7. 22. «?. 42. £>. 24. Art. 2. 4. 5. 7. 4. 27. 5. 2. 6. 1. 7. 13. 171. 8. 13. P. 113. i^. 17; 87. 11. 124 ; 2. i^. 12 ft. i.?. 3 bu. U, 4329 bags. 75. 5; 9; 11 hr. m, 8162 rails. Art. 177. 2. 2856. .?. 120. 4. 450. 5. 30030. ^. 13860. 7. 1680. <5. 5280. Art. 178. 2, 4896. ,?. 16800. 4. 51282. 5. 1560. 6. 7200. 7. 3060. ¥^- 112 5, 6, 7. 5. n. U, 15. ¥, 21 . 32 . 12 ¥8 » T¥ > ?B 63 . 50 . 35 TIT » TIF » ?TF 86 . 35 . 44 g^O" » ¥11 » t?F- 12 . 33 . 34 fF> ¥«•» TF- fi 't yf J tI« 160 . 27 . r?8 f TB^F » !%• 502 ANSWERS. Art. 221. -? i IS ^. 82i|. ^. 3t3\^. s, 9i|. ,/a 106^^. u, 693V 13. 25i^v y^- i-4. 9A yd. 15. 201f|. 17. 19i|?. Art. 224. A- is 5. 6. 26 j\. 7. 164||. S. 9. ri' JO. 11. 'tI- 12. 13tV 13. 7|f. U> 10/5. 15. ■^ilF- 16. 16:A%. 17. 183.Vf. 18. 248^1,. 19. 115|f. Art. 226 1, 885»j, tl greater. 2. 69|§. 3. ISSii. 4. 1014||. 5. A- 6. If. 7. m^. 8. mi S. IrVs- the ^6?. 6^. ii. 34i|. 13. 158Jf. i>4. 32811. i5 265i%. Art. 22a ^. 7f. «?. 2f . 5. 1|. e. 13^. 7.61. 8.1 9. 16|. i6>. 24 i^. 15. i.7. 2|. 14. 20. i5. 126. i6'. 128f . 17. 255. ic?. 5|-f. 19. 72. ^^. 119^. ^^. 1532. 23. 1287. ^4. 5386. 25. 3949i. ^g. 15099. ^7. 1275a^'5. 29. 1212. ■* ^^. 3624. 31. 7429. ^^. 3729. 33. 13272. ^4. 10200|. ^5. 23586. 208993k 322. 36. $580. 37. $15f ^<5. $1769i-V. ^^. $940f. 40. $7i; $221; $60 ; $11^1 . Art, 232. 3. |. 4. «. 5. f. ^- sir. ^. 9ili. 10. 512. ii. 32. i-^. 1530. 13. 62|. i^. 5278. 15. 18^. i7. $69|. 18. $211 J. i5. $345. 20. $12/xF. :^i. $63;V ^^. $145|. 23. |23|. ^4. $11.65i. 25. $2293|. -g'^?. $7196. ;^7. $5734. 28. $47|-J. ^o »5t 1 3 " ^^. $28J. ^i. $199|f. 32. $73:rV- ^:?. $11 Off. S4^ 424|f . Art. 235. ^ 4 / 18 ^- A- 10. 16AV- i5. $i^» i^. 5i|. i^. tItt a. 5lOi|. /5. 20. $85|. ^i. 39^1^. ^^. 6|} lb.. 23, 176jig lb. Art. 238< 2. 117. ^. 126. 4. 205f . 5. 408ifo 6. 877|. 7. 1486U. T^» T^> 1. 72- ;^. 84. 5. 3043J. 6. 9072. 7. $4613. <9. $l0588rV. 9. 7yV. m $10946. 11. |4. i^. $.75. i.?. $4577/ff. U, $4103^1 iJ. 4^ tons 16, 17. 18. 19. 146 A miles. 3?. 13J days. $192^1. 21. $5625. 22. Inc'd^Q. I'e?. Dim'd 3«j. 21,. 9|| bu. ^5. 15. 26. %\. 27. 878 bbl. 28. $3829^; 47 acres. 29. 31. 20 bbl. 32. $1}. 33. $1840. ^4. 152i ft. 35. 1,Vt1 SJ. ^6\ $3224cotton. $24 18 sugar. $1488mores. $9o72 total. 4i. 45. Of?-* 1 16 22jl,. Art. 267. 18. .598. i5. .0825. ^6>. .0012. 21. .000074. ^i'. .0000105. 23. .000099010. 21,. .437549. , 25. i^04001SJ>'. ^^. 1600.00000- 024. 27. 495705000.- 0043075. 28. 4735000.- 00903624. Art. 283. m. $|. 17. $tV. 18 -rV ^^. A- ^^. if. ^7. $1. 22. %%. 23. %\. 25. #. 27. $15|. ,f^. $36J. ^S>. $9|. 30. $27f. ^i. 24j-V 32. 84^V ^^. 38iV ^.^. 104y^^. Art. 285. 3. $.75. 4. $.875. 5. .56. 6'. .9375. 7. $.8. 6\ $.495. 9. .024. i^*. .8125. ii. .83333-1-. 12. .25925 -f-. 13. .78785-1-. U. .24666 + . 15. .60625. 16. .050784-. 17. .003125. 18. .005825. 19. .7. 20. .082. ,^i. $1,875. 22. $.066. ^.5. 101.75. 2 J,. 225.625. ^5. 11.125. -^6\ 8.6825. 27. $.934375. 28. $4,008. 29. 12.69. Art. 288. ;^. 1.703326. 3. 599.007. 4. $206,874. 5. .058815. 6. 51.180608. 7. $275,215. 8. 150.0680325. .9. 79.9992. 10. 111.233 A. $70.03. 1.5547 + . $7062.15. 387.33 rods. $598480. Art. 290. 2. 253 86319. 3. $533.06. $26 6875. .376118. $161,085. 1.99655. 10.040174. 103.5. 4.9999875. $.25. 12. $.0625. 13. 6.8045. U. .238517 + . 1.878125. $129.0625. .35|. .57675. .09. 2194.85 A. $6.458i. $411.58. 1.6625.' 4.1375. $95. $47.07. 5. 6. 7. 8. 9. 10. 11. SOi AKS WEBS. Art. 293. ^. .33615. S, 14.21628. 4. .00087. 5. 24.5470625. 6. $105,138. 7. $36.0062. S. 572.8. 9. 620.7. 10. 1.375. ii. 676. i^. 20.496. IS. .04765825. U. .0431388. 15. 7.03125. 16. 15 015. 17. .0084375. 18. 1252.6875. 19. $53.5. ^0. 114.75. n. .0815. ;^^. $155.8475. ;g',^. 556.718bu.+ U. $446.25. ^5. $438. ^6. $14891.925. ^7. $53.696 4-. £8. $12300.75. ^9. $113,235 + . 50. $101,175. 51. $389.49. S^. $242,937 + . 53. 402.788976. 54. 4.437. 55. 1.69064. 56. 7.03175. Art. 296. f. 22.68|. S. 4500. /,. .2. 5. 1.25. ^. 36.4. 7. 4602. ^. 73.73; 24.5766+; 5898.4 ; 85.0730 + . 10. $15,125. 14 i^ times ; 80 - 6.42 + '^ 11. 8|; 6.661; .075. 1£. 3.131331; 313.133"! ; 3131.33|; 31313.31; IS. 10. 387.5 ; 38.75; 3.875 ; .3875. U. .6455. 15. 50000 times. 16. $8000. 17. .4. i^. .16. 19. .1344. f^. .0175. <^7. .00734. ^2. $.72. .00001. 100000. 121.875 (5. .00331 ; 7. 23 tons. 8. 12 coats. 17 horses. 136 bbl. $65,406 + . S^. 550 lb. 55. $6.25. S4. 135 lb. J5. .831. 56. 1554. .^7. 688. .?«?. 2. 07887 + . S9. 1744.0598 + 40. $295. Art. 805. 1. .4375. ^. .8; 875; .36; .8125 ; .575 ; .088 ; .385. ^S. 25. 29. SI. 4. .857142 ;^ 7; .8i; .324; .476190; .17073. ^ 6. .416 ; .53 ; .590 ; .36 ; .313. 7. .12; .125; 2941176470- 588235 ; 484375 ; .6 ; 28125; .088; 238095 ; .2288. Art. 306. O 5 5. f f . /? 1 2 J 42 /^ .'JO i^. «. ^^. If. Art. 307. P 26 . 11 . /T 523 ^' 6^B^' e. If. 7. if 95 J. yg-. Art. 313. .4. $172. 5. $19.83 J. 7. $856. 8. $384. Art. 315. ^. $157,875. S. $3986.722. 4. $44.83.V. 5. $4696.30. 6. $65,875. 7. $438.75. 8. $9.1875. 5. $40,176. 10. $17.71. ii. $325.80 + . i-^. $183.15. 13. $212.75. 14. $85.93 + . 15. $23.96 + . Art. 316. 2. $6.68 + . S. $30.34 + . 4. $630.70 + . 5. $39.65. 6. $20,173 + . Art. 320. 1. Dr. $812.72. 2. Cr. $21788.- 16. Art. 327. 1. $448.07. f. $1489.46. S. $1489.84. 4. $6053.50. 5. $81.80. 6. $258.85. 7. Cr. Bal., $169,675. 8. Note to Bal., $176.16. ANSWERS. 505 6, 6. 7. Art. 328. 1. $60. 2. $59.57. 3. $21,375. A. $7.50. $3228.34. $3.40. $.50. 8. $1165. .9. $83.531J. 10. $191.10. ii. $4.6Sf. 12. $122.50. 13. $.33^. U. $.15. i5. $176,475. ie. $104.10. 17. 23 bu. i^. $21,125. 19. 134? tons. ;^^. 21557.47343. 21. 48 lb. each. 22. $1581, gain. $450. $5.25. $1.25 per C. .15. $4.50. $5.06 + . $196.21 + . 11001b. 80 bu. 1.69 + . $3.40 + . 188 bu. $74, cost. $59, selling price. $232,745 g'n, 24. 25. 26. 31 32. 33. 34. 35. m Art. 425. 2. 45515 gr. 3. 105948 oz. 4. 910 in. 5. 68245 min. 6. 63984 ft. 7. 2046 in. ^. 222 eighths. 11. 12. 13. 9. 43695 sq. ft. 10. 224800 P. 8960 A. 29106 1. 9696 cu. ft. 14. 6216 pt. 15. 792 qt. 16. 3 4800. 17. 1008 gi. 18. 14918 lb. 19. 5480 pwt. 20. S4785. 21. 525600 min. ^^. 7948800 sec. 23. 8784 hr. 24. 19325. ^5. 200 quires. 26. 864 doz. ^7. 78360 d. 28. 2650 ct. <^P. 8280 d. $1045.50. 960 rd. 563 bbl. 80 boxes. $29.25. 35. 13440 times. 36. 1440 min. 37. 2160 sheets. 38. $432. 39. 2419200. 40. 11082240. 41. 414.96 St. mi. 42. 36 of each. 45. 876576 hr. 44- 1485 vols. y^. 256 pp. 46. 3625 lb. 47. 8344 lb. 48. 29501b. 4^. 32620. 5(?. 6325 lb. 51. 31501b.N.T. 52. 7545 *' '' 5.?. 461824 '* 54. 1400 lb. 55. 1800 lb. 56. $136,262. 57. $124,095 + . 58. $4,825. 59. $10.13|. Art. 428. 2. 15 w. 4 da. 9 hr. 40 min. 3. 10 mi. 8 ch. 20 1. 4. 2031 lb. 9 oz. 10 pwt 5. 50 mi. 6. 1605 A. 7. 1 sq. mi. 8. 125 cu. ft. 840 cu. in. 9. 297 C. 26 cu. ft. 10. Id hhd. 19 gal. 3 qt. 1 pt 11. 846 bu. 12. 264 bbl. 26 gal. 3 qt. 13. 12965 gal. 14. Cong. 63, O. 2, ? 10. 15. 14 lb. 10 oz. 18 pwt 22 gr. 16. 25 T. 15 cwt. 70 lb. 17. 25 cwt. 37 lb. 15 oz. 18. 12 lb. 6 oz. 19. 201 bu. 20. 15| bbl. 21. 203 bu. ;^-^. 31.72 quin. 23. 5 w. 1 da. 1 hr. 1 min. 1 seG 24. 191 mo. 8 da. 11 hr. 40 i^h^ 25. 557° 33' 20". 26. 87 deg. 50 naut. mi. 27. 836 gro. 1 doz. 4 pens. 28. 227| doz 29. 251 sc. 30. 22 Rm. 7 Qu. 10 sh. 57. 151 Bund. 8 Qu. ,5^. 411 Cr. 2s. 33. 2038 fl. 34. 80 half-sov. 55. £44 2s. 2d. 2 far. 36. 450 f r. 57. 46sov. 6s. 3.9 + d 38. 200 marks. 39. $1689600. 40. 4725 lb. 41. $32.55 N. Y. 42. T 3'. 43. 456 da. 12 hr. 45 mii^ 44' $1.87^. 45. $48. .4^. $56.ia 506 ANSWERS. Art. 431. f . i ft. 3. 4 gr. 4. f pt. 5. .24s. I sq. rd. .32 pt. If yd. .83 ft. I oz. I yd. .252 inin. 3% sq- rd. 6. 7. 8. 9. to. 11. 12. 13. U. 15. Art. 483. ^. 10s. lOd. 3. §4 3 1 31 gr. 16. Jf.. 85 rd. 5 ft. 6 in. 6. 9 oz. 6'. 3 ft. 9 in. 7. 8.8 oz. <9. 17 da. 3? lir. 0. lis. 1.2d. 10. 86 P. 4 sq. yd. 5 sq. ft. 127^j sq. in. 11. 6 1 oz. Avoir. 12. 14 cu. ft. 691^ cu. in. 13. 5" 48' 7.2". i.^. f 3 5 TU 36. 15. 18cwt.961b.14oz. 16. 55 gal. 1 pt. 17. 16 sq. yd. 7 sq. ft. 3() sq. in. IS. |8 3 1 3lgr.7A i5. 6 gro. 1 Of doz. fO. 2 mi. 101 rd. 6 ft. 6i in. ^i. 3 gal. 3 qt. 1 pt. 2gl. 22. 2 pR. 2 qt. 1 pt. 23. 212 rd. 2J^. 4 T. 5 cwt. 55 1 lb, 25. 1 A. 60 P. 2G. 2 Cd. 89.6 cu. ft. bi. 57 rd. 9 ft. 10| in, 28. 6 Qu. 6 sheets. ^P. $54.16|. Art. 435. ^. tV gal. 3. £^V 4. .01 bu. 5. .0001 lb. ^. .0004 ton. .018 ton. 7. ^\ cord. 8. .00045 oz. ^- Wiri) *on. i^. ^1^ da. ; .005 da. 11. .02 rd. 12. fl pt. less. 13. r^T^ A. Art. 437. I- bbl. .4375 Cd. .22+ lihd. 5 6 9 IK 6. 7. 1 .092. 8. 9. .005489 10. .3. 11. 12. if. .581 lea. Tp. LO. T(T. U. .001625. 15. .09. /6\ If. 27. £1-|. Art. 438. i. 44352 steps. 2. 18 li. 45 min. 3. $53.1665125. J^. $199.25. 5. $111.94. ^. £60. 7. $128.23|. 8. $34,574. 5>. 7083^3 bu., 111. 657 f bu., La. 634|f bu., N. Y. 10. 57141 bu., Ct. 5333^ bu., N. J. 11. $332,679. 12. 15 carats. 13. 424.98775 A. U. $90. 15. $13457.46 + c 16. $3,525. 17. $87,815. 18. $2.54+. 19. I. 20. $138.95. 21. 3125 bu. 22. 720 centals. 23. 5 bbl. 152 lb. 2^. $21,988 + . 26. 3.02. 27. 487^- Rm. 28. 307^V Rni- Art. 440. 3. 22 yd. 2 ft. 10 in. 4. 19 Cd. 3 cd. ft. 13 cu. ft. 5. 2 hhd. 17 gal. 2 qt. 3 gi. 6. 15 h. 28 min. 7. 60 gal. 1 qt. 8. 22 cwt. 84 lb. 141 oz. 9. 10 Pch. 9|1 cu. ft. 10. lb 11 32 32gr.5. ii. $133.24. 12. $61.50. i«?. 31 yr. 11 mo. 3 da. Art. 441. 3. 25 Cd. 6 cd. ft. 4 cu. ft. 4. 239 rd. 11 ft. /). 8 cwt. 41 lb. 10 oz 6. 10s. 7id. 7. 5 lb. 3 oz. 10 pwt 8. 1 w. 6 da. 5 lir. 17 min. 16.8 seG 9. 29 gal. 1 qt. 1 gi 10. 1 mi. 193.7 rd. AN^SWERS, 507 u. 12. 13, U. 15, 16, 17. 18. 6. 6, 8, 9. 10. 11. 5. 6, 7. 8. to, 11. IS. 14- 6| doz. 13 3 5 3 3. 20 h. 24 min. 65.44 P. 8 gal. 3 qt. $7300.71 + . 13 cwt. 38 lb. 30 Cd. 5 cd. ft. 14 cu. ft. Art. 442. 7 yr. 9 mo. 1 da. 3yr. 11 mo. 28 da. 2 yr. 5 mo. 24 da. 258 da. 1 yr. 10 mo. 12 da. 6h. 204 da. 157 da. 21 h. 2 yr. 5 mo. 8 da. 9 h. 22 min. 2 yr. 8 mo. 10 da. 4 hr. 14 min. 59 sec. Art. 443. 26 bu. 1 pk. 6 qt. 39 Cd. 3 cd. ft. 13 hlid. 42 gal. 3qt. 2715 bu. Cong. 25 O. 6 § 11 3 5 TTl 36. £120 18s. 6d. 189 A. 40 P. 16 sq. yd. 6 sq. ft. 89 T. 11 cwt. 1 qr. 19 lb. 14 oz. 557 yd. 2 ft. lU in. 13 T. 3 cwt. 67.85 lb. $196,796. $125.25. Art. 444. 51 A. 31 P. 8 sq.ft. £7 Is. lid. £5 Is. 4f d. ^. 31 bu. 1 pk. 5 qt. Ipt. 28 bu. lpk.l.9pt. 23 bu. 2 pk. 2 qt. ipt. 5. 12 yd. 5|f in. 6 yd. 2f f in. 6,2Cd.5 cd. ft. 13J cu. ft. 7. 17160 rails. 8. 1 sq. mi. 42 A. 112 P. 26 sq. yd. 8 sq. ft. 9. 337 yd. 1ft. 7i in. 10, 70 times. 11, 243 boxes. IS. 9 cwt. 42 lb. 14^ 165 A. 25 P. 24.4 sq. yd,, nearly. Art. 448. 2. 107° 19' 48f". S, 122" 26 45 ' W. 4. 71° 12' 15" W. 5. 90th W.; 90th E.; 180th E. Art. 450. 2. 50 min. 21^ sec. S, 1 h. 17 min. 24 sec. A.M., or next day. 4, 5 hr. 57 min. 49 sec. 5. 5 h. 59 min. 51 sec. 6. 1 hr. 3 min. 58 sec. 7, 1 h. 13 min. 32| sec. 8. 51 min. 18 sec. 9, 1 hr. 40 rain. 8 sec. 10. 6 hr. 28 min. 27 sec. 11. 1 h. 33 min. 27 12. 5 hr. 6 min. 15^ sec. A.M. at Cinn. 4 hr. 53 min. 43 sec. A.M. at Chi. 4 h. 43 min. 13 sec. A.M. at St. Louis. 13. 12 h. 7 min. 41 sec, at night, B. 4 h. 53 min. 4&i sec. P.M., St. P. 2 h. 58 min. 6 sec P.M., Ast. Or. 14' 5 h. 46 min., latei Rome. 5 h. 5 min. 32 sec, later, Paris. IS. 10 h. 58 min. 37 sec., gains. Art. 454. 1. 52 ft. 9'. 2. 319 ft. 4' 3". 3. 23 ft. 10' 9'. Art. 456. 2, 68 ft. 3, 55 ft. 10' 3" 2'" 8"". 4. 240 ft. 9' 4". 5. 50 ft. 9 10 " 6'". Art. 465. 1. 63 sq. yd. 2. 7i ft. wide. 3. 61 ft. long. 4. 152sq. yd. 1 sq.ft. 5. 348 sq. yd. 4 sq.ft. 6. 379i sq. rd. 7. 32 sq. ch. 2 P. 8. 427i sq. ft. 9. 7 sq. rd. 1 sq. yd. 6 sq. ft. 88 sq. in. 10. 18 ft. 3 in. width. 11. 7 ch. 25 1. length 12. 18 yd. 2 ft. 13. 58^ sq. yd. 14. 48. planks. 15. 44 yd. 16. 260 yd. 17. 88| yd. 18. 81f yd. 508 Aiq^SWERS. 19. $98.54 + . ^0. $106.48. 21, $81.87. 22, $146.40. 23, $60.95. 24,, 1080 tiles. 25. $608.40. 26. $65,475. 27. 21.29fi squares. 28. $139.57. 29. $198. 30. $27,378. 31. 840 sods. 32. 28f yd. 33. 13f rolls. 34^. $71.60. 35. $13,525 + . 36. 11316 shingles. 37. $25.55. 38. $447,989 + . Ai-t. 467. 1. 90 A. 2. 32 rd. wide. 3. 190| farms, .^. .025 A. 5. 264 rd. 6. $7504.80 + . ^. 25 rd. 9. $220 less. 10. $4000 gain. Art. 468. 1. 80 A. ; i Sec. 2. 5760 rails ; $230.40. 3. $340 gain. 4. 240 A. ; -I Sec. 5. 120 A. left ; $27.20 gain. 6. 420 A. left ; $635 gain. Art. 474. 1. 96 cu. ft. 2. 108 cu. ft. 3. 84 ft. 4. 221 cu. ft. 6. 208 cu. yd. 6. 3 cu. yd. 26 cu. ft. 297 cu. in. 7. 7 cu. yd. 11 cu. ft. 200 cu. in. 8. 5 cu. yd. 25 cu. ft. 9. 1^ in., height. 10. 8 in., height. 11. 9 it. 2 in., length. 12. 4840 cu. ft. 13. 12y\ Cd. 1/h 8 ft. 15. $13,182 + . 16. $166.60. 17. 8 ft. 18. 80 cans. 19. $410,156 + . 20. 24 ft. Art. 477. 2. 31278x\ bricks. 3. 60 Pch. 4. 49y\ Pch. 5o 62006+ bricks. 6. $1607.82 + . 7. $471.66|. 8. 667xV. * P. $3276. 10. $4-33.53. ii. 2142 ; $333.20. Art. 481. 3. 53i. 12. $614 + .^. 93i. 13. $1,064. 6'. $4:20. i5. 13^ ft. 7. $15.75. i. 23%. ii. 50%. 12. 37|-%, ^«^. 60%. Art. 533. 3. $9375. .^. $8.80. 5. $1.50. e. $14.14. 7. $16666.66|. 5. A. $16000 ; B. $10000. Art. 534. 2. 6.86. 3. .75. >^. $4.91. 5. $.20. (?. $244,094. 7. $183.33^. 5. $586.66f. S>. $6553.60. Art. 535. 2. $1.47. 3. $150. 4. $1.06|e Art. 547» ^. $378,125. c?. $82.11. 4. $379.40. 5. $285.19. 6. $20.18. 7. $584.17^. ^. $96.90. Art. 548. 2. difo. 3. 5%. ^. H%. 5. 2f %. 6. 5%. 7. 6i%. Art. 549. 2. $2784. ^. $3500. 4. $9600. 5. $9000. 6. $960.40. Art. 550. 2. $3750. 3. $588.33io 4. $25372. Art. 551 2. $4696.65. 3. $3182.55. 4. $1500. 5. $10648. 6. $6400.76 Inv. ; $320.04 Com. 510 AKS WEBS. 7? 31000 lb. 8. $10623.44. 9. $44231.71 Inv.; $1105.79 Com. 10. 1640 yd. Art. 553. 1. 48 bu. S, $1700, 1st yr. ; $1785, 2d yr. S, 24^3%. 4. $67.50 gain ; 12% gain. 5. $3640. 6. $40842 cost. $6807 gain. 7. $30000. 8. 40§%. 9. $468.75. JO. Loses 25%. IL 25ffh nearly. 13. 5%. IS. $4948.125. 14- $2964 whole gain; 21|av. gain %. 15. Prints® $.15; Cassim.@$4.06}; Ticking @ $.25 ; Shawls @ $9.20 ; Thread @ $.875 ; Buttons @ $1.25; Amt. @ $729.96. 16. $705.12. 17. $155.09. 18. 61788.6 lb. +. 19. $.50 ^0. $10532; $132 Com. ^1. 5«%. ^^. $8.875;loss4§%+. ^S. $3049.20 wiiole gain; 50% gain + . Art. 507. e. $101 25 int. ; $551.25 amt.; $21 int.; $471 amt, 3. $71.32 int. ; $318.32 amt.; $16.47 int. ; $263.47 amt. 4. $208.33 int. ; $708.33 amt. ; $22.92 int. ; $522.92 amt. 5. $3.46 int. at 6%; $4.03 int. at 7%; $4.32 int. at7i%. G. $115.70 at 5%"; $185.12 int. at 8%; $208.26 int. at 9%; 7. $196.41int.at6^%; $235.70int.at7^%. 8. $58.97 int. at 10%; $73.71int.atl2i%. 9. $886.40 amt. 10. $71.37 amt. 11. $1176.50 amt. 12. $442.50. Art. 569. 2. $12.58 int. at 6%; $8.39 at 4%. S. $92.53 @ 5%; $148.04(5)8%. 4. $269.47(^7%; $288.72 @7i. 5. $61.12 int. 6. $292.50 int. 7. $1204.12 amt. 8. $276.52 amt. 9. $41.27 int. 10. $421.99 amt. 11. $85.72 int. 12. $13227.50. Art. 573. 2. $22.70 int. S. $3.84. 4^ $38.34. 5. $242.94. 6. $318. 7. $269.34. Art. 574. 2. $120. S. $.04. 4. $10.58. 5. $82.36. 6. $10.96. Art. 5753 1. $58.93. 2. $8.40. S. $67.67. $159,745. $67.09. $38.11. $8.63. $3647.61. $115.20. $1066.36. $2010.42. $142.45+. $1886.17. $131.40. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. $8.93. 17. $263.83. 18. $828.07. 19. $1986.60. 20. $3925.17. 21. $1120.69. 22. $76.67. 23. $1931.40 loss. Art. 577. S792. $6936.09; $6069.08. $516.71. $669.12; $334.56. $10000. Art. 579« 2. $1000. 3. $1403.08. 4. $1500. 5. $889.25. 6. $650.80. ANSWERS. 511 10%. 40%; 16f%; Art. 581. ^. 7%. 3. 7%. ^. 8i% o. 6%. e. 2% a month. 7. lOA-%. ^.25%; 161 9. 100% ■ 28 A % 10%. ?a 7|%. Zi. The 2d is 1J|% better. Art. 588. 2. 7 mo. 10 d. 3. 6 yr. 8 mo. 4. 7 mo. 6 da. 6. 3 yr. 4 mo. 24 da. 6. 33i ; 20 ; 16| ; 13|; 10 yr. 7. 50 ; 40 ; 28f ; 25; 16 yr. 8. 12|; 6^; 25 yr. Art. 586. 2. $428.76. 3. $189.15. 4. $1176.14. 5. $100.32. 6. $41.99 + . 7. $1495.77. 8. $53.38. i(? $1525.64. ii. $1540.79. 12. $987.23. 13. $1934.84. 14. $18142.81. Art. 589. S. $464.10. 3. $7308. 4. $11.30. 5. $1161.04.. 6. $1047.52. Art. 597. 3. $659.94. 4. $30.14. 5. $162.25. Art. 598o ^. $312.47. 3. $355.16. Art. 003. 2. $281. 83o 3. $102.90. 4. $1137.61. • 5. $43.65 in favor of dis. 6. $931.20. 7. $333.26. 8. i^jWfo. 9. $931.83. 10. $.05 per bbl. more profitable to buy at S8.75 on 6 mo. 11. $3677.75. Art. 615. 2. $6.27 Bk. dis. $591.23 proceeds. 3. $1614.48. 4. $10839.83. 5. Mat. Oct. 30 ; 81 days term of dis. ; $940.38 proceeds. 6. Mat. April 8 ; 46 days term of dis. ; $917.37 proceeds. 7. Mat. Aug. 2 ; 79 days term of dis. ; $1295.82 proceeds. 8. Mat. Dec. 15 ; 30 da. term of dis. $1281. 77 proceeds, Art. 617. 2, $1434.20. 3. $719.61. Jf. $1951.03. 5. $2291.44. 6. $821.46. 7. $659.88. 8. $368.25. Art. 619. 2. $188 43 bal. Julj 1st. 3. $4.90. 4. $869.36. 5. $327,927. Art. 648. 2. $34256.25. 3. $16856.25. 4. $15843.75. Art. 649. 2, 250 shares. 3. 220 " 5. 220 " G. 480 " 7. 200 " Art. 650. 2. $25500. 3, $21100. 4- Art. 651. 2. 8|%. ^.8% bonds at 110 f|% better. 4. 6% bonds at 84 1^% better. 6. 9H%. 6. 5if %. 7. 3|i%. Art. 652. 2. 62i. 3. 33i^. 4. 71f. 5. $40. 6. 75 ; 66f . 512 ANSWERS. Art. 653. 2, $5466.28. 3. $268.20. ^. $262.66 better to pay in currency. Art. 654. 2. $4000 ; $4035.87 ; $4109.59. 3. $74000. 4. $1755800. 5. Dim. $26.25. G, $113 per annum. 7, Stock invest, is $50 better, or f-?-% yearly. 8, $21384 in N. Y. S. 6's; $42768 U. S. 5's of 81. 9, $792. Art. 664. 2. $42.75. 3. $24.06. 4. $187.50. 5. $156.25. Art. 665. 2, 1}%. 3. i%. >^. !/.. Art- 666. 2. $13600. S. $8960. 5, $22220.77. (>. $49147.91. 7. $24500. ^. $24766.58. P. $9.90. Art. 675. 2, $284.78. 3, $1055.30. 5. $527.65. 6. $5888.57. 7. $4416.57. 8. $3263.93. 9. $1131.12 loss. 10. $7200. Art. 685. 2. $11350. 3. $19072.16. J^. $401920. 7, $25.09. 8, $87.38. 9, $112.50. m $226.50. 11. .0228 tax rate. $214.65. 12. $410.95. 13. $224.37. $178.13. 15. $420000. Art. 700. 2. $1566.15. 4. $4764.84. 5. $5153.24. 6. $6388 80. 7. $5632.20. Art. 701. 2. $787.46. 3 $720. Jf. $31645. 5. 451 shares. 6. 97J%. 7. $20108.35. Art. 706. 2. $2303.25. 3. $3317.63. 5. $134.78. 6. $352.67. 8. $421.09. 9. $566.50. 7. $801.94. '2. $4621,16. $5243.80. 13, U. $3500.40. Art. 707. 2. £1543 4s. 2d. If. 2318.84 marks. 5. 1664.13 marks. 7. 31888.83 francs. 8. 12918.75 francs. Art. 711. 2. $179.21. 5. 5.31 francs. 6. $4 987. 7. £1055 12s. 4d. ; £21 9s. 9.7d. 8. $32.78 ind. ex. 9. 696.374 ^uild.loss. 10. $12617.08. Art. 726. 2. $437.50. 3. $1706.25. J^. $1843.75. 5. $1234.38. e. $63.18. 7. $5775. 8. $2376.28 duty. $8815.75 cost in currency. 9. $1755.89. 10. $987.08. Art. 733. 2. 3 mo. 25 da. 3. 6 mo. 26 da. time of Cr. : June 27, '77 Eq. time. J^. May 5, 1875. 5. 5 yr. 20 da. from date of last paym't. 7. Nov. 26, Eq. time. 8. 73 da. term of Cr.- Feb 26, Eq. time 9. Mar. 7, Eq. time.; $1178.01 cash value. Art. 734. 2. Aug. 19, 1875. Eq. time. 3. June 7, 1876. ANSWERS. 513 June 27, 1874 ; Dis. $149.28. Apr. 23, 1874 $2337.02. May 20, 1875. Art. 737. 2, Dec. 13, Eq. time. S. Dec. 19. 4 Jan. 24, 1879. Art. 738. 2. May 18 ; $1486.17 due. 8, Dec. 5, 1875. 4. $2069.59. 5, Oct. 27 ; $2102.58. 6. $1272.33. Art. 739. 2. $2331.65 Sales ; $762.83 Charges ; $1568.82 Net pro- ceeds ; Bal. due, Dec. 27. 3, $3966.25 Sales ; $412.98 Charges ; $3553.27 Net pro- ceeds ; Eq. time Apr. 14, 1875. Art. 767. 2. 60 bu. 8. $100. U, $4.05. 5. 44J-bbl. Art. 770. 3 9 horses. Jf, 100 yd. 5. 16 men. e. 96 sheep. 7. $5355. 8, 7 hr. 135- niiii- 9, 855 bu. m 112|mi. 11. 591 da. 15. 16. 17. 18. $7320. 9 yd. 46 A. 134 P. $63. $10958.90. $3.25. 19. $89.60. 20. $120. 21. 2 yr. 6 mo. Art. 772. 2. 43 1- tons. 8. 5^ weeks. 4. 432 mi. 5. 15 da. Art. 774. 2. $498.08. 4. 1120 bu. 5. $6428.57. 6. 114/^ ream. 7. 220f Cd. 8. $52.79. 9. 9 men. 10. 546 bbl. 11. 2080 lb. 12. $100. 18. 26660^ U. $236.25. 15. 694| yd. 16. $1728. 17. 5 da. 18. 150 yd. 19. 3 yr. 4 mo. 24 da. 20. $11.66|. 21. 9 men. 22. 8.116 ft. 28. $48. -^.^. $53.08. 25. 1.6 mo. + Art. 782. 8. A's share $320. B's '* $216. Cs " $184. 4. A. $303.45. B. $337.17. C. $404.61. D. $682.77. 5. A. $1710. B. $870.20. 6. A. $6000. B. $8402.25. C. $5055.75. D. $3042. 7. $5785.20, the first-, $5142.40, the second. 8. $3516.80 A's gain; $5861.33^ B's '' $8205.861 C's '* 9. $269559.55 Ke- sources ; $26434.55 Lia. bilities ; $243125 Stock ; $125000 Origi. nal capital ; $118125 net gains $56700 Ames' share ; $37800 Lyon's share * $23625 Clark's share. Art. 783. 2. $2400 Barr ; $2666.661 Banka I $2933.33J- Butts. 8. $388,704+ A.; $249,109+ B.; $112,122 C. 4. $1344.164 A.; $2027.836 B. 5. $5700 A., • $3760 B.; $1340 C. 6. $1088.434 Crane ; $3868.862 Childs ; $2012.703 Coe. 514 ANSWERS. Art. 787. 2. $.32. 3. $.30 per bushel. 4. $6 gain. 6. $6.16. Art. 788. 5. 2 lb. of first ; 2 lb. of second ; 3 lb. of third. 4. 1 at $4 ; 5 at \ Sat \ 1 at f 5. 3 bbl. at $5^ ; 3 bbl. at $6 ; 2 bbl. at $7f . 6. 3 gal. at $1.20 ; 3 gal. at $1.80 ; 15 gal. at $2.30 ; 8 gal. water. Art. 789. 2, 10 cows at $32 ; 10 cows at $36 ; GO cows at $48. S, 10 lb. at $.80 ; 10 lb. at $1.20 ; 70 lb. at $1.80. 4, 12 yd. at $3^ ; 16 yd. at %U. 5. 150 acres. Art. 790. S, 30 men, 5 women, 20 boys. S, 331 gal. water. J^., 16, 24, 4, and 12 da. respectively. 1, Art. 792. 72 and 48. ^. D's age 16 ; E's age 24 ; F's age 84. 3. 15 bu. 4. 18 da. 6. 8f da. 6. Starch $2 a box ; Soap $3. 7. ^ da. ; First in 26| da.; Second in 40 da. ; Third in 20 da.; $180 share of 1st; $120 share of 2d ; $240 share of 3d. 8, 14 bbl. at $10 ; 6 bbl. at $7. 9, 16 min. 21yV sec. past 3 o'clock. 10, Wheat $1.33 J per bu. ; Oats $.50 per. bu. 11, 8 da. 12. $347.71. 13. 50 bu. 15. 27% nearly. 16. $7384-/V younger ; $11076i-| elder. 17. 146f ft. 18. $800. 19. $960 first ; $720 second ; $840 third. 20. $1570.31. 21. 506 lb. 22. Oct. 26, 1875. 23. $37439.998; $33345; $27359.999; $25108.82. 24. $1.60. 25. 42 geese ; 58 turkeys. 26. $5700. 27. $282.24 Sim. Int.; $2202.24 Amt.; $295.56 Com. Int. ; $2215.56 '' Amt.; $1673.93+ Pres- ent Worth ; $246.07 True Dis.; $283.20 Bk. Dis. ; $1636.80 Proc'ds; $2252.199 Face. 28. $315.79. $473 69. $710.52. 29. $900, July 28. 30. $.97|. 31. $10665.80 in U. S 6's, 5-20. $21331.60 in U.S. 5's of '81. 32. A. 3600 bu.; B. 1200 bu.; C. 1200 bu. 33. $1.72. 3 A. ^- 35. $5614.27 Net Proceeds. July 10, Eq. time. 36. $6400 M.'s Cap.; 15 mo. N.'s time. 37. $2023.22; Apr. 24 38. $2244.66. Art. 802. 2. 1369; 1764; 8136 ; 5625. 3. 3375 ; 5832 ; 74088 ; 157464. 4. 3969 ; 110592 ; 1048576 ; 24883^ n 49 . 1728 7 G501 .^125 8. 645.16. 9. 1191016. 10. 1958tV. / / 14 6 4 1 12. .00116964. 13. .015625. U, 46733.803208. 15. .065528814274496. 16. 33169 If. 17. 16.6056^. 18. 24.76099. 19. .000000250047. 20. 1520875. 21. 2023^V,.. 22. 5.887. 23. 640000. 24. 2540.0390625. 25. 125. 26. 1200, AIS^SWERS. 515 Art. 803. 3. 1764. 4. 2304. 5. 3136. 6. 9604. 7. 15625. 8. 11025. 9. 50625. 6>. 38809. L 116964. Art. 804. g'. 39304 4. 110592. 5. 262144. 3. 857375. ?-. 1953125. Art. 810. ?. 8 ; 16 ; 24 ; 81. ?. 9 ; 14 ; 21 ; 15. Art. 813. ?. 85. ^. 242. K 98. ?. 115. ^ 109. ?. 997. ^ 1432. K 5464. > 13 • Ti' > 25 •• 8T- •. .035. '. 14.0048 + . '. 1.5005 + . '. 7.625. '. 4.213 + . '. 103.9. . 59049. '. 3.00001 65-i-. '. 5.656854 + . . 1.5411. : .91287 + . . .04419. ^7. 36.37. ^8. 1.50748 + . ^9. 64. 31. 1. 32. 1.78 +. 33. 72. 34. 90. 35. 480.8827. Art. 815. 1. 1008 ft. 2. 240.33 rd. 3. 52 rd. 4. 200.56 rd. 5. 145ird. 6. $187.20. Art. 819. 3. 25. 4^ 55. 5. 101. e. 165. 7. 1015. 8. 1598. 11' «. i^. 1.42 + . i^. 34. U- .45. i5. 2.34. 16. 4624. i7. .0809. i^. .7936. 19. 5.73 + . ^^. «. -^i. .5569. S2, 1. ^^. 14.75. X 24, 60.8. Art. 821. • 1. 3 ft. ^. 8 ft. 3. 2 ft. .4. 12150 sq. ft. 5. 5 ft. 8+ in. 6. 9 ft. 5.3 + in. 7. 8 ft. 1.4 in. 2. 274. 3. 32. 4- 543. 5. 1.05 + . Ai-t. 829. 3. 8. 6. 149, 4' 17. 7. 16. 5. 33. 8. m^ Art. 822. Art. 843. 3, 765. 6. li. 7. 2. ^. 280. P. $1023. 10. $5314.40. Art. 853. 3. $3819.75. 4. $1292.31. 5. $3625. 6. 6 yr. 7. 7%. ^. $375.30. Art. 854. 3. $300. .4. $3907.665+. 5. $1182.05 + . G. $3725.87 + . 7. $629,426 + . Art. 882. 2. 600 sq. ft. 3. 42 j-\ sq. ft. 4. 22 A. « sq. ch. 13.45 P. 5. $449.07. e. $147. 7. 210 sq. ft. Art. 883. 2. ^ ft. ^. 13 in. 4. 28 rd. 5. 672 rd. 5A yd. ch. Art. 830. 2. 2. .5. 4. .^. 2. ^. 7^ ^. f. ^. A. Art. 831. 2. 9. c?. 15. 4.4. 5. 27. e. 11 yr. Art. 832. 2. 600. 3. 154. .4. 125000. 5. 78. 6. 57900 ft. Art. 840. 4. 6144 5. 3. 6. $524288. 7. $315,619 + . 8. $10485.76. Art. 841. 2. i. 4. 5. «?. 5. 5. 3. Art. 842. 2. 9. 5. 7. 4. 8. 6. 8i 7. 50 rd. Art. 884. 2. 111.80 sq. ft 3. 3 sq. ft. 1.7 sq. in. 4. 13 A. 41.76 P. 5. 849.07 sq.ft. 516 AJ^SWERS, Art. 886. j^. 39 ft. 3, 25 ft. 7.34 in. 4, 33.97 ch. 6, 28 ft. 3.36 in. Art. 887. 5. 45 yd. 3. 19 ft. 2.5 in. ^. 360 ft. 6f in. 6. 20 ft. Art. 898. f . 84 sq. ft. 3, 5J A. Art. 899. 2. 11178 sq. ft. 3. 28| sq. ft. 4. 2 A. Art. 900. e. 213 sq. ft. 5. 17 A. 8 ch. 3.4 p. Art. 904:. 15 ft. 10.98 in. 5 x«. 10.67 in. 5 ft. 7 ft. 3.96 in. Art. 905. 4. 318.3 A. + 5. 114.59 A. Art. 906. 3. 7 rd. 4. 19.098 ft. Diara. 59.998 ft. Circum. Art. 907. 2, 141.42 ft. 3, 23.4 yd. 4- 4, 7.07 ft. + Art. 908. ^. 32.98 sq. ft. + 3. 796.39 sq. ft. ^. 1 A. 75.62 P. land • 78.54 P. water. Art. 2, 84 3' 28. ^' -A- 5. 32 lb. 909. 13.7 oz. Art. 910. 5. 369 rd. L. ; 123 rd. W. 6. 3.5 in. 7. 221 ; 238 ; and 255 ft. 8. $75. 9. 126.78 rd. Art. 911. 1. $185.53. £. 35.35 ft. + 3. 403.7 rd. + 4. $5812.50. 5. $32.40. 6. 28.66 P. + 7. 5 A. ; or twice as large. ■8. $724.75. 9. 20 ft. 10. 98 A. 28 P. 11. 14.645 ft. 12. 294 rd.; 45.36 rd. 13. 14 A. 150.04 P. U, 6 in. Art. 918. 4. 207.34 sq. ft. 5- 168| sq. ft. 6. 263.89 sq. ft. 7. 301.177 sq. ft. Art. 919. 3. 274| en. ft. 4. $27. 5. 73.63 cu. ft. 6. $53.63. Art. 925. f . 834 67 sq. ft. 3. 4291 sq. ft. 4. 512.9 sq. ft. 5. $25. Art. 926. 3, 39.27 cu. ft. 4, $29.23. 5, 192000 cu. ft. vol 22284.6 sq. ft. surface. Art. 927. 2. 345 sq. ft. 3. 256| sq. yd. Art. 928. 2. 58.1196 cu. ft. 3. 38i cu. ft. 4. 64.99 cu. ft. Art. 932. 2. 28.27 sq. ft. 12.57 sq. ft. Art. 933. 2. 8cu.ft.313.2cu.in. 523.6 cu. yd. Art. 934. 2. 10 ft. ; 15 ft. ; and 20 ft. 3. 24 ft. ; 32 ft. ; and 40 ft. Art. 936. 1. 13.228 ft. edge. 2315.03 cu. ft. vol. 2. 11 ft. 7 in. 3. 1494.257 gal. 4. $5.46. 5. 576 ft. 6. 17.32 in. 7. 40 sq. ft. 7f . 8. 1 cu.ft. vol.of cnbi> 1 cu. ft. 659 cu. in. vol. of sphere. 9. 9 lb. 10. 5 hr. 26.4 min/ 11. 12 ft. 6.79 in. 12. 53.855 bu. Art. 937. 2. 99.144 gal. 3. 120.09 gal. % \ ^ UNIVERSITY OF CALIFORNIA LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW .915 Sf P 21 1920 aslarfeaito J* . .S3 1 30m-l,'15 W i fooZ 165925