U N I T UNIT 1 INCREASED BY THE SCALE OF TENS. DIVIDED, ACCORDING TO SCALE OF TENS. &c. 10000 1000 100 10 I1.1.01.001.0001.00001 &c. 1 INCREASED BY VARYING SCALES. DIVIDED, ACCORDING TO VARYING SCALES..~ & far. 1 1 1 1 1 1 6lb. L-oz. ~cwt. } Denominate. &c., for all denominate numbers. PROPORTION OF TO ALL NUMBERS. For explanation and of the numbers to each other. of the Diagram, see "GRAMMAR OF- ARITHMETIC." APPLICATIONS. ARITHMETICAL -DIAGRAM. L' —----- - -_ THE UNIVERSITY ARITHMETIC, RMBRACING THE SCIENCE OF NUMBERS, AND THEIR NUMEROUS APPLICATIONS tt Y CHARLES DAVIES, LL. D., AUTHOR OF FIRST LESSONS IN ARITHMETIC; ARITHMETIC; ELEMENTARY ALGEBRA: ELEMENTARY GEOMETRY; ELEMENTS OF DRAWING AND MENSURATION; ELEMENTS OF SURVEYING; ELEMENTS OF ANALYTICAL GEOMETRY; DESCRIPTIVE GEOMETRY; SHADES, SHADOWS, AND PERSPECTIVE; AND DIFFERENTIAL AND INTEGRAL CALCULUS REVISED AND IMPROVED EDITION. NEW YORK: PUBLISHED BY A. S. BARNES & CO., No 51 JOIIN STREET. 1850. Entered according to Act of Congress, in the year Eighteen Hundred and Fifty, BY CHARLES DAVIES, In the Clerk's Office of the District Court of the United States for the Southern District of New York. Stereotyped by RICHARD C. VALENTINE, New York. F. C. GUTIERREZ, PRINTER, Cor. John and Dt)utch-streets, N. Y. PRE FACE. SCIENCE, in its popular signification, means knowledge reduced to order; that is, knowledge so classified and arranged, as to be easily remembered, readily referred to, and advantageously applied. ARITHMETIC is the science of numbers. It lies at the foundation of the exact and mixed sciences, and a knowledge of it is an important element either of a liberal or practical education. While Arithmetic is a science in all that concerns the properties of numbers, it is yet an art in all that relates to their practical application. It is the first subject in a well-arranged course of instruction to which the reasoning powers of the mind are applied, and is the guide-book of the mechanic and man of business. It is the first fountain at which the young votary of knowledge drinks the pure waters of intellectual truth. It has seemed to the author of the first importance that this subject should be well treated in our Elementary Text Books. In the hope of contributing something to so desirable an end, he has prepared a series of arithmetical works, embracing three books, entitled First Lessons in Arithmetic; Arithmetic; and University Arithmeticthe latter of which is the present volume The First Lessons in Arithmetic are designed for beginners. The subjects treated are divided into separate lessons, each lesson embracing one combination of numbers, or one set of combinations. The Arithmetic is designed for the use of schools and Vi PREFACE. academies, and contains all that is usually taught in a course of academical instruction. The University Arithmetic is intended to answer another object.' In it, the entire subject is treated as a science. The scholar is supposed to be familiar with the operations in the four ground rules, which ate now taught to small children either orally or from elementary treatises. This being premised, the language of figures, which are the representatives of numbers, is carefully taught, and the different significations of which the figures are susceptible, depending on the manner in which they are written, are fully explained. It is shown, for example, that the simple numbers in which the value of the unit increases from right to left according to the scale of tens, and the Denominate or Compound numbers in which it increases according to a different scale, belong in fact to the same class of numbers, and that both may be treated under a common set of rules. Hence, the rules for Notation, Addition, Subtraction, Multiplication, and Division, have been so constructed as to apply equally to all numbers. This arrangement, which the author has not seen elsewhere, is deemed an essential improvement in the science of Arithmetic. In developing the properties of numbers, from their elementary to their highest combinations, great labor has been bestowed in classification and arrangement. It has been a leading object to present the entire subject of airithmetic as forming a series of dependent and connected propositions: so that the pupil, while acquiring useful and practical knowledge, may at the same time be introduced to those beautiful methods of exact reasoning, which science alone can teach. Great care has also been taken to demonstrate fully a} PREFACE. Vii the rules and to explain the reason of every process from the most simple to the mnost difficult. It has been thought that the Teachers of the country would like to possess a work of this kind, and that it might be studied advantageously as' a text book in our advanced schools and academies. To adapt it to such a use, a large number of practical examples has been added, many of which have been selected from an English work by Keith. In the preparation of the work, another object has been kept constantly in view, viz., to adapt it to the business wants of the country. For this purpose much pains have been bestowed in the preparation of the articles on Weights and Measures, foreign and domestic; on Banking, Bank Discount, Interest, Coins and Currency, and Exchanges. Although by law the hundred weight is estimated at 100 pounds, and consequently the quarter at 25 pounds, in the United States, yet the old hundred of 112 pounds is still much used; and in all our intercourse with Great Britain, goods and wares are so estimated. Hence, it was thought best in this arithmetic, intended for general instruction, to retain the old standard. In fine, it has been the aim of the author to publish both a scientific and practical treatise on the subject of Arithmetic, and one which shall in some measure correspond to the higher qualifications of teachers and the improved methods of communicating instruction. Several excellent works, of an elementary character, having recently been published on Book-keeping, it has seemed best to omit, in the present edition, the article on that subject, and to supply its place by matter of a practical character. FISHKILL LANDING, JANUARY, 1850. DAVIES' COURSE OF MATHEMATICS. DAVIES' FIRST LESSONS IN ARITHMETIC-For beginners. DAVIES' ARITHMETIC.-Designed for the use of Academies and Schools. KEY TO DAVIES' ARITHMETIC DAVIES' UNIVERSITY ARITHMETIC-Embracing the Science of Numbers, and their numerous applications. KEY TO DAVIES' UNIVERSITY ARITHMETIC DAVIES' ELEMENTARY ALGEBRA-Being an Introduction to the Science, and forming a connecting link between ARITHMETIC and ALGEBRA. KEY TO DAVIES' ELEMENTARY ALGEBRA. DAVIES' ELEMENTARY GEOMETRY.-This work embraces the elementary principles of Geometry. The reasoning is plain and concise, but at the same time strictly rigorous. DAVIES' ELEMENTS OF DRAWING AND MENSURATION -Applied to the Mechanic Arts. DAVIES' BOURDON'S ALGEBRA-Including Sturms' Theorem,Being an Abridgment of the work of M. Bourdon, with the addition of practical examples. DAVIES' LEGENDRE'S GEOMETRY AND TRIGONOMETRY. -Being an Abridgment of the work of M. Legendre, with the addition of a Treatise on MENSURATION OF PLANES AND SOLIDS, and a Table of LOGARITHMS and LOGARITHMIC SINEs. DAVIES' SURVEYING-With a description and plates of the THEODOLITE, COMPASS, PLANE-TABLE, and LEVEL: also, Maps of the ToPoGRAPHICAL SIGNS adopted by the Engineer Department-an explanation of the method of surveying the Public Lands, and an Elementary Treatise on NAVIgATION. DAVIES' ANALYTICAL GEOMETRY —Embracing the EQUA. TIONS OF THE POINT AND STRAIGHT LINE — f the CONIC SECTIONS-Oi the LINE AND PLANE IN SPACE-also, the discussion of the GENERAL EQUATION of the second degree, and of SURFACES of the second order. DAVIES' DESCRIPTIVE GEOMETRY,-With its application- to SPHERICAL PROJECTIONS. DAVIES' SHADOWS AND LINEAR PERSPECTIVE. DAVIES' DIFFERENTIAL AND INTEGRAL CALCULUS, CONTENTS. FIRST FIVE RULES. PAGE N umeration and Notation............................................. 13:-19 Of the Signs...................................................... 14-15 Of the Denomination of Numbers.................................... 19-20 Tables of Money, Weights, Measures, &c.,-American and Foreign.............................................................. 20 —39 Remarks on the Formation of Numbers........................... 40-41 Of Reduction............................................................. 41-45 Addition.............................................. 46-60 Subtraction....................61-68 Multiplication................................................................. 69-80 Division.................................................................. 81-93 Of the Properties of thie 9'... 93-97 Remarks........................................................ 98 Divisions of Arithmetic............................................. 99 VULGAR FRACTIONS. Definition of, and First Principles........................ 100-103 The six Kinds of Fractions........................................... 103-104 Six Propositions......................... 105-109 Greatest ommon Divisor........................................... 109-114 Second Method of finding.............................................. 112-114 Least Common Multiple.............................................. 114-117 First Method of finding................................................. 115-116 Second Method......................... 116-117 Reduction of Vulgar Fractions....................................... 117-128 Reduction of Denominate Fractions..................... 128-134 Addition of Vulgar Fractionis........................................ 135-139 Subtract on of Vulgar Fractions...................................... 139-141 Multiplication of Vulgar Fractions........................ 141-145 Division of Vulgar Fractions...................................... 145-148 X CONTENTS. DECIMAL FRACTIONS. PAGE Definition of Decimals, &c,.............................................149-150 Decimal Numeration Tabte-First Principles, &c........... 150-154 Addition of Decimals.............................. 155-156 Subtraction of Decimals............156-157 Multiplication of Decimals............................................ 158-159 Contraction in Multiplication.......................................... 159-161 Division of Decimals................................................... 162-165 Applications in the Four Rules............................... 165-166 Contraction in Division....................................... 167-168 Reduction of Vulgar Fractions to Decimals...................... 168-170 Reduction of Denominate Decimals............................. 170-174 Circulating or Repeating Decimals-Definition of, &c......... 175-178 Reduction of Circulating Decimals................................. 178 —184 Addition of Circulating Decimals..................................... 184-185 Subtraction of Circulating Decimals......................... 185 Multiplication of Circulating Decimals.............................. 186 Division of Circulating Decimals..................................... 187 RATIO AND PROPORTION OF NUMBERS. Ratio Defined and Illustrated....................................... 188-189 Proportion Defined and Illustrated.................................. 190-192 Of Cancelling............................................. 193-195 Rule of Three-Defined, Proof, &c.................................. 196-202 Rule of Three by Analysis............................................ 203-204 Rule of Three by Cancelling.......................................... 205 —206 Examples involving Fractions....................................... 207-208 Of Questions requiring two Statements................ 209-210 Double Rule of Three-Definition, Demonstration, &c......... 210-214 PRACTICE-TARE AND TRET. Practice-Definition of, &c........................................... 215 Table of Aliquot Parts....................................... 215 Examples illustrating Principles, &c.......... 216. -219 Tare and Tret.. 220-222 PERCENTAGE-INTEREST. Percentage................................................ 223-224 Interest-Definition of............... 225 Principles, and ways of finding Interest............................ 226 —237 Applications................................................ 237-238 Partial Payments-New York Rule............................ 238-241 Questions in Interest................................................... 241-243 CONTENTS. Xi PAGE Table showing the legal Rate of Interest in each State, Number of Shillings to the Dollar, Value of the Dollar in Pounds, &c............................................... 244 Reduction of Currencies............................... 244-245 Compound Interest-Definition of, &c............................. 246-247 Table showing the Interest of ~1El or $1, &c...................... 247-248 APPLICATIONS TO BUSINESS. Loss and Gain............................................................ 249-252 Stocks and Coorations................................................ 252-253 Commission and Brokerage........................................... 253-256 Banking.................................................................. 256-257 Forms of Notes.......................................................... 257 Remarks relating to Notes.................. 258-259 Bank Discount............................................................ 259-262 Discount...2...................................2................... -262-2fi4 Insurance................................................................. 265-266 Assessing Taxes......................................................... 266-269 Equation of Paynments.................. 269-272 Partnership or Fellowship.............................................. 273-274 Double Fellowshi p.............................. 274-276 Alligation Medial..................................................... 276-277 Alligation Alternate..................................................... 2777-262 Custom House Business................................... 282-285 Forms relating to Business in General.......................... 285-287 Forms of Orders.......................................................... 285 Forms of Receipts........................................................ 285-286 Forms of Bonds......................................................... 286-287 General Average............................ 288-290 Tonnage of Vessels.291-292 Custom House Charges on Vessels............................ 291 Government Rule-Carpenter's Rule............................... 292 Gauging-Varieties of Casks, &c.................................. 293-296 Life Insurance............................ 296-299 Table showing the Expectation of Life........................... 297 Rates of Insurance on Life............................................ 298 Endowments and Annuities.................................... 299-300 Coins and Currencies —Definition of................... 301 Values of Foreign Coins............................................ 301-303 Exchange-Definition of.......................... 303 Bills of Exchange....................................................... 304-305 Endorsing Bills..........- - 306 Acceptance-Liabilities of the Parties.............................. 306-307 Par of Exclihalge-Course of Exchange............................ 307-,308 X11 CONTENTS. PAGE Examples................................................................. 309-312 Arbitration of Exchange............................... 312-314 DUODECIMALS. Definition of, &c......................................................... 315-316 Multiplication of Duodecimals........................................ 316-318 INVOLUTION. Definition of, &c..................................... 318-319 EVOLUTION. Definition of, &c.................... -320 Extraction of the Square Root........................... 320-326 Extraction of the Cube Root......................................... 326-331 ARITHMETICAL PROGRESSION. Definition of, &c...................................................... 331-332 Different Cases......................................... 333-335 General Examples.................................. 335 GEOMETRICAL PROGRESSION, ETC. Definition of, &c................................... 336 —337 Cases......................................... 337-338 MENSURATION) ETC. To find the area of a Triangle............................ 339 — 340 To find the area of a Square, Rectangle, &c......... 340-341 To find the area of a Trapezoid...........................341-342 To find the circumference and diameter of a Circle...........343-344 To find the area of a Circle............................... 344 To find the surface of a Sphere........................... 344-345 To find the solidity of a Sphere........................... 346 To find the convex surface of a Prism......................346-347 To find the solid contents of a Prism..................... 347 To find the convex surface of a Cylinder................... 348 To find the solidity of a Cylinder.............348-349 To find the solidity of a Pyramid..........................349-350 To find the solidity of a Cone.............. 350-351 Right-Angled Triangle...............3................ 351-352 Mechanical Powers...................................... 353-360 Promiscuous Questions...................................361-367 ARITHMETIC. NOTATION AND NUMERATION. 1. SCIENCE in its popular sense, is knowledge reduced to order: that is, knowledge so classified and arranged, as to be easily remembered, readily referred to, and' advantageously applied. In a strictly technical sense, it refers to the laws which connect the facts and principles of any subject of knowledge with each other. 2. ARITHMETIC is both a science and an art. It is a science in all that concerns the properties, laws and proportions of numbers; and an art in all that relates to their uses and applications. 3. NUMBERS are expressions for one or more things of the same kind: thus, the words one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, &c., are called numbers. 4. The unit of a number is one of the equal things which the number expresses. Thus, if the number express six apples, one apple is the unit; if it express five pounds of tea, one pound of tea is the unit; if ten feet of length, one foot is the unit; if four hours of time, one hour is the unit. 5. In common language numbers are expressed by words: in the language of arithmetic they are generally expressed by figures. In our language there are twenty-six different QUEST.-1. What is Science? 2. What is Arithmetic? When is it a science and when an art? 3. What are numbers? Give an example. 4. What is thle unit of a number? What is the unit of six apples 8 Of five pounds of tea? Of ten feet in length? Of four hours of time? 5. How are numbers expressed in common language? How are they expressed in the language of arithmetic? How many characters are there in our language? ( 13 14 NOTATION AND NUMIERATION. characters called letters: in the language of arithmetic there are but ten characters which represent numbers; they are called figures. They are naught, one, two, three, four, five, six, seven, eight, nine. 0 1 2 3 4 5 6 7 8 9 The character 0 is used to denote the absence of a thing. As, if we wish to express by figures that there are no apples in a basket, we write, the number of apples in the basket is 0. The nine other figures are called significant figures, or digits. 6. Besides the figures which represent numbers, there are certain other characters used, called signs, which indicate the operations to be performed on numbers. They are the following: The sign + is called plus, and when placed between two numbers, indicates that they are to be added together: thus, 3 -+ 2 shows that 3 and 2 are to be added, and is read, 3 plus 2. The sign - is called minus, and when placed between two numbers, indicates that the one on the right is to be taken from the one on the left: thus, 4 - 3 shows that 3 is to be taken from 4, and is read, 4 minus 3. The sign = is called the sign of equality, and when placed between two numbers, indicates that they are equal to each other: thus, 2 + 3 = 5 shows that 2 added to 3 gives a sum equal to 5, and is read, 2 plus 3 equals 5. The sign X is called the sign of multiplication, and when placed between two numbers, indicates that they are to be multiplied together: thus, 12 x 3 shows that 12 is to be multiplied by 3, and is read, 12 multiplied by 3. The sign - is called the sign of division, and when placed between two numbers, indicates that the one on the left is to QuEST.-What are the characters called? In arithmetic, how many characters are there which represent numbers? What are they called? Name them. What is the 0 used for? What are the other nine figures called? 6. What signs are lsecd to indicate the operations to be perfolned on nlumbers? Name each, and explain iia use. NOTATICON AND NUAMErATION. 15 be divided by the one on the right: thus, 8. 4 shows that 8 is to be divided by 4: and is read, 8 divided by 4. The parenthesis is used to indicate that the sum of two or more separate numbers is to be multiplied by a single number: thus, (3 + 5) X 6 shows that the sum of 3 and 5 is to be multiplied by 6. 7. We have now learned-the alphabet of the arithmetical language, and understand that A single thing, or a unit of a number, may be expressed by 1, two things of the same kind, or two units, " " by 2, three " " or three units " " by 3, four " " or four units " " by 4, five " " or five units " " by 5, six c" " or six units " " by 6, seven " " or seven units" " by 7, eight " " or eight units " " by 8, nine " " or nine units " " by 9. The units of the numbers expressed above are called simple units, or units of the first order. 8. The next step, in the arithmetical language, is to write the 0 on the right of the 1; thus, 10. This sign is the arithmetical expression for the word ten. The character 1 still expresses a single thing, viz., one ten. This ten, however, is ten times as great as the simple unit, and is called a unit of the second order. 9. We next write two O's on the right of the 1; thus, 100. This is the arithmetical expression for one hundred, that is, for ten tens. Here, again, the 1 expresses but a single thing, viz., one hundred; but this one hundred is equal to ten units of the second order, or to one hundred units of the first order. In a similar manner we may form as many QUEsT.-7. What character stands for four things? What for eight? What are the units of such numbers called? 8. What is the next step in the language of figures? What does 1 still express? What is the single thing called? What is it equal to? 9. What is the next step? What does 1 still express? To how many units of the second order is it equal? To how many of the first? 16 NOTATION AND NUMERATION. orders of units as we please: thus, a single unit of the first order is expressed by - 1, a unit of the second order by 1 and a 0; thus, 10, a unit of the third order by 1 and two O's; thus, 100, a unit of the fourth order by 1 and three O's; thus, 1000, a unit of the fifth order by 1 and four O's; thus, 10000, a unit of the sixth order by 1 and five O's; thus, 100000, and so on for the units of higher orders. When units simply are named, units of the first order are always meant. 10. We see, from the language of figures, that units of the first order always occupy the place on the right; units of the second order the second place from the right; units of the third order, the third place; and so on for places still to the left. We also see that ten units of the first order make one of the second; ten of the second, one of the third; ten of the third, one of the fourth; and so on for the higher orders. Hence, the language expresses that, When figures are written by the side of each other, ten units of any one place make one unit of the place next to the left. 11. For the purpose of reading figures, they are often separated into periods of three figures each. The units of the first order are read, simply, units; those of the second order are generally read, tens; those of the third, hundreds; those of the fourth, thousands, &c., according to the following QUEST.-How is a single unit of the first order expressed? How do you express one unit of the second order? One of the third? One of the fourth? One of the fifth? 10. What places do units of different orders occupy? When figures -are written by the side of each other, how many units of one order make one unit of the place next to the left? 11. How are figures separated for the purpose of reading? How are units of the first order read? Those of the second? Those of the third? Those of-the fourth, &c.? NOTATION AND NUMERATION. 17 NUMERATION TABLE.* b 00O COQ o 00 o us o d4 0 0 C O 0 - -4 &O -4 --- o co a _ t {3e = ct.: H._ m O t.~ H = 7 C X cl k o o o 0 C2 5) o 0 9 (J2I 5 2EO O SH k OaH. 52 tens, hundreds, &c., are equally applicable to all numbers, Wand must be committed to memory. The table may be continued to any extent. The higher periods take the names of Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions. D)uodecillions. 12. Expressing or writing numbers in figures is called NOTATION Reading the signification of the figures correctly, when written, is called NUMERATION. EXAMPLES IN READING FIGURES. 1. In how many ways may the figures 658 be read? 1st. The common way, six hundred and fifty-eight. 2d. We may read, six hundreds, five tens, and eight units. 3d. We may read, sixty-five tens and eight units. * NOTE.-This table is formed according to the French method of numeration. The English method gives six places to thousands, &c. QUEST.-Are the words at the head of the table applicable to all numbers? May the table be continued? After what method is the table formed? What is the difference between it and th6 old English method? 12. What is notation? What is numeration? In how many ways may the figures 658 be read? m~~~~~ Th od tteh~ ftehmrto al,~ SOTA~O Redin th sinfcatio of~ thC figrscrety when w r ttn isclld~UEATO 1 I NOTATION AN'D N'U:A1RATION. 2'. How may the figures 8046 be -read? 1st. Eight thousand and forty-six. 2d. Eight thousand, no hundreds, four tens, and six units. 3d. Eighty hundreds find forty-six, or eighty hundreds, four tens, and six units. 4th. Eight hundred and four tens, and six units. 3. Give all the readings of the number 49704. 4. Give all the readings of the number 740692. 5. Give all the readings of the number 99800416. 6. Give all the readings of the number 80741047. NOTE.-The pupil should be much exercised in these readings. Ile should remark that the lowest order of units used in any reading, whether it be units, tens, hundreds, &c., &c., gives the name or denomination to the part or whole of the number used in the reading. We are now able to express any number whatever in the language of figures. EXAMPLES. 1. Write, in figures, six units of the first order. Ans. 6. 2. Write, in figures, eight units of the second order. Ans. 80. 3. Write, in figures, nine units of the third order. Ans. 900. 4. Write, in figures, seven units of the fifth order. Ans. 70000. 5. Write, in figures, nine units of the first order, three of the third, and none of the second. Ans. 309. 6. Write, in figures, eight units of the eighth order, six of the fifth, seven of the seventh, five of the sixth, none of the fourth, none of the third, one of the second, and one of the first, and read the number. An.s. 87560011. 7. Write, in figures, six quintillions, four hundred and fifty-one billions, sixty-five millions, forty-seven ten thousands, and one hundred and four. 8. Write, in figures, nine hundred and ninety-nine oc.illions, sixty-five millions, eight hundred and forty-one billions, four trillions, and eleven nonillions. QUEST.-How may the figures 8046 be read? Also, 49704? 740692? What gives the name or denomination to the number? OF THE DENOMIN'ATION OF NUMBERS. 19 9. Write, in figures, sixty-five decillions, eight hundred quadrillions, seven hundred and fifty billions, seven hundred and fifty-one trillions, nine hundred and seventy-five thousand, three hundred and ten. OF THE DENOMINATION OF NUMBERS. 13. A SIMPLE NUMBER is one which expresses a collection of'units of the same kind, without expressing the particular value of the unit. Thus, 6 and 25 are simple numbers. 14. A DENOMINATE NUMBER expresses the kind of unit which is considered. For example, 6 dollars is a denominate number, the unit 1 dollar being denominated or named. 15. When two numbers have the same unit, they are said to be of the same denomination: and when two numbers have different units, they are said to be of different denominations. For example, 10 dollars and 12 dollars are of the same denomination; but 8 dollars and 20 cents express numbers of different denominations, the unit of 8 dollars being 1 dollar, and of 20 cents, I cent. The kind of unit always indicates the denomination. In simple numbers, the unit in the place of units is different from the unit of the second order in the place of tens, and this last is different from that of the third order in the place of hundreds, and so on for places still to the left. These units, as we have seen, have differentnames or denominations, viz., simple units, or units of the first order; tens, or units of the second order; hundreds, or units of the third order, &c., and considered in this relation to each other, may be regarded as denominate numbers. The following tables show the various kinds of denominate QUEST.-13. What is a simple number? 14. What is a denominate number? 15. When are two numbers said to be of the same denomination? When of different denominations? What indicates the denomination? In simple numbers, how are the units of the different places? How do they compare in value. with each other? 20 OF THE DENOMINATION OF NUMBERS. numbers in general use, and also the relative values of their different units. OF FEDERAL MONEY. 16. Federal money is the currency of the United States. Its denominations, or names, are Eagles, Dollars, Dimes, Cents, and Mills. The coins of the United States are of gold, silver, and copper, and are of the following denominations. 1. Gold: eagle, half-eagle, quarter-eagle, dollar. 2. Silver: dollar, half-dollar, quarter-dollar, dime, halfdime. 3. Copper: cent, half-cent. If a given quantity of gold or silver be divided into 24 equal parts, each part is called a carat. If any number of carats be mixed with so many equal carats of a less valuable metal, that there be 24 carats in the mixture, then the compound is said to be as many carats fine as it contains carats of the more precious metal, and to contain as much alloy as it contains carats of the baser. For example, if 20 carats of gold be mixed with 4 of silver, the mixture is called gold of 20 carats fine, and 4 parts alloy. 17. The standard, or degree of purity, of the gold coin, is fixed by Congress. Nine hundred equal parts of pure gold, are mixed with 100 parts of alloy, of copper and silver, (of which not more than one half must be silver,) thus forming 1000 parts, equal to each other in weight. The silver coins contain 900 parts of pure silver, and 100 parts of pure copper. The copper coins are of pure copper. The eagle contains 258 grains of standard gold; the dollar 41 24 grains of standard silver; the cent 168 grains of copper. QUEsT.-16. What is Federal Money? What are its denominations? Of what are the coins of the United States made? What are their de_ nominationsa What is a oarat? What do you understand by'carats fine' What would be 20 carats fine of gold? 17. What is the standard of gold coin in the United States? What of silver? What of copperd What is the weight of the eagle? What of the dollar? What of the cent OF THE DENOMINATION OF NUMBERS. 21 TABLE. Mills. Cents. Dimes. Dollars. Eagle. m. cts. d. $ E. 10 =1 100 10 1 1000 100 10 1 I 10000 1000 100 10 1 This table is read, ten mills make one cent, ten cents one dime, ten dimes one dollar, ten dollars one eagle. In this table, ten units of each denomination make one unit of the denomination next higher, the same as in simple numbers. In expressing Federal Money in the language of figures, the dollars are separated from the cents and mills by a comma: thus, 36,645 is read, 36 dollars, 64 cents, 5 mills; but may also be read, 36 dollars, 6 dimes, 4 cents, 5 mills; 375,043 is read 375 dollars, 4 cents, 3 mills. ENGLISH CURRENCY. 18. The relative proportion between gold and silver in the English coins, according to the mint regulations, both for the old and new coinage, is as follows: in the old coinage, a pound of gold is worth 15.2096 times a pound of silver. In the new coinage, a pound of gold is worth 14.2878 times a pound of silver. A standard gold coin is composed of 22 parts of pure gold and 2 parts of copper. A standard silver coin is composed of 224 parts of pure silver and 18 parts of copper. In the copper coin 24 pence make one pound avoirdupois. QvEsT.-Repeat the table of Federal money. How many units of each denomination make one of the next higher? In expressing Federal money in figures, how are the dollars separated from the cents? What place do the mills occupy, counting from the comma? 18. What is the relative proportion between gold and silver in the old and new coinage of English money? What is the standard of the English gold? Of the silver? What is the weight of the English penny? 22 OF THE DENOMINATION OF NUMBERS. TABLE. a S i' a 2 ~ W far. d. s. I I I 4 21 16 8 41-1 24 12 6 1 =1 48 24 12 3 j 2=1 120 60 30 711 5 21=1 240 120 60: 151 10 5 2 =1 336168 84 21 14 7 2 1 =1 480 240 120; 30 20 10 4 2 1 —=1 504 252 126 311 21 101 41 21- 1 1 - =1 960 480 240; 60 40 20 8 4 2 2 1 -=1 1008 504 252i 63 42 21 8Q 41 3 2-1 2 = AVOIRDUPOIS WEIGHT. 19. The standard avoirdupois pound of the United States, as determined by Mr. Hassler, is the weight of 27.7015 cubic inches of distilled water weighed in air. By this weight are weighed all coarse articles, such as hay, grain, chandlers' wares, and all the metals, excepting gold and silver. In this weight the words gross and net are used. Gross is the weight of the goods, with the boxes, casks, or bags in which they are contained. Net is the weight of the goods only; or what remains after deducting from the gross weight the weight of the boxes, casks, or bags. An hundred weight, in its general sense, means 112 pounds, as appears in the table. But by the laws of the United States it is fizred at 100 pounds, (See Preface.) QvEsT.-Repeat the table of English money. 19. What is the standard avoirdupois pound of the United States? For what is this weight used? What is the meaning of the terms gross and net? What is a hundred weight? How are goods now generally bought and sold? OF THE DENOMINATION OF NUMBERS. 23 TABLE. Drams. Ounces. Pounds. Quarters. Hundreds. Tons. dr. oz. lb. qr. cWt. T. 16 i 1 256 16 1 —1 7168 448 28 = 1 28672 1792 112 4 = 1 573440 35840 2240 80 20 = 1 TROY WEIGHT. 20. By this weight are weighed gold, silver, jewels, and some liquids. The standard troy pound of the United States, as determined by Mr. Hassler, is the weight of 22.794377 cubic inches of distilled water weighed in air. Hence, the pound is less than the pound avoirdupois. TABLE. Grains. Pennyweights. Ounces. Pounds. gr. pwt. oz. lb. 24 -- 1 480 20 = 1 5760 240 12 - 1 COMPARISON WITH AVOIRDUPOIS WEIGHT. 7000 troy grains = 1 lb. avoirdupois. 175 troy pounds = 144 lbs. " 175 troy ounces = 192 oz. " 437- troy grains = 1 oz. c; QvEsT.-Repeat the table of avoirdupois weight. 20. What articles are weighed by troy weight? What is the standard troy pound of the United States? Is it greater or less than the avoirdupois pound? Repeat the table of troy weight. How does it compare with avoirdupois weight? 24 OF THE DENOMINATION OF NUMBERS. APOTHECARIES WEIGHT. 21. This weight is used by apothecaries and druggists in mixing their medicines. They, however, buy and sell their drugs by avoirdupois weight. The pound and ounce are the same as the pound and ounce in troy weight. The difference between the two weights consists in the different divisions and subdivisions of the ounce. TABLE. Grains. Scruples. Drams. Ounces. Pound. gr. 9 3 Ib 20 =1 60 3 -1 480 24 8 = 1 5760 288 96 12 -- 1 FOREIGN WEIGHTS. 22. The foreign weights differ somewhat from ours. i pound avoirdupois, English = 27.7274 cubic inches distilled water. 1 pound troy, English = 22.815689 cubic inches distilled water. OLD FRENCH SYSTEM. 1 livre = 16 onces = 1.0780 lb. avoirdupois. 1 once = 8 gros = 1.0780 oz. " 1 gros = 72 grains = 58.9548 grains troy. 1 grain, = 0.8188 " QUEST.-21. By whom is apothecaries weight used? By what weight do druggists buy and sell their drugs? In what respects is the weight sinmilar to troy? In what is the difference? Repeat the table of apothlecaries weight. 22. What is the value of the English pound avoirdupois'! Of the English pound troy? What is the old French system of weights? OF THE DENOMINATION OF NUMBERS. 25 NEW FRENCH SYSTEM. 23. The basis of this system of weights is the weight in vacuo of a cubic decimetre of distilled water. This weight is called a kilogramme, and is the unit of the Frenchsystem. It is equal to 2.204737 pounds avoirdupois. (For the value of a decimetre, see table of linear measure, French, page 29.) The one-thousandth part of a kilogramme is called a gramme. and the one-thousandth part of a gramme is called a milligramme. The divisions are made on the decimal principle, and are of the following denominations: TABLE. Milligramme. Centi- Deci- Gramme. Deca- Hecto- Kilo-' Milgramme. gramme. gramme. gram'e. gr'me. lier. 100 10 — 1 1000 100 10 1 10000 1000 100 10 - 1 100000 10000 1000 100 10 = 1 1000000 100000 10000 1000 100 10 - 1 10000000 1000000 100000 10000 1000! 100 10 100000000 10000000 1000000 100000 10000 1000 1001 1 1000000000 100000000 10000000 1000000 100000 100 100 00ooo 1 =1 COMPARISON OF WEIGHTS. English, 1 pound = 1.000936 pounds avoirdupois. French4 1 kilogramme = 2.204737 " " Spanish, I pound = 1.0152 " " Swedish, 1 pound =0.9376 " c Austrian, 1 pound 1.2351 " " Prussian, 1 pound = 1.0333 " " QuIaST.-23. What is the basis of the new French system? Repeat the table. How do the weights of different countries compare with ours? 2 26 OF THE DENOMINATION OF NUMBERS. ENGLISH WOOL WEIGHT. 24. The following is the table of wool weight in England. As yet, many of the denominations have not been much used in this country; but as we are now exporting wool to England, they must soon be generally introduced. TABLE. Pounds. Cloves. Stones. Tods. Weys. Sacks. Last. 7 1 14 2 =1 28 4 2 1 182 26 13 6 - 1 364 52 26 13 2 1 4368 624 312 156 24 12 — 1 CLOTH MEASURE. 25. Cloth measure is used by woollen and linen drapers. Hollands are measured in English ells, and tapestry by the French ell; woollens, linens, silks, tape, &c., by the yard. TABLE. Inches. Nails. Quarters. Ells Flemish. Yards. Ells English. Ell French. in. na. qr. E. Fl. yd. E. E. E. Fr. 2t -1 9 4 — 1 27 12 3 — 1 36 16 4 1 — 1 45 20 5 12 1 - 1 54 24 6 2 1~ 1j- 1 QuEST.-24. Is English wool weight yet in use in this country? Repeat the table of wool weight. 25. By whom is cloth measure used? Howr are hollands measured? Tapestry? Repeat the table. OF THE DENOMINATION OF NUMBERS. 27 LONG MEASURE. 26. This measure is used to measure distances, lengths, breadths, heights, depths, &c. Gunter's chain is generally used by surveyors in measuring land. A standard measure has been adopted by the United States, copies of which are distributed in various parts of the country. This standard is a brass rod, one yard or 3 feet long. TABLE. Inches. Gunter's Feet. Yards. Fathoms. Rods. Gunter's Fur- Mile. Link. chain. longs. 7a. 1. ft. yd. rd. c. f ~r. mi. 12 il" =1 36 4T~ 3 =z1 72 97-, 6 2 -1 198 25 16-1 51 23 =1 792 100 66 22 11 4 =1 7920 1000 660 220 110 40 10 = —1 63360 8000 5280 1760 880 320 80 8 =1 FOREIGN MEASURES OF LENGTH. 27. The imperial standard yard of Great Britain is the one from which our yard is taken. It is referred to a natural standard, viz., to the distance between the axis of suspension and the centre of oscillation of a pendulum which shall vibrate seconds in vacuo, in London, at the level of the sea. This distance is declared to be 39.1393 imperial inches; that is, 1 imperial yard and 3.1393 inches. QuEST.-26. For what is long measure used? For what is Gunter's chain used? Repeat the table of long measure. 27. What is the standard of the English imperial yard? What is its length? 28 OF THE -DENOMINATION OF NUMBERS, OLD FRENCH SYSTEM. 1 point = 0.0074 U. S. inches. 1 line = 12 points = 0.08884 " 1 inch = 12 lines = 1.06604 " 1 foot = 12 inches = 12.7925 " 1 ell = 43 in. 10 lines = 46.728 " = ]..298 yd 1 toise = 6 feet = 76.755 " = 2.132 " 1 perch (Paris) = 18 feet. 1 perch (royal) = 22 feet. 1 league (common) 25 to a degree = 2280 toises = 4861 yards = 2.76 miles. 1 league (post) = 2000 toises = 4264 yards = 2.42 miles. 1 fathom (Brasse) = 5 feet French = 63.963 inches, or 51 feet English, nearly. 1 cable length = 100 toises,= 120 fathoms French = 1062 fatho/is English. TABLE FOR REDUCING OLD FRENCH MEASURES TO UNITED STATES MEASURES. (According to Mr. Hassler's comparison.) French U. States French ft. U. S. feet French U. States French U. States feet. inches. or inches. or inches. lines. inches. points. inches. 1 l 12.7925 1 1.0660 1 0.0888 1 0.0074 2!25.5850 2 2.1321 2 0.1777 2 0.0148 3 38.3775 3 3.1981 3 0.2665 3 0.0222 4 51.1700 4 4.2642 4 0.3554 4 0.0296 5 63.9625 5 5.3302 5 0.4442 5 0.0370 6 76.7550 6 6.3963 6 0.5330 6 0.0444 7 89.5475 7 7.4623 7 0.6219 7 0.0518 8 102.3400 8 8.5283 8 0.7107 8 0.0592 9 1 15.1325 9 9.5944 9 0.7995 9 0.0666 10 127.9250 10 10.6604 10 0.8884 10 0.0740 11 140.7175 11 11.7265 11 0.9772 11 0.0814 NEW FRENCH SYSTEM. 28. The basis of the new French system of measures is the Quesr.-What is the old French long measure? 28. What is the basis of the new French system? OF THE DENOMINATION OF NUMBERS. 29 measure of the meridian of the earth, a quadrant of which is 10,000,000 metres, measured at the temperature of 32~ Fahr. The multiples and divisions of it are decimals, viz.: 1 metre = 10 decimetres - 100 centimetres = 1000 millimetres = 39.3809171 United States inches, or 3.28174 feet. Road measure. Myriametre = 10,000 metres. Kilometre = 1000metres. Decametre = 10 metres. Metre = 0.51317 toise. TABLE FOR REDUCING METRES TO INCHES. (According to Mr. Hassler's comparisons; 1 metre = 39.3809171 inches.) Metres.- Inches. Metres. Inches. |Metres. Inches. Metres. Inches. 0.001 0.039381 0.026 1.023904 0.051 2.008427 0.076 2.992950 2 0.078762 27 1.063285 52 2.047808[ 77 3.032331 3 0.118143 28 1.102666 53 2.0871891 78 3.071712 4 0.157524 29 1.142047 54 2.126570' 791,3.111093 5 0.196905 0.030 1.181428 55 2.1659501 0.080 3.150474 6 0.236286 31 1.220809 56 2.205331 81 3.189855 7 0.275666 32 1.260189 57 2.244712 82 3.229236 8 0.3150471 33 1.299570 58 2.284093 i 83 3.268617 9 0.354428 34 1.338951 59 2.3234741 84 3.307998 0.010 0.393809] 35 1.378332! 0.060 2.3628551 851 3.347379 11 0.4331901 36 1.417713! 61 2.402236 86 3.386759 12 0.4725711 37 1.457094! 62 2.441617 87 3.426140 13 0.511952' 381.496475 63 2.480998[ 88 3.465521 14 0.551333 39' 1.535856 64 2.520379' 89 3.504902 15 0.5907141 0.040 1.575237 65 2.559760, 0.090'3.544282 16 0.6300951 41 1.6146181 66 2.599141! 91'3.583663 17 0.6694761- 42i 1.6539991 671 2.6385221 92 3.623044 1 0.708855/ 43' 1.6933791 68 2.677903' 93 3.662425 19 0.7482371 4411.732760, 6912.717283!1 94' 3.701806 0.020 0.787618' 451 1.772141 0.070, 2.7566641 95 3.741187 21 0.826999/ 46 1.811522, 71 2.796045/ 96 3.780568 22 0.8663801 47 1.850903' 72 2.835426 97 3.819949 23 0.905761 48 1.890284 73, 2.8748071 98 3.859330 24 0.9451421 49! 1.929665 74 2.9141881 99 3.898711 25 0.984593i 0.050 1.969046 75 2.9535691 0.100 3.938092 QuEsT. —What are the multiples and divisions of it? Repeat the table of road measure 30 OF THE DENOMINATION OF NUMBERS. Austrian, 1 foot = 12.448 U. S. inches - 1.03737 foot. Prussian, Rhineland, nRhinelanld,1 foot 12.361 " " - 1.0300 " Swedish, 1 foot = 11.690 " " - 0.974145 " 1 foot = 11.034 " " = 0.9195 " Spanish, league (royal) = 25000 Span. ft. = 4~ miles ( "(common)= 19800 " = 3 " SQUARE MEASURE. 29. Square measure is used for measuring all kinds of superficies, such as land, paving, flooring, plastering, and every thing which has length and breadth. TABLE. Square inches. Square Square Squa e Poles or.A c links. feet. yards. perches. ] Q M, Sq. in. Sq. 1. Sq. ft. Sq. yd. P. Sq. c. R... JM. 1296 20 8 9 o 9' 39204 625 2721 30 =' 1 627264 10000 4356 484 16 = 1 1568160 25000 10890 1210 40 21- = 1 6272640 100000 43560 48401 160 10 4 -1 4014489600i64000000 27878400i3097600 102400 6400 1560 640 =1 FRENCH SUPERFICIAL MEASURE, OLD SYSTEM. 1 square inch = 1.1364 U. S. square inches. 1 arpent (Paris) = 100 square perches, (Paris,) or 900 square toises = 4088 square yards, or 5ths of an acre, nearly. I arIent (woodland) = 100 square perches (royal) = 6108 square yards, or 1 acre, 1 rood, 1 perch. QUEST.-29. For what is square measure used? Repeat the table. What is the old French system of square measure? OF THE DENOMINATION OF NUMBERS. 31 NEW SYSTEM. 1 are = 100 square metres = 119.665 square yards.. 10 ares = 1 decare. 10 decares = 1 hecatare. CUBIC OR SOLID MEASURE. 30. Forty cubic feet of round timber, or 50 solid feet of square timber, make 1 ton. A cord of wood is a pile 4 feet high, 4 feet wide, and 8 feet long, and consequently contains 128 solid feet. A cord foot is one foot in length of the pile which makes a cord. It contains 16 solid feet. TABLE. Cubic inches. Cubic feet. Cubic yards. Cubic rods. fu Cub. S. in. S. ft. S. yd. S. rd. S.fur. S. mi. 1728 = 1 46656 27 = 1 7762392 4492 166 3 = 1 496793088000 287496000 10648000 64000 = 1 254358061056000 14719795200f0 5451776000 327680001 512 =1 FRENCH SOLID MEASURE. cubic foot = 2093.470 cubic inches of the U. States. 1 cubic decimetre = 61.074664 " " " I stere = 1 cubic metre = 61074.664 cubic inches = 35.375 cubic feet = 1.309 cubic yards. LIQUID MEASURE OF THE UNITED STATES. 31. The standard gallon of the United States is the wine gallon, which measures 231 cubic inches, and contains, as QUEST.-What the new system? 30. How much timber makes a ton I What is a cord of wood? How many solid feet does it contain? What s'a cord foot? How many solid feet does it contain? Repeat the table of cubic or solid measure. Repeat the table of French solid measure. 31 What is the standard gallon of the United States? 32 OF THE DENOMINATION OF NUMBERS. determined by Mr. Hassler, 8.3388822 pounds avoirdupois of distilled water. TABLE. Cubic inches. Gills. Pints. Quarts. Gallon. S. in. gi. pt. qt. gal. 77 -1 287 4 =1 573 8 2 =1 231 32 8 4 — 1 DRY MEASURE. 32. The standard bushel of the United States is the "Win chester bushel, which measures 2150.4 cubic inches, and con tains 77.627413 pounds avoirdupois of distilled water. TABLE. Cubic inches. Pints. Quarts. Gallons. Pecks. Bushels. S. in. pt. qt. gal. pk. bu. 33 =21 67 2 1 — 268-4 8 4 -- 1 5373 16 8 2 — =1 21502 64 32 8 4 =1 FOREIGN MEASURES. 33. The British imperial gallon contains 10 pounds avoirdupois of distilled water weighed in air, and measures 277.274 cubic inches. The same measure is now used foi liquids as for dry articles which are not measured by heaped measure. QvEsT.-Repeat the table of liquid measure. 32. What is the standard bushel of the United States? How many cubic inches does it contain? Repeat the table of dry measure. 33. What is the standard of the British imperial gallon? OF THE DENOMINATION OF NUMBERS. 33 For the latter, the bushel is heaped in the form of a cone, not less than 6 inches high, the base being 9- inches. French, 1 litre = 1 cubic decimetre = 61.074 U. S. cubic inches = 1.057 U. S. quarts, wine measure = 1.761 imperial pints of Great Britain. 1 boisseau = 13 litres = 793.364 cubic inches = 3.4349 gallons. 1 pinte = 0.931 litre = 56.817 cubic inches = 0.98397 quarts. Spanish, 1 wine arroba = 4.2455 gallons. 1 fanega (corn measure) = 1.593 bushels. ENGLISH ALE AND BEER MEASURE. 34. The following is the English beer measure. By it all malt liquors and water are measured. TABLE. 5 0 Cubic inches.,,, Q)_ CsW C a a a S. in. pt. qt. gal. fir. K. bar. khd. pun. B. tun. 34.6591 — 1 69.3181 2 -1 277.274 8 4 =1 2495.466 72 36 9 = 1 4990.93-2 144 72 18 2 =1 9981.864 288 144 36 4 2 =1 14972.796 432 216 54 6 3 11-=1 19963.728 576 288 72 8 4 2 11 =1 29945 592 864 432 108 12 6 3 2 1~ =1 59891.184 1728 864 216 24 12 6 4 3 2 =1 QUEST. —What is heaped measure? What is the French measure? 34. What is measured by English beer measure? Repeat the table 2* 34 OF THE DENOMINATION OF NUMBERS. ENGLISH WINE MEASURE. 35. The following is the English wine measure. All the denominations of it are not generally used in this country. By this measure all wines, brandy, rum, and distilled liquors are bought and sold. TABLE. o' 0 69.3181 8 1 Cubic inches. A _ i0 S. in. gi. pt. |t. arnk. run. bar. tier. khkd. un. pi. tun. 34.6591 4 1= 1 277.274 32 8 4=1 2772.740 320 80 40 10 =1 4990.932 5761 1441 72 18 14 —=1 8734.131 1008 252 126311 3- 1 =1 11645.508 13441 336 168 42 44 21 11 =1 17468.262 20161 504 252 63 6-3- 31 2 11 =1 23291.016 2688 672 336 84 8 4 2 2 11 =1 34936.524 40321008, 504126 123 7 4 3 2 11 =1 69873.048 1806420161008252 251*14 8 6 4 3 2=1 ENGLISH CORN OR DRY MEASURE. 36. Dry measure is used for all dry commodities, such as wheat, barley, beans, coal, oysters, &c. The following is the English table, all the denominations of which are not in general use in this country. The standard bushel is a cylinder 18.789 inches in the interior diameter, and 8 inches in depth, and consequently contains 2218.192 cubic inches. QUEST.-35. What liquids areaneasured by wine measure? Repeat the table. 36. What articles are measured by corn measure? What is the standard bushel? How many cubic inches does it contain? OF THE DENOMINATION OF NUMBERS. 35 TABLE. Cubic inches. A S S. in. pt. qt. gal. pk. bu. qr. 34.659 — = 1 69.318' 21-1 138.637 4121 I 277.274 8i 4 2 =1 554.548 161 8 4 2-1 2218.192 64! 32 16 8 4=1 4436.384 1281 64 32 16 8 2 =1 8872.768 256 128 64 32 16 4 2 =1 17745.536 5121 256 128 64 32 8 4 2=1 88727.680 -2560 1280 640 320 160 40 20 10 5=1 177455.360 5120'2560'1280 640 320 80 40 201 10 2=1 OLD AND NEW ENGLISH COAL MEASURE. 37. By act of Parliament passed in 1831, all- coals sold within 25 miles of the Post Office in London, are to be sold by weight. One sack weighs 2 cwt. or 224 lbs.; consequently, 10 sacks make 1 ton. Twelve sacks make a London chaldron of 36 bushels, while it takes 791 bushels to make a Newcastle chaldron, as shown by the table. TABLE. Pounds Pecks. Bushels. Sacks. Vats, or London Newc. eels. Scows. 1 ship weight. strikes. chadron. chald. load. 18 =1 74.2 4 1 224 12 3 =1 672 36 9 3 -1 2688 144 36 12 4 = 1 5936 318 79- 26- 8# 2 =-1 47488 2544 636 212 702 17 I 8 =1 56448 3024 756 252 84 i 21 9X7 1= =1 949760 50880 12720 4240 4131- 353~ 160 20 1 133 -- QuEsT.-Repeat the table. 37. Wjat was established by act of Parliament? Repeat the table of coal measure. 36 -OF THE DENOMINATION OF NUMBERS. MEASURE OF TIME. Thirds. Seconds. Minutes. Hours. Days. Weeks. Months. Year. thirds. sec. m. hr. da. wk. mo. yr. 60 — 1 3600 60 — 1 216000 3600 60 - 1 5184000 86400 1440 24 = 1 36288000 604800 10080 168 7 =1 145152000 2419200 40320 672 28 4 =1 1893456000 31557600 525960 8766 365 52 13 = 1 38. The whole days only are reckoned. The odd six hours, by accumulating for 4 years, make one day, so that every fourth year contains 366 days. This is called the Bissextile, or Leap year. The leap years may always be known by this, that the numbers which express them are exactly divisible by 4. Thus, 1840, 1844, 1848, &c., are all leap years. Although the year is reckoned at 365da. 6hr., it is in fact but 365da. 5hr. 48m. 48sec., and the difference by acumulating for 100 years makes about 1 day, so that the centennial years, though divisible by 4, are not leap years. The year is also divided into 12 calendar months, which contain an unequal number of days. NVames. JVo of Days. 1 month January - - - 31 2 " February - - - 28 3 " March - - - 31 4 " April - - 30 A5 " May- - - -31 6 " June - - 30 7 " July- - - -31 8 " August - - - 31 9 " September - - 30 10 " October - - - 31 11 " November - - - 30 12 " December - - - 31 Total 365 QUEsT.-38. Repeat the table of time. What is the length of a yearl What is done with the quarter o.a& day? How do you determine the leap years? What years that are divisible by four are not leap years? OF THE DENOMINATION OF NUMBERS. 37 The. additional day, when it occurs, is added to the month of February, so that this month has 29 days in the Leap year. Thirty days hath September, April, June, and November; All the rest have thirty-one, Excepting February, twenty-eight alone. But Leap year coming once in four, February then has one day more. TABLE, SHOWING THE NUMBER OF DAYS FROM ANY DAY OF ONE MONTH TO THE SAME DAY OF ANY OTHER MONTH IN THE SAME YEAR. To the same day. From any From any day of day of Jan. Feb. Mar. April May June July Aug. Sept. Oct. Nov. Dec. dy of January 365 31 59 90 120 151 1811212243 27313041334 Jan. February 334 365 28 59 89 120 150 181 212 242 273 303 Feb. March 30613371365 31 61 92 1221153184'214124512751March April 275{306 334{365 30 61 91/122 153183 2141244/April May 245[2761304[335 365 31 61 92[123 153]184i2141May June 214 24512731304/334365 30 61 92 122/153183}June July 184[215'243l27413041335t365 31 62 92123 153 July August 153]18421224312731304334/365 31 62 921221August Sept. 122 153 181 212 422731303334[365 30 61 91 Sept. October 92 1231151 182 112 243273,304335!365 31 61 Oct. Nov. 61 92 120 151 181!212 2421273 30413341365 30 Nov.!Dec. 31 62 901121 151 182 2121243 274 304 3351365 De. 39. The months counted from any day of, are arranged in the left hand vertical column; those counted to the same day, are in the upper horizontal line: the days between these periods are found in the angle of intersection, in the same way as in a common table of multiplication. If the end of February be included between the two points of time, a day must be added in leap years. Suppose, for example, it were QUEST.-38. What are the calendar months? How many days does each contain? What is done with the odd day in leap year? Repeat the verse which indicates the number of days in each month of the year. 39. What is the object of this table? 38 OF THE DENOMINATION OF NUMBERS. required to know the number of days from the fourth of.March to the fifteenth of August. In the left hand vertical column find March, and then referring to the intersection of a horizontal line drawn from March, with the column under August. we find 153, which is the number of days from the fourth (or any other) day of March to the fourth (or same) day of August; but as we want the time to the fifteenth of August, 11 days (the difference between 4 and 15) must be added to 153, which shows that 164 is the number of days between the fourth of March and the fifteenth of August. Again, required the number of days between the tenth of October and the third of June, in the following year. Opposite to October and under June, we find 243, which is the lumber of days from the tenth of October to the tenth of June; but as we sought the time to the third only, which is 7 days earlier, we must deduct 7 from 243, leaving Q236, the number of days required; and so of others. DIVISION OF THE CIRCLE-MEASURE OF TIME. The geographical division of any line drawn round the cir- Diurnal motion of the cumference of the Earth. Earth reduced to time. 60 seconds, 1 minute - -= 4 seconds. 60 minutes, I degree - -= 4 minutes. 15 degrees, X sign of the zodiac - 1 hour. 30 degrees, 1 sign of the zodiac - 2 hours. 90 degrees, 1 quadrant - - 6 hours. 4 quadrants, or 360 degrees, 1 great circle - = 24 hours. 40. Every circle is supposed to be divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, and each minute into.60 equal parts called seconds. For astronomical purposes, the circumference of the circle is also supposed to be divided into 12 equal parts, each of which QuvsT. —How do you find the number of days from the fourth of March to the fifteenth of August? What is the number of days from the tenth of October to the third of June? Also the same in a leap year? 40. How is any circle -supposed to be divided? What is a sign, or sign of the zodiac? Repeat the table. OF THE DENOMINATION OF NUMBERS. 39 is called a sign. The characters which mark these divisions are as follows: C s / // circumference, sign, degree, minutes, seconds. TABLE OF PARTICULARS. 41. For various things 12 things make 1 dozen. 12 dozen - - - - 1 gross. 12 gross, or 144 dozen - 1 great gross. ALSO, 20 things make 1 score. 112 pounds - - - - 1 quintal of fish. 24 sheets of paper - - 1 quire. 20 quires - - - - 1 ream. 2 reams - - - - 1 bundle. BOOKS. A sb- et folded in two leaves is called a folio. folded in four leaves - a quarto, or 4to. " folded in eight leaves - an octavo, or 8vo. " folded in twelve leaves - a duodecimo, or 12mo. " folded in eighteen leaves - an 18mo. DIMENSIONS'OF DRAWING PAPER. Demy, 1 ft. 71 in. by 1 ft. 3- in. Medium, 1 ft. 10 in. by 1 ft. 6 in. Royal, 2. ft. 0 in. by I ft. 7 in. Super Royal, 2 ft.'3 in. by I ft. 7 in. Imperial, 2 ft. 5 in. by 1 ft. 94 in. Elephant, 2 ft. 34 in. by 1 ft. 104 in. Columbia, 2 ft. 93 in. by 1 ft. 11 in. Atlas, 2 ft. 9 in. by 2 ft. 2 in. Double Elephant, 3 ft. 4 in. by 2 ft. 2 in. Antiquarian, 4 ft. 4 in. by 2 ft. 7 in. QUE6T.-41. Repeat the table of particulars. Also for books. What arc the dimensions of drawing paper? 40 OF THE FORMATION OF NUMBERS. REMARKS ON THE FORMATION OF NUMBERS. 42. We have seen (Art. 10) that when figures are written by the side of each other, thus, 8562041304723 the language implies that the unit in each place is equal to ten units of the place next to the right; or that ten units of any one place make one unit of the place next to the left. 43. When figures are written thus, ~ s. d. far. 4 17 10 3 the language implies, that four units of the lowest denomination make one of the second; twelve of the second, one of the third; and twenty of the third, one of the fourth. 44. When figures are written thus, T. cwt. qr. lb. oz. dr. 27 17 2 27 11 10 the language implies, that 16 units of the lowest denomination make one of the second; 16 of the second, one of the third; 28 of the third make one of the fourth; four of the fourth, one of the fifth; and 20 of the fifth, one of the sixth. All the other denominate numbers are formed on the same principle; and in all of them we pass from a lower to the next higher denomination by considering how many units of the one make one unit of the other. 45. In.our written language, each of its elementary letters has a particular signification, which must be learned as a first step. We next learn to place these letters in the form of words, and then what may be done by using these words in connection with each other. QUEST.-42. When figures are written by the side of each other, what does the language imply? 43. When figures are written with the mark ~: s. d. far. placed over them, what does that language imply? 44. When figures are written with T. cwt. qr. lb. oz. dr. placed over them, what relation exists between the orders of their units? How do we always pass from one denomination of denominate numbers to another? 45. How do we learn a commonlanguage? OF REDUCTION. 41 So in figures: we first learn what each figure expresses by itself, and then what it is made to express in all the various ways in which it may be written. We thus learn the language of figures. 46. Let us give a few examples of the changes which are produced in the signification, by changing the places of letters and figures. In common language, was, is a known word. But the same letters also give saw, an instrument. Also, 375 expresses, three hundred and seventy-five; but 573 expresses, five hundred and seventy-three. It may be well to observe that the same letter has the same name, and generally represents the same sound wherever it may fall in a word. So, likewise, the same figure always expresses the same number of units, wherever it may be placed. Thus, in the example above: in the first number, 5 expresses Jive units of the first order, and 3, three units of the third. In the second number, 5 expresses five units of the third order, and 3, three units of the frst order. The value of the unit, however, always depends on the place of the figure. OF REDUCTION. 47. REDUCTION is changing the denomination of a number from one unit to another, without altering the value of the number. Thus, if we have 2 tens, and wish to reduce them to the denomination of units of the first order, we multiply by 10, or add one 0; this gives 20 units of the first order, which are equal to 2 tens. QUEST.-How must we learn the language of figures? 46. Give some examples of the changes in signification which are produced by altering the places of letters and figures. Has the same letter always the same name and sound? Has the same figure always the same name? Does it always express the same number of units? Does the value of the unit expressed remain the same? On what does it depend? 47. What is reduction? How are tens reduced to units of the first order? 42 OF REDUCTION. If, on the contrary, we wish to reduce 300 to units of the second order, we divide by 10, and the quotient is 30 units of the second order, which are equal to 300. Had we wished to reduce to units of the third order, we should have divided by 100, giving 3 for the quotient: hence, reduction of denominate numbers is divided into two parts; 1st. To reduce a number from a higher denomination to a lower; and -2d. To reduce a number from a lower denomination to a higher. The first reduction is effected by beginning with the number in the highest denomination. Multiply this number by the value of its unit expressed in units of the next lower denomination, and add to the product the number in that denomination. Proceed in the same manner through all the denominations to the lowest. The second reduction is effected thus: Divide the given number by so many as make one of the denomination next higher; set aside the remainder, if any, and proceed in the same manner through all the denominations to the highest. Thus, in the first, if we wish to reduce ~ s. d. 3 14 4 to pence, we first multiply the ~3 by 20, which gives 60 shillings. We then add 14, making 74 shillings. We next multiply by 12, and the product is 888 pence. To this we add 4d., and we have 892 pence, which are of the same value as ~3 14s. 4d. If, on the-contrary, we wished to change 892 pence to pounds, shillings, and pence, we should first divide by 12: the -quotient is 74'shillings, and 4d. over. We again divide by 20, and the quotient is ~3, and 14s. over: hence, the result is ~3 14s. 4d., which is equal to 892 pence. QUEsT. —How will you reduce units of the first order to those of the seo. ond? How to those of the third? To those of the fourth? Into how many parts is reduction of denominate numbers divided? How do you effect the first reduction? How do you effect the second? OF REDUCrION. 43 The reductions, in all the denominate numbers, are made in the same manner. EXAMPLES. 1. In ~5 5s., how many In 5040 farthings, how shillings, pence, and far- many pence, shillings, and things? pounds? ~ s. 4)5040 farthings. 20 12)1260 pence. 20 2105)105 shillings. 105 shillings. 1~5 5s. 1260 pence. 4 In this example, the reduc5040 tion is from a less to a greater unit. Here the reduction is from a greater to a less unit. 2. In 55 guineas, how many shillings, pence, and farthings? 3. Reduce ~54 1ls. 91d. to farthings. 4. Reduce ~77 1ls. 10d. to halfpence. 5. Reduce ~94 14s. 8d. to pence. 6. Reduce ~47 14s. 4d. to twopences. 7. Reduce ~34 lls. 9d. to threepences, and to pence. 8. In ~108 11s. 6d., how many sixpences? 9. How many crowns, half-crowns, shillings, sixpences, pence, and farthings are there in ~54? 10. Reduce ~74 13s. 9d. into shillings, threepences, and farthings. 11. In 11520 farthings, how many pence, shillings, and pounds? 12. In 17880 pence, how many pounds? 13. Reduce 100000 farthings into guineas. 14. In 50400 halfpence, how many pounds? 15. In.12050 shillings, how many crowns and pounds? 16. Reduce 311040 pence into shillings, crowns, and pounds. 44 OF REDUCTION. 17. Reduce 171b. 5oz. troy weight to grains. 18. Reduce 6720 grains to ounces. 19. In 14 ingots, or bars of silver, each weighing 27oz. l0pwt., how many grains? How many in one? 20. How many grains of silver in 41b. 6oz. 12pwt. ard 7gr.? 21. How many pounds, ounces, pennyweights, and grains of gold in 704121 grains? 22. How many of each denomination in 351262 grains? 23. In 25t, apothecaries weight, how many ounces, drams, scruples, and grains? 24. In 907920 grains, how many ounces and pounds? 25. In 15tb 13 13 13 2gr., how many grains? 26. In 174947 grains, how many pounds? 27. In 16tb 103 13 149 7gr., how many grains? 28. In 12 tons, avoirdupois weight, how many pounds? 29. In 313601b. of iron, how many tons? 30. In 375cwt. 2921b. of copper, how many pounds? 31. Reduce 740900oz. into cwts. and tons. 32. In 9T. 19cwt. 3qr. 271b. 14oz., how many ounces? 33. In 14478406oz., how many tons, cwt., qrs., lb., oz., and drs.? 34. In 314 yards of cloth, how many nails? 35. In 576 French ells, how many yards? 36. Reduce 97yds. 3qrs. to English ells. 37. In 57 pieces of cloth, each 35 ells Flemish, how many ells English and nails? 38. In 14 bales of cloth, each 17 pieces, each piece 56 ells Flemish, how many yards, ells English, and ells French? 39. In 471 miles, long measure, how many furlongs and poles? 40. In 123200 yards, how many miles? 41. In 50 miles, how many yards, feet, inches, and barleycorns? 42. Reducc 37mi. 7fur. 37rd. 6yd. 5ft. to feet. 43. How many barleycorns will reach round the earth, each degree being 691 miles? and how many quarters of OF REDUCTION. 45 barley are contained in such a number of barleycorns, admitting 7922 barleycorns to fill.a pint? 44. In 77A. 1R. 14P., land measure, how many perches? 45. In 17280 perches, how many acres? 46. In 50A. 3R. 10P. 9sq. yd. 789sq.ft., how many square feet? 47. In 175 square chains, how many square rods? 48. In 14976 perches, or square rods, how many acres? 49. In 83789263P., how many square miles? 50. In 28 tons of round timber, how many solid inches? 51. In 155 cords of wood, how many solid feet? 52. In 17 cords of wood, how many solid inches? 53. In 56320 solid feet, how many cords? 54. Reduce 349938 cord feet to cords. 55. In 32hhds., wine measure, how many quarts? 56. In 3276 gallons, how many tuns? 57. In 75hhds., how many pints? 58. In 77hhds. of brandy, how many half-ankers? 59. In l0tuns 2hhds. 18gals. of wine, how many gills? 60. In 98 hogsheads of ale, how many pints? 61. In 38 butts of porter, how many pints? 62. In 516 barrels of beer, how many half-pints? 63. How many gallons of beer are contained in 50 barrels? 64. In 44 quarters of corn, how many pecks? 65. In 30720 quarts, how many lasts? 66. How many sacks in 103 London chaldrons and 12 bushels of coal? 67. How many seconds in a year of 365da. 6hr.? 68. How many seconds in 6 years of 365da. 23hr. 57m. 39sec. each? 69. In 7569520118 seconds, how many years of 365 da each? 70. In 5927040 minutes, how many weeks? 46 ADDITION. ADDITION. 48. The sum of two or more numbers, is a number which contains as many units, and no more, as are found in all the numbers added; and ADDITION is the process of finding the sum or sum total of two or more numbers. If 3 be added to 5 their sum will be 8, and the unit of the number 8 will be the same-as the unit of the numbers 3 and 5. The numbers 3 and 5, which are thus added, must have the same unit; for, if 3 denoted tens, and 5 expressed units of the first order, their sum would neither be 8 tens nor 8 simple units. So if 3 expressed yards, and 5 feet, their sum would neither be 8 yards nor 8 feet. 49. Small numbers may be added mentally; but it is not convenient to add large numbers without first writing them down. How are they to be written? If we place one above the other, units OPERATION. of the same kind will fall in the same ver- 3 tical line, and the units of the same order 5 will fall directly under each other in the Sum 8 sum. Again, let it be required to add together 324 and 635. In the first number there are 4 units, 2 OPERATION. tens, and 3 hundreds. In the second, 5 units, 3 tens, and 6 hundreds. Let the figures of each order of units be placed under 324 those of the same order, and added: their 635 sum will be 9 units, 5 tens, and 9 hundreds, Snm 959 or nine hundred and fifty-nine. QUEsT.-48. What is thesum of two or more numbers I What is addition What numbers can be blended into one sum? 49. How may small numbers be added? How are numbers written down for addition? ADDITION. 47 50. Add together the numbers 894 and 637. OPERATION. Write the numbers thus - - 894 637 And draw a line beneath them - sum of the column of units - - - 11 sum of the-column of tens - - - 12 sum of the column of hundreds - - - 14 Sum total 1531 In this example, the sum of the units is 11, which, cannot be expressed by a single figure. But 11 units are equal to 1 ten and 1 unit; therefore, we set down 1 in the place of units, and 1 in the place of tens. The sum of the tens is 12. But 12 tens are equal to 1 hundred, and 2 tens; so that 1 is set down in the hundred's place, and 2 in the ten's place. The sum of the hundreds is 14. - The 14 hundreds are equal to 1 thousand, and 4 hundreds; so that 4 is set down in the place of hundreds, and 1 in the place of thousands. The sum of these numbers, 1531, is the sum sought. The example may be done in another way, thus: Having set down the numbers, as before, we say, 7 and 4 are 11: we set down 1 OPERATION. 894 in the units place, and write the 1 ten 637 under the 3 in the column of tens. We 11 then say, 1 to 3 is four, and 9 are 13. We 1531 set down the 3 in the tens place, and write the 1 hundred under the 6 in the column of hundreds. We then add the 1, 6, and 8 together, for the hundreds, and find the entire sum 1531, as before. When the sum in any one of the denominations exceeds 10, or an exact number of tens, the excess must be written down, and a number equal to the number of tens added to the next higher denomination. This is called carrying to- the next column or higher denomination. The number to be carQfEST. —50. What is the sum of the units? What of the tens? What of the hundreds?'What the entire sum? 48 ADDITION. ried may be written under that column or remembered and added in the mind. 51. What is the sum of the numbers 375, 6321, and 598? In this example, the small figure placed OPERATION. under the 4, shows how many are to be 375 carried from the first denomination to the 6321 second, and the small figure under the 9, 598 how many are to be carried from the sec- 7294 ond to the third. In like manner, in the examples below, the small figure under each column shows how many are to be carried to the next higher denomination. Beginners had better set down the numbers to be carried as in the examples. (2.) (3.) (4.) 96972 9841672 81325 3741 793139 6784 9299 888923 2130 Sum 110012 Sum 11523734 Sum 90239 2221 221111 1110 52. Let it be required to fiird the sum of ~14 7s. 8d. 3far., and ~6 18s. 9d. 2far. We write down the numbers, as before, so that units of the same value shall fall under each other. Beginning with the lowest denomination, we find the OrERATION. sum to be 5 farthings. But as 4 far- O AIN ~s. d. far. things make a penny, we set down 14 7 8 3 14 7 8 3 the 1 farthing over, and carry 1 to 6 18 9 2 the column of pence. The sum of 21 6 6 1 the pence then becomes 18, which make 1 shilling and 6 over, Set down the 6, and carry I to the column of shillings, the sum of which becomes 26; that is, 1 pound and 6 shillings. Setting down the 6 shillings and QuEsT. —How may the units to be carried be disposed of? 51. How will you remember how many are to be carried from one column to an other? 52. Explain the manner of adding pounds, shillings, and pence, and of passing from one denomination to another. ADDITION. 49 carrying I to the column of pounds, we find the entire sum to be ~21 6s. 6d. lfar. 53. Hence, for the addition of all numbers, Write down the numbers so that units of the same denomination shall fall in the same column, and draw a line beneath them. Add up the units of the lowest denomination, and divide their sum by so many as make one of the denomination next higher. Set down the remainder and carry the quotient to the next higher denomination, and proceed in the same manner through all the denominations to the last. PROOF OF ADDITION. 54. The proof of an arithmetical operation is a second operation, by means of which the first is shown to be correct. Addition may be proved by adding all the columns downward. It may also be proved by dividing the numbers to be added into two parts, adding each of the parts separately, and then adding their sums. If the last sum is the same as that of all the numbers first found, the work may be considered right. EXAMPLES. 182796 182796 32160 143274 143274 47047 32160 Partial sums 326070 79207 47047 Sum 405277 326070 1st partial sum. 79207 2d " Proof 405277 QuEST.-53. What is the general rule for addition? 54. What is the proof of an arithmetical operation? What is the first method of proving addition? What the seoond? 3 50 ADDITION. (1.) (2.) (3.) 34578- 22345- 234563750 67890 78901 87 8752 23456 328 340 78901 17 350 23456 327 78 78901 Sums total Partial sums 4509J 77410 - 283615Proofs 39087 99755 307071 (4.) (5.) (6.) 672981043 1278976 8416785413 67126459 7654301 6915123460 39412767 876120 31810213 7891234 723456 7367985 109126 31309 654321 84172 4871 37853 72120 978 2685 7. Add together six tens, fourteen hundreds, seven th,sands, nine ten thousands, forty-five millions, and six thousand seven hundred and fifty-one. 8. What is the sum of six hundreds, eight units of the fifth order, thirteen of the sixth, twenty of the second, forty of the third, and two billions, three millions, four trillions, two hundred and twenty-one thousand seven hundred and fifty-five? 9. What is the sum of eight hundred units of the first order, sixty of the second, one thousand of the third, ninety-nine of the folirth, one hundred of the fifth, six trillions, one billion, forty-nine thousand eleven hundred and sixty-one? 10. What is the sum of three hundred and forty units of the third order, seven thousand six hundred and fifty of the fourth, three millions of the second, and six trillions seven hundred and ninety-nine of the first? H1. Collect together into one sum, two hundred and seventy-eight millions four thousand six hundred and sixty-nine; seventy-six billions four hundred and fifty-eight millions four ADDITION. 51 hundred and seventy-five thousand five hundred and two; fifty billions three hundred millions; four hundred and seventytwo millions four thousand five hundred and fifty-five; nine millions seven hundred thousand three hundred and two; twelve millions three hundred thousand four hundred and sixty-one; two hundred millions four hundred thousand and four; eight hundred millions seven hundred and forty-nine thousand seven hundred and ninety-nine; two hundred and six millions four hundred and forty thousand and thirty-four. 12. Find the sum total of five billions six hundred and forty-nine millions three hundred and seven thousand and sixty; nine hundred and forty millions three hundred and seventy-four thousand six hundred and eighty-one; nine billions eight hundred and seventy-six millions five hundred and forty-three thousand two hundred and ten; one hundred and twenty-three millions four hundred and fifty-six thousand seven hundred eighty-nine; five billions three hundred millions seven hundred and seventy-seven thousand seven hundred and seven. 13. Add together seven hundred and four billions three hundred and sixty-millions five hundred and thirteen thousand and forty.two; sixty-four billions seven hundred and'ninetythree millions six hundred and twenty-nine thousand five hundred and forty-eight; six hundred and ninety-nine billions six hundred and ninety-nine millions eight hundred and sixtyfive thousand seven hundred and seventy-five. 14. Collect together and find the sum of fifty-eight billions nine hundred and eighty-two millions four hundred and eightyseven thousand six hundred and fifty-four; seven hundred and forty billions three hundred anc fifty millions five hundred and forty thousand seven hundred and sixty; four hundred and twenty-five billions seven hundred and three millions four hundred and two thousand six hundred and three; thirtyfour billions twenty millions forty thousand and twenty; five hundred and sixty billions eight hundred millions seven hundred thousand and four hundred. 52 ADDITION. (15.) (16.) (17.) (18.) $87,046 $950,60 $109,049 $8704,067 21,846 107,27 691,027 7504,61 19. What is the sum of 6 eagles 15 dollars 75 cents 5 mills, + 4 eagles 100 dollars 30 cents 8 mills, + 607 dollars 8 cents 1 mill, + 407 eagles 604 dollars 89 cents 9 mills? 20. What is the sum of 47 eagles 207 dollars 51 cents 8 mills, + 4 eagles 49 dollars 1 cent 1 mill, + 1000 eagles 40009 dollars 16 cents 9 mills, + 691 eagles 9791 dollars 14 cents 2:nlills? (21.) (22.) (23.) (24.) ~ s. d. ~ s. d. ~ s. d. ~ s. d. 149 14 7 14 11 31 14 19 44 14 10 41 37 11 93 19 18 10 17 11 10 77 18 3 64 14 7 77 11 31 39 18 11~ 14 13 91 104 19 11k 4914 7 19 14 9 67 12 43 64 13 10 16 18 41 19 15 114 9 1110 174 19 11- 17 15 10 18 19 10 18 10 5 47 14 104 1 14 94 77 19 114 17 19 4 39 15 11 6 1810 14 11 104 19 10 4 (25.) (26.) (27.) (28.) lb. oz.pwt. oz. pwt. gr. lb. oz. pwt. oz.pwt. g. 174 11 19 174 19 23 71 11 19 74 19 23 74 10 13 714 11 14 64 8 14 64 14 17 944 9 14 714 b 18 77' 0 0 74 19 11 74 11 19 74 1 22 14 3 11 -66 13 9 944 10 13 -948 2 21 64 2 9 74 14 11 74 11 3 74 1 12 74 1 14 14 10 3 14 9 4 715 2 14 77 2 13 19 11 14 77 10 11 714 18 16 19 2 14 17 10 13 ADDITION. 53 (29.) (30.) (31.) (32.) DI I 3 3 33 3 3 gr. lb 3 47 11 7 149 7 2 749 2 19 84 11 7 94 10 6 714 3 0 607 1 18 74 10 6 74 10 4 619 2 1 714 2 17 37 5 4 75 9 3 74 6 2 400 0 0 19 4 3 69 0 2 169 5 2 74 1 13 74 1 2 57 1 2 74 1 2 715 2 14 79 2 6 18 2 1 777 6 1 64 1 18 19 2 4 74 1 2 948 5 2 174 2 19 74 9 5 (33.) (34.) (35.) (36.) T. cwt. qr. cwt. qr. lb. qr. lb. oz. lb. oz. dr. 174 19 3 174 3 27 44 27 15 17 15 15 74 14 2 714 2 24 74 26 14 27 14 11 714 13 1 149 1 14 19 14 13 16 13 9 718 16 2 719 2 16 74 19 14 74 14 14 734 15 2 407 1 23 66 27 13 70 0 0 714 14 1 149 2 17 74 19 10 64 13 10 70 13 2 714 2 18 14 18 11 74 14 11 (37.) (38.) (39.) (40.) yd. qr. na. E. E. qr. na. E. FT. qr. na. E. Fl. qr. na. 74 3 3 77 4 3 749 5 3 714 2 3 64 2 1 14 3 2 704 4 2 615 1 2 74 1 3 74 2 1 108 3 1 714 1 3 49 2 1 49 1 2 705 4 0 724 2 2 74 1 2 74 2 1 708 3 1 149 1 2 44 3 1 74 3 2 474 5 2 718 2 3 74 2 0 44 1 2 174 0 1 419 1 1 14 1 2 74 2 3 194 3 2 710 1 2 54 ADDITION. (41.) (42.) (43.) (44.) L. mi. fur. Fur. rd. yd. Rd. yd. ft Ft. in. bar 17 2 7 147 39 5- 177 54 2 174 11 2 14 1 6 614 37 4- 714 443 1 49 10 1 74 1 7 714 19 3- 714 1~ 2.74 11 2 69 2 4 674 17 1- 615 0 1 64 9 1 74 1 0 719 27 24 714 12- 2 74 10 1 69 2 1 197 19 1 719 1~ 1 -64 11 2 74 1 2 714 14 34 437 23 1 74 10 0 94 0 3 704 19 44 614 1~ 2 64 9 1 (45.) (46.) (47.) (48.) A. R. P. A. R. P. A. R. P. A. R. P. 77 3 39 714 3 39 14 3 39 174 3 39.64 2 37 619 1 18 74 1 19 714 1 27 74 1 24 714 2 27 64 2 14 618 2 12 64 2 19 619 1 34 74 1 18 719 1 14 74 1 18 719 2 37 47 2 24 734 2 11 64 2 17 719 1 24 18 1 14 715 1 24 14 1 13 615 2 14 74 2 19 639 2 14 174 2 11 74 1 18 34 1 14 714 3 24 (49.) (50.) (51.) (52.) Tun hhd. gal. Pun. gal. qt. Tierce gal. qt. Gal. qt. pt. 714 3 62 714 83 3 74 41 3 14 3 1 614 2 61 615 81 2 64 40 2 74 2 1 174 1 39 714 74 1 74 19 1 39 2 1 164 2 47 614 18 2 64 39 2 17 1 0 274 1 49 713 75 0 74 40 1 19 2 0 175' 2 37 614 17 ] 69 19 1 77 1 1 375 1 49 715 14 3 17 39 2 39 3 1 714 2 61 719 28 2 18 41 1 14 1 1 ADDITION. b5 (53.) (54.) (55.) (56.) Bar. fir. gal. Bar.Jir. gal. HM-d. gal. qt. Hhd. gal. qt. 7 1 3 8 73 3 7 714 47 3 714 53 3 14 2 7 69 2 6 614 44 1 415 47 2 16 1 4 14 1 7 374 43 2 714 19 1 17 1 3 39 2 2 157 41 1 614 27 1 29 2 2 19 1 6 719 42 1 715 51 2 17 1 7 49 2 6 374 41 2 714 37 2 41 2 6 37 1 4 174 12 1 -615 19 1 37 1 5 19 1 2 19 13 2 714 18 2 (57.) (58.) (59.) (60.) L. ch. bu. pk. Weys qr. bu. Qr. bu. pk. Scows. L. ch. bu 14 31 3 174 3 7 149 7 3 74 20 35 74 31 2 375 1 6 715 3 2 49 19 33 64 30 1 400 0 5 649 1 3 64 17 35 74 27'2 371 1 4 479 2 1 74 14 10 64 19 2 634 2 3 675 1 3 39 13 9 74 31 1 719 1 2 149 2 1 47 16 3 64 11 1 149 2 1 375 1 2 19 17 4 95 10 2 375 1 3 649 1 3 37 18 34 (61.) (62.) (63.) (64.) Yr. mo. wk. Mo. wk. da. Da. hr. min. Hr. min. sec. 737 12 3 64 3 6 714 23 59 647 59 59 347 11 2 74 1 5 74 14 54 137 54 54 618 10 1 34 2 3 64 21 55 375 56 56 374 92 74 1 4 74 13 53 714 17 19 175 3 1 63 2 1 69 12 14 615 54 54 714 12 3 74 1 2 74 12 19 714 17 13 615 10 1 64 2 1 37 11 17 613 34 56 714 3 1 74 1 3 16 12 19 624 27 39 56 ADDITION. APPLICATIONS. 1. In 1843, the number of acres of the public lands solo in the several states and territories was as follows:-In Ohio, 13338 acres, Indiana 50545, Illinois 409767, Missouri 436241, Alabama 178228, Mississippi 34500, Louisiana 102986, Michigan 12594, Arkansas 47622, Wisconsin 167746, Iowa 143375, Florida 8318. What was the whole number of acres sold in the United States? 2. The number of acres of the public lands sold in 1834 was 4658218; in 1835, 12564478; in 1836, 25167833. THe number sold in 1840 was 2236889; in 1841, 1164796; in 1842, 1129217. How many acres were sold in the first three, and how many in the last three years? 3. In 1844, the school fund of Connecticut was invested as follows: in bonds and mortgages, $1695407,44; in bank stock, $221700; in cultivated lands, and buildings, $78367; in wild lands, $52493,75; in stock in Massachusetts, $210; in cash, $3245,58. What was the whole amount of the fund? 4. Thb salaries of the English cabinet ministers are as follows:' of the First Lord of the Treasury, ~5000; of the Lord High Chancellor, ~1.4000; of the Lord President of the Council, ~2000; of the Lord Privy Seal, ~2000; of the Secretaries of State for the Home, Foreign, and Colonial Departments, ~15000; of the Chancellor of the Exchequer, ~5000; of the First Lord of the Admiralty, ~4500; of the Paymaster-general, ~2500; of the President of the Board of Control, ~2000. Required the sum of the salaries of the cabinet. 5. What was the whole number of pieces coined in the United States'mint in 1835, there having been 371534 halfeagles, 131402 quarter-eagles, 5352006 half-dollars, 1952000 quarter-dollars, 1410000 dimes, 2760000 half-dimes, 3878000 cents, and 141000 half-cents? Required also the value of the whole number of coins executed in that year. 6. The value of the imports during Mr. Monroe's second ADDITION. 57 administration was, in 1821, $62585724; in 1822, $83241541; in 1823, $77579267; in 1824, $80549007., The value of the exports in 1821, was $64974382; in 1822, $72160281; in 1823, $74699030; in 1824, $75986657. What was the amount of imports and the amount of exports in that term? V 7. What was the population of the British provinces in North America in 1834, the population of Lower Canada being stated at 549005, of Upper Canada 336461, of New Brunswick 152156, of Nova Scotia and Cape Breton 142548, of Prince Edward's Island 32292, of Newfoundland 75000?' 8. What was the population of Brazil in 1819, there having been of whites 843000; of free people of mixed blood, 426000; of Indians, 259400; of free negroes, 159500; of negro slaves, 1728000; of slaves of mixed blood, 202000 t 9. The imports into France, in 1826, were valued at 564728392 francs; in 1827, at 565804228 francs; in 1828, 607677321 francs; in 1829, 616353397 francs; in 1830, 638338433 francs; in 1831, 512825551 francs; in 1832, 652872341 francs; in 1833, 693275752 francs. What was the value of the imports for those years? 10. The number of emigrants in 1837, from Great Britain to British North America, was: from England, 5027; from Scotland, 2394; and from Ireland, 22463. The number to the United States the same year was, from England, 31769; from Scotland, 1130; from Ireland,33871. Required the number of emigrants to each place, and the entire number. 11. The consumption of coffee in Great Britain is stated to be 10500 tons a year; in the Netherlands and Holland, 40500 tons; in Germany and the countries round the Baltic, 32000 tons; in France, Spain, Italy, Turkey in Europe, and the Levant, 35000 tons; ip America, 20500 tons. What is the entire consumption of coffee in these countries? 12. The numbei of regular troops furnished by each of the states in the revolution, was as follows: New Hampshire, 12497T; Massachusetts, 67907; Rhode Island, 5908; Con necticut, 31939; New York, 17781; New Jersey, 10726;. 3* 58 ADDITION. Pennsylvania, 25678; Delaware, 2386; Maryland, 13912; Virginia, 26678; North Carolina, 7263; South Carolina, 6417; Georgia, 2679. What was the number of regular troops engaged during the war? 13. The revenue of the post-office at Albany, for the fourth quarter of 1845, was $2697; at Baltimore, $10339; at Brooklyn, N. Y., $1279; at Bangor, Me., $1107, at Buffalo, $2339; at Cincinnati, $6103; at Detroit, $1007; at Hartford, $1239; at Louisville, $1946; at Mobile, $4199; at Nashville, $1194; at Newark, N. J., $1026; at Norfolk, Va., $1175-; at Petersburg, Va., $1090; at Philadelphia, $21642; at Pittsburg, Pa., $3612; at Providence, $3046; at Rochester, N. Y., $2606; at Springfield, Mass., $1031; at Troy, N. Y., $1883. What was the total amount of revenue received from these post-offices? 14. The list of vessels in the British navy, on the 1st of January, 1846, was as follows: sailing vessels in' commission' and in'ordinary,' 361; sail vessels building, 42; stean frigates, 11;. steam frigates building, 12; other steam vessels, 88; steam vessels building, 8; steam packets, 25; receiving and quarantine vessels, transports, &c., 134. What is the whole number of vessels, and what the number of each kind? 15. The deposites of gold for coinage at the mint in Philadelphia, in 1842, were: from mines in the United States, $273587; coins of the United States, old standard, $27124; foreign coins, $497575; foreign bullion, $158780; jewellery, $20845. The deposites of silver were.: bullion from North Carolina, $6455; foreign bullion, $153527; Mexican dollars, $1085374; South American dollars, $26372; European coins, $272282; plate, $23410. What was the amount of gold deposited? What of silver? And what the entire sum? 16. Of the public lands, there were ceded by the states of Virginia, New York, Massachusetts, and Connecticut, 169609819 acres; by Georgia, 58898522 acres; by North and South Carolina, 26432000 acres; and 987852332 acres were purchased of France and Spain. Required the number of acres ceded and purchased. ADDITION. 59 17. The population of New York city in 1840 was 312710; of Philadelphia, 258037; of Baltimore, 134379; of New Orleans, 102193; of Boston, 93383; of Cincinnati, 46338; of Brooklyn, 36233; of Albany, 33721; of Charleston, 29261; of Louisville, 21210; of Richmond, 20153; of St. Louis, 16469. What was the whole number of inhabitants in these twelve cities? 18. The following table exhibits the population of the several states and territories, at the taking of each census to 1840. What was the population of the United States in each of those years? States. 1790. 1800. 1810. 1820. 1830. 1840. Maine - - - 965401151719 228705 298335 399955 501793 New Hampshire 1418991183762 214360 244161 269328 284574 Vermont - 854161154465 217713 235764 280652 291948 Massachusetts 378717423245 472040 523287 610408 737699 Rhode Island - 691101 69122 77031 83059 97199 108830 Connecticut - 23814112510022 62042 275'202 297665 309978 New York - 34012015867561959949 137282 1918608 2428921New Jersey - 18413921194J9249555 2775751 320823 373303 Pennsylvania - 434373 602365 810091 1049458 1348233 1724033 Delaware - - 59098 64273 72674 72749 76748 78085 Maryland - - 3197281341548 380546 407350 447040 470019 Virginia - - 748308 880200 974642 1065379 1211405 1239797 North Carolina 3937511478103 555500 638829 737987 753419 South Carolina 249073 3455911415115 502741 581185 594398! Georgia - - 825481162101 252433 340987 516823 691392 Alabama - - - - 20845 127901 309527.590756 Mississippi - - 8850 40352 75448 136621 375651 Louisiana - - - - 76556 153407 215739 352411 Arkansas - - - - - 14273 30388 97574 Tennessee - - 30791 1056021261727 422813 681904 829210 Kentucky - - 730772209551406511 564317 687917 779828 Ohio - - - - 453651230760 581434 9379031519467 Michigan - - - _ 4762 8896 31639 212267 Indiana - - - 4875 24520 -147178 343031 685866 Illinois - - - 12282 55211 157455 476183 Missouri - - 20845 66586 140445 383702 Dist. Columbia - 14093 24023 33039 39834 43712 Florida - - - - - - 34730 54477i Wisconsin - - _ _ _ _ - 309451 Iowa.- - - _ _ _ _ 4311 1 l J~~~~~~~~~309 60 ADDITION. 19. The slave population of the states and territories, according to each census, is shown in the following table. Required the number of slaves in the United States at each enumeration. States. 1790. 1800. 1810. 1820. 1830. 1840. Maine - - 0 0 O 0 New Hampshire - 158 8 0 0 Q 1 |Vermont - 17 0o 0 0 0 Massachusetts - 0 0 0 0 0 0 IRhode Island - 952 381 103 48 17 5 Connecticut - - 2759 951 310 97 25 17 New York - - 21324 20343 15017 10088 75 4 New Jersey - - 11423 12422 10851 7657 2254 674 Pennsylvania - 3737 1706 795 211 403 64 Delaware - - 8887 6153 4177 4509 3292 2605 Maryland -- 103036 105635 111502 107398 102294 89737 Virginia - - 203427 345796 392518 425153 469757 448987 North Carolina - 100572 133296 168824 295017 235601 245817 South Carolina - 107094 146151 196365 258475 315401 327038 IGeorgia - - - 2964 59404 105218 149656 217531 280944 Alabama - - - - - 41879117549 253532 Mississippi - - - 3489 17088 32814 65659 19521-1 Louisiana - - 34660 69064 109588 168452 Arkansas - - - - - 1617 4576 19935 Tennessee - - 34-17 13584 44535 80107 141603 183059 Kentucky - - 11830 40343 80561126732 165213182258 Ohio - - -. 3 Michigan - - 24 - 32 0 Indiana - - - - 135 237 190 0 3 Illinois - - - - - 168 117 *747 331 Missouri - - - 3011 10222 25081 58240 Dist. Columbia - - 3244 5395 6377 6119 4694 Florida - " - - - - - 15501 25717 Wisconsin - - - _ - 11 Iowa - - - I - - - 16 * Indented colored servants. SUBTRACTION. 61 SUBTRACTION. 55. SUBTRACTION is the process of finding the difference between two numbers. When the numbers are unequal, the larger of the two is called the minuend, and the less is called the subtrahend, and their difference, whether they are equal or unequal, is called the remainder. When the numbers are small, their difference is apparent, and the subtraction may be made mentally. EXAMPLES. 1. From 869 subtract 327: that is, from 8 hundreds 6 tens and 9 units, it is required to take 3 hundreds 2 tens and 7 units. We begin at the right hand figure of the lower line, and say, 7 from 9 869 Minuend. leaves 2: set down the 2 under the 7. 327 Subtrahend. Proceeding to the next column, we 542 Remainder. say, 2 from 6 leaves 4; set down the 4, and then say, 3 from 8 leaves 5. Hence, the remainder or true difference between the numbers is 542. 2. From 843 subtract 562. Beginning with the lowest denomina- OPERATION. tion, we say, 2 from 3 leaves 1. At the 843 next step we meet a difficulty, for we can- 562 not subtract 6 from 4. If, now, we add 10 281 tens to the 4, (which are written in small figures above,) and 10 tens to the 6 directly under it, it is plain that the difference will not be affected, since both the QUEST.-55. What is subtraction? How many numbers are considered in subtraction? What are they called? When the numbers are small, how may the subtraction be made? Ex. 2. How do we get over the difficulty in subtracting the tens? If equal numbers be added to the minuend and subtrahend, will their difference be changed? C2 SUBTRACTION. numbers are equally increased. But adding 10 tens to the 6 is the same thing as adding 1 to the 5 hundreds: hence, we may consider 10 to be added to any figure of the minuend, provided we add 1 to the next figure of the subtrahend to the left. We can now go on with the subtraction; for we say, 6 from 14 leaves 8. Then, 1 carried to 5 makes 6: and 6 from 8 leaves 2. Hence the remainder is 281; and all similar examples are done in the same manner. 3. T: cwt. qr. lb. oz. 20 28 From 6 14 2 20 12 take 4 17 1 21 10' 1 Remainder 1 17 0 27 2 In this example we say, 10 ounces from 12 leaves 2. At the next denomination we meet a'difficulty, for we cannot subtract 21 from 20. We add to the 20 so many units as make 1 unit of the next higher denomination-that is, 28, and suppose at the same time 1 unit to be added to that denomination in the subtrahend. We then say, 21 from 48 leaves 27: then 2 from 2 leaves 0. In the hundreds we again have to add, after which we say, 17 from 34 leaves 17; then we take 5 from 6, and have the true remainder. 56. Hence, to find the difference between two numbers: Set down the less number under the greater, so that units of the same denomination shall fall in the same column, and beginning with the lowest denomination, subtract each from the one above it. When the units in any one denomination of the subtrahend exceed those of the saine denomination in the minLuend, suppose so many units added in the minuend as make one unit of the next higher denomination; after which add one to the next denomination of the subtrahend, and subtract as before. QUEST.-If you add 10 to any figure of the minuend, what will you ard] to the subtrahend? Ex. 3. How is the subtraction made in this example! 56. What is the rule for subtraction? SUBTRACTION. 63 PROOF. 57. Add the remainder to the subtrahend, and if the sum is equal to the minuend, the work may be regarded as right. Or, subtract the remainder from the minuend, and the remainder thus found should be equal to the subtrahend. EXAMPLES. (1.) (2.) (3.) 1010 10101010 20 124 From 87407 27431 ~14 16s. 7~d. take 6079 19872 6 17 94 11 1 1 11 1 I 1 Rem. Proof 4. From 47348406051320047 take 13456507031079054. 5. From 19493899900056075 take 14954298990056076. 6. From 500714960079690650 take 742350986470501. 7. From 149348761340526465 take 48973024012394. (8.)) (9.) (10.) (11.) From $374,674 $270,604 $137,04 $9496,004 take 195,097 191,280 127,97 8496,049 Rem. 12. What is the difference between $487,25 and $379,674? 13. What is the difference between $670,04 and one hundred and four dollars and 6 mills? 14. What is the difference between $1000 and $14,003? (15.) (16.) (17.) (18.) lb.,oz. pwt. oz. pwt. gr. lb. oz. pwt. oz.pwt. gr. 14 11 9 74 12 13 175 3 10 17 10 20 11 10 14 64 14 17 159 11 14 14 11 23 QUEST.-57. What is the first method of proving subtraction? What is the second 7 64 SUBTRACTION. (19.) (20.) (21.) (22.) lb 3 3 3 3 9gr. lb 3. 3 144 10 5 27 4 1 27 1 14 74 10 5 64 11 7 14 7 2 14.0 19 65 11 6 (23.) (24.) (25.) (26.) T. cwt. qr. Cwt. qr. lb. Qr. lb. oz. lb. oz. dr. 14 12 2 17 1 25 143 22 12 174 11 10 1 14 3 14 2 27 74 19 14 39 12 13 (27.) (28.) (29.) (30.) Yd. qr. na. E.E. qr. na. E. Fr. qr. na. E. Fl. qr. na. 174 2 1 174 3 1 171 1 3 12 1 1 39 3 2 49 4 2 74 5 2 10 2- 3 (31.) (32.) (33.) (34.) L. mi. fur. Fur. rd. yd. Rd. yd. ft. Ft. in. bar. 21 2 4 13 34 3- 14 33 1 17 11 2 3 2 6 12 39 51 9 42 2 14 11 - 1 (35.) (36.) (3-7.) (38.) A. R. P. A. R. P. A. R. P. A. R. P. 12 1 32 112 1 31 12 1 25 19 1 20 1 3 14 74 2 37 10 3 39 14 2 21 (39.) (40.) (41.) (42.) Tun hhd. gal. Pun. gal. qt. -Tier. gal. qt. Gal. qt. pt. 27 2 54 147 14 2 14 1 2 24 3 0 19 3 62 79 83 3 12 41 3 17 0 1 SUBTRACTION. 65 (43.) (44.) (45.) (46.) Bar.jir. gal. Bar. fir. gal. HAd. gal. qt. Hhd. gal. qt. 14 3 5 147 1 3 271 1 2 143 1 2 12 3 7 39 3 8 49 47 3 79 52 3 (47.) (48.) (49.) (50.) L.ch. bu. pk. Weys qr. bu. Qr. bu. pk. Scows.L.ch. bu. 74 31 3 17 3 1 147 6 2 47 1 12 47 31 2 14 3 7 94 7 3 14 20 35 (51.) (52.) (53.) (54.) Yr. mo. wk. Mo. wk. da; Da. hr. min. Hr. min. sec. 17 11 2 147 2 3 167 21 50 147 50 51 14 12 3 19 2 4 19 23 54 94 59 57 PROMISCUOUS EXAMPLES. 55. A horse in his furniture is worth ~52 10s.; out of it, ~24 10s. 6d. How much does the price of the furniture exceed that of the horse? 56. What sum added to ~11 14s. 9-d. will make ~133 1Is. and 91d.? 57. A tradesman failing, was indebted to A ~105 19s. ld., to B 150 guineas, to C ~34 18s. 10d., to D ~500 19s., to E ~700 14s. 9d. When this happened, he had cash by him to the amount of ~50, goods to the amount of ~350 14s. 9d., his household furniture was worth ~24 1 s., his bookdebts amounted to ~94 14s. 8d. If these things were faithfully given up to his creditors, what did they lose by him? 58. The great bell at Oxford, the heaviest in England, weighs 7T. llcwt. 3qr. 41b.; St. Paul's bell at London weighs 5T. 2cwt. lqr. 221b.; and 7Tom of Lincoln weighs 4T. 1.6cwt t66 SUBTRACTION..3qr. 181b. How much are these bells, together, inferior in weight to the great bell at Moscow, the largest in the world, which weighs 198T. 2cwt. 1qr.? 59. An apprentice, who is 14 years, 11 months, 13 weeks, 14 hours, 38 minutes old, is to serve his master till he is 21 years of age. How long has he to serve'? 60. What is the difference of latitude and longitude between Calcutta in the East Indies, (lat. 22~ 34' N., long. 880 34' E.,) and Lima, in South America, (lat. 12~ 1/ S., long. 76~ 44' W.)? 61. NEWTON (Sir Isaac) was born at'Woolsthorp, a hamlet in the parish of Colsw6rth, in Lincolnshire, on $unday, the 25th December, 1642; and died at Kensington, in Middlesex, on Monday, the 20th March, 1727. EULER (Leonard) was born at Basil, in Switzerland, on Tuesday, the 15th April, 1707; and died at Petersburg, in Russia, on Sunday, the 7th September, 1783. LAGRANGE (Joseph Louis) was born at Turin, in Italy, on Friday, the 30th January, 1736; and died at Paris, on Saturday, the 10th April, 1813. LAPLACE (Pierre Simon, marquis of) was born at Beaumont-en-Auge, in France, on Thursday, the 23d March, 1749; and died at Paris, on Tuesday, the 27th March, 1827. How old was each of these eminent philosophers and mathematicians at the time of his decease? and how many years was it from the time each died to January 1st, 1846. 62. In 1840 the amount of tobacco sent from the United States to England, was 26255 hogsheads, and to Holland, 29534 hogsheads. How much more was sent to Holland than to England? 63. The population of the northern district of New York in 1840 was 1683068, and the population of the southern district was 745853. How many more inhabitants were there in the northern than in the southern district, and what was the population of the state? 64. The population of England in 1841 was 14995508, the population of Scotland 2628957, and of Wales 911321. How SUBTRACTION. 67 much did the population of England and Wales combined exceed that of Scotland, and what was the entire population of great Britain? 65. The value of the gold coined at the mint in Philadelphia in 1842 was $960017,50; the value of that coined at Charlotte, N. C., was $159005; at Dahlonega, Ga., $309648; and at New Orleans, $405500. How much more was coined at Philadelphia than at the three other places? 66. The whole amount received for the public lands to 1843, was $170940942,62. There have been paid for the Indian title, the Florida and Louisiana purchase, including interest, $68524991,32; and for surveying and selling, including salaries of officers, $9966610,14. Required the net amount derived from the sale of the public lands. 67. The revenue of Great Britain for the year 1843 was ~50071943, and for the previous year, ~44329865. Required the increase. 68. The value of the merchandise imported into the United States during the year ending June 30th, 1844, was $108435035; of which $24766881 was admitted free of duty, $31352863 paid specific duties, and the remainder paid duties ad valorem. What amount paid ad valorem duties? 69. The value of the products of the sea exported from the United States in 1844, was $3350501; the value of the products of the forest, exported the same year, was $5808712. How much more was exported of the products of the forest than of the sea? 70. The imports from England to the United States in 1844, amounted to $41476081, from Scotland $527239, and from Ireland $88084.' The value of the exports to England, the same year, was $46940156, to Scotland $1953473, and to Ireland $42591. How much did'our exports to Great Britain and Ireland exceed the imports? 71. What was the balance in the treasury of the state of Tennessee, in October, 1844, the income for the year ending 68 SUBTRACTION. that month having been $271823,08; a surplus had been left the preceding year of $38875,21; and the expenditure was $261416,26? 72. The cost of the internal improvements of the state of Ohio, was $15283783,64, of which the Ohio canal cost $4695203,69; the Miami canal, $1237552,16; the Miami Extension, $2856635,96; and the Wabash and Erie canal, $3028340,05. What was the cost of the other works of the state? 73. St. Augustine was founded Sept. 8th, 1565. Jamestown was founded May 13th, 1607. The Battle of Princeton was fought Jan. 3d. 1777. Cornwallis surrendered, Oct. 19th, 1781. Washington was first inaugurated April 30th,' 1789: he died, Dec. 14th, 1799. The French Berlin de-, cree was issued Nov. 21st, 1806, and the British orders in council, Nov. 11th, 1807. The United States declared war against Great Britain June 18th, 1812. The Guerriere was captured by the Constitution Aug. 19th, 1812. The frigate United States captured the Macedonian, Oct. 25th, 1812. York in Upper Canada was captured by the Americans, and General Pike killed, April 27, 1813. Fort George was captured May 27th, 1813. The British were repulsed from Sackett's Harbor by the Americans commanded by General Brown, May 28th, 1813. The Battle of Lake Erie was fought Sept. 10th, 1.813. The Battle of Chippewa was gained by a detachment of the American army under General Scott, July 5th, 1814. The Battle of Niagara, or Lundy's Lane, was fought July 25th, 1814. General Brown conducted the sortie from Fort Erie, Sept. 17th, 1814. The battle of New Orleans Was fought Jan. 8th, 1815. Adams and Jefferson died July 4th, 1826. The compromise bill was introtroduced into the senate Feb. 12th, 1833. General Lafayette died May 20th, 1833. The Cherokees began to remove May 26th, 1838. What time has elapsed from the date of each of these events to March 17th, 1846? MULTIPLICATION. G9 MULTIPLICATION. 58. IF the number 1 be multiplied by 2, that is, taken Qo times, the result will be 2; and 2 is said to be two times greater than 1. If 1 be multiplied by 3, that is, taken three times, the result will be 3; and 3 is said to be three times greater than 1. If 2 be multiplied by 2, that is, taken 2 times, the result will be 4; and 4 is said to be two times greater than 2. If 3 be multiplied by 4, the result will be 12; and 12 is said to be four times greater than 3. In the first case, 1 was taken 2 times; in the second it was taken 3 times; in the third 2 was taken 2 times; and in the fourth 3 was taken 4 times. MULTIPLICATION iS a short process of taking one number as many times as there are units in another. Hence, it is a short method of performing addition. The number to be taken is called the multiplicand. The number denoting how many times the multiplicand is to be taken, is called the multiplier. The number arising from taking the multiplicand as many times as there are units in the multiplier, is called the product. The multiplicand and multiplier, together, are called factors, or producers of the product. There are three-numbers in every multiplication. First, the multiplicand; second, the multiplier; and third, the product. QUEST.-58. If 1 be multiplied by 2, what is the result? How many times greater is this result than 1? If 3 be multiplied by 4, what is the result? How many times greater is the result than 3? What is multiplication? What is the number to be taken called? What is the number showing how many times the multiplicand is to be taken, called? WVhat is the result called? What are the multiplier and multiplicand taken together called? How many numbers are there in every multiplication? What are they called? 70 MULTIPLICATION. 59. Now, since the product is the result which arises from taking the multiplicand as many times as there are units in the multiplier, it follows that, 1st. If the multiplier is unity, the product will be equal to t]e multiplicand. 2d. If the multiplier contains several units, the product will be as mary times greater than the multiplicand, as the multiplier is greater than unity. 3d. If the multiplier be less than unity, that is, if it be a proper fraction, then the product will be as many times less than the multiplicand as the multiplier is less than unity. 60. Let it be -required to multiply any two numbers together, say 6 by 4. If we make, in a horizontal line, as 6 many stars as there are units in the multiplicand, and make as many such j * * lines as there are units in the multi- 4* * * * * plier, it is evident that all the stars will l * * * represent the number of units which result from taking the multiplicand as many times as there are units in the multiplier. Let us now change the multiplier into the multiplicand, and let the multiplicand become the multiplier. Then make, in a vertical line, as many stars as there are units in the new multiplicand, and as many vertical lines as there are units in the new multiplier, and it will be again evident that all the stars will represent the number of units in the product. Hence, Either of the factors may be used as the multiplier without altering the product. For example, 3 X 7 = 7 X 3 = 21: also, 6 X 3 =3 X 6-=18. 9 x 5 = 5 X 9 = 45: also, 8 X 6 = 6 X 8 = 48. and, 8 X 7 = 7 x 8 = 56: also, 5 X 7 = 7 5 = 35. QUEST. —59. If the multiplier is unity, how will the product compare with the multiplicand? How will it compare if the multiplier is greater than unity? How when it is less? 60. If the multiplicand be made the multiplier, will the product be-altered? MULTIPLICATION. 71 61. A composite number is one that may be produced by the multiplication of two or more numbers, which are called the components orfactors. Thus, 2 X 3 = 6. Here 6 is the composite number, and 2 and 3 are the factors, or components. The number 16 = 8 X 2: here 16 is a composite number, and 8 and 2 are the factors; and since 4 x 4 = 16, we may also regard 4 and 4 as factors or components of 16. Let it be required to multiply 8 by the composite number 6, of which the factors are 2 and 3. 8 3o2 evident that the product of 8 x 6 = 48, the number of units in all the lines. But let us first connect the lines -in sets of 2 each, as on the right; there will then beWin each set 8 x 2 = 16, or 16 units -in each set. But there are 3 sets; hence, 16 x 3 48, the number of units in all the sets. umber of units in each set will be equal to 8 x 3 = 24, and there being 2 sets, 24 x 2 = 48, the whole number of units. As the same may be shown for any composite number, we may conclude that, When the multiplier s a composite number, we may multi-on ply by each of the factors in succession, and the last produdt will be the entire product sought. QuEsr.-61. What is a composite number? What are the separate parts called? dividhat are the components or factors in the number 12? In 16? In 20? How do you proceed when the multiplier is a composite number? 72 MULTIPLICATION. 62. Let it be required to multiply 236 by 4; that is, to take 6 units, 3 tens, and 2 hundreds, each 4 times. First set down the 236, then place the OPERATION. 4 under the unit's place 6, and draw a 236 line beneath it. Then multiply the 6 4 units by 4: the product is 24 units; set 24 its them down. Next multiply the 3 tens by 12 tens. 4: the product is 12 tens; set down the 8 hundreds. 2 under the tens of the 24, leaving the 1 944 to the left, which is the place of the hundreds. Next multiply the 2 by 4: the product is 8, which being hundreds, is set down under the 1. The sum of these numbers, 944, is the entire product. The product can also be found, thus: say 4 times 6 are 24; set down the 4, and 236 then say, 4 times 3 are 12 and'2 to carry 4 are 14; set down the 4, and then say, 4 944 times 2 are 8 and i to carry are 9. Set down the 9, and the product is 944 as before. 63. Let it be required to multiply 627 by 84. Multiply by the 4 units, as in the last OPERATION. example. Then multiply by the 8 tens. 627 The first product 56, is 56 tens; the 6, 84 therefore, must be set down under the 0, 2508 which is the place of tens, and the 5 car- 5016 ried to the product of the 2 by 8. Then 52668 multiply the 6 by 8, carry the 2 from the last product, and set down the result 50. The sum of'the numbers, 52668, is the required product. 64. Let it be required to multiply ~3 8s. 6d. 3far. by 6, in which each of the denominate numbers is to be taken 6 times. QUEST.-62. Explain the manner of multiplying 236 by 4. 63. Explain the manner of multiplying 627 by 84. MULTIPLICATION. 73 We first say, 6 times 3 are 18; that is, 18 farthings, which by dividing by 4 3 8 6 3 are found equal to 4d., and 2 farthings 6 over. Set down the 2 farthings, and 20 11 4 2 then say, 6 times 6 are 36, and 4 to carry make 40; that is, 40 pence, which after dividing by 12, are found equal to 3 shillings and 4 pence, Set down the 4d., and then say, 6 times 8 are 48 and 3 are 51; that is, 51 shillings, which are equal to ~2 and 11 shillings over. Set down the 11 shillings, and say, 6 times 3 are 18, and 2 to carry make 20, which write under the pounds. 65. Hence, to multiply one number by another, Mu7tiply every order of units in the multiplicand, in succession, beginning with the lowest, by each figure in the multiplier, and divide each product so formed by so many as make one unit of the next higher denomination: write down each remainder under units of its own order, and carry the quotient to the next product. PROOF OF MULTIPLICATION. 66. Write the multiplier in the place of the multiplicand, and find the product as before; if the two products agree, the work may be supposed right: Or, Divide the product by one of the factors, and the quotient will be the other factor. EXAMPLES. (1.) (2.) (3.) (4.) 847046 9807602 570409 216987 8 7 8 6 QUEsT. —64. Explain the manner of multiplying'.3 8. 6d. 3far. by 6 65. What is the general rule for multiplication? 66. What is the first proof of multiplication? What is the second? 4 74 MULTIPLICATION. (5.) (5.)'Multiply 471493475 471493475 by 4395 4395 2-357467375 1885973900 4243441275 1414480425 1414480425 424344127. 1885973900 2357467375 2072213822625 2072213822625 NOTE 1. Although we generally begin the multiplication by the figure of the lowest denomination,- yet we may multiply in any order, if we only preserve the places of the different orders of units. In the example to the right, we began with the order of thousands. NOTE 2. Although either factor may be used as the multiplier, (Art. 60,) still it is best to use that one which contains the fewest places of figures, as is shown in the last example. For, if we change the process and use the multiplicand as the multiplier, there will be nine multiplications instead of four. 6. Multiply 430714934 by 743. Ans. 7. Multiply 37157437 by 14972. Ans. -- 8. Multiply 47157149 by 37049. Ans. - 9. Multiply 57104937 by 40709. Ans. - 10. Multiply 79861207 by 890416. Ans. —. 11. Multiply 9084076 by 9908807. Ans. ---- 12. Multiply 2748 by 200. Ans. 549600. When there are naughts on the right hand of the significant figures of the multiplier or multiplicand, we may at first neglect them in the multiplication; but then the first significant figure of the product will be of a higher order than the first, and all the ciphers must be added in order to reduce the product to units of the first order. 13. Multiply 67046 by 10: also by 100. 14. Multiply 57049 by 100: als6 by 1000. 15. Multiply 4980496 by 1000: also by 10000 16. Multiply 90720400 by 100: also by 10000 17:. Multiply 74040900 by 1: also by 10 18. Multiply 674936 by 100: also bly 100000. MULTIPLICATION. 7519. Multiply 478400 by 270400. Ans. 20. Multiply 367000 by 37409000. Ans. 21. Multiply 7849000 by 84694000. Ans. 22. Multiply 89999000 by 97770400. Ans. 23. Multiply 9187416300 by 274987650000. 24. Multiply 86543291213456 by 12637482965. 25. Multiply 76729835645873 by 217834569. 26. Multiply 92413627858476 by 90587963412. 27. Multiply 87956743982714 by 819254837609. -- 28. Multiply 23869572491872 by 4007865347912. 29. Multiply 68 by the composite number 72. In this exaimple xWe multiply in succession by the factors 9 and 8. 30. Multiply 3657 by the factors of 64. 31. Multiply 37046 by the factors of 121. 32. Multiply 2187406 by the factors of 144. 67. In multiplying Federal money care must be taken to point off as many places for cents and mills as there are in the multiplicand 1. Multiply 14 dollars 16 cents and 8 mills, by 5, 6, and 7. $14,168 $14,168 $14,168 5 6 7 (2.) (3.) (4.) $870,46 $894,120 $2141,096 9 14=7 x 2 36=6 x 6. 5. What will 95 pounds of tea cost, at $1,04 per pound? 6. What will 105 yards of cloth cost, at $3,25 per yard? 7. What will four firkins of butter cost, each containing 97 pounds, at 251 cents per pound? QvUrT.-67 What precaution is necessary in multiplying Federal money? 76. MULTIPLICATION. 8. What will five casks of wine cost, each containing 59 gallons, at $2,756 per gallon? 9. A bale of goods contains 106 pieces, costing $55 and 37~ cents each: what is the cost of the entire bale? 10. What is the value of 695 hats, at $3,654 each? 11. What will be the cost of 97046 oranges, at 24 cents each? 12. What will be the cost of 6742 sheep, at $2~ each? 13. What will be the cost of 59 barrels of apples, at $~2 per barrel? 14. What will be the cost of 6741 barrels of corn, at $3,254 per barrel? BILLS OF PARCELS. 15. New York, May 1st, 1846. Mr. James Spendthrift Bought of Benj. Saveall. 18 pounds of tea at 85 cents per pound 35 pounds of coffee at 15~ cents per pound 27 yards of linen at 66 cents per yard - - - - Rec'd payment, Benj. Saveall. 16. Albany, June 2d, 1846. Mr. Jacob Johns Bought of Gideon Gould. 48 pounds of sugar at 9~ cents per pound - 6 hogsheads of molasses, 63 gals. each,. at 27 cents a gallon - - - 8 casks of rice, 285 pounds each, at 5 cts. per pound 9 chests of tea, 86 pounds each, at 96 cts. per pound Total cost Rec'd payment, For Gideon Gould, Chares Clark. MULTIPLICATION. 77 17. Hartford, November 21st, 1846. Gideon Jones Bought of Jacob Thrifty. 78 chests of tea, at $55,65 per chest - 251 bags of coffee, 100 pounds each, at 122 cts. per pound - - - - 317 boxes of raisins, at $2,75 per box - 1049 barrels of shad, at $7,50 per barrel - - - - 76 barrels of oil, 32 gallons each, at $1,08 per gal. Amount Received the above in full, Jacob Thrifty. (18.) (19.) (20.) s. d. T. qr. lb. oz. yds. ft. in. Multiply 20 6 8~ 3 3 27 15 16 2 9 by 4 8 9 21. What will 4 yards of 25. 8oz. at 7s. 10d. cloth cost at 7s. 6~d. per 26. 81b. at 7s: 5~d. yard? 27. 10 gallons' at 16s. 41d. 22. 5 bushels at 5s. IOd. 28. llcwt. at ~1 9s. 104d. 23. 6 yards at 6s. 9d. 29. 12 sheep at ~1 17s. 9d. 24. 7 ells at 5s. 11d. 30. In 9 pieces of kersey, each 14yds. 3qrs. 2na., how many yards? 31. What is the weight of 12 tankards, each weighing lloz. 10pwt. 19gr.? 32. In 11 pieces of cloth, each 17yds. 3qrs. 3na., how many yards? 68. In multiplying denominate numbers, if the multiplier is a composite number, and greater than 12, it is best to multiply by the factors in succession. QUEST. —68. If the multiplier is a composite number, how should you multiply ii'denominate numbers? 78 MULTIPLICATION. 33. What will 15 gallons 37. 36T. at. ~5 15s. 1lad. of wine cost at 5s. 31d. per 38. 84 chaldrons at~1 16s. gallon? 93d. 34. 18hhds. at ~3 14s. 5d. 39. 108 bushels at 7s. 91d. 35. l24yds. at 7s. 5~d. 40. 132 ells at 18s. 91d. 36. 35cwt. at ~1 17s. 8~d. I 41. 144 butts at ~5 13s. 9~d. 42. In 32 wedges of gold, each 211b. 7oz. 14gr., how many pounds? 43.-In 21 fields, each 3A. 2R. 19P., how many acres? 69. When the multiplier is greater than 12 and is not a composite number, Take the nearest composite number to the given multiplier, and multiply by its factors in succession. Then multiply by the difference, and add the product when the composite number is less than the multiplier, and subtract it when greater. 44. What is the cost of 23 yards of cloth, at 14s. 9d. per yard? OPERATION. s. d. s. d. (14 9) X (7 X 3)+2 Or this, (14 9) X (6 X 4) —1 7 6 5 3 3 price of 7yds. 4 8 6 price of 6. 3 4 4 15 9 9 price of 21. 17 14 0 price of 24. Add 1 9 6 price of 2. Subtract 14 9 price of 1. Ans. X16- 19 3 price of 23. Ans. ~16 19 3 23 45. What is the cost of 31 yards at 12s. 7~d.? 46. 39 dozen of handkerchiefs at 16s. 9{d. 47. 139 pairs of stockings at 4s. 9-d. 48. 861b. of silk at 19s. 4d. 49. 111 sacks of flour at ~1 4s. 9d. 50. 156cwt. at ~4 9s. 6d. QvuEsr.-69. How do you multiply when the multiplier is greater than 12 and not a composite number? MULTIPLICATION. 79 51. In 57 years, each 13 months, 1 day, 6 hours, how many months? 52. What is the weight of 29hhds. of sugar, each weighing 7cwt. 2qr. 181b.? 53. In 67 parcels of tea, each 257b. 7oz. 13dr., how many cwt., &c.? 54. What will 394 yards cost at 17s. 5~d. per yard? OPERATION. 55. 357 calves at ~7 lOs. 7d. s. d. 56. 549 yards at 12s. 9 —d. 17 5~ 57. 7541b. of tea at 6s. 10d. 10 58. 1981b. of indigo at 6s. 9 X8 14 7 price of lOyds. 3d. 10 4 59. 754 weys at ~20 5s. 87 5 10 priee of 100. ld. 60. 178 ells at 5s. 9-1d. 261 17 6 price of 300. 61. 198bbls. at ~1 14s. 9d. 78 11 3 price of 90. 62. 744 -chaldrons at~1 18s. 3 9 10 price of 4. 8d. ~343 18 7 price of 394. 70. When the multiplier has a fraction annexed to it, multiply first by the whole number, and then add such a part of the multiplicand as the fraction is of unity. 63. What will 56~ chal- 64. What will be the cost of drons cost at ~L 14s. 9d. per 4- yards at 7s. 6d. per yard? chaldron? s. d. ~; s. d 7 6 1 149 4 _ 7I 1 10 0 price of 4. 12 3 3 price of 7. 4 2 price of-. s. d. 8 1 14 2 price of 4. 7 6 97 6 0 price of 56. 5-5. 17 4- price of~. 9)37 6 Ans. ~98 3 4 price of 56A. 4 2 QuEST.-70. How do you multiply when the multiplier has a fraction annexed? EO MULT1PLICATION. 65. 17883 gallons at 6s. 4d. Ans. 66. 37 14-cwt. at ~4 11 s. 9d. Ans. --- 67. 7149A chaldrons at ~1 14s. 9d. Ans. 68. 547w- lasts at ~5 5s. Ans. - 69. 1749~ firkins at lqs. 9kd. Ans. - 70. 7543cwt. at 17s. 51d. Ans. - BILLS OF PARCELS. 71. New Orleans, Jan. 2d, 1846. James Lamb, Esq. Bought of John Simpson. ~ s. d. 71 lbs. of green tea at 10s. 4d. per lb. - 141 do. finest bloom at 14s. 8d. per lb. - - - 103 dfo. fine green at 16s. 5d. per lb. - - - 21 do. hyson at 10s. 104d. per lb. - - - 19 do. good hyson at 13s. 9~d. per lb. - 81 do. bohea at 6s. 9d. per lb. 72. Louisville, March 19th, 1846. George Veres, Esq. Bought of Charles West. ~ s. d. A loin of lamb, weight 71 lb., at 10d. per lb. - A fillet of veal, weight 161 lb., at 64d. per lb. - A buttock of beef, weight 374 lb., at 4~d. per lb. A pig, weight 121 lb., at 7-d. per lb. A leg of pork, weight 16- lb., at 5~d. per lb. A leg of mutton, weight 131 lb., at 4-d. per lb. It 4~ ~ ~, DIVISION. 81 DIVISION. 71. DIVISION 1s the process of finding how many times one number called the. dividend is greater or less than another munubet called the divisor; and the number which expresses how many times the dividend is greater or less than the divisor, is called the quotient. Hence, the quotient is as many times greater or less than unity, as the dividend is greater or less than the divisor. 72. When the entire quotient can be expressed by a whole number, the dividend is said to contain the divisor an exact number of times; but when it cannot be so expressed, the part of the dividend which remains undivided is called the remainder. 73. Since the quotient shows how many times the dividend exceeds the divisor, it follows, that if the divisor be taken as many times as there are units in the quotient, the product will be equal to the dividend. And hence, if the divisor and quotient be multiplied together, and the remainder, if any, added, to the product, the result will be equal to the dividend. EXAMPLES. 1. Divide 86 by 2. Place the divisor on the left of the divi- OPERATION. dend, draw a curved line between them, a and a straight line under the dividend. Now, there are 8 tens and 6 units to be divided by 2. We say, 2 in 8, 4 times, 2)86 43 quotient. which being 4 tens we write the 4 under 43 quotient QUEsT.-71. What is division? What is the quotient? How many times is it greater or less than unity?. 72. When can the entire quotient be expressed by a whole number? When it cannot, what do you call the part of the dividend which is over? 73. If the divisor and quotient be multiplied together, what will the product be equal to? 4* 82 DIVISION. the tens. We then say, 2 in 6, 3 times, which are three units, and must be written under the 6. The quotient, therefore, is 4 tens and 3 units, or 43. Remark that each order of units in the dividend, on being divided, gives the same order of units in the quotient. 2. Divide 729 by 3. In this example there are 7 hundreds, OPERATION. 2 tens, and 9 units, all to be divided by 3. 3)729 Now, we say, 3 in 7, 2 times; that is, 2 243 hundreds, and 1 hundred over. Set down the 2 hundreds under the 7. Now of the 7 hundreds there is 1 hundred or 10 tens not yet divided. We put the 10 tens with the 2 tens, making it 12 tens, and then say, 3 in 12, 4 times; that is, 4 tens times; therefore write the 4 in the quotient, in the ten's place; then say, 3 in 9, 3 times. The quotient, therefore, is 243. 3. Divide 729 by 9. In this example we say, 9 in 7 we can- OPERATION. not, but 9 in 72, 8 times, which are 8 9)729 tens: then, 9 in 9, 1 time. 81 The quotient is therefore 81. 4. Divide 8040 by 8. In this example we say, 8 in 8, 1 time, OPERATION. and set 1 in the quotient. We then say, 8)8040 8 in 0, 0 times, and set the 0 in the quo- 1005 tient: then say, 8 in 4, 0 times, and set the 0 in the quotient: then say, 8 in 40, 5 times; that is, 5 units times, and therefore we set the 5 in the unit's place of the quotient. Therefore the true quotient is 1005. 5. Let it be required to divide 36458 by 5. In this example, we find the quo- OPERATION. tient to be 7291 and a remainder 3. 5)36458 This 3 ought in fact to be divided 7291-3 remain by the divisor 5, but the division cannot be effected, since 3 does not contain 5. The division must then be indicated by placing 5 under the 3, thus, ]. DIVISION. 83 The entire quotient, therefore, is 7291, which is read, seven thousand two hundred and ninety-one, and three divided by five. Therefore, Where there is a remainder after the division, it may be written after the quotient, and the divisor placed under it. 74. When the divisor is 12 or less than 12, the operation may be performed as in the last examples, and this method of dividing is called SHORT DIVISION. 6. Divide 2756 by 26. oPERATION. We first say, 26 in 27 hundreds, once, and set down 1 in the quotient, in the hundred's place. Multiplying' >:' by 1, subtracting, and bringing down I 26)2756(106 the 5, we say, 26 in 15 tens, 0 tens 26 times, and place the 0 in the quo- 156 tient. Bringing down the 6, we 156 find that the divisor is contained in 156, 6 times. Hence, the entire quotient is 106. 7. Divide 11772 by 327. Having set down the divisor on OPERATION. the left of the dividend, it is seen 327)11772(36 that 327 is not contained in the first 981 three figures on the left, which are 1962 117 hundreds. But by observing 1962 that 3 is contained in 11, 3 times 0000 and something over, we conclude that the divisor is contained at least 3 times in the first four figures of the dividend, which are 1177 tens. Set down the 3, which are tens, in the quotient, and multiply the divisor by it: we thus get 981 tens, which being less than 1177, the quotient figure is not too great: we subtract the 981 tens from the first four figures of the dividend, and find a remainder 196 tens, which being less than the divisor, the quotient figQUEST.-74. When the divisor is 12 or less than 12, what'is the division called? 84 DIVISION. ure is not too small. Reduce. this remainder to units and add in the 2, and we have 1962. As 3 is contained in 19, 6 times, we conclude that the divisor is contained in 1962 as many as 6 times. Setting down 6 in the quotient and multiplying the divisor by it, we finld the product to be 1962. Therefore the entire quotient is 36, or the divisor is contained 36 times in the dividend. 8. Divide ~133 9s.,8d. by 4. Here we again take the least num- OPERATION. ber of units of the highest order which 4)~133 9s. 8d. will contain the divisor, viz., 13 tens of ~33 7s. 5d. the denomination of tens of pounds. Dividing by 4, we find the quotient to be 3 tens of the same denomination, and 1 ten over. We reduce these tens to units and add in the 3, and thus obtain 13 pounds, which being divided by 4 gives 3 pounds and 1 over. Reducing this ~1 to shillings and adding in the 9, gives 29, which being divided by 4 gives 7 shillings and 1 over. Reducing this to pence and adding in 8d., and again dividing by 4, we have ~33 7s. 5d for the entire quotient. 9. Divide ~6 8s. 8d. by 8. 8)~6 8s, 8d. Here we have to pass to shillings be- 16s. ld fore making the first division. 75. Combining the principles illustrated in the foregoing examples we have, for the division of numbers, the following: Beginning with the highest order of units of the dividend, pass on. to the lower orders until the fewest numbe," of figures be found that will contain the divisor: divide these figures by it for the first figure of the quotient: the unit'of this figure will be the same as that of the lowest order used in the dividend. Multiply the divisor by the quotient figure so found, and subtract the product from the dividend, observing to place units of the same order under each other. Reduce the remainder to units of the next lower order, and add in the units of that order QU;EST.-75. What is the rule for the division of numbers? DIVISION. 85 found in the dividend: this will furnish a new dividend. Proceed in a similar manner until unzts of every order shall have been divided. 76. There are always three nujnbers in every operation of division, and sometimes four. First, the dividend; second, the divisor; third, the quotient; and fourth, the remainder, when the numbers are not exactly divisible. 77. There are five operations in division. First, to write down the numbers; second, find how many times; third, multiply; fourth, subtract; and fifth, reduce to the next lowel order. EXAMPLES. 1. Divide 1203033 by 3679. By the first operation, 300 times OPERATION. the divisor is taken from the divi- 3679)1203033(327 dend; or, what is the same thing, 11037 the divisor is taken from the divi- 9933 dend 300 times. By the second, it 7358 is taken 2 tens or twenty times; 25753 and by the third, it is taken 7 units 25753 times; therefore, it is taken in all 327 times: hence, 78. DIVISION is a short method of performing subtraction; and the quotient found according to the rules always shows how many times the divisor may be subtracted from the dividend. Prove the above work by multiplying the divisor and quotient together. 2. Divide 714394756 by 1754. Ans. 3. Divide 47159407184 by 3574. Ans. 4. Divide 5719487194715 by 45705. Ans. 5. Divide 4715714937149387 by 17493. Ans. - 6. Divide 671493471549375 by 47143. Ans. - 7. Divide 571943007145 by 37149. Ans. QUEST.-76. How many numbers are considered in division? What are they? 77. How many operations are there in division? Name them. 78. How may division be defined; and what does the quotient show? b6 DIVISION. 8. Divide 17i4347149347 by 57143. Ans. - 9. Divide 49371547149375 by 374567. Ans. - 10. Divide 171493715947143 by 571007. Ans. - 11. Divide 6754371495671594 by 678957. 12. Divide 7149371478 by 121. 13. Divide 71900715708 by 57149. 14. Divide 15241578750190521 by 123456789. 15. Divide 121932631112635269 by 987654321. 16. Divide 14714937148475 by 123456. 17. Divide 8890896691492249389482962974 by 987675. PROOF OF MULTIPLICATION. 79. When two numbers are multiplied together, the multi plicand and multiplier are both factors of the product; and if the product be divided by one of the factors, the quotient will be the other factor. Hence, if the product of two numbers be divided by the multiplicand, the quotient will be the multiplier; or, if it be divided by the multiplier, the quotient will be the multiplicand. 1. The multiplicand is 61835720, the product 8162315040: what is the multiplier? 2. The multiplier is 270000, the product 1315170000000: what is the multiplicand? Ans. 3. The product is 68959488, the multiplier 96: what is the multiplicand? 4. The multiplier is 1440, the product 10264849920: what isethe multiplicand? Ans. 5. The product is 6242102428164, the multiplicand 6795634: what is the multiplier? Ans. 80. When the? divisor is a composite number. Divide the dividend by one of the factors of the divisor, and then divide the quotient thus arising by the other factor: the last quotient will be the one sought. QUEST. —79. How may multiplication be proved by division? 80. How do you divide when the divisor is a composite number? DIVISION. 87 EXAMPLES. 1. Let it be required to divide 1407 dollars equally among 21 men. Here the factors of the divisor are 7 and 3. Let the 1407 dollars be first divi- OPERATION ded equally among 7 men. Each 7)1407 share will be 201 dollars. Let each 3)201 1st quotient. one of the 7 men divide his share 67 quotient sought into 3 equal parts, each one of the three equal parts will be 67 dollars, and the whole number of parts will be 21; here the true quotient is found by dividing continually by the factors. 2. Divide.18576 by 48 = 4 x 12. Ans. 3. Divide 9576 by 72 = 9 X 8. Ans. 4. Divide 19296 by 96 = 12 X 8. Ans. - 81. It sometimes happens that there are remainders after division-they are to be treated as follows: The first remainder, if there be one, forms a part of the true remainder. The product of the second remainder, if there be one, by the first divisor, forms a second part. Either of these parts, when the other does not exist, forms the true remainder, and their sum is the true remainder when they both exist together, and similarly when there are more than two remainders. 1. What is the quotient of 751 grapes, divided by 16? (4)751 4 x 4=16 4)187... 3 46. 3 X 4 - 12 -— ~~- 3 15 the true remainder. Ans. 4615 In 751 grapes there are 187 sets, (say bunches,) with 4 grapes or units in each bunch, and 3 units over. In the 187 bunches there are 46 piles, 4 bunches in a pile, and 3 bunches over. But there are 4 grapes in each bunch; therefore, the Qvs. —81. How do you dispose of the remainders, if there are any, after division? 88 DIVISION. number of grapes in the 3 bunches is equal to 4 X 3 = 12, to which add 3, the grapes of the first remainder, and we have the entire remainder 15. 2. Divide 4967 by 32. 4)4967 4 X 8 = 32 8)1241... 3, 1st remainder. 155... 1 X 4 + 3 = 7 the true remainder. Ans. 155-. 3. Divide 956789 by 7 x 8 = 56. 4. Divide 4870029 by 8 x 9 72. 5. Divide 674201 by 10 x 11 = 110. 6. Divide 445767 by 12 X 12 = 144. 7. Divide 375197351937 by 349272 -12 X 11 X 9 x 7 x 7 x 6. 12)375197351937 11)31266445994....9 9 9)2842404181...3 -- 12 X3 - - - = 36 7)315822686....7 - - 12 X 11 X 7 - - 924 7)45117526... 4 - -12 X 11 X 9 X4 - = 4752 6)6445360...6 — 12xll1x9x7x6 = 49896 Quotient -- 1074226...4 - -12 x 11 x 9 x 7 x7 x4= 232848 Remainder = 288465 8. Divide 7349473857 by 27. Ans. - 9. Divide 749347549 by 144. Ans. - 10. Divide 649305743 by 55. Ans. - 11. Divide 4730715405 by 121. Ans. - 12. Divide 3704099714 by 108. Ans. - 13. Divide 4710437154 by 132. Ans. 14. Divide 1071540075 by 99. Ans. - 15. Divide 468248 by 3 x 4 x 2 x 5 x 6. 16. Divide 98765432101234567890 by 12 X 11 X 10 x 9 x8 x 7 x 6 x 5 x 4 x 3 x 2. DIVISION. 89 82. When the divisor has one or more O's at the right, it may be regarded as a composite number, of which one factor is 1 with as many O's on the right as there are O's at the right of the divisor, and the remaining figures express the other factor. Strike off the O's and the same number of figures from the right of the dividend-this is dividing by one of the factors; then proceed to divide by the other. 1. Divide 14715967899 by 145000. 145000)14715967899(10148 62so9 Quotient. 145 215 145 Or thus, 709 145000)14715967899(101489 4y%%s 580 2'15 1296 709 1160 1296 1367 1305 ~1305 ~62899 Rem. 62899 Rem. NOTE. In the second operation of this example, the products of the divisor by each quotient figure are subtracted mentally, and the remainders only written down. Let the pupil perform many examples in division in this way. 2. Divide 571436490075 by 36500. 3. Divide 194718490700 by 73000. 4. Divide 795498347594 by 47150. 5. Diviae 1495070807149 by 31500. 6. Divide 6714934714934 by 754000. 7. Divide 1071491471430715 by 754000. 8. Divide 14714937493714957 by 157900. 9. Divide 7149374947194715 by 1749000. 10. Divide 714947349 by 90. 11. Divide 1714937148 by 14400. 12. Divide 69616103498721931800 by 975005700. QuEST.-82. What is the process when the divisor has O's annexed t 90 DIVISION. 13. Divide 656458931996524171800 by 700489070. 14. Divide 7149437149547 by 3714900. EXAMPLES IN DENOMINATE NUMBERS. 1. A gentleman's income is ~1260 15s. 5d. a year: what is that per day, 365 days being contained in one year? ~ s. d. ~ s. d. 365)1260 15 5 ( 3 9 1 = Ans. 1095 10 165 34 10 10 X 6 20 -10 365) 3315(9s. 345 8 4 3285 3 30 1036 5 0 12 207 5 0 365)365(1d. 17 5 5 365' 1260 15 5 Proof. 0 2. Divide ~47 19s. 4d. by 3. Ans. - 3. Divide ~37 14s. 10d. by 24. Ans. 4. Divide ~49 19s. 11 d. by 66. Ans. 5. Divide ~34 14s. 94d. by 149. Ans. 6. Divide ~1774 19s. 10~d. by 179. Ans. - 7. Divide 47yd. 3qr. 2na. by 5. Ans. 8. Divide 37A. 3R. 14P. by 9. Ans. - 9. Divide 7141b. 10oz. 12gr. by 89. Ans. 10. Divide 374cwt. 3qr. 101b. by 48. Ans. 11. Divide 374E.E. 2qr. 3na. by 142. Ans. - 12. If 60 sheep be sold for;~112 10s., what is the value of 1 sheep? 13. If 1121b. of cheese cost ~2 18s. 8d., what is that per pound? 14. If 17cwt. of lead cost ~15 5s. 73d., what costs lcwt.? 15. Bought 7 yards of cloth for 16s. 4d.; what is that per yard? DIVISION. 91 16. If 63 oxen cost ~2553 Is. 6d., what costs 1 ox? 17. If 661b. of butter cost ~5 15s. 6d., what costs llb.? 18. If 5281b. of tobacco cost ~23 13s., what costs llb.? 19. If a tun, or 252 gallons, of wine cost ~60, what costs 1 gallon? 20. A prize of 1000 guineas-is to be divided among 150 sailors; what is each man's share, after deducting 6 part for the officers? 21. If 125 ingots of silver, each of an equl weight, weigh 1347oz. 11lpwt. 14gr., what is the weight. of 1 ingot? 22. If 475cwt. 1qr. 141b., be the -weight of 27hhds. of tobacco, what is the weight of lhhd.? 23. Bought 6 pieces of tapestry, containing 237E. Fl. 2qr. 2na.; what is the length of 1 piece? APPLICATIONS. 1. In 1842, nine mills in Lowell manufactured 434000 pounds of cotton per week. How much was manufactured by each mill, supposing the amount was exactly the same? 2. The number of inhabitants in the city of New York in 1840 was 312710, and the expenses of the city government $1645779,30. If this was raised by an equal tax upon every inhabitant, how much would each have to pay? 3. The number of hogsheads of tobacco exported from the United States in the 20 years preceding 1841, was 1792000, and their estimated value was $131346514. What was the average value by the hogshead? 4. The amount of coffee imported in 1840 was 94996095 pounds, and its value was estimated at $8546222. What was its worth per pound'? 5. The number of scholars attending the public schools of the state of Maine,-in 1839, was 201024, and the amount expended for the support of the schools was $258113,43. What was the cost to the state for the tuition of each scholar? 92 DIVISION. 6. The militia force of the United States, according to the Army Register for 1845, was 1426868, and the number of commissioned officers belonging to it was 69450. How many soldiers did that allow to each officer? 7. The whole coinage of the United States for the 51 years preceding 1845, amounted to $110177761,38. Suppose an equal amount had been coined each year, what would it have been? 8. In 1843 there were sold 1605264 acres of the public lands. The siffn received for them was- $2016044,30, and the sum' paid into'the national treasury, after deducting expenses, was $1997351,57.' What was the average cost per acre to the purchasers, and what was the average price per acre received by the government? 9. The net amount of duties on imports for eighteen years preceding 1843, was $452539360,81. How much was collected in each year, supposing the sums to have been equal? 10. There was inspected in Onondaga county, N. Y., in 1844, 4003554 bushels of salt. The duties collected on these amounted to $240305. What was the duty on each bushel? Il. There were thirty-five banks in New Hampshire in 1844,- whose whole resources were $5836014,07. If this sum was equally divided, how much would belong to each? 12. The population of Europe in 1837 was estimated at 233884800, and the number of square miles at 3708871. How many inhabitants would this give to each square mile? 13. In 1843, there were 3173 public schools in Massachusetts, which were attended during the winter by 119989 scholars. How many would this allow to each school? 14. The number of male scholars attending the public schools of Pennsylvania was reported, in 1843, to be 161164 arnd the number of female scholars 127598. The number of male teachers employed by the state was 5264, and the number of female teachers 2330. How many'scholars would this give to each teacher? PROPERTIES OF THE 9'3. 93 15. The value of the exports from the United States in 1841, was $104691534. If an equal amount had been exported each day of the year excepting Sundays, what would it have been? OF THE PROPERTIES OF THE 9'S. 83. Besides the methods already explained of proving the operations in figures, there is yet another called the method by casting out the 9's. That method we will now explain. -84. An excess of units over exact 9's, is the remainder after the number has been divided by 9: hence, any number less than 9 must be treated as an excess over exact 9's. Let us write down the numbers OPERATION. to be added, as at the right. Now, X. if we divide each number by 9, rt and place the quotients to the right, H and the remainders in the column c still to the right, we shall have, in c3 a the middle column, the exact num- 3870... 30... 8 ber of 9's contained in each num- 33. 7 ber, and in the column at the right, 9)6882 0 the excesses over exact 9's. By 2 764 6 adding these columns, we.find 1,5 764-6 in the column of remainders, which is equal to one 9 and 6 over: hence, there are 764 exact 9's and 6 over. But it is evident that the sum of all the numbers, viz., 6882, must contain exactly the same number of 9's and the same excess over exact 9's, as are found in the numbers taken separately, since a whole is equal to the sum of all its parts any way taken: therefore, in the sum of any numbers whatever, the number of exact 9's and the excess over 9's are equal, respectively, to the aggregate of exact 9's and the excess of 9's in the numrbers taken separately. QuesT.-83. What other methods of proof are there for arithmetical operations, besides those already explained? 84. What is an excess of 9's? How do the- exact number of 9's and the excess of 9's in any sum compare with the exact 9's and the exces of 9's in the several aumbers? 94 PROPERTIES OF THE 9'S. 85. We will now explain a short process of finding the excess over an exact number of 9's in any number whatever; and to do this, we must look a little into the formation of numbers. In any number, written with a single significant figure, as 4, 50, 600, 8000, &c., the excess over exact 9's will always be equal to the number of units expressed by the significant figure; for, in any such number we shall always have 4 = 4 Also, - - - - - 50 = (9 +l)x5. - - - 600 = (99 +l )x6 - - - - - - 8000 = (999+ 1) x 8 &c. &c. &c. Each of the numbers 9, 99, 999, &c., expresses an exact number of 9's; and hence, when multiplied by 5, 6, 8, &c., the several products will each contain an exact number of 9's; therefore, the excess over exact 9's, in each number, will be expressed by 4, 5, 6, 8, &c. If, then, we write-any number whatever, as 6253, we may read it 6 thousand 2 hundred 50 and 3. Now, the excess of 9's in the 6 -thousand is 6; in 2 hundred it is 2; in 50 it is 5; and in 3 it is 3: hence, in'them all, it is 16, which makes-one 9 and 7 over: therefore, 7 is the excess over exact 9's in the number 6253. Hence, the excess over exact 9's in any number whatever, may be found by adding together the significantfigures, and rejecting the exact 9's from the sum. -NOTE. —It is best to reject or drop the 9 as soon as it occurs: thus we say, 3 and 5 are 8 and 2 are 10; then dropping the 9, we say, 1 to 6 is 7, which is the excess; and the same for all similar operations. 1. What is the excess of 9's in 48701? In 67498? 2. What is the excess of 9's in 9472021? In 2704962? 3. What is the excess of 9's in 87049612? In 4987051? QUEsT. —-85. What will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whatever? PROPERTIES OF THE 9's. 95 PROOF OF ADDITION BY CASTING OUT THE 9'S. 86. —1. in the first of these num- OPERATION. bers we find the excess of 9's to be 5; Excess of 9's. 94874... 5 in the second 2; in the third 8; in 46073 2 the fourth 2; and in the fifth 8: 50498... 8 hence, in them all it is ~5, which 3674... 2 leaves 7 for the excess over exact 341... 8 9's. We also find 7 to be the excess 195460-7 7 over exact 9's in the sum 195460: hence the work is supposed to be right. Notwithstanding this proof, it is possible, after all, that the work may be erroneous. For example, if either figure in the sum is too large by one or more units, and any other figure is too small by the same number of units, the excess over exact 9's wjll not be affected. But as it would seldom happen that one error would be exactly balanced by another, the work when proved may be relied on as correct. Similar sources of error exist i the proof of all arithmetical operations. 2. Add together, 8754608, 490872i, 6027983, 89704543, 3142367, and-28949760, and prove the result by rejecting ~he 9's. 3. Add together 40799903, 874162, 32704931, 6704192, 2146748, 94004169, and prove the result by casting out the 9's. PROOF OF SUBTRACTION BY CASTING OUT THE 9'S. 87.-1. Since the sums of the re- OPERATION. mainder and subtrahend must be equal 874136.. 2 to the minuend, it follows that the ex- 45302.. 5 cess of 9's in these two numbers must 828834... 6 be equal to the excess of 9's in the minuend: hence, to the excess of 9's in the remainder add the excess of 9's in the subtrahend, and the excess of 9's zn the sum will be equal to the excess of 9's in the minuend. QursT. —86. Explain the proof of addition by casting out the 9's. In what is the proof defective 7 87. Explain the proof of subtraction by casting out the 9's 96 PIROPERTIES OF THE 9'S. 2. From 874096 take 370494, and prove the work by re. jecting the 9's. 3. From 47096702 take 1104967, and prove the work by rejecting the 9's. PROOF OF MULTIPLICATION BY CASTING OUT THE 9'S. 88. We will first remark, that if any number containing an exact number of 9's be multiplied by another whole number, the product will also contain an exact number of 9's. Let it be required to multiply any two numbers together, as 641 and 232. We first find the excess over exact OPERATION. 9's in both factors, and then separate 641 = 639 + 2 each factor into two parts, one of 232 = 225 + 7 which shall contain exact 9's, and 4473 + 14 the other the excess, and unite the 450 two together by the sign plus. It is 3195 now required to take 639 + 2 = 641, 1278 is many times as there are units in 1278 225 + 7 = 232. 148698 +- 14 Beginning with the 7, we have 14 for the product of 2 by 7, and 4473 for the product of 639 by 7; and this last contains an exact number of 9's.'We then take 2, 225 times, which gives 450, which also contains an exact number of 9's. We next multiply 639 by the figures of 225, and each of the several products contains an exact number of 9's, since 639 contains an exact number. Hence, the entire sum 148698 contains an exact number of 9's, to which if we add. the one 9 from the 14, we shall find tile excess of 9's in the product to be 5; and as the same may be show'n for any numbers, we conclude that, the excess of 9's in any product must arise from the product of the excess of 9's in the factors.'Qusr.-88. Explain the proof of multiplication by casting out the 9'a. What does the excess of 9's in any product arise from? PROPERTIES OF THE 9's. 97 But since the product of two numbers found in the ordinary way must contain the sanle number of 9's, and the same excess of 9's as a product found above, it follows that, if the excesses of 9's in any number offactors be multiplied together, the excess of 9's in such product will be equal to the excess of 9's in the product of the factors. EXAMPLES. (1.) (2.) Multiply 87603... 6 818327... 2 by 9865 1 9874 1 Prod. 864203595... 6 8080160798... 2 3. By multiplication we have Ex. 4. Ex. 8. Ex. 4. Ex. of product, 2. 7285 X 143 X 976 - 1016752880. Ex. 5. Ex. 4. Ex. 0. Ex. 0. 4. We also have 869 X 49 X 36 = 1532916. When the excess of 9's in any factor is a, the excess of 9's in the product is always 0. PROOF OF DIVISION BY CASTING OUT THE 9'S. 89. Since the divisor multiplied by the quotient must produce the dividend, it follows that if the excess of 9's in the divisor be multiplied by the excess of the 9's in the quotient, the excess of 9's in the product will be equal to the excess of 9's in the dividend. 1. The dividend is 8162315040, the divisor 61835720, and the quotient 132: is the work right? 2. The dividend is 10264849920, the divisor 1440, and the quotient 7128368: is the work right? 3. The dividend is 7A4855092410, the quotient 78795, and the divisor 949998: is the work right? Let the pupils apply the property of the 9's to other examples. QUEST. —If the excess of 9's in any number of factors be multiplied together, what will the excess of 9's in the product be equal to? 89. How do you prove division by casting out the 9's? 5 98 REMARKS. REMARKS. 90. —1. Numeration, Addition, Subtraction,' Multiplication, and Division are called the five ground rules, because all the other operations of arithmetic are performed by means of them. Multiplication, however, is but a short method of performing addition, and division but an abridged method of subtraction. 2. A prime number is one which cannot be exactly divided by any number except itself and unity. Thus, 1, 3, 5, 7, 11, 13, 19, 23, &c., are prime numbers. 3. The product of two or more prime numbers will be exactly divisible only by one or the other of the factors. 4. If an even number be added to itself any number of times, the sum will be even; hence, if one of the factors of a product be an even number, the product will be even. 5. An odd number is not divisible by an even number; nor is a less number exactly divisible by a greater. 6. The quotient arising from the division of the sum of two or more numbers, by any divisor, is equal to the sum of the quotients which arise from the division of the parts separately. 7. Any number is divisible by 2, if the last significant figure is even; and is divisible by 4, if the last two figures are divisible by 4. 8. Any number whose last figure is 5 or 0, is exactly divisible by 5; and any number whose last figure is 0, is exactly divisible by 10. QUEST.' —90.-1. What are the five ground rules of arithmetic? What other rule in fact embraces the rule of multiplication? How may division be performed? 2. What is a prime number? 3. By what numbers only will the product of prime factors be divisible? 4. If an even number be multiplied by a whole number, will the product be odd or even? 5. Is an odd number divisible by an even number? 6. What is the quotient arising from the division of the sum of two or more numbers by any divisor equal to? 7. When is a numnber exactly divisible by 2? When by 4? 8.. If the last figure of a number be 5 or 0, by what numbers may it be divided? DIVISIONS OF ARITHMETIC. 99 DIVISION -o0 ARITHMETIC. 91. The science of arithmetic, which treats of numbers, may be. divided into four parts: 1st. That which treats of the properties of entire units, called the Arithmetic of Whole Numbers; 2d. That which treats of the parts of unity, called the Arithmetic of Fractions; 3d. That which treats of the relations of the unit 1 to the numbers which come from it, whether they be integers or fractions, and the relations of these numbers to each other; and 4th. The application of the science of numbers to practical and useful purposes. A portion of the First part has already been treated under the heads of Numeration, Addition, Subtraction, Multiplication and Division. The Second part comes next in order, and naturally divides itself into two branches; viz., Vulgar or Common Fractions, in which the unit is divided into any number of equal parts, and Decimal Fractions, in which the unit is divided according to the scale of tens. The Third part relates to the comparison of numbers, with respect either to their difference or quotient. The Rule of Three, and Arithmetical and Geometrical Progression, make up this branch of Arithmetic. The Fourth part embraces the applications of rules deduced from the science of numbers, to the ordinary transactions and business of life. 106. Of what does the science of arithmetic treat Into how many parts may it be divided? Of what does the first part treat? Of what does the second part treat? What is it called? Of what does the third part treat? What does the fourth part embrace? Which part has been treated I'Under how many heads? Into how many heads is the second part divided What are they called? What distinguishes them? To what does the third part relate? What does the fourth part embrace I 100 OF VULGAR FRACTIONS. OF VULGAR FRACTIONS. 92. THE unit 1 represents an entire thing, as I apple, I chair, 1 pound of tea. If we suppose 1 thing, as 1 apple, or 1 pound of tea, to be divided into two equal parts, each part is called one half of the thing. If the unit be divided into 3 equal parts, each part is called one third. If the unit be divided into 4 equal parts, each part is called one fourth. If..the unit be divided into 12 equal parts, each part is called one twelfth; and when it is divided into any number of equal parts, we hlave a similar expression for each of the parts. These equal parts of a unit are called Fractions. How are these fractions to be expressed by figures? They are expressed by writing one figure under another. Thus; a is rdad one half. 1- is read one seventh. 2 7 1 " " one third. 1 " " one eighth. 1 "' " one fourth. 1 " " one tenth. 4 1 " " one fifth. 1 al " one fifteenth. A1 " " c one sixth. 1 " " one fiftieth. I - - It should, however, be observed, that 2 is an entire half; $, an entire third, and the same for all the other fractions. Now, these fractions being entire things, may be regarded as units, and each is called a fractional unit. 93. It is thus seen that every fraction is expressed by two numbers. The number which is written above the line is called the numerator, and the one below it, the denominator, because it gives a denomination or name to the fraction. For example, in the fraction I, 1 is the numerator, and 2 QuEsr.-92. What does the unit 1 represent? If we divide it into two equal parts, what is each called? If it be divided into three equal parts, what is each part? Into 4, 5, 6, &c., parts? What are such expressions called? How may the fractions be regarded. What are they called? 93. Of how many numbers is each fraction made up? What is the one above the line called 8 The one below t5le line? OF VULGAR FRACTIONS. 101 the denominator. In the fraction 1, 1 is the numerator, and 3 the denominator. The denominator in every fraction shows into how many equal parts the unit, or single thing, is divided. For example, in the fraction 1, the unit is divided into 2 equal parts; in the fraction 1, it is divided into three equal parts; in the fraction x, it is divided into four equal parts, &c. In each of the above fractions one of the equal parts is expressed. But suppose it were required to express 2 of the equal parts, as 2 halves, 2 thirds, 2 fourths, &c. We should then write, 2 they are read two halves. 2 " " " two thirds. 2 " " " two fourths. 2 " " " two fifths, &c. If It were required to express three of the equal parts, we should place 3 in the numerator; and generally, the niumerator shows how many of the equal parts are expressed in the fraction. For example, three eighths are written, 3 and read three eighths. 94 " " four ninths. 6 " " six'thirteenths. 9 " " nine twentieths. 94. When the numerator and denominator are equal, the numerator will express all the equal parts into which the unit has been divided: and, the value of the fraction-is then equal to 1. But if we suppose a second unit, of the same kind, to be divided into the same number of equal parts, those parts QuEST.-What does the denominator show? What does the numerator show? In the fraction three-eighths, which is the numerator? Which the denominator? Into how many parts is the unit divided? How many parts are expressed? In the fraction nine-twentieths, into how many parts is the unit divided? How many parts are expressed? 94. When the numerator and denolhminator are equal, what is the value of the fraction? 102 OF VULGAR FRACTIONS. may also be expressed in the same fraction with the parts of the first unit. Thus, a- is read three halves. 7 " " seven, fourths. 16 ~' "t sixteen fifths. 15 " " IS " eighteen sixths. 275 " " twenty-five sevenths. If the numerator of a fraction be divided by its denominator, the integer part of the quotient will express the number of entire units which have been used in forming the fraction, and the remainder will show, how many fractional units are over. The unit, or whole thing, which is divided, in forming a fraction, is called the unit of the fraction; and one of- the equal -parts is called the unit of the expression. Thus, in the fiaction I, 1 is the unit of the fraction, and - the unit of the expression. In every fraction, we must distinguish carefully, between the unit of the fraction and the unit of the expression. The first, is the whole thing from which the fraction is derived; the second, one of the equal parts of the fractional expression. From what has been said, we conclude: 1st. That a fraction is the expression of one or more equal parts of unity. 2d. That the denominator of a fraction shows into how many equal parts the unit or single thing has been divided, and the nume'rator expresses how many such parts are taken in the fraction. 3d. That the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator. 4th. That, when the numerator is less than the denominator, the value of the fraction is less than 1. QUEST.-What is the value of the fraction three-halves? Of sevenfourths? Of sixteen-fifths? Of eighteen-sixths? Of twenty-five-sevenths? What is the first conclusion? What the 2d? What the 3d? What the 4th? What the 5th? What the 6th? What the Ith? What is the unit of the fraction three-fourths? What is the unit of the expression? OF VULGAR FRACTIONS. 103 5th. Thiat, when the numerator is equal to the denominator, the vac 1.?i of the fraction is equal to 1. 6tl. That, when the numerator is greater than the denominzator, the value of the fraction will be greater -than 1. 7th. That, the unit of every fraction is the whole thing from which it was derived; and the unit of the expression, one of the equal parts taken. 95. There are six kinds of Vulgar Fractions: Proper, Improper, Simiple, Compound, Mixed, and Complex. A PROPER FRACTION is one in which the numerator is less than the denominator. The value of every proper fraction is less than 1, (Art. 94). The following are proper fractions: 1 1 3 3 5 9 8 5 An IMPROPER FRA6TION is one in which the numerator is equal to, or exceeds the denominator. Such fractions are called improper fractions because they are equal to, or exceed unity. When the numerator is equal to the denominator the value of the fraction is 1; in every other case the value of an improper fraction is greater than 1. The following are improper fractions: 3 5 6 8 9 12 14 19 2' 3 5X 7' 8y 6 7' 7 A SIMPLE FRACTION is a single expression. A simple fraction may be either proper or improper. The following are simple fractions: 1 3 5 8 9 8 6 7 4X, I 6X 7 2XY' 3X 3Y 5 A COMPOUND FRACTION is a fraction of a fraction, or several fractions connected together with the word of between them. QUEsT.-Write the fraction nineteen-fortieths:-also, 60 fourteenths18 fiftieths-16 twentieths-17 thirtieths-41 one thousandths-85 millionths-106 fifths. 95. How many kinds of vulgar fractions are there? What are they? What is a proper fraction? Is its value greater or less than 1? What is an improper fraction? Why is it called irpproper? When is its value equal to 1? What is a simple fraction? What is a cdmpound fraction? Give an example of a proper fraction. Of an improper fraction. Of a simple fraction. 104 OF VULGAR FRACTIONS. The following are compound fractions: of+ 4, of Iof, 6-of 3, 1 of 8 of 4. A MIXED NUMBER is made up of a whole number and a fraction. The whole numbers are sometimes called integers. The following are mixed numbers: 3, 41~, 62, 5-, 65, 31 A COMPLEX FRACTION is one having a fraction or a mixed number in the numerator or denominator, or in both. The following are complex fractions: 3 2 3 425 4 _ 5 6 14' 471 4' 874' 96. The numerator and denominator of a fraction, taken together, are called the terms of the fraction. Hence, every fraction has two terms. 97. A whole number may be expressed fractionally by writing 1 below it for a denominator. Thus, 3 may be written A- antiiis read, 3 ones. 5, is " 5 ones. 6 6",, " " 6 ones. 8 4 8 4LT i" 8 ones. But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6. and 8 ones to 8. Hente, the value of a number is not changed by placing 1 under it for a denominator. QuEsT. — What is a mixed number? Give an example of a compound fraction. Of a mixed fraction. Is four-ninths a proper or improper fraction? What kind of a fraction is six-thirds? What is its value? What kind of a fraction is nine-eighths? What is its value? What kind of a fraction is one-half of a third? What kind of a fraction is two and onesixth? Four and a seventh? Eight and a tenth? What is a complex fraction? 96. What are the termns of a fraction? What are the terms of the fraction three-fourths? Of five-eighths? Of six-sevenths? 97. How may a whole number be expressed fractionally? Does this alter its value? Give an example. OF VULGAR FRACTIONS. 105 98. If an ajple be divided into 6 equal parts, I will express one of the parts, 2 i" " two of the parts, 3 " " three of the parts, &c., &c., &c., and generally, the denominator shows into how many equal parts the unit is divided,-and the numerator how many of the parts are taken. Hence, also, we may conclude that, x 2; that is, - taken 2 times - 2 x 3; that is, - taken 3 times -3 x 4; that is, 6 taken 4 times -6 &c., &c., &c., and consequently we have, PROPOSITION I. If the numerator of a fraction be multiplied by any number, the denominator remaining unchanged, the value of the fraction will be increased as many times as there are units in the multiplier. Hence, to multiply a fraction by a whole number, we simply multiply the numerator by the number. EXAMPLES. 1. Multiply 7 by 5. 5. Multiply 1-7 by 11. 2. Multiply -T by 7. 6. Multiply -19 by 12. 3. Multiply I- by 9. 7. Multiply ~1 by 14. 4. Multiply II by 12. 8. Multiply 13 by 15. 99. If three apples be each divided into 6 equal parts, there will be ] 8 parts in all, and these parts will be expressed by the fraction 1?8. If it were required to express but onethird of the parts, we should take, in the numerator, but oneQUEST.-98. If an apple be divided into six equal parts, how do you express one of those parts? Two of them? Three of them? Four of them? Five.of them? Repeat the proposition. How do you multiply a fraction by a whole number? 99. If 3 apples be each divided into 6 equal parts, how many parts in all? If 4 apples be so divided, how many parts in all? If 5 apples be so divided, how many parts? How many parts in 6 apples? In 7? In 8? In 9? In 10? I 06 OF VULGAR FRACTIONS. third of the eighteen parts; that is, the fraction - would ex lress one-third of 1. If it were required to express onesixth of the 18 parts, we should take one-sixth of 18, and 3 would be the required fraction. In each case the fraction 1? has been diminished as many imes as there were units in the divisor. Hence, PROPOSITION II. If the numerator of a fraction be divided by any number, the denominator remaining unchanged, the value of the fraction will be diminished as many times as there are units in the divisor. Hence, a fraction may be divided by a whole number by dividing its numerator. EXAMPLES. 1. Divide 24 by 6. 5. Divide 75 by 5. 2. Divide II2- by 8. 6. Divide -360 by 12. 3. Divide 2641 by 12. 7. Divide 2-5 by 32. 734 b y 136 4. Divide 4- 7 by 7. 8. Divide 3-2 by 36. 100. Let us again suppose the apple to be divided into 6 equal parts. If, now, each part be divided into 2 equal parts, there will be 12 parts of the apple, and consequently each part will be but half as large as before. Three parts in the first case will be expressed by 3, and in the second by 3. But since the parts in the second are only half the parts in the first fraction, it follows that, = one half of 3. If we suppose the apple to be divided into 18 equal parts, QuEsT. —What expresses all the parts pf the three apples? What expresses one-half of them? One-third of them? One-sixth of them? Oneninth of them? One-eighteenth of them? What expresses all the parts of four apples? One-half of them? One-third of them? One-fourth of them? One-sixth of them? O.ne-eighth of them? One-twelfth of them? One-twenty-fourth of them? Put similar questions for 5 apples, 6 apples, &c. Repeat the proposition. How may a fraction be divided? 100. If a unit be divided into 6 equal parts and then into 12 equal parts, how does one of the last parts compare with one of the first? If the second division be into 18 parts, how do they oompare? If into 24? OF VULGAR FRACTIONS. 107 three of the parts will be expressed by 4, and since the parts are but one-third as large as in the first case, we have -= one third of 3: and since the same may be said of all fractions, we have PROPOSITION III. If the denominator of a fraction be multiplied by any number, the numerator remaining unchanged, the value of the fraction will be diminished as many times as there are units in the multiplier. Hence, a fraction may be-divided by any number, by multiplying the denominator by that number. EXAMPLES. 1.Divide 67 by 6. 5. Divide 37 by 14. 4 7 4-6 2. Divideoby9. 6. Divide 14 74 by 15. 3. Divide 127 by 12. 7. Divide -2 by 5. 4 Divide 327 by 11. 8. Divide 94 by 8. 101. If we suppose the apple to be divided into 3 parts instead of 6, each part will be twice as large as before, and three of the parts will be expressed by 3 instead of a. But this is the same as dividing the denominator 6 by 2; and since the same is true of all fractions, we have PROPOSITION IV. If the denominator of a fraction be divided by any number, the numerator remaining unchanged, the value of the fraction will be increased as many times as there are units in the divisor. Hence, a fraction may be multiplied by a whole number, by dividing the denominator by that number. QUEST.-What part of 24 is 6? If the second division be into 30 parts, how do they compare? If into 36 parts? Repeat the proposition. How may a fraction be divided by a whole number? 101. If we divide 1 apple into three parts and another inAo 6, how much greater will the parts of the first be than those of the second? Are the parts. larger as you decrease the denominator? If you divide the denominator by 2, hew do you affect the parts? If you divide it by 3? By 4? By 5? By 6? By 7? By 8? Repeat the proposition. How may a fraction be multiplied by a whole number? 108 OF VULGAR FRACTIONS. EXAMPLES. 1. Multiply - by 2, by 4. 5. Multiply 12 by 7. 2. Multiply J6 by 4, 8, 16. 6. Multiply by 5,10,20. 3. Multiply 934 by 4, 6, 12. 27. Multiply 3236 by 8, by 16. 4. Multiply -14 by 16, 56. 8. Multiply 449 by 7, by 21. 102. It appears from Prop. I. that if the numerator of a fraction be multiplied by any number, the value of the fraction will be increased as many times as there are units in the multiplier. It also appears from Prop. III., that if the denominator of a fraction be multiplied by any number, the value of the fraction will be diminished as many times as there are units in the multiplier. Therefore, when the numerator and denominator of a fraction are both multiplied by the same number, the increase from multiplying the numerator will be just equal to the decrease from multiplying the denominator; hence we have, PROPOSITION V. If both terms of a fraction be multiplied by the same number, the value of the'fraction will remain unchanged. EXAMPLES. I. Multiply the numerator and denominator of -5- by 7. 5 5x7 35 We have, 7 - 7 x 7 -49' 2. Multiply the numerator and denominator of 7 by 3, by 4, by 6, by 7, by 9, by 15, by 17. 3. Multiply both terms of the fraction 3 by 9, by 12, by 16, by 7, by 5, by 11. QUEST.-102. If the numerator of a fraction be multiplied by a number, how many times is the fraction increased? If the denominator be multi-plied by the same number, how many times is the fraction diminished? If then the numerator and denominator be both multiplied at the same time, is the value changed? Why not? Repeat the'proposition. GREATEST COMMON -DIVISOR. 109 103. It appears from Prop. II. that if the numerator of a fraction be divided by any number, the value of the fraction will be diminished as many times as there are units in the divisor. It also appears from Prop. IV., that if the denominator of a fraction be divided by any number, the value of the fraction will be increased as many times as there are units in the divisor. Therefore, when both terms of a fraction are divided by the same number, the decrease from dividing the numerator will be just -equal to the increase from dividing the denominator: hence we have, PROPOSITION VI. If both terms of a fraction be divided by the same number, the value of the fraction will remain unchanged. EXAMPLES. 1. Divide both terms of the fraction 8 by 4: this gives 4) 8 - 2 Ans. 2. 2'. Divide each term by 8: this gives = I 3. Divide each term of the fraction 32 by 2, by 4, by 8, by 16, by 32. 4. Divide each term of the fraction 6ToI by 2, by 3, by 4, by 5, by 6, by 10, by 12, by 15, by 20, by 30, by 60. GREATEST COMMON DIVISOR. 104. Any number greater than unity that will divide two or more numbers without a remainder, is called their common divisor: and the greatest number that will so divide them, is called their GREATEST COMMON DIVISOR. QUEST. —103. If the numerator of a fraction be divided by a number, how many times will the value of the fraction be diminished? If the denominator be divided by the same number, how many times will the value of the fraction be increased? If they are both divided by the same number, will the.value of the fraction be changed? Why not? Repeat the proposition. 104. What is a common divisor? What is the greatest common divisor of two or more numbers? 110 GREATEST COMMON DIVISOR. Before explaining the manner of finding this divisor, it is necessary to explain some principles on which the method depends. One number is said to be a multiple of another when it contains that other an exact number of times. Thus, 24 is a multiple of 6, because 24 contains 6 an exact number. of times. For a like reason 60 is a multiple of 12, since it contains 12 an exact number of times. FIRST PRINCIPLE. Every number which exactly divides another number will also divide without a- remainder any multiple of that number. For example, 24 is divisible by 8 giving a quotient 3. Now, if 24 be multiplied by 4, 5, 6, or any other number, the product so arising will also be divisible by 8. SECOND PRINCIPLE. If a number be separated into two parts, any divisor which will divide each of the parts separately, without a remainder, will exactly divide the given number. For, the sum of the two partial quotients must be equal to the entire quotient; and if they are both whole nunm bers, the entire quotient must be a whole number; for the sum of two whole numbers cannot be equal to a fraction. For example, if 36 be separated into the parts 16 and 20, the number 4, which will divide both numbers 16 and 20, will also divide 36; and the sum of the quotients 4 and 5 will be equal to the entire quotient 9. THIRD PRINCIPLE. If a number be decomposed into two parts, then any divisor which will divide the given number and one of the parts, will also divide the other. For, the entire quotient is equal to the sum of the two partial quotients; and if the entire quotient and one of the partial quotients be whole numbers, the other must also be a whole number; for no proper fraction added to a whole number can give a whole number. QUEsT.-When is one number said to be a multiple of another? What is the first principle? What is the second? What is the third? GREATEST COMMON DIVISOR. 1ii 1. Let it be required to find the OPERATION. greatest common divisor of the num- 216)408(1 bers 216 and 408. 216 It is evident that the greatest cornm- 192)216(1 mon divisor cannot be greater than 192 the least number 216. Now, as 216 24)192(8 will divide itself, let us see if it will 192 divide 408; for if it will, it is the greatest common divisor sought. Making this division, we find a quotient 1 and a remainder 192; hence, 216 is not the greatest common divisor. Now we say, that the greatest common divisor of the two given numbers is the common divisor of the less number 216 and the remainder 192 after the division. For, by the second principle, any number which will exactly divide 216 and 192, will also exactly divide the number 408. Let us now seek the common divisor between 216 and 192. Dividing the greater by the less, we have a remainder of 24; and from what has been said above, the greatest common divisor of 192 and 216 is the same as the greatest common divisor of 192 and 24, which we find to be 24. Hence, 24 is the greatest common divisor of the given numbers 216 and 408; and to find it Divide the greater number by the less, and then divide the divisor by the remainder, and continue to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought. EXAMPLES. 1. Find the greatest common divisor of 408 and 740. 2. Find the greatest common divisor of 315 and 810. 3. Find the greatest common divisor of 4410 and 5670. 4. Find the greatest common divisor of 3471 and 1869. 5. Find the greatest common divisor of 1584 and 2772. QUEST. —Give the rule for finding the greatest common divisor. How do you find the greatest common divisor of more than two numbers? 112 GREATEST COMMON DIVISOR. NOTE.-If it be required to find the greatest common divisor of more than two numbers, find first the greatest common divisbr of two of them, then of that common divisor and one of the remaining numbers, and so on for all the numbers: the last common divisor will be the greatest common divisor of all the numbers. 6. What is the greatest common divisor of 492, 744, and 1044? Ans. 7. What is the greatest common divisor of 944, 1488, and 2088? 8. What is the greatest common divisor of 216, 408, and 740? 9. What is the greatest common divisor of 945, 1560, and 22683? 10. What is the greatest common divisor of 204, 1190, 1445, and 2006? SECOND METHOD. 105. It has already been remarked (Art. 90), that a prime number is one which is only divisible by itself or unity, and that a composite number is the product of two or more factors (Art. 61). Now, every composite number may be decomposed into two or more prime factors. For example, if we have the composite number 36, we may write 36 = 18 x 2 =9 x 2 x 2 = 3 x 3 x 2 x 2; in which we see there are four prime factors, viz., two 3's and two 2's. Again, if we have the composite number 150, we may write 150 = 15 X 10 = 3 X 5 X 10 = 3 X 5 x 5 x 2; in which there are also four prime factors, viz., one 3, two 5's, and one 2. Hence, to decompose a'number into its prime factors, QUj3ST.-105. What is a prime number? What is a composite number? Into what may it be decomposed? What are the prime factors of 36? GREATEST COMMON DIVISOR. 113 Divide it continually by any prime number which will divizde it without a remainder, and the last quotient, together with the several divisors, will be the prime factors sought. EXAMPLES. 1. What are the prime factors of I OPERATION. 180? 1 2)180 We first divide by the prime num- 3)90 ber 2, which gives 90; then by 3, 530 then by 5, then by 3, and find the 3)6 six prime factors 2, 3, 5, 3, and 2. 2 2x3x5x3x2=180 2. What are the prime factors of 645? Ans. 3. What are the prime factors of 360? Ans. - 106. It is plain that the greatest common divisor of two or more numbers, will always be the greatest common factor, and that such factor must arise from the product. of equal prime numbers in each. Hence, to find the greatest common divisor of two or more numbers, Decompose them into their prime factors, and the product of those factors which are common'will be the greatest common divisor sought. EXAMPLES. 1. What is the greatest common divisor of 1365 and 1995? 3)1365 3)1995 5)455 5)665 7)91 7)133 13 19 Hence, 3, 5, 7, and 13 are Hence, 3, 5, 7, and 19 are prime factors. the factors. Hence, 3 x 5 X 7 = 105 = the greatest common divisor. QUEsT.-How do you decompose a number into its prime factors? 106. What is the greatest common divisor of two or more numbers? What does such factor arise from? How then do you find the greatest common divisor? 114 LEAST COMMON MULTIPLE. 2. What is the greatest common divisor of 12321 and 54345? 3. What is the greatest common divisor of 3775 and 1000? 4. What is the greatest common divisor of 6327 and 23997? 5. What is the greatest common divisor of 24720 and 4155? LEAST COMMON MULTIPLE. 125. A number is said to be a common multiple of two or more numbers, when it can be divided by each of them, separately, without a remainder. The least common multiple of two or more numbers, is the least numbedr which they will separately divide without a remainder. Thus, 6 is the least common multiple of 3 and 2, it being the least number which they will separately divide without a remainder. A factor of a number, is any number greater than 1 that will divide it without a remainder; and a prime factor is any prime number that will so divide it. Now, it is plain, that a dividend will contain its divisor an exact number of times, when it contains as factors, every factor of that divisor: and hence, the question of finding the least common multiple of several numbers is reduced to finding a number which shall contain all the prime factors of each number and none others. If the numbers have no common prime factor their product will be their least common multiple. EXAMPLE 1. —Let it be required to find the least common multiple of 6, 8 and 12. 125. When is one number said to be a common multiple of two or more numbers? What is the least common multiple of two or more numbers? Of what numbers is 6 the least common multiple? What is the difference between a common multiple and the least common multiple? What is a factor of a number? What is a prime factor? What is a prime number? When will a dividend exactly contain its divisor? To what is the question of finding the least common multiple reduced LEAST COMMON MULTIPLE. 115 We see, from inspection, that the prime factors of 6, are 2 and 2x3 2X2x2 2- 2 3 3;-of 8; 2, 2 and 2;-and of 6.... 8.. 12 12; 2, 2 and 3. Now, every factor, of each number, must appear in the least common multiple, and none others: hence, we must have all the factors of 8, and such other prime factors of 6 and 12 as are not found among the prime factors of 8, that is, the factor 3. Hence 2 X 2 X 2 X 3 =24, the least common multiple. To separate the prime factors, or to find the least common multiple of two or more numbers, FIRST METHOD. I. Place the numbers on the same line, and divide by any prime number that will divide two or more of them without a remainder, and set down in a line below, the quotients and the undivided numbers. II. Divide as before, until there is no number greater than 1 that will exactly divide any two of the numbers: then multiply together the numbers of the lower line and the divisors, and the product will be the least common multiple. If, in comparing the numbers together, we find no common divisor, their product is the least common multiple. 2. Find the least common multiple of 3, 8, and 9. We arrange the numbers in a line, OPERATION. and see that 3 will divide two of 3 2X2X2 3X3 them. We then write down the quo- 3)3 8 tients 1 and 3, and also' the 8 which cannot be divided. Then, as there 72 is no common divisor between any two of the numbers 1, 8, QUEsT.-G-ive the rule for finding the least common multiple. If the numbers have no common divisor, what is the least common multiple? 1 16 LEAST COMMON MULTIPLE. and 3, it follows that their product, multiplied by the divisor 3, will give the least common multiple sought. 3. Find the least common multiple of 6, 7, 8, and 10. 4. Find the least common multiple of 21 and 49. 5. Find the least common multiple of 2, 7, 5, 6, and 8. 6. Find the least common multiple of 4, 14, 28, and 98. 7. Find the least common multiple of 13 and 6. 8. Find the least common multiple of 12, 4, and 7. 9. Find the least common multiple of 6, 9, 4, 14, and 16. 10. Find the least common multiple of 13, 12, and 4. 11. What is the least common multiple of 11, 17, 19, 21, and 7? SECOND METHOD. 108. To find the least common multiple by this method. Decompose each number into its prime factors; after which, select from. each number so decomposed the Jactors which are common to them all, if there be such: then select those which are common to two or more of the numbers, and so on until all the factors common to- any two of them shall have been selected. Then multioly these seiteral factors together, and also the factors which are not common, and the product will be the least common multiple. EXAMPLES. 1. What is the least common multiple of 99 and 468? The prime factors of 99 OPERATION. are 3, 3, and 11; and of 99=3x3x11 468, 3, 3, 2. 2, and 13: 468=3x3x2x2x13. hence, the common factors are 3 and 3, which are to be 3 x3 X 11 X2x2x13 —5148. multiplied by 11,2, 2, and 13. QuEsT. —108. How do you find the least common multiple by the second method? REDUCTION OF VULGAR FRACTIONS. 117 2. What is the least common multiple of 12, 14, and 36? Having decomposed the OPERATION. numbers into their prime fac-12 =,X\ tors, we see that 2 is common 14 = x7 to them all. We then set it 36 =\x, x,'x 3. aside as a multiplier, and cross it in each number. We 2 3 X 2 X3 X 7 252. next set 3 and 2 aside, and cross them in a contrary direction. We then have 7 and 3 remaining, which we use as factors. It-is plain- that, this method introduces into the common multiple every prime factor of each number. 3. What is the least common multiple of 4, 9, 10, 15, 18, 20, 21? 4. What is the least common multiple of 8, 9, 10, 12, 25, 32, 75, 80? 5. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9? 6. What is the least common multiple of 9, 16, 42, 63, 21, 14, 72? 7. What is the least common multiple of 7, 15, 21, 28, 35, 100, 125? 8. What is the least common multiple of 15, 16, 18, 20, 24, 25, 27, 30? REDUCTION OF VULGAR FRACTIONS. 109. Reduction of Vulgar Fractions is the method of changing their forms without altering their value. A fraction is said to be in its lowest terms, when there is no number greater than 1 that will divide the numerator and denominator without a remainder. The terms of the fraction have then no common factor. QuEsT.-109. What is reduction? When is a fraction said to be in its lowest terms? Is one-half in its lowest terms? Is the fraction two-fourths? Is three-fourths? 118 REDUCTION OF VULGAR FRACTIONS. CASE I. 110. To reduce an improper fraction to its equivalent whole or mixed number. Divide the numerator by the denominator; the quotient will be the whole number; and the remainder, if there be one, placed over the given denominator will form the fractional part. _ It was shown in Art. 94, that the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator: hence the value of the fraction is not changed by the reduction. EXAMPLES. 1. Reduce 84 and %7 to their equivalent whole or mixed numbers. OPERATION. OPERATION. 4)84 9)67 Ans. 21 Ans. 74 2. Reduce 99 to a whole or mixed number. Ans. 3. In 179 of yards of cloth, how many yards? Ans. 4. In %1' of bushels, how many bushels? Ans. -- 5. If I give I of an apple to each one of 15 children, how many apples do I give? 6. Reduce 327 3672 50287 987625 to their whole or T-~2,' 5r? t 6941-, 7230o, mixed numbers. 7. If I distribute 878 quarter-apples among a number of boys, how many whole apples do I use? 8. Reduce 4 7 9 to a whole or mixed number. 9. Reduce 1512 to a whole or mixed number. 10. Reduce 3 7-59941 to a whole or mixed number. 11. Reduce 3745174 to a wholeor mixed number. QUEsT. —1 10. How do you reduce a fraction to its equivalent whole or mixed number? Does this reduction alter its value? Why not? What are four-halves equal to? Eight-fourths? Sixteen-eighths? Twenty-fifths? Thirty-six-sixths? Four-thirds? What arenine-fourthsequal to? Nine'fifths? Seventeen-sixths? Eighteen-sevenths? REDUCTION OF VULGAR FRACTIONS. 119 CASE II. 111. To reduce a mixed number to its equivalent improper fraction. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and place the sum over the given denominator. EXAMPLES. 1. Reduce 44 to its equivalent improper fraction. Here, 4 x 5 = 20: then 20 + 4 =.24; hence, 24 is the equivalent fraction. This rule is the reverse of Case I. In the example 4 we have the integer number 4 and the fraction -. Now 1 whole thing being equal to 5 fifths, 4 whole things are equal to 20 fifths; to which, add the 4 fifths, and we obtain the 24 fifths. 2. Reduce 253- to its equivalent improper fraction. 3 25 x 8+ 3 203 25- 8 8 Ans. 3. Reduce 475- to its equivalent improper fraction. 4. Reduce 67653, 874 39, 690 4 7 367 9, to their equivalent improper fractions. 5. Reduce 8473 6, 874-?7o, 67426368-, to their equivalent improper fractions. 6. How many 200ths in 6751o? Ans. 7. How many 15 ths in 1 87W1r? Ans. 8. Reduce 1495 to an improper fraction. Ans. 9. Reduce 3759A to an improper fraction. Ans. QUEsT. —111. How do you reduce a mixed number to its equivalent improper fraction? How many fourths are there in one? In two? In three? How many sixths in four and one-sixth? In eight and two-sixths? In seven and three-sixths?. In nine and five-sixths? In ten and five-sixths? How many eighths in two and one-eighth? In three and three-eighths? In four and four-eighths? In five and six eighths? In seven and seveneighths? In eight and seven-eighths? 120 REDUCTION OF VULGAR FRACTIONS. 10. Reduce 174949-543- to an improper fraction. 11. Reduce 4834-75 to an improper fraction. 12. Reduce 1789- to an improper fraction. 13. Place 4 sevens in such a m0anner that they may be equal. to 78. CASE III. 112. To reduce a fraction to its lowest terms. I. Divide the numerator and denominator by any number that will divide them both without a remainder, and then divide the quotients arising in.the same way, until there is no number greater than 1 that will divide them without a remainder. II. Or, find the greatest common divisor of the numerator and denominator, and divide them by it, The value of the fraction will not be altered by the reduction. EXAMPLES. 1. Reduce o7-T to its lowest terms. IST METHOD. 5) 70 7)14 2 5)175 - 7)35= — 5 which are the lowest terms of the fraction, since no number greater than 1 will divide the numerator and denominator without a remainder. 2D METHOD, BY THE COMMON DIVISOR. 70)175(2 140 3.5) 70 2. Ans. Greatest common div. 35)70(2 35) 175 5 70 2. Reduce -O4 to its lowest terms. Ans. 3. Reduce 1049 to its lowest terms. Ans. - QUEST.-112. When is a fraction in its lowest terms? (see Art. 109.) How do you reduce a fraction to its lowest terms by the first method? By the second? What are the lowest terms of two-fourths? Of six-eighths? Of nine-twelfths? Of sixteen-thirty-sixths? Of ten-twentieths.? Of fifteentwenty-fourths? Of sixteen-eighteenths? Of nine-eighteenths? REDUCTION OF VULGAR FRACTIONS. 121 4. Reduce 275 to its lowest terms.. Ans. 5. Reduce d-f to its lowest terms. Ans. - 6. Reduce ME72H to its lowest terms. Ans. 6. Reduce TTV As.;T- - -.7. Reduce 6j to its lowest terms by the 2d method. 8.' Reduce 31k to its lowest terms by the 2d method. 9. Reduce 1-17 to its lowest terms by the 2d method. 10. Reduce 792 to its lowest terms by the 2d method. 11. Reduce 37 to its lowest terms. Ans. 12. Reduce to its lowest terms. Ans.03132. Reduce A 4 o to its lowest terms. Ans. 14. Reduce 3s7 to its lowest terms. Ans. 15. Reduce 73 4 to its lowest terms. Ans. CA Reduce SE IV. 15. Reduce 54- tO it6 lowest terms. Ans. CASE Iv. 113. To reduce a whole number to an equivalent fraction having a given denominator. Multiply the whole number by the given denominator, and set the product over the said denominator. EXAMPLES. 1. Reduce 6 to a fraction whose denominator shall be 4. Here 6 x 4 = 24; therefore U4 is the required fraction. It is plain that the fraction will in all cases be equal to the whole number, since it may be reduced to the whole number by Case I. 2. Reduce 15 to a fraction whose denominator shall be 9. 3. Reduce 139 to a fraction whose denominator shall be 175. 4. Reduce 1837 to a fraction whose denominator shall be 181. Qtum. —113. How do you reduce a whole number to an equivalent fraction having a given denominator? How many thirds in 1? In 2? In 3S In 4? If the denominator be 5, what fraction will you form of 5? Of 4 Of 9? Of 7? Of 8? With the denominator 6, what fraction Vwill y~i formnof3? Of 4? Of? Of 6? Of 7? Of 9? 6 122 REDUCTION OF VULGAR FRACTIONS. 5. If the denominator be 837, what fractions will be formed from 327? From 889?~ From 575? 6. Reduce 167 to a fraction whose denominator shall be 89. 7. Reduce 3074 to a fraction whose denominator shall be 17. CASE V. 114. To reduce a compound fraction to its equivalent simple one. I. Reduce all mixed numbers to their equivalent improper fractions. II. Then multiply all the numerators together for a numerator and all the denominators together for a denominator: their product will form the fraction sought. EXAMPLES. 1. Let us take the fraction 4 of -. First, 3= 3 x 1: hence the fractions may be written 4- of 5 = 3 x I of 5-; that is, three times one-fourth of 5. But 1 of 5 = -__s: hence we have, -4 7 28 - of —3X fs -- i5' a result which is obtained by multiplying together the nume rators and denominators of the given fractions. When the compound fraction consists of more than two simple ones, two of them can be reduced to a simple fraction as above, and then this fraction may be reduced with the next, and so on. Hence, the reason of the rule is manifest. 2.'Reduce 21 of 64 of 7 to a simple fraction. Ans. QUEsT. —114. What is a compound fraction? How do you reduce a compound fraction to a simple one? Does this alter the value of the fraction? What is one-half of one-half? One-half of one-third? One-third of onefourth One-sixth of one-seventh? Three-halves of one-eighth? Six-thirds of two-ones? REDUCTION OF VULGAR FRACTIONS. 123 3. Reduce 5 of I of 7 of 6 to a simple fraction. METHOD BY CANCELLING. 115. The work may often be abridged by striking out or cancelling common factors in the numerator and denominator. EXAMPLES. 1. Reduce 5 of A of 6 to a simple fraction. 5 X, 5 Here, X by cancelling or striking out the 3's and 6's in the numerator and denominator. By cancelling or striking out the 3's we only divide the numerator and denominator of the fraction by 3; and in canceiling the 6's we divide by 6. Hence, the value of the fraction is not affected by striking out like figures, which should always be done when they multiply the numerator and denominator. 2. Reduce 6 of. of 9 to its simplest terms. 8 eu o'2 2 Here,,XX i=; Ans. 5 Besides cancelling the like factors 8 and 8 and 9 and 9, we also cancel the factor 3 common to 6 and 15, and write the quotients 2 and 5 above and below the numbers. 3. Reduce 3 of A of a of 2'7 of 5 to its simplest terms. 4. Reduce - of'" Of Aio of 7 to its simplest terms. 5. Reduce 3- of 5- of 2o7 of 49 to its simplest terms. CASE VI. 116. To reduce complex fractions to simple ones. Reduce the numerator and denominator, when necessary, to stmple fractions: then the numerator multiplied by the denominator with its -terms tnverted, will give the equivalent sunple fraction. Qusr. —115. How may the work often be abridged? 116. latS: a complex fraction? How do you reduce a complex fraction to a simple one? 124 REDUCTION OF VULGAR FRACTIONS. EXAMPLES. 4 1. Reduce the complex fraction i to a simple fraction. 9 Now, if we multiply the numerator and denominator of this fraction by any number whatever, the value of the fraction will not be altered (Art. 102). Let us then multiply them by the denominator with it terms inverted. This will give, 4 X9 36 7 2>T' 36 It is plain that when the denominator is multiplied by the fraction which arises from inverting its terms, the product will be equal to unity. Hence, the required simple fraction will always be equal to the numerator of the given fraction multiplied by the denominator with its terms inverted. All the cases in the reduction of fractions of this class are embraced in the following eight forms. First. 1 8 8 79 7 63 4 8 - 32 Second. 4 x = 7 7 1 7 Third. x5=0 5 To- X = - x6 Four 4th. x x8 7 1 39 13 Fourth. X 9 8 9 724 7 7 7 9 63 9 Fifth.7X F Fifth. = (72 + 5) -77 77 - 7 x 11 9 9 j f 7 2 14 7 Sixth =9 = 8 x = 72 =36 Seveth 23 9 207= 6 e'igenth. - X = 8 =. B 4 8 ~- 32 9.1 88 88 7 616 308.Eigh = - = -X - x-= 74l Eighth. 3+ 9 26 234 1-17 REDUCTION OF VULGAR FRACTIONS. 125 47O 2. Reduce - to a simple fraction. Ans. —95 345 3. Reduce 47- to a simple fraction. Ans. 84 44 4. Reduce to a simple fraction. Ans. - 247 5. Reduce i to a simple fraction. Ans. -- 147 6. Reduce 504 to a simple fraction. Ans. - 1789 7. Reduce 94~- to a simple fraction. Ans. - 894547 CASE VII. 117. To reduce fractions of different denominators to equivalent fractions having a common denominator. I. Reduce complex and compound fractions to simple ones, and all whole or mixed numbers to improper fractions. II. Then multiply the numerator and denominator of each fraction by the product of the denominators of all the others. EXAMPLES. 1. Reduce 4, 7, and 4 to a common denominator. 1 X 3 X 5 = 15 the new numerator of the 1st. 7 x 2 x 5 - 70 " " " 2d. 4 X 3 x 2 = 24 " " " 3d. and 2 x 3 x 5 = 30, the common denominator. Therefore, -0, and 30 are the equivalent fractions. It is plain that this reduction does not alter the values of the several fractions, stnce the numerator and denominator of each are multiplied by the same number (see Prop. V). QUEST. —117. What is the first step in reducing fractions to a common denominator? What is the second? Does the reduction alter the values of the several fractions? Why not? 126 REDUCTION OF VULGAR FRACTIONS. 2. When the numbers are small the work may be performed mentally. Thus, 1, 1 -= 420 10 16 Here we find the first numerator by multiplying I by 4 and 5; the second, by multiplying I by 2 and 5; the third, by multiplying 2 by 4 and 2; and the common denominator by multiplying 2, 4, and 5. together. 3. Reduce 2- and 1 of 1 to a common denominator. 21 = 7; and I of =1-: consequently, - and - = 9 and 3 are the answers. 4. Reduce 51, 7 of 1 and 4 to a common denominator. 5. Reduce 7, 135, and 37 to a common denominator. 6. Reduce 4, 3, 62 to a common denominator. 7. Reduce 72, 31, 64-to a common denominator. 8. Reduce 41-, 81, and 2~ of 5 to a common denomi nator. 9. Reduce 1,, 5 and 11 to a common denominator. 5, VI, an 16 10. Reduce 3 of 2 of 5 and 4 of A of 3 to a common denominator. 11. Reduce 51, 357 4-, and 65 to fractions having a common denominator. 12. Reduce,, 1 and I to a common denominator. 13. Reduce 4,, 3,, and 19 to a common denominator. 14. Reduce,' 33' 9' and -to simple fractions having a common denominator. 118. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator in each by such a number as shall make the denominators the same in all. Qusr —When the numlners are small, how may the work be performed? 118. By what second method may fractions be reduced to a common denominator? REDUCTION OF VULGAR FRACTIONS. 127 EXAMPLES. 1. LJet it be required to reduce ~ and A to a common denominator. We see at once that if we multiply the numerator and de nominator of the first fraction by 3, and the numerator and denominator of the second by 2, they will have a common denominator. The two fractions will be reduced to 94 and 1O. 2. Reduce 4 and 5 to a common denominator. If we multiply both terms of the first fraction by 3, and both terms of the second by 5, we have 4 = 12 and 5= 25 S. Rdc 3 t S. Reduce 1,1 and 4 to a common denominator. 4.- Reduce, 3, 4 to a common denominator. 5. Reduce 5s 35, and 3 to a common denominator. 5' 6' 4 6. Reduce 6-~, 89, and 5-2 to a common denominator. 7. Reduce 75, 4 3 and 2 to a common denominator. 6' I T'2 TV8 119. To reduce fractions to their least common denominator. T. Find the least common multiple of the denominators as in Art. 107, and it will be the least denominator sought. II. Multiply the numerator of each fraction by the quotient which arises from dividing the common multiple by the denominator, and the products will be the numerators of the required fractions; under which write the least common multiple. EXAMPLES. 1. Reduce 3,- 5, and 2 to their least common denominator. OPERATION. 2)7..8..6 7.. 4.. 3 and 3 x 4 x7 X 2 = 168 the leas common denominator. QUEsT. —119. How do you reduce fractions to their least common drnominator'! Does this reduction affect the values of the fractions? 128 REDUCTION OF DENOMINATE. FRACTIONS. 168 1 X 3 = 24 X 3 = 72 1st numerator. 168 X 5 = 21 X 5 = 105 2d numerator. 168 - X 2 = 28 X 2 = 56 3d numerator. 6 Ans. 7W2, a68, and i6 2. Reduce, 9 and 3 to their least common denomi. nator. 3. Reduce 144, 63, and 54 to their least common de nominator. 4. Reduce A3, 4 and - to their least common denominator. 5. Rjduce i, Ao, - to their least common denominator. 6. Reduce 41, 34k, 41, and 8 to a common denominator. 7. Reduce 3., 44, 8A-, 147 to their least common denominator. 8. Reduce 1, 2, A, and -6 to fractions having the least common denominator. 9. Reduce 2, 4, and 7 to fractions having the least common denominator. 10. Reduce 1 3 5, 1 and 17 to equivalent fractions having.the least common denominator. REDUCTION OF DENOMINATE FRACTIONS. 120. We have seen (Art. 14), that a denominate number is one in which the kind of unit is denominated or expressed. For the same reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus, I of a ~ is a denominate fraction. It expresses that one ~ is the unit which has been divided. QUErr. —120 What is a denominate number? What is a denominate fraction? What is the unit called? In two-thirds of a pound, what is the unit of the fraction? REDUCTION OF DENOMINATE FRACTIONS. 129 The fraction 8 of a shilling is also a denominate fraction, in which the unit is one shilling. The two fractions, 32 of a ~ and 83 of a shilling, are of different denominations, the unit of the first being one pound, and that of the second, one shilling. Fractions. therefore, are of the same denomination when they express parts of the same unit, and of different denominations when they express parts of different units. REDUCTION of denominate fractions consists in changing their denominations without altering their values. CASE I. 121. To reduce a denominate fraction from a lower to a higher denomination. I. Consider how many units of the given denomination make one unit of the next higher, and place 1 over that number form. ing a second fraction. II. Then consider how many units of the second denomination make one unit of the denomination next higher, and place 1 over that number forming a third fraction, and so on to the ienomination to which you would reduce. Then reduce the compound fraction to a simple one (Art. 114). EXAMPLES. 1. Reduce -1 of a penny to the fraction of a ~. The given fraction is I- of a OPERATION. penny. But one penny is equal.1 of 1 of - = - to - of a shilling: hence I- of a penny is equal to I- of 1 of a shilling. But one shilling is QUEsT. —In three-eighths of a shilling, what is the unit? In one-half of a foot, what is the unit? When are fractions of the same denomination? When of different denominations? Are one-third of a ~ and one-fourth of a ~ of the same or different denominations? One-fourth of a ~ and one-sixth of a shilling? One-fifth of a shilling and one-half of a penny? What is reduction? How many shillings in a ~? How many in ~2? In 3? In 4? How many pence in ls.? In 2? In 3? In 2s. 8d.? In 38. 6d.? In 5s. 8d.? How many feet in 3 yards 2ft.? How many inches? 121. How do you reduce a denominate fraction from a lower to a higher denomination? What is the first step? What the second? What th. third? 6* 130 REDUCTION OP DENOMINATE FRACTIONS. equal to <1 of a pound: hence ~- of a penny is equal to I of of - of a ~ = ~-. The reason of the rule is therefore evident. 2. Reduce 3 of a barleycorn to the denomination of yards. Since 3 barleycorns OPERATION. make an inch,- we first 2 of I of 1 of 1 -= yards. place 1 over 3: then as 12 inches make a foot, we place 1 over 12, and as 3 -feet make a yard, we next place 1 over 3. 3. Reduce -oz. avoirdupois to the denomination of tons. 4. Reduce - of a pint to the fraction of a hogshead. 5. Reduce 4 of a shilling to the fraction of a ~. 6. Reduce I of a farthing to the fraction of a ~. 7. Reduce 8 of a gallon to the fraction of a hogshead. 8. Reduce A of a shilling to the fraction of a ~. 9. Reduce x8-7 of a minute to the fraction of a day. 10. Reduce 1 of a pound to the fraction of a cwt. 11. Reduce 6 of an ounce to the fraction of a ton. 12. Reduce 2-87~_o of a farthing to the fraction of a pound. 13. Reduce 57 of a penny to the fraction of a pound 14. What part of a lb. troy is - of a pwt.? 15. What part of a cwt. is, A of a lb. avoirdupois? 16. What part of a hhd. of wine is I of a gallon? CASE II. 122. To reduce a denominate fraction from a higher to a lower denomination. I. Consider how many units of the next lower denomination make 1 unit of the given denomination, and place 1 under that numberforming a second fraction. II. Then consider how many units of the denomination still lower make one unit of the second denominationr and place 1 QEST. —122. How do you reduce a denominate fraction from a higher to, lower denomination?'What is the first step? What the second? What the third? REDUCTION OF DENOMINATE FRACTIONS, 131 uz(ler /t.at number forming a third fraction, and so on to the denomination to which you. would reduce. III. Connect all the fractions together, forming a compound fraction. Then reduce the compound fraction to a simple one (Arn. 114.) EXAMPLES. 1. Reduce 1 of a ~ to the fraction of a penny. In this example 1 of a pound OPERATION. is equal to I of 20 shillings. But 1 of 20 of 12 = Z40d. 7 ~f')n c~h~ll;~rr~ Rl 7 1 117 1 shilling is equal to 12 pence; hence, 7 of a ~ =- of 20 of 1 =- 24_d. Hence the reason of the rule is manifest. 2. Reduce 4-cwt. to the fraction of a'pound. 3. Reduce ~-5 of a ~ to the fraction of a penny. 4. Reduce I of a day to the fraction of a minute. 5. Reduce a of an acre to the fraction of a pole. 6. Reduce 6 of a ~ to the fraction of a farthing. 7. Reduce T34 of a hogshead to the fraction of a gallon. 8. Reduce 14 of a bushel to the fraction of a pint. 9. Reduce IL of a day to the fraction of a second. 10. Reduce 5 of a tun to the fraction of a gill. 11. Reduce 3 of a pound to the fraction of a farthing. 12. Reduce 7 of a pound to the fraction of a penny. 13. Reduce - of a lb. troy to the fraction of a pwt. 14. Reduce -49- of a cwt. to the fraction of a lb, 15. Reduce -7 of a week to the fraction of a second& 16. Reduce A of, a ton to the fraction of am oxycge 17. Reduce A7 of a yard to the fraction of a nail, 18. Reduce ~ of a league to the fraction of a foot, 19. Reduce: of a tb to the fraction of a scruple. QUEsT.-123. How much is one-half qf a,? One-third of a shilling? One-half of a penny? How much is one-halfof. a' lb. avoirdupois? Onefourth of a ton? One-fourth of A cwt.?' One-half of a quarter? Onefourth of a quarter? One-seventh of a quarter? One-fourteenth of a quarter? One-twenty.eighth of a quarters? 132 REDUCTION OF DENOMINATE FRACTIONS. CASE III. 123. To find the value of a fraction in integers of a less denomination. I. Reduce the numerator to the next lower denomination, and then divide the result by the denominator. II. If there be a remainder, reduce it to the denomination still less, and divide again by the denominator. Proceed in the same way to the lowest denomination. The several quotients, being connected together, will form the equivalent denominate number. EXAMPLES. 1. What is the value of 2 of a ~? OPERATION. 2 We first bring the pounds to 20 shillings. This gives,the frac- 3)40 tion 45o of shillings, which is equal 13s.... 1 Rem. to 13 shillings and 1 over. Redu- 12 cing this to pence gives the frac- 3)12 tion 1? of pence, which is equal 4d. to 4 pence. Ans. 13s. 4d. 2. What is the value of 4lb. troy? Ans. - 3. What is the value of -15 of a cwt.? Ans. 4. What is the value of 5 of an acre? Ans. 5. What is the value of - of a ~? Ans. --- 6. What is the value of -5 of a hogshead? Ans. 7. What is the value of of a hogshead? Ans. - 8. What is the value of 9 of a guinea? Ans. --- 9. What is the value of ] of a lb. troy? Ans. --- 10. What is the value of Z of a tun of wine? Ans. 11. What is the value of of of a lb. troy? Ans. - QUEST.-How do you find the value of a fraction in terms of integers of a law denomination? REDUCTION OF DENOMINATE FRACTIONS. 133 12. What is the value of 5 of a league? Ani. -- 13. What is the value of 5 of - of an acre? Ans. - 14. What is the value of 9- of 15 yards of cloth? 15. What is the value of 7 of a tun of wine? 16. What is the value of -3 of a butt of beer? 17. What is the value of 7 of a year? 18. What is the value of 5 of a chaldron of coal? 19. What is the value of 2 of 13s. 4d.? 20. What is the value of A of 15cwt. 3qr. 141b.? 21. What is the value of A of a cubic yard? 22. What quantity of ale is contained in - of 15228 cubic inches, English measure? CASE IV. 124. To reduce a denominate number to a denominate fraction of a given denomination. Reduce the hnumber to the lowest denomination mentioned in it: then if the reduction is to be made' to a denomination still less, reduce as in Case II.; but if to a higher denomination, reduce as in Case I. EXAMPLES. 1. Reduce 4s. 7d. to the fraction of a ~. We first reduce the OPERATION. given number to the 4s. 7d. =55d. lowest denomination Then, 55 of I of I =. named in it, viz., pence. Then, as the reduction is to be made to pounds, a higher denomination, we reduce by Case I. 2. What part of a bushel is 2pk. 3qt.? We first reduce to quarts, this OPERATION. being the lowest denomination. 2pk. 3qt. = 19qt. We then reduce to bushels by 19 of - of I = 9bu Case I. QUEST.-124. How do you reduce a denominate number to a fraction at a given denomination? 134 REDUCTION OF DENOMINATE FRACTIONS. 3 Reduce 2 feet 2 inches to the fraction of a yard. Ans. 4. Reduce 3 gallons 2 quarts to the fraction of a hogshead. A.Its. 5. Reduce lqr. 71b. to the fraction of a hundred. Ans. cwt. 6. What part of a hogshead is 3qt. lpt.? Ans. 7. What part of a mile is 6ft. 7in.? Ans. - 8. What part of a mile is 1 inch? Ans. 9. What part of a month of 30 days, is 1 hour 1 minute 1 second? Ans. 10. What part of 1 day is 3hr. 3m.? Ans. 11. What part is 3hr. 3m. of 2 days? Of 3? Of 4? Of 10? Of 25? 12. Reduce 15s. 11d. to the'fraction of a pound. 13. Reduce 5}d. to the fraction of a shilling. 14. Reduce 1cwt. 2qr. 6lb. 3oz. 8~dr. to the fraction of a cwt. 15. Reduce 5oz. 3}gr. to the fraction of a lb. troy. 16. Reduce 3qr. 3Rna. to the fraction of an English ell. 17. Reduce 147da. 15hr. to the fraction of a -year. 18. What part of a pound is 15s. 9~d.? 19. What part of a groat is 2 of three halfpence? 20. Reduce 4bu. 227-pk. of corn to the fraction of a quarter. 21. Reduce lqr. 3na. to the fraction of a yard. 22. Reduce 2R. 15P. to the fraction of an acre. 23. Reduce 23 llgr. to the fraction of a ib. 24. Reduce 3qt. lpt. 2gi. to the fraction of a hogshead. 25. Reduce 184 cubic inches to the fraction of a cubic yard. 26. Reduce 17bu. 3pk. to the fraction of a London chal. dron. 27. Reduce 24' 33" to the fraction of a degree. 28. Reduce 27gal. 3qt. lpt. to the fraction of a hogshead, beer measure ADDITION OF VULGAR FRACTIONS. 135 ADDITION40F VULGAR FRACTIONS. Addition of integer numbers is the process of finding a single number which shall express all the units of the numbers added (ART. 48.) Addition of fractions is the process of finding a single fraction which shall express the value of all the fractions added. It is plain that we cannot add fractions so long as they have different units: for, 2 of a ~ and ~ of a shilling make neither ~1 nor 1 shilling. Neither can we add parts of the same unit unless they are like parts; for. of a ~ and I of a ~ make neither 1 of a ~ nor 2 of a ~. But I of a ~ and I of a ~ may be added: they make 2 of a ~. So, of a ~ and 2 of a ~ make 3 of a ~. Hence, before fractions can be added, two things are necessary. 1st. That the fractions be reduced to the same denomi. nation; that is, to the same unit: 2d. That they be reduced to a common denominator; that is, to the same fractional unit (ART. 94). CASE I. 126. When the fractions to be added are of the same denomination and have a common denominator. Add the numerators together, and place their sum over the common denominator: then reduce the fraction to its lowest terms, or to its equivalent mixed number. QUEST.-125. What is addition of integer numbers? What is addition of fractions What two things are necessary before fractions can be added a Can one-half of a ~ be added to one-half of a shilling without reduction? Can one-half be added to one-fourth without reduction? 126. When the fractions are of the same denomination and have a common denominator, how do you find their sum? What is the sum of one third and two-thirds? Of three-fourths, one-fourth, and fourfourths? Of three-fifths, six-fifths, and two-fifths? Of three-sixths, sevensixths, and nine-sixths? Of one-eighth, three-eighths, and four-eighths? 136 ADDITION OF VULGAR FRACTIONS. EXAMPLES. 1. Add 1 3 6. and 3 together. It is evident, since all the parts OPERATION. are halves, that the true sum will I + 3 + 6 + 3 = 13; be expressed by the number of hence, 3 = sum. halves; that is, by thirteen halves. 2. Add 1 of a ~, 5 of a ~, and 9 of a ~ together. 3. What is the sum of + - + + 13 + 16? 4. What is the sum of -3 + + _- +- + - ~? 5. What is the sum of 9 + - + 1-4 - 1-1 + 1-5? 6. What is the sum of 3 + + 9 + A + -13 1+ 7 CASE II. 127. When the fractions are of the same denomination but have different denominators, Reduce complex and compound fractions to simple ones, mixea numbers to improper fractions, and all the fractions to a common denominator. Then add them as in Case I. EXAMPLES. 1. Add 4,,a and 2 together. By reducing to a corn- OPERATION. mon denominator, the new 6 X 3 X 5 = 90 1st numerator fractions are 4 X 2 X 5 - 40 2d numerator. 3+ ~o - = 130 - 2 X 3 X 2 =- 12 3d numerator. which, by reducing to the which, by reducing to'the -2 x 3 X5 = 30 the denominator lowest terms becomes 411 2. Add 1 of a ~, 2 of a ~, and 5 of a ~ together. 3. Add together -1, 1, 4-1, and 61. Ans. - 4. Find thk least common denominator (Art. 119) and add the fractions 1 3 2, and 4 Ans. - QUEST.-127. How do you add fractions which have different denominators? How do you reduce fractions of different denominators to equiva. lent fractions having a common denominator? ADDITION OF VULGAR FRACTIONS. 137 5. Finl the least common denominator and add 6 3 4,21 5, and 6 Ans. 30' 6. Find the least common denominator and add 1 of 3, Q of 19, and 5 of 12 together. Ans. 7. Add 3, 9 of.1 of 6, and 6 of 3 of 11 together. 128. When there are mixed numbers, instead of reducing them to improper fractions we may add the whole numbers and the fractional parts separately and then add their sums. EXAMPLES. 1. Add 191, 6*, and 4-4 together. OPERATION. Whole numbers. Fractional parts. 19 +6+ 4= 29. 7 + + 4 1 = 1AAs. Hence, 29 + 13 =- 306o 4, the sum. 2. Add together 31, 6.5, 8A-9, and 659. Ans. —3. Add together A, A, 13, and 18. Ans. - 4. Add together -A, 172, 1, and 96. Ans. 5. Add together 381, 133, and 95. Ans. - 6. Add together 62, 133, 1835, and 1322. Ans. 7. Add 35, 45, and -5 together. Ans. 8. Add 3 and 9 of J of 153 together. Ans. - 45 47-~ 9. Add I, 7, -, and 314 together. Ans. - CASE III. 129. When the fractions are of different denominations. Reduce the fractions to the same denomination. Then reduce all the fractions to a common denomznator, and then add them as in Case I. QuEsT.-128. How may you proceed when there are mixed numbers? 129. When the fractions are of different denominations, how are they added? What is the second method? 138 ADDITION OF VULGAR FRACTIONo. EXAMPLES. 1. Add of a to Qf a shilling 2 of a ~ =- 2 of 2-0 4=0 of a shilling.'T 3 1 3 Then, 4 + 6240o + s -,,5s. =?s. = 14s. 2d. Or, the 5- of a shilling might have been reduced to the fraction of a ~ thus, 5 of -= 5 oof a - of a o~. Then, 3 7 =4 7 + -2 = + 5 of a ~: which being reduced gives 1.4s. 2d. 2. Add - of a yard to s of an inch. 3. Add 3 of a week, I of a day, and I of an hour together. 4. Add 4 of a cwt., 8l1b., and 39o9 z. together. 5. Add 11 miles, 7 furlongs, and 30 rods together. NOTE.-The value of each of the fractions may be found separately, and their several values then added. 6. Add -! of a year, - of a week, and ~ of a day together. 3 of a year = 53 Of 3 65 days = 219 days. I of a week =- of 7 days = 2 days 8 hours. 3 3 I of a day = 3 hours. Ans. 221da. 1 1hr. 7. Add 2 of a yard, 3 of a foot, and 7 of a mile together. 3 4 8. Add 3 of a cwt., 42 of a lb. 13oz., and I of a cwt. 61b. together. 9. Add A- of a pound, - of a shilling, and -5 of a penny together. 10. What is the sum of 1 of ~1 10s., l of ~1 10s., and 4 4 g0 of a hundred guineas? 11. Add X of a lb. troy to - of an ounce., Ans. - 12. Add 4 of a ton to 5 of a cwt. Ans.9 12 13. Add 4 of 3 ells English to - of a 5ard. Ans. 14. Ad of a yardoffoot, 7and x of a mile together. 15. Add t of an acre,! of 19 square feet, and 37 of a square inch together. SUBTRACTION OF VULGAR FRACTIONS. 139 16. What is the sum of 3 of a tun of wine and 3 of a hhd.? 4. 17. Add - of a chaldron to 37 of a bushel. 18 Add { of a week, 3 of a day, and 1 of an hour together. 19. Add i of I of a year, of f f, a day, and f of 5 of 194 hours together. SUBTRACTION OF VULGAR FRACTIONS. 130. It has been shown (Art. 125), that before fractions can be added together they must be reduced to the same unit and to a common denominator. The same reductions must be made before subtraction. SUBTRACTION Of Vulgar Fractions is the process of finding the difference between two fractional numbers. CASE I. 131. When the fractions are of the same denomination and have a common denominator. Subtract the less numerator from the greater, and place the difference over the common denominator. EXAMPLES. 1. What is the difference between 5 and 3? Here we have 5 - 3 = 2: hence, 2 = the difference. 2. From 5 take {69 Ans. 3. From 4977 take 1697 Ans. 4. From 1~906 take 909 Ans. CASE II. 132. When the fractions are of the same denomination, but have different denominators. Quesr.-130. Can one-third of a shilling be subtracted from one-third of a ~ without reduction? Can one-fourth of a shilling be subtracted from one-fifth of a shilling? What reductions are necessary before subtraction? What is subtraction? 131. How do you subtract fractions of the same denomination and denominator? 140 SUBTRACTION OF VULGAR FRACTIONS. Reduce mixed numbers to improper fractions, compound and complex fractions to simple ones, and all the fractions to a common denominator: then subtract them as in Case I. EXAMPLES. 1. What is the difference between 56 and? Here, - = - = I = ~ answer. 2. What is the difference between 121 of I and 2? 2 6 3. What is the difference between 24 of a ~, and 3 of a~? 4. From - of 6, take 16 of 1. Ans. - 5. Frd tn of A of 7, take 3 of. Ans. - 6. From 371-1, take 35 of I Ans. 7. What is the difference between 3 and -A2? Ans. - 8. What is the difference between 35 and 2 of 5? 498 34? 9. What is the difference between 97 and 146? 10. From 1155 take 397. Ans. - 11. Subtract 754 from a unit. Ans. 12. Subtract 11 from 365. Ans. 13. What is the difference between a of 15 and ] of 72? 14. To what fraction must I add - that the sum may be 8? 15. What number isha that to which if 72 be added, the sum will be 173? 16. What number is that from which if yo:vsubtract I of 9_ of a unit, and to the remainder add X of of'- aunit, the sum will be 9? CASE III. 133. When the fractions are of different denominations. Reduce the fractions to the same denomination. Then re duce them to a common denominator; after which subtract as in Case I. QUEsT. —132. How do you subtract fractions of different denominators? What is the difference between one-half and one-third? 133. How do you subtract fractions which are of different denominations? MULTIPLICATION OF VULGAR FRACTIONS. 141 EXAMPLES. 1. What is the difference between i of a ~ and k of a shilling? i of a shilling = ~ of -- =- of a ~. Then, ~-6-o- =- -- of aZ = 9s.d. 2. What is the difference between a of a day and 2 of a second? 3. From 13 of a lb., troy weight, take 1 of an ounce. 4. What is the difference between 4 of a hogshead and 6- of a quart? 5. From i of a ~ take 3 of a shilling. Ans. - 6. From 3oz. take Ipwt. Ans. - 7. From 4A.cwt. take 4 olb. Ans. 8. What is the difference between3 of a pound and 5of a shilling? 9. From 5 of a lb. troy take A- of an ounce. Ans. 10. From 3 of a ton take 2 of a of a lb. Ans. 11. From 2- of 3 of a hhd. of wine take 2 of I of a pint. 12. From 3 of a league take of a mile. Ans. 13. From 5 of 365]- days take 3- of A of an hour. 14. A pound avoirdupois is equal to 14oz. 1 lpwt. 16gr. troy; what is the difference, in troy weight, between the ounce avoirdupois and the ounce troy? MULTIPLICATION OF VULGAR FRACTIONS. 134. MULTIPLICATION is a short method of taking one number, called the multiplicand, as many times as there are units in another number, called the multiplier. Hence, when the multiplier is less than 1 we do not take the whole of the multiplicand, but only such a part of it as the mulQuasr.-134. What is multiplication? What is required when the mul. tiplier is less than 1 Does multiplication then imply increase? What is th.- prodact of 8 multiplied by one-half? By one-fourth? 4By oneeighth? By three-halves? By six-halves? What is the product of 9 multiplied by one-half? By one-third? By one-sixth? By one-ninth? 142 MULTIPLICATION OF VULGAR FRACTIONS. tiplier is of unity. For example, if the multiplier be one-half of unity, the product will be half the multiplicand; as, for example, the product of 8 multiplied by ~ is 4. If the multiplier be I of unity, the product will be one-third of the multiplicand. Hence, to multiply by a proper fraction does not imply increase, as in the multiplication of whole numbers. CASE I. 135. To multiply a fraction by a whole number. Multiply the numerator, or divide the denominator by the whole number. EXAMPLES 1. Multiply the fraction - by 4. When it is required to mul- OPERATION. tiply a fraction by a whole { X 4-2 = = - = 2; number, it is required to in- or by dividing the denomcrease the fraction as many inator by 4, we have times as there are units in the 5 X 4 = 4 = 2. multiplier, which may be done by multiplying the numerator (Art. 98), or by dividing the denominator -(Art. 101). 2. Multiply 347 by 12. Ans. 3. Multiply 49 by 7. Ans. 4. Multiply 1-7- by 9. Ans. 5. Multiply 1127 by 5. Ans. 6. Multiply 6 by 49. Ans. - 7. Multiply 271 by 357. Ans. - 8. Multiply 1s23 by 198. Ans. - 9. Multiply 4291 by 2433. Ans. - QuErT. —When the multiplier is less than 1, how much of the multipli. cand is taken? Does the multiplication by a proper fraction imply increase? 135. -How do you multiply a fraction by a whole number? 136. What Is the product of one-sixth by one-seventh? Of three-fourths by one-half? Of six-niths by three-fifths? Give the general rule for the' multiplieation of fractions. MULTIPLICATION OF VULGAR FRACTIONS. 143 CASE II. 136. To multiply one fraction by another. Reduce all the mixed numbers to improper fractions, and all compound and complex fractions to simple ones: then multiply the numerators together for a numerator, and the denominators together for a denominator. EXAMPLES. 1. Multiply 3 by 5. In- this example ~ is to be OPERATION. taken -. times; that is, 3 is first A X x5 = X 5 X -g to be multiplied by 5 and the product divided by 7, a result which is obtained by multiply ing the numerators and denominators together. 2. Multiply I of 3 by 8. We first reduce the com- OPERAT'ON. pound fraction to the simple _ of 3 3 — one 3, and the mixed num- 8~- 25. ber to the equivalent fraction Hence, -a X 23 = 75 = 2. 25 after which, we multi- 2 ply the numerators and denominators together. 3. Multiply 5{ by I. Ans. - 4. Multiply'l2 by 3 of 9. Ans. - 5. Multiply.1 of 3 of - by 151. Ans. 6. Multiply - by 2 of 6. Ans. - 7. Required the product of 6 by 2 of 5. Ans. - 8. Required the product of - of by 5- of 32. 9. Required the product of 32 by 41. Ans. - 10. Required the product of 5, 2,. of 3, and 41. 11. Required the product of 4, 3 of +, and 181-. 12. Required the product of 14, of 9, and 6. 13. What is the product of 162, 9 1 and? 6j 94' 19' 9.A. 144 MULTIPLICATION OF VULGAR FRACTIONS. 137. In multiplying by a mixed number, we may first multiply by the integer, then multiply by the fraction, and then add the two products together. This is the best method when the numerator of the fraction is 1. EXAMPLES. 1. Multiply 26 by 31. OPERATION. 26 We first multiply 26 by 3: the 3 product is 78. Afterwards we multiply 26 by': the product is 13: 26 x i = 13 hence the entire product is 91. 91 Ans. 2. Multiply 48 by 8-. OPERATION. 48 X 8 = 384 We first multiply by 8, and then 48 1 = 8 add a sixth. 392 Ans. 392 Ans. 3. Multiply 67 by 9. Ans. - 4. Multiply 842 by 7T. Ans. - 5. Multiply 3756 by 31. Ans. - 6. Multiply 2056 by 56. Ans. - GENERAL EXAMPLES. 1. What is the product of A of., 2 of 15-, and of 2? 4 71 2. What is the continued product of 143, 5, 5_' 4, and _.? Ans. - 9 3. What is the product of, -8 73. 4, 8' p4-' ~ 7T4~ and 20? Ans. -.4. What is the product of I of -17 of 15, and 14- of 114? Ans. 5. What will 7 yards of cloth cost, at $4 per yard? Qv. —!137. How mayyou multiply by a mixed number? When is this the best method? DIVISION OF VULGAR FRACTIONS. 145 6. What will 32 gallons of brandy cost, at $1 - per gallon 7 7. If lb. of tea cost $14, what will 61lb. cost? 8. What will be the cost of 17~ yards of cambric, at 2~ shillings per yard? 9. What will 15 1 barrels of cider come to, at $3 per barrel? 10. What will 33 boxes of raisins cost, at $2~ per box? 11. What will 15~ barrels of sugar cost, at 171 dollars per barrel? DIVISION OF VULGAR FRACTIONS. 138. We have seen that division of entire numbers explains the manner of finding how many times a less number is contained in a greater. In division of fractions the divisor may be larger than the dividend, in which case the quotient will be less than 1. For example, divide 1 apple ito 4 equal parts. Here it is plain that each part will be 1; or that the dividend will contain the divisor but - times. Again, divide 2 of a pear into 6 equal parts. If a whole pear were divided into 6 equal parts, each part would be expressed by 6-. But since the half of the pear was divided, each part will be expressed by I of 6, or -A. In the division of fractions we should note the following principles: 1st. When the dividend is just equal to the divisor, the quotient will be 1. 2d. When the dividend is greater than the divisor, the quotient will be greater than 1. QUEsT.-138. What does division of whole numbers explain? In division of fractions, may the divisor exceed the dividend? How will the quotient then compare with 1? If an apple be divided in 2 equal parts, what will express each part? If half an apple be divided into 4 equal parts, what will express one of the parts? What is one-half of one-half? What-isonesixth of one-half? What principles do you note in the divisianof-asetinsns When will the quotient be 1? When greater than 1? 7 146 DIVISION OF VULGAR FRACTIONS. 3d. When the dividend is less than the divisor, the quotient will be less than 1. 4th. The quotient will be just so many times greater than 1, as the dividend is greater than the divisor. 5th. The quotient will be just as many times less than, I as the dividend is less than the divisor. CASE I. 139. To divide a fraction by a whole number. Divide the numerator or multiply the denominator by the whole number. EXAMPLES. 1. Divide 4 by 2. OPERATION. In the first operation we 4 4 4 2 divide the fraction by mul- 3' 3 X 2 - 6- 3 tiplying the denominator 4 2)4 2 (Art. 100): in the second or 2 = we divide the numerator (Art. 99), giving the same result in both cases. 2. Divide I 1 by 9. Ans. 3. Divide 4 096 by 15. Ans. - 4. Divide 27s5 by 19. Ans. 5. Divide — 9 by 15. Ans. - 6. Divide a7 by 8. Ans. - 7. Divide 61 by 37. Ans. CASE II. 140. To divide one fraction by another. EXAMPLES. 1. Let it be required to divide 0 by 5 The true quotient will be expressed by the complex fraction 2. QuzET.-When will the quotient be less than 1? When greater than 1, how many times greater? When less than 1, how many times less? 139. In how many ways may a fraction be divided by a whole number? 140 How do you divide one fraction by another? DIVISION OF VULGAR FRACTIONS. 147 Let the terms of this fraction be now multiplied by the denominator with its terms inverted: this will not alter the value of the fraction (Art. 102), and weshall then have. 10 8 100 8 _5 4...= - X 5 = quotient. 8 8 X 1 It will be seen that the quotient is obtained by simply multiplying the numerator by the denominator with its terms inverted. This quotient may be further simplified by cancelling the common factors 5 and 8, giving 2 for the true quotient SECOND METHOD OF PROOF. Let us first divide the dividend by OPERATION. 5. This is done by multiplying the __ 5 10 denominator (Art. 100), which gives 120 X 8 10 But the divisor being but I1 of 5, 120. 8 this quotient is 8 times too small, since the eighth of a number will be contained in the dividend 8 times more than the number itself. Therefore, by multiplying 1 by 8, we obtain - for the true quotient. Hence, to divide one fraction by another, Reduce compound and complex fractions to simple ones, also whole and- mixed numbers to improper fractions: then multiply the dividend by the divisor with its terms inverted, and the product reduced to its simplest terms will be the quotient sought. EXAMPLES. 1. Divide ~ by 4. Ans. 2. Divide 31 by j-. Ans. 3. Divide 161 of f- by 41. Ans. -- 4. Divide 441, by.. Ans. - 5. Divide 3714 by 4T' Ans. 6. Divide 6 4 by 1 3 Ans. 7. Divide I of 2 by 2 of 3. Ans. 8. Divide 5 by 7. Ans. - 9. Divide 52051 by 4 of 91. Ans. - 148 DIVI'SION O VULGAR FRACTIONS. 10. Divide 100 by 4. Ans.l1. Divide, of by. Ans. - 12. Divide - of 50 by 4X. Ans. 13. Divide 14-l of I by 31 of 6. Ans. 14. Divide 34~ by 5. Ans. 9361 15. Divide - by 71. Ans. - 95 1 149-" 16. What number multiplied by - will give 153 for the product? 17. What part of 108 is -? Ans. i8. What number is that which, if multiplied by -8 of A of 154N, will produce 5? 19. If 71b. of sugar cost 47 of a dollar, what is the price per pound? 20. If 7- of a dollar will pay for 101b. of nails, how much is the price per pound? 21. If A of a yard of cloth cost $3, what is the price per yard? 22. If $214- will buy 71] barrels of apples, how much are they per barrel? 23. If 41 gallons of molasses cost $2-., how much is it per quart? 24. If 14hhd. of wine tost $2501, how much is the wine per quart? 25. If 8 pounds- of tea cost 7- of a dollar, how much is it per pound? 26. In 8j4 weeks a family consumes 1653 pounds of butter: how much-do they consume a week? 27. If a piece of cloth containing 176] yards costs $3754, what do -:it cost per yard? 2 1 f of 7 42 28. Divide 15~ of I- Of ofT by of- of of. 4 — DECIMAL FRACTIONS. 149 DECIMAL FRACTIONS. 141. IF the unit 1 be divided into 10 equal parts, the parts are called tenths, because each part is one-tenth of unity. If the unit 1 be divided into one hundred equal parts, the parts are called hundredths, because each part is one-hundredth of unity. If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, because each part is one-thousandth of unity: and we have similar expressions for the parts when the unit is divided into ten thousand, one hundred thousand, &c., equal parts. The division of the unit into tenths, hundredths, thousandths, &c., forms a'system of numbers called Decimal Fractions. They may be written, Four-tenths, - - - 0 Six-tenths, -... 6 Forty-five hundredths, - - - 125 thousandths, - - - - - 1047 ten thousandths, - - - - o From which we see, that in each case the denominator gives denomination or name to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c. 142. The denominators of decimal fractions are seldom set down. The fractions are usually expressed by means of QUEST.-141. When-the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided into 100 equal parts? When into 1000? Into 10,000, &c.? How are decimal fractions formed? What gives denomination to the fraction? 142. Are the denominators of decimal fractions generally set down?. How are the fractious expressed? 150 DECIMAL FRACTIONS. a comma, or period, which is called the decimal point, and is placed at the left of the numerator. Thus, 1 - is written - -.4 o "- -.45 125.125 1047 ~- -.1047. &c &c., &c. This manner of expressing decimal fractions is a mere language, and is used to avoid the inconvenience of writing the denominators. The denominator, however, of every decimal fraction is always understood. It is a unit 1, with as many ciphers annexed as there are places of figures in the numerator. The place next to the decimal point, is called the place oT tenths, and its unit is 1 tenth. The next place to the right is the place of hundredths, and its unit is 1 hundredth; the next is the place of thousandths, and its unit is 1 thousandth; and similarly for places still to the right. a DECIMAL NUMERATION TABLE. 4" ". r.4 is read 4 tenths..6 4 " "C 64 hundredths..0 6 4 " " 64 thousandths..6 7 5 4 " " 6754 ten-thousandths..0 1 2 3 4 " "'1234 hundred-thousandths..0 0 7 6 5 4 " " 7654 millionths..0 0 4 3 6 0 4 " " 43604 ten-millionths. QUEsT. —Is the denominator understood? What is it What is the place next the decimal point called? What is its unit? What is the next place called What is its unit The next Its unit Which way are decimals numerated? DECIMAL FRACTIONS. 151 Decimal fractions are numerated from the left hand to the right, beginning with the tenths, hundredths, &c., as in the to'Ale. 143. Let us now write and numerate the following decimals. Four-tenths,.4. Four hundredths, - - -.0 4. Four thousandths, - - -.0 0 4. Four ten-thousandths, - -.0 0 0 4. Four hundred thousandths, - -.0 0 0 0 4. Four millionths, - - -.0 0 0 0 0 4. Four ten-millionths, - - -.0 0 0 0 0 0 4. Here we see, that the same figure expresses units of different values, according to the place which it occupies. But I of 4 is equal to 4 =.04. 10 100 100 ~ ~f P 6 " " o'o o' —b-ff - i of 4- " A- 4 =.0004. fo~ Of 1000 10000 I o L=.0o004. - of - " " _ -.000004. oo — o-' —00000 -I I of 4, 4 -4 -.0000004. 1 of — 00o "_.0000004. Therefore the value of the units of the different places, in passing from the left to the- right, diminishes according to the scale of tens. Hence, ten of the parts in any one of the places are equal to one of the parts in the place next to the left; that is, ten thousandths make one hundredth, ten hundredths make onetenth, and ten tenths a unit 1. This law of increase from the right hand towards the left, is the same as in whole numbers. Therefore, whole numbers and decimal fractions may be united by placing the decimal point between them. Thus, QUEST. —143. Does the value of the unit of a figure depend upon the place which it occupies? How does the value of the unit change firom the left towards the right? What do ten parts of any one place make? How do the units of place increase from the right towards the leftf? How may whole numbers be joined with decimals 152 DECI-MAL FRACTIONS. Whole numbers. Decimals. 83630641. 0478976. of a~decillal, is called'd mixed number. s 0 P hem.. Sixteen3 and 064 1t 04 4 976. 3. Fiveia is called and nin mixedths. 5number.09 4. SWrite the following numbers ifigurefteen thousandths., and numerate 1. Eighty, and three millionths. 6. Sixteenwo, and three hundred millionths. 16.000003 7. Fivour hundre d an d nine y-two thousahundredths. 5.09 8. hree thousandixty-five, and fiftetwenty- neten-thousandths. 9. Eighty-seven, and thrwety-oe millionte-th)usanths. 60. FifTwo, and three hundred and three millionths. 71. FouThirty-nine, and six hundred and nifornety-two thousandths. 82. Three thousand, eight hundred and twforenty millionths. 13. FoSirt hundredven, and twfifty-one tet-thousandtdths. 14. Fifteeny thousandred and four hundrede millionths. 15. Six hundred, and eighteen ten-thousandths. 16. Three millionths. 17. Thirty-nine hundred-thousandths. 144. The denominations of Federal Money will corresporfd to the decimal division, if we regard 1 dollar as the unit. QUEST.-What is a number called when composed partly of whole numbers and partly of decimals? 144. If the denominations of Federal Money bo expressed decimally, what is the niit? DECIMAL FRACTIONS. 153 For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar. EXAMPLES. 1. Express $17, 3 dimes 8 cents and 9 mills decimally. 2. Express $92, 8 dimes 9 cents 5 mills decimally. 3. Express $107, 9 dimes 6 cents 8 mills decimally. 4. Express $47 and 25 cents decimally. 5. Express $39, 39 cents and 7 mills decimally. 6. Express $12 and 3 mills decimally. Ans. - 7. Express $147 and 4 cents decimally. Ans. - 8.- Express $148, 4 mills decimally. Ans. - 9. Express four dollars, six mills decimally. Ans. - 10. Express $14, 3 cents 9 mills decimally. Ans. -- 11. Express $149, 33 cents 2 mills decimally. 12. Express $1328, 5 mills decimally. Ans. -- 13. Express 9 dimes 4 mills decimally. Ans. -- 14. Express 5 cents 8 mills decimally. Ans. - 15. Express $3856, 2 cents decimally. Ans.145. A cipher is annexed to a number when it is placed on the right of it. If ciphers be annexed to the numerator of a decimal fraction, the same number of ciphers must also be annexed to the denominator; for there must always be as many ciphers in the denominator as there are places of figures in the numerator (Art. 142). The numerator and denominator will therefore be multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 102). Hence, Annexing ciphers to a decimal fraction does not alter its value. Quvsr.-What part of a dollar is one dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill? 145. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value? Why not? What do threetenths become by annexing a cipher? What by annexing two ciphers? 7* 154 DECIMAL FRACTIONS. We may take as an example the decimal.3 = 3' If, now, we annex a cipher to the numerator, we must, at the same time, annex one to the denominator, which gives.30` 10% by annexing one cipher,.300 -=.0~oo by annexing two ciphers,.3000 = 3-oo0o all of which are equal to 13 =.3. Alo,.5- -=.50 5 - 500- 5 00 Al -f - * -loo - * 1000Also,.8 =.80 =.800 =.8000 -.80000. 146. Prefixing a cipher is placing it on the left of a number. If ciphers be prefixed\to the numerator of a decimal fraction, that is, placed at the left hand of the significant figurbs, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged while the denominator will be increased ten times for every cipher which is annexed, and the value of the fraction will be decreased in the same proportion (Art. 100). Hence, Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed. Take as an example the fraction.2 = 2..02 -= 02 by prefixing one cipher,.002 -= -a6 by prefixing two ciphers,.0002 = 0-o~%%2~ by prefixing three ciphers: in which the fraction is diminished ten times for every cipher prefixed. Also,.03 becomes.003 by prefixing one cipher; and.0003 by prefixing two. QuzsT.-What does.8 become by annexing a cipher? By annexing two ciphers? By annexing three ciphers? 146. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the denominator? What effect then has it on the value of the fraction? What does.5 become by prefixing a cipher? By prefixing two ciphers? By prefixing three?- What does.07 become by prefixing a cipher? By prefixing two? By prefixing three? By prefixing four ADDITION OF DECIMAL FRACTIONS. 155 ADDITION OF DECIMAL FRACTIONS. 147. It must be recollected that only like parts of unity can be added together, and therefore in setting down the numbers for addition, the figures occupying places of the same value must be placed in the same column. The addition of decimal fractions is then made in the-same manner as that of whole numbers. Add 37.04, 704.3, and.0376 together. In this example, we place the tenths OPERATION under tenths, the hundredths under hun- 37.04 dredths, and this brings the decimal points 704.3 and the like parts of the unit directly un-.0376 der each other. We then add as in whole 741.3776 numbers. Hence, for addition of decimals, I. Set down the numbers to be added so that tenths shall fall under tenths, hundredths under hundredths, fc. This will bring all the decimal points under each other. II. Then add as in simple numbers and point off in the sum, from the right hand, so many places for decimals as are equal to the greatest number of places in any of the added numbers. EXAMPLES. 1. Add 6.035, 763.196, 445.3741, and 91.5754 together. 2. Add 465.103113,.78012, 1.34976,.3549, and 61.11. 3. Add 57.406 + 97.004 + 4 +.6 +.06 +.3. 4. Add.0009 + 1.0436 +.4 +.05 +.047. 5. Add.0049 + 49.0426 + 37.0410 + 360.0039. 6. Add 5.714, 3.456,.543, 17.4957 together. QUEST.-147. What parts of unity may be added together? How do you set down the numbers for addition? How will the decimal points fall? How do you then add? How many decimal places do you point od in the sum? 156 SUBTRACTION OF DECIMAL FRACTIONS. 7.-Add 3.754, 47.5,.00857, 37.5 together. 8. Add 54.34,.375, 14.795, 1.5 together. 9. Add 71.25, 1.749, 1759.5, 3.1 together. 10. Add 375.94, 5.732, 14.375, 1.5 together. 11. Add.005,.0057, 31.008,.00594 together. 12. Required the sum of 9 tenths, 19 hundredths, 1,8 thousandths, 211 hundred-thousandths, and 19 millionths. * 13. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. 14. Required the sum of two hundred dollars one dime three cents and nine mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill. 15. What is the sum of one tenth, one hundredth, and one thousandth? 16. What is the sum of 4, and 6 ten-thousandths? 17. What is the sum of 3 thousandths, 9 millionths, 5 hundredths, 6 hundredths, 3 tenths, and 2 units? 18. Required, in dollars and decimals, the sum of one dollar one dime one cent one mill, six dollars three mills, four dollars eight cents, nine dollars six mills, one hundred dollars six dimes, nine dimes one mill:, and eight dollars six cents. 19. What is the sum of 4 dollars 6 cents, 9 dollars 3 mnills, 14 dollars 3 dimes 9 cents 1 mill, 104 dollars 9 dimes 9 cents 9 mills, 999 dollars 9 dimes 1 mill, 4 mills, 6 mills, and 1 mill? SUBTRACTION OF DECIMIAL FRACTIOIN:-;. 148. Subtraction of Decimal Fractions is the proiess of finding the difference between two decimal numbers. 1. From 3.275 take.0879. In this example a cipher is annexed to OPERATION. the minuend to make-the number of deci- 3.2750 mal places equal to the number in the.0879 subtrahend. This does not alter the value 3.1871 of the minuend (Art. 145). SUBTRACTION OF DECIMAL FRACTIONS. 157 Hence, for the subtraction of decimal numbers, I. Set down the less number under the greater, so that figures occupying places of the same value shall fall in the same column. II. Then subtract as in simple numbers, and point of irn the remainder, from the right hand, as many places for decimals as are equal to the greatest number ofplaces in either of thegiven numbers EXAMPLES. 2. From 3278 take.0879. Ans. 3. Fromn 291.10001 take 41.496. Ans. - 4. From 10.00001 take.111111. Ans. 5. Required the difference between 57.49 and 5.768. 6. What is the difference between.3054 and 3.075? 7. Required the difference between 1745.3 and 173.45. 8. What. is the difference between seven-tenths and 54 ten-thousandths? 9. What is the difference between.105 and 1.00075? 10. What is the difference between 150.43 and 754.355? 11. From 1754.754 take 375.49478. Ans. 12. Take 75.304 from 175.01. Ans. 13. Required the difference between 17.541 and 35.49. 14. Required the difference between 7 tenths and 7 millionths. 15. From 396 take 8 ten-thousandths. Ans. 16. From 1 take one-thousandth. Ans. - 17. From 6374 take one-tenth. Ans. 18. From 36,5.0075 take 5 millionths. Ans. 19. From 21.004 take 98 ten-thousandths. Ans. 20. From 260.3609 take 47 ten-millionths. Ans. 21. From 10.0302 take 19 millionths. Ans. 22. From 2.03 take 6 ten-thousandths Ans. QUEsr. —148. What does subtraction teach? How do you set down the numbers for subtraction? How do you then subtract? How many decimal places do you point off in the remainder? 158 MULTIPLICATION OF DECIMAL FRACTIONS. MULTIPLICATION OF DECIMAL FRACTIONS. 149.-1. Multiply.37 by.8. We may first write.37 - 37 arid.8- 8 If, now, we multiply the fraction 3-!Z6 OPERATION. by o, we find the product to be 29o6o;.37 1. the number of ciphers in the denomina-.8 = 8. tor of this product is equal to the number.296 9 of decimal places in the two factors, and =.296. the same will be true for any two factors whatever. 2. Multiply.3 by.02. OPERATION..3 X.02 -= X - =r 60 =-.006 answer. Now, to express the 6 thousandths decimally, we have to prefix two ciphers to the 6, and this makes as many decimal places in the product as there are in both multiplicand and multiplier. Therefore, to multiply one decimal by another, ]fultiply as in simple numbers, and point off in the,product, from the right hand, as many figures for decimals as are equal to the number of decimal places in the multiplicand and multiplier; and if there be not so many in the product, supply the deficiency by prefixing ciphers. EXAMPLES. 1. Multiply 3.049 by.012. Ans..036588. (2.) (3.) Multiply 365.491 Multiply 496.0135 by.901 by 1.496 Ans. Ans. QUEST.-149. After multiplying, how many decimal Iplaces will you point off in the product'! When there are not so many in the product, what do you do? Give the rule for the multiplication of decimals. MULTIPLICATION OF DECIMAL FRACTIONS. 159 4. Multiply one and one millionth by one thousandth. 5. Multiply 473.54 by.057. Ans. 6. Multiply 137.549 by 75.437. Ans. - 7. Multiply 3.7495 by 73487. Ans. - 8. Multiply.04375 by.47134. Ans. - 9. Multiply.371343 by 75493. Ans. - 10. Multiply 49.0754 by 3.5714. Ans. -- 11. Multiply.573005 by.000754. Ans. - - 12. Multiply.375494 by 574.375. Ans. 13. Multiply two hundred and ninety-four millionths, by one millionth. 14. Multiply three hundred, and twenty-seven hundredths by 62. Ans. -- 15. Multiply 93.01401 by 10.03962. Ans. - 16. What is the product of five-tenths by five-tenths? 17. What is the product of five-tenths by five thousandths? 18. Multiply 596.04 by 0.000012. Ans. - 19. Multiply 38049.079 by 0.000016. Ans. 20. Multiply 1192.08 by 0.000024. Ans. - 21. Multiply 76098.158 by 0.000032. Ans. - CONTRACTION IN MULTIPLICATION. 150. CONTRACTION in the multiplication of decimals is a short method of finding the product of two decimal numbers in such a manner, that it shall contain but a given number of decimal places. 1. Let it be required to find the product of 2.38645 multiplied by 38.2175, in such a manner that it shall contain but four decimal places. In this example it is proposed to take the multiplicand 2.38645, 38 times, then 2 tenths times, then 1 hundredth times, thef 7 thousandth times, then 5 ten-thousandth times, QUEST.-150. What is contraction in the multiplication of decimals? What is proposed in the example? How are the numbers written down for multiplication? 160 MULTIPLICATION OF DECIMAL FRACTIONS. and the sum of these several products-will be the product sought. OPERATION. Write the unit figure of the multiplier OPERATION. directly under that place of the multi- 5712.83 plicand which is to be retained in the 715935 product, and the remaining places of in- 190916 teger numbers, if any, to the right, and 4773 then write the decimal places to the left 239 in their order, tenths, hundredths, &c. 167 12 When the numbers are so written, the 91.2042 product of any figure in the multiplier by the figure of the multiplicand directly over it, will be of the same order of value as the last figure to be retained in the product. Therefore, the first figure of each product is always to be arranged directly under the last retained figure of the multiplicand. But it is the whole of the multiplicand which should be multiplied by each figure of the multiplier, and not a part of it only. Hence. to comnipensate for the part omitted, we begin with the figure to the right of the one directly over any multiplier, and carry one when the product is greater than 5 and less than 15, 2 when it falls between 15 and 25, 3 when it falls between 25 and 35, and so on for the higher numbers. For example, when we multiply by the 8, instead of saying 8 times 4 are 32, and writing down the 2, we say first, 8 times 5 are 40, and then carry 4 to the product 32, which gives 36. So, when we multiply by the last figure 5, we first say, 5 times 3 are 15, then 5 times 2 are 10 and 2 to carry make 12, which is written down. EXAMPLES. 1. Multiply 36.74637 by 127.0463, retaining three decimal places in the product. QUEST.-When the numbers are so written, what will be the order of value of the product of any figure of the multiplier by the figure directly over it? Where then is the first figure by each product to be written I How do you compensate for the part omitted? MULTIPLICATION OF DEJCIMAL FRACTIONS. 161 CONTRACTION. COMMON WAY. 36.74637 36.74637 3640.721 ]27.0463 3674637 11023911 734927 22047822 257224 14698548 1470 25722459 220 7349274 11 3674637 4668.489 4668.490346931 2. Multiply 54.7494367 by 4.714753, reserving five places of decimals in the product. 3. Multiply 475.710564 by.3416494, retaining three decimal places in the product. 4. Multiply 3754.4078 by.734576, retaining five decimal places in the product. 5. Multiply 4745.679 by 751.4549; and reserve only whole numbers in the product. 151. NOTE.-When a decimal number is to be multiplied by 10, 100, 1000, &c., the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier; and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers. 10 67.9 14100 679. Thus, 6.79 multiplied by 1000 6790. 10000 67900. 100000 L679000. 10'3700.36 100 37003.6 Also, 370.036 multiplied by 1000 = 370036. 10000 3700360. 100000 37003600. QuEsT.-151. How do you multiply a decimal number by 10, 100, 1000, &c.? If there are not as many decimal figures as there are ciphers in the multiplier, what do you do? 162 DIVISION OF DECIMAL FRACTIONS. DIVISION OF DECIMAL FRACTIONS. 152. Division of Decimal Fractions is similar to. that of simple numbers. We have just seen that, if one decimal fraction be multiplied by another, the product will contain as many places of decimals as there were in both the factors. Now, if this product be divided by one of the factors, the quotient will be the other. factor (Art. 79). Hence, in division, the dividend must contain just as many decimal places as the divisor and quotient together. The quotient, therefore, will contain as many places as the dividend, less the number in the divisor. EXAMPLES, 1. Divide 1.38483 by 60.21. There are five decimal places in OPERATION. the dividend, and two in the divi- 60.21)1.38483(23 sor: there must therefore be three 1.2042 places in the quotient: hence one 18063 0 must be prefixed to the 23, and 18063 the decimal point placed before it. Ans..023. Hence, for the division of decimals, Divide as in simple numbers, and point of in the quotient, from the right hand, so many places for decimals as the decimal places in the dividend exceed those in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers. 2. Divide 4.6842 by 2.11. Ans. -- 3. Divide 12.82561 by 1.505. Ans. 4. Divide 33.66431 by 1.01. Ans. - QUEST.-152. If one decimal fraction be multiplied by another, how many decimal places will there be in the product? How does the number of decimal places in the dividend compare with those in the divisor and quotient? How do you determine the number of decimal places in the quotient? If the divisor contains four places and the dividend six, how many in the quotient? If the divisor contains three places and the dividend five, how many in the quotient? Give the rule for the division of decimals. DIVISION OF DECIMAL FRACTIONS. 163 5. Divide.010001 by.01. An. - 6. Divide 24.8410 by.002. Ans. - 7. What is the quotient of 75.15204, divided by 3? By.3? By.03? By.003? By.0003? 8. What is the quotient of 389.27688, divided by 8? By.08? By.008? By.0008? By.00008? 9. What is the quotient of 374.598, divided by 9? By.9? By.09? By.009? By.0009? By.00009? 10. What is the quotient of 1528.4086488, divided by 6? By.06? By.006? By.0006? By.00006? By.000006? 11. Divide 17.543275 by 125.7. Ans. 12. Divide 1437.5435 by.7493. Ans. - 13. Divide.000177089-by.0374. Ans. - 14. Divide 1674.35520 by 960. Ans. - 15. Divide 120463.2000 by 1728. Ans. - 16. Divide 47.54936 by 34.75. Ans. - 17. Divide 74.35716 by.00573. Ans. 18. Divide.37545987 by 75.714. Ans. - 153. NOTE 1. —When any decimal number is to be divided by 10, 100, 1000, &c., the division is made by removing the decimal point as many places to the left as there are O's in the divisor; and if there be not so many figures on the left of the decimal point, the deficiency must be supplied by prefixing ciphers. 10 2.769 27.69 divided by 100.02769 1000.02769 10000.002769 10 r 64.289 100 6.4289 642.89 divided by 1000 =.64289 100000.064289 _ 100000.0064289 QuErsT.-153. How do you divide a decimal number by 10, 100, 1000, &c.? If there be not as many figures to the left of the decimal point as there are ciphers in the divisor, what do you do? 164 DIVISION OF DECIMAL FRACTIONS. ],54. NOTE 2. —When there are more decimal places in the divisor than in the dividend, annex as many ciphers to the dividend as are necessary to make its decimal places equal to those of the divisor; all the figures of the quotient will then be whole numbers. Always bear in mind that the quotient is as many times greater than unity, as the dividend is greater than the divisor. EXAMPLES. 1. Divide 4397.4 by 3.49. OPERATION. 3.49)4397.40(1260 We annex one 0 to the dividend. Had it contained no decimal place 698 698 we should have annexed two. 2094 Ans. 1260. 2. Divide 1097.01097 by.100001. Ans. 3. Divide 9811.0047 by.1629735. Ans. 4. Divide.1 by.0001. Ans. - 5. Divide 10 by.1. Ans. 6. Divide 6 by.6. By.06. By.006. By.2. By.3 By.003. By.5. By.005. By.000012. 155. NOTE 3.-When it is necessary to continue the division farther than the figures of the dividend will allow, we may annexciphers to it, and consider them as decimal places. EXAMPLES. 1. Divide 4.25 by 1.25. OPERATION. In this example, after having ex- 1.25)4.25(3.4 hausted the decimals of the dividend, 3.75 we annex an 0, and then the decimal 500 places used in the dividend will exceed 500 those in the divisor by 1. Ans. 3.4 QuEST.-154. If there are more decimal places in the divisor than in the dividend, what do you do. What will the figures of the quotient then be! 155. How do you continue the division after you have brought down all the figures of the dividend? DIVISION. OF DECIMAL FRACTIONS. 165 2. Divide.2 by.06. OPERATION. We see in this example that the di-.06).20(3.333 + vision will never terminate. In such 18 cases the division should be carried to 20 the third or fourth place, which will 18 give the answer true enough for all 20 practical purposes, and the sign + 18 should then be written, to show that 20 the division may still be continued. Ans. 3.333 +. 3. Divide 37.4 by 4.5. Ans. -- 4. Divide 586.4 by 375. Ans. 5. Divide 94.0369 by 81.032. Ans. REMARKS. 156. The unit of Federal Money, the currency of the United States, is one dollar, and all the lower denominations, dimes, cents, and mills, are decimals of the dollar. Hence, all the operations upon Fed'eral Money are the same as the corresponding operations on decimal fractions. APPLICATIONS IN THIE FOUR PRECEDING RULES. 1. A merchant s-old 4 parcels of cloth; the 1st contained 239 and 3 thousandths yards; the 2d, 6 and 5 tenths yards; the 3d, 4 and one hundredth yards; the 4th, 90 and one millionth yards: how many yards did he sell in all? 2. A merchant buys three chests of tea; the first contains 70 and one thousandth lb.; the second, 49 and one ten-thousandth lb.; the third, 36 and one-tenth lb.: how much did he buy in all? 3. What is the sum of $20 and three hundredths; $44 and one-tenth, $6 and one thousandth, and $18 and one hundredth? QuEsT.-When the division does not terminate, what sign do you place after the quotient? What does it show? 156. What is the unit of the cur. rency of {'fh United Stitcs? What parts df this unit are the inferibr de nominations, dimes, cents, and mills? 166 DIVISION OF DECIMAL FIoACTIONS. 4. A puts in trade $1504.342; B puts in $350.1965; C puts in $100.11; D puts in $99.334; and E puts in $9001.31: what is the whole amount put in? 5. B has $936, and A has $5, 3 dimes, and 1 mill: how much more money has B than A? 6. A merchant buys 112.5 yards of cloth, at one dollar twenty-five cents per yardc how much does the whole come to? 7. A farmer sells to a merchant 13.12 cords of wood at $4.25 per cord, and 17 bushels of wheat at $1.06 per bushel: he is to take in payment 13 yards of broadcloth at $4.07 per yard, and the remainder in cash: how much money did he receive? 8. If 11 men had each $339 1 dime 9 cents and 3 mills, what would be the total amount of their money? 9. If one man can remove 5.91 cubic yards of earth in a day, how much could 38 men remove? 10. What is the cost of 24.9 yards of cloth, at $5.47 per yard? 11. If a man earns one dollar and one mill per day, how much will he earn in a year? 12. What will be the cost of 675 thousandths of a cord of wood, at $2 per cord? 13. A farmer purchased a farm containing 56 acres of woodland, for which he paid $46.347 per acre; 176 acres of meadow land at the rate of $59.465 per acre; besides which there was a swamp on the farm that covered 37 acres, for which he was charged $13.836 per acre. What was the area of the land; what its cost; and what the average price per acre? 14. A person dying has $8345 in cash, and 6 houses valued at $4379.837 each; he ordered his debts to be paid, amounting to $3976.480, and $120 to be expended at his funeral; the residue was to be divided among his five sons in the following manner: the eldest was to have a fourth part, and each of the other sons to have equal shares. What was the share of each son? DIVISION OF DECIMAL FRACTIONS. 167 CONTRACTION IN DIVISION. 157. Contraction in division is a short method of obtaining the quotient of one decimal number divided by another. EXAMPLES 1. Divide 754.347385 by 61.34775, and let the quotient contain three places of decimals. COMMON METHOD. 61.34775)754.34738500(12.296 CONTRACTED METHOD. 61347175 61.34775)754.347385(12.296 140861988 61348 12269550 14086 1817 4385 12269 1226 9550 1817 59048350 1227 552 12975 590 38'353750 552 36!808650 38 1545100 37 It is plain that all the work by the common method, which stands on the right of the vertical line, does not affect the quotient figures. On what principle is the work omitted in the contracted method? In every division, the firstfigure of the quotient will always be of the same order of value as thatfigure of the dividend under which is written the product of the first figure of the quotient by the unit's figure of the divisor. Having determined the order of value of the first quotient figure, make use of as many figures of the divisor as you wish places of figures in the quotient. Let each remainder be a new dividend, and in each following division omit one figure to the right hand of the divisor, QurST-157. What is contraction in division? In every division, what will be the order of the first quotient figure? How many figures of the divisor will you use? How will you then make the division? 168 REDUCTION OF DECIMAL FRACTIONS. observing to carry for the increase of the figures cut off, as in contraction of multiplication. In the example above, the order of the first quotient figure was obviously tens; hence, as there were three decimal places required in the quotient, five figures of the divisor must be used. 2. Divide 59 by.74571345, and let the quotient contain four places of decimals. 3. Divide 17493.407704962 by 495.783269, and let the quotient contain four places of decimals. 4. Divide 98.18743-7 by 8.4765618, and let the quotient contain ten places of decimals. 5. Divide 47194.379457 by 14.73495, and let the quotient contain as many decimal places as there will be integers in it. REDUCTION OF VULGAR FRACTIONS TO DECIMALS. 158. The value of every vulgar fraction is equal to the quotient arising from dividing the numerator by the denominator (Art. 94). EXAMPLES. 1. What is the value in decimals of 9? We first divide 9 by 2, which OPERATION. gives a quotient 4, and 1 for a re- 9 - 41; but mainder. Now, 1 is equal to 10 4 41- = 4.5. tenths. If, then, we add a cipher, 2 will -divide 10, giving the quotient 5 tenths. Hence, the true quotient is 4.5. 2. What is the value of 1?3? We first divide by 4, which gives OPERATION. a quotient 3 and a remainder 1. 1-3 = 31; but 4 4 But 1 is equal to 100 hundredths. 3 = 3100 = 3.25. If, thet, we add two ciphers, 4 will divide the 100, giving a quotient of 25 hundredths. QvusE. —What is the order of the first quotient figure in Ex. 2? In 3? In 4? 158. Whit is the Clue of a fraction equal to? What is the value of four-halves? REDUCTION OF DECIMAL FRACTIONS. 169 Hence, to reduce a vulgar fraction to a decimal, I. Annex one or more ciphers to the numerator and then divilde by the denominator. II. If there is a remainder, annex a cipher or ciphers, and divide again, and continue to annex ciphers and to divide until there is no remainder, or until the quotient is sufficiently exact: the number of decimal places to be pointed off in the quotient is the same as the number of ciphers used; and when there are not so many, ciphers must be prefixed to supply the deficiency. EXAMPLES. 1. Reduce 6 5 to its equivalent decimal. OPERATION. 125)635(5.08 We here use two ciphers, and there- 625 fore point off two decimal places in the 1000 quotient. 1000 2. Reduce -1 and is to decimals. Ans. 3. Reduce 4 o, 7Y o, and 8-b6o to decimals. 4. Reduce I and TY58 to decimals. Ans. - 5. Reduce 231495a712 to a decimal. Ans. - 6. Reduce 6 1 3 3 2 574 to decimals. Ans. 7. Reduce 30 to decimals. Ans. - 8. Reduce o to decimals. Ans. 9. Reduce To7 to decimals. Ans. 10. Reduce 34 to decimals. Ans. - 11. Reduce 2oIb a-v to decimajs. Ans. - 12. Reduce I to decimals. Ans. 13. Reduce 17 to decimals. Ans. 14. Reduce 9 to decimals. Ans. 15. Reduce - to decimals. Ans. 16. Reduce 1 to decimals. Ans. - 17. Reduce 1412 to decimals. Ans. QuvST.-What is the decimal value of one-half? Of three-fourths? Of six-fourths? Of nine-halves? O seven-halves? Of five-fourths? Of onefourth? Give the rule for reducing a vulgar fraction to a decimal 8 170 REDUCTION OF DENOMINATE DECIMALS: 18. What is the decimal value of 3 of 5 multiplied by - 19. What is the value in decimals of 2 of 2 of 7 divided by ] of 1? 20. A man owns 8- of a ship; he sells ~4 of his shate what part is that of the whole, expressed in decimals? 21. Bought 11 of 87 3 bushels of wheat for 9, of 7 dollars a bushel: how much did it come to, expressed in decimals? 22. If a man receives - of a dollar at one time, 7~ at another, and 83 at a third: how many in all, expressed in decimals? 23. What decimal is equal to A of 18, and y1 of 1I of 7 4 added together? 24. What decimal is equal to 2 of 6 taken from ] of 83? 25. What decimal is equal to 21 1, added together? 222 T I i,,~,added together REDUCTION OF DENOMINATE DECIMALS. 159. We have seen that a denominate number is one in which the kind of unit is denominated or expressed (Art. 14). A denominate decimal is a decimal fraction in which the kind of unit that has been divided is expressed. Thus,.5 of a ~, and.6 of a shilling are denominate decimals:- the unit that was divided in the first fraction being ~1, and that in the second 1 shilling CASE I. 160. To find the value of a denominate number in decimals of a higher denomination. 1. Reduce 9d. to the decimal of a ~. We first find that there are 240 CPERATION. pence in ~1. We then divide 9d. by 240d.=~1 240, which gives the quotient.0375 240)9(.0375 of a ~. This is the true value of 9d. Ans. ~.0375. in the decimal of a ~. QUEzsT.-159. What is a denominate number? What is a denominate decimal? In the decimal five-tenths ofa ~, what is the unit? In the decimal six-tenths of a shilling, what is the unit? REDUCTION OF DENOMINATE DECIMALS. 171 Hence, to make the reduction, I. Consider how many units of the given denomination make one unit of the denomination to which you would reduce. II. Divide the given denominate number by the number so found, and the quotient will be the value in the required denomination. EXAMPLES. 1. Reduce 14 drams to the decimal of a lb. avoirdupois. 2. Reduce 78d. to the decimal of a ~. 3. Reduce.056 poles to the decimal of an acre. 4. Reduce,42 minutes to the decimal of a day. 5. Reduce 63 pints to- the decimal of a peck. 6. Reduce 9 hours to the decimal of a day. 7. Reduce 375678 feet to the decimal of a mile. 8. Reduce 72 yards to the decimal of a rod. 9. Reduce.5 quarts to the decimal of a barrel. 10. Reduce 4ft. 6in. to the decimal of a yard. 11. Reduce 7oz. 19pwt. of silver to the decimal of a pound. 12. Reduce 9! months to the decimal of a year. 13. Reduce 62 days to the decimal of a year of 365- days. 14. Reduce ~25 19s. 6~d. to the decimal of a pound. 15. Reduce 3qr. 211b. to the decimal of a cwt. 16. Reduce 5fur. 36rd. 2yd. 2ft. 9in. to the decimal of a mile. 17. Reduce 4cwt. 21qr. to the decimal of a ton. 18. Reduce 3cwt. 71b. 8oz. to the decimal of a ton. 19. Reduce 17hr. 6m. 43sec. to the decimal of a day. CASE II. 161. To reduce denominate numbers of different denominations to an equivalent decimal of a given denomination. QussT. —160. How do you find the value of a denominate number in a decimal of a higher denomination? 172 REDUCTION OF DENOMINATE DECIMALS. 1. Reduce ~1 4s. 93d. to the denomination of pounds. We first reduce 3 farthings OPERATION. to the decimal of a penny, by'Id. =-.75d.; hence, dividing by 4. We then annex 93d. = 9.75d. the quotient.75 to the 9 pence. 12)9.75d. We next divide by 12, giving.8125s., and.8125, which is the decimal 20)4.8125s. of a shilling. This we annex X.240625; therefore, to the shillings, and then di- 1 4s. 9d. = ~1.240625. vide by 20. Hence, to make the reduction, Divide the lowest denomination named, by that number which makes one of the denomination next higher, annexing ciphers if necessary; then annex this quotient to the next higher denomination, and divide as before: proceed in the same manner through all the denominations to the last: the last result will be the answer sought. EXAMPLES. 1. Reduce ~19 17s. 31d. to the decimal of a ~. 2. Reduce 46s. 6d. to the denomination of pounds. 3. Reduce 7~d. to the decimal of a shilling. 4. Reduce 21b. 5oz. 12pwt. 16gr. troy to the decimal of a lb. 5. Reduce 7 feet 6 inches to the denomination of yards. 6. Reduce llb. 12dr. avoirdupois to the denomination of pounds. 7. Reduce 10 leagues 4 furlongs to the denomination of leagues. 8. Reduce 7s. 5~d. to the decimal of a pound. 9. What decimal part of a pound is three halfpence? 10. Reduce 4s. 7r9d. to the decimal-of a pound. 11. Reduce loz. 1lpwt. 3gr. to the decimal of a pound troy. QuEvsr.-161. How do you reduce denominate numbers of different de. nominations to equivalent decimals of a given denomination? REDUCTION OF DENOMINATE DECIMALS. 173 12. Reduce 24 grains to the decimal of an ounce troy. 13. Reduce 5oz. 4dr. avoirdupois to the decimal of a pound troy. 14. Reduce 3cwt. lqr. 141b. to the decimal of a ton. 15. Reduce 2qr. 151b. to the decimal of a hundred-weight. 16. Reduce 51b. 10oz. 3pwt. 13gr. troy to the decimal of a hundred-weight avoirdupois. 17. Reduce lqr. Ina. to the decimal of a yard. 18. Reduce 2qr. 3na. to the decimal of an English ell. 19. Reduce 2yds. 2ft. 6~in. to the decimal of a mile. 20. What decimal part of an acre is 1R. 37P? 21. What decimal part of a hogshead of wine is 2 quarts 1 pint? 22. Reduce 3 bushels 3 pecks to the decimal of a chaldron of 36 bushels.. 23. What decimal part of a year is 3w*. 6da. 7hr., reckoning 365da. 6hr. a year? 24. Reduce 2.45 shillings to the decimal of a ~. 25. Reduce 1.047 roods to the decimal of an acre. 26. Reduce 176.9 yards to the decimal of a mile. CASE III. 162. To find the value of a denominate decimal in terms of integers of inferior denominations. 1. What is the value oP.832296 of a ~? We first multiply the decimal by OPERATION. 20, which brings it to shillings, and.832296 after cutting off from the right as 20 many places for decimals as in the 16.645920 given number, we have 16s. and 12 the decimal.645920 over. This 7.751040 we reduce to pence by multiplying 4 by 12, and then reduce to farthings 3.004160 by multiplying by 4. i Ans. 16s. 7d. 3far. 174 REDUCTION OF DENOMINATE DECIMALS. Hience, to, make the reduction, I. Consider how many in the next less denomination make one of the given denomination, and multiply the decimal by this number. Then cut off from the right hand as many places as there are in the given decimal. II. Multiply the figures so cut off by the number which it takes in the next less denomination to make one of a higher, and cut off as before. Proceed in the same way to the lowest denomination: the figures to the left will form the answer sought EXAMPLES. 1. What is the value of.625 of a cwt.? Ans. 2. What is the value of.625 of a gallon? Ans. -- 3. What is the value of.0041681b. troy? Ans. -- 4. What is the value of.375 hogshead of beer? 5. What is the value of.375 of a year of 365 days? 6. What is the value of.085 of a ~? Ans. 7. What is the value of.258 of a cwt.? Ans. 8. What is the difference between.82 of a day and.64 of an hour? 9. What is the value of 2.078 miles? Ans. 10. What is the value of ~.3375? Ans. 11. What is the value of.3375 of a ton? Ans. 12. What is the value of.05 of an acre? Ans. 13. What is the value of.875 pipes of wine? 14. What is the value of.046875 of a pound, avoirdupois?'15. What is the value of.56986 of a year of 365+ days? 16. What is the value of ~2.092? Ans. 17. What is the value of ~5.64? Ans. 18. What is the value of.36974 of a last, wool weight? 19. What is the value of.827364qr., corn measure? 20. What is the value of.0937651b.? Ans. QUEsT.-162. How do you find the value of a denominate decimal in in tegers of inferior denominations? What is the value in shillings of one. half of a f? In pence of one-half of a shilling? CIRCULATING OR REPEATING DECIMALS. 175 CIRCULATING OR REPEATING DECIMALS. 163. WE have seen that in changing a vulgar into a decimal fraction, cases will arise in which the division does not terminate, and then the vulgar fraction cannot be exactly expressed by a decimal (Art. 158). Let it be required to reduce - to its equivalent decimal. We find the equivalent decimal to be.4166 + &c., giving 6's, as far as we OPERATION. choose to continue the division. The further the division is continued.4166 + the nearer the decimal will approach to the true value of the vulgar fraction; and we express this approach to equality of value by saying, that if the division be continued withoutrlimit, that is, to infinity, the value of the decimal will then be equal to that of the vulgar fraction. Thus, we also say,.999 +, continued to infinity = 1, because every annexation of a 9 brings the value nearer to 1. 164. Let us now examine the circumstances under which, in the reduction of a vulgar to a decimal fraction, the division will not terminate. If the vulgar fraction be first reduced to its lowest terms, (which we suppose to be done in all the cases which follow,) there will be no factor common to its numerator and denominator. Now, by the addition of O's to the numerator we may increase its value ten titnes for every 0 annexed; that is, we introduce into the numerator the two factors 2 and 5 for every QUEST.-163. Can a vulgar fraction always be exactly expressed by a decimal? Can five-twelfths? If we continue the division, does the quotient approach to the true value? By what language do we express this fact? 164. In annexing a 0 to the numerator, what factors do we in-. troduce into it? 1.76 CIRCULATING Oft REPEATING DECIMALS. additional 0. But the numerator can never be exactly divided by the denominator, if the denominator contains any prime factor not found in the niimerator (Art. 107): hence it can never be so divided, if the denominator contains any prime factor other than 2 or 5. Hence, to determine whether a vulgar fraction in its lowest terms can be expressed by an exact decimal, Decompose the denominator into its prime factors, and if there are any factors other than 2 or 5, the exact division cannot be made. EXAMPLES. 1. Can 2' be exactly expressed by decimals? OPERATION. 25 = 5 x 5; hence, the fraction 25) 70 (.28 can be exactly expressed by a deci- 50 mal. 200 200 2. Can.-5 be exactly expressed by decimals? OPERATION. 36 = 18 x 2 = 9 x 2 x 2 _ 36) 50 (.1388+ 3 X 3 x 2 x 2; in which we see that 36 the denominator contains other fac- 140 tors than 2 and 5, and hence the 108 fraction cannot be exactly expressed 320 by decimals. 320 288 3. Can Ti9 be exactly expressed by decimals? 4. Can T- be exactly expressed by decimals? 5. Can ~-13 be exactly expressed by decimals? 6. Can 12 8 be exactly expressed by decimals? QUEST.-Under what circumstances will the numerator be exactly divisible by the denominator? When not so? How do you determine whether a vulgar fraction can be exactly divisible by a decimal? CIRCULATING OR REPEATING DECIMALS. 177 NOTE.-165. When there are no prime factors in the denominator other than 2 or 5, the division will always be exact, and she number of decimal places in the quotient will be equal to the greatest number of factors among the 2's or 5's. 7. What is the decimal corresponding to the fraction 2-7-? 8. What is the decimal corresponding to -? 9. What is the decimal corresponding to 1L7L? 166. The decimals which arise from vulgar fractions, where the division does not terminate, are called circulating decimals, because of the continual repetition of the same figures. The set of figures which repeats, is called a repetend. 167. A SINGLE REPETEND is one in which only a single figure repeats, as =.22222 +, or - =;3333 +. Such repetends are expressed by simply putting a mark over the first figure; thus,.2222+ is denoted by.2 +, and.3333 + by.3+. 168. A COMPOUND REPETEND has the same figures circulating alternately: thus 19.5757+ and -9 -=.57235723 + are compound repetends, and are distinguished by marking the first and last figures of the circulating period. Thus.5757 +. is written.57' +, and.57235723 + is written.5723'+. 169. A PURE REPETEND is an expression in which there are no figures except the repeating figures which make up the repetend; as.3 +,.5 +,.473 +, &c. 170. A MIXED REPETEND is one which has significant figures or ciphers between the repetend and the decimal QuEST. —165. If there are no prime factors in the denominator other than 2 and 5, will the division be exact? How many decimal places will there be in the quotient? 166. What are the decimals called when the division does not terminate? What is the set of figures which repeats called? 167. What is a single repetend? How is it expressed? 168. What is a compound repetend? How is it expressed? 169. What is a pure repetend? 170. What is a mixed repetend? 8* 178 REDUCTION OF CIRCULATING DECIMALS. point, or which has whole numbers at the left hand of the decimal point: such figures are called finite figures. Thus,.0 4-+,.0 733+-,.4 73' +,.3 573 +-, 6.5, and 4.375 + are all mixed repetends,.0,.4,.3, and 6 are the finite figures. 171. SIMILAR REPETENDS are such as begin at an equal distance from the decimal points; as.3 54 -+, 2.7 534 +. 172. DISSIMILAR REPETENDS are such as begin at different places from the decimal point; as.253 +,.4752 +. 173. CONTERMINOUS REPETENDs are such as end at the same distance from the decimal points; as.1f25/+,.354 +, &c. 174. SIMILAR AND CONTERMINOUS REPETENDS are such as begin and end at the same distance from the decimal point: thus, 53.2"753/+, 4.6'325 +, and.4632'+, are similar and conterminous repetends. REDUCTION OF CIRCULATING DECIMALS. CASE I. 175. To reduce a pure repetend to its equivalent vulgar fraction. Since -.1 +, and 3 =.3 +, and 5.54 +; and since all repetends may be placed under similar forms i therefore, to find the finite value of a pure repetend, Make the given repetend the numerator, and write a denominator containing as many 9's as there are places in the repetend, and this fraction reduced to its lowest terms will be the equivalent fraction sought. QUEsT.-What are such figures called? 171. What are similar repetends? 172. What are diisimilar repetends? 173. What are conterminous repetends? 174. What are similar and conterininous repetenis? 175 How do you reduce a pure repetend to its equivalent vulgar fraction? REDUCTION OF CIRCULATING DECIMALS. 179 EXAMPLES. 1. What is the equivalent vulgar fraction of the repetend 0.3+? Now, = - = 0.33333 +. = 0.3 +. 2. What is the equivalent vulgar fraction of the repetend.162'+? We have, - 62 = 8 Ans. 3. What are the simplest equivalent vulgar fractions of the repetends.6 +,.162 +, 0.769230'+,.945 +, and.09+? 4. What are the least equivalent vulgar fractions of the repetends.594405/+,.36 +, and.142857'+? CASE II. 176. TQ.reduce a mixed repetend to its equivalent vulgar fraction. A mixed repetend is composed of the finite figures which precede, and of the repetend itself; and hence its value must be equal to such finite figures plus the repetend. Hence, to find such value, To the finite figures add the repetend divided by as many 9's as it contains places of figures, with as many O's annexed to them as there are places of decimal figures which precede the repetend; the sum reduced to its simplestform will be the equivalent fraction sought. EXAMPLES. 1. Required the least equivalent vulgar fraction of the mixed repetend 2.4 18'+. Now, 2.418 + =2 + - + 1-+ =2 + -+99 — =2253 Ans. QUEST.-176. How do you reduce a mixed repetend to its equivalent vulgar fraction? 180 REDUCTION OF CIRCULATING DECIMALS. 2. Required the least equivalent vulgar fraction of the mixed.repetend.5925- +. We have,.5 925 + _= -0_.2s = 16 Ans. 3. What is the least equivalent vulgar fraction of the repetend.008 497133/+? We have,.008 497133 + = 100 + 999999000 4. Required the least equivalent vulgar fractions of the mixed repetends.13 8 +, 7.5 43 +,.04354 +, 37.54+,.6 75 +, and.7 54347 +. 5. Required the least equivalent vulgar fractions of the mixed repetends 0.75 +, 0.4 38'+,.093 +, 4.7 543 +,.009 87 +, and.4 5 +. 177.'here are some properties of the repetends which it is important to remark. 1. Any finite decimal may be considered as a circulating decimal by making ciphers to recur: thus,.35 =.35 0 + =.35 00'+ =.35 000+ =.350000'+, &c. 2. If any circulating decimal have a repetend of any number of figures, it may be reduced to one having twice or thrice that number of figures, or any multiple of that number. Thus, a repetend 2.3 57/+, having two figures, may be reduced to one having 4, 6, 8, or 10 places of figures. For, 2.3 57 + _ 2.3 575757+ = 2.3 575757 + = 2.3 57575757/+ &c.; so, the repetend 4.16316/+ may be written 4.16 316'+ = 4.16 316316'+ =4.16 316316316 + &c. &c., and the same may be shown of any other. Hence, two or more repetends, having a different number of places in each, may be reduced to repetends having the same number of places, in the following manner: QUEST. —177. What is the first property of the circulating decimals? How do you-reduce several repetends having different places in each, to repetends having the same number of places? REDUCTION OF CIRCULATING DECIMALS. 181 Find the least common multiple of the number of places in each repetend, and reduce each repetend to such number of places. 1. Reduce.138+, 7.543 +,.04354'+ to repetends having the same number of places. Since, the number of places are now 1, 2, and 3, the common multiple will be 6, and hence each new repetend will contain 6 places. Hence,.13 8 + =.13 888888 -+; 7.5 43'+ - 7.5 434343'+; 0.4'354'+ - 0.4 354354'+. 2. Reduce 2.4 18+-,.5925-'+,.008497133 + to repetends having the same number of places. 3. Any circulating decimal may be transformed into another having finite decimals and a repetend of the same number of figures as the first. Thus,.57 + -.5 75 + -.57 7'+ =.575 75'+-.5757757'; and 3.4 785 -- = 3.47 857/+ - 3.478 578 + - 3.4785 85~ +; and hence, any two repetends may be made similar. These properties may be proved by changing the repetends into their equivalent vulgar fractions. 4. Having made two or more repetends similar by the last article, they may be rendered conterminous by the previous one: thus, two or more repetends may always be made similar and conterminous. 1. Reduce the circulating decimals 165.164 +,.04 +,.03 7 + to such as are similar and conterminous. 2. Reduce the circulating decimals.5 3 +,.4 75 +, and 1:.757/+, to such as are similar and conterminous. 5. If two or more circulating decimals, having several QUEST.-When a repetend has more than one figure, may it be transformed into a circulating decimal having finite decimals? How many places must there be in the repetend? What are similar and conterminous repetends? May all circulating decimals be made simnilar and.cnterminols? 182 REDUCTION OF CIRCULATING DECIMALS. repetends of equal places, be added together, their sum will have a repetend of the same number of places; for every two sets of repetends will give the same sum. 6. If any circulating decimal be multiplied by any number, the product will be a circulating decimal having the same number of places in the repetend; for, each repetend will be taken the same number of times, and consequently must produce the same product. CASE III. 178. To find the number of places in the repetend corresponding to any vulgar fraction which cannot be expressed by a finite decimal. Let the fraction be first reduced to its lowest terms, after which find all the prime factors 2- and 5 of the denominator. Then separate the fraction into two factors, viz., 1st. The numerator divided by the product of all the prime factors 2 and 5; and 2d. Unity divided by the remaining factor of the denorninator. As an example, let us decompose the fraction 2 into the two factors named above. They'are, 3 3 1 280 2x2x2x5 7 If, now, we add a 0 to the 1 and proceed to make the division, every remainder will be less than the divisor, and hence we cannot make more divisions than there are units in the divisor less 1, without reducing the remainder to unity, when the first quotient figures will repeat. And observe carefully when any remainder becomes the same as a remainder previously used, for at this point the repeating figures begin. QUEST. —What is the fifth property named? What is the sixth? 178. What is the first operation in finding the form of the decimal corresponding to a given vulgar fraction? Into how many factors is it then divided? What are these factors? How many divisions may be performed in the second factor? REDUCTION OF CIRCULATING DECIMALS. 183 If, now, we suppose the remainder 1 to be subtracted from thle dividend so used, there would remain as many 9's as there were divisions. Hence, If, after' having taken out the 2's and 5's from the denominator, we divide a succession of 9's by the result until there is no remainder, the number of 9's so used will be equal to the number of places of the repetend, which can never exceed the number of units in the denominator less one. Having found the number of finite decimals which precede the repetend, and the number of places in the repetend, as above, Divide the numerator of the vulgar fraction, reduced to its lowest terms, by the denominator, and point off in the quotient the finite decimals, if any, and the repetend. EXAMPLES. 1. Required to find whether the decimal equivalent to 249 is finite or circulating; the number of places in the repetend and the place at which the repetend begins; and, also, the equivalent circulating decimal. We first reduce the fraction OPERATION. to its lowest terms. giving 7 3 2 49 9'768' 2'9304 I- -— f 96 We then search for the factors 2)9768 2 and 5 in the denominator, and 2)4884 find that 2 is a factor 3 times: hence we know that there are 1224 three finite decimals preceding the repetend. We next divide 1221)999999(819 99999, &c., by the factor 1221 3 =.008497133 - of the denominator, and find 7 - that we use six nines before the remainder becomes 0: hence, we know that there are six places of figures in the QUEsT.-What will determine the highest limit of the number of figures in the repetend? What will determine the number of finite decimals? How then will you find the equivalent decimal? 184 ADDITION OF- CIRCULATING DECIMALS. repetend. We then divide 83 by 9768, and point- off the proper piaces in the quotient. 2. Find the finite decimals, if any, and the repetend, if any, of the fraction 2 1 0 3. Find the finite decimals, if any, and the repetend, if any, of the fraction 11T-46 4. Find the finite decimals, if any, and the repetend, if any, of the fractions 1 2,~, 3 172 ADDITION OF CIRCULATING DECIMALS. 179. To add circulating decimals. Make the repetends similar and conterminous, and then write the places of like value under each other, and so many figures of the second repetends to the right as shall indicate with certainty how many are to be carried from one repetend to the other; after which add them as in whole numbers. If all the figures of a repetend be 9's, omit them and add one to the figure next to the left. EXAMPLES 1. Add.125 +, 4.163'+, 1.7143, and 2.54' together. Dissimilar. Similar. Similar and conterminous..125 + -.125 +.12s555555553555+.. 5555 4.163'+ -4.16316'+ =4.16'316316316316'+.. 3163 1.7143+ -=1.714371+ - 1.71 437143714371'+.. 4371 2.54 + =2.54 54 + =2.54545454545454 +.. 5454 The true sum =8.54854470131697- +, lto carry. 2. Add 67.3 45+, 9.651'+,.25 +, 17.47 +,.5 + together. 3. Add.475'+, 3.7543'+, 64.75 +,.57 +,.1788'+ together. QUEST -179. How do you add circulating decimals? SUBTRACTION OF CIRCULATING DECIMALS. 185 4. Add.5 -, 4.3 7 +, 49.4 57+,.4 954 +,.7345 + together. 5. Add.175 +, 42.57 +-,.3753 +,.5 945'+. 3.7 54 + together. 6. Add 165,.164 +, 147.04 +, 4.95 +, 94.37 +, 4.7123456'+ together. SUBTRACTION OF CIRCULATING DECIMALS. 180. To subtract one finite decimal from another. Make the repetends similar and conterminous, and subtract as in finite decimals, observing that when the repetend of the lower line is the largest its first right hand figure must be increased by unity. EXAMPLES. 1. From 11.4"75'+ take 3.45735~+. Dissimilar. Similar. Similar and conterminous. 11.4 75'+ = 11.47 57'+ = 14.47 575757/'... 575 3.45735'+ = 3.45 735'+ = 3.45 735735/+... 735 The true difference = 8.01_840021/+, 1 to carry. 2.. From 47.53 + take 1.757/+. Ans. 3. From 17.573 - take 14.57 -. Ans. 4. From 17.4 3 + take 12.34 3 +. Ans. 5. From 1.12 754/+ take.4"7.384/+. Ans. 6. From 4.75 take.375 -. Ans. 7. From 4.794 take.1744/'+. Ans. - 8. From 1.457 + take.3654. Ans. 9. From 1.4'937/'+ take.1475. Ans. QuEsT.-180. How do you subtract circulating decimals? 186 MULTIPLIC TION OF CIRCULATING DECIMALS MULTIPLICATION OF CIRCULATING DECIMALS. 181. To multiply one circulating decimal by another. Change the circulating decimals into their equivalent vulgar fractions, and then multiply them together; after which reduc6 the product to its equivalent circulating decimal, as in Art. 178. EXAMPLES. 1. Multiply 4.25 3 + by.257. OPERATION. 4.25 3 - - 4.To 9 00 4 +225 900 9-00 _1914 - 957 - 450 - 225' Also,.257 = - 2057: hence, 957 257 245949 1.09310 6 + and since 225000 = 5 X 5 X 5 X 5 X 5 X 2 X 2 X 2 X 9, there will be five places of finite decimals, and one figure in the repetend (Art. 178). NOTE. Much labor will be saved in this and the next rule by keeping every fraction in its lowest terms, and when two fractions are to be multiplied together, cancelling all the factors common to both terms before making the multiplication. 2. Multiply.375 4 + by 14.75. Ans. —3. Multiply.4253 + by 2.57. Ans. 4. Multiply.437 by 3.7 5 +. Ans. 5. Multiply 4.573 by.3 75t+. Ans. 6. Multiply 3.45 6 + by.42 5 +. Ans. - 7. Multiply 1.456'+ by 4.2:3 +. Ans. 8. Multiply 45.1 3 + by.245 +. Ans. 9. Multiply.4705 3 + by 1.7 35 +. Ans. 10. Multiply 3.4573 + by 54.753 +. Ans. - QuEsT.-181. How do you multiply circulating decimals? What is to be observed in regard to keeping fractions in their lowest terms? DIVISION OF, CIRCULATING DECIMALS. 187 DIVISION OF CIR.CULATING DECIMALS. 182. To divide one circulating decimal by.another. Change the decimals into their equivalent vulgar fractions, andfind the quotient of these fractions. Then change the quotient into its equivalent decimal, as in Art. 178. EXAMPLES. 1. Divide 56.6 + by 137. OPERATION. 56.6 + = 56 + - o = o. Then, Ao —. 137 0 =.41362530_+. 2. Divide 24.3 18 + by 1.792. Ans. 3. Divide 8.5968 by.2 45 +. Ans. 4. Divide 2.295 by.297/+. Ans. 5. Divide 47.345 by 1.76'+. Ans. 6. Divide 13.5 169533'+ by 4.297 +. Ans. 7. Divide.45 + by.118881 +. Ans. 8. Divide.475/+ by.3753/+. Ans. 9. Divide 3.753' by.24'+. Ans. - QUEsT.-182. How do you divide circulating decimals? 188 RATIO AND PROPORTION OF NUMBERS. OF TIHE RATIO AND PROPORTION OF NUMBERS 183. Two numbers having the same unit may be compared together in two ways. 1st. By considering how much one is greater or less than the other,'which is shown by their difference; and 2d. By considering how many times one is greater or less than the other, which is shown by their quotient. Thus, in comparing the numbers 3 and 12 together with respect to their difference, we find that 12 exceeds 3 by 9; and in comparing them together with respect to their quotient, we find that 12 contains 3 four times, or that 12 is four times as great as 3. The quotient which arises from dividing the second number by the first, is called the ratio of the nutfibers, and shows how many times the second number is greater than the first, or how many times it is less. Thus, the ratio of 3 to 9 is 3, since 9 - 3 = 3. The ratio of 2 to 4 is 2, since 4 4 2 = 2. We may also compare a larger number with a less. For example, the ratio of 4 to 2 is 2-; for, 2 - 4 = ~.'IThe ratio of 9 to 3 is 1, since 3 - 9 EXAMPLES. 1. What is the ratio of 9 to 18? Ans. 2. What is the ratio of 6 to 24? Ans. QUEST.-183. In how many ways may two numbers having the same unit be compared? How do you determine how much one number is greater than another? How do you determine how many times it is greater or less? How much does 12 exceed 3? How many times is 12 greater than 3? What is the quotient called which arises from dividing the second number by the first? What does it show? When the second number is the least, what does it show? RATIO AND PROPORTION OF NUMBERS. 189 3. What is the ratio-of 12 to 48? Ans. 4. What is the ratio of 11 to 13? Ans. 5. What part of 20 is 4? Or what is the ratio of 20 to 4? 6. What part of 100 is 30? Or what is the ratio of 100 to 30? 7. What part of 6 is 3? Ans. 8. What part of 9 is 3? Ans. 9. What part of 12 is 4? Ans. 10. What part of 50 is 5? Ans. 11. What part of 75 is 3? Ans. - NOTE.-In determining what part one number is of another, it is plain that the number to be measured, must be written in the numerator; while the standard, or number with which it is compared, and of which it forms a part, is written in the denominator. This fraction; reduced to its lowest terms, will express the part. 184. If one yard of cloth cost $2, how many dollars will 6 yards of cloth cost at the same rate? It is plain that 6 yards of cloth will cost 6 times as much as one yard; that is, the cost will contain $2 as many times as 6 contains 1. Hence the cost will be $12. In this example there are four numbers considered, viz., 1 yard of cloth, 6 yards of cloth, $2, and $12: these numbers are called terms. 1 yard of cloth is the 1st term, 6 yards of cloth is the 2d term, $2 is the - - - 3d term, $12 is the - - - 4th term. Now the ratio of the first term to the second is the same as the ratio of the third to the fourth. This relation between four numbers is called a proportion; and generally Four numbers are said to be in proportion when the ratio of QUEST.-How do you determine what part one number is of another? 184. If one yard of cloth cost $2, what will 6 yards cost? Htow many numbers are here considered? What are they called? What is the ratio of the first to the second equal to? What is this relation between numbers called? When are four numbers said to be in proportion? 190 RATIO AND PROPORTION OF NUMBERS. the first to the.second is the same as that of the third to the fourth. Hence, A PROPORTION is an equality of ratios between numbers compared together two and two. 185. We express that four numbers are in proportion thus: 1 6:: 2: 12. That is, fe write the numbers in the same line and place two dots between the 1st and 2d terms, four between the 2d and 3d, and two between the 3d and 4th terms. We read the proportion thus, as 1 is to 6, so is 2 to 12. The 1st and 2d terms of a proportion always express quantzties of the same kind, and so likewise do the 3d and 4th terms. As in the example, yd. yd. $ $ 1 ~ 6: 2 12. This is implied by the definition of a ratio; for, it is only quantities of the same kind which can be divided the one by the other. The ratio of the first term to the second, or of the third to the fourth, is called the ratio of the proportion. 1. What are the ratios of the proportions 3: 9:: 12 36? Ans. 2: 10:: 12 60? Ans. 4 2:: 8:. 4? Ans. 9 1:: 90 10? Ans. 16: 15:: 48 45? Ans. 186. When two numbers are compared together, the first is called the antecedent, and the second the consequent; and when four numbers are compared, the first antecedent and consequent are called the first couplet, and the second antecedent and consequent the second couplet. Thus, in the last QUEST.-How do you define proportion? 185. How do you indicate that four numbers are in proportion? How is the proportion read? What do you remark of the first and second terms? Also of the third and fourth? 186. When two numbers are compared together, what is the first called? What the second? When four numbers are compared, what are the two first called? What the two second? RATIO AND PROPORTION OF NUMBERS. 191 proportion, 16 and 48 are the antecedents, and 15 and 45 the consequents; also, 16 and 15 make the first couplet, and 48 and 45 the second. 187. We have said that proportion is an equality of ratios (Art. 184). Besides the method above, we may express that equality thus: 4 6 2 3' and we may then write the proportion thus: 2 4:: 3 6. Put the following equal ratios into proportion. 8 16 21 105 i. 5. - s 9 18' 16 80 17 19 42 252 2' 51 = 57- -{' 35 = 210 9 27 29 232 16 — 4' — 37 - 296' 19 76 45 405 413 52 23- 207' 188. If 41b. of tea cost $8, what will 121b. cost at the same rate? OPERATION. lb. lb. $ $ As 4: 12:: 8: Ans. 12 12 4)6 1 x 8 = 3 X 8 =-24. $24 the cost of 121b. of tea. Ans.'$24. It is evident that the 4tli term, or cost of 121b. of tea, must be as many times greater than $8, as 1216b. is greater than 41b. But the ratio of 41b. to 121b. is 3; hence, 3 is the number of times which the cost exceeds $8: that is, the cost is Qua sT.-187. What has proportion been called? By what second method rmy this equality be expressed? 188. Explain this example mentally. 192 RATIO AND PROPORTION OF NUMBERS. equal to $8 x 3 = $24. But instead of writing the numbers 12 12 x 8 = 24, we may write them (12 X 8)- 4 = 24: and as the same may be shown for every proportion, we conclude, That the 4th term of every proportion may be found hy multiplying the 2d and 3d terms together, and dividing their product by the 1st term. EXAMPLES. 1. The first three terms of a proportion are 1, 2, and 3: what is the fourth? Ans. 2. The first three terms are 6, 2, and 1: what is the 4th? Ans. 3. The first three terms are 10, 3, and 1: what is the 4th? A n7s. 189. The 1st and 4th terms of a proportion are called the two extremes, and the 2d and 3d terms are called the two means. Now, since the 4th term is obtained by dividing the product of the 2d and 3d terms by the 1st term, and since the product of the divisor by the quotient is equal to the dividend, it follows, That in every proportion the product of the two extremes is equal to the product of the two means. Thus, in the example, Art. 184 we have 1: 6::: 12; and 1 X 12 = 2 x 6; also, 4: 12:: 8: 24; and 4 x 24 = 12 X 8; " 6: 9:: 10 15; and 6 X 15 = 9 X 10; " 7: 15:: 14: 30; and 7 X 30 = 15 x 14. QUEST. —How may the fourth term of every proportion be found? 189. What are the first and fourth terms of a proportion called? What are the second and third terms called? In every proportion, what is the product of the extremes equal to? OF CANCELLING. 193 OF CANCELLING. 190. When one number is to be divided by another, the operation may often be shortened by striking out or cancelling the factors common to both, before the division is made. 1. For example, suppose it were required to divide 360 by 120. We first write the i OPERATION. dividendkabove a ho- 1 360 12 x 30 /,Z x 3 x ff. rizontal line, and the 120 12 x 10,1- x divisor beneath it, after the form of a fraction. We next separate both of them into factors, and then cancel the factors which are alike. 2. Divide 630 by 35. We separate the divi- OPERATION. dend and divisor into like 630 3 X x/ 6 X / factors, and then cancel 35 -- = 18. those which are common in both. 3. Divide 1860 by 36. 7. Divide 1768 by 221. 4 Divide 7920 by 720. 8. Divide 2856 by 238. 5. Divide 1890 by 210. 9. Divide 3420 by 285. 6. Divide 1260 by 504. 10. Divide 9072 by 1512. 191. If two or more numbers are to be multiplied together and their product divided by the product of other numbers, the operation may be abridged by striking out or cancelling any factor which is common to the dividend and divisor. For example, if 6 is to be multiplied by 8 and the product divided by 4, -we have 6x8 48 6x8 64 x =48 12; or, 6 X 2 = 12: 4 4 4 QuEnT. —190. How may the division of two numbers be often abridged? Explain the example mentally. Also the second example. 191. When two numbers are multiplied together and their product divided by a third. how may the operation be abridged? 9 194 OF CANCELLING. in the latter case we cancelled the factor 4 in the numer~ or and denominator, and multiplied 6 by the quotient 2. 1. Let it be required to multiply 24 by 16 and divide the product by 12. v OPERATION. Having written the product of the fig- 2 ures, which form the dividend, above 92A X 16 the line,'and the product of the figures -, -32. which form the divisor below it, then 1 We cancel the common factors in the numerator and denominator, and write the quotients over and under the numbers in which such common factors are found, and if the quotients stib have a common factor, they Pray be again divided. 2. Reduce the compoun0 fraction 4 of 6 of 3 of 56 to a simple fraction. Beginning with the first nu- OPERATION. merator, we find it is once a 1 I 1 1 factor of itself and 4 times in A,3' f 16; 6 is twice a factor in 15; a ] X 2'A X 3 3 three times a factor in 9; 1 3 2 4 and 5, once a factor in the denominator 5. 3. What is the product of 3 X 8 X 9 X 7 x 15 divided by 63 x 24 x 3 X 5? This example presents a i OPERATION. case that often arises, in 2' x Xf x,9' x i X which the product of two - X X 1. factors may be cancelled. Thus, 3 x 8 is 24: then cancel the 3 and 8 in the numerator and the 24 in the denominator. Again, 9 times 7 are 63; therefore cancel the 9 and 7 in the numerator and the 63 in the denominator. Also, 3 X 5 in the denominator cance s the 15 remaining in the numerator: hence, the quotient is unity. 4. What is the product of 126 X 16 X 3 divided. by 7'< 12 OF CANCELLING. 195 We see that 7 is a factor of OPERATION. 126, giving a quotient 18, which 3 we place over 126, crossing at the P~ 8 same time 126 and the 7 below. I X (fi X 3 72 We then divide 18 and 12 by 6, = 72. crossing them both and writing 1 down the quotients 3 and 2. We next divide 16 and 2 by 2, giving the quotients 8 and 1. Hence, the result is 72. EXAMPLES. 1. What is the product of 1 x 6 x 9 X 14 X 15 X 7 X 8 divided by 36 x 128 X 56 X 20? 2. What is the value of 18 X 36 X 72 X 144 divided by 6 x 6 x 8 x 9 x 12 x 8? 3. What is the product of 3 X 9 X 7 X 3 X 14 X 36 divided by 252 x 81 X 2 x 7? 4. What is the product of 19 X 17 X 16 X 8 X 9 X 6 divided by 32 x 4 x 27 x 2? 5. What is the product of 4 X 12 x 16 X 30 x 16 X 48 x 48 divided by 9 X 10 x 14 x 24 X 44 X 40? 192. The process of cancelling may be applied to the terms of a proportion. If we have any proportion, as 6: 15:: 28 70, We may always cancel like factors in either couplet. Thus, 2 5 14 35 6 15: 28: 70, in which we divide the terms of the first couplet by 3, and those of the second by 2, and write the quotients above. EXAMPLES. 1. What is the simplest form of 18: 72:: 100: 400? 2. What is the simplest form of 14: 49:: 42: 147? 3. What is the simplest form of 365: 876:: 140: 336? QUErST. —192.- How else may the process of cancelling be applied? What may be cancelled in each couplet? 196 RuLE OF THR]EE. RULE OF THREE. 193. THE Rule of Three takes its name from the circumstance that three numbers are always given to find a fourth, which shall bear the same proportion to one of the given numbers as exists between the other two. The following is the manner of finding the fourth term: I.- Reduce the two numbers which have different names from the answer sought, to the lowest denomination named in either of them. II. Set the number which is of the same kind with the answer sought in the third place, and then consider from the nature of the question whether the answer will be greater or less than the third term. III. When the answer is greater than the third term, write the least of the remaining numbers in the first place, but when zt is less place the greater there. IV. Then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term or answer sought, and will be of the same denomination as the third term. EXAMPLES. 1. If 48 yards of cloth cost $67,25 what will 144 yards cost at the same, rate? QuEsr.-193. From what does the Rule of Three take its name? What is the first thing to be done in stating the question? Which number do you make the third term? How do you determine which to put in the first? After stating the question, how do you find the fourth term? What will be its denomination 7 RULE OF THREE. 197 In this example, as the OPERATION. answer is to be dollars, yd. yd. $ $ we place the $67,25 in 48: 144:: 67,25: Ans the third place. Then, as 144 144 yards of cloth will 26900 cost more than 48 yards, 26900 6725 the fourth term must be 6 greater than the third, and 48)9684,00($201,75 therefore, we write the least of the two remain- 84 ing numbers in the first place. The product of 360 the second and third terms is $9684,00: dividing this 240 by the first term, we obtain $201,75 for the cost of 144 yards of cloth. 2. If 6 men can dig a certain ditch in 40 days, how many days would 30 men be employed in digging it? As the answer must be days, OPERATION. the 40 days are written in the men men. days days third place. Then, as it is 30: 6:: 40: Ans. evident that 30 men will do I 6 the same work in a shorter 310)2410 days. time than 6 men, it is plain Ans. 8 days. that the fourth term must be less than the third; therefore, 30 men, the greater of the remaining numbers, is taken as the first term. Besides, it is plain that the fourth term must be just so many times less than 40, as 6 is less than 30. 3. If 25 yards of cloth cost ~2 3s. 4d., what will 5 yards cost at the same rate? QUEsT.-In the first example which is greater, the third or fourth term? Which number must then be in the first term? How many times will the fourth term be greater or less than the third? 198 RULE OF THREE. OPERATION. When we come to yd. yd. X s. d. divide the product of 25: 5:: 2 3 4 Ani the second and third 1 5 terms by the first, it is 25).10 16s. 8d. found the ~ 10 does not 20 contain 25. We then 25)216(8s. reduce to the next low- 200 er denomination, and 16 divide as in division of 12 denominate numbers. 25)200(8d. 200 Ans. 8s. 8d. 4. If 3cwt. of sugar cost ~9 2s. Od., what will 4cwt. 3qr 261b. cost at the same rate? 3cwt. 4cwt. 3qr. 261b. 4 4 ~9 2s. Od. 12 19 20 f 7 7 182s. 4X7= 28 84 133 12 4 4 2184 3361b.: 5581b. 2184d.: Ans. 558 17472 We first reduce the 10920 first and second terms 10920 to pounds, then the 336)1218672(3627d. third term to pence. 1008 The answer comes out I 2106 12)3627 in pence, and is af- 2016 20) 302s. 3d. terwards reduced to 907 ~15 2s. pounds, shillings, and 672 pence. 2352 2352 Ans. ~15 2s. 3d. RUJLE OF THREE. 199 PROOF. 1 ~ 1. The product of the two means is equal to the product of the extremes (Art. 189). Hence, if either of these equal products be divided by one of the mean terms the quotient will be the other. Therefore, Divide the product of the extremes by one of the mean terms, and if the work is right the quotient will be the other mean term. EXAMPLES. 1. The first term is 4, the second 8, the third 12, and the answer 24: is the answer true? The product of the extremes OPERATION OF PROOF. is 96. If this be divided by 8 24 X 4 = 96 the quotient is 12; if by 12 the 8)96(12; quotient is 8: hence, the an- t or, 12)96(8 swer is right'. APPLICATIONS. 1. If 8 hats cost $24, what will 110 cost at the same rate? 2. What is the value of 4cwt. of sugar at 5d. per pound? 3. If 80 yards of cloth cost $340, what will 650 yards cost? 4. If 120 sheep yield 330 pounds of wool, how many pounds will be obtained from 1200? 5. If 6 gallons of molasses cost $1,95, what will 6 hogsheads cost? 6. If 16 men perform a piece of work in 24 days, how many men would it take to perform the work in 12 days? 7. Suppose a cistern has two pipes, and that one can fill it in 8~ hours, the other in 4-: in what time can both fill it together? 8. If a man travels at the rate of 630 miles in 12 days, how far will he travel in a year, supposing him not to travel on Sundays? QUEST. —194. How do you prove the Rule of Three? 200 RULE OF THREE. 9. If 2.yards of cloth cost $3,25, what will be the cost of 3 pieces, each containing 25 yards? 10. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 25 men? 11. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 200 men? 12. If 50 persons consume 600 bushels of wheat in a year, how much will they consume in 7 years? 13. What will be the cost of a piece of silver weighing 731b. 5oz. 15pwt., at 5s. 9d. per ounce? 14. If the penny loaf weighs 8 ounces when the bushel of wheat costs 7s. 3d., what ought it to weigh when the wheat is 8s. 4d. per bushel? 15. If one acre of land costs ~2 15s. 4d., what will be the cost of 173A. 2R. 14P. at the same rate? 16. A gentleman's estate is worth ~4215 4s. a year: what may he spend per day and yet save ~1000 per annum? 17. A father left his son a fortune, { of which he ran through in 8 months, - of the remainder lasted him 12 months longer, when he had barely ~820 left: what sum did his father leave him? 18. There are 1000 men besieged in a town with provisions for 5 weeks, allowing each man 16 ounces a day. If they are reinforced by 500 more and no relief can be afforded till the end of 8 weeks, how many ounces must be given daily to each man? 19. A father gave 7 of his estate to one son, and 7 of the remainder to another, leaving the rest to his widow.:lhe difference of the children's legacies was ~514 6s. 8d.: what was the widow's portion? 20. If 14cwt. 2qr. of sugar cost $129,92, what will be the price of 9cwt.? 21. The clothing of a regiment of foot of 750 men amounts to ~2831 5s.: what will it cost to clothe a body of 10500 men? RULE OF THREE. 201 22. How many yards of carpeting, that is 3 feet wide, will cover a floor that is 40 feet long and 27 feet broad? 23. After laying out I of my money, and I of the remainder, I had 114 guineas left: how much had I at first? 24. A reservoir has. three pipes, the first can fill it in 24 days, the second in 22 days, and the third can empty it in 28 days: in what time will it be filled if they are all running together? 25. If the freight of 80 tierces of sugar, each weighing 3-cwt., 150 miles, cost $84, what must be paid for the freight of 30hhd. of sugar, each weighing 12cwt., 50 miles? 26. If 1500 men require 45000 rations of bread for a month, how many rations will a garrison of 3600 men require? 27. The quick step in marching is 2 paces per second, at 28 inches each: at what rate is that per hour, and how long will a troop be in reaching a place 60 miles distant, allowing a halt of an hour and a half for refreshment? 28. Two persons A and B are on the opposite sides of a wood which is 536 yards in circumference; they begin to travel in the same direction at the same moment; A goes at the rate of 11 yards per minute, and B at the rate of 34 yards in 3 minutes: how many times must the quicker one go round the wood before he overtakes the slower? 29. Two men and a boy were engaged to do a piece of work, one of the men could do it in 10 days, the other in 16 days, and the boy could do it in 20 days: how long would it take the three together to do it? 30. A certain amount of provisions will subsist an army of 9000 men for 90 days. If the army be increased by 6000, how long will the same provisions subsist it? 31. Four thousand soldiers were supplied with bread for 24 weeks, each man to receive 14oz, per day; but by some accident 210 barrels containing 2001b. each were spoiled: what must each man receive in order that the remainder may last the same time? 9* 202 RULE OF TiTIREE. 32. Let us suppose the 4000 soldiers having one-fourteenth of their bread spoiled, to be put on an allowance of 13oz. of bread per day for 24 weeks: required the weight of their bread, good and spoiled, and the amount spoiled. 33. If 56 yards of cloth cost 40 guineas, how many ells Flemish can be bought for ~ 1135 1Os.? 34. If a pack of wool weighs 2cwt. 2qr. -141b., what is it woith at 17s. 6d. per tod? 35. A merchant bought a quantity of broadcloth and baize for ~124; there was 117~ yards of broadcloth at 17s. 9d. per yard; for every 5 yards of broadcloth he had 1~ yards of baize; how many yards of baize did he buy, and what did it cost him per yard? 36. If - of a pole stands in the mud, 1 foot in the water, and -5 in the air, or above the water, what is the length of the pole? 37. A bankrupt's effects amount to 1000~ guineas. His debts amount to ~2547 14s. 9d.: what will his creditors receive in the pound? 38. If 12 dozen copies of a certain book cost $54,72, what will 297 copies cost at the same rate? 39. Suppose 4000 soldiers after losing 210 barrels of bread, each containing 2001b., were to subsist on 13oz. each a day for 24 weeks; had none been lost they might have received 14oz. a day: what was the whole weight, and how much did they receive? 40. Let us now suppose 4000 soldiers to lose one-fourteenth of their bread, then to receive 13oz. each a day for 24 weeks: what was the whole weight of their bread including the lost, and how much would each have received per day had none been spoiled? 41. Provisions in a garrison were sufficient for 1800 men for 12 months; but at the end of 3 months it was reinforced by 600 men, and 4 months after that a second reinforcement of 400 men was sent in. How long did the provisions last? RULE OF THREE BY ANALYSIS. 203 RULE OF THREE BY ANALYSIS. 195. The solution of questions in the Rule of Three by analysis consists in finding the ratio of two of the given terms, and multiplying this ratio by the other terin. The ratio of two of the terms will generally express the value or cost of a single thing. EXAMPLES. 1. If 3 barrels of flour cost $24, what will 7 barrels cost? By dividing the $24 by 3 we get the OPERATION cost of 1 barrel. For, if $24 will buy 3 3)24 barrels, it is plain that - of it will buy 1 8 barrel. This, multiplied by 7, gives $56 8 x 7 = 56 the cost of 7 barrels. I Ans. $56. 2. If in 20 days a man travels 58 miles, how far will he travel in 30 days? 3. If 6 men consume 1 barrel of flour; in 30 days, how much would 48 men consume? It is evident that I of a barrel would be- [ oPERATION. 6 the amount consumed by 1 man: hence, X 48 = 8. 48 times I is the amount consumed by 48 i Ans. 8. men. 4. If 2 barrels of flour cost $13, what will 12 barrels cost? 5. If I walk 168 miles in 6 days, how far should I walk at the same rate in 18? 6.'If 81b. of sugar cost $1,28, how much will 131b. cost? What is 16 x 13? 7. If 3 of a piece of cloth cost $8,25, what will 9 cost? 8. If 300 barrels of flour cost $570, what will 200 cost? What is 3 X 570? 9. If 87 of a barrel of cider cost -9 of a dollar, what will 3 cost? What is 2 X -9? ~ QUEST.-195. In what does the solution of questions by analysis consist? What does the ratio of the two terms express? If this ratio be multiplied by the other term, what will be the product? 04 RULE OF TIHEE BY ANALYSIS. 10 If 90 bushels of oats will feed 40 horses for 6 days, how long would 450 bushels feed them? 11. If 5 oxen, or 7 colts, eat up a certain quantity of grass. in 87 days, in what time will 2 oxen and 3 colts eat up the same quantity of grass? 12. A person's income is ~146 per annum: how much is that each day? 13. If 3 paces of common steps be equal to 2 yards, how many yards will 160 paces make? 14. A certain work can be done in 12 days by working 4 hours each day: how long would it require to do the work by working 9 hours a day? 15. A pasture of a certain extent having supplied a body of horse, consisting of 3000, with forage for 18 days, how many days would the same pasture have supplied a body of 2000 horse? 16. The governor of a besieged city has provisions for 54 days, at the rate of 21b. of bread per day, but is desirous of prolonging the siege to 80 days, in expectation of succor: in that case what must be the allowance of bread per day? 17. If a person pays half a guinea a week for his board, how long can he be boarded for ~21? 18. If a person drinks 80 bottles of wine per month, when it costs 2s. per bottle, how much can he drink, without increasing the expense, when it costs 2s. 6d. per bottle? 19. How long will a person be in saving ~100, if he saves is. 6d. per week? 20. A merchant bought 21 pieces of cloth, each containing 41yards, for which he paid $1260; he sold the bloth at $1,75 per yard: did he gain or lose by the bargain? 21. A grocer bought a puncheon of rum for ~41 14s. 6d., to which he added as much water as reduced its cost to 5s. 6d. per gallon; how much water did he put in? 22. If one pound of tea be equal in value to 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound of tea when a lemon is worth two cents? RULE OF TIIREE BY CANCELLING. 205 RULE OF THREE BY CANCELLING. 196. The cancelling process may be applied to all ques tions in the Rule of Three, where the second or third terms have a factor common to the first. Let the second and third terms be written above the line, with the sign of multiplication between them, and the first term below it. and then cancel the common factors. EXAMPLES. 1. If 48 yards of cloth cost $67,25, what will 144 yards cost? The process here is obvious, OPERATION. neing entirely similar to that 6725 X3 explained in Art. 191. 48 201X75. 2. If 25 yards of cloth cost ~2 3s. 4d., what will 5 yards cost? 3. If 24 hats cost $120, how much will 80 cost? 4. If 90 barrels of flour will subsist 100 men for 120 days, how long will it subsist 75? 5. If 60 sheep yield 1801b. of wool, how many pounds will be obtained from 900? 6. If a man travel 210 miles in 6 days, how far will he travel in 120 days? 7. If the freight of 40 tierces of sugar, each weighing 3~cwt.,- 150 miles, cost $42, what must be paid for the freight of 10 hogsheads, each weighing 12cwt., 50 miles? 8. A certain amount of provisions will subsist an army of 9000 men for 90 days: if the army be increased by 4500, how long would the same provisions subsist it? 9. If 50 persons consume 600 bushels of wheat in one year, how much will 278 persons consume in 7 years? 10. If 3 yards of cloth cost 18s., what will 24 yards cost? QUEST. —196. To what questions may the cancelling process be applied? How are the numbers written? What factors do you cancel? 206 RULE OF THREE BY CANCELLING. 11. If 112 pounds of sugar cost 56s., what will 1 pound cost? 12. If 4 men can do a piece of work in 80 days, how many days of the same length will 16 men require to do the same work? 13. If 21 pioneers make a trench in 18 days, how many days of the same length will 7 men require to make a similar trench? 14. If a field of grass be mowed by 10 men in 12 days, in how many days would it be mowed by 20 men? 15. A certain piece of grass was to be mowed, by 20 men in 6 days; an extraordinary occasion calls off half the workmen. It is required to find in what time the rest will finish it. 16. If the penny loaf weighs 5oz. when flour is 2s. a peck, what should it weigh when flour is sold for 2s. 6d. a peck? 17. Provisions in a garrison are found sufficient to last 1800 soldiers for three months; but a reinforcement being wanted that the provisions may last for one month only, what number of soldiers must be added to the garrison? 18. If 3yd. 2qr. of cloth of lyd. 3qr. wide will make a suit of clothes, how many yards of stuff of 3qr. wide will make a suit for the same person? 19. If I lend my friend ~200 for 12 months on condition of his returning the favor, how long ought he to lend me ~150 to requite my kindness? 20. If an acre be 220 yards long, the breadth will be 22 yards; but if the breadth of an acre be 40 yards, what then will be the length? 21. How mapy pounds of sugar at 12d. per pound are equal in value to 241b. of tea, worth 9s. 6d. per pound? 22. A tax of ~225 10s. was laid upon four villages A, B, C, and D; it had been the custom with these villages, that whenever any taxes were to be levied, as often as A, B, and C paid each 3d., D paid only 2d.: how much did each village pay? EXAMPLES INVOLVING FRACTIONS. 207 EXAMPLES INVOLVING FRACTIONS. 1. If 3 of a yard of cloth cost $3,20, what will 2- yards cost? We state the ques- OPERATION. tion exactly as in 3. 21 3,20: Ans. whole numbers. In 21 multiplying the sec- 6,40 ond and third terms by multiplying by ~ 1,60 together, we observe 8,00 the rules for multi- 8,00 = 8,00 X 3 = 6430 plying fractions, and = $21,33 +. in dividing by the first term, the rules for division. Thus, in this example, we invert the terms of the divisor and multiply. 2. If 5oz. cost ~X-, what will l -oz. cost? Ans. 3. If -13 of -a ship cost ~273 3s. 6d., what will 32 of her cost? 4. If 375 yards of cloth cost.164 9d. what will 257~ yards cost? 5. If 14 yards of cloth can be bought for 10 guineas, how many ells Flemish can be bought for ~283.875? 6. If i]oz. of plate cost 10s. 11~-d., what will a service weighing 327.61875oz. cost? 7. If 142lb. of sugar cost $13, what will 121b. cost? 8. If 4 of a yard of cloth cost $1-, what will 7- yards cost? 9. If 2 men can do 125 rods of ditching in 65 days, in how many days can 18 men do 24254 rods? 10. If a wedge of gold weighing 1731b. troy, be worth ~6795-, what is the value of l- 3qr. of that gold? 11. If the carriage of 2.5 tons of goods 2.9 miles cost 0.75 guinea, what is that per cwt. for a mile? 12. If 1cwt. of tobacco cost ~4 18s., how much may be bought for ~7 7 208 EXAMPLES INVOLVING FRACTIONS 13. If 14I yards of cloth cost $191, how much will 393 yards cost? 14. If.3 of a house cost $201.5, what would.95 cost? 15. A man receives 3 of his income, and finds it.equal to $3724.16: how much is his whole income? 16.. If 3.5 yards of cloth cost ~2 14s. 3d., what will 27.75 yards cost at the same rate? 17. If 12 men and a boy can perform a piece of work in.100 days, the boy doing ~ as much work as one man, in how many days will 20 men perform the same? 18. A mercer bought 10- pieces of silk, each containing 244 yards; he paid 6s. ~d. per yard: what does the whole come to? 19. If 41b. of beef cost 3 of a dollar, what will 601b. cost? 20. If a traveller perform a journey in 35.5 days when the days are 13.625 hours long; in how many days of 11.9 hours would he perform the same journey? 21. If 5400 bricks be required to pave a yard, when the bricks are.5 foot long and.25 broad, how many will be required of.75 foot long and 3- foot broad? 22. A man with his family, consisting of 5 persons, usually drink 7.8 gallons of beer in a week: how much would they drink in 23.5 weeks, if the family was to be increased by 3 persons? 23. If 248 men in 601 hours dig a trench containing 13924- solid yards of earth, how long would it take the same number of men to dig a similar trench containing 26460 solid yards of earth? 24. The earth turns round on its axis from west to east in 23 hours 56 minutes, and the circumference of every circle on its surface is supposed to be divided into 360 degrees. At the equator a degree is 69.07 miles; at Madras, Barbadoes, &c., 67.21 miles; at Madrid, Philadelphia, &c., 52.85 miles; and at Petersburg, 34.53 miles. How many miles per hour are the inhabitants in each of these places carried from west to east by the revolution of the earth on its axis? QUESTiONS REQUIRING TWO STATEMENTS. 209 OF QUESTIONS REQUIRING tWO STATEMENTS. 197. The answer to each of the questions heretofore considered, has been found by a single statement. Questions, however, frequently occur in which two or more statements will be necessary, if the question be resolved by the principles above explained. EXAMPLES. 1. If a family of 6 persons expend $300 in 8 months, how much will serve a family of 15 persons for 20 months? First question. If 1ST OPERATION. $300 will support a perbsons. persons. $ $ family of 6 persons BY CANCELLING. for 8 months, how 2 5::300 Ans. 5 many dollars will sup- 2)1500 port 15 persons for 2)1500 the same time? Ans. $750. Second question. If 2D OPERATION. $750 will support a months. months. $ $ family of 15 persons 2 5 750 Ans. for 8 months, how I much will serve them 2)3750 for 20 months? Ans. $1875 2*. If 16 men build 18 feet of wall in 12 days, how many men must be employed to build 72 feet in 8 days, working at the same rate? The first question OPERATION. is, how long would it feet. feet. days. days. I 4 12 Ans. take the 16 men to 4: 12: Ans. build the 72 feet of ~~~~~wall ~~? i48 days. It is evident that 18 feet of wall, is to 72 feet, as 12 days, QUEST.-197. How many statements have been necessary in the questions heretofore considered? What other questions frequently occur'. 21-0 DOUBLE RULE OF THREE. the time necessary to build 18 feet, is to 48 days, the time necessary to build 72 feet. The second ques- OPERATION. tion Is, if 16 men can days. days. men. men. build 72 feet of wall 8 48:: 16 Ans. in 48 days, how many 48 men are necessary to 128 64 build it in 8 days? I 64 Make 16 men the 8)768 third term. Then as Ans. 96 men. the same work is to be done in less time, more men will be necessary; therefore, the fourth term will be greater than the third, and hence 8 days are placed in the first term (Art. 193). 3. If a man travel 217 miles in 7 days, travelling 6 hours a day, how far would he travel in 9 days, if he travelled 11 hours a day? 1ST OPERATION. 2D OPERATION. days. days. miles. miles. hours. hours. miles. miles. 7 9: 217: 279 6: 11:': 279: 5116 9 11 7)1953 6)3069 279 5113Ans. 511Ai miles DOUBLE RULE OF THREE. 198. The last three questions, and all similar ones involving five, seven, or even nine terms, have generally been classed under a separate rule, called the DOUBLE RULE OF THREE, or COMPOUND PROPORTION. They may be thus stated and resolved: I. Make the first statement as though the question were to be solved by two or more statements by the Single Rule of Three, and suppose the fourth term to be found. QUEST.-198. Under what rule have questions similar to the last three been classed? How may they be stated and resolved? Give the rule. DOUBLE RULE OF THREE. 211 II. If it is of the same name with the answer sought, mark its place blank under the third term; if not, mark its place under the second term, and in either case arrange the two remaining terms as though it were a question in the Single Rule of Three. If there are more than five terms in the question, suppose the fourth term of the second proportion to be found, and make the third statement in the same manner as the second was made. II. Then multiply the second and third terms together, and divide their product by the product of the first terms, and the quotient will be the answer sought. EXAMPLES. Let us first resolve each of the last three questions by this rule. 1. If a family of 6 persons expend $300 in 8 months, how much will serve a family of 15 persons for 20 months? OPERATION. persons. persons. $ 6 15 300: st answer months. months. 8 2: 0:: 1st ans. true aiswer 25 5'64 15 X,.o' x ( _ 15 x 5 x 25 - $1875. Having made the first statement, we see that the- fourth term is of the same name with the answer sought, and that if this term be placed in the second proportion, the true answer will be found. But since the first answer is equal to the product of the second and third terms divided by the first, it is plain that the true answer will always be equal to the continued product of the second and third terms divided by the product of the first terms, and similarly when there are more than five terms. In the operation we first cancel the 6 in the 300, then the 4 from 20, and then the 2 from the 50 over the 300. 212 DOUBLE RULE OF THREE. 2. If 16 men build 18 feet of wall in 12 days, how many men must be employed to build 72 feet in 8 days, working at the same rate? OPERATION. feet. feet. days. 18: 72:: 12: 1st answer. days. days. men. 8:' - 16: true answer. 4 2 Then, X,g 4 X 12 X 2 = 96 Ans. 3. If a man, travel 217 miles in 7 days, travelling 6 hours a day, how far would he travel in 9 days, if he travelled 11 hours a day.? OPERATION. days. days. miles. miles. 3 7 ~:: 217 1st answer. f: 11:.:.- true answer. 2 4. If 4 compositors, in 16 days of 12 hours long, can compose 14 sheets of 24 pages each sheet, 44 lines in a page, and 40 letters in a line; in how many days of 10 hours long will 9 compositors compose a volume consisting of 30 sheets, 16 pages in a sheet, 48 lines in a page, and 45 letters in a line? The number of letters set by the first compositors is expressed by 14 X 24 X 44 X 40; and the letters to be set by the second by 30 x 16 x 48 x 45. OPERATION. com. com. days. Ans. 9 4:: 16 1st answer. hours. hours. 10: 12:: 2d answer. 14x24x44x40: 30X16X48X45: -: true answer. DOUBLE RULE OF THREE. 213 4X12x16x30x 16x48x45 4x3x16x3x2 9x 10X 14x24x44x40 7X 11 - 1152 77 = 1474 days = Ans. Let us now analyze this statement. Had the compositors worked the same number of hours per day, and had the same work to do, the first would have been the true answer; and the second would have been the true answer had the time only been different and the work to be done been the same. The third proportion.accounts for the inequality of the work done, and gives the answer under all the suppositions. It is evident the same answer would have been obtained, had the first answer been substituted in the second proportion, and the second answer in the third proportion. Hence, the reason of the rule is obvious. 5. If a pasture of 16 acres will feed 6 horses for 4 montfhs, how many acres will feed 12 horses for 9 months? 6. If 25 persons consume 300 bushels of corn in 1 year, how much will 139 persons consume in 7 years at the same rate? 7. If 32 men build a wall 36 feet long, 8 feet high, and 4 feet wide in 4 days; in what time will 48 men build a wall 864 feet long, 6 feet high, and 3 feet wide? 8. If a regiment of 1878 soldiers consume 702 quarterslof wheat in 336 days, how many quarters will an army of 22536 soldiers consume in 112 days? 9. If 12 tailors in 7 days can finish 13 suits of clothes, how many tailors in 19 days of the same length, can. finish the clothes of a regiment of soldiers consisting of 494 men? 10. An ordinary of 100 men drank ~20 worth of wine at 2s. 6d. per bottle; how many men, at the same rate of drinking, will ~7 worth suffice, when wine is rated at Is.-9d. per bottle? 11. If 60 bushels of oats will serve 24 horses for 40 days, how long will 30 bushels serve 48 horses at the same rate? 12. If a garrison of 3600 men, in 35 days, at 24oz. per 214 DOUBLE!RULE OF TIHREE. day each man, eat a certain quantity of bread, how many men in 45 days, at the rate of 14oz. per day each man, will eat double the quantity? 13. A garrison of 3600 men has just bread enough to allow 24oz. a day to each man for 35 days; but a siege coming on, the garrison was reinforced to the number of 4800 men. How many ounces of bread a day must each man be allowed, to hold out 45 days against the enemy? 14. If 336 men, in 5 days of 10 hours each, dig a trench of 5 degrees of hardness, 70 yards long, 3 wide, and 2 deep, what length of trench of 6 degrees of hardness, 5 yards wide, and 3 deep, may be dug by 240 men in 9 days of 12 hours each? 15. If 12 pieces of cannon, eighteen-pounders, can batter down a castle in an hour, in what time would nine twentyfour-pounders batter down the same castle, both pieces of cannon being fired the same number of times, and their balls flying with the same degree of velocity? 16. If 15 weavers by working 10 hours a day for 10 days, can make 250 yards of cloth, how many must work 9 hours a day for 15 days, to make 6072- yards? 17. If ~3~ be the wages of 13 men for 71 days, what will be the wages of 20 men for 153 days? 18. If a footman travel 294 miles in 73- days, of 12} hours long, in how many days, of 102 hours long each, will he travel 147- miles? 19. Bought 5000 planks, of 15 feet long and 2- inches thick; how many planks are they equivalent to, of 12} feet long and 14 inches thick? 20. If 248 men, in 51 days of 11 hours each, dig a trench of 7 degrees of hardness, 232~ yards long, 32 wide;, and 23 deep; in how many days, of 9 hours long, will 24 men dig a trench of 4 degrees of hardness, 3371 yards long, 53 wide, and 3} deep? PRACTICE.. 215 PRACTICE. 199. PRACTICE is an easy and concise method of applying the rules of arithmetic to questions which occur in trade and business. It is only a contraction of the RULE OF THREE when' the first term is unity. For example, if 1 yard of cloth cost half a dollar, what will 60 yards cost? This is. ajquestion which may be answered by the rule called Practice. The cost is obviously $30. 200. One number is said to be an aliquot part of another, when it forms an exact'part of it: that is, when it is contained in that other an exact number of times. Hence, an aliquot part is an exact or even part. For example, 25 cents is an aliquot part of a dollar. It is an exact fourth part, and is contained in the dollar four times. So also, 2 months, 3 months, 4 months, and 6 months, are all aliquot parto of a year. TABLE OF ALIQUOT PARTS. Its |Parts Mo. Parts of a Days. Parts of Parts of XI. Parts of of $1. year. mo. I shilling 50 1 6= 1 15= = 10s. 1 6 d. = 333- i 4= - 10 = 6s. 8d.- 4 d. - 25 3= I 5s. - 1'20 2= - 4s. - 2' — { 1= 71 G -5 2 d. = ~121-=-= 6 = A 3s. 4d.- 6a= -' va-or Aof 5 - 2s. 6d.= 1d= 45 -19s 3 mo. 3 =is. 8d.t I d.I 1 QuEsT.-199. What is Practice? If one yard of cloth cost $8, what will half a yard cost? What will one quarter of a yard cost? 200. Whens is one number said to be an aliquot part of another? What is an aliquot part? What are the aliquot parts of a dollar expressed in the- table? What the aliquot parts of a year? What the aliquot parts of a month? What the aliquot parts of a pound? What are the aliquot parts of a shilling? 2 1[ PRACTICE.'EXAMPLES. 1. What is the cost of 376 yards of cloth at $0,75, or 3 ot a dollar per yard? Had the cloth cost $1 OPERATION. per yard. the cost of the cts. 376 yards would have 50 a 376 been $376. Had it cost 188 cost at 50cts. 50cts. per yard, the cost 25 1 94 cost at 25cts. would have been % of 75 3 $282 cost at 3doll. $376, or $188: had it 4 4 been 25cts. per yard, the cost would have been I of $376, or $94; but the price being 75cts. per yard, the cost is 188 + 94 = $282. 2. What is the cost of 196 yards of cotton, at 9d. per yard? 196yd. at 6d. or Is. = 98s. 196yd. at 3d. or Is. = 49s. Therefore, 196yd. at 9d. or Is. = 147s. = ~7 7s. Ans. 3. What is the cost of 4. What is the cost of 425 4715 yards of tape, at 1d. yards at 1 penny per yard? Or yard? 1d.=11s. - 12)425 Id. -' 4)4715 20)35s. 5d. 12)1178-4d. = cost. Ans. 2~1 15s. 5d. 20)98s. 2ad. Ans. = ~4 18s. 2Vd. 5. What will be the cost 6. What will be the cost of 354 yards at 1~ per yard? of 4756 yards of cotton shirtld.=-s. - 12)354 ing, at 121 cents per yard? 4)29s. 6d. 12{cts.=- of 1$. 8)4756 id. - - - 7s. 4~d. 594} Cost 36s. 10d. Ans. $594,50. - ~1 16s. 10~d. PRACTICE. 217 7. What will be the cost 8. What will be the cost of 3754 pairs of gloves, at of 5320 bushels of wheat, at 2s. 6d. per pair? 3s. 6d. per bushel? 2s. 6d.=-~. - - 8)3754 5320 469 3 Anb. ~469 5s. at 3s. - - - - 15960 at Is. - - - - 2660 at 3s. 6d. - - 18620s. Ans. ~9310 9. What will be the cost 10. What will be the cost of 435 yards of cloth, at ~2 of 660 yards at 2s. 6d? 7s. per yard? 2s. =1~. 10)660 435 6d. = of 2s. 4)66 cost at 2s. 2 16 10s. Cost at ~2 - - ~870 Ans. ~82 10s. 5s.-=- of~. 108 15s. 2s. 1= of ~. 43 10s. Total cost ~1022 5s. 201. When the price in shillings is less than 20, Multiply by half the number of shillings, and the figures to the left of the right hand figure will express the pounds, and this Jigure doubled will be the shillings. 11. What is the cost of 56 yards of cloth, at 16s. per yard? 16s. -= 6 of a ~: Hence 56 x - = the amount in pounds. But 56 x = 56 x - = ~44.8, in which the right hand figure 8 expresses tenths of pounds, and by doubling it,, we obtain twentieths of pounds, or shillings: therefore, the reason of the rule is manifest. Qus8Tr.-201. When-the price is in shillings and less than 20, how will you find the cost? What is the reason of the rule? 10 218 PRACTICE. 12. What will be the cost of 4514 yards of cloth at ~2 17s. 7~d. per yard? 4514 2 Cost at ~2 9028 4514 x 8~ gives - - - 3836 lBs. at 6d. = of~ - - - 112 17s. lad. = 1 of 6d. - - - - 28 4s. 3d. Total cost ~13.,05 19s. 3d. GENERAL EXAMPLES. 13. What will 19cwt. 3qr. 11 lb. of hops cost, at ~4 11s. 9d. per cwt.? 14. 19cwt. 3qr. 191b. of sugar, at ~2 4s. 8d. per cwt.? 15. l1cwt. lqr. 161b. of soap, at ~3 7s. per cwt.? 16. 9cwt. 3qr. 101b. of treacle, at ~1 18s. 9d. per cwt.? 17. What is the cost of 401h. of soap, at 63cts. per pound? 18. What is the cost of 70 yards of tape, at 2-cts. per yard? 19. What is the cost of 876 bushels of apples, at 62-cts. per bushel? 20. What is the cost of 1000 quills, at a cent per quill? 21. What is the cost of 1800 lead pencils, at 6 cents apiece? 22. What is the cost of 9T. 13cwt. 191b. of pewter, at ~14 15s. 9d. per ton? 23. 3qr. 197b. 10oz., at ~11 12s. 5~d. per cwt.? 24. 74oz. 2pwt. 12gr. of silver, at 4s. 1 lid. per oz.? 25. A pair of chased silver salts, weight 7oz. 11lpwt., at 8s. 1 1-d. per oz.? 26. 571oz. 14pwt. 16]-gr., at ~3 11s. 9-d. per oz.? 27_ What will be the cost of 851 yards of cloth, at $9j per yard? 28. What will be the cost of 1848 yards of linen, at 871 cents per yard? PRACTICE. 219 29. What is the cost of 51~ tons of hay, at $12,50 per ton? 30. What is the cost of 693 yards of linen, at 75cts. per yard? 31. What is the rent of 725A. 2R. 19P. of land, at ~2 1 s. 9d. per acre? 32., 51A. 3R. 15P. at ~4 10s. per acre? 33. 97A. 14P. at ~3 Ils. 10d. per acre? 34. What is the cost of 281 yards of cloth, at $43 per yard? 35. What will be the cost of 2000 quills, at A cent per quill? 36. What will 1541 tons of hay come to, at $12 per ton? 37. Wba' is +he cost of 514yd. 3qr. 2na., at 17s. 9~d. per vard? 38. 125E. E. iqr. Ina. at ~1 11s. 9 per ell? 39. What will be the cost of 1752 bushels of apples, at 621 cents per bushel? 40. What is the cost of 280 yards of tape, at 24 cents per yard? 41. What is the cost of 120 pounds of soap, at 63 cents per pound? 42. What cost 17E.Fr. lqr. 3na. of Brussels lace, at ~3 19s. 11-d. per ell? 43. 349E. Fl. lqr. 3na. of holland, at ~1 Ils. 6d. per ell? 44. 475yd. 3qr. 2na. at~1 14s. 91d. per ell English? 45. 3753E. E. at 18s. 11 1d. per yard? 46. What will be the cost of 2hhd. 5gal. 3qt. 2gi. of molasses, at 124 cents per quart? 47. What will be the cost of 376 yards of cloth, at $1~ per yard? 48. What will be the cost of Ihhd. 2gal. 3qt. lpt. lgi. of brandy, at 56- cents per quart? 49. What will be the cost of 27bu. 3pk. 6qt. lpt. of wheat at $1,75 per bushel 220 TARE AND TRET. TARE AND TRET. 202. Tare and Tret are allowances made in selling goods by weight. Draft is an allowance on the gross weight in favor of the buyer or importer: it is always deducted before the Tare. Tare is an allowance made to the buyer for the weight of the hogshead, barrel or bag, &c., containing the commodity sold. Gross TWeight is the whole weight of the goods, together with that of the hogshead, barrel, bag, &c., which contains them. Suttle. is what remains after a part of the allowances have been deducted from the gross weight. Net Weight is what remains after all the deductions are made. EXAMPLES. 1. What is the net weight of 25 hogsheads of sugar, the gross weight being 66cwt. 3qr. 141b.; tare 111b. per hogshead? cwt. qr. lb. 66 3 14 gross. 25 X 11 = 2751b. - - 2 1 23 tare. Ansq net. 2. If the tare be 41b. per hundred, what will be the tare on 6T. 2cwt. 3qr. 141b.? - Tare for 6T. or 120cwt. = 4801b. 2cwt. = 8 3qr. = 3 141b. = 0O Tare -. 4911 QuEs. —202. What are Tare and Tret? What is Draft? What is Tare? What is Gross Weight? What is Suttle? What is Net Weight? TARE _AND TRET. 221 3. What is the tare on 32 boxes of soap, weighing 315501b., allowing 41b. per box for draft and 121b. in every hundred for tare? 31550 gross. 31422 32 X 4= 128 draft. 12 31422 3770.64 Ans. 4. What will be the cost of 3 hogsheads of tobacco at $9,47 per cwt. net, the gross weight and tare being of cwt. qr. Ib. lb. No. 1 - - 9 3 25 - - tare 146 " 2 - - 10 2 12 -- " 150 " 3 - - 11 1 25 - - " 158 Ans. -- 5. At ~1 5s. per cwt. net; tare 41b. per cwt.: what will be the cost of 4 hogsheads of sugar, weighing gross, cwt. qr. lb. No. 1 - - - 10 3 6 2 - - - 12 5 19 " 3 - - - 13 1 10 " 4 - - - 11 2 7 49 0 14 gross. Tare 41b. per cwt 1 3 0 80z. 47 1 13 8oz. net. Ans. - 6. At 21 cents per lb., what will be the cost of 5hhd. of coffee, the tare and gross weight being as follows: cwt. qr. Ib. lb. No. 1 - - 6 2 14 - - tare 94 " 2 - - 9 1 20 - - " 100 " 3 - - 6 2 22 - - " 88 " 4 - - 7 2 25 - -" 89 " 5 - - 8 0 13 - - " 100 Ans. 7. At ~7 5s. per cwt. net, how much will 16hhd. of sugar come to, each weighing gross 8cwt. 3qr. 71b.; tare 121b. per cwt.? 222 TARE AND TRET. 8. What is the net weight of 18hhd. of tobacco, each weighing gross 8cwt. 3qr. 141b.; tare 161b. to the cwt.? 9. In 4T. 3cwt. 3qr. gross, tare 201b. to the cwt., what is the net weight? 10. What is the net weight and value of 80 kegs of figs, gross weight 7T. 11cwt. 3qr., tare 141b. per cwt., at $2,31 per cwt.? 11. A merchant bought 19cwt. lqr. 271b. gross of tobacco in leaf, at $24,28 per cwt.; and 12cwt. 3qr. 191b. gross in rolls, at $28,56 per cwt.; the tare of the former was 1491b., and of the latter 481lb.: what did the tobacco cost him net? 12. A grocer bought 17~hhd. of sugar, each 10cwt. lqr. 141b., draft 71b. per cwt., tare 41b. per 1041b. What is the value at $7,30 per cwt. net? 13. In 29 parcels, each weighing 3cwt. 3qr. 141t. gross, draft 81b. per cwt., tare 41b. per 1041b., how much net'weight, and what is the value at $7,50 per cwt. net? 14. A merchant bought 7 hogsheads of molasses, each weighing 4cwt. 3qr. 141b. gross, draft 171b. per cwt., tare 8lb. per hogshead, and damage in the whole 9931b.- What is the value at $8,45 per cwt. net? 15. The net value of a hogshead of Barbadoes sugar was $22,50; the custom and fees $12,49, freight $5,11, factorage $1,31; the gross weight was llcwt. 1qr. 151b.,-tare 11lb. per cwt. What was the sugar rated at per cwt. net. in the bill of parcels? 16. In 7hhd. of oil, each weighing 3cwt. 2qr. 141b. gross, tare 211b. per cwt., how many gallons net, and what is the value at $1,24 per gallon? 17. I have imported 87 jars of Lucca oil, each containing 47 gallons: what came the freight to at $1,19 per cwt. net reckoning lb. in 111b. for tare, and 91b. of oil to the gallon? PERCENTAGE. 223 PERCENTA GE. 203. THE term per cent comes from per centum, and means by the hundred. The term is generally used to express the interest on money, but may also be employed to designate hundredth parts of other things. Thus, when we say twenty per cent of a bushel of wheat, we mean twenty hundredths, or one-fifth of it. 204. The rate per cent may always be expressed by a decimal fraction. Thus, five per cent may be expressed by.05, eight per cent by.08, fifteen per cent by.15, &c. Hence, to find the amount of percentage on any number, Miultiply the number by the rate per cent, expressed in a decimalfraction, and the product will be the percentage. EXAMPLES. 1. A has $852 deposited in the bank, and wishes to draw out 5 per cent of it: how much must he' draw for? 2. A merchant has 120) barrels of flour; he shipped 64 per cent of it and sold the remainder: how much did he sell? 3. A merchant bought 1200 hogsheads of molasses. On getting it into his store, he found it short 32 per cent: how many hogsheads were wanting? 4. Two men had each $240. One of them spends 14 per cent, and the other 18~ per cent: how many dollars more did one spend than the other? 5. What is the difference between 5~ per cent of $800 and 6~ per cent of $1050? 6. A trader laid out $160qas follows: he pays 24 per ct. of his money for broadcloths; 30 per ct. of what is left for linens; 12 per ct. of what is left for calicoes; and then 5 per ct. of the residue for cottons: how much did he pay for cottons? QUEST.-203. What do you understand by the term per cent? For what is the term generally used? What do you understand by twenty per cent? What by eight per cent? 204. How may the rate per cent be expressed? How do you express five per cent? Eight per cent? How do you find the amount of percentage on any given number? 224 Pf13cWNTAGx. 7. A man purchased 250 boxes of oranges, and found that ne had lost in bad ones 18 per cent: to how many full boxes were his good oranges equal? 8. If I buy 895 gallons of molasses and lose 17 per cent by leakage, how much have I left? 205. To find the per cent which one number is of another. If I buy 6 hogsheads of molasses for $200 and sell them for $220, what do I gain per cent, on the money expended? It is plain that $20 is the amount made. What per cent is $20 of $200; that is, how many hundredths of $200? If we add two ciphers to the first, and then divide it by the second, the quotient will express the hundredths. Thus, 2000 10; 200 that is, 20 is ten per cent of 200. Hence, to determine the per cent which one number is of another, I. Bring the number which determines the per cent to hundredths by annexing two ciphers or removing the decimal point two places to the right. II. Divide the number so formed by the number on which the percentage is estimated, and the quotient will express the per cent. EXAMPLES. 1. A man has $550 and purchases goods to the amount of $82,-75: what per cent of his money does he expend? 2. A merchant goes to New York with $1500; he first lays out 20 per ct., after which he expends $600: what per ct. was his last purchase of the money that remained after his first? 3. Out of a cask containing 300 gallons, 60 gallons are drawn: what per cent is this? 4. If I pay $698,33 for 3 hhds. of molasses and sell them for $837,996, how much do I gain per ct. on the money laid out? 5. If I pay $698,33 for 3 hhds. of sugar and sell them for $837,996, how much do I make per ct. on the amount received? QUEST.-205. How do you find the per ct. which one number is of another? SIMPLE INTEREST. 225 SIMPLE INTEREST. 206. INTEREST is an allowance made for the use of money that is borrowed. For example, if I borrow $100 of Mr. Wilson for one year, and agree to pay him $6 for the use of it, the $6 is called the interest of $100 for one year, and at the end of the time Mr. Wilson should receive back his $100 together with the $6 interest, making the sum of $106. The money on which interest is paid, is called the Principal. The money paid for the use of the principal, is called the Interest. The principal and interest, taken together, are called the Amount. In the above example, $100 is the principal, $ 6 is the interest, and $106 is the amount. The interest of $100 for one year, determines the rate of interest, or rate per cent. In the example above, the rate of interest is 6 per cent, or $6 for the use of the hundred. Had $8 been paid for the use of the $100, the rate of interest would have been 8 per cent; or had $3 only been paid, the rate of interest would have been 3 per cent. Legal interest is the rate of interest established by law. In the New England States, and indeed in most of the other states, the legal interest is 6 per cent per annum, that is, 6 per cent by the year. QUEST.-206. What is Interest? What is the money called on which interest is paid? What is the money called which is paid for the use of the principal? What is the amount? What determines the rate of interest? What is legal interest? What is meant by per annum? How much is the interest per annum in most of the states? What is it in New York? In Alabama? 10# 226 SIMPLE INTEREST. In New York, however, it is 7, and in Alabama 8 per cent. CASE I. 207. To find the interest on any given principal for one or more years. The interest of each dollar, for a single year, will be so many hundredths of itself as are expressed by the rate of interest. Thus, if the rate of interest be 4 per cent, each dollar will produce annually an interest of.04 of a dollar, or 4 cents: if the rate be 5 per cent, it will produce.05 of a dollar, or 5 cents: if 6 per cent,.06, or 6 cents, &c. Hence, to find the interest on any given sum for one or more years, Multiply the principal by the decimal fraction which expresses the rate of interest, and the product so arising by the numbe7 of years. Or, Multiply the decimal fraction which expresses the rate of interest by the number of years, and then multiply the principal by this product. EXAMPLES. 1. What is the interest on $1960 for four years, at 7 per cent per annum? The rate of interest being OPERATION. 7 per cent, each dollar will $1960 produce.07 of a dollar, or 7.07 cents, in one year: hence, $137,20 int. for 1 year. $137,20 will be the interest 4 number of years. On the sum for one year, and $548,80 Ans. $548,80 for 4 years. QUEST.-207. What will be the interest of one dollar for one year? What will express decimally the interest on one dollar for one year at 4 per cent? What will express it at 5 per cent? At 6? At 7? At 8? How do you find the interest on any sum for one or more years? What will be the multiplier when the rate of interest is 4 per cent, and the time 3 years? When the rate is 6 per cent and the time 5 years? When the rate is 8 per cent and the time 3 years? SIMPLE- INTEREST. 227 2. What is the interest on $78,457 dollars for three years, at 5 per cent per annum? Since there are three places OPERATION. of decimals in the multipli- 78,457 cand and two in the multi-.05 X 3=.15 plier, there will be five in the 392285 product (Art. 149). Observe 78457 that the two first, counting from Ans. $1 1,76855 the comma to the right, are cents, the third mills, the fourth tenths of mills, &c. 3. What is the interest on $365,874 for one year, at 5per cent? We first find the interest at OPERATION. 5 per cent, and then the in- $365,874 terest for 2 per cent: the.05 sum is the interest at 51 per 18,29370 cent. 1,82937 1 per cent. $20,12307 Anis. 4. What is the interest on $2871,24 for 6 years, at 7 per cent? 5. What is the interest on $535,50 for 25 years, at 6 per cent? 6. What is the interest on $1125,819 for 5 years, at 8 per cent per annum? 7. What is the interest on $8089,74 for 12 years, at 5 per cent? 8. What is the interest on $1226,35 for 7 years, at 71 per cent? 9. What is the interest on $3153,82 for 9'years, at 41 per cent? 10. What is the interest on $982,35 for 4 years, at 6 per cent? 11. Whiat is the interest on $1914,16 for 18 years, at 3i per cent? 12. What is the interest on $2866,28 for 6 years, at 8 per cent? 228 SIMPLE INTEREST. 13. What is the interest on $16199,48 for 16 years, at 5 per cent? 14. What is the interest on $897,50 for 21 years, at 6 per cent? CASE IT. 208. To find the interest for any number of months, at the rate of 6 per cent per annum. At the rate of 6 per cent per annum, one month produces a per cent on the principal; and hence, every two months produces one per cent on the principal. Therefore to find the interest for months, Divide the number of months by 2 and regard the quotient as hundredths. Then multiply the principal by the decimal so found, and the product will be the interest. EXAMPLES. 1. What is the interest on $651 for 8 months, at 6 per cent per annum? Trhe decimal corre- OPERATION. sponding. to 8 months, $651 which gives 4 per cent,.04 half the number of months is.04: hence, the in- - $26,04 terest is $26,04. 2. What is the interest on $614,364 for 9 months, at 6 per cent per annum? OPERATION. $614,364 The decimal corresponding to 9 $614 4 months is.041, and hence the in- * a 2457456 terest is $27,64638. 307182 I $27,64638 QUEST.-208. At the rate of 6 per cent, what will be the interest on any principal for one month? What time will produce one per cent? How do you find the interest on ally principal for any number of months? What is the multiplier for 4 months? What for 6 months? What for 7? What for 8? For 9? What for 10? For ll? What for 12? SIMPLE INTEREST. 229 3. What is the interest on $17507,30 for 14 months, at 6 per cent? 4. What is the interest on $982,41 for 9 months, at 6 per cent? 5. What is the interest on $75192,84 for 16 months, at 6 per cent? 6. What is the interest on $7953,70 for 9 months, at 6 per cent? 7. What is the interest on $15907,40 for 27 months, at 6 per cent? 8. What is the interest on $4918,50 for 11 months, at 6 per cent? 9. What is the interest on $84377,91 for 7 months, at 6 per cent? 10. What is the interest on $91358,24 for 17 months, at 6 Ter cent? 11. What is the interest on $31573,25 for 10 months, at 6 per cent? 12. What is the interest on $959875,45 for 18 months, at 6 per cent? CASE III. 209. To find the interest at 6 per cent per annum, for any number of days. In computing interest the month is reckoned at 30 days. Hence, 60 days, which make two months, will give an interest of one per cent on the principal, and consequently, 6 days will give an interest of one mill on the dollar, or onethousandth of the principal. If, therefore, the days be divided by 6, the quotient will show how many thousandths of the principal must be taken on account of the days. Hence, to find the interest for any number of days less than 60, QUEST. —209. In computing interest for days, at what is the month reckoned? How many days give one per cent? What part of the principal is one, per cent? How many days will give one-thousandth of the principal? How will you find how many thousandths of the principal must be taken for the days? 230 SIMPLE INTEREST. Divide the days by 6, and multiply the principal by the quotient, considered as thousandths. EXAMPLES. 1. What is the interest on $297,047 for 28 days, at 6 per cent per annum? OPERATION. We find that the 28 days give $297,047 42 thousandths.'We multiply the 286 -=43..0043 principal by.004, and then add I 1188188 of the principal multiplied by one- Add 99015 96 1 99015 thousandth for the fractional part. s $1,386218 210. To avoid the fractions which sometimes appear in the multipliers, we may, if we please, first multiply the principal by the number of days, and then divide the product by 6, which will give the same quotient as found above. Hence, to find the interest for any number of days, Multiply the principal by the number of days, divide the product by 6, and then point off in the quotient three more places for decimals than there are decimals in the given principal. 2. What is the interest on $657,87 for 13 days, at 6 per cent per annum? OPERATION. We first multiply the given prin- $657,87 cipal by 13; we then divide the. 13 product by 6; and since there are 197361 two places of decimals in the prin- 65787 cipal, we point off five in the quo- 6)855231 tient. $1,42538 NOTE —Let each of the following examples be worked by both methods; though, when the days exceed 60, the second method is preferable. QUEsT.-H-ow do you find the interest for less than 60 days? What is the maltiplier for 6 days? For 9 days? For 10 days? For 15 days? For 20 days? For 25 days? 210. How may the interest for days be found by the second method? SIMPLE INTEREST. 231 3. Find the interest on $785,469 for 25 days. Also, the interest on $8709,27 for 100 days. 4. What is the interest on $2691,12 for 150 days? 5. What is the interest on $1151,44 for 29 days? 6. What is the interest on $136,25 for 19 days? 7. What is the interest on $981,90 for 70 days? 8. What is the interest on $757,06 for 9 days? 9. What is the interest on $864 for 95 days? 10. What is the interest on $11268,75 for 17 days? 11. What is the interest on $4428,10 for 165 days? 12. What is the interest on $975,95 for 14 days? 13. What is the interest on $28793,28 for 127 days? 211. NoTE. —The above method of computing interest for days, is the one in general use. It, however, considers the year as made up of 360 instead of 365 days; and hence the result is too large by 5 of the 365 parts into which the interest found may be divided. Hence, the interest found will be too large by its A part, by which it must be diminished when entire accuracy is desired. CASE IV. 212. To find the interest at 6 per centper annum for years, months, and days. Find the interest for the years by Case I., for the months by Case II., and for the days by Case III.; then add the several results together, and their sum will be the answer sought. Or, Form a single multiplier for the years, months, and days, and then multiply the principal by it. EXAMPLES. 1. What is the interesS on $1597,27 at 6 per cent, for 3 years 9 months and 11 days? QuEST.-211. How many days does the above method give to the year? Is the result obtained too great or too small? By how much is it too great? How will you find the exact interest'! 212. How do you find the interest at 6 per cent per annum for years, months, and days? What is the multiplier for 1 year 4 months and 12 days? What for 2 years 8 mouths and 18 days? For 3 years 10 months and 24 days? 232 SIMPLE INTEREST. 1ST METHOD. $1597,27 $1597,27 $1,59727 for 6 days..06x3=.18.04,79863 for 3 days. 1277816 638908,53242 for 2 days. 159727 79863k — $2,92832 for 1 days. $287,5086 $71,87711 Interest for 3 years $287,5086 " " 9 months 71,8771+ s" " 11 days 2,9283+ Total interest $362,3140+ 2D METHOD. Multiplier for 3 years =.06 x 3 =.18. 99" " 9 months -.045. " 11 days = 11 -.0015 Entire multiplier 0.2265. Then, $1597,27 x 0.2265 = $362,3140+. 2. What is the interest on $252803,87 for 1 year 1 month and 1 day? 3. What is the interest on $3195,54 for 7 years 6 months and 22 days? 4. What is the interest on $1352,25 for 4 years and 7 months? 5. What is the interest on $23518,20 for 9 years, 11 months, and 16 days? 6. What is the interest on $2420,70 for 1 year and 10 months? 7. What is the interest on $195e74 for 12 years and 1 day? 8. What will be the amount of $1947,66 after 21 years and 8 months? 9. What is the interest on-$1330,50 for 14 years, 4 months, and 24 days? 10. What is the interest on $3227,60 for 2 years, 8 months, and 20 days? SIMPLE INTEREST. 233 11. What is the interest on $79265,375 for 8 years 7 months and 6 days? 12. What will be the amount of $9537,15 after 11 years, 2 months, and 18 days? CASE V. 213. To find the interest when there are months and days, and the rate of interest is greater or less than 6 per cent. Find the interest at 6 per cent. Then add to it or subtract from it such a part of the interest so found as the given rate exceeds or falls short of six per cent per annum. EXAMPLES. 1. What is the interest on $179,25, at 7 per cent per annum, for 3 years and 4 months? Multiplier for 3 years =.06 X 3 =.18 " 4 months =.02 Entire multiplier.20 Hence, $179,25 x.20 = $35,8500 interest-at 6 per cent. Add - 5,9750 $41,8250 interest at 7 per cent. 2. What is the interest on $974,50 for 9 years, 6 months, and 18 days, at 4 per cent per-annum? Multiplier for 9 years at 6 per cent = 9 x.06 =.54 6" "6 6 months =.03 " "S 18 days = 18. 6= 3 =.003 Entire multiplier -.573 Hence, $974,50 x.573 = $558,3885 Subtract one-third 186,1295 Int. at 4 per cent $372,2590 3. What is the interest on $874,42, at 3 per cent, for 19 years and 6 months? QUEsT.-213. How do you find the interest when there are months and days, and the rate greater than 6 per cent? How do you find the mterest when it is less? 234 SIMPLE INTEREST. 4. What is the interest on $358,50, at 7 per cent, for 6 years and 8 months? 5. What is the interest on $1975,98, at 5 per cent, for 10 years 4 months and 18 days? 6. What is the interest on $1461,75 for 4 years and 9 months, at 8 per cent? 7. What is the interest on $45000 for 1 year and 4 months, at 7 per cent? 8. What will be the total amount of $2238,96 after 2 years and 7 months, at 7 per cent? 9. What is the interest of $1200 for 1 month and 12 days, at 5 per cent? 10. What is the interest on $1064,82 for 6 years and 6 months, at 42 per cent? 11. What is the interest on e1752,96, at 7 per cent, for 4 years 9 months and 14 days? 12. What is the interest on $17518,54, at 7 — per cent, for 3 years and 9 days? 13. What is the interest on $15138,22 for 3 years, 4 months and 18 days at 6~ per cent per annum? 1'4. What is the interest on $4187,635, at 5 per cent, for 5 years 5 months and 5 days? 15. What is the interest on $167,50 for 7 months and 17 days, at 7 per cent per annum? 16. What is the interest on $2934,25 for 2 years 8 months and 19 days, at 8~ per cent? 17. What is the interest on $19345,31, at 4~ per cent, for 5 years 6 months and 15 days? 214. NOTE.-In computing interest, it is often very convenient to find the interest for the months by considering them as aliquot parts of a year, and the interest for days by considering them as aliquot parts of a month. QUEST.-214. Explain the second method of computing interest for months and days. What part of a year are 3 months? Four months? Six'! Eight? Nine? What part of a month are 6 days? Five days? Ten days t SIMPLE INTEREST. 235 EXAMPLES. 1. What is the interest of $806,90 for 1 year 10 months and 10 days, at 6 per cent? $806,90.06 6)$48,4140 = int. for 1 year. $8,069 2)8,069 = int. for 2 months. 5 3)4,034+ int. for 1 month. $40,345 int. for 10 mo. 1,344+ int. for 10 days. Interest for 1 year - - $48,4140 "' " 10 months - 40,345'L" " 10 days - - 1,344+ Total interest $90,103 + 2. What is the interest of $200 for 10 years 3 months and 6 days, at 7 per cent? $200.07 4) 14,00 = int. for 1 year. $14,00 3)3,50 = int. for 3 months. 10 5)1,16 + = int. for 1 month. $140,00 for 10 years.,23+ = int. for 6 days. $140,00 interest for 10 years. 3,50 interest for 3 months.,23+ interest for 6 days. Ans. $143,73+ 3. What is the interest of $264,52 for 2 years 8 months and 20 days, at 6 per cent per annum? 4. What is the interest of $76,50 for 1 year 9 months and 12 days, at 6 per cent? 5. What is the interest of $1041,75 for 1 year 1 month and 6 days, at 4 per cent per annum? Also, at 5 per cent? At 5~ per cent? At 6 per cent? At 7 per cent? At 71 per cent? At 8 per cent? At 8 per cent? And at 9 per cent? 236 SIMPLE INTEREST. 6. What is the interest, at 6 per cent per annum, on $241,60, for 3 years 3 months and 15 days? 7. What is the interest, at 8 per cent per annum, on 1351,74, for 3 years 6 months and 6 days? 8. What is the interest, at 7 per cent, on $1761,75, for 5 years 5 months and 5 days? 9. What is the interest on $135178,40 for 3 years 9 months and 12 days, at 5 per cent per annum? CASE VI. 215. To find the interest, when the sum on which the interest is to be cast is in pounds, shillings, and pence. I. Reduce the shillings and pence to the decimal of a pound (Art. 161). I1. Then find the interest as though the sum were dollars and cents; after which reduce the decimal part of the answer te shillings and pence (Art. 162). EXAMPLES. 1. What is the interest, at 6 per cent, of ~27 15s. 9d. for 2 years? OPERATION. ~27 15s. 9d. = ~27,7875.03 1.667250 We first find the interest 1667 for one year. We then mul- 3.334500 tiply by 2, which gives the 20 interest for two years. We 6.690000 then reduce to pounds, shil- 12 lings, and pence. 8.280000 4 1.120000 Ans. ~3 6s. 8~d QuEsT-215. How do you determine the interest when the sum is in pounds, shillings, and pence? SIMPLE INTEREST. 237 2. What is the interest on ~203 18s. 6d., at 6 per cent, hir 3 years 8 months 16 days? 3. What is the interest on ~255 10s. 8d. at 6 per cent, for 3 years and 3 months? 4. What is the interest of ~215 13s. 8d., at 6 per cent, for 3 years 6 months and 6 days? 5. What will ~559 7s. 4d. amount to in 3 years and a half, at 5~ per cent per annum? 6. What is the interest of ~1543 10s. 6d. for 3 years and a half, at 4 per cent? 7. What is the interest of ~1047 3s. for 3 years and a half, at 6 per cent? 8. What is the interest on ~511 is. 4d., at 6 per cent per annum, for 6yr. 6mo.? 9. What is the interest on ~161 15s. 3d., at 6 per cent, for 7yr. 13da.? APPLICATIONS. 216. For computing the interest on notes, thre time may be found by the table in Art. 38. The day on which a note is dated and the day on which it falls due, are not both reckoned in determining the time; but one of them is always excluded. Thus, a note dated on the 1st of May, and falling due on the 16th of June, will bear interest but one month and 15 days. Calculate the interest on the following notes. $382,50 Philadelphia, January 1st, 1846. 1. For value received I promise to pay on the 10th day of June next, to C. Hanford or order, the sum of three hundred and eighty-two dollars and fifty cents with interest from date, at 7 per cent. John Liberal. QuEsr.-216. How may the time be found for computing interest on notes? What days named in a note are reckoned and what excluded, in reckoning the time? If a note is dated on the first and payable on the 15tb, how many days will the interest run? 238 SIMPLE INTEREST. $612 Baltimore, January 1st, 1833. 2. For value received I promise to pay on the 4th of July, 1835, to Winm. Johnson or order, six hundred and twelve dollars with interest at 6 per cent from the 1st of March, 1833. John Liberal. $3120 Charleston, July 3d, 1846. 3. Six months after date, I promise to pay to C. Jones or order, three thousand one hundred and twenty dollars with interest from the 1st of January last, at 7 per cent Joseph Springs. $786,50 Mobile, July 7th, 1845. 4. Twelve months after date, I promise to pay to Smith & Baker or order, seven hundred and eighty-six o dollars for value received with interest from December 3d, 1845, at 8 per bent. Silas Day. $4560,72 Cincinnati, March 10th, 1846. 5. Nine months after date, for value received, I promise to pay to Redfield, Wright, & Co. or order, four thousand five hundred and sixty 7A2 dollars with interest after 6 months, at 7 per cent. Frederick Stillman. $1854,83 Boston, July 17th, 1846. 6. Eleven months after date, for value received, we promise to pay to the order of Fondy, Burnap, & Co., one thousand eight hundred and fifty-four a dollars with interest from Mhy 13th, 1846, at 6 per cent. Palmer ~ Blake. PARTIAL PAYMENTS. 217. We shall now give the rule established in New York (See Johnson's Chancery Reports, Vol. I. page 17) for computing the interest on a bond or note, when partial payments nave been made. The same rule is also adopted in Massachusetts, and in most of the other states. SIMPLE INTEREST. 239 I. Compute the interest on the principal to the time of the first payment, and if the payment exceed this interest, add the interest to the principal and from the sum subtract the payment: the remainder forms a new principal. II. But if the payment is less than the interest, take no notice of it until other payments are -made, which in all, shall exceed the interest computed to the time of the last payment: then add the interest, so computed, to the principal, and from the sum subtract the sum of the payments: the remainder will form a new principal on which interest is to be computed as before. EXAMPLES. $349,998 Richmond, Va., May 1st, 1826. 1. For value received I promise to pay James Wilson or order, three hundred and forty-nine dollars ninety-nine cents and eight mills with interest, at 6 per cent. James Paywell. On this note were endorsed the following payments: Dec. 25th, 1826 received $49,998 July 10th, 1827 " $ 4,998 Sept. 1st, 1828 " $15,008 June 14th, 1829 " $99,999 What was due April 15th, 1830? Principal on int. from May 1st, 1826 - - $349,998 Interest to Dec. 25th, 1826, time of first payment, 7 months 24 days -13,649+ Amount - - $363,647+ Payment Dec. 25th, exceeding interest then due - $ 49,998 Remainder for a new principal - - - - $313,649 Interest of $313,649 from Dec. 25th, 1826, to June 14th, 1829, 2 years 5 months 20 days $ 46,524+ Amount - - $360,173 QUEsr. —217. What is the rule in regard to partial payments? 240 SIMPIE INTEREST. Payment, July 10th, 1827, less $ 4,998 than interest then due - - Payment, Sept. 1st, 1828 - - - 15,008 Their sum -20 — - less than interest then due - Payment, June 14th, 1829 - - 99,999 Their sum exceeds the interest then due - $120,005 Remainder for a new principal, June 14th, 1829 - - 240,168 Interest of $240,168 from June 14th, 1829, to April 15th, 1830, 10 months,1 day - - $ 12,048 Total due, April 15th, 1830 - - $252,216 + $6478,84 New Haven, Feb. 6th, 1825. 2. For value received I promise to pay William Jenks or order, six thousand, four hundred and seventy-eight dollars and eighty-four cents with interest from date, at 6 per cent. John Stewart. On this note were endorsed the following paymentsMay 16th, 128, received $545,76 May 16th, 1830, " $1276 Feb. lst, 1831, " $2074,72. What remained due August 11th, 1832? 3. A's note of $7851,04 was dated Sept. 5th, 1837, on which were endorsed the following payments, viz: —Nov. 13th, 1839, $416,98; May 10th,-1840, $152: what was due March 1st, 1841, the interest being 6 per cent? $8974,56 New York, Jan. 3d, 1842. 4. For value received I promise to pay to James Knowles or order, eight thousand nine hundred and seventy-four dollars and fifty-six cents, with interest from date at the rate of 7 per cent. Stephen Jones. On this note were endorsed the following payments: SIMPLE INTEREST. 241 Feb. 16th, 1843, received $1875,40 Sept. 15th, 1844, " $3841,26 Nov. 11th, 1845, " $1809,10 June 9th, 1846, " $2421,04. What was due July 1st, 1846? QUESTIONS IN INTEREST. 218. In all the questions relating to interest four things have been considered, viz.: 1st. The principal; 2d. The rate of interest; 3d. The time; and 4th. The amount of interest. Now, these four quantities are so connected with each other, that if three of them be known the fourth can always be found. CASE I. 219. The principal, the rate of interest, and the time being known, to find the interest. This case has already been considered. CASE II. 220. Having given the interest, the time, and the rate of interest, to find the principal. When the time and rate are the same, the interest on any principal, is to the principal, as any other interest is to its principal; that is, Interest of $1: $1:: given interest: principal Hence, to find the principal, Cast the interest on one dollar for the given time and divide the given interest by the interest so found, and the quotient will be the principal. QUEsT.-218. How many things are considered in all questions relating to interest? How many of these must be given before the remaining ones can be determined? 219. What are given in Case I.? What required? 220. What are given in Case II.? What required? How do you find the principal? 11 242 SIMPLE INTEREST. EXAMPLES. 1. The interest on a certain sum for 1 year and 4 months, at 6 per cent, is $3007,7136: what is the principal? The interest on $1 for the same time is $0,08. Hence, $3007,7136 - 0,08 = $37596,42 = principal. 2. The interest on a certain sum for nine months, at 6 per cent, is $178,9582: what is the principal? 3. The interest for 29 days is $2,78, at 6 per cent: what is the principal? 4. The interest for 17 days, at 6 per cent, is $4,0366: what is the principal? 5. The interest on a certain sum for 1 year 1 month and 6 days, at 7 per cent, is $26,7381: what is the sum? If the interest for the same time be $22,9184 at the rate of 6 per cent, what will be the sum? For the same time, what will' be the principal, when the rate is 4 per cent and interest $15,2790? When the rate is 5 per cent and interest $19,0987? When the interest is $21,0086 and rate 51 per cent? When the rate is 7~ per cent and interest $28,6479? When the rate is 8 per cent and interest $30,5578? CASE III. 221. Having given the interest, the principal, and time, to find the rate per cent of interest. If interest be cast at different rates, on the same sum and for the same time, the amounts of interest will be proportional to the rates. Therefore, cast the interest on the principal for the given time, at 1 per cent per annum; then, Interest at 1 per cent: given interest:: 1 per cent: rate. Hence, to find the rate of interest, Cast the interest on the principal for the given time at 1 per cent: then divide the given interest by the interest so found, and the quotient will be the rate of interest. QUEST.-221. What are given in Case III? What are required? How do you find the rate of interest? SIMPLE INTEREST. 243, EXAMPLES. 1. The interest on $437,21 for 9 years and 9 months is $127,8840: what is the rate of interest? Interest on $437,21 for 9 years and 9 months, at 1 per cent, is $42,6280: hence, $127,8840 - 42,6280 = 3 per cent, the rate. 2. The interest on $987,99, for 5 years 2 months and 9 days, is $256,4657: what is the rate of interest? NoTE.-In examples similar to the above, and to those of the following section, the fractions of a per cent less than a quarter, or of a day, may be omitted. Such small fractions may arise from the different methods of computation. CASE IV. 222. Having given the principal, the interest, and the rate of interest, to find the time. If interest be cast at the same rate and on the same principal for different times, the amounts of the interest will be proportional to the times. Hence, if the interest on the principal be cast for.1 year, we shall have, Interest for 1 year: given Interest:: 1 year: time. Hence, to find the time, Cast the interest on the given principal at the given rate for one year: then divide the given interest by the interest so found, and the quotient will be the time. EXAMPLES. 1. The interest on $15000, at 7 per cent per annum, is $700: what is the time? Interest on $15000 for 1 year at 7. per cent = $1050: hence, W5s- = F6% = _' of a year = 8 months. 2. The inteirst on $1119,48, at 7 per. cent per annum, is $195,909; what is the time? Quewi.-222. What am given in Case IV.? What are required? How do you find the time? 244 REDUCTION OF CURRENCIES. A TABLE, showing the number of shillings in a dollar in each State, and the rate of interest: also, the value of a dollar expressed in parts of a pound, which is found by dividing the number of pence in a dollar by the number in a pound. STATES. NO. of shillings Value of the dollar in Legal rate of to the dollar. pounds. interest. 1 Maine 6 shillings $I= — 7X=2 C3 6 per cent. 2 N. Hampshire 6 shillings $1= —C72 --- 6 per cent. 3 Vermont 6 shillings $1=~ —C7 -=- — 3 6 per cent. 4 Massachusetts 6 shillings $= l~ —72=-~= 6 per cent. 5 Rhode Island 6 shillings $1 7 — =~130 6 per cent. 6 Connecticut 6 shillings $1~-X.'12 3 6 per cent. 7 New York 8 shillings $1=-246O = 7 per cent. 8 Ohio 8 shillings $1=~ — ~ X 6 per cent. 9 New Jersey 7s. 6d. $1= —.C9-= 3 6 per cent. 10 Pennsylvania 7s. 6d. $1-=~-o -.=~ 6 per cent. II1 Delaware 7s. 6d. $1=.-290. =; 6 per cent. 12 Maryland 7s. 6d. $1-=-.CE4 —~ 3 6 per cent. 13 Michigan 8 shillings $1=-94-.- =&~ 2 7 per cent. 14 Indiana 6 shillings $l =72=-C 3 6 per cent. 15 Illinois 6 shillings $1 4 -- ~C =~6 per cent. 16 Missouri 6 shillings $1-= —72 =C —3 6 per cent. 17 Virginia 6 shillings $1 =~ -*-; =~ 6 per cent. 18 Kentucky 6 shillings $1=- X2 C-~ 6 per cent. 19 Tennessee -6 shillings $1=-72E C 3 6 per cent. 20 North Carolina 10 shillings $1 = —L2 0 ~: 6 per cent. 21 South Carolina 4s. 8d. $1=f- 5-~ 7 per cent. 22 Georgia 4s. 8d. $1 =~ —5 =~- 7' per cent. 23 Alabama Fed. money 8 per cent. 24 Mississippi 6 shillings $1=-72 = 3 6 per cent. 25 Louisiana Fed. money 6 per cent. 26 Arkansas Fed. money 6 per cent. 27 Florida Fed. money 6 per cent. 28 Texas 6 shillings $1- - = 8 per cent. 2 Nov. Scotia 5ha 29 and Canada 5 shillings $1= —-- =g-~ J 6 per cent. I I....., REDUCTION OF CURRENCIES. It has already been shown (Art. 16), that Federal Money is the currency of the United States; the pound, however) is occasionally used. REDUCTION OF CURRENCIES. 24 5 There are two principal reductions: 1st. To change any sum expressed in Federal money into pounds shillings and pence. 2d. To change any sum expressed in pounds shillings and pence, into Federal money. For the first, Multiply the sum in dollars cents and mills, by the value of $1 expressed in the fraction of a pound, and the product will be the corresponding value in pounds and the decimal of a pound. For the second, Reduce the shillings and pence to the decimal of a pound by Art. 161, and annex the decimal to the entire pounds. Then multiply by the fraction with its terms inverted, which expresses the value of $1 in terms of a pound, and the product will be dollars cents and mills. EXAMPLES. 1. What is the value of $375,87, in pounds shillings and pence, New York Currency? We first multiply by 2, and OPERATION. then reduce the decimal of a 375,87 x - =~ 50.348 pound to shillings and pence. -=~150 6s 11d.+ 2. What is the value of ~127 18s. 6d., in Federal money, if the currency be 6 shillings to the dollar? We first reduce the shillings and pence to the fraction of a OPERATION. ~, and then multiply by the ~127 18s. 6d.=127.925 fraction of a dollar with its 127.925 X 1- = $426,416+. terms inverted. 3. What is the value of $2863,75 in pounds shillings and pence, Pennsylvania currency? 4. What is the value of ~459 3s. 6d., Georgia currency, in dollars and cents? 5. What is the value of $9763,28, in pounds shillings and pence, North Carolina currency? 6. What is the value, in dollars and cents, of ~637 18s.8d., Nova Scotia currency? 246 COMPOUND INTEREST. COMPOUND INTEREST. 223. COMPOUND Interest is when the interest on a sum of money becoming due, and not being paid, is added to the principal, and the interest then calculated on this amount, as on a new principal. For example, suppose I were to borrow of Mr. Wilson $200 for one year, at 6 per cent. If at the end of the year Mr. Wilson should add the interest, $12, to the principal, $200, making $212, and charge interest on this sum till paid, this would be Compound Interest, because it is interest upon interest. Hence, Calculate the interest to the time at which it becomes due: then add it to the principal and calculate the interest on the amount as on a new principal: add the interest again to the principal and calculate the interest as before: do the same for all the times at which payments of interest become due: from the last result subtract the principal, and the remainder will be the compound interest. EXAMPLES. 1. What will be the compound interest, at 7 per cent, of $3750 for 4 years, the interest being added yearly? $3750,000 principal for 1st year. $3750 x.07 _ 262,500 interest for 1st year. 4012,500 principal for 2d " $4012,50 X.07 - 280,875 interest for 2d 4293,375 principal for 3d $4293,375- x.07 = 300,536+ interest for 3d " 4593,911+ principal for 4th" $4593,911 x.07 = 321,573-+ interest for 4th 4915,484+ amount at 4 years. 1st principal 3750,000 Amount of interest $1 165,484 + QuEST.-223. What is compound Interest? How do you find the com. pound interest on any sum? COMPOUND INTEREST. 247 2. If the interest be computed annually, what will be the interest on $300 for three years, at 6 per cent? 3. What will be the compound interest on $590,74, at 6 per cent, for 2 years, the interest being added annually? 4. What will be the compound interest on $500 for 1 year, at 8 per cent, the interest being computed quarterly? 5. What will be the compound interest on $3758,56 for 3 years, at 7 per cent, the interest being added every 6 months? 6. What will be the compound interest on $95637,50 for 7 years, at 6 per cent, the interest being added annually? 7. What will be the compound interest on $75439,75 for 4 years, at 41 per cent, the interest being added annually? A TABLE, 224. Showing the interest of ~1, or $1, compounded annually, for any number of years not exceeding 20. Years. 3 per cent. per cent. 4 pe cent. 34 per cent. 5 per cent. per cet. 1.030000'.035000!.040000.045000.050000.060000! 2.060900'.071225.081600,.0920251.102500~.123600 3.09:-272.108718.124864.141166i.157625'.191016 4.125509.147523.169859.1925191.215506.26-2477! 5.159274.187686.216653.2461821.2752821.3382%261 6.1940521.229255.265319.302260'.340096'.4185191 7.2298741.272279.315932.260862.407100.5036301 8.266770i.316809.368569.422100.477455.593848! 9.3047731.362897.423312.486095.551328.689479 10.343916.410599.480244.552969.628895.-790848 11.3842341.459970.539454.622853.710339.898299 1 2.425761.511069.601032.6958811.79585611.012196 13.468534.563956.665074.772196!.88564911.132928 14.512590.618695.731676.8519451.9799321.260904 15.5579671.675349.800944.935282 1.0789281.396558m 16.604706.733986.87298 1 1.02237011. 182875 1. 540352! 17.6528481.794676.94790011.113377 1.2920181.692773 18.7024331.857489.0258171.208479 1.406619 1.854339 19..7535061.922501.106849 1. 307860 1.5269502.025600 20.806111'.989789.191123 1.411714' I.653298 2.207135 *11 4SlA COMJPOUND INTEREST. We will now explain the method of finding the compound interest on any sum, for any time, by means of the above table. Take from the table the interest-of ~1 or $1 for the same time, and at the same rate, and then multiply the number so found by the principal, and the product will be the compound interest. EXAMPLES. 1. What will be the compound interest on $350 for three years, at 6 per cent per annum, the interest being computed annually? Interest from the table on $1 = $0.191016; then, $0.19~016 x 350 $66.8556. 2. What will be the compound interest on $856,95 for 15 years, at 3~- per cent per annum? 3..What will be the compound interest on $9864,05 for 16 years, the interest being computed annually, at 4 per cent? 4. What will be the compound interest on $1675.20 for 20 years, at 4~ per cent, the interest being computed annually? 5. What will be the compound interest on $5463,25 for 17 years, at 5 per cent, the interest being computed annually? 6. Wlht will be the compound interest on $3769,75 for 18 years, at 3 per cent, the interest being computed annually? 7. What will be the compound interest on ~24 17s. 6d. for 10 years, at 4 per cent, the interest being computed annually? 8. What will be the compound interest on $9854,50 for 12 years, at 6 per cent, the interest being computed annually? QUEST.-224. How do you find the compound interest on any sum by the table? LOSS AND GAIN. 249 LOSS AND GAIN. 225. Loss and Gain is a rule by which merchants discover the amount lost or gained in the purchase and sale of goods. It also instructs them how much to increase or diminish the price of their goods, so as to make or lose so ~nuch per cent. EXAMPLES. 1. Bought a piece of cloth containing75yd. at $5,25 per yard, and sold it at $5,75 per yard: how much was gained in the trade? OPERATION. $5,75 price of 1 yard. $5,25 cost of 1 yard. We first find the profit 50cts. profit on 1 yard. on a single yard, and then yd. yd. cts. the profit on the 75 yards. 1: 75:: 50: Ans. 75 $37,50 Ans. $37,50. 2. A merchant bought a bale of goods containing 125 yards for $687,75, and sold it at auction for $4,50 per yard: how much did he lose in all, and how much per yard? $687,75 = cost of the bale. 562,50 = price of 125 yd. at $4,50 per yd. $125,25 -= total loss. 125)$125,25(1,00,2. Ans I Total loss $125,25. Loss on each yd. $1,00,2. QuEsT.-225. What is the rule of loss and gain? 250 LOSS AND GAIN. 2. Bought a piece of calico containing i50yd. at 2s. 6d. per yard: what must it be sold for per yard to gain ~1 Os. 10d.? 50yd. at 2s. 6d. = ~6 5s. Profit = ~1 Os. 1Od. It must sell for ~7 5s. 10d. 50)~7 5s. 10d.(2s. ld. Ans. 2s. lid. 3. Bought a hogshead of brandy at $1,25 per gallon, and sold it for $78: was there a loss or gain? 4. A merchant purchased 3275 bushels of wheat for which he paid $3517,10, but finding it damaged is willing to lose 10 per cent: what must it sell for per bushel? 226. In the sale of goods, knowing the per cent of gain, and the amount received, to find the principal or cost. I sold a parcel of goods for $195,50, on which I made 15 per cent: what did they cost me? It is evident that the cost added to 15 hundredths of the cost will be equal to what the goods brought, viz., $195,50. If we call the cost 1, then 1 plus 1-5o of the cost will be equal to what they bring: that is, 1 + 15 = l- = $195,50; or, cost equals $195,50 X 100 - 115 = $170. Hence, to find the cost, Multiply the amount by 100 and divide the product by 100 plus the per cent of gain, and the quotient will be the cost. 227. When there is a loss, we have the following method: If I sell a parcel of goods for $170, by which I 4ose 15 per cent, what did they cost? It is evident that the cost, less 15 per cent, that is, less 15 hundredths of the cost, is equal to $170. Hence, 85 hunQuEsT.-226. Knowing the per cent of gain and the amount received, how do you find the cost? 227. Knowing the per cent and the amount lost, how do you find the cost? LOSS AND GAIN. 251 dredths of the cost is equal to $170; and consequently, the cost is equal to $170 x 100.- 85 = $200 cost. Hence, to find the cost when there is a loss, Multiply the amount received by 100 and divide the product by the difference between 100 and the per cent lost, and the quotient will be the cost. EXAMPLES. 1. Sold cloth at $1,25 per yard and lost 15 per cent: for what should I have sold it to have gained 12 per cent? 2. Sold cloth at $1,25 per yard and lost 15 per cent: what per cent should I have gained'had I sold it at $1,6470o-7 per yard? 3. Sold cloth at $1,6470-10 per yard and gained 12 per cent: for what ought I to have sold it to lose 15 per cent? 4. A bought a piece of cotton containing 80 yards, at 6 cents per yard; he sold it for 71 cents per yard: how much did he gain, and how much per cent? 5. Bought a piece of cloth containing 150 yards for $650: what must it be sold for per yard, in order to gain $300? 6. Bought a quantity of wine at $1,25 per gallon, but it proves to be bad and I am obliged to sell it at 15 per cent less than I gave: how much must I sell it for per gallon? 7. A farmer sells 375 bushels of corn for 75cts. per bushel: the purchaser sells it at an advance of 20 per cent: how much did he receive for the corn? 8. A merchant buys one tun of wine for which he pays $725, and wishes to sell it by the hogshead at an advance of 20 per cent: what must he charge per hogshead? 9. A merchant buys 316 yards of calico for which he pays 20 cents per yard; one-half is so damaged that he is obliged to sell it at a loss of 6 per cent; the remainder he sells at an advance of 19 per cent: how much did he gain? 252 STOCKS AND CORPORATIONS. 10. If I buy coffee at 16 cents and sell it at 20 cents, how much do I make per cent on the money paid? 11. If I buy tea at 4s. per pound and sell it at 4s. 9d. per pound, how much should I gain on a purchase of ~100? 12. A merchant bought 650 pounds of cheese at 10 cents per pound, and sold it at 12 cents per pound: how much did he gain on the whole, and how much per cent on the money laid out? 13. Bought cloth at $2,50 per yard, which proving bad, I wish to sell it at a loss of 18 per cent: how much must I ask per yard? 14. Bought 150 gallons of molasses at 75 cents a gallon, 30 gallons of which leaked out. At what price per gallon must the remainder be sold that I may clear 10 per cent on the cost? STOCKS AND CORPORATIONS. 228. STOCK is a general name for the money contributed by individuals for the establishment of banks and manufacturing companies, and the individuals who contribute the money are called Stockholders. 229. The individuals so associated are called, in their collective capacity, a Corporation; and the law which defilles their rights and powers, is called the Charter of the Bank or Company. 230. The amount of money paid in by the stockholders to carry on the business of the corporation, is called the Capital. The capital is generally divided into a certain numbei of equal parts called shares, and the written evidences of ownership of such shares, are called certificates of stock. QUtET.-228. What is stock? What are individuals called who own the stock? 229. What are they called in their associated capacity? What is the law called which incorporates them? 230. What is the amount of money paid in by the stockholders called? How is the capital generally divided? What is the evidence of ownership called? COMMISSION AND BROKERAGE. 253 231. When the General Government, or any of the states borrows money for public purposes, an evidence is given to the lender in the form of a bond, bearing a given interest. Such bon'ds, when given by the United States, are called United States Stock; and when given by any one of the states, are called State Stocks. These bonds or stocks are generally made transferable from one person to another. 232. The nominal or par value of a stock is its original cost; that is, the amount named in the certificate or bond. The market value is what it will bring when sold. If the market value is above the par value, the. stock is said to be at a premium, or above par; but if the market value is below the par value. it is then said to be at a discount, or below par. For example, if $100 of stock will bring in the market $110, the stock is 10 per cent above par; if, on the contrary, it will bring but $90, it is 10 per cent below par: the percentage of premium or discount being always estimated on the par value. COMMISSION AND BROKERAGE. 233. A person who buys or sells goods for another, receiving therefor a certain rate per cent, is called a factor or commission merchant; and the percentage on any purchase or sale, is called the commission. 234. Dealers in money or stocks are called Brokers, and the amount of their commissions on any purchase or sale, is called the brokerage. The commission for goods or moneys is generally a certain per cent'or rate per hundred on the moneys paid out or received, and the amount may be determined by the rules of simple interest. QUEST.-231. What is United States stock? What are state stocks? 232. What is the nominal or par value' of a stock? What is the market value? What do you understand by a stock's being at a premium? What by its being at a discount? 233. What is the business of a commission merchant? 234. What is the business of a broker? How is the commission on goods and moneys generally estimated? 254 COMMISSION AND BROKERAGE. The commission for the purchase and sale of goods varies froin 2~ to 12~ per cent, and under some circumstances even higher rates are paid. The brokerage on the purchase and sale of stocks in Wall-street, in the city of New York, is generally one-fourth per cent on the par value of the stock. EXAMPLES. 1. What is the commission on $4396 at 6 per cent? We here find the corn- OPERATION. mission, as in simple in- $4396 terest, by multiplying by the.06 decimal which expresses $263,76 the rate per cent. Ans. $263,76. 2. A factor sells 120 bales of cotton at $425 per bale, and is t0receive 2~ per cent commission: how much must he pay over to his principal? 3. A sent to B, a broker, $382'5 to be invested in stock: B is to receive 2 per cent on the amount paid for the stock: what was the value of the stock purchased? OPERATION. As B is to receive 2 per 100 cent, it follows that $102 2 of A's money will purchase 102: 100:: 3825: Ans but $100 of stock: hence, 100 100 + the commission, is to 102)382500(3750 100, as the given sum to 306 the value of the stock which 765 it will purchase. Hence, 714 to find the value of the stock 510 purchased, 510 Ans. $3750. Multiply the amount to be invested by 100 and divide the product by 100 plus the brokerage. QEsT. —What is the general commission on the purchase and sale of goods? How may it be determined? What is the customary brokerage on the purchase and sale of stocks? COMMISSION AND BROKERAGE. 255 PROOF. Amount paid - $3750 Brokerage on $3750, at 2 per cent- 75 Total sum - - $3825 4. I have $5000 to be laid out in stocks which are 15 per cent below par: how much will it purchase at the par or nominal value? It is plain that every 85 dollars will purchase stock of the par value of $100: hence, $85: $100:: $5000: Ans. Therefore, to find how much will be purchased at the par value, when the stock is below par, Multiply the sum to be invested by 100 and divide the product by 100 minus the discount. 5. A person has $7000 which he wishes to invest: what will it purchase in 5 per cent stocks, at 3~ per cent below par, if he pays 4 per cent brokerage? 6. How much 6 per cent stock can be purchased for $8500, at 8~ per cent premium, 1 per cent being paid to the broker? 7. A factor receives $708,75, and is directed to purchase iron at $45 per ton: he is to receive 5 per cent on the money paid: how much iron can he purchase? S. Messrs. P, W, & K buy 200 shares of United States stock for Mr. A. The par value is $1000 dollars a share, the stock is at a premium of 61 per cent, and their brokerage is one-fourth per cent. How much must A pay them for his stock? 9. Messrs. P, W, & K receive $28750 to be invested in stock. They charge I per cent commission on the amount paid: what is the value of the stock purchased? 10. The par value or first cost of 257 shares of bank stock was $200 per share: what is the present value, if the stock is at a premium of 15 per cent, that is, 15 per cenlt above par? 256 BANKING. 11. What would be the value of the stock named in the last example, if it were at a discount of 10 per cent? 12. One hundred shares of United States Bank stock was worth 18~ per cent premium: the par value being $200 per share, what was the value of the 100 shares? 13. A bank fails, and has in circulation bills to the amount of $267581. It can pay 9~ per cent: how much money is there on hand? 14. Sixty-nine shares of bank stock, of which the par value is $125, is at a discount of 8 per cent: what is its value? 15. My commission merchant sells goods to the amount of $1000, on which I allow him a commission of 2 per cent-; and as he pays over before the money becomes due, I allow him 11 per cent: how much am I to receive? 16. My broker receives from me $2000 to be laid out in stocks: what will be the value of my stocks after allowing him I per. cent commission? 17. I sold $13921,60 worth of goods for a merchant at a commission of 2-1 per cent: how much ought I to pay over to my principal? 18. I remitted to my agent $14760 to lay put in the purchase of iron. He takes 3! per cent on the whole sum for his commission, and then buys iron at 95 dollars per ton: how much does he purchase? BANKING. 235. BANKS are corporations created by law for the purpose of receiving deposites, loaning money, and furnishing a paper circulation represented by specie. The notes made by a bank circulate as money, because they are payable in specie on presentation at the bank. They are called bank notes, or bank bills. QUEs?.-235. What are banks? Why do the notes of a bank circulate as money? What are they called? BANKING. 257 236. The note of an individual, or as it is generally called, a promissory note or note of hand, is a positive engagement, in writing, to pay a given sum at a time specified, and to a person named in the note, or to his order, or sometimes to the bearer at large. FORMS OF NOTES. Negotiable Note. No. 1. $25,50. Providence, May 1, 1846. For value received I promise to pay on demand, to Abel Bond, or order, twenty-five dollars and fifty cents. REUBEN HOLMES. Note Payable to Bearer. No. 2. $875,39. St. Louis, May 1, 1845. For value received I promise to pay, six months after date, to John Johns, or bearer, eight hundred and seventy-five dollars and thirty-nine cents. PIERCE PENNY. Note by two Persons. No. 3. $659,27. Buffalo, June 2, 1846. For value received we, jointly and severally, promise to pay to Richard Ricks, or order, on demand, six hundred and fifty nine dollars and twenty-seven cents. ENos ALLAN. JOHN ALLAN. Note Payable at a Bank. No. 4. $20,25. Chicago, May 7, 1846. Sixty days after date, 1 promise to pay John Anderson, or order, at the Bank of Commerce in the city of New York, twenty dollars and twenty-five cents, for value received. JESSE STOKES. QUEST.-236. What is a promissory note? 258 BANKIN G. Remarks relating to Notes. 1. The person who signs a note, is called the drawer or maker of the note; thus Reuben Holmes is the drawer of note No. 1. 2. The person who has the rightful possession of a note, is called the holder of the note. 3. A note is said to be negotiable when it is made payable to A B, or order, who is called the payee, (see No. I.) Now, if Abel Bond, to whom this note is made payable, writes his name on the back of it, he is said to endorse the note, and he is called the endorser; tnd when the note becomes due, the holder must first demand payment of the maker, Reuben Holmes, and if he declines paying it, the holder may then require payment of Abel Bond, the endorser. 4. If the note is made payable to A B, or bearer, then the drawer alone is responsible, and he must pay to any person who holds the note. 5. The time at which a note is to be paid should always be named, but if no time is specified, the drawer must pay when required to do so, and the note will draw interest after the payment is demanded. 6. When a note, payable at a future day, becomes due, it will draw interest, though no mention is made of interest. 7. In each of the States there is a rate of interest established by law, which is called the legal interest, and when no rate is specified, the note will always draw legal interest. If a rate higher than legal interest be taken, the drawer, in most of the States, is not bound to pay the note. 8. If two persons jointly and severally give their note, (see No. 3,) it may be collected of either of them. 9. The words "For value received," should be expressed in every note. QUEST.-1. What is the person called who signs a note? 2. What is the person called who owns it? 3. When is a note said to be negotiable? What is the person called to whom a note is made payable? When the payee writes his name on the back, what is he said to do? What is he then called? 4. If a note is made payable to A B, who is responsible for its payment? 5. If no time is specified, when is a note to be paid? 6. Will a note draw interest after it falls due, if not stated in the note? 7. If the rate of interest named in a note is higher than the legal rate, can- the amount of the note be collected? 8. If two persons jointly and severally give a note, of whom may it be collected? 9. What words should be put in every note? BANK DISCOUNT. 259 10. When a note is given, payable on a fixed day, and in a specific article, as in wheat or rye, payment must be offered at the specified time, and if it is not, the holder can demand the value in money. 237. By mercantile usage and the custom of banks, a note does not really fall due until the expiration of 3 days after the time mentioned on its face. For exampTe, Note No. 1 would be due on the 4th of November, and the three additional days are called days of grace. When the last day of grace happens to be a Sunday, or a holiday, such as New Year's or the 4th of July, the note must be paid the day before; that is, on the second day of grace. BANK DISCOUNT. 238. BANK DISCOUNT is the charge made by a bank for the payment of money on a note before it becomes due. By the custom of banks, this discount is the interest on the amount named in the note, from the tine the note is discounted to the time when it falls due, in which time the three days of grace are always included. The amount named in a note is called the face of it. The PRESENT VALUE of a note is the difference between the face of the note and the discount. 239. There are two kinds of notes discounted at banks: ist. Notes given by one individual to another for property actually sold-these are called business notes, or business paper. 2d. Notes made for the purpose of borrowing money, QUEST.-10. If a note is made payable on a fixed day and in a specified article, and is not paid, what may be done? 237. How long is the time for the payment of a note extended by mercantile usage? What are these days called? When the last day of grace falls on a Sunday, or holiday, when must the note be paid? 238. What is bank discount? fhow is it estimated? How is it estimated by the custom of banks? What is the face of a note? What is the present value of a note? 239. How many kinds of notes are discounted at banks? What distinguishes one kind from the other, and what are they called? 260 BANK DISCOUNT. which are called accommodation notes, or accommodation paper. The first class of paper is much preferred by the banks, as more likely to be paid when it falls due, or in mercantile phrase, " when it comes to maturity." Hence, to find the bank discount-on a note, -Add 3 days to the time which the note has to run before it becomes due, and calculate the interest for this time at the given rate per cent. EXAMPLES. 1. What is the bank discount of a note of $1000 payable in 60 days, at 6 per cent interest? This note will have 63 days to run. 2. A merchant sold a cargo of cotton for $15720, for which he receives a note at 6 months: how much money will he receive at a bank ibr this note, discounting it at 6 per cent interest? 3. What is the bank discount on a note of $556,27 payable in 60 days, discounted at 6 per cent per annum? 4. A has a note against B for $3456, payable in three months; he gets it discounted at 7 per cent interest: how much does he receive'? 5. What is the bank discount on a note of $367,47, having i year, 1 month, and 13 days to run; as shown by the face of the note, discounted at 7 per cent? 6. For value received I promise to pay to John Jones, four months from the 17th of July next, six thousand five hundred and seventy-nine dollars and 15 cents. What will be the discount on this, if discounted oil the 1st of August, at 6 per cent per annum? 240. It is often necessary to make a note, of which the present value shall be a given amount. For example, if I wish to receive at bank the sum of two hundred dollars, for what amount must I give my note payable in three months? QuESTr.-Which kind is preferred? How do you find the bank discount on a note? 240. What is often necessary in bank business? BANK DISCOUNT. 261 If we calculate the interest on one dollar for the time, which will be 3 months added to the 3 days of grace, and at the same rate per cent, this will be the bank discount on $1 payable in 3 months; and if this discount be subtracted from one dollar, the remainder will be the present value of one dollar, to be paid at the end of 3 months. Therefore, Pres. val. of $1: pres. val. of note:: $1: amount of note. Hence, to find the face of a note, due at a future time and bearing a given interest, that shall have a.known present value, Find the present value of $1 for the same time and at the same rate of interest, by which divide the present value of the note, and the quotient will be the face of the note.* EXAMPLES. 1. For what sum must a note be drawn at 3 months, so that when discounted at a bank, at 6 per cent, the amount received shall be $500? Interest on $1 for the time, 3mo. and 3da.=$0,0155, which taken from $1, gives present value of $1 = 0,9845; then $500 - 0,9845 = 507,872 - = face of note. PROOF. Bank interest on $507,872 for 3 months, including 3 days of grace, at 6 per cent - 7,872, which being taken from the face of the note, leaves $500 for its present value. 2. For what sum must a note be drawn, at seven per cent, payable on its face in 1-year 6 months and 14 days, so that when discounted at bank it shall produce $307,27? 3. A note is to be drawn having on its face 8 months and 12 days to run, and to bear an interest of 7 per cent, so that it will pay a debt of $5450: what is the amount? * The first rule founded on the above well-known principle, was, it is believed, first published by Roswell C. Smith, in his New Arithmetic. Qvr. —Wh"st will be the pres,.nt value of one dollar due in 3 months? How-will you find the face of a note, of a given present value, that shall be payable at a future time? 262 DISCOUNT. 4. What sum, 6 months and 9 days from July 18th; 1846, drawing an interest of 6 per cent, will pay a debt of $674,89 at bank, on the 1st of August, 1846? 5. Mr. Johnson has Mr. Squires' note for $874,57, having 4 months to run, from July 13th, and bearing an interest of 5 per cent. On the 1st of October he wishes to pay a debt at bank of $750,25. and gives the note in payment: how much must he receive back from the bank? 6. What must be the amount of a note discounted at 6 pr ct. having 4 months and 7 days to run, to pay a debt of $1 475,50! 7. Mr. Jones, on the 1st of June, desires to pay a debt at bank by a note dated May 16th, having 6 months to run and drawing 7 per cent interest: for what amount Inust the note be drawn the debt being $1683,75? 8. What amount at the end of one year, with grace, interest at 5 per cent, will pay $1004,20 at bank? DISCOUNT. 241. If I give my note to Mr. Wilson for $106, payable in one year, the true present value of the note will be less than $106 by the interest on its present value for one year; that is, its true present value will be $100. The true present value of a note is that sum which being put at interest until the note becomes due, would increase to an amount equal to the face of the note. Thus, $100 is the true present value of the note to Mr: Wilson. The discount is the difference between the face of a note and its true present value. Thus, $6 is the discount on the note to lMr. Wilson. To find the true present value of a note due at a future time, find the interest of $1 for the same time; then, $1 + its interest: $1: given sum: its present value. QUEST. —241. What is the true present value of a note? What is, the true discount? How do you find the true present value of a note due at a future time? DISCOUNT. 263 Hence, to find the present value of any sum, Add one dollar to its interest bfor the given time and divide the given amount by this number, and the quotient will be the present- value. EXAMPLES. 1. What is the present value of a note for $1828,75, due in one year, without grace, and bearing an interest of 4~ per cent per annnm? $1 + its interest for the given time - $1,045: Hence, $1828,75. $1,045 = $1750 the present value. PROOF. Int. on $1750 for 1 year, at 41 per cent = $78,75 Add principal - - 1750 Amount - - $1828,75 2. A note of $1651,50 is due in 11 months, without grace, but the person to whom it is payable sells it with the discount off at 7 per cent: how much shall he receive? 3. How much ought Mr. Ready to pay in cash for his note of ~36, due 15 rrmoths hence, without grace, it being discounted at 5 per cerit? 242. NoTE. —When payments are to be made at different times, find.the present value of the sums separately, and their sum will he the present value of the note. 4. What is the present value of a note for $10500, on which $900 are to be paid in six months; $2700 in one year; $3900 in eighteen months; and the residue at the expiration of two years, all without grace, the rate of interest being 6 per cent per annum? -5. What is the discount of ~4500, one-half payable in 6 months and the other half at the expiration of a year, without grace, at 7 per cent per annum? QUETr.-242. When payments are made at different times, how do you find the true present value? 264 DISCOUNT. 6. What is the present value of $5760, one-half payable in 3 months, one-third in 6 months, and the rest in 9 months, without grace, at 6 per cent per annum? 7. Mr. A gives his note to B for $720, one-half payable in 4 months and the other half in 8 months, without grace: what is the present value of said note, discount at 5 per cent per annLm? 8. What is the present value of ~825 payable as follows: one-half in 3 months, one-third in 6 months, and the rest in 9 months, without grace, the discount being 6 per cent per annum? 9. Bought goods for ~750 ready money, and sold them for ~900 payable by a note at 6 months, without grace: now, if I discount the note at 6 per cent per annum, will I make or lose? 10. What is the present value of $4000 payable in 9 months, without grace, discount 4~ per cent per annum? 11. How much corn must I carry to a miller that I may receive a bushel of meal, -I- being allowed for toll and waste? 12. Mr. Johnson has a note against Mr. Williams for $2146,50, dated August 17th, 1838, which becomes due Jan. 11th, 1839: if the note is- discounted at 6 per cent, what ready money must be paid for it September 25th, 1838? 13. C owes D $3456, to be paid October 27th, 1842: C wishes to pay on the 24th of August, 1838, to which D consents: how much ought D to receive, interest at 6 per cent? 14. What is the present value of a note of $4800, due 4 years hence, without grace, the interest being computed at 5 per cent per annum? 15. A man having a horse for sale, offered it for $225 cash in hand, or 230 at 9 months, without grace; the buyer chose the latter: did the seller lose or make by his offer, supposing money to be worth 7 per cent? INSURANCE. 265 INSURANCE. 243. INSURANCE is an agreement, generally in writing, by which an individual or company bind themselves to exempt the owners of certain property, such as ships, goods, houses, &c., from loss or hazard. The written agreement made by the parties, is called the policy. The amount paid by him who owns the property to those who insure it, as a compensation for their risk, is called the premium. The premium is generally so much per cent on the property insured, and is faund by the rules for simple interest. EXAMPLES. 1. What would be the premium for the insurance of a house valued at $8754 against loss by fire for 1 year, at a per cent? By multiplying by.01, we have the insurance $ 87,54 at 1 per cent - The half, is the insurance at half per cent - $43,77. 2. What would be the premium for insuring a ship and cargo, valued at $147674, from New York to Liverpool, at 3~ per cent? 3. What would be the insurance on a ship valued at $47520, at I per cent? -Also at I per cent? 4. What would be the insurance on a house valued at $16800, at 1 per cent? Also at 3 per cent? At per cent? At X per cent? At - per cent? 5. What is the insurance on a store and goods valued at $47000, at 21 per cent? At 2 per cent? At 1~ per cent? At j. per cent? At ~ per cent? At i per cent? At X per cent? At i per cent? QuEsT.-243. What is insurance? What is the written agreement called? What is the amount paid for the insurance called? How are the premiums generally estimated? How are they found? 12 266 ASSESSING TAXES. 6. A merchant wishes to insure on a vessel and cargo at sea, valued at $28800: what will be the premium at 13 per cent? 7. What is the premium on $2250 at 14 per cent? S. What is the premium on $8750 at 32 per cent? 9. A merchant owns three-fourths of a ship valued at $24000, and insures his interest at 2- per cent:.what does he pay for his policy? 10. A merchant learns that his vessel and cargo, valued at $36000, have been injured to the amount of $12000; he effects an insurance on the remainder at 5~ per cent: what premium does he pay? 11. What is the insurance on my house, valued at $7500. at l per cent? ASSESSING TAXES. 244. A TAX is a certain sum required to be paid by the inhabitants of a town, county, or state, for the support of government. It is generally collected from each individual, in proportion to the amount of his property. In some states, however, every white male citizen over the age of twenty-one years is required to pay a certain tax. This tax is called a poll-tax; and each person so taxed is called a poll. 245. In assessing taxes. the first thing to be done is to make a complete inventory of all the property in the town on which the tax is to be laid. If there is a poll tax, make a full list of the polls and multiply the number by the tax on each poll, and subtract the product fronm the whole tax to be raised by the town; the remainder will be the amount to be raised on the -property. Having done this, divide the whole QuEsT.-244. What is tax? How is it generally collected? What is a poll-tax? 245. What is the first thing to be done in assessing a tax? If there is a poll-tax, how do you find the amount? How then do you find the per cent of tax to be levied on a dollar? ASSESSING TAXES. 267 tax to be raised by the amount of taxable property, and the quotient will be the tax on $1. Then multiply this quotient by the inventory of each individual, and the product will be the tax on his property. EXAMPLES. 1. A certain town is to be taxed $4280; the property on which the tax is to be levied is valued at $1000000. Now there are 200 polls, each taxed $1,40. The property of A is valued at $2800, and he pays 4 polls, B's at $2400, pays 4 polls, E's at $7242, pays 4 polls, C's at $2530, pays 2 " F's at $1651, pays 6 " D's at $2250, pays 6 " G's at $1600,80 pays 4" What will be the tax on one dollar, and what will be A's tax, and also that of each on the list? First, $1,40 x 200 = $280 amount of poll-tax. $4280 - $280 = $4000 amount to be levied on property. Then, $4000. $1000000 = 4 mills on $1. Now, to find the tax of each, as A's, for example, A's inventory - - - $2800,004 11,20 4 polls at $1,40 each - 5,60 A's whole tax - - $16,80 In the same manner the tax of each person in the township may be found. 246. Having found the per cent, or the amount to be raised on each dollar, form a table showing the amount which certain sums would produce at the same rate per cent. Thus, after having found, as in the last example, that four mills are to be raised on every dollar, we can, by multiplying in succession by the numbers 1, 2, 3, 4, 5, 6, 7, 8, &c., form the following QUEsT.-How do you then find the amount to be levied on each individual? 246. How do you form an asseasment table? 268 ASSESSING TAXES. TABLE. $ $ $ $ $ $ - 1 gives 0,004 20 gives 0,080 300 gives 1,200 2 " 0,008 30 " 0,120 400 " 1,600 3 " 0,012 40 " 0,160 500 " 2,000 4 " 0,016 50 " 0,200 600 " 2,400 5 " 0,020 60 " 0,240 700 " 2,800 6 " 0,024 70 " 0,280 800 " 3,200 7 " 0,028 80 " 0,320 900 " 3,600 8 " 0,032 90 " 0,360 1000 " 4,000 9 " 0,036 100 " 0,400 2000 " 8,000 10 "y 0,040 200 " 0,800 3000 " 12.000 This table shows the amount to be raised on each sum in the columns under $'s. 1. To find the amount of B's tax from this table. B's tax on $2000 - is - $8,000 B's tax on 400 - - is - $1,600 B's tax on 4 polls, at $1,40 - $5,600 B's total tax - - is - $15,200 2. To find the amount of C's tax from the table. C's tax on $2000 - - is - $8,000 C's tax on 500 - - is - $2,000 C's tax on 30 - - is - $0,120 C's tax on 2 polls - - is - $2,800 C's total tax - - is - $12,920 In a similar manner, we might find the taxes to be paid by D, E, &c. 2. In a county embracing 350 polls, the amount of property on the tax list is $318200; the amount to be raised is as follows: for state purposes $1465,50; for county purposes $350,25; and for town purposes $200,25. By a vote of the county, a tax is levied on each poll of $1,50: how much per cent will be laid upon the property? 3. In a county embracing a population of 98415 persons, a tax is levied for town, county, and state purposes, amount E9UATION OF PAYMENTS. 20 ing to $100406. Of this sum, a part is to be raised by a tax of 25 cents on each poll, and the remainder by a tax of two mills on the dollar: what was the amount of property on the tax list? EQUATION OF PAYMENTS. 247. 1 OWE Mr. Wilson $2 to be paid in 6 months; $3 to be paid in 8 months; and $1 to be paid in 12 months. I wish to pay his entire dues at a single payment, to be made at such a time, that neither he nor I shall lose interest: at what time must the payment be made? The method of finding the mean time of payment of several sums due at different times, is called Equation of Payments. Taking the example above, Int. of $2 for 6io. = int. of $1 for 12mo. 2 x 6 = 12' of $3 for 8mo. = int.'of $1 for 24mo. 3 x 8 = 24 " of $1 for 12mo. = int. of $1 for 12mo. 1 X 12 = 12 $6 48 48 The interest on all the sums, to the times of payment, is equal to the interest of $1 for 48 months. But 48 is equal to the sum of all the products which arise from multiplying each sum by the time at which it'becomes due: hence, the sum of the products is equal to the time which would be necessary for $1 to produce the same interest as would be produced by all the sums. Now, if $1 will produce a certain interest in 48 months, in what time will $6 (or the sum of the payments) produce the same interest? The time is obviously found by dividing 48 (the sum of the products) by $6, (the sum of the payme nts.) QUEST.-247. What is Equation of Payments? What is the sum of the products, which arise from multiplying each payment by the time to which it becomes due, equal to? 270 EQUATION OF PAYMENTS. Hence, to find the mean time, Multiply each payment by the time before it becomes due, and divide the sum of the products by the sum of the payments: -the quotient will be the mean time. EXAMPLES. 1. B owes A $600: $200 is to be paid in two months, $200 in four months, and $200 in six months: what is the mean time for the payment of the whole? OPERATION. We here multiply each sum 200 X 2 = 400 by the time at which it be- 200 x 4 = 800 comes due, and divide the sum 200 X 6 = 1200 of the products by the sum of 6oo00 )24100 the payments. 4 Ans. 4 months 2. A merchant owes $1200, of which $200 is to be paid in 4 months, $400 in 10 months, and the remainder in 16 months: if he pays the whole at once, at what time must he make the payment? 3. A merchant owes $1800 to be paid in 12 months, $2400 to be paid in 6 months, and $2700 to be paid in 9 months: what is the equated time of payment? 4. A owes B $2400; one-third is to be paid in 6 months, one fourth in 8 months, and the remainder in 12 months: what is the mean time of payment? 5. A merchant has due him $600 to be paid in 30 days, -$1000 to be paid in 60 days, and 1500 to be paid in 90 days: what is the equated time for the payment of the whole? 6. A merchant has duethim $4500; one-sixth is to be paid in 4 months, one-third in 6 months, and the rest in 12 months: what is the equated time for the payment of the whole? QvUET. —How do you find the mean time of payment? When you reckon the time from the date at which the first payment becomes due, do you include the first payment? EQUATION OF PAYMENTS. 271 NOTE 1. —If one of the payments is due on the day from which the eqlated time is reckoned, its corresponding product will be nothing, bvit i!e' payment must still be added in finding the sum of the paynellCts. 7. I owe $1000 to be paid on the 1st of January, $1500 on the 1st of February, $3000 on the 1st of March, and $4000 on the 15th of April: reckoning from the 1st of January, and calling February 28 days, on what day must the money be paid? NOTE 2.-In finding the equated time of payments for several sums, due at different times, any day may be assumed as the one from which we reckon. Thus, if I owe Mr. Wilson $100 to be paid on the 15th of July, $208 on the 15th of August, and $300 on the 9th of September, and we require the mean time of a single payment, it would be most convenient to estimate from the 1st of July. From 1st of July to 1st payment 14 days " " " to 2d payment 45 days c" " " to 3d payment 70 days. 100 x 14 = 1400 200 X 45"= 9000 Then, by rule given above, we 300 x 70= 21000 600 6100)314100 521 Hence, the amount will fall due in 525 days from the 1st of July; that is, on the 22d day of August. But we may, if we please, demand at what time the payment would be due from the 1st of June. From June 1 st to 1st payment 44 days - " " "a to 2d payment 75 days " " " to 3d payment 100 days. Thus, 100 x 44 = 4400 200 x 75 - 15000 300 x 100 = 30000 600 6100)494100 821 272 EQUATION OF PAYMENTS. Hence, the payment becomes due in 824 days from June ist, or on the 22d of August-the same as before. Any day may, therefore, be taken as the one from which the mean time is estimated. 9. Mr. Jones purchased of Mr. Wilson, on a credit of six months, goods to the following amounts: 15th of January, a bill of $3750, 10th of February, a bill of 3000, 6th of March, a bill of 2400, 8th of June, a bill of 2250. He wishes, on the 1st of July, to give his note for the amount; at what time must it be made payable? 10. Mr. Gilbert bought $4000 worth of goods: he wvs to pay $1600 in five months, $1200 in six months, and the remainder in eight months: what will be the time of credit, if he pays the whole amount at a single payment? 11. A owes B $1200, of which $240 is to be paid in three months, $350 in five months, and the remainder in ten nionths what is the mean time of payment? 12. A merchant bought several lots of goods, as follows: A bill of $650, June 6th, Do. of 890, July 8th, Do. of 7940, August 1st. Now, if the credit is 6 months, at what time will the whole become due? 13. Mr. Swain bought goods to the amount of $3840, to be paid for as follows, viz.: one-fourth in cash, one-fourth in 6 months, one-fourth in 7 months, and the remainder in one year: what is the average time of payment? 14. Mr. Johnson sold, on a credit of 8 months, the following bills of goods: April 1st, a bill of $4350, May 7th, a bill of 3750, June 5th, a bill of 2550. At what time will the whole become due? PARTNERSHIP OR FELLOWSHIP. 273 PARTNERSHIP OR FELLOWSHIP. 248. PARTNERSHIP or Fellowship is the joining together of several persons in trade, with an agreement to share the losses and profits according to the amount which each one puts into the partnership. The money employed is called the Capital Stock. The gain or loss to be shared is called the Dividend. It is plain that the whole stock which suffers the gain or loss, must be to the gain or loss, as the stock of any individual to his part of the gain or loss. Hence, As Me whole stock is to each man's share, so is the whole gain or loss to each man's share of the gain or loss. PROOF. Add all the separate profits or shares together; their sum should be equal to the gross profit or stock. EXAMPLES. 1. A and B buy certain merchandise amounting to ~160, of which A pays ~90, and B ~70: they gain by the purchase ~32: what is each one's share of the profits? A - - ~90 B - - ~70 X1 6 90 32 ~18 A's share. ~ 70$ ~14 B's share. 2. A and B have a joint stock of $4200, of which A owns $3600, and B $600: they gain in a year $2000: what is each one's share of the profits? 3. A, B, C, and D have ~40,000 in trade: at the end of six months their profits amount to ~16,000: what is each one's share, supposing A to receive ~50 and D ~30 out of the profits, for extra services? QUEsT. —248. What is Partnership, or Fellowship? What is the gain or loss called? What is the rule for finding each one's share? 274 DOUBLE FELLOWSIIP. 4. Five persons, A, B, C, D, and E, have to share between them an estate of $20,000: A is to have one-fourth, B oneeighth, C one-sixth, D one-eighth, and E what is left: what will be the share of each? DOUBLE FELLOWSHIP. 249. WHEN several persons who are joined together in trade, employ their capital for different periods of time, the partnership is called Double Fellowship. For example, suppose A puts $100 in trade for 5 years, B $200 for 2 years, and C $300 for 1 year: this would make a case of double fellowship. Now it is plain that there are two circumstances which should determine each one's share of the profits: 1st, The amount of capital he puts in; and 2dly, The time which it is continued in the business. Hence, each man's share should be proportional to the capital he puts in, multiplied by the time it is continued in trade. Therefore, to find each share, Multip7y each man's stock by the time he continues it in trade; then say, as the sum of the products is to each particular product, so is the whole gain or loss to each man's share of the gain or loss. EXAMPLES. 1. A and B enter into partnership: A puts in ~840 for 4 months, and B puts in ~650 for 6 months: they gain ~300' what is each one's share of the profits? A's stock ~840 X 4=3360 B's stock ~650 X 6=3900 ~ s. d. ~7260: 3360 I:: 300: 138 16 10 ~7260 3900 164 3 1 QUEST. —249. What is Double Fellowship? What two circumstances determine each one's share of the profits? Give the rule for fiding each one's share. DOUBLE FELLOWVSHIP.. 275 2. A put in trade ~50 for 4 months, and B ~60 for 5 months: they gained ~24: how is it to be divided between them? 3. C and D hold a pasture together, for which they pay ~54: C pastures 23.horses for 27 days, and D 21 horses for 39 days: how much of the rent ought each one to pay? GENERAL EXAMPLES IN FELLOWSHIP. 1. A bankrupt is indebted $2729, viz.: to A $509,37; to B $228; to C $1291,23; and to D $709,40; but his estate is only worth $2046,75. How much can he pay on the dollar, and. how much will each creditor receive? 2. A, B, and C send a ship to sea, which together with her cargo was worth $15000. A and B owned each onefifth, and C the rest. They gained $1250: how much did each pay towards the ship and cargo, and what did each receive of the profits? 3. A man bequeathed his estate to his four sons in the following manner, viz.: to his first $5000; to his second $4500; to his third $4500; and to his fourth $4000. But on settling his estate, it was found that after paying debts, charges, &c., only $12000 remained to be divided: how much must each receive? 4. A widow and her two sons have a legacy of $4500, of which the widow is to have one-half and the sons each onefourth. Now suppose the eldest son to relinquish his share, and the whole to be divided in the above proportions between the mother and youngest son, what will each receive? 5. Suppose premiums to the value of $12 are to be distributed in a school in the following manner. The premiums are-&'vided into three grades. The value of a premium of the rst grade is twice the value of one' of the second; and the value of one of the second grade twice that of the third. Now' there are 6 to receive premiums of the first grade, 12 of the second, and 6 of the third: what will-be the value of a single premium of each grade? 276 ALLIGATION MEDIAL.4 6. Four traders form a company: A puts in $300 for 5 months, B $600 for 7 months, C $960 for 8 months, D $1200 for 9 months. In the course of trade they lost $750: how much falls to the share of each? 7. A and B lay out certain sums in merchandise amounting to $320, of which A pays $180 and B $140; they gain by the purchase $64: what is each one's shate? ALLIGATION MEDIAL. 250. A MERCHANT mixes 81b. of tea, worth 75cts. per pound, with 161b. worth $1,02 per pound: what is the value of the mixture per pound? The manner of finding the price of this mixture is called Alligation Medial. Hence, ALLIGATION MEDIAL teaches the method of finding the price'of a mixture when the simples of which it is composed, and their prices, are known. In the example above, the simples 81b. and 161b., and also their prices per pound, 75cts. and $1,02, are known. 81b. of tea at 75cts. per lb. 6,00 161b. " " $1,02 per lb. - -. - - 16,32 24 sum of simples. Total cost $22,32 -Now, if the entire cost of the mix- OPERATION. ture, which is $22,32, be divided by 24)22,32(93cts. 24, the number of pounds or sum of 216 the simples, the quotient 93cts. will be 72 the price per pound. Hence, to find 72 the price of the mixture, Divide the entire cost of the whole mixture by the sum of the simples, and the quotient will be the price of the mixture. QUEST.-250. What is Alligation Mediar? How do you find the price of the mixture? ALLIGATION ALTERNT'AE. 277 EXAMPLES. 1. A farmer mixes 30 bushels of wheat worth 5s. per bushelf, with 72 bushels of rye at 3s. per bushel, and with 60 bushels of barley worth 2s. per bushel: what is the value of a bushel of the mixture? 30 bushels of wheat at 5s. - - 150s. 72 " rye at 3s. - - 216s. 60 " " barley at 2s. - - 120s. 162 162)486(3s. 486 Ans. 3s. 2. A wine merchant mixes 15 gallons of wine at $1 pergallon with 25 gallons of brandy worth 75 cents per gallon: what is the value of a gallon of the compound? 3. A grocer mixes 80 gallons of whiskey worth 3lects. per gallon with 6 gallons of water, which'costs nothing: what is the value of a gallon of the mixture? 4. A goldsmith melts together 21b. of gold of 22 carats fine, 6oz. of 20 carats fine, and 6oz. of 16 carats fine-: what is the fineness of the mixture? 5. On a certain day the mercury in the thermometer was observed to average the following heights: from 6 in the morning to 9, 64~; from 9 to 12, 740; from 12 to 3, 84~; and from 3 to 6, 700: what was the mean temperature of the day? ALLIGATION ALTERNATE. 251. A FARMER would mix oats worth 3s. per bushel with wheat worth 9s. per bushel, so that the mixture shall be worth 5s. per bushel: what proportion must be taken of each sort? The method of finding how much of each sort must be taken is called Alligation Alternate. Hence, 278 ALLIGATION ALTERNATE. ALLIGATION ALTERNATE teaches the method 6f finding what proportion must be taken of several simples, whose prices are known, to form a compound of a given price. Alligation Alternate is the reverse of Alligation Medial, and may be proved by it. For a first example, let us take the one above stated. If oats worth 3s. per bushel be mixed with wheat worth 9s., how much must be taken of each sort that the compound may be worth 5s. per bushel? If the price of the mixture were 3 — 4 Oats. 6s., half the sum of the prices of' the 5 2 heat. simples, it is plain that it would be ( 9 2 Wheat. necessary to take just as much oats as wheat. But since the price of the mixture is nearer to the price of the oats than to that of the wheat, less wheat will be required in the mixture than oats. Having set down the prices of the simples under each other, and linked them together, we next set 5s., the price of the mixture, on the left. We then take the difference between 9 and 5 and place it opposite 3, the price of the oats, and also the difference between 5 and 3, and place it opposite 9, the price of the wheat. The difference standing opposite each kind shows how much of that kind is to be taken. In the present example, the mixture will consist of 4 bushels of oats and 2 of wheat; and any other quantities, bearing the same proportion to each other, such as 8 and 4, 20 and 10, &c., will give a mixture of the same value. PROOF BY ALLIGATION MEDIAL. 4 bushels of oats at 3s. - - 12s. 2 biushels of wheat at 9s. - - 18s. 6 6)30 Ans. 5s. QUEST. —251. What is Alligation Alternate? How do you prove kligation Alternate? ALLIGATION ALTERNATE. 279 CASE I. 252. To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price. I. Set down the prices of the simples one under the other, in the order of their values, beginning with the lowest. II. Link the least price with the greatest, and the one next to the least with the one next to the greatest and so on, until the price of each simple which is less than the price of the mixture is linked with one or more that is greater;and every one that is greater with one or more that is less. III. Write the difference between the price of the mixture and that of each of the simples opposite that price with which the particular simple is linked; then'the difference standing opposite any one price, or the sum of the differences when there is more than one, will express the quantity to be taken of that price. EXAMPLES. 1. A merchant would mix wines worth 16s., 18s., and 22s. per gallon in such a way, that the mixture may be worth 20s per gallon: how much must be taken of each sort? 16 12 at 16s. 20 18- 2 at 18s. 22' 4+2=6 at 22s. Ans. i 2gal. at 16s., 2 at 18s., and 6 at 22s.: or any other quantities bearing the proportion of 2, 2, and 6. 2. What proportions of coffee at 8cts., lOcts., and 14cts. per lb. must be mixed together so that the compound shall be worth 12cts. per lb.? 3. A goldsmith has gold of 16, of 18, of 23, and of 24 carats fine: what part must be taken of each so that the mixture shall be 21 carats fine? -QuEsr. —252. How do you find the proportions so that the compound may be of a given price? 280 ALLIGATION ALTERNATE. 4. What portion of brandy at 14s. per gallon, of old Madeira at 24s. per gallon, of new Madeira at-21s. per gallon, and of brandy at 10s. per gallon, must be mixed together so that the mixture shall be worth 18s. per gallon? CASE II. 253. When a given quantity of one of the simples is to be taken. I. Find the proportional quantities of the simples as in Case I. II. Then say, as the number opposite the simple whose quantity is given, is to either proportional quantity, so is the given quantity, to the proportional part of the corresponding simple. EXAMPLES. 1. How much wine at 5s., at 5s. 6d., and 6s. per gallonmust be mixed with 4 gallons at 4s. per gallon, so that the mixture shall be worth 5s. 4d. per gallon? 66] 4. proportional quantities. 72 16 Then 8 2 4 1 8: 4:: 4 2 8 16:: 4 8 Ans. lgal. at 5s., 2 at 5s. 6., and 8 at 6s PROOF BY ALLIGATION MEDIAL. 4gal. at 4s. per gallon - - 192d. 1 " 5s. " - - - 60 2 " 5s. 6d. " - - - 132 8 " 6s. " - - - 576 15 15)960(64d. price of mixture. QuEsT.-253. How do you find the proportion when the quantity of one of the simples is given? ALLIGATION ALTERNATE. 281 2. A farmer would mix 14 bushels of wheat at $1,20 per bushel, with rye at 72cts., barley at 48cts., and oats at 36cts.: how much must be taken of each sort to make the mixture worth 64 cents per bushel? 3. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at 18d. per bushel: how much has been taken of each sort when the mixture is worth 3s. 6d.? 4. A distiller would mix 40gal. of French brandy at 12s. per gallon, with English at 7s. and spirits at 4s. per gallon: what quantity must be taken of each sort, that the mixture may be afforded at 8s. per gallon? CASE III. 254. When the quantity of the compound is given as well as the price. 1. Find the proportional quantities as in Case I. II. Then say, as the sum of, the proportional quantities, is to each proportional quantity, so is the given quantity, to the corresponding part of each. EXAMPLES. 1. A grocer has four. sorts of sugar worth 12d., 10d., 6d., and 4d. per pound; he would make a mixture of 1441b. worth 8d. per pound: what quantity must be taken of each sort? 4 12 4:: 144: 48 6 2 12 2:: 144: 24 10] 2 12 12:: 144: 24 12~~' 4 12: 4:: 144: 48 Sum of the proportional parts 12 Ans. 481b. at 4d.; 241b. at 6d.; 241Ans. at 10d.; and 481b. at 12d. QuEr. —254. How do you determine the proportion when the quantity of the compound is given as well as the price? 282 CUSTOM HOUSE BUSINESS. PROOF BY ALLIGATION MEDIAL. 481b. at 4d. - - - - 192d. 241b. " 6d.' - - - 144d. 241b. " lOd. - - - - 240d. 481b. " 12d. - - - - 576d. 144 144)1152(8d. Hence, the average cost is 8d. 2. A grocer having four sorts of tea worth 5s., 6s., 8s., and 9s. per lb., wishes a mixture of 871b. worth 7s. per lb.: how much must be taken of each sort? 3. A vintner has four sorts of wine, viz., white wine at 4s. per gallon, Flemish at 6s. per gallon, Malaga at 8s. per galoin, and Canary at 1 Os. per gallon: he would make a mixture of 60 gallons to be worth 5s. per gallon: what quantity must be taken of each? 4. A silversmith has four sorts of gold, viz., of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine: he would make a mixture of 42oz. of 17 carats fine: how much must be taken of each sort? CUSTOM HOUSE BUSINESS. 255. PERSONS who bring goods, or merchandise, into the United States, from foreign countries, are required to land them at particular places or ports, called Ports of Entry, and to pay a certain amount on their value, called a Duty. This duty is imposed by the General Government, and must be the same on the same articles of merchandise, in every part of the United States. Besides the duties on merchandise, vessels employed in commerce are required, by law, to pay certain sums for the privilege of entering the ports. These sums are large or QUEST.-255. What is a port of entry? What is a duty? By whom are duties imposed? What charges are vessels required to pay? What are the moneys frising from duties and tonnage called? CUSTOM HOUSE BUSINESS. 283 small, in proportion to the size or tonnage of vessels. The moneys arising from duties and tonnage, are called revenues. 256. The revenues of the country are under the general direction of the Secretary of the Treasury, and to secure their faithful collection, the government has appointed various officers at each port of entry or place where goods may be landed. 257. The office established by the government at any port of entry, is called a Custom House, and the officers attached to it are called Cust6m House Officers. 258. All duties levied by law on goods imported into the United States, are collected at the various custom houses, and are of two kinds, Specific and Ad valorem. A specific duty is a certain sum on a particular kind of goods named; as so much per square yard on cotton or woollen cloths, so much per ton weight on iron, or so much per gallon on molasses. An ad valorem duty is such a per cent on the actual cost of the goods in the country from which they are imported. Thus, an ad valorem duty of 15 per cent on English cloths, is a duty of 15 per cent on the cost of cloths imported from England. 259. The laws of Congress provide, that the cargoes of all vessels freighted with foreign goods or merchandise, shall be weighed or gauged by the custom house officers at the port to which-they are consigned. As duties are only to be paid on the articles, and not on the boxes, casks, and bags which contain them, certain deductions are made from the weights and measures, called Allowances. Gross Weight is the whole weight of the goods, together QvuEsr.-256. Under whose direction are the revenues of the country? 257. What is a custom house? What are the officers attached to it called? 258. Where are the duties collected? How many kinds are there, and what are they called? What is a specific duty? An ad valorem duty? 259. What do the laws of Congress direct in relation to foreign goods? Why are deductions made from their weight? What are these deductions called? What is gross weight? 284 CUSTOM HOUSE BUSINESS. with that of the hogshead, barrel, box, bag, &c., which contains them. Draft is an allowance from the gross weight on account of waste, where there is not actual tare. lb. lb. On 112 itis 1; From 112 to 224 " 2, "C 224 to 336 c" 3, "4 336 to 1120 " 4,' 1120 to -2016 " 7, Above 2016 any weight" 9; consequently, 91b. is the greatest draft allowed. Tare is an allowance made for the weight of the boxes, barrels, or bags containing the commodity, and is of three kinds. 1st. Legal tare, or such as is established by law; 2d. Customary tare, or such as is established by the custom among merchants; and 3d. Actual tare, or such as is found by removing the goods and actually weighing the boxes or casks, in which they,are contained. On liquors in casks, customary tare is sometimes allowed on the supposition that the cask is not full, or what is called its actual wants; and then an allowance of 5 per cent for leakage. A tare of 10 per cent is allowed on porter, ale, and beer, in bottles, on account of breakage, and 5 per cent on all other liquors in bottles. At the custom house, bottles of the common size are estimated to contain 23- gallons the dozen.. For tables of Tare and Duty, see Ogden on theTariff of 1842. EXAMPLES. 1. What will be the duty on 125 cartons of ribbons, each containing 48 pieces, and each piece weighing 3oz. net, and paying a duty of $2,50 per lb.? QUEST.-What is draft? What is the greatest draft allowed? What is tare? What are the different kinds of tare? What allowances are made on liquors? FORMS RELATING TO BUSINESS IN GENERAL. 285 2. What will be the duty on 225 bags of coffee, each weighing gross 16Qlb., invoiced at 6 cents per lb.-; 2 per cent being the legal rate of tare, and 20 per cent the duty? 3. What duty must be paid on 275 dozen bottles of claret, estimated to contain 23 gallons per dozen, 5 per cent being allowed for breakage, and the duty being 35 cents per gallon? 4. A merchant imports 175 cases of indigo, each case weighing 1961b. gross: 15 per cent is the customary rate of tare, and the duty 5 cents per lb. What duty must he pay on the whole? 5. What is the tare and duty on 75 casks of Epsom salts, each weighing gross 2cwt. 2qr. 271b., and invoiced at 17 cents per lb., the customary tare being 11 per cent, and the rate of duty 20 per cent? FORMS RELATING TO BUSINESS IN GENERAL. FORMS OF ORDERS MESSRS. M. JAMES & CO. Please pay John Thompson, or order, five hundred dollars, and place the same to my account, for value received. PETER WORTHY. Wilmington, N. C., June 1, 1846. MR. JOSEPH RIcH, Please pay, for value received, the bearer, sixty-one dollars and twenty cents, in goods from your store, and charge the sanme to the account of your Obedient Servant, JOHM PARSONS. Savannah, Ga., July 1, 1846. FORMS OF RECEIPTS. Receipt for Money on Account. Received, Natchez, June 2d, 1845, of John Ward, sixty dollars on account. $60,00 JowN P. FAY. 286 FORMS RELATING TO BUSINESS IN GENERAL. Receipt for Money on a Note. Received, Nashville, June 5, 1846, of Leonard Walsh, six hundred and forty dollars, on his note for one thousand dollars, dated New York, January 1, 1845. $640,00 J. N. WEEKS. A BOND FOR ONE PERSON, WITH A CONDITION. KNOW ALL MEN BY THESE PRESENTS, THAT, I James Wilson of the City of Hartford and State of Connecticut, am held and firmly bound unto John Pickens of the Town of WVaterbury, County of New Haven and State of Connecticut, in the sum of Eighty dollars lawful money of the United States of America, to be paid to the said John Pickens, his executors, administrators, or assigns: for which payment well and truly to be made I bind myself, my heirs, executors, and administrators, firmly by these presents. Sealed with my Seal. Dated the Ninth day of March one thousand eight hundred and thirty-eight. THE CONDITION of the above obligation is such, that if the above bounden James Wilson, his heirs, executors, or adininistrators, shall well and truly pay or cause to be paid, unto the above named John Pickens, his executors, administrators, or assigns, the just and full sum of Here insert the condition. then the above obligation to be void, otherwise to remain in full force and virtue. Sealed and delivered in the presence of John Frost, James Wilson, Joseph Wiggins, NOTE.-The part in Italic to be filled up according to circumstance. If there is no condition to the bond, then all to be omitted after and including the words " THE CONDITION, &c." FORMS RELATING TO BUSINESS IN GENERAL. 287 A BOND FOR TWO PERSONS, WITH A CONDITION. KNOW ALL MEN BY THESE PRESENTS, THAT, WE James Wilson and Thomas Ash of the City of Hartford and State of Connecticut, are held and firmly bound unto John Pickens of the Town of WlVaterbury County of New Haven and State of Connecticut, in the sum of Eighty dollars lawful money of the United States of America, to be paid to the said John Pickens, his executors or assigns: for which payment well and truly to be made We bind ourselves, our heirs, executors, and administrators, firmly by these presents. Sealed with our Seal. Dated the Ninth day of March one thousand eight hundred and thirty-eight. THE CONDITION of the above obligation is such, that if the above bounden James Wilson and Thomas Ash, their heirs, executors, or administrators, shall well and truly pay or cause to be paid, unto the above named John Pickens, his executors, administrators, or assigns, the just and full sum of Here insert the condition. then the above obligation to be void, otherwise to remain in full force and virtue. Sealed and delivered in the presence of John Frost, James Wilson, 13 Joseph Wigfglns,I Thomas Ash. 4E NoTE. —The part in Italic to be filled up according to circumstance. If there is no condition to the bond, then all to be omitted after and including the words " THE CONDITION, &c." 288 GENERAL AVERAGE GENERAL AVERAGE. 260. AVERAGE is a term of commerce and navigation, to signify a contribution by individuals, where the goods of a particular merchant are thrown overboard in a storm, to save the ship from sinking, or where the masts, cables, anchors, or other furniture of the ship are cut away or destroyed for the preservation of the whole. In these and like cases, where any sacrifices are deliberately made, or any expenses voluntarily incurred, to prevent a total-loss, such sacrifice or expense is the proper subject of a general contribution, and ought to be rateably borne by the owners of the ship, the freight, and the cargo, so that the loss may fall proportionably on all. The amount sacrificed is called the jettison. 261. Average is either general or particular; that is, it is either chargeable to all the interests, viz., the ship, the freight, and the cargo, or only to some of them. As when losses occur from ordinary wear and tear, or frow the perils incident to the voyage, without being voluntarily incurred; or when any particular sacrifice is made for the sake of the ship only or the cargo only, these losses must be borne by the parties immediately interested, and are consequently defrayed by a particular average. There are also some small charges called petty or accustomed averages, one-third of which is usually charged to the ship and two-thirds to the cargo. No general average ever takes place, except it can be shown that the danger was imminent, and that the sacrifice was made indispensable, or supposed to be so by the captain and oficers, for the safety of the ship. 262. In different countries different modes are adopted of valuing the articles which are to constitute a general average. In general, however, the value of the freight is held to be the clear sum which the ship has earned after seamen's QUEST.-260. What does the term average signify? 261. How many kinds of average are there? What are the small charges called? Under what circumstances will a general average take place? 262. How is the freight valued? How much is charged on account of the seamen's wages I GENE1RAL AVERAGE. 289 wages, pilotage, and all such other charges as came under the name of petty charges, are deducted; one-third, anrl in some cases one-half, being deducted for the wages of the crew. The goods lost, as well as those saved, are valued at the price they would have brought in ready money at the place of delivery, on the ship's arriving there, freight, duties, and all other charges being deducted: indeed, they bear their proportions, the same as the goods saved. The ship is valued at the price she would bring on her arrival at the, port of delivery. But when the loss of masts, cables, and other furniture of the ship is compensated by general average, it is usual, as the lnew articles will be of greater value than the old, to deduct one-third, leaving two thirds only to be charged to the amount to be contributed. EXAMPLES. 1. The vessel Good Intent, bound from New York to New Orleans, was lost on the Jersey beach the day after sailing. She cut away her cables and masts, and cast overboard a part of her cargo, by which another part was injured. The ship was finally got off, and brought back to New York. AMOUNT OF LOSS. Goods of A cast overboard - - - $500 Damage of the goods of B by the jettison - 200 Freight of the goods cast overboard - - 100 Cable, anchors, mast, &c., worth $300 200 Deduct one-third - 100 Expenses of getting the ship off the sands 56 Pilotage and port duties going in and out 100 of the harbor, commissions, &c. -100 Expenses in port - 25 Adjusting the average - 4 Postage - 1 Total loss $1186 QUEST.-How is the cargo valued? Does the part lost bear its part of the loss? How is the ship valued? When parts of the ship are lost, how are they compensated for? How-do you explain the example? 13 290 GENERAL AVERAGE. ARTICLES TO CONTRIBUTE. Goods of A cast overboard $500 Value of B's goods at N. O., deducting freight, &c. 1000 -" of C's i' " S 500 " of I)'s " " " " 2000 " Of E's. " " 5000 Value of the ship -. - - 2000 Freight after deducting one-third - - 800 $11,800 Then, total value: total loss:: 100: per cent of loss. $11800: 1180:: 100: 10; hence, each loses 10 per cent on the value of his interest in the cargo, ship, or freight. Therefore, A loses $50, B $100, C $50, D $200, E $500, the owners of the ship $280-in al $1180. Upon this calculation the owners are to lose $280; but they are to receive their disbursements from the contribution, viz., freight on goods thrown overboard $100, damages to ship $200, various disbursements in expenses $I80, total $480; and deducting the amount of contribution, they will actually receive $200. Hence, the account will stand: The owners are to receive - $200 A loses $500, and is to contribute $50; hence,! 450 he receives B loses $200, and is to contribute $100; hence, ~ 100 he receives - ---- Total to be received - - - $750 C $ 50 C, D, and E have lost nothing, and are to pay D 200 E 500 Total actually paid - - - $750; do that the total to be paid is just equal to the total loss, as it should be, and A and B get their remaining and' injured goods, and the three others get theirs in a perfect state, after paying their rateable proportion of the loss. TONNAGE OF VESSELS. 291 TONNAGE OF VESSELS. 263. THERE are certain custom house charges on vessels, which are made according to their tonnage. The tonnage of a vessel-is the number of tons weight she will carry, and this is determined by measurement. [From the " Digest," by Andrew A. Jones, Esq., of the N. Y. Custom House.] Custom house charges on all ships or vessels entering from any foreign port or place. Ships or vessels of the United States, having three-fourths of the crew and all the officers American citizens, per ton $0,06 Ships or vessels of nations entitled by treaty to enter at the same rate as American vessels,06 Ships or vessels of the United States not having threefourths the crew as above,50 On foreign ships or vessels other than those entitled by treaty -,50 Additional tonnage on foreign vessels, denominated light money,50 Licensed coasters are also liable once in each year to a duty of 50 cents per ton, being engaged in a trade from a port in one state to a port in another state, other than an adjoining state, unless the officers and three-fourths of the crew are American citizens; to ascertain which, the crews are always liable to an examination by an officer. A foreign vessel is not permitted to carry on the coasting trade; but having arrived from a foreign port with a cargo consigned to more than one port of the United States, she may proceed coastwise with a certified manifest until her voyage is completed. 264. The government estimate the tonnage according to one rule, while the ship carpenter who builds the vessel uses another. QUEsT.-263. What is the tonnage of a vessel? What are the custom house charges on the different classes of vessels trading with foreign coun tries? To what charges are coasters subject? 292 TONNAGE OF VESSELS. GOVERNMENT RULE. I. Measure, in feet, above the u/pper deck the length of the vessel, from the fore pairt of te main stem to the after part of the stern post. Then measure the breadth taken at the widest part above the main wale on the outside, and tike depth from the under side of the deck plank to the ceiling in the hold. II. From the length take three-ffths of the breadth and multiply the remainder by Ihe breadth and depth, and the product divided by 95 will give the tonnage of a single decker; and the same for a double decker, by merely making' the depth equal to half the breadth. CARPENTERS? RULE. Multiply together the length of the keel, the breadth of the main beam, and the depth of the hold, and the product divided by 95 will be the carpenters' tonnage for a single decker; and for a double decker, deduct from the depth of the hold half the distance between decks. EXAMPLES. 1. What is the government tonnage of a single decker, whose length is 75 feet, breadth 20 feet, and depth 17 feet? 2. What is the carpenters' tonnage of a single decker, the length of whose keel is 90 feet, breadth 22 feet 7 inches, and depth 20 feet 6 inches? 3. What is the carpenters' tonnage of a steamship, doubledecker, length 154 feet, breadth 30 feet 8 inches, and depth after deducting half between decks, 14 feet 8 inches? 4. What is the government tonnage of a double decker, the length being 103 feet, breadth 25 feet 6 inches? 5. What is the carpenters' tonnage of a double decker, its length 125 feet, breadth 25 feet 6 inches, entire depth 34 feet, and distance between decks 8 feet? QUEST. —264. What is the government rule for finding the tonnage? What the ship-builders' rule. GAUGING 293 GAUGING. 265. CASK-GAUGING is the method of finding the number of gallons which' a cask contains, by measuring the external dimensions of the cask. 266. Casks are divided into' four varieties, according to the curvature of their sides. To which of the varieties any cask belongs, must be judged of by inspection. 1. Of the least curvature. 2d Variety. 3d Variety........... 4th Variety. 267. The first thing to be done is to find the mean diameter. To do this, Divide the head diameter by the bung diameter, and find the quotient in the first column of the following table, marked, Qu. Then if the buing diameter be multiplied by the number on the same line with it, anld in the column answering to the proper QuEST.-265. What is cask-gauging? 266. Into how many'varieties are casks divided? 267 How do you find the mean diameter? 294 GAUGING. variety, the product will be the -true mean diameter, or the diarneter of a cylinder of the same content with the cask proposed, cutting offfour figures for decimals. Qu. lst Var. 2d Var. 3d Var. 4th Var. Qu. 1st Var. 2d Var 3d Var. 4th Var. 50 8660 8465 7905 7637 76 9270 9227 8881 8827 51 8680 8493 7937 7681 77 9296 9258 8944 8874 52 8700 8520 7970 7725 78 9324 9290 8967 8922 153 8720 8548 8002 7769 79 9352 9320 9011 8970 54 8740 8576 8036 7813 80 9380 9352 9055 9018 55 8760 8605 8070 7858 81 9409 9383 9100 9066 56 8781 8633 8104 7902 82 9438 9415 9144 9114 57 8802 8662 8140 7947 83 9467 9446 9189 9163 58 8824 8690 8174 7992 84 9496 9478 9234 9211 59 8846 8720 8210 8037 85 9526 9510 9280 9260 60 8869 8748 8246 8082 86 9556 9542 9326 9308 61 8892 8777 8282 8128 87 9586 9574 9372 9357 62 8915 8806 8320 8173 88 9616 9606 9419 9406 63 8938 8835 8357 8220 89 9647 9638 9466 9455 64 8962 8865 8395 8265 90 9678, 9671 95i3 9504 65 8986 8894 8433 8311 91 9710 9703 9560 9553 66 9010 8924 8472 8357 92 9740 9736 9608 9602 67 9084 8954 8511 8404 93 9772 9768 9656 9652 68 9060 8983 8551 8450 94 9804 9801 9704 9701 69 9084 9013 8590 8497 95 9836 9834 9753 9751 70 9110 9044 8631 8544 96 9868 9867 9802 9800 71 9136 9074 8672 8590 97 9901 9900 9851 9850 72 9162 9104 8713 8637 98 9933 9933 9900 9900 73 9188 9135 8754 8685 99 -9966 9966 9950 9950 74 9215 9166 8796 8732 100 10000 10000 10000 10000 75 9242 9196 8838 87 80. EXAMPLES. 1. Supposing the diameters to be 32 and 24, it is required to find the mean diameter for each variety. Dividing 24 by 32, we obtain.75; which being found in the column of quotients, opposite thereto stand the numbers,.9242 9 47441.91] 96 1which being each mul- 2 272 for the corresond i.883899 tiplied by 32, produce 2816 4 ing mean dianse-.8780 respectively 28.0960 ters required. rset.8780 iL28.0960 GAUGING. 295 2. The head diameter of a' cask is 26 inches, and the bu,-g diameter 3 feet 2 inches: what is the mean diameter, tl,-,a:jk being of the third variety? 3. The head diameter is 22 inches, the bung diameter 34 inches: what is the mean diameter of a cask of the fourth variety? 268. Having found the mean diameter, we multiply the square of the mean diameter by the decimal.7854, and the product by the length; this will give the solid content in cubic inches. Then if we divide by 231, we have the content in wine gallons (see Art. 31), or if we divide by 282, we have the content in beer gallons. For wine measure we multiply OPERATION. the length by the square of the I x d2 x.7235_ = mean diameter, then by the deci- 1 x d2 x.0034. mal.7854, and divide by 231. If, then, we divide the decimal.7854 by 231, the quotient carried to four places of decimals is.0034, and this decimal multiplied by the square of the mean diameter and by the length of the cask, will give the content in wine gallons. For similar reasons, the con- OPERATION. tent is found in beer gallons by I X d' X *7854 multiplying together the length, X da X.0028. the square of the mean diameter, and the decimal.0028. Hence, for gauging or measuring casks, Multiply the length by the square of the mean diameter; then multiply by 34 for wine, and by 28 for beer measure, and point of in the product four decimal places. The product will then express gallons and the decimals of a gallon. 1. How many wine gallons in a cask, whose bung diameter is 36 inches, head diameter 30 inches, and length 50 inches; the cask being of the first variety? QUEST.-268. How do you find the solidity? How do you find the content in wine gallons? In beer gallons? 296 LIFE INSUR'ANCE. 2. What is the number of beer gallons in the last example? 3. How many wine, and how many beer gallons in a cask whose length is 36 inches, bung diameter 35 inches, and heat diameter 30 inches, it being of the first variety? 4. How many wine gallons in a cask of which the head diameter is 24 inches, bung diameter 36 inches, and length 3 feet 6 inches, the cask being of the second variety? LIFE INSURANCE. 269. INSURANCE for a term of years, or for the entire continuance of life, is a contract on the part of an authorized association to pay a certain sumn,'pecified in the policy of insurance, on the happening of an event named therein, and for which the association receives a certain premium, generally in the form of an annual payment. 270. To enable the company to fix their premiums at such rates as shall be both fair to the insured and safe to the association, they must know the average duration of life from its commencement to its extreme limit. This average is called the "Expectation of Life," and this is determined by collectinlg from many sources the most authentic information in regard to births and deaths. The " Carlisle Table," which is subjoined, and which shows the expectation of life from birth to 103 years, is considered the most accurate. It is much used in England, and is in general use in this country. By the "Expectation of Life," must be understood the average age of any number of individuals. Thus, if 100 infants be taken, some dying in infancy, some in childhood, some in youth, some in middle life, and some in old age, theaverage ages of all will be 38.72 yeais. So from 10 years old, the average age is 48 82 years. QUEST. —269. What is an insurance? 270. WhVat is necessary to enable a company to fix their premiums? tHow is the expectation determined? What Table is generally used in this country? What do you understand by the expectation of;fe? LIFE INSUIRANCE. 297 TABLE SHOWING THE EXPECTATION OF LIFE. Age. Expectation. Age. Expectation. Age. Expectation. Age. Expectation. 0 38.72 26 37.14 52 19.68 78 6.12 1 44.68 27 36.41 53 18.97 79 5.80 2 47.55 28 35.69 54 18.28 80 5.51 3 49.82 29 35.00 55 17.58 81 5.21 4 50.76 30 34.34 56 16.89 82 4.93 5 51.25 31 33.68 57 16.21 83 4.65 6 51.17 32 33.03 58 15.55 84 4.39 7 50.80 33 32.36 59 14.92 85 4.12 8 50.24 34 31.68 60 14.34 86 3.90 9 45.57 35 31.00 61 13.82 1 87 3.71 10 48.82 36 30.32 | 62 13.31 88 3.59 11 48.04 37 29.64 63 12.81 89 3.47 12 47.27 38 28.96 1 64 12.30 I 90 3.28 13 46.51 39 28.28 65 11.79 91 3.26 14 45.75 40 27.61' 66 11.27 i 92 3.37 15 45.00 41 26.97 67 10.75 93 3.48 16 44.27 42 26.34 68 10.23 94 3.53 17 43.57 43 25.71 69 9.70 1 95 3.53 18 42.87 44 25.09 70 9.19 96 3.46 19 42.17 45 24.46'!71 8.65 97 3.28 20 41.46. 46 23.82 72 8.16 98 3.07 21 40.75 47 23.17 { 73 7.72 99 2.77 22 40.04 48 22.50 i 74 7.33 100 2.28 23 39.31 49 21.81 17 5 7.01 101 1.79 24 38.59 50 i21.11 | 76 6.69 102 1.30 25 37.86 51 1 20.39 77 6.40 103 0.83 271. From the above table, and the value of money, which is shown by the rate of interest, a company can calculate with great exactness the amount which they shopld receive annually, for an insurance on a life for any number of years, or during its entire continuance. Among the principal life insurance companies in the United States, are the New York Life Insurance and Trust Company, the Girard Life Insurance, Annuity, and Trust Company of Philadelphia, and the Massachusetts Hospital Life Insurance and T'rust Company of Boston. QUEsT.-Explain the table showing the expectation of life. 271. What must be known besides the expectation of life in order to find the premium? What are the principal life insurance companies in the United States? How do you find the amount which must be paid for the insuranbe of -$100 19* 298 LIFE INSURANCE. NEW YORK AND PHILA. COMPANIES. MASSACHUSETTS. Age. 1 year. | 7 years. 1 For life. 1 year. 7 years. For life. 14.72.86 1.53.89 1.08 1.88 15.77.88 1.56.90 1.15 1.93 1.6.84.90 1.62.96 1.23 1.99 17.86.91 1.65 1.06 1.30 2.04 18.89.92 1.69 1.16 1.38 2.09 19.90.94 1.73 1.25 1.43 2.14 20.91.95 1.77 1.36 1.48 2.18 21.92.97 1.82 1.44 1.50 2.23 22.94.99 1.88 1.46 1.53 2.26 23.97 1.03 1.93 1.49 1.55 2.31 24.99 1.07 1.98 1.51 1.58 2.35 25 1.00 1.12 2.04 1.53 1.60 2.40 26 1.07 1.17 2.11 1.55 1.63 2.45 27 1.12 1.23 2.17 1.58 1.66 2.50 28 1.20 1.28 2.24- 1.60 1.69 2.55 29 1.28 1.35 2.31 1.64 1.71 2.61 30 1.31 1.36 2.36 1.66 1.75 2.66 31 1.32 1.42 2.43 1.69 1.78 2.73 32 1.33 1.46 2.50 I 1.71 1.81 2.79 33 1.34 1.48 2.57 1.75 1.84 2.85 34 1.35 1.50 2.64 1.79 1.89 2.93 35 1.36 1.53 2.75 1.81 1.94 2.99 36 1.39 1.57 2.81 1.85 1.98 3.06 37 1.43 1.63 2.90 1.89 2.05 3.14 38 1.48 1.70 3.05 1.93 2.09 3.23 39 1.57 1.76 3.11 1.96 2.15 3.31 40 1.69 1.83 3.20 2.04 2.20 3.40 41 1.78 1.88 3.31 2.10 2.26 3.49 42 1.85 1.89 3.40 2.18 2.33 3.59 43 1.89 1.92 3.51 2.23 2.39 3.69 44 1.90 1.94 3.63 2.28 2.46 3.79 45 1.91 1.96 3.73 2.34 2.54 3.90 46 1.92 1.98 3.87 2.39 2.63 4.01 47 1.93 1.99 4.01 2.45 2.71 4.13 48 1.94 2 02 4.17 2.51 2.81 4.25 49 1.95 2.04 4.49 2.61 2.93 4.39 50 1.96 2-09 4.60 2.75 3.04 4.54 51 1.97 2.20 4.75 2.86 3.14 4.68 52 2.02 2 37 4.90 2.95 3.24 4.83 53 2.10 2.59 5.24 3.05 3.35 4 98 54 2.18 2 89 5.49 3.15 3.48 5.14 55 2 32 3.21 5.78 3.25 3.60 5 31 56 2.47 3.56 6.05 3.36 3.74 5.50 57 2 70 4.20 6.27 3.49 3.88 5.70 58 3.14 4 31 6.50 3 61 4.03 5.91 59 3.67 4.63 6.75 3.75 4.19 6.14 60 4.35 4.91 7.00 3.90 4.35 6.36 ENDOWMENTS AND ANNUITIES. 299 The above table shows the rates at which they insure the amount of $100 for 1 year, for 7 years, or for life. It should be observed, that when a person insures for 7 years or for life, he pays annually the premium set opposite the age. PIaving found the premium for $100, it is easily found for arly other amount, by simply multiplying- by the amount and dividing by 100. EXAMPLES. 1. What will be the premium per annum on the insurance of a life for 7 years, for $4500, the person being at the age of 40 years, in the New York or Philadelphia companies? Premium per annum for 7 years on $100 = 1,83; then, 1,83 x 4500- 100 = 82,35: hence, $82,35 is the premium per annum. 2. What would be the premium per year if insured for life? 3. A person at 21 wishes to insure at his death $8500 to his friends: how much must he pay per annum to insure that amount at his death, in the Boston Company? ENDOWMENTS AND ANNUITIES. 272. AN ENDOWMENT iS a certain sum to be paid at the expiration of a given time, in case the person on whose life it is taken shall live till the expiration of the time named. 273. ANNUITIES are certain annual or periodical payments made to individuals by incorporated companies or associaLions, for a given sum paid in hand. 274. The following table shows the value of an endowment purchased for $100, at the several periods mentioned on the column of ages, the endowment to be paid if the person attains the age of 21 years. QUEST.-272. What is an endowment? 273. What is an annuity' 274. What does the table of endowments show? 13* ,'00 ENDOW1EN'TS AND ANNUITIES. TABLE OF ENDOWMENTS. Sum to be paid Age. Sum to be paid Ag Sum to be paid ge. at 21, if alive. ge. at 21, ifalive, e. at 21, if alive. Birth.... $376,84 5....... $210,53- 13.'1.... $144,12 3 months.. 344,28 6....... 198,83 114....... 137,86 6 ".. 331,46 7....... 188,83 15....... 131,83 9 ".. 318,90 8....... 179,97 116....... 125,97 1 year.... 306,58 9....... 171,91 17....... 120,31 2 ".... 271,03 10....... 164,46 18....... 114,89 3 ".... 243,69 i11... 157,43 19....... 109,70 4 ".... 225,42 12.. 150,64 20....... 104,74 275. The following table exhibits the sums which must be paid, at the several ages named, to purchase an annuity of $100 a year in the Massachusetts Life Insurance Co., and in the Girard Life Insurance, Annuity, and Trust Company, Philadelphia. Age. - Age. Age. 20....... $1836,30 39.. $1527,20 58...... $1125,00 21...... 1823,30 40..... 1507,40 59...... 1100,00 22...... 1809,50 41....... 1488,30 60....... 1070,00 23...... 1795,10 42....... 1469,40 61....... 1045,00 24....... 1780,10 43. 1450,50 62 fi....... 1020,00 25...... 1764,50 44....... 1430,80 63...... 995,00 26...... 1748,60 45.... 1410,40 ij64...... 970,00f 27....... 1732,00 46.... 1388,90 65........ 940,00 28...... 1715,40 47...... 1366,20 66... 910,00'29...... 1699,70 48....... 1341,90 67...... 880,00 30...... 1685,20 49.... 1315,30 68...... 850,00 31...... 1670,50 50....... 1300,00 69...... 820,00 32....... 1655,'20 51....... 1280,00 70....... 790,00 33....... 1639,00 5....... 1260,00'71... 780,00 34....... 1621,90 53...... 1240,00 72....... 770,00 35...... 1604,10 54...... 1220,00 73...... 760,00 36...... 1585,60 55...... 1200,00 74...... 750,00 37...... 1566,60 56...... 1175,00 75....... 740,00 38...... 1547,10 57....... 1150,00 EXAMPLES. 1. What sum at birth will purchase an endowment at 91 of $859,61? 2. What sum at the age of 30 years will purchase an annuity of $3150? QUEST.-275. What does the table of annaities show 1 COINS AND CURRENCIES. 301 COINS AND CURRENCIES. 276. COINs are pieces of metal, of gold, silver, or copper, of fixed values, and impressed with a public stamp prescribed by the country where they are made. These are called specie, and are generally declared to be a legal tender in payment of debts. The Constitution of the United States provides, that gold and silver only shall be a legal tender. The coins of a country and those of foreign countries having a fixed value established by law, together with bank notes redeemable in specie, make up what is called the Currency. 277. A foreign coin may be said to have four values: 1st. The intrinsic value, which is determined by the amount orf pure metal which it contains. 2d. The custom house or legal value, which is fixed by law. 3d. The mercantile value, which is the amount it will sell for in open market. 4th. The exchange value, which is the value assigned to it in buying and selling bills of exchange between one country and another. Let us take, as an example, the English pound sterling, which is represented by the gold sovereign. Its intrinsic value, as determined at the Mint in Philadelphia, compared with our gold eagle, is $4,861. Its legal or custom house value is $4-,84. Its commercial value, that is, what it will bring in Wall street, New York, varies from $4,83 to $4,86, seldom reaching either the lowest or highest limit. The QUEST.-276. What are coins? What are they called? What is declared in regard to them? What is provided by the Constitution of the United States? What do you understand by Currency? 277. How many values may a coin be said to have? What is the intrinsic value? What is the mercantile value? What is the exchange value? 302 COINS AND CURRENCIES. exchange value of the English pound, is $4,444, and was the legal value before the change in our standard. This change raised the legal value of the pound to $4,84, but merchants and dealers in exchange preferred to retain the old value, which became nominal, and to add the difference in the form of a premium on exchange, which is explained in Art. 292. TABLE OF FOREIGN COINS WHOSE VALUES ARE FIXED BY LAVW. $ cts. Franc of France and Belgium...................... 18i, Florin of the Netherlands..40 Guilder of do. 40 Livre Tournois of France..18. Milrea of Portugal................................1 12 Milrea of Madeira................................ 00 Milrea of the Azores.............................. 831 Marc Banco of Hamburg.. 35 Pound Sterling of Great Britain.................... 4 84 Pagoda of India.................................. 84 Real Vellon of Spain.. 05 Real Plate of do. 10 Rupee Company................. 44q Rupee of British India 44 Rix Dollar of Denmark..................1......... 00 Rix Dollar of Prussia. 681 Rix Dollar of Bremen.. 78 Rouble, silver, of Russia. 75 Tale of China................................... 1 48 Dollar of Sweden and Norway..................... 1 06 Specie Dollar of Denmark...................1..... 05 Dollar of Prussia and Northern States of Germany.... 69 Florin of Southern States of Germany.............. 40 Florin of Austria and city of Augsburg.. 481 Lira of the Lombardo Venetian Kingdom 16 Lira of Tuscany................................. 16 Lira of Sardinia.. 18.i Ducat of Naples................... 80 Ounce of Sicily.................... - 2.. 40 Pound of Nova Scotia, New Brunswick, Newfoundland, and Canada..................................... 4 04 QuEsr. —Give the different values of the English sovereign. How came the value of the sovereign to be altered? How is the difference now made up? EXCHANGE. 303 TABLE OF FOREIGN COINS WHOSE VALUES ARE FIXED BY USAGE, When a Consular's certificate of the real value or rate of exchange is not attached to the invoice. $ cts. Berlin Rix Dollar................................. 69' Current Marc..................................... 28 Crown of Tuscany................................ 1 05 Elberfeldt Rix Dollar............................. 693 Florin of Saxony................................. 48'" Bohemia............................... 48 " Elberfeldt................ 40 " Prussia................................. 22 Trieste............. 48 " Nuremburg............................ 40 "c Frankfort......................... 40 " Basil.................................. 41 " St. Gaul................................ 40 -6-0 " Creveld................................. i 40 Florence Livre................................... 15 Genoa do.................................... 183 Geneva do..................................... 21 Jamaica Pound................................... 5 00 Leghorn Dollar................................... 90 Leghorn Livre (6L to the dollar).................... 15L Livre of Catalonia................................ 53Neufchatel Livre................................. 26Pezza of Leghorn................................. 90 Rhenish Rix Dollar................................ 603 Swiss Livre..........................2......... 27 Scuda of Malta................................... 40 Turkish Piastre.................................. 05 [The above Tables are taken from a work on the Tariff, by E. D Ogden, Esq., of the New York Custom House.] EXCHANGE. 278.. EXCHANGE is a term which denotes the payment of money by a person residing in one place to a person residing in another. The payment is generally made by means of a bill of exchange. QUEST.-278 What is exchange? How is the payment generally made? 304 EXCHANGE. 279. A BILL OF EXCHANGE is an open letter of request from one person to another, desiring the payment to a third party named therein, of a certain sum of money to be paid at a specified time and place. There are always three parties to a bill of exchange, and generally four. 1. He who writes the open letter of request, is called the drawer or maker of the bill. 2. The person to -whoml it is directed is called the drauwee. 3. The person to whom the money is ordered to be paid is called the payee; and 4. Any person who purchases a bill of exchange is called the buyer or remitter. 280. Bills of exchange are the proper money of' commerce. Suppose Mr. Isaac Wilson of the city of New York, ships 1000 bags of cotton, worth ~96000, to Samuel Johns & Co. of Liverpool; and at about the same time William James of New York orders goods from Liverpool, of Ambrose Spooier, to the amount of eighty thousand pounds sterling. Now, Mr. Wilson draws a bill of exchange on Messrs. Johns & Co. in the following form: viz., Exchange for ~80000. New York, July 30th, 1846. Sixty days after sight of this my first Bill of Exchange (second and third of the same date and tenor uripaid*) pay to David C. Jones or order, eighty thousand pounds sterling, with or without further advice. ISAAC WILSON. Messrs. Samuel Johns & Co., Merchants, Liverpool. I Let us now suppose that Mr. James purchases this bill of David C. Jones for the purpose of sending it to Ambrose * Three bills are generally drawn for the same amount, called the first, second, and third, and together they form a set. One only is paid, and then the other two are -of no value. This arrangement avoids the accidents and delays incident to transmitting the bills. QUEST.-279. What is a bill of exchange? How many parties are there to a bill of exchange? Name them. 280. How do bills of exchange aid commerce? Name all the parties of the bill in this example. EXCHANGE, 305 Spooner of Liverpool, whom he owes. We shall then have all the parties to a bill of exchange;. viz., Isaac Wilson, thd maker or drawer; Messrs. Johns & Co., the drawees; David C. Jones, the payee; and William James, the buyer or remitter. 281. A bill of exchange is called an inland bill, when the drawer and drawee both reside in the same country; and when they reside in different countries, it is called a foreign bill. Thus, all bills in which the drawer and drawee reside ill the United States, are inland bills; but if one of them resides in England or France, the bill is a foreign bill. 282. The time at which a bill is made payable vari&s, and is a matter of agreement between the drawer and buyer. They may either be drawn at sight, or at a certain number of days after sight, or at a certain number of days after date. 283. DAYS OF GRACE are a certain numberof days granted to the person who pays the bill, after the time named in the bill has expired. In the United States and Great Britain three days are allowed. 284. In ascertaining the time when a bill payable so many days after sight, or after date, actually falls due. the day of presentment, or the day of the date, is not reckoned. When the time is expressed in months, calendar months are always understood. If the month in which a bill falls due is shorter than the one in which it is dated, it is a rule not to go on into the next month. Thus a bill drawn on the 28th, 29th, 30th, or 31st of December, payable two months after date, would fall due QUEST.-281. What is an inland bill? What is a foreign bill? Are bills drawn between one state and another inland or foreign? 282. Iow is the time determined at which a bill is made payable? fHow are bills always drawn? 283. What are days of grace? How many days of grace are allowed in this country and in Great Britain? 284. In ascertaining the time when a bill is payable, what days are reckoned? When the time is exlressed in months, what kind of months is understood? If the month in which the bill falls due is shorter than that in which it is drawn, what rule is observed'? 306 EXCHANGE. on the last of February, except for the days of grace, and would be actually due on the third of March. ENDORSING BILLS. 285. In examining the bill of exchange drawn by Isaac Wilson, it will he seen that Messrs. Johns & Co. are requested to pay the amount to David C. Jones or order; that is, either Mr. Jones or to any other person named by him. If Mr. Jones simply writes his name on the back of the bill, he is said to endorse it in blank, and the drawees must pay it to any rightful owner who presents it. Such rightful owner is called the holder, and Mr. Jones is called the endorser. If Mr. Jones writes on the back of the bill, over his signature, " Pay to the order of William James," this is called a special endorsement, and William James- is the endorsee, and he may either endorse in blank or write over his signature " Pay to the order of Ambrose Spooner," and the drawees, Messrs. Johns & Co., will then be bound to pay the amount to Mr. Spooner. A bill drawn payable to bearer, may be transferred by mere delivery. ACCEPTANCE. 286. When the bill drawn on Messrs. Johns & Co. is presented to them, they must inform the holder whether or not they will pay it at the expiration of the time named. Their agreement to pay it is signified by writing across the face of the bill, and over their signature the word " accepted," and they are then called the acceptors. LIABIIITIES OF THE PARTIES. 287. The drawee of a bill does not become responsible for its payment until after he has accepted. On the presentaQUEsT.-285. What is an endorsement in blank? What is the pe rso making it called? What is a special endorsement? What is the etrect of an endorsement? How may a bill drawn to bearer be transferred? 286. What is an acceptance? How is it made? 287. When does the drawee of a bill become responsible for its payment? ,EXCHANGE. 307 tion of the bill, if the drawee does not accept, the holder should immediately take means to have the drawer and all the endorsers notified. Such notice is called a protest, and is given by a public officer called a notary, or notary public. If the parties are not notified in a reasonable time, they are not responsible for the payment of the bill. If the drawer accepts the bill and fails to make the payment when it becomes due, the parties must be notified as before, and this is called protesting the bill for non-payment. If the endorsers are not notified in a reasonable time, they are not responsible for the amount of the bill. PAR OF EXCHANGE-COURSE OF EXCHANGE. 288. The intrinsic par of exchange, is a term used to compare the coins of different countries with each other, with respect to their intrinsic values, that is, with reference to the amount of pure metal in each. Thus, the English sovereign, which represents the pound sterling, is intrinsically worth $4,861 in our gold, taken as a standard, as determined at the Mirit in Philadelphia. This, therefbre, is the value at which the sovereign must be reckoned, in estimating the par of exchange. 289. The commercial par of exchange is a comparison of the coins of different countries according to their market value. Thus, the market value of the English sovereign, varying from $4,83 to $4,85 (Art. 277), the commercial par of exchange will fluctuate. It is, however, always determined when we know the value at which the foreign coin sells in our market. QUEsT.-If the drawee does not accept, what must the holder do? What is such notice called? By whom is it made? If the parties to the bill are not notified, what is the consequence? If the drawee accepts the hill and fails to make the payment, what must then be done? If the bill is not protested, what will be the consequence? 288. What do you understand by the intrinsic par of exchange? What is the intrinsic value of the English sovereign? 289. What is the commercial par of exchange? What is the commercial value of the English sovereign? 308 EXCHANGE. 290. The course of exchange is the variable price which is paid at one place for bills of exchange drawn on another. The course of exchange differs from the intrinsic par of exchange, afid also from the commercial par, in the same way that the market price of an article differs from its natural price. The commercial par of exchange would at all times determine the course of exchange, if there were no fluctuations in trade. 291. When the market price of a foreign bill is above the commercial par; the exchange is said to be at a premium, or in favor of the foreign place, because it indicates that the foreign place has sold more than it has bought, and that specie must be shipped to make up the difference. When the market price is below this paf, exchange is said to be below par, or in favor of the place where the bill is drawn. Such place will then be a creditor, and the debt must be paid in specie or other property. It sh9uld be observed that a favorable state of exchange is advantageous to the buyer but not to the seller, whose interest, as dealer in exchange, is identified with that of the place on which the bill is drawn. 292. It was stated in Art. 277 that the exchange value of the pound sterling is $4,44- = 4,4444+; that is, this value is the basis on which the bills of exchange are drawn. Now this value being below both the commercial and intrinsic value, the drawers of bills increase the course of exchange so as to make up this deficiency. For example, if' we add to the exchange value of the pound, 9 per cent, we shall have its commercial value, very nearly. Thus, exchange value = $4,4444 + Nine per cent --- =,9999+ which gives - - 4,8 143 QuEr.sT.-290. What do you understand by the course of exchange? How does it differ from the intrinsic par and the commercial par? What causes it to differ from the commercial par? 291. Wihat is said when the price of a foreign bill is above the commercial par? When it is below it? Towhiom is a favorable state of exchange advantageous? To whom is it injurious? 292. What is the exchange value of the pound sterling? EXCHANE. 309 and this is the average of the commercial value, very nearly. Therefore, when the course of exchange is at a premium of 9 per cent, it is at the commercial par, and as between England and this country it would stand near this point, but for the fluctuations of trade and other accidental circumstances. INLAND BILLS. 293. We have seen that inland bills are those in which the drawer and drawee both reside in the same country (Art. 281). EXAMPLES. 1. A merchant at New Orleans wishes to remit to New York $8465, and exchange is 1~- per cent premium. How much must he pay for such a bill? 2. A merchant in Boston wishes to pay in Philadelphia $8746,50; exchange between Boston and Philadelphia is 1- per cent below par. What must he pay for a bill? 3. A merchant in Philadelphia wishes to pay $9876,40 in Baltimore, and finds exchange to be 1 per cent below par: what must he pay for the bill? ENGLAND. 294. It has already been stated that the exchanges between this country and England are made in pounds, shillings and pence, and that the exchange value of the pound sterling is $4,444, and that the premiums are all reckoned from this standard. EXAMPLES. 1. A merchant in New York wishes to remit to Liverpool ~1167 10s. 6d., exchange being at 8~ per cent premium. How much must he pay for the bill in Federal money? QUEsr.-293. What are inland bills? 294. In what currency are the exchanges betwreen this country and England made? What is the exchange value of the pound sterling? 310 {EXCHANGE. First, ~1167 l10s. 6d. - - - = ~1167.525 For 8~ per cent multiply by -.085 the product is the premium - = 99.239625 this being added gives - - ~1266.764625 which reduced to.dollars and cents at the rate of $4,444 to the pound, gives the amount which must be paid for the bill in dollars and cents. 2. A merchant has to remit ~36794 8s. 9d. to London, how much must he pay for a bill in dollars and cents, ex change being 73 per cent premium? 3. A merchant in New York wishes to remit to London $67894,25, exchange being at a premium of 9 per cent. What will be the amount of his bill in pounds, shillings and pence? NOTE.-Add the amount of the premium to the exchange value of the pound, viz. $4,444, which in this case gives $4,84443, and then divide the amount in dollars by this sum, and the quotient will be the amount of the bill in pounds and the decimal of a pound. 4. A merchant in New York owes ~1256 18s. 9d. in London; exchange at a nominal premium of 7~ per cent: how much money in Federal currency will be necessary to purchase the bill? 5. I have.947,86 and wish to remit to London ~364 18s. 8d., exchange being at 81 per cent: how much additional money will be necessary? FRANCE. 295. The accounts in France, and the exchange between France and other countries, are all kept in francs -and centimes, which are hundredths of the franc. We see from the table that the value of the franc is 18.6 cents, which gives, very nearly, 5 francs and 38 centimes to the dollar. The rate of exchange is computed on the value 18.6 cents, but is often quoted by stating the value of the dollar in francs. QUEzT.-295. In wfiat currency are the exchanges with France conducted? What is a centime? What is the value of a franc? EXCHANGE. 311 Thus, exchange on Paris is said to be 5 francs, 40 centimes, that is, one dollar will buy a bill on Paris of 5 francs and 40 hundredths of a fratlc. EXAMPLES. 1. A merchant in New York wishes to remit 167556 francs to Paris, exchange being at a premium of 1~ per cent. What will be the cost of his bill in dollars and cents? Commercial value of the franc - - 18.6 cents Add 11 per cent ----- 279 Gives value for remitting 18.879 cents; then, 167556 X 18.879 = $31632,89724, which is the amount to be paid for the bill. 2. What amount in dollars and cents will purchase a bill on Paris for 86978 francs, exchange being at the rate of 5 francs and 2 centimes to the dollar? First, 86978-. 5.02 -$17326,274 + the amount. Is this bill above or below par? What per cent? 3. How much money must be paid to purchase a bill of exchange on Paris for 68097 francs, exchange being 3 per cent below par? 4. A merchant in New York wishes to remit $16785,25 to Paris; exchange gives 5 francs 4 centimes to the dollar: how much can he remit in the currency of Paris? HAMBURG. 296. Accounts and exchanges with Hamburg are generally made in the marc banco, valued, as we see in the table, at 35 cents. EXAMPLES. 1. What amount in dollars and cents will purchase a bill of exchange on Hamburg for 18649 marcs banco, exchange being at 2 per cent premium? QUEST. —What is meant when exchange on Paris is quoted at 5 francs t0 centimes? 296. In what are accounts kept at Hamburg? What is.he value of the marc banco? 312 ARBITRATION OF EXCHANGE. 2. What amount will purchase a bill for 3678 marcs banco reckoning the exchange value of the marc banco at 34 cents. Will this be above or below the par of exchange? ARBITRATION OF EXCHANGE. 297. Arbitration of exchange is the method by which the currency of one country is changed into that of another, through the medium of one or more intervening currencies, with which the first and last are compared. 98. When there is but one intervening currency it is called simple arbitration; and when there is more than one it is called compound arbitration..The method of performing this is called the Chain Rule. 299. The principle involved in arbitration of exchange is simply this: To pass from one system' of values through several others, and find the true proportion or relation between the first and last. For example, suppose we wish to exchange 109150 pence into dollars by first changing them into shillings, then into pounds, and then into, dollars. For this we have, 12: 109150:: 1s.: 109150 x - - = number of shillings. 20: 109150xx 1 ~1: ~109150x - x2 —=No. of pounds. ~1: $4,444:: 109150 x X: 109150 x X -_I X4 4-14: hence the Chain Rule may be stated as follows: Multiply the sum to be remitted by the following quotients, after having cancelled the common factors, viz., by a certain amount at the second place divided by its equivalent at the first; a certain amount at the third place by its equivalent at the second; a certazn amount at the fourth place divided by its equivalent at the third, and so on to the last place. QUEsT.-297. What is arbitration of exchange? 298. When there is but one intervening currency, what is the exchange called? When there is more than one, what is it called? 299. What principle is involved in the arbitration of exchange? What is the Chain Rule? Give the rule. ARBITRATION OF EXCHANGE. 313 NOTE.-In the above rule the amounts named are supposed to be expressed in the currency of the place from which the remittance is made. If in any case an amount is expressed in the currency of the place to which the remittance is made, the terms of the corresponding multiplier must be inverted. The example wrought above may be thus stated: Required to transmit 109150 pence to a second place where one piece of coin is worth 12 at the first place; thence to transmit it to a third where one piece is worth 20 at the second; thence to a fourth place where 4.444 pieces are equal to 1 at the third. EXAMPLES. 1. A merchant wishes to remit $4888,40 from New York to London, and the exchange is 10 per cent. He finds that he can remit to Paris at 5 francs 15 centimes to the dollar, and to Hamburg at 35 cents per marc banco. Now, the exchange between Paris and London is 25 francs 80 centimes for ~1 sterling, and between Hamburg and London 133 marcs banco for ~1 sterling. How had he better remit? 1st. To London direct. The amount to be remitted is $4888,40. The exchange value of ~1 is $4,444, and since the exchange is at a premium of 10 per cent, the value of ~1 is $4,444+,4444=$4,8884: hence, $4888,40 X i = ~1000: hence, if he remits direct he will obtain a bill for ~1000. 2d. Exchange through Paris. 1.03 4888,40 X - ~975,7852 = ~975 15s. 8jd. 5.16 Since 5,15 francs are equal to 1 dollar, the first multiplier will be this amount divided by $1; and since ~1 is equal to 25.80 francs, the second multiplier will be ~1 divided by this amount. Then by dividing by 5 and multiplying, we find that the amount remitted by the second method would b1 ~975 15s. 8-d. 14 314 ARBITRATION OF EXCHANGE. 3d. Method through Hamburg. $4888,40 x I.3 x = 1 015.771 = ~1015 15s. 5d. Since 1 marc bancois equal to 35 cents, it is 35 hun dredths of a dollar: hence, the first multiplier is 1 marc banco divided by.35, and the second 1 divided by 13.75. The result shows that the best way to remit is through Hamburg, the next best direct, and the most unfavorable through Paris. 2. A merchant in London has sold goods in Amsterdam to the amount of 824 pounds Flemish, which could be remitted to London at the rate of 34s. 4d. Flemish per pound sterling. He orders it to be remitted circuitously at the following rates, viz., to France at the rate of 48d. Flemish per crown; thence to Vienna at 100 crowns for 60 ducats; thence to Hamburg at 100d. Flemish per ducat; thence to Lisbon at 50d. Flemish per crusado of 400 reas; and lastly, from Lisbon to England at 5s. 8d. per milrea: does he gain or lose by the circular exchange? 48d. Flemish -= 1 crown, 100 crowns = 60 ducats, 1 ducat = 100d. Flemish, 50d. Flemish = 400 reas, 1 milrea or 1000 reas = 68d. sterling. My01 Tg 17 1,'i,''Bs,'k. 824 X 17 14008 q824xx1 XxX X~Pk- X25 25 25 = ~560 6s. 44d. The direct exchange would give, 34s. 4d. Flemish = 824 x 2 = ~480 sterling. Hence, the amount gained by circuitous exchange would be ~80 Gs. 41d. DUODE CIMALS. 315 DUODECIMALS. 300. DUODECIMALS are denominate fractions in which 1 foot is the unit that is divided. The unit 1 foot is first supposed to be divided into 12 equal parts, called inches or primes, and marked'. Each of these parts is supposed to be again divided into 12 equal parts, called seconds, and marked ". Each second is divided, in like manner, into 12 equal parts, called thirds, and marked ///. This division of the foot gives 1' inch or prime - - - = L2 of a foot. 1" second is =12 of 1 - - 11 of a foot. 1T/ so12 144 1///third is - 1 of 1 of 1 - 1 of a foot. 12 12 12 172W Hence, in duodecimals, the divisions of the foot increase from the lower denominations to the higher, according to the scale of twelves. 301. Duodecimals are added and subtracted like otherdenominate numbers, 12 of a lesser denomination making one of a greater, as in the following TABLE. 12"' make 1" second. 12" " 1' inch or prime. 12' " 1 foot. EXAMPLES. 1. In 185', how many feet? Ans. - 2. In 250", how many feet and inches? Ans. 3. In 4367"', how many feet? Ans. 4. In 847", how many feet? Ans. QUEST.-300. In Duodecimals, what is the unit that is divided? How is it divided? How are these parts again divided? What are the parts called? 301. How are duodecimals added and subtracted? How many of one denomination make 1 of the next greater? 316 MULTIPLICATION OF DUODECIMALS. EXAMPLES IN ADDITION AND SUBTRACTION. 1. What is the sum of 3ft. 6' 3" 2"' and 2ft. 1 10// 11"'? 2. What is the sum of 8ft. 9' 7" and 6ft. 7/ 3// 4"'/// 3. What is the difference between 9ft. 3/ 5' 6///' and 7ft. 34 6" 7///? 4. What is the difference between 40ft. 6/ 6" and 19ft. 7"'///? 5. What is the sum of 18ft. 9/ 11" 5/' and 17ft. 6/ 7//? 6. What is the difference between 27ft. 7/// and 4ft. 9' 10" 9"'? MULTIPLICATION OF DUODECIMALS. 302. It is known that feet multiplied by feet give square feet in the product. It is now required to show what fractions of the square foot will arise from multiplying feet by the divisions of the foot, and the divisions of the foot by each other. EXAMPLES. 1. Multiply 6ft. 7/ 8" by 2ft. 9/. Set down the multiplier under OPERATION. the multiplicand, so that feet shall ft. fall under feet, and the correspond- 6 7' 8" ing divisions under each other. It is found most convenient to begin 2 X 8"_ 1' 4" 2 X7'/- 1 2/ with the highest denomination of 2 X 6 1 21 the multiplier, and multiply it by 9/x 68" 6" the lowest denomination of the mul- 9' X 7/ - 5' 3// tiplicand. Recollecting that 7/' ex- 9/X 6 = 4 6/ presses j7- of a foot, and that 8// Prod. 18 3' 1 expresses iz of 1 of a foot, we see that 2 x 8" will give 16-twelfths of twelfths of a square foot; that is, one-twelfth and four twelfths of one twelfth, or 4". The 2 feet multiplied by 7' give 14 twelfths of a square foot; that is, 1 square foot and two twelfths, or 2/. The feet multiplied by 6 give 12 square feet. QUEsT.-302. In multiplication how do you set down the multiplier? Where do you begin to multiply? How do you carry from one denomination to another? MULTIPLICATION OF DIUODECIMALS. 317 Again, 9 inches or 9 of a foot multiplied by 8 twelfths of I of a foot, will give 72 twelfths of twelfths of twelfths of a square foot, which are equal to six twelfths of twelfths, or to 6"/. Then 9' x 7' gives 63 twelfths of twelfths of a square foot, equal to 5' and 3": and 9' X 6 gives 4 square feet and 6'. 303. Hence we see, 1st. That feet multiplied by feet give square feet in the product. 2d. That feet multiplied by inches give twelfths of square feet in the product. 3d..That inches multiplied by inches give twelfths of twelfths of square feet in the product. 4th. That inches multiplied by seconds give twelfths of twelfths of twelfths of square feet in the product. 2. Multiply 9ft. 4in. by oft. 3in. Beginning with the 8 feet, we OPERATION. say 8 times 4 are 32', which is 9 4/ equal to 2 feet 8': set down the 8 3/ 8'. Then say 8 times 9 are 72 74 8/ and 2 to carry are 74 feet: then 2 4/' 0// multiplying by 3/ we say, 3 times 77 0/ O" Ans. 4/ are 12/, equal to 1 inch: set down 0 in the second's place: then 3 times 9 are 27 and 1 to carry make 28/, equal to 2ft. 4/. Therefore the entire product is equal to 77ft. 3. How many solid feet in a stick of timber which is 25ft. 6in. long, 2ft. 7in. broad, and 3ft. 3in. thick? 4. Multiply 9ft. 2in. by 9ft. 6in. Ans. - 5. Multiply 34ft. loin. by 6ft. 8in. Ans. 6. Multiply 70ft. 9in. by 12ft. 3in. Ans. - 7. How many cords and cord feet in a pile of wood 24 feet long, 4 feet wide, and 3ft. 6in. high? 8. Multiply 6ft. 9' by 8ft. 6'. Ans. QUEST.-303. Repeat the four principle& 3818.INV:ovM' )IN. 9. How many cord feet in a pile of wood 25 feet long, 6 feet wide, and 5 feet high? 10. Multiply 16ft. 9' by l Ift. 1 ". Ans. - NOTE.-It must be recollected that 16 solid feet make 1 cord foot, (Art. 30). INVOLUTION. 304. IF a number be multiplied by itself, the product is called the second power, or square of that number. Thus, 4 X 4 = 16: the number 16 is the 2d power or square of 4. If a number be multiplied by itself, and the product arising be again multiplied by the number, the second product is called the 3d power, or cube of the number. Thus, 3 x 3 x 3 = 27: the number 27 is the 3d power, or cube of 3. The term power designates the product arising from multiplying a number by itself a certain number of times, and the number multiplied is called the root. Thus, in the first example above, 4 is the root, and 16 the square or 2d power of 4. In the 2d example, 3 is the root, and 27 the 3d power or cube of 3. The first power of a number is the number itself. 305. Involution teaches the method of finding the powers of numbers. The number which designates the power to which the root is to be raised, is called the index or exponent of the power. It is generally written on the right, and a little above the root. QUEST.-How many solid feet make a cord foot? 304. If a number be multiplied by itself once, what is the product called? If it be multiplied by itself twice, what is the product called? What does the term power mean?'What is the root? What is the first power of a number? 305. What is Involution? What is the number called which designates the power? Where is it written? I NVOL'UTlloN. 319 Thus, 42 expresses the 2d power of 4, or that 4 is to be multiplied by itself once: hence, 42 = 4 X 4 =- 16. For the same reason 33 denotes that 3 is to be raised to the 3d power, or cubed: hence, 33 = 3 X 3 x 3 = 27: we may therefore write 4 = 4 the 1st power of 4. 4a —4 X 4 = 16 the2d power of 4. 43= 4 X 4 X 4 = 64 the 3d power of 4. 44= 4 X 4 X 4 x 4-= 256 the 4th power of 4. 45 = 4 X 4 X 4 X 4 x 4 = 1024 the 5th power of 4. &c., &c., &c. Hence, to raise a number to any power, Multiply the number continually by itself as many times less 1 as there are units in the exponent, and the last product will be the- power sought. EXAMPLES. 1. What is the 3d power of 125? Ans. 2. What is the cube of 7? Ans. 3. What is the square of 60? Ans. - 4. What is the 4th power of 5? Ans. 5. What is the 5th power of 18? Ans. 6. What is the cube of 1? Ans. 7. What is the square of I? Ans. 8. What is the cube of.1? Ans. 9.'Vrhat is the cube of 3? Ans. 10. What is the square of.01? Ans. - 11. What is the square of 6.12? Ans. 12. What is the 6th power of 10? Ans. 13. What is the cube of 3~? Ans. 14. What is the 4th power of 36? Ans. 15. What is the cube of 8733? Ans. QUEsT.-What is the exponent of the square of a number? Of the cube? Of the fourth power? How do you raise a number to any power? 320 EVOLUTION. EVOLUTION. 306. WE have seen that Involution teaches how to find the power when the root is given. Evolution is the reverse of Involution: it teaches how to find the root when the power is known. The root -is that number which being multiplied by itself a certain number of times, will produce the given power. The square root of a number is that number which being multiplied by itself once, will produce the given number. The cube root of a number is that number which being multiplied by itself twice, will produce the given number. For example, 6 is the square root of 36, because 6 X 6 =36; and 3 is the cube root of 27, because 3 x 3 x 3 = 27. "he sign 5/- placed before a number denotes that its square root is to be extracted. Thus, 7/36 = 6. The sign V' is called the radical sign, or the sign of the square root. When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root: thus, -8 = 2 and {-" = 3, and 3 is called the index of the root. EXTRACTION OF THE SQUARE ROOT. 307. To extract the square root of a number, is to find a number which being multiplied by itself once, will produce the given number. Thus, VT4 2; for 2 X 2 4; V_9=3; for 3 X 3=9. QUEST.-306. What is Evolution? What does it teach? What is the root of a number? What is the square root of a number? What is the cube root of a number? Make the sign denoting the square root. How de you denote the cube root? 307. What is required when we wish to ex tract the square root of a number? EXTRACTION OF THE SQUARE ROOT. 321 Before proceeding to explain the rule for extracting the square root, let us first see how the squares of numbers are formed. The first ten numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Roots. 1 4 9 16 25 36 49 64 81 100 Squares. The numbers in the second line are the squares of those in the first; and the numbers in the first line are the square roots of the corresponding numbers of the second. Now, it is evident that, the square of a number expressed by a single figure will not contain any figure of a higher order than tens; and also, that if a number contains three figures, its root must contain tens and units The numbers 1, 4, 9, &c., of the second line, are called perfect squares, because they have exact roots. Let us now see how the square of any number may be formed, say the number 36. This number is made up of 3 tens or 30, and 6 units -Let the line AB represent F 30 I D 30 the 3 tens or 30, and BC the 6 6 six units. H 180 36 Let AD be a square on AC, and AE a square on the tens line AB. Then ED will be a square, 30 30 30 on the unit line 6, and the s180 rectangle EF Will be the product of HE which is equal to the tells line, by IE which' A 30 B C is equal to the unit line. Also, the rectangle BK will be the product of EB which is equal to the tens line, by the unit line BC. But the whole square QuEsT.-What is the greatest square of a single figure? What is the highest order of units that can be derived from the square of a single figure? How many perfect squares are there among the numbers that are less than sne hundred? 14# 322 E I.' o. on AC is made up of the square AE, the two rectangles FE and EC, and the square ED. Hence, The square of two figures is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. Let it now be required to extract the square root of 1296. Since the number contains more than two places, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the ten's place of the required root must be found in the figures on the left of 96. Hence, we point off the number into periods of two 12 96(36 figures each. 9 We next find the greatest square con- 6 6 tained in 12, which is 3 tens or 30. We 6)396 then square 3 tens which gives 9 hundred, and then place 9 under the hundred's place, and subtract. This takes away the square AE and leaves the two rect- F 30 I D 30 6 angles FE and BK, together 6 6 with the square ED on the H l unit line. Now, since tens multiplied 900+180180+36=1296. by units will give at least c tens in the product, it follows 30 30 that the area of the two rect- 900 180 angles FE and EC must be expressed by the figures of the given number at the left A 30 B C of the unit's place 6, which figures may also express a part of the square ED. If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BC or EK, the unit of the root. QUEST.-What is the square of a number expressed by two figures equal to? In what places of figures will the square of the tens be found? In what places will the product of the tells by the units be found? EXTRACTION OF THE SQUTARE ROOT. 323 Then, placing BC or 6, in the root, and also in the divisor, and then multiplying the whole divisor 66 by 6, we obtain for a product the two rectangles FE and CE, together with the square ED. Hence, the square root of 1296 is 36; or, in other words, 36 is the side of a square whose area is 1296. CASE I. 308. To extract the square root of a whole number, I. Point off the given number into periods of two figures each, countedfrom the right, by setting a dot over the place of units, another over the place of hundreds, and so on. II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second pericd for a dividend. III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure in the root and also at the right of the divisor. IV. Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. But if the product should'exceed the dividend, diminish the last figure of the root. V. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought down. EXAMPLES. 1. What is the square root of 263169? QUEST. —308. What is the first step in extracting the square root of numbers? What the second? What the third? What the fourth? What the fifth? Give the entire rule. 324 EVOLUTION. We first place a dot over the 9, OPERATION. making the right hand period 69. 26 31 69(513 We then put a dot over the 1 and 25 also over the 6, making three pe- 101)131 riods. 101 The greatest perfect square in 1023)3069 26, is 25, the root of which is 5. 3069 Placing 5 in the root, subtracting its square from 26, and bringing down the next period 31, we have 131 for a dividend, and by doubling the root we have 10 for a divisor. Now 10 is contained in 13, 1 time. Place 1 both in the root and in the divisor: then multiply 101 by 1; subtract the product and bring down the next period. We must now double the whole root 51 for a new divisor, or we may take the first divisor after having doubled the last figure 1; then dividing we obtain 3, the third figure of the root. 309. NOTE 1.-There will be as many figures in the root as there are periods in the given number. NOTE 2. —4f the given number has not an exact root, there will be a remainder after all the periods are brought down, in which case ciphers may be annexed, forming new periods, each of which will give one decimal place in the root. 2. What is the square root of 36729? OPERATION. 3 67 29(191.64+. 1 29)267 In this example there 261 are two places of deci- 381)629 mals, which give two pla- 381 ces of decimal in the root. 3826)24800 22956 38324)184400 153296 31104 Rem. QUErr. —309. How many figures will there be in the root? If the given number has not an exact root, what may be done? EXTRACTION OF THE SQUARE ROOT. 3-25 3. What is the square root of 213444? Ans. - 4. What is the square root of 2268741? Ans. 5. What is the square root of 15193592? Ans. 6. What is the square root of 36372961? Ans. - 7. What is the square root of 22071204? Ans. - CASE II. 310. To extract the square root of a decimal fraction, I. Annex one cipher, if necessary, so that the number of deczmal places shall be even. II. Point off the decimals into periods of two figures each, by putting a point over the place of hundredths, a second over the place of ten thousandths, 4c.: then extract the root as in whole numbers, recollecting that the number of decimal places in the root will be equal to the number of periods in the given decimal. EXAMPLES. 1. What is the square root of.5? OPERATION. We first annex one cipher to.50(.707+ 49 make even decimal places. We then extract the root of the first 140)100 000 period, to which we annex ciphers, forming new periods. 1407)10000 9849 151 Rem. NOTE. — When there is a decimal and a whole number joined together the same rule will apply. 2. What is the square root of 3271.4207? Ans. -I 3. What is the square root of 4795.25731? Ans. - 4. What is the square root of 4.372594? Ans. - 5. What is the square root of.00032754? Ans. - QuEST.-310. How do you extract the square root of a decimal fraction? When there is a decimal and a whole number joined together, will the same rule apply? 326 EVOLU'rION. 6. What is the square root of.00103041? Ans. - 7. What is the square root of 4.426816? Ans. - 8. What is the square root of 47.692836? Ans. CASE IlI. 311. To extract the square root of a vulgar fraction, I. Reduce mixed numbers to improper fractions, and compound fractions to simple ones, and then reduce the fraction to its lowest terms. II. Extract the square root of the numerator and denbminator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal and extract the root as in Case II. EXAMPLES. 1. What is the square root of 2 of of 4-? 2. What is the square root of 42 724? Ans. 3. What is the square root of 2-2Vt346? Ans. - 4. What is the square root of 2 75? Ans. 5. What is the square root of 4 5? Ans. - 6. What is the square root of 544? Ans. EXTRACTION OF THE CUBE ROOT. 312. To extract the cube root of a number is to find a second number which being multiplied into itself twice, shall produce the given number. Thus, 2 is the cube root of 8; for, 2 X 2 X 2 = 8: and 3 is the cube root of 27; for, 3 x 3 x 3 = 27. Roots 1, 2, 3, 4, 5, 6, 7, 8, 9. Cubes I 8 27 64 125 216 343 512 729. QUEST.-311. How do you extract the square root of a vulgar fraction? EXTRACTION OF THE CUBE ROOT. 327 From which we see, that the cube of units will not give a higher order than hundreds. WTe may also remark, that the cube of one ten or 10, is 1000: and the cube of 9 tens or 90, 729000; and hence, the cube of tens will not give a lower denomination than thousands, nor a higher denomination than hundreds of thousands. Hence also, if a number contains more than three figures its cube root will contain more than one; if the number contains more than six figures the root will contain more than two; and so on, every three figures from the right giving one additional place in the root, and the figures which remain at the left hand, although less than three, will also give one place in the root. Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written 10 + 6. Now to find the cube of 16 or of 10 + 6, we must multiply the number by itself twice. To do this we place the numbers thus 10 + 6 10 + 6 Product by the units - —.- 60 + 36 Product by the tens - - - - 100 + 60 Square of 16, -- 100 + 120 + 36 Multiply again by 16 - - - - - - 10 + 6 Product by the units - - - - 600 + 720 + 216 Product by the tens - - 1000 + 1200 + 360 Cube of 16 - - - 1000 + 1800 + 1080 + 216 1. By examining the composition of this number it will be found that the first part 1000 is the cube of the tens; that is, 10 X 10 X 10 =- 1000. 2. The second part 1800 is equal to three times the square of the tens multiplied by the units; that is, 3 x (10)2 x 6 = 3 X 100 X 6 = 1800. 3. The third part 1080 is equal to three times the square of the units multiplied by the tens; that is, 3 X 62 x 10 = 3 x 36 x 10 = 1080. 328 EVOLUTION. 4. The fourth part is equal to the cube of the units; that is, 6- = 6 x 6 x 6 = 216. Let it now be required to extract the cube root of the number 4096. Since the number con- OPERATION. tains more than three fig- 4 096(16 ures, we know that the root 1 will contain at least units 12 X 3 =3)3 0 (9-8-7-6 and tens. 16' = 4 096. Separating the three right hand figures from the 4, we know that the cube of the tens will be found in the 4. Now, 1 is the greatest cube in 4. Hence, we place the root I on the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, andto the remainder we bring down the first figure 0 of the next period. Now, we have seen that the second part of the cube of 16, viz., 1800, being three times the square of the tens multiplied by the units, will have no significant figure of a less denomination than hundreds, and consequently will make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If this were not the case, the 30 hundreds divided by three times the square of the tens wuutia give the unit figure exactly. Forming a divisor of three times the square of the tens, we find the quotient to be ten; but this we know to be too large. Placing 9 in the root and cubing 19, we find the result to be 6859. Then trying 8 we find the cube of 18 still too large; but when we take 6 we find the exact number. Hence, the cube root of 4096 is 16. CASE I. 313. To extract the cube root of a whole number, I. Point of the given number into periods of three figures each, by placing a dot over the place of units, a second over the EXTRACTION OF THE CUBE ROOT. 329 place of thousands, and so on to the left: the left hand period will often contain less than three places of figures. II. Seek the greatest cube in the first period, and set its root on the right after the manner of a quotient in division. Subtract the cube of this figure from the first period, and to the remainder bring down the first figure of the next period, and call this number the dividend. III. Take three times the square of the root just foundfor a divisor and see how often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the Jirst two periods of the given number, diminish the last figure, but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period, for a new dividend. IV. Take three times the square of the whole root for a new divisor, and seek how often it is contained in the new dividend: the quotient will be the third figure of the root. Cube the whole root and subtract the result from the first three periods of the given number, and proceed in a similar way for all the periods. EXAMPLES. 1. What is the cube root of 99252847? 99 252 847(463 43 - 64 42 X 3 = 48)352 dividend First two periods - - 99 252 (46)3 = 46 X 46 X 46 = 97 336 3 X (46)2 = 6348 ) 19168 2d dividend The first three periods - - 99 252 847 (463)3 = 99 252 847 Ans. 463. QUEsT. —312. What is required when we are to extract the cube root of a number? 313 How do you extract the cube root of a whole number? 330 EVOLUTION. 2. What is the cube root of 389017? Ans. 3. What is the cube root of 5735339? Ans. - 4. What is the cube root of 32461759? Ans. - 5. What is the cube root of 84604519? Ans. 6. What is the cube root of 259694072? Ans. 7. What is the cube root of 48228544?- Ans. 8. What is the cube root of 27054036008? Ans. CASE II. 314. To extract the cube root of a decimal fraction, Annex ciphers to the decimals, if necessary, so that it shall consist of. 3, 6, 9, 4c., places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; after which extract the root as in whole numbers. NOTE 1.-There will be as many decimal places in the root as there are periods in the given number. NOTE 2.-The same rule applies when the given number is composed of a whole number and a decimal. NOTE 3.-If in extracting the root of a number there is a re mainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. EXAMPLES. 1. What is the cube root of.127464? Ans. 2. What is the cube root of.870983875? Ans. 3. What is the cube root of 12.977875? Ans. 4. What is the cube root of 75.1089429? Ans. 5. What is the cube root of.353393243? Ans. 6. What is the cube root of 3.408862.625? Ans. 7. What is the cube root of 27.708101576? Ans. QUEST.-314. How doyou extract the cube root of a decimal fractio a How many decimal places will there be in the root? Will the same rule apply when there is a whole number and a decimal? In extracting the root if there is a remainder, what may be done? ARITHMETICAL PROGRESSION, 331 CASE III. 315. To extract the cube root of a vulgar fraction, I. Reduce compoundfractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms. II. Then extract the cube root of the numerator and denominator separately, if they have exact roots; but if either of them has not an exact root, reduce the fraction to a decimal, and extract the root as in the last Case. EXAMPLES. 1. What is the cube root of 2 5 -? Ans'. 2. What is the cube root of 2? Ans. 2. What is the cube root of 1219? Ans. 6. What is the cube root of 3? Ans. - 7. What is the cube root of 311? Ans. 4. What is the cube root of -~? Ans. 6. What is the cube root of 4? Ans. 7. What is the cube root of 5 7 Ans. ARITHMETICAL PROGRESSION. 316. IF we take any number, as 2, we can, by the con tinued addition of any other number, as 3, form a series of numbers: thus, 2, 5, 8, 11, 14, 17, 20, 23, &c., in which each number is formed by the addition of 3 to the preceding number. This series of numbers may also be formed by subtracting 3 continually from the larger number: thus, 23, 20, 17, 14, 11, 8, 5, 2. A series of numbers formed in either way is called an Arithmetical Series, or an Arithmetical Progression; and the.QuEsT.-315. How do you extract the cube root of a vulgar fraction? 316. How do you form an Arithmetical Series? 332 ARITHMETICAL PROGRESSION. number which is added or subtracted is called the common difference. When the series is formed by the continued addition of the common difference, it is called an ascending series; and when it is formed by the subtraction of the common difference, it is called a descending series; thus, 2, 5, 8, 11, 14, 17, 20, 23, is an ascending series. 23, 20, 17, 14, 11, 8, 5, 2, is a descending series. The several numbers are called terms of the progression: the first and last terms are called the extremes, and the intermediate terms are called the means. 317. In every arithmetical progression there are five things which are considered, any three of which being given or known, the remaining two can be determined. They are 1st, the first term; 2d, the last term; 3d, the common difference; 4th, the number of terms; 5th, the sum of all the terms. 318. By considering the manner in which the ascending progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term; the 3d, by adding the common difference to the 2d; the 4th, by adding the common difference to the 3d, and so on; the number of additions being 1 less than the number of terms found. But instead of making the additions, we may multiply the common difference by the number of additions, that is, by 1 less than the number of terms, and add the first term to the product. Hence, we have QUEST.-What is the common difference? What is an ascending series? What a descending series? What are the several numbers called? What are the first and last terms called? What are the intermediate terms called? 317. In every arithmetical progression how many things are considered? What are they? 318. How do you find the last term when the first term and common difference are known? ARITHMETICAL PROGRESSION. 333 CASE I. Having given the first term, the common difference, and the number of terms, to find the last term. Multiply the common difference by 1 less than the number of terms, and to the product add the first term. EXAMPLES. 1. The first term is 3, the common difference 2, and the number of terms 19: what is the last term? OPERATION. We multiply the number 18 number of terms less 1. of terms less 1, by the cornm- 2 common difference mon difference 2, and then 36 add the first term. 3 1st term. 39 last term. 2. A man bought 50 yards of cloth; he was to pay 6 cents for the first yard, 9 cents for the 2d, 12 cents for the 3d. and so on increasing by the common difference 3: how much did he pay for the last yard? 3. A man puts out $100 at simple interest, at 7 per cent; at the end of the first year it will have increased to $107, at the end of the 2d year to $114, and so on, increasing $7 each year: what will be the amount at the end of 16 years? 319. Since the last term of an arithmetical progression is equal to the first term added to the pro4uct of the common difference by 1 less than the number of terms, it follows, that the difference of the extremes -will be equal to this product, and that the common difference will be equal to this product divided by 1 less than the number of terms. Hence, we have CASE II. Having given the two extremes and the number of terms of an arithmetical progression, to find the common difference. Subtract the less extreme from the greater and divide the reQUEST.-319. How do you find the common difference, when you bnow the two extremes and number of terms? 334 ARITHIMETICAL PROGRESSION. mainder by 1 less than the number of terms: the quotient will be the common difference. EXAMPLES. 1. The extremes are 4 and 104, and the number of terms 26: what is the common difference? We subtract the less ex- OPERATION. treme from the greater and 104 divide the difference by one 4 less than the number of 26 - 1 = 25)100(4 terms. 100 2. A man has 8 sons, the youngest is 4 years old and the eldest 32, their ages increase in arithmetical progression: what is the common difference of their ages? 3. A man is to fravel from New York to a certain place in ]2 days; to go 3 miles the first day, increasing every day by the same number of miles; so that the last day's journey may be 58 miles: required the daily increase. 320. If we take any arithmetical series, as 3 5 7 9 11 13 15 17 19, &c. 19 17 15 13 11 9 7 5 3 by reversing the order of 22 22 22 22 22 22 22 22 22 1 the terms. Here we see that the sum of the terms of these two series is equal to 22, the sum of the extremes, multiplied by the number of terms; and consequently, the sum of either series is equal to the sum tf the two extremes multiplied by half the number of terms; hence, we have CASE III. To find the sum of all the terms of an arithmetical progression, Add the extremes together and multiply their sum by half the number of terms: the product will be sum of the series. EXAMPLES. 1. The extremes are 2 and 100, and the number of terms 22: what is the sum of the series? QUEST.-320. How do you find the sum of an arithmetical series? ARITHMETICAL PROGRESSION. 335 OPERATION. We first add together 2 1st term the two extremes, and 100 last term then multiply by half the 102 sum of extremes number of terms. 11 half the number of terms 1122 sum of series. 2. How many times does the hammer of a clock strike in 12 hours? 3. The first term of a series is 2, the common difference 4, and the number of terms 9: what is the last term and sum of the series? 4. If 100 eggs are placed in a right line, exactly one yard from each other, and the first one yard from a basket, what distance will a man travel who gathers them up singly, and places them in the basket? GENERAL EXAMPLES. 1. What is the 18th term of an arithmetical progression of which the first term is 4' and the common difference 5? 2. The 18th term of an arithmetical progression is 89 and the common difference 5: wbat is the first term? 3. A flight of stairs has 18 steps; the first ascends but 12 inches in a vertical line, and each of the others 18: what is the entire ascent in a vertical line? 4. A debtor has 18 creditors; he owes to the largest creditor 89 dollars, and 5 dollars less to each of the others in succession: how much does he owe to the least? 5. A person travelled from Boston to a certain place in 8 days; he travelled 2 miles the first day, and every succeeding day he travelled farther than he did the preceding by an equal number of miles: the last day he travelled 23 miles: how much did he travel each day, and how much in all? 6. The number of terms is 22, the common difference 5, and the sum of the terms 1221: what is the least term? 7.. A man is to receive $3000 in 12 payments, each succeeding payment to exceed the previous by $4: what will the last payment be? 336 GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION 321. IF we take any number, as 3, and multiply it continually by any other number, as 2, we form a series of numbers: thus, 3 6 12 24 48 96 192, &c., in which each number is formed by multiplying the number before it by 2. This series may also be formed by dividing continually the largest number 192 by 2. Thus, 192 96 48 24 12 6 3. A series formed in either way, is called a Geometrical Series, or a Geometrical Progression, and the number by which we continually multiply or divide, is called the common ratio. When the series is formed by multiplying continually by the common ratio, it is called an ascending series; and when it is formed by dividing continually by the common ratio, it is called a descending series. Thts, 3. 6 12 24 48 96 192 is an ascending series. 192 96 48 24.12 6 3 is a descending series. The several numbers are called terms of the progression. The first and last terms are called the extremes, and the intermediate terms are called the means. 322. In every Geometrical, as well as in every Arithmetical Progression, there are five things which are considered, any three of which being given or known, the remaining two can be determined. They are, QUEST.-321. How do you form a Geometrical Progression? What is the common ratio? What is an ascending series? What is a descending series? What are the several numbers called? What are the first and last terms called? What are the intermediate terms called? 322. Ir every geometrical progression, how many things are considered? What are they? GEOMETRICAL PROGRESSION. 3W7 1st, the first term, 2d, the last term, 3d, the common ratio, 4th, the number of terms, 5th, the sum of all the terms. By considering the manner in which the ascending progression is formed, we see that the second term is obtained by multiplying the first term by the common ratio; the 3d term by multiplying this product by the common ratio, and so on, the number of multiplications being one less than the number of terms. Thus, 3 = 3 1st term, 3X2 —=6 2d term, 3 X 2 x 2 = 12 3d term, 3 X 2 X 2 x 2 - 24 4th term, &c. for the other terms. But 2 x2=-22, 2 X 2 x2=-23, and 2 x2 x2 x2 = 24. Therefore, any term of the'progression is equal to the first term multiplied by the ratio raised to a power 1 less than the number of the term. CASE I. Having given the first term, the common ratio, and the number of terms, to find the last term, Raise the ratio to a power whose exponent is one less than the number of terms, and then multiply the power by the first term: the product will be the last term. EXAMPLES. 1. The first term is 3 and the ratio 2: what is the 6th term? 2 X 2 X 2 X 2 X 2 =2 = 32 3 1st term. Ans. 96 QusT.-I-How many must be known before the remaining ones can bs found? What is any term equal to? How do you find the lst term? 15 338 GEOMETRICAL PROGRESSION. 2. A man purchased 12 pears: he was to pay 1 farthing for the 1st, 2 farthings for the 2d, 4 for the 3d, and so on doubling each time: what did he pay for the last? 3. A gentleman dying left nine sons, and bequeathed his estate in the following manner: to his executors ~50; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger: what was the eldest son's portion? 4. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2d, 12 for the 3d, &c.: what did he pay for the last yard? CASE II. 323. Having given the ratio and the two extremes to find the sum of the series. Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme: the sum will be the sum of the series. EXAMPLES. 1. The first term is 3, the ratio 2, and last term 192; what is the sum of the series? 192 - 3 = 189 difference of the extremes, 2 -1 = 1) 189(189; then 189 + 192 = 381 Ans. 2. A gentleman married his daughter on New Year's day, and gave her husband Is. towards her portion, and was to double it on the first day of every month during the year: what was her portion? 3. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last: what did he pay for the last bushel and for the 10 bushels? 4. A man has six children; to the 1st he gives 815, to the 2d $300, 4to the 3d $600, and so on, to each twice as much as the last: how much did the eldest receive, and what was the amount received by them all? QuEsT. —223. How do you find the sum of the series? MENSURATION. 33g MENSURATION. 324. Mensuration is the process of determining the contents of geometrical figures, and is divided into two parts, the mensuration of surfaces and the mensuration of solids. MENSURATION OF SURFACES. 325. Surfaces have length and breadth. They are measured by means of a square, which is called the unit of surface. A square is the space included between 1 Foot. four equal lines, drawn perpendicular to each other. Each line is called a side of the square. foouare If each side be one foot, the figure is called a. square foot. If the sides of a square be each four feet, the square will contain sixteen square feet. For, in the large square there are sixteen small squares, the sides of which are each one foot. Therefore, the square whose side is four feet, contains sixteen square feet. The number of small squares that is contained in any large square is always equal to the product of two of the sides of the large square. As in the figure, 4 X 4=16 square feet. The number of square inches contained in a square foot is equal to 12X 12=144. 326. A triangle is a figure bounded by three straight lines. Thus, BAC is a triangle. QUEsT.-324. What is mensuration? 325. What is a surface? What is a square What is the number of small squares contained in a large square equal to? 326. What is a triangle? 340 MENSURATION. The three lines BA, AC, BC, are call-;,,, C ed sides: and the three corners, B, A, and C, are called angles. The side AB is called the base. When a line like CD is drawn making A D B the angle CDA equal to the angle CDB, then CD is said to be perpendicular to AB, and CD is called the altitude of the triangle.> Each triangle CAD or CDB is called a right-angled triangle. The side BC, or the side AC, opposite the right angle, is called the hypothenuse. The area or contents of a triangle is equal to half the product of its base by its altitude (Bk. IV. Prop. VI).* EXAMPLES. 1. The base, AB, of a triangle is 50 yards, and the perpendicular, CD, 30 yards: what is the area? We first multiply the base OPERATION. by the altitude, and the pro- 30 duct is square yards, which 2)1500 we divide by 2 for the area. Ans. 750 square yards. 2. In a triangular field the base is 60 chains, and the perpendicular 12 chains: how much does it contain? 3. There is a triangular field, of which the base is 45 rods and the perpendicular 38 rods: what-are its contents? 4. What are the contents of a triangle whose base is 75 chains and perpendicular 36 chains? 327. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are aot equal. * All the references are to Davies' Legendre. QuEsT.-326. What is the base of a triangle? What the altitude What is a right-angled triangle? Which side is the hypothenuset What is the area of a triangle equal to t 327. What is a rectangle? MENSURATION. 41 328. A parallelogram is a four-sided D figure which has its opposite sides equal and parallel, but its angles not rightangles. The line DE, perpendicular to the base, is called the altitude. E 329. To find the area of a square, rectangle, or parallelogram, Multiply the base by the perpendicular height, and the product will be the area (Bk. IV. Prop. V). EXAMPLES. 1. What is the area of a square field of which the sides are each 66.16 chains? 2. What is the area of a square piece of land of which the sides are 54 chains? 3. What is the area of a square piece of land of which the sides are 75 rods each? 4. What are the contents of a rectangular field, the length of which is 80 rods and the breadth 40 rods? 5. What are the contents of a field 80 rods square? 6. What are the contents of a rectangular field 30 chains long and 5 chains broad? 7. What are the contents of a field 54 chains long and 18 rods broad? 8. The base of a parallelogram is 542 yards, and the perpendicular height 720 feet: what is the area? 330. A trapezoid is a four-sided figure D E C ABCD, having two of its sides, AB, DC, parallel. The perpendicular EF is called. the altitude. A F B QUEST.-328. What is a parallelogram? 329. How do you find the area of a square, rectangle, or parallelogram? 330. What is a trapezoid I 842 MENSURATION. 331. To find the area of a trapezoid, Mlfultiply the sum of the two parallel sides by the altitude, and divide the product by 2, and the quotient will be the area (Bk. IV. Prop. VII). EXAMPLES. 1. Required the area or contents of the trapezoid ABCD, having given AB=643.02 feet, DC=428.48 feet, and EF =342.32 feet. We first find the sum of OPERATION. the sides, and then mul- 643.02 + 428.48 = 1071.50 = tiply it by the perpendi- sum of parallel sides. Then, cular height, after which, 1071.50 x 342.32=366795.88; we divide the product by and, 3626795. 18339794= 2, for the area. the area. 2. What is the area of a trapezoid, the parallel sides of which are 24.82 and 16.44 chains, and the perpendicular distance between them 10.30 chains? 3. Required the area of a trapezoid whose parallel sides are 51 feet, and 37 feet 6 inches, and the perpendicular distance between them 20 feet 10 inches. 4. Required the area of a trapezoid whose parallel sides are 41 and 24.5, and the perpendicular distance between them 21.5 yards. 5. What is the area of a trapezoid whose parallel sides are 15 chains, and 24.5 chains, and the perpendicular height 30.8 chains? 6. What are the contents when the parallel sides are 40 and 64 chains, and the perpendicular distance between them 52 chains? QusT. —831. How do you find the area of a trapezoid MENSURATION. 343 332. ~ circle is a portion of a plane bitndle(l by a curved line, every part of wiicii is equally distant from a certain point within, called the centre. C The curved line AEBD is called the circumference; the point C the centre; the line AB passing through the centre, a diameter; and CB the radius. The circumference AEBD is. 3.1416 times as great as the diameter AB. Hence, if the diameter is 1, the circumference will be 3.1416. Therefore, if the diameter is known, the circumference is found by multiplying 3.1416 by the diameter (Bk. V. Prop. XIV). EXAMPLES. 1. The diameter of a circle is 8: what is the circumference? OPERATION. The circumference is found by OPERATION. simply multiplying 3.1416 by the 8 diameter. Ans. 25.1328 2. The diameter of a circle is 186: what is the circumference? 3. The diameter of a circle is 40: what is the circumference? 4. What is the circumference of a circle whose diameter is 57? 333. Since the circumference of a circle is 3.1416 times as great as the diameter, it follows, that if the circumference is known, we may find the diameter by dividing it by 3.1416. QUEST.-332. What is a circle? What is the centre? What is the circumference? What is the diameter? What the radius? How many times greater is the circumference than the diameter? How do you find the circumference when the diameter is known? 333. How do you find the diameter when the circumference is known 8 *4A MENSURATION. EXAMPLES. 1. What is the diameter of a circle whose circumference is 157.08? We divide the circumference OPERATION: by 3.1416, the quotient 50 is the 3.1416)157.080(50 diameter. 157.080 2. What is the diameter of a circle whose circumference is 23304.3888? 3. What is the diameter of a circle whose circumference is 13700? Ans. - 334. To find the area or contents of a circle, Multiply the square of the diameter by the decimal.7854 (Bk. V. Prop. XII. Cor. 2). EXAMPLES. 1. What is the area of a circle whose diameter is 12? We first square the diam- OPERATION. eter, giving 144, which we 2= then multiply by the decmal 12 =113.0976 144 X.7854z-113.0976 7854: the product is thens. 113.0Y76 area of the circle. 2. What is the area of a circle whose diameter is 5? 3. What is the area of a circle whose diameter is 14? 4. How many square yards in a circle whose diameter is 31 feet? 335. A sphere is a solid termina- - ted by a curved surface, all the points of which are equally distant from a _Ad certain point within, called the centre. The line AD, passing through its centre C, is called the diameter of the sphere, and AC its radius. QUEST.-334. How do you find the area of a circle? 835. What is a sphere? What is a diameter? What is a radius MENSURATION. 345 336. To find the surface of a sphere,,iultiply the square of the diameter by 3.1416 (Bk. VIm. Prop. X. Cor.) EXAMPLES. 1. What is the surface of a sphere whose diameter is 6? We simply multiply the deci- OPERATION. mal 3.1416 by the square of 3.1416 -the diameter: the product is the surface. Ans. 113.976 2. What is the surface of a sphere whose diameter is 14 3. Required the number of square inches in the surface of a sphere whose diameter is 3 feet or 36 inches. 4. Required the area of the surface of the earth, its mean diameter being 7918.7 miles. MENSURATION OF SOLIDS. 337. A cube is a body, or solid, having six equal faces, which are squares. If the sides of the cube be each one foot long, the solid is called a cubic or solid foot. But when the sides of the cube are one yard, as in the figure, the cube is 3 feet=1 yard. called a cubic or solid yard. The base of the cube, which is the face on which it stands, contains 3 x 3=9 square feet. Therefore 9 cubes, of one foot each, can be placed on the base. If the solid were one foot high it would contain 9 cubic feet; if it were 2 feet high it would contain two tiers of cubes, or 18 cubic feet; and if it were 3 feet high, it would contain QrEsT.-336. How do you find the surface of a sphere? 337. What is a cue? What is a cubic or solid foot? What is a cubic yard I How many cubic feet in a cubic yard? 15* 346 MENSURATION. three tiers, or 27 cubic feet. Hence, the contents of a solid are equal to the product of its length, breadth, and height. 338. To find the solidity of a sphere, Multiply the surface by the diameter and divide the product by 6, the quotient will be the solidity (Bk. VIII. Prop. XIV. Sch. 3). EXAMPLES. 1. What is the solidity of a sphere whose diameter is 12? OPERATION. We first find the surface by OPERATION. multiplying the square of the multiply by 3.14 diameter by 3.1416. We then surface =452.3904 surface =452.3904 multiply the surface by the dia- diameter 12 meter, and divide the product 6)54281.6848 by 6. solidity =904.7808 2. What is the solidity of a sphere whose diameter is 8? 3. What is the solidity of a sphere whose diameter is 16 inches? 4. What is the solidity of the earth, its mean diameter being 7918.7 miles? 5. What is the solidity of a sphere whose diameter is 12 feet? 339. A prism is a solid whose ends are equal plane figures and whose faces are parallelograms. The sum of the sides which bound the base is called the perimeter of the base, and the sum of the parallelograms which bound the solid is called the convex surface. 340. To find the convex surface of a right prism, QUEsT. —What are the contents of a solid equal to? 838. How do you find the solidity of a sphere 339. What is a prism What is the perimeter of the base? What is the eonvex surface? MENSURATION..347 Multiply the perimeter of the base by the perpendicular height, and the product will be the convex surface (Bk. VII. Prop. I). EXAMPLES. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the altitude being 52~ feet? 2. What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet? 341. To find the solid contents of a prism, Multiply the area of the base by the altitude, and the product will be the contents (Bk. VII. Prop. XIV). EXAMPLES. 1. What are the contents of a square prism, each side of the square which forms the base being 16, and the altitude of the prism 30 feet? OPERATION. We first find the area of the -1626 square which forms the base, and 30 then multiply by the altitude. Ans. 7680 2. What are the solid contents of a cube, each side of which is 48 inches? 3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 5 feet? 4. How many gallons of water will a cistern contain, whose dimensions are the- same as in the last example? 5. Required the solidity of a triangular prism, whose height is 20 feet, and area of the base 691. QuEST.-340. How do you find the convex surface of a prism? 241. How do you find the solid contents of a prism! 3448 MENSURATION. 342. A cylinder is a round body with i circular ends. The line EF is called the axis or altitude, and the circular surface i'the convex surface of the cylinder. 343. To find the convex surface of a cylinder, Multiply the circumference of the base by the altitude, and the product will be the convex surface (Bk. VIII. Prop. I). EXAMPLES. 1. What is the convex surface of a cylinder, the diameter of whose base is 20 and the altitude 40? We first multiply 3.1416 by OPERATION. the diameter, which gives the cir- 20 cumference of the base. Then 62.8320 multiplying by the altitude, we 40 obtaih the convex surface. Ans. 2513.2800 2. What is the convex surface of a cylinder whose altitude is 28 feet and the circumference of its base 8 feet 4 inches? 3. What is the convex surface of a cylinder, the diameter of whose base is 15 inches and altitude 5 feet? 4. What is the convex surface of a cylinder, tl:e diameter of whose base is 40 and altitude 50 feet? 344. To find the solidity of a cylinder, Mrlultiply the area of the base by the altitude: the product will be the solid contents (Bk. VIII. Prop. II). QUEST.-342. What is a cylinder? What is the axis or altitude? What is the convex surface?'34. How do you find the convex surface? 844. How do you find the solidity of a cylinder? MENSURATION. 349 EXAMPLES. 1. Required the solidity of a cylinder of which the altitude is 11 feet, and the diameter of the base 16 feet. OPERATION. We first find the area of the 16 =-256 base, and then multiply by the.7854 altitude: the product is the soli- area base 201.0624 dity. 11 2111.6864 2. What is the solidity of a cylinder, the diameter of whose base is 40 and the altitude 29? 3. What is the solidity of a cylinder, the diameter of whose base is 24 and the altitude 30? 4. VWhat is the solidity of a cylinder, the diameter of whose base is 32 and altitude 12? 5. What is the solidity of a cylinder, the diameter of whose base is 25 and altitude 15 S 345. A pyramid is a solid formed by several triangular planes united at the same point S, and terminating in the different sides of a plane figure, as ABCDE. The altitude of the pyramid is the line SO, drawn perpendicular to E D the base. A 0 o B 346. To find the solidity of a pyramid, Multiply the area of the base by the altitude, and divide the product by 3 (Bk. VII. Prop. XVII). QUEST.-345. What is a pyramid I What is the altitude of a pyramid I 846. How do you find the solidity of a pyramid? 350 MENSURATION. EXAMPLES. 1. Required the solidity of a pyramid, of which the area of the base is 86 and the altitude 24. OPERATION. 86 We simply multiply the area of the 24 base 86, by the altitude 24, and then 344 divide the product by 3. 172 3)2064 Ans. 688 2. What is the solidity of a pyramid, the area of whose base is 365 and the altitude 36? 3. What is the solidity of a pyramid, the area of whose base is 207 and altitude 36? 4. What is the solidity of a pyramid, the area of whose base is 562 and altitude 30? 5. What are the solid contents of a pyramid, the area of whose base is 540 and altitude 32? 6. A pyramid has a rectangular base, the sides of which are 50 and 24; the altitude of the pyramid is 36: what are its solid contents? 7. A pyramid with a square base, of which each side is 15, has an altitude of 24: what are its solid contents? 347. A cone is a round body with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CB is called the axis or altitude. Qu.EST.-347. What is a cone? What is the vertex? What is the axis? $48. How do you find the solidity of a cone? MENSURATION. 351 348. To find the solidity of a cone, Multiply the area of the base by the altitude, and divide the product by 3; or, multiply the area of the base by onethird of the altitude. (Bk. VIII. Prop. V.) EXAMPLES. 1. Required the solidity of a cone, the diameter of whose base is 6 and the altitude 11. We first square the diameter, OPERATION. 62 —= 36 and multiply'it by.7854, which 36x.7854-28.2744 gives the area of the base. We 11 next multiply by the altitude, and 3)311.0184 then divide the product by 3. Ans. 1-03.6728 2. What is the solidity of a cone, the diameter of whose base is 36 and the altitude 2 7? 3. What are the solid contents of a cone, the diameter of whose base is 35 and the altitude 27? 4. What is the solidity of a cone, whose altitude is 27 feet and the diameter of the base 20 feet? RIGHT ANGLED TRIANGLE. 349. The properties of the right angled are so important as to be worthy of particular notice. In every right angled triangle, the square described on the hypothenuse, is equal to C the sum of the squares de- F E scribed on the other two sides. Thus, if ABC be a right angled triangle, right angled at C, then will the square D de- D scribed on AB be equal to the sum of the squares E and F, described on the sides CB and AC. This is called the carpenter's theorem. 352 MENSURATION. Hence, to find the hypothenuse when the base and perpendicular are known, 1st. Square each side separately. 2d. Add the squares together. 3d. Extract the square root of the sum, and the result will be the hypothenuse of the'triangle. EXAMPLES. 1. The wall of a building, on the brink of a river is 120 feet high, and the breadth of the river 70 yards: what is the length of a line which would reach from the top of the wall to the opposite edge of the river? 2. The side roofs of a house of which the eaves are of the same height, form a right angle with each other at the top. Now, the length of the rafters on one side is 10 feet, and on the other 14 feet: what is the breadth of the house? 3. What would be the width of the house, in the last example, if the rafters on each side were 10 feet? 350. When the hypothenuse and one side of a right angled triangle are known, to find the other side. Square the hypothenuse and also the other given side, and take their difference: extract the square root of their difference, and the result will be the required side. 1. The heioht of a precipice on the brink of a river is 103 feet, and a line of 320 feet in length will just reach from the top of it to the opposite bank: required the breadth of the river. 2. The hypothenuse of a triangle is 53 yards, and the perpendicular 45 yards: what is the base? 3. A ladder 60 feet in length, will reach to a window 40 feet from the ground on one side of the street, and by turninff it over to the other side, it will reach a window 50 feet from the ground: required the breadth of the street. QUEST.-349. What is the property of a right angled triangle 8 When can you find the hypothenuse How X 360. How do you find a side? OF THE MECHANICAL POWERS. 353 OF THE MECHANICAL POWERS.* 351. There are six simple machines, which are called Mechanical powers. They are, the Lever, the Pulley, the Wheel and Axle, the Inclined Plane, the Wedge, and the Screw. 352. To understand the power of a machine, four. things must be considered. 1st. The power or force which acts. This consists in the efforts of men or horses, of weights, springs, steam, &c. 2d. The resistance which is to be overcome by the power. This generally is a weight to be moved. 3d. We are to consider the centre of motion, or fulcrum, which means a prop. The prop or fulcrum is the point about which all the parts of the machine move. 4th. We are to consider the respective velocities of the power and resistance. 353. A machine is said to be in equilibrium when the resistance exactly balances the power, in which case all the parts of the machine are at rest. We shall first examine the lever. 354. The Lever, is a straight bar of wood or metal, which moves around a fixed point, called the fulcrum. There are three kinds of levers. Ist. When the fulcrum, is be- I tween the weight and the power. * This article is taken from a Practical Work for mechanics, entitled "Mensuration and Drawing." QUnET.-351. How many simple machines are there? What are they called? 352. What things must be considered in order to understand the power of a machine? 353. When is a machine said to be in equilibrium? 354. What is a lever? How many kinds of levers are there? Describe the first kind. 354 OF THE MECHANICAL POWERS. 2d. When the weight is between the power and the A fulcrum. 3d. When the power is between the fulcrum and the weight. The parts of the lever from the fulcrum to the weight and power, are called the arms of the lever. 355. An equilibrium is produced in all the levers, when the weight multiplied by its distance from the fulcrum is equal to the product of the power multiplied by its distance from the fulcrum. That is, The weight is to the power, as the distance from the power to the fulcrum, is to the distance from the weight to the fulcrum. EXAMPLES. 1. In a lever of the first kind, the fulcrum is placed at the middle point: what power will be necessary to balance a weight of 40 pounds? 2. In a lever of the second kind, the weight is placed at the middle point: what power will be necessary to sustain a weight of 50 lbs.? 3. In a lever of the third kind, the power is placed at QUEsT.-Where is the weight placed in the second kind? Where is the power placed in the third kind? 355. When is an equilibrium produced in all the levers? What is then the proportion between the weight and power? OF THE MECHANICAL POWERS. 355 the middle point: what power will be necessary to sustain a weight of 25 lbs.? -4. A lever of the first kind is 8 feet long, and a weight of 60 lbs. is at a distance of 2 feet from the fulcrum: what power will be necessary to balance it? 5. In a lever of the first kind, that is 6 feet long, a weight of 200 lbs. is placed at I foot from the fulcrum: what power will balance it? 6. In a lever of the first kind, like the common steelyard, the distance from the weight to the fulcrum is one inch: at what distance from the fulcrum must the poise of 1 lb. be placed, to balance a weight of 1 lb.? A weight of 1- lbs.? Of 2 lbs.? Of 4 lbs.? 7. In a lever of the third kind, the distance from the fulcrum to the power is 5 feet, and from the fulcrum to the weight 8 feet: what power is necessary to sustain a weight of 40 lbs.? 8. In a lever of the third kind, the distance from the fulcrum to the weight is 12 feet, and to the power 8 feet: what power will be necessary to sustain a weight of 100 lbs.? 356. REMARKS.-In determining the equilibrium of the lever, we have not considered its weight. In levers of the first kind, the weight of the lever generally adds to the power, but in the second and third kinds, the weight goes to diminish the effect of the power. In the previous examples, we have stated the circumstances under which the power will exactly sustain the weight. In order that the power may overcome the resistance, it must of course be somewhat increased. The lever is a very important mechanical power, being much used, and entering indeed into all the other machines. QUEST.-356. Has the weight been considered in determining the equilibrium of the levers? In a lever of the first kind, will the weight increase or diminish the power? How will it be in the two other kinds I 356 0F THE MECHANICAL POWERS. OF THE PULLEY. 357. The pulley is a wheel, having a groove cut in its circumference, for the purpose of receiving a cord which passes over it. When motion is imparted to the cord, the pulley turns around its axis, which is generally supported by being attached to a beam above. 358. Pulleys are divided into two kinds, fixed pulleys and moveable pulleys. When the pulley is fixed, it does not increase the power which is applied to raise the weight, but merely changes the direction in which it acts. 359. A moveable pulley gives a mechanical advantage. Thus, in the moveable pulley, the hand which sustains the cask does not actually support blut one-half the weight of it; the other half is supported by the hook to which the other end of the cord is attached. 360. If we have several moveable pulleys, the. advantage gained is still greater, and a very heavy weight may be raised by a small power. A longer time, however, will be required, than with the single pulley. It is indeed a general principle in machines, that what is gained in power, is QUEST.-357. What is a pulley? 358. How many kinds of pulleys are there? Does a fixed pulley give any increase of power? 359. Does a moveable pulley give any mechanical advantage? In a single moveable pulley, how much less is the power than the weight? 860. Will an advantage be gained by several moveable pulleys OP THE MECHANICAL POWERS. 357 lost in time; and this is true for all machines. There is also an actual loss of power, viz. the resistance of the machine to motion, arising from the rubbing of the parts against each other, which is called the friction of the machine. This varies in the different machines, but must- always be allowed for, in calculating the power necessary to do a given work. It would be wrong, however, to suppose that the loss was equivalent to the gain, and that no advantage is derived from the mechanical powers. We are unable to augment our strength, but, by the aid of science we so divide the resistance, that by a continued exertion of power, we accomplish that which it would be impossible to effect by a single effort. If in attaining this result, we sacrifice time, we cannot but see that it is most advantageously exchanged for power. 361. It is plain, that in the moveable pulley, all the parts of the cord will be equally stretched, and hence, each cord running from pulley to pulley, will bear an equal part of the weight; consequently the power will always be equal to the weiqght, divided by the number of cords which reach from pulley to pulley. EXAMPLES. 1. In a single immoveable pulley, what power will support a weight of 60 lbs.? 2. In a single moveable pulley, what power will support a weight of 80 lbs.? 8. In twq moveable pulleys with 5 cords, (see last fig.,) what power will support a weight of 100 lbs.? QUEST. —State the general principle in machines. What does the actual loss of power arise from? What is this rubbing called? Does this vary in different machines? 361. In the moveable pulley, what proportion exists between the cord and the weight? 358 OF THE MECHANICAL POWERS. WHEEL AND AXLE. 362. This machine is cornposed of a wheel or crank -firmly attached-to a cvl- i indrical axle. The axle is supported at its ends by wo pivots, ivhich are of less diameter than the axle around which the rope is coiled, and which turn freely about the points of support. In order to balance the weight, we must have The power to the weight, as the radius of the axle, to the length of the crank, or radius of -the wheel. EXAMPLES. 1. What must be the length of a crank or radius of a wheel, in order that a power of 40 lbs. may balance a weight of 600 lbs. suspended from an axle of 6 inches radius? 2. What must be the diameter of an axle that a power of 100 lbs. applied at the circumference of a wheel of 6 feet diameter may balance 400 lbs.! INCLINED PLANE. 363. The inclined plane is nothing more than a slope or declivity, which is used for the purpose of raising weights. It is not difficult to see that a weight can be forced up an inclined plane, more easily than it can be raised in a vertical line. But in this, as in the other machines, the advantage is obtained by a partial loss of power. QUEST.-362. Of what is the machine called the wheel and axle, composed How is the axle supported? Give the proportion between the power and the weight. 3863. What is an inclined plane? OF THE MECHANICAL POWERS. 351. Thus, if a weight W, W F be supported on the in- B dined plane ABC, by a P cord passing over a pul- A C ley at F, and the cord from the pulley to the weight be parallel to the length of the plane AB, the power P, will balance the weight W, when P W: height BC: length AB. It is evident that the power ought to be less than the weight, since a part of the weight is supported by the plane. EXAMPLES. 1. The length of a plane is 30 feet, and its height 6 feet: what power will be necessary to balance a weight of 200 lbs.? 2. The height of a plane is 10 feet, and the length 20 feet: what weight will a power of 50 lbs. support? 3. The height of a plane is 15 feet, and length 45 feet: what power will sustain a weight of 180 lbs.? THE WEDGE. 364. The wedge is composed of two inclined planes, united together along A their bases, and forming a solid ACB. It is used to cleave masses of wood or \ stone. The resistance which it overcomes is the attraction of cohesion of the body which it is employed to separate. The wedge acts principally by being struck with a hammer, or mallet, on its head, and very little effect can be produced with it, by mere pressure. All cutting instruments are constructed on the principle QUEsT.-What proportion exists between the power and weight when they are in equilibrium? 864. What is the wedge I What is it used for? What resistance is it used to overcome 360 OF THE MECHANICAT, POWERS. of the inclined plane or wedge. Such as have but one sloping edge, like the chisel, may be referred to the inclined plane, and such as have two, like the axe and the knife, to the wedge. THE SCREW. 365. The screw is composed S of two parts-the screw S, and the nut N. The screw S, is a cylinder with a spiral projection winding around it, called the thread. The nut N is perforated to admit the screw, and within it is a groove into which the thread of the screw fits closely. The/ handle D, which projects from the nut, is a lever which works the nut upon the screw. The power of the screw depends on the distance between the threads. The closer the threads of,the screw, the greater will be the power; but then the number of revolutions made by the handle D, will also be proportionably increased; so that we return to the general principle-what is gained in power is lost in time. The power of the screw may also be increased by lengthening the lever attached to the nut. The screw is used for compression, and to raise heavy weights. It is used in cider and wine-presses, in coining, and for a variety of other purposes. QuEsT. —366. Of how many parts is the screw composed? Describe the screw. What is the thread? What the nut? What is the handle used for? To what uses is the screw applied? PROMISCUOUS QUESTIONS. 361 PROMISCUOUS QUESTIONS. 1. Two persons have put in trade each a certain sum; that which the first contributed is to that of the second as 11 to 15: the first put in $1359: what did the second contribute? 2. Twelve workmen working 12 hours a day have made in 12 days 12 pieces of cloth, each piece 75 yards long. How many pieces of the same stuff would have been made, each piece 25 yards long, if there had been 7 more workmen? 3. A workman earns $18,50 by working 12 days in 14, during these 14 days he spends 50 cents a day for his board and gives 4 cents a day to the poor; on Sunday he triples the alms. How long will it take him at this rate to pay his rent, which is $56, and a debt of $11,50? 4. How much time would it require to receive $80 of interest with a capital of $400, knowing that $600 placed at the same rate would produce an interest, of $90 every three years? 5. If $100 at interest gains $3 every nine months, what capital would be necessary to gain $800 every two years? 6. Four partners have gained $21175; the first is to have $4250 more than the second; the second $1700 more than the third; the third $1175 more than the fourth: whdt is the share of each? 7. The sum of two numbers is 5330, their difference 1999: what are the two numbers? 8. A person was born on the 1st of October, 1792, at 6 o'clock in the morning; what was his age on the 21st of September, 1839, at half past 4 in the afternoon? 9. A merchant bought 80 yards of cloth, then sold 140 yards, after which there remained to him one half the quan16 3F'2 PROMISCUOUS QUESTIONS. tity he had in the store before his last purchase: what was this quantity? 10. Sourid travels about 1142 feet in a second If then the flash of a cannon be seen at the moment it is fired, and the report heard 45 seconds after, what distance would the observer be from the gun? 11. A person having a certain sum borrowed $65,50, and then paid a debt of $94,90; he received $56,75 which was due him, and found that he had $49,30 after having expended $9,30. How much had he at first? 12. A house which was sold a second time for $7180, would have given a profit of $420 if the second proprietor had purchased it $130 cheaper than he did: at what price did he purchase it? 13. A person purchased 78000 quills, for half of which he gave $4,50 per thousand, and for the rest 87-1 cents per handred; he sells them at 1- cents each: what is his profit supposing he takes 265 for his-own use? 14. In order to take a boat through a lock from a certain river into a canal, as well as to descend from the canal into the river, a body of water is necessary 46~ yards long, 8 yards wide, and 2-~ yards deep. How many cubic yards of water will this canal throw into the river in a year, if 40 boats ascend and 40 descend each day except Sundays and eight holidays? 15. How many scholars are there in a class, to which if 11 be added the number will be augmented one-sixteenth? 16. A person being asked the time, said, the time past noon is equal to - of the time past midnight: what was the. hour? 17. What number is that which being augmented by 85, and this sum divided by 9, will give 25 for the quotient? 18. Three travellers have 1377 miles to go before they reach the end of their journey; the first goes 30 miles a day, the second 27, and the third 24: how many days should one set out after another that they may arrive together? PROMISCUOUS QUESTIONS. 363 19. A company numbering sixty-six shareholders have constructed a bridge which cost $200000: what will be the gain of each partner at the end of 22 years, supposing that 6400 persons pass each day, and that each pays one cent toll, the expense for repairs, &c., being $5 per year for each shareholder? 20. The entire length of the walls of a fort is 495 yards, their height 8F yards, and their thickness 3 yards: how many years has it taken to construct them., each cubic yard having cost 16 francs, and the expenses having been 20086 francs per year; and what will this sum amount to in dollars and cents, at the custom house value? 21. One-fifth of an army was killed in battle, 1- part was taken prisoners, and Ir died by sickness: if 4000 men were left, how many men did the army at first consist of? 22. A person delivered to another a sum of money to receive interest for the same at 4 per cent per annum. At the end of three years he received for principal and interest ~176 8s. What was the sum lent? 23. A snail in getting up a pole 20 feet high, was observed to climb up 8 feet every day, but to descend 4 feet every night: in what time did he reach the top of the pole? 24. Four merchants A, B, C, and D, trade together; A clears ~76 4s. in 6 months, B ~57 10s. in 5 months, C 100 guineas in 12 months, and D, with a stock of 200 guineas, clears ~78 15s. in 9 months. Required each man's stock. 25. Three merchants traded together as follows: A put in $2500 for 3 months, B $1750 for 5 months, and C $2000 for 2 months: C's gain was $147,50. What must A and B receive for their respective shares, and what was the whole gain? 26. Three different kinds of wine were mixed together in such a way that for every 3 gallons of one kind there were 4 of another, and 7 of a'Third: what quantity of each kind was there in a mixture of 292 gallons? 27. Divide ~500 among four persons, so that when A has ~~, B shall have i, C ~, and D i. 2' 364 PROMISCUOUS QUESTIONS. 28. Two partners have invested in trade $1600, by which they have gained $300; the gain and stock of the second amount to $1140. What is the stock and gain of each? 29. How many planks 15 feet long and 15 inches wide will floor a barn 60~ feet long and 331 feet wide? 30. A merchant bought a quantity of wine for $430. He sold 55 quarts of it for $24,50, and gained 5 cents a quart: how much wine had he at first? 31. Twenty-five.workmen have agreed to labor 12 hours a day for 24 days, to pay an advance made to them of $900; but having lost each an hour per day, five of them engage to fulfil the agreement by working 12 days: how many hours per day must these labor? 32. If a person receives $1 for 7 of a day's work, how much is that a day? 33. If 141o pieces of ribbon cost $26,50, how much is that a piece? 34. What number is that of which, a and 4 added together, will make 48? 35. A landlord being asked how much he received for the rent of his property, answered, after deducting 9 cents from each dollar for taxes and repairs, there remains $3014,30. What was the amount of his rents? 36. A person traded 360 yards of linen for cloth worth $1,62 per yard: how many yards of cloth has he received, and for how much has he sold the linen per yard, knowing that the price of a yard of cloth is equal to that of 23 yards of linen? 37. If 165 pounds of soap cost $16,40, for how much will it be necessary to sell 390 pounds, in order to gain the price of 36 pounds? 38. What is the height of a wall which is 14~i yards in length, and 7 of a yard in thickness, and which has cost $406, it having been paid for at the rate of $10 per cubic yard? 39. If the tare of a quantity of merchandise is 541b. 7oz., what is the gross weight, the tare being 41b. in 100? PROMISCUOUS QUESTIONS. 365 40. At what rate'per cent will $1720,75 amount to $2325,86 in 7 years? 41. In what time will $2377,50 amount to $2852,42, at 4 per cent per annum? 42. What principal put at interest for 7 years, at 5 per cent per annum, will amount to $2327,89? 43. What difference is there between the interest of $2500 for 41 years, at 6 per cent, and half that sum for twice the time, at half the same rate per cent? 44. If, when I sell cloth at 8s. 9d. per yard I gain 12 per cent, what will be the gain per cent when it is sold for 10s. 6d. per yard? 45. A tea-dealer purchased 1201b. of tea, 2 of which he sold at 10s. 6d. per. lb.; but the rest being damaged, he sold it at a loss of ~3 12s., after whichkhe found he had neither gained nor lost. What did it cost him per lb., and what was the damaged tea sold for? 46. A piece of cloth containing 5000 ells Flemish was sold for $21250, by which the gain upon every yard was equal to 1 of the prime cost of an English ell. What was the first cost of the whole piece? 47. A person lent a certain sum at 4 per cent per annum; had this remained at interest 3 years, he would have received for principal and interest $9676,80. What was the principal? 48. Three persons purchased a house for $9202; the first gave a certain sum; the second three times as much; and the third one and a half times as much as the two others together: what did each pay? 49. A piece of land of 165- acres was cleared by two companies of workmen; the first numbered 25 men and the second 22; how many acres did each company clear, and what did the clearing cost per acre, knowing that the first company received $86 more than the second? 50. The greatest of two numbers is 15 and the sum of their squares is 346: what are the two numbers? 366 PROMiSCUOUS QUESTIONS. 51. A water tub holds 147 gallons; the pipe usually brings in 14 gallons in 9 minutes: the tap discharges, at a medium, 40 gallons in 31 minutes. Now, supposing these to be left open, and the water to be turned on at 2 o'clock in the morning; a servant at 5 shuts the tap, and is solicitous to know in what time the tub will be filled in case the water continues to flow. 52. A thief is escaping from an officer. He has 40 miles the start, and travels at the rate of 5 miles an hour; the officer in pursuit travels at the rate of 7 miles in an hour: how far must he travel before he overtakes the thief? 53._Five merchants were in partnership for four years; the first put in $60, then, 5 months after, $800, and at length $1500, 4 months before the end of the partnership; the second put in at first $600, and 6 months after $1800; the third put in $400, and every six months after he added $500; the fourth did not contribute till 8 months after the commencement of the partnership; he then put in $900, and repeated this sum every 6 m-onths; the fifth put in no capital, but kept the accounts, for which the others agreed to pay him $1,25 a day. What is each one's share of the gain, which was $20000 f 54. A traveller leaves New Haven at 8 o'clock on Monday morning, and walks towards Albany at the rate of 3 miles an hour; another traveller sets out from Albany at 4 o'clock on the same evening, and walks towards New Haven at the rate of 4 miles an hour: now supposing the distance to be 130 miles, where on the road will they meet? 55. An employer has 45 workmen, by each of whom he gains 15 cents a day: how long a time would it require for them to gain him $468,93, and what must he pay them during this time, he paying each $1,25 a day? 56. When it is 12 o'clock at New York, what is the hour at London, New York being 750 of longitude west of London? Since the circumference of the earth is supposed to be divided into 360 degrees (Art. 40), and since the sun appa* PROMISCUOUS QUESTIONS. 337 rently passes through these 360~ every twenty-four hours, it fol wws that in a single hour it will pass through one twentyfourth of 360~, or 15~. Hence, there are 15~ of motion in, 1 hour of time, 10 of motion in 4 minutes 1/ of motion in 4 seconds. If two places, therefore, have different longitudes, they will have different times, and the difference of time will be one hour for every 150 of longitude, or 4 minutes for each degree, and 4 seconds for each minute. It must be observed that the place which is most easterly will have the time first, because the sun travels from east to west. To return then to our question. The difference of longitude between London and New York being 750, the difference of time will be found in minutes OPERATION. by multiplying 750 by 4, giving 300 750 minutes, or 5 hours. Now since 4 New York is west of London, the time will be later in London; that is, when it is twelve o'clock at New York, it will be 5, P. M. in London; or when it is 12 at London, it will be 7, A. M. at New York. 57. Boston is 60 40' east longitude from the city of Washington: when it is 6 o'clock P. M. at Washington, what is the hour at Boston? The 6 degrees being mul- OPERATION. tiplied by 4 give 24 minutes 6 X 4 = 24m. of time, and the 40 minutes 40 X 4 = 160"= 2m. 40sec. being multiplied by 4 give 26mn. 40sec. 160 seconds, or 2 minutes Ans. 26m. 40sec. past 6. 40 seconds. The sum is 26m. 40sec., and since Boston is east of Washington the time is later at Boston. 58. The difference of longitude of two places is 85~ 20/: what is the difference of time? 368 ( PAGES 43, 44. ) ANS WE RS. REDUCTION. ET. A ns. Ex. Ans. ( 1155s 15. 241 Ocr. 2. 13860d.- * ~602 10Os. (55440far. i25920s. 3. 52405far. 16. 5184cr. 4. 37245hf. d. 1296. 17. 5. 6. 5726two d. 18. 14z. 7d67tr. d. 19. I 84800gr. 13200gr. 2767tr. d. 13200gr. 8301d. 20. 26215gr. 8. 4343six d. r 21fcr. 21. 1221b. 2oz. 18pwt. 9gr. 432/i. cr. 22. 9. J 1080s. 3003 2160six d. 24003 12960d. 72009 51840far. 144000gr. 1493s. 24. 1571b 73 43. 10. 5975tr. d25. 7l700far 86962gr. 2 820ds. 126. 301b 43 33 29 7gr. 2880d. 27,.1)~1. ~240s. ~12. 28. 268801b. 12. ---- 29. 14T. 13. 99gu. 4s. 4d. 30. 422921b. 14. ~105. 31. 20T. 13cwt. lqr. 221b. 4oz. (PAGES 44,45, 50, 51. ) 39 PEx. Ans. Ex. Ans. 32. - 48. 93A. 2R. 16P. 403T. 19cwt. 1qr. 241b. 49. 818M. 162A. 3R. 23P. 6oz. 231654496dr. 50. 34. 5024na. -51. 19840S. ft. 35. 864yds. 52. 3760128S. in. 36. 78EE.. 1qr. 53. 440cords. 37. 1197E. E. 54. 43742cords 32S.ft. 23940na. 55. i 9996yd. 56. 13tuns 38. ~ 7996E. E. lyd. 37800pt. 6664E. Fr. 39. 5 3768fur. 58. 9704hf. ank..150720rd. 59. 85248gi. 40. 60. 88000yd. 61. 32832pt. 41. 264000ft. 62. 297216hf. pt. 3168000in. 9504000bar. 63. 1800gal. 42. 200613fl. 6in 64. 1408pk. (4755801600bar. 65. 43. 1172qrs. 4bu. lgal. I 66. 1240sacks. (3184bar. over. 67. 31557600sec. 44. 12374P. 68. 189733554sec. -45. 69 240yr. 10da. 4hr. 28m. 46. 2214262sq. ft. 72sq. in. 38sec. 47, 2800P. 70. ADDITION. 4. 787676921. 9. 6001001250561. 5. 10570011. 10. 6000037684799. 6. 15371781930. 11. 128738075326 7. 45105211. 12. 21890459447. 8. - 13. 16* 870 ( PAGES 51-57. ) Ex. Ans. Ex. Ans. 14. 1819857171437. 47. 403A. IR. IP. 15. $108,892. 48. 16. $1057,87. 49. 3209tun. Ohhd. 27gal. 17. $800,076. 50. 5422pun. 57gal. 2qt. 18. 5- 1. 460tier. 29gal. 1qt. 19. $5498,043. 52. 297gal. 2qt. 20. $67476,840. 53. 21. ~684 5s. 7d. 54. 323bar. lfir. 4gal. 22. ~205 3s. 10 Od. 55. 3150hhd. 16gal. lqt. 23. 56. 5220hhd. 4gal. 2qt. 24. ~240 6s. 83d. 57. 528L. ch. 13bu. 2pk. 25. 23821b. loz. 16pwt. 58. 26. 3441b. loz. 19pwt. 20gr. 59. 3842qr. 6bu. 2pk. 27. 4621b. 9oz. 14pwt. 60. 409scows 12L. ch. l9bu. 28. 61. 4299yr. 7T7Z mo. 2wk. 29. 511b 113 33. 62. 525mo. Owk. 4da. 30. 294th 03 73. 63. 31. 36tb 53 63 19 18gr. 64. 4444hr. 23m. 50sec. 32. 464th dO 53. 33. 34. 3030cwt. lqr. 271b. APPLICATIONS. 35. 92cwt. 2qr. 151b. 10oz. 1. 1605260acres. 36. 34716. 7oz. 6dr. 36. 3471yb. 7OZ. 6dr. C 16dr. lst 3 yrs. 42390529A. 37. 471yd. 2qr. 1na. 2. last " 4530902A. 38. _ 3. $2051423,77. 39. 3821E. Fr. 4. 40. 4768E. Fl. Oqr. 2na. 5 15995942 coins. 41. 489L. lmi. 6fur. 5. $5668663=value. 42. 4487fur. 35rd. 5yd. 6 Imports, $303955539. 43. Exports, $287820350. 44. 644ft. Oin. lbar. 7. 1287462. 45. 509A. 2R. 18P. 8. 3617900. 46. 4797A. 2R. 11P. - (PAGES 57-64. ) 371 LEx. Ans. Ex. Ans. 29884 to Br. N. Amer. 17. 1104087. 10..66770 to U. S. In 1790, 3924829. 96654 entire number. 1790, 5305941. 11. 138500tons. 1810, 7265579. 18. 12. 231771men. 1820, 9638191. 13. $70560. 1830, 12861192. 681 No. of vessels. 1840, 17063350. 14. 403 sail vessels. In 1790, 607897 144 steam vessels. 1800, 893041. $ 977911 of gold. 19. 1810, 1191359. 15. 1567420 of silver. 1820, 1627428. 2545331 entire sum. 1830, 1998318. 16. - 1840, 2487355. SUBTRACTION. 1. 81328. 19. 791b 103 63. 2. 7559. 20. - 3. ~7 18s. 93d. 2-1. 133 09 15gr. 4. 33891899020240993. 22. 81b 103 73. 5. 23. 12T. 17cwt. 3qr. 6. 499972609093220149. 24. 2cwt. 2qr. 261b. 7. 149299788316514071. 25. 8. $179,577. 26. 1341b. 14oz. 13dr. 9. $79,324. 27. 134yds. 2qr. 3na. 10. 28. 124E. E. 3qr. 3na. 11. $999,955. 29. 96E. Fr. 2qr. Ina. 12. $107,576. 30. 13. $566,034. 31. 17L. 2mi. 6fur. 14. 8985,997. 32. 34rd. 4yd. 15. 33. 4rd. 33yd. 2ft. 16. 9oz. 17pwt. 20gr. 34. 3ft. Oin. lbar. 17. 151b. 3oz. 16pwt. 35. 18. 2oz. 18pwt. 2lgr. 36. 37A. 2R. 34P. 372 (PAGES 64-68. ) Ex. Ans. Ex. Ans. 37. 1A. 1R. 26P. 51. 2yr. 11ymo. 3wk. 38. 4A. 2R. 39P. 52. 127mo. 3wk. 6da. 39. 7tun 2hhd. 55gal. 53. 147da. 21hr. 56min. 40. 54. 52hr. 50min. 54sec. 41. 1tier. lgal. 3qt. 42. 7gal. 2qt. lpt. PROMISCUOUS EXAMPLES. 43. lbar. 3fir. 7gal. 55. ~3 9s. 44. 107bar. lfir. 4gal. ~3 9s. 45. " 56. ~121 17s. 0-d. 57. 4980 2s. 4d. 46. 63hhd. 2gal. 3qt. ~980 2s. ld. 47. 27L. ch. Obu. Ipk. 180T. lcwt. 121b. 5r. II11 -mo. 2wk. 48. 2weys 4qr. 2bu. 59. 5yr. 11 l-Tmo. 2wk. 6da. 9hr. 22min. 49. 52qr. 6bu. 3pk. 340 35'-dif of lat 50. * 165~ 18'/ " " long Newton's age was 84yr. 2mo. 26da. Euler's " " 76yr. 4mo. 22da. Lagrange's " 77yr. 2mo. llda. I~aplace's " " 78yr. 4da. 61. From Newton's death to Jan. 1st, 1846, was - - -. 118yr. 9mo. 12da. I" Euler's " " " 62yr. 3mo. 24da. " Lagrange's " " " 32yr. 8mo. 21da. " Laplace's " " " 18yr. 9mo. 5da. 62. 3279hhd. 67. ~5742078. 6 937215diff. 68. $52315291. 2428921 pop. of state. 69. 13277872 d458211. 18535786 whole pop. 65. 71. $49282,03. 66. $92449341,16. 72. $3466051,78. PAGES 68-75. ) 373 Ex. Ans. rFrom the founding of St. Augustine, 280yr. 6mo. 9da "' " " Jamestown, 238yr. lOmo. 4da. " " Battle of Princeton, 69yr. 2mo. 14da. " " Surrender of Cornwallis, 64yr. 4mo. 29da. " Washington's Inauguration, 56yr. Omo. 17da. " Washington's Death, 46yr. 3mo. 3da. " the French Berlin Decree, 39yr. 3mo. 26da. " " Orders in Council, 38yr. 4mo. 6da. " " Declaration of War, 33yr. 8mo. 29da. " Capture of the Guerriere, 33yr. 6mo. 29da. " " " " Macedonian, 33yr. 4mo. 23da. " York, 32yr. 10mo. 20da.' ".... " Fort George, 32yr. 9mo. 21da. Defeat at Sackett's Harbor, 32yr. 9mo. 20da. " " Battle of Iake Erie, 32yr. 6mo. 7da. of Chippewa, 31yr. 8mo. 12da. o" f Niagara, 31yr. 7mo. 23da. " Sortie from Fort Erie, 31yr. 6mo. " Battle of New Orleans, 31yr. 2mo. 9da. " " Death of Adams and Jefferson, 19yr. 8mo. 13da. " Compromise Bill, 13yr. 1mo. 5da. " Death of Lafayette, 12yr. 9mo. 28da. " " Removal of the Cherokees, 7yr. 9mo. 22da. MULTIPLICATION. 11 6776368. 9. 2324684880333. 2. 68653214. 10. 71109696492112. 3. 4563272. 11. 4. 1301922. 19. 129359360000. 6. 20. 13729103000000. 7. 556321146764. 21. 664763206000000. 8. 1747125213301. 22. 8799238229600000. 23. 2526426017908695000000. 24. 1093689368445084378777040. 25. 26. 8371562339213807802080112 374 ( PAGES 75-79. ) Ex. Ans. 27. 72058988008174745973090826 28. 95666032459647278072171264. 29. 4896. 23. ~2 Os. 6d. 30. 234048. 24. ~2 is. 8-d. 31. 25. 32. 314986464. 26. ~2 19s. 6d. 27. ~8 3s. 9d ART. 67. 28. ~16 8s. 7~d. $70,840. 29. ~22 1]3s. 1. ~ 85,008. 30. ( 99,176. 31. 1llb. 6oz. 9pwt. 12gr. 2. $7834,14. 32. 197yd. lqr. lna. 3. $12517,68. 33. ~3 19s. 4~d. 4. $77079,456. 34. ~66 19s. 6d. 5. 35. 6. $341,25. 36. ~65 19s. 9.d. 7. $98,94. 37. ~208 13s. 9d. 8. $813,020. 38. ~154 12s. 3d. 9. $5869,75. 39. ~42 is. 6d. 10. 40. 11. $2426,15. 41. ~819 6s. 12. $15169,50. 42. 6901b. 8oz. 18pwt. 16gr. 13. $162,25. 43. 75A. 3R. 39P. 14. $21935,214. 45. ~19 lOs. 8Md. 15. 46. 16. $963,66. 47. ~33 3s. lad. 17. $18844,01. 48. ~83 2s. 8d. 18. ~81 6s. 10d. 49. ~137 7s. 3d. 19. 24T. 7cwt. 3qr. 271b. 8oz. 50. ~698 2s. 20. 51. 21. ~1 10s. 2d. 52. 222cwt. 181b. 22. ~1 9s. 2d. 53. 15cwt. 271b. loz. 7dr. (PAGES 79-88. ) 375 Ex Ans. Ex. Ans. 55. ~2687 18s. 3d. 65. ~566 5s. 31d. 56. ~351 2s. 7~d. 66. ~17038 10s. Ild. 2!2far. 57. 67. ~12422 2s. 7d. 2-afar. 58. ~62 is. 7~d. 68. ~2875 Os. 7~d. 59. ~15299 18s. 4d. 69. 60. ~51 7s. 2~d. 70. ~658 Os. lid. 13far. 61. ~344 Os. 6d. 71. ~50 12s. 9d. 62 - 72. ~2 10s. 2d. 2 far DIVISION 2. 4072948100. 9. 1318096553~4990 3. 131951331f4B-2. 10. 4. 12513920113010 1 99481579776I8I6 12. 59085714834 13. 1258127578-. 6. 142437577483541 57149' 6. 14243757748 14. 123456789. 7. 153959191221 5. 3 7 ITs 15. 8. 30001000 6147. 16. 119191753 9~0107 17. 9001844424018274624226548424 987675' ART. 79. ART. 81. 1. 132. 3. 17085g2. 2. 4871000. 4. 676392~. 3. - 5. 6129irlo. 4. 7128368. 6. 3095 874. 5. 918546. 8.ART. 80. 9. 5203802A-1-. 2. 387. 10. 118055585. 3. 133. 11. 390968216 4-. 4. 201. 12. 34297219i62 376 ( PAGES 68-93. ) Ex. Ans. Ex. Ans. 13. - 11. 2E. E. 3qr. ln,,a. 14. 1082363712. 12. ~1 17s. 6d. 15 65024 8 13. 6d. 11-' far. 16. 206190192477 473964 9006- 14. 14 MaId. 15. ART. 82. 116. ~40 10 s. 6d. 2. 15655794 975-. 17. 1 s. 9d 3. 18. 10 4d. 4. 168716512944. 19. 4s. 9-36~d. 5. 47462565 9649 20. 6. 8905748 - 0 -. 21. 1 Ooz. 1 5pwt. 14-8 —8~-gr. 7. 142]076222o427. 2 78 142o,07622274754000 22. ) 17cwt. 2qr. 1lb. l4oz 9. 4087692937 -3 a7 g1 5% 23. 39E. Fl. lqr. 3na. 1lin. 10. 79438599. 11. 11909223_4_ 1... 14 400' APPLICATIONS. 12. 71400714374. 13. 1. 21 T. l0cwt. 2qr. 621b. 14. 1924530 52. 4t 7 2. 9 3. 873,296+. 4. $0,0899+. EXAMPLES IN DENOMINATE NUMBERS. 5. $1 283 -f. 6. 20-37868 renm. 2..~15 19s. 9d. lfazr. 7.3. ~1 ls. 5d. ] 2far. 8 $1,255+. 4. 15s. 1d. 3-%5far. 8. $1,244+ 5. -_ 9. $25141072,267+. 6. ~9 18s. 3d. 39 —far. 10. $0,06+. 7. 9yd. 2qr. 1-na. 11. $166743,259+. 8. 4A. 32P. 20sq. yd. 12. lsq.ft. 72sq. in. 13. 37-2588 rem. 9. 81b. 7pwt. 15l5-gr. 14. 38-190 rem 10. - 15. $334477,744+. (PAGES 93-108. ) 377 PROPERTIES OF THE 9'S. ART. 85. En. Ans. Ex. Ans. 3. 177234105-3. 1. 2. 7. 2. ART. 87. 3. 1. 7. 2 503602-7. Excess in minuend, 7. ART. 86. ~3ART. 86.I 45991735-7. 82 140487982-7. iExcess n minuend, 8. VULGAR FRACTIONS. ART. 98. ART. 100. 1. 3 5 1. 6 1 262 2. 21 2. 34 3. 162 3. 127 13 4 418T 4. 1_3_2 4. 5. --- 1 5, 327 5. 5. 5684' 6. 1 078 6. 1249 17 4815' 7. 126 7. 327 f ~1 1 825' 8. 195 8. 94 ART. 99. ART. 101. 1. 4 7 1. 2. 7242. 2. 1,6 16 16 ~3. 22 3 ~ 34 34 34 3.'~~4i * 24, 16, 8'~ 4. - 6 4. 1 _7 1 4 7 4356' 75. 6 5. 127 6. 630. 7. --— 1 7. 24_6 246 7.~' * 4 2 21 8. 9 8. 449 449 O. 492'. T2 X -4~X ART. 102. 2. h51 68 102 119 153 255 289 5 7 76 11 ]-4 33' 171Y' 285' -32 117 156 208 91 65 143 3. 153' 204' 272' 119', 85, o87 378 ( PAGES 109 — 11. ) ART. 103. Ex. Ans. 3. 1 6 8 -4 2 1 64' 32' 16'1 6' 4 4. 30 20 15 1 2 10 6 5 4 3 21 90' 6-0~ 45' 36' 3-0' 18' T5' 12' 9' 6 3 GREATEST COMMON DIVISOR. 1. 4. 2D METHOD. 2. 45. ART. 105. 3. 630. 2., 43, 3, 5. 4. 267. 3. 3, 3, 2, 2, 2, 5. 5. 6. 12. ART. 106. 7. 8. 2. 3. 8. 4. 3. 25. 9. 3. 4. 10. - 5. 15 LEAST COMMON MULTIPLE. 3. 840. 11. 4. 147. 2D METHOD. 5. 840. 3. 1260. 6. - 4. 7200. 7. 78. 5. 2520. 8. 84. 6. 1008. 9. 1008. 7. 10. 156. 8. 10800. REDUCTION OF VULGAR FRACTIONS. ART. 110. 6. 27 24, 76 I 2. 123 72301' 8' 7. 219. 3. 25-yd. 8. 191 5 4. 55bu. 9. 14. 5. - 10. (PAGES 118-126. ) 379 Ex. Ans. Ex. A ns. 11. 10731 55 4. 332491 5. ART. 111. 6. A-?8. 3. 27 52258 6 7. 5_( 34513 7899 69047. 5 L48261 91772 59267822 2. I T7 5 104 879 * 15 6. 7 ~7~ 2s827s 4. 8. 1346 ART. 115. 9 37219 3 375 9 9 T —9-1504 10. 1749383049 4. 1681 11. - 5. 3445 = 8294. 12. 16106 12. 161069 ART. 116. 13. 777=78. 382 ART. 112. 3. 2. 1 4. -99 3. 1. 5. 3454 4. 6. 7 5. iI 7. 28096520 6368996 7 9 4. 7 ~8. 7~9- 5. 525 1080 222 00 9. 6 600' 6 600 6. 200 62 1 550 10. 0 11. 7- 7. 1080 248 900 11' 187 T6046 T44 4 13~ 9. 7T7- 8. 518 1026 1575 13. 69 9 14-'.!F2. 4 9 14. 20664 14976 16576.3Q 1 2 15. 3811 4032 - ART. 113. 12. 126 140 30 125 2. 2304 - 13. 40328 40832 4036' 3. 2T4 4 32I T. 141 5o7. 380 (PAGES 126-132. ) LE. Ans. Ex. A ns. 14. - 3. 122 51 44 ART. 118. 4. 72 60 320 360, 360, 360' 32 2 1 9 7,' TI' TI~ 5, 67 18 300 4. 42 24 2420 20' 84' T4' 84' 6. 5 15 92 18 ~g, ~g, Ig' 6. 154 228 127 7. 450, 24 1204 2749 7. - 8. 6 8 9 10 7, 1-2, 9T, YI' ART. 119. 9 3 6 60 50 63 50'' 9f, 9-6, 2. 36 4 9 10. 16 3'6 40 42 33 34 451 45, 45' 148 48, 48, 4 38 X -f 4-8REDUCTION OF DENOMINATE FRACTIONS. ART. 121. 6. 3. 7. 3gal. 4. 45 3hhd. 8. -lApt. 5. ~40. 9. _-_o7-sec. TOO 7 6. ~2. 10. 504Cgi. 7. /Ahhd. 11. 8. 12. -d. 9. r87 da. 13. 4 pt. 1'82'80 i'Pa 10. s cwt. 14. 7lb. 11. I T. 15. 4233600sec. 12. ~37. 16. 13. 17, 2-24na. 14. 2srlb. 18. ]7~2432~t. 15. 1 lb. 19. 2 4 4 8 16 1 hhld. F16 81h/zd ART. 123. ART. 122. 2. 9oz. 12pwt. 2. 641b. 3. 3. 32d. 4. 2R. 20P. 4. 480m. 5. 3s. 4d. 5. -L.. 6. 52gad. 2qt. (PAGES 132-136. ) 381 JEx. Ans. Ex. Ans. 7. 24gal. 2qt. 9. 2339626 da. 8. ___0o. 2od. 9. 7oz. 4pwt. 6 of 2da. 61 of 3da. 10. lpi. lhhd. 31gal. 2qt. of 3d. 11. 7oz. 4pwtda. 12. Imi. 6fur. 16rd. 4 of lOda 13. L~'~6 of 25da. 14. 8yd. lqr. l na. 12. 15. lpt. lhhd. 7gal. 13 2. 16. 29gal. lqt. 1lP. 17. 98da. 8hr. 4m. 36-3sec. 15., b. 18.. 16. T4E.E. 19. 5s. 4d. 17. 20. 6cwt. 3qr. 61b. 18. 480' 21. 10S.ft. 216S. in. 19. 4-groat. 22. 30gal. 2 6217t. 20. 4quarter. 21. 76-yd. ART. 124. 22. 3. 23. 51760 4. I tl hhd. 24. 3 6hhd. 5. 5rcwt. 25. s2 3S.yd. 6. 7S1hd. 26. T74L4L. ch. 7. 7930mi. 27. 8. 28. 228.hd. 4 3 s —hhd ADDITION OF VULGAR FRACTIONS. ART. 126. 6. 18. 2. ~21. ART. 127. 3. 15 2. ~X5. 4. 2. 3. o494 -5.- 4. — r 382 ( PAGES 137-141. ) Ex. Ans. Ex. Ans. 5. 4. 2qr. 171b. loz. 311d. 6. 13 0 9 5. lmi. 3fur. 18rd. 7. 4 6353 7. ART. 128. 8. lcwt. lqr. 271b. 13oz. 2. 843 9. 11s. 642d. 2. 84193 -' 4-6' 3. 311o0 10. ~3 12s. 4. 11. 2oz. 10pwt. 12gr. 5. 61 -2 3 12. 6. 170.5 3 13. 2E. E. 4qr. 02na. 7. 489- 14. 3fur. 25rd. 3yd. 13-9in. 7. 6'1 6' 3-. f;__7 6_3_ 15. 2R. 20P. 1;Sq.ft. 8. 6' 78 3 15S ___0 58- Sq. in. 9. 16. 3hhd. 37gal. 3~1t. ART. 129. 17. 2. 141-8in. =25 8yd. 18. 2da. 2hr. 12nm. 3. 2da. 14~hr. 19. 55da. 10hr. 81m. SUBTRACTION OF VULGAR FRACTIONS. ART. 131. 10. 753. 2. 66 11. 3. 3 2 1 12. 364A. 4. - 13. 473. 14. ART. 132. 14. 15. 914 21. l — --- 3. 2. 16. 4... ART. 133. 5.. 2. lhr. 59m. 591sec. 6. ---- 3. 1lb. 8oz. 16pwt. 16gr. 7. Yu 4. 16gal. 2qt. lpt. 3;Sgi. 8. 7. 5. 9s. 3d. 9. jxjjj-a. 6. 6242 ~' (PAGES 141-146. ) 383 Ex. Ans. Ex. Ans. 7. 4cwt. lqr.; 151b. loz. 93dr. I1. 8, 14s. 3ld. 12. 1mi. lfur. 16rd. 13. 202da. 21 hr. 45m. 321sec. 9. 9oz. 7pwt. 12gr. 7 14 Troy oz. 1pwt. 18~gr. 10. 7cwt. 1qr. 271b. 8oz. 14. greater. MULTIPLICATION OF VULGAR FRACTIONS. ART. 135. 13.1 7 5 3.. 3-12. 3I. ART. 137. 3. 4. 4. 334~2 7.3. 608k. 4. 5 9 87 5. 421 9 6 5. 6. 124io1 I 149 6. 1062237. 493119 3' 8. 9. 11969629 GENERAL EXAMPLES. 87146' ART. 136. 1. 11~8. 3. 7 2. 3n3 4-8-0'. 84 3.5. 53 4. 6. 5. $51. 7. 20. 6. $36 8. 23 7. $713 84'' 16" 9. 14124. 8. ~2 3s. 9d. 10. 2-. 9. 11. - 10. $84g. 12. 540. 11. $267-. DIVISION OF VULGAR FRACTIONS. ART. 139. 4. 2. 2 5. 779 3. l. 6. 67N. 384 (PAGES 146-153. ) Ex. Ans. Ex. Ans. 7. 61 514. a31758 ART. 140. 15. 180 —3 1. ~ 16. 21. 8', 5 2. 29. 17. 5 2.-i ~ 3. - 18. 4. 68 5s7 7 19. 8 1 cts. 5. 153801. 20. 4-4-Acts. 6. 301~-4 2 21. $5,25. 7. -2 22. $29-4. 8. 23. 9. 7 1 9. 24. 8ss8512TCts. 10. 2020. 25. 95 cts. 11. 63 26. 192-lb. 12. 9. 27. $2-45 4 13. 28. DECIMAL FRACTIONS. ART. 143. 15. 1. 41.3. 16..000003. 2. 16.000003. 17..00039. 3. 5.09. 4. 65.015. ART. 144. 5. 6. 2.00000003. 1. $17.389. 7..492. 2. $92.895. 8. 3000.0021. 3. 9, 47.0021. 4. $47.25. 10. - 5. $39.397. 11. 39.640. 6. $12.003. 12, 300.000840. 7. $147.04. 13..650. 8. 14 50000.04. 9. $4.006. (PAGES 153-159. ) 385 Ex. Ans. Ex. Ans. 10. $14.039. 13. 11. $149.332. 14. $0.058. 12. $1328.005. 15. $3856.02. ADDITION OF DECIMAL FRACTIONS. 1. 1306.1805. 11. 31.02464. 2. 528.697893. 12. 1.110129. 3. 13. 4. 1.5415. 14. $641.249. 5. 446.0924. 15..111. 6. 27.2()87. 16. 4.0006. 7. 88.76257. 17. 2.413009. 8. 18. 9. 1835.599. 19. $1-132.365. 10. 397.547. SUBTRACTION OF DECIMAL FRACTIONS. 2. 3277.9121. 13. 17.949. 3. 249.60401. 14..699993. 4. 9.888899. 15. 5. - 16..999. 6. 2.7696. 17. 6373.9. 7. 1571.85. 18. 565.007497. 8..6946. 19. 20.9942. 9..89575. 20. 10. 21. 10.030181. 11. 1379.25922. 22. 2.0294. 12. 99.706. MULTIPLICATION OF DECIMALtr FRACTIONSB 2. 329.307391-.' 4. 3. 742.036196. 5. 26.99178 17 38; (PAGES 159-164.) Ex. Ans. Ex. Ans 6. 10376.283913. 14. 7. 275539.5065. 15. 933.8253150762. 8..020621125. 16..25. 9. 17..0025. 10. 175.26788356. 18..00715248. 11..00043204577. 19. 12. 215.67436625. 20..02860992. 13..000000000294. 21. 2.435141056. CONTRACTION IN MULTIPLICATION. 2. 258.13005. 4. 3. 162.525. 5. 3566163. DIVISION OF DECIMAL FRACTIONS. ART. 152. f 254.7347748. 2. 2.22. 25473.47748. 254734.7748. 3. 8.522. 10. 254734.748. 2547347.748. 4. 33.331. 25473477.48. 5. L254734774.8. 6. 12420.5. 11. 25.05068. 12. 1918+. 250.5068. 13..00473 +-.'7. 2505.068. 7.2505.068. 14. 1.74412. 25050.68. 250506.8. 15. 69.7125. 48.65961. 16. 4865.961. 17. 12976+. 8,.~ 48659.61. 8X48659.6118..0049589+. 486596.1. ART. 154. 41.622. 416.22. 2. 10970, 4162.2. 89., k41622. 3. 60200. 41622. 416220. 4. 4162200. 5. 100. ( PAGES 164-170. ) 387 Ex. Ans. Ex. Ans. [10. 4. $1'1055.2925 100. 5 1000. 1000. 6. $140.625. 6.. 20. 7. $20.87. 2000. 8. $3731.123. 12. 1200. 9. 224.58S. yd. 1200. 500000. 10. 11. $365.61525. ART. 155. 12. $1.35 3. 8.3111+. r 269 acres = area. 4. 1.563-+{. 13. $13573.204 = cost. 5 $50.458+ = average price. ART. 156. $7631.8855 = eldest 1. 339.513001yd. son's sare. 14. $5723.914125 -= share 2. 155.10111lb. of each of the other 3. $88.141. sons. CONTRACTION IN DIVISION. 2. 4. 11.5834036625-6 rem, 3. 35.2843-3 rem. 5. 3202.8870-1 rem. REDUCTION OF VULGAR FRACTIONS TO DECIMALS. 2..125..0159+. 11..000000488+. 3. 12..8571 +. 4..5..0028+.13. 5. 1.496+. 14..2571+. 6. j 1.333+-.1629+. 6.792+. 4.666+. 15..8947+. 7..02343+. 16..008033+. 8. 17..23903+. 9..0003. 18. 10..22244-. 19. 1.5555+. 388 (PAos 170-174. ) Ex. Ans. Ex. Ans. 20..15909+. 23. 21. $100.8. 24. 1.25. 22. $17.85. 25. 3.0339+. REDUCTION OF DENOMINA.TV D1CIMALS. ART. 160. 7. 10.16666+. I..05468751b. 8. ~.3729+. 2. ~.325. 9. 3 10. ~.2325757+-. 4..029166da.+...129681b.+. 5. 3.9375pk..5oz 6..375da. 13. 3987631b.- + troy. (See Arith., Art. 20, p. 23.) 7. 71.151mi.+. 14. 8.. 15..633928cwt. +. 9..00396bar. +. 109..003965ar. ~ 1..042965cwt. +. (See 10. 1.5yd6. Arith., Art. 20, p. 23.) 11..66251b. 17..3125yd. 12..7282yr.+. 18..55E. E. 13. 19. 14. ~25.977+. 20..48125A. 15..9375cwt. 21..00992hd. +. 16..7391mi. +. 22..104166ch.+. 17..2325T. 23..07472yr.+. 18. 24. 19..7129da.+. 25..26175A. ART. 161. 26..1005113mi. +.1. ~19.8635+. ART. 102. 2. ~2.325. 1. 2cr. 141b. 3..625s. 2Z 2qt. lpt. 5. 2.5yd. 4. 20gal. lqt. 6. 1.046875/b, 51; 136da. 21hr. (PAGES 174-181 ) 389 Ex. Ans. Ex. Ans. 6. is. 8~d.+. 14. 12dr. 7. lqr. 14oz. 5dr.+. 15. 208da. 3hr. 23m. 33sec.+. 8. 16. ~2 Is. 10d.+ 9. 2mi. 24rd. 5yd. 10in.+. 17. ~5 12s. 9-d.+. 10. 6s. 9d. 18. 11. 6cwt. 3qr. I lcIoom 1strike 2pk. 12. 8P. 3qt. lpt.+. 13. -- 20. 13 13+ CIRCULATING OR REPEATING DECIMALS. ART. 164. 3. Factors of den. 5x5x2. decimal value.06. 4. 37X5X2..0878+. 5. 6. 2X2X2X2x2x2X2X2X5..01328125. ART. 165. 7. 5X5X5X2..028. 8. 5X5X5x5..0176. 9. 2X2x2x2x2x2x2..1328125. REDUCTION OF CIRCULATING DECIMALS. ART. 175. ART. 177. Sec. 2. 3. 2.4 1818184..5 925925'+. 1, 4. 3 T 7'.008 497133_+. Sec. 4. ART. 176. 165.16416416 +. 1. Q.04 040404 +. 4. -72969, 2, 37-9,.03 7777777/. 223 75434.5 3733333 —. 34 217 21 4125_6 2. ) 163 41 T-w — -— 6,6' 1.7 577577'+. 390 (PAGES 184-187. ) ART. 178. Ex. Ans. 2..1875. 3..0 0344827586206896551724137931/+. 4. ADDITION OF CIRCULATING DECIMALS. 2..95.2 829647+. 5. 47.4 754481'+. 3. 69.74 203112'+. 6. 4. 55.6 209780437503 +. SUBTRACTION OF CIRCULATING DECIMALS. 2. 45.7 755'. 6. 3. 2.9 957'+. 7. 4.619'525' +. 4. 5.09. 8. 1.0923 7+. 5..65 370016280906'+. 9. 1.3462 937'+. MULTIPLICATION OF CIRCULATING DECIMALS. 2. 7. 3. 1.093 086'+. 8. 11.068735402 +. 4. 1.6411 7+. 9..81654 168350/+. 5. 1.7183 39'+. 10. 189.301 977 + 6. 1.4710 037'+. DIVISION OF CIRCULATING DECIMALS. 2. 13.570413'961038'+. 6. 3.145+-. 3. 7. 3.8235294117647058-+. 4. 7.719'54'+. 8. 5. 26.7837 428571/+. 9. 15.48423'+. ( PAGES 188-199. ) 391 RATIO AND PROPORTION OF NUMBERS. ART. 183. ART. 187. Ea Ans. Ex. Ans. 1. 2. 1. 9: 8:: 18: 16. 2. 4. 2. 3. 4. 3. 16: 9:: 48: 27. 4. - 4. 13: 19:: 52: 76. 5. 5. 16: 21:: 80:. 105. 6. a0. 6. 35: 42::210: 252. 7. i. 7. 8. X' 8. 23: 45:: 207: 405 9. 1 0. X. ART. 188. 11. X. 1. 6. ART. 185. 2. I.. 3. 5. 3. 5.... OF CANCELLING. ART. 190. ART. 191. 3. 1. 62. 4. 2. 27. 5. 9. 3. 4. 323. 6. 23. 5. 153369 7. 8. ART. 192. 8. 1. 2: 8:: 1: 4. 9. 12. 2. 2:7::6:21. 10. 6. 3. RULE OF THREE. APPLICATIONS. 3. $2762,50. 1. $330. 4. 33001b. 2. ~9 6s. 8d1 5. 892 ( PAGEs 199-204. ) Ex Ans. Ex. Ans. 6. 32 men. 17. ~1913 6s. 8d. 7'. 3Mr. 2m. 49~4sec. 18. 62oz. 8. 16432{- miles. 19. ~1270 is. 9-1d. 9. $121,875. 20. 10. - 21. ~39637 10s..11. 20days. 22. 120 yards. 12. 4200bu. 23. 190 guineas. 13. ~253 10s. 3d. 24. 19 4da. 14. 6oz. 15-95dr. 25. 15. 26. 108000. 16. ~8 16s. 2-3-2-d. 27.; Rate 3mz. lfur. 18r2rd. per hour. Time 20hr. 21m. 255sec. 28. 17 times round. 36. 29. 4{- days. 37. 8s. 2d. 324545 far. 30. 291 31. 13oz. per day. 38. $112,86. 32. 5880001b. total weight. 39 whole weight, 5880001b. 420001b. spoiled. they received, 5460001b. 33. 2018E. Fl. 2qr. 34. f~9 3s. 9d. 40. 5 whole weight 9408000oz. 35{yd. baize. 14oz per day. 35. ~ cost 1ls. 2d. 1 3-Tfar. 41. per yard. RULE OF THREE BY ANALYSIS. 2. 87 miles. 11. 105 days. 4. $78. 12. 5. 504 miles. 13. 106yd. 2ft. 6. $2,08. 14. 5- days. 7. - 15. 27 days. 8. $380. 16. llb. 5oz. 9!d'. 9. $0,429+-. 17. 10. 30 days. 18. 64 bottles. (PAGES 204-203. ) 393.Ex. Ans. Ex. Ans. 19. 25yr. 202da. 2hr. 21. 67 11gal. 20. He gained $246,75. 22. RULE OF THREE BY CANCELLING 2. 8s. 8d. 13. 54 days. 3. $400. 14. 6 days. 4. 160 days. 15. 12 days. 5. 27001b. 16. 6. 1 7. 3600. 7. $12. 18. 81 yards. 8. 60 days. 19. 16 months. 9. 23352bu. 20. 121 yards. 10. ~74s. 21. (1 41, D's part. 1-..22. <~61 lOs. paid by each 12. 20 days. of the others. EXAMPLES INVOLVING FRACTIONS. 2. ~1 18s. 6d. 13. $52.50. 3. ~682 18s. 9d. 14. $638.08+ 4. ~112 12s. 95o-d. 15. 5. - 16. $21 10s. 1d. 6. ~102 7s. 7d.+-. 17. 62.734375 days. 7. $1.431-h9J. 18. ~77 3s. 7ad. 8. $}4.58~. 19. $5.625. 9. 14 days. 20. 10. - 21. 2700. 11..005172 guineas. +. 22. 293.28gals. 12..71428cwt. +. 23. 1143hrs. At the equator, 1038s- miles. 24" Madras, &c. 101034- miles. " Madrid, " 79444 miles. "Petersburg" 5191 - miles. ]17* 394 ( PAGES 213-219. ) DOUBLE RULE OF THREE..EX. Ans. Ex. Ans. 5 13. 14oz. 6. 11676bu. 14. 540S.yd. 36yd. long. 7. 36 days. 15. 8. 2808qrs. 16. 27 men. 9. 168. 17. ~11 2A2d. 10. - 18. 4da. 1lhr. 54 3'-193m. 11. 10 days. 19. 8571A. 12. 9600 men.! 20. PRACTICE. 13. ~91 is. 92 d. 32. 14. ~44 9s. 8l3d. 33. ~348 14s. 1l-7d. 15. ~38 3s. 36d. 34. $135,375. 16. ~19 Is. 3 —d. 35. $15. 17. 36. $1854. 18. $1,575. 37. 19. $547,50. 38. ~199 ls: 107-d. 20. $5. 39. $1095. 21. $108. 40. $7. 22. - 41. $8,10. 23. ~10 15s. Mi5d. 42. 24. ~18 7s. 6A-7d. 43. ~550 lls. 10~d. 25. ~3 7s. 559d. 44. ~661 17s. 3-A-d. 26. ~2051 13s. 10A95-9d. 45. ~445 5s. 4 -7A3d. 27. 46. $65,90625. 28. $1617. 47. 29. $643,75. 48. $148,2890625. 30. $519,75. 49. $48,91796875. 31. ~1877 10s. 9 -d. (PAGES 220-224. ) 395 TARE AND TRET. Ex. Ans. Ex. Ans. 1. 65cwt. lqr. 191b. 6. $808,71. 2. 4cwt. 1qr. 1516b. 8oz. 7. ~912 14s. 57d. 3. ____ ~ 8. 9. 3T. 8cwt. 3qr. 51b. 5. $265, 16d. 10 i 6T. 12cwt. 3qr. 311b. 5. ~59 4s. 3596d. Value, $306,724685. 11. In leaf, $26,0586+ per cwt. In rolls, $29,5504+ " " 12. Net weight, 161cwt. 1qr. 8.961lb.+. Value, $ 1177,709 +. 13. 14. r Net weight, 27cwt. 2qr.' 6-lb. Value, $232.837+. is5 Net weight, 10cwt. 272lb. *15.~~~~?Value, $4,0417+. Net weight, 20cwt. 2qr. 13-lb. 16. 358.312gal. + (Freight, $444,306+. 17. Net weight, 298cwt. 2qr. 23,4541b.+ * Freight, $355,464+. PERCENTAGE. ART. 204. Ex. Ans. Ex. Ans. 8. 742gal. 3qt. 31gi. 1. 2. 432bar. ART. 205. 3. 42hAd. 1. 15 per cent. 4. $10,80. 2. 55 " " 5. $24,25. 3. 6. 4. 20 per cent 7. 205 boxes. 5. 16*" " 396 ( PAGES 227-234. ) SIMPLE INTEREST. ART. 207. Ex. Ans. Ex. Ans. 10. $31,928125. 4. 11. $121,77275 5. $803,25. 12. 6. $450,32760. 13. $609,45776. -7. $4853,844. ART. 212. 8. $643,83375. 2. $16474,3855+. 9. - 3. $1449,70998. 10. $235,764. 4. $371,86875. 1.1. $1205,9208. 5. 12. $1375,8144. 6. $266,277. 13. $12959,584. 7. $14096,5+. 14. - 8. $4479,618. ART. 208. 9. $1149,552. 3. $1225,511. 10. 4. $44,20845. 11. $40900,9335. 5. $6015,4272. 12. $6418,50195. 6. $357,9165. ART. 213. 7. - 3. $511,5357. 8. $270,5175. 4. $167,30. 9. $2953,22685. 5. 10. $7765,4504. 6. $555,465. 11. $1578,6625. 7. $4200. 12. 8. $2643,8386. ART. 210. 9. $7,00. $3,2727875. 10. $145,1545. 11. $587,6311+. 4. $67,278. 12. $3974,5187 +. 5. $5,56529+. 13. $3329,1468+. 6. $0,4314 +. 14. $1137,0592 -. 7. - 15. 8. $1,13559. 16. $678,2599+. 9. $13,68. 17. $4824,2366. A PAGES 234 —245. ) 397 ART. 214. Ex. Ans. Ex. Ans. 3. $43,2049+. 4. $8,1855. Interest at 4 per cent, $45,837. C 5 CC C; $57,29625. " 5" " " $63,025875. " 6 " " $6q,7555. 5. " 7 " " 6" $80,21475. " 72:":" $85,944375. " 8 c" " $91,674. " 82 " " $97,4036-25. L " 9 " " $103,13325. 6. - 4. 7. $380,28952. 5. $4640,5326. 8. $669,7096875. 6. $1976,6305 +. 9. $25571,2473+ AR'r. 217. ARnr. 215. 2. $5359,:363. 2. ~45 8s. 1id. 3. $8921,618+. 3. 4. 4. ~45 10s. 2d.+ ART. 220. 5. ~662 3s.~+ 2. $3976,848 +. 6. ~216 Is. 101(l.+ 3. $575,569~-+. 7. ~219 18s. Old.+ 4. $1424,84'9+. 8. 5. 9. ~68 5s. 10d..+ ARr. 221. ARr. 216. ARr. 2 16. 2. 5 per c:nt. 1. $394,3256-+. 2. $697,986. Ar. 222. 3. $3339,6+. 2. 2yr. Gnm. REDUCTION OF CURRENCIES. 3. ~1073 18s. 1ld. 5. 4. $1967,892+. 6.:$2551,733 +. 398 ( PAGES 247-256. ) COMPOUND INTEREST. ART. 223. ART. 224. 2. $57,3048+. 2. $578,740+. 3. $73,015+. 3. $8611,128+-. 4. $41,216+. 4. _5_ 5. $7058,617+. 6. $2647,996+. 6. $48165,938+. 7. ~11 18s. 1ld.+. 7. $14523,553+. 8. $9974,685+. LOSS AND GAIN. ART. 225. 6. 3. Loss of $0,75. 7. $337,50. 4. $0,966+. 8. $217,50. 9. $4,108. ART. 227. 10. 25 per cent. 1. - 11. 2. 12 per cent. 2. 12 per cent. 12 Whole gain, $13,00. 12. 3. $1,25. Gain, 20 per cent. 4. $1,20. 13. $2,05. 5. $6,331. 14. $1,0314. COMMISSION AND BROKERAGE. 2. $49725. 11. $46260. 4. - 12. $23700. 5. $7235,752 f+. 13. $25420,195. 6. $7814,516+. 14. 7. 15 tons. $965,30. 21520.16. $1995. 8. $213532,50. 13573,56 17. $13573,56. 0. i 149T. 18cwt. 2qr. 121b. ~~10. ~ $59110. 18 6oz. I1dr. $59110.~V (PAGES 260-268. ) 399 BANK DISCOUNT. ART. 239.. ART. 240. Ex. Ans. Ex. Ans. 1. 2. $344,59+. 2. $15240,54. 3. $5734,32+.'4. $695,64-. 3. $5,840+.. $65,4+. 5. $118,85+ -. 4. $3393,504. 6. 5. $29,0097+-. 7. $1740,61+. 6. 8. $1057,51 + DISCOUNT. 2. $1551,918+. 9. 3. ~33 17s. 73d. +.. 10. $3869,407+. 4. 11. lbu. 2 2 qts. 5. ~223 5s. 8d.+-. 12. $2109,236 +. 6. $5620,175+. 13. $2763,694 +. 7. $702,485+. 14. 8. ~804 19s. 5d.-+-. 15. He lost $6,473+-. INSURANCE. 2. $5168,59. 5. 3. $237,60. 6. $504. $158,40. 7. $39,375. $252. 8. $306,25. $126. 9 $450. $56. $42. 11. $18,75. ASSESSING TAXES. 2..4685- per cent. 1 3. $37901125. 4e0l ( PAGES 270-276. ) EQUATION OF PAYMENTS. Ex. Ans. Ex. Ans. 2. 12 months. 9. 3. - I 10. 6mo. 6days. 4. 9 months. 11. 7mo. 3da. 5. 6832 days. 12. Jan. 25th. 6. 8- months. 13. 8mo. Oda. 7. 671-9 days. i14. PARTNERSHIP OR FELLOWSHIP. 2i A's share, $1714,285+. - A's share, $5000. B's share, $285,714+. B's " $2500. A's share, ~4030. 4. C's " $3333,33~+. B's " ~3980. D's " $2500. 3's " ~3980. E's " $6666,67+. tD's " ~4010. DOUBLE FELLOWSHIP. 2. A's share, ~9 12s. r 1st son's share, * B's ".14 8s. $3333,33-. 3. 3. 2d " $3000. 3d " $3000. GENERAL EXAMPLES IN FEL- 4th " $2666,660. LOWSHIP. r $750 widow's gain. 75 cents on the dollar. J $375 younger son's gain. A's part, $375,2714. $3000 widow's share. 1.. B's " $171. 4$1500 younger son's do. C's " $968,42-. 5. paid's $530005. 6. A's loss, $46,526+.A paid $3000. B's $130,273+. B "'$3000. 6. B " $3000. 6 C's $238,213~. 2.'s " 250. D's" $334,988+. A's gain, $250. C's " $750. B's " $28. (PAGES 277-292. ) 401 ALLIGATION MEDIAL.. Ex. A ns. Ex. Ans. 2. $0,84375. 4. 3. $0,283. 5. 730~ ALLIGATION ALTERNATE. ART. 252. r 12bu. of oats. J 12bu. of barley. ( 21b. at 8cts. 12bu. of rye. 2. t 21b. at ] Octs. 96bu. of wheat. (61b. at 14c of spirits. 4. 32gal. of Eng. brandy. 2 " of 18 3. 3 of 23 ART. 254. 5 " of 24 "291. at 5s. 6gal. at 10s. 2. 142 at 6s. 43gal. at 14s. 2 at at8s. 4. 4gak. at 21s. 29 at9s. 8gal. at 24s. 45gal. at 4s. 3. gal. at 6s. 5gal. at 8Os. ART. 253. 3 45gal at 2. 4. CUSTOM HOUSE BUSINESS. 1. $2812,5. 4. $1442,875. 2. $418,068. 5. - 3. $251,45+. TONNAGE OF VESSELS. 1. 225T9 tons. 4. 300.14 tons+. 2. 438.59 tons+. 5. 3. 7299 7- tons. 402 (PAGES 295-316. ) GAUGING. ART. 267. Ex. Ans. Ex. Ans. 2. 162.613 beer gal. +-. 2. 32.4938in. 3. 28.1010in. ART. 268. 4. 147.384 wine gal. ~. 1. 197.459 wine gal. +. LIFE INSURANCE. 2. $144. I 3. $189,55. ENDOWMENTS AND ANNUITIES. 1. $228,11+. 1 2. EXCHANGE. ART. 293. 5. $807,873 +. 1. $8591,975. ART. 295. 2. $8637,168+. 2. 7 per cent* above par. 3. $9777,636. 3. ART. 294. 4. 84597 francs 66 centimes. 1. $5630,065. ART. 296. 2. 1. $6657,693. 3. ~14014 18s. 2d+. ( $1250,52, 3 per cent* 4. $6005,368+. 2. nearly below par. DUODECIMALS. ART. 301. EXAMPLES IN ADDITION AND SUBTRACTION. 1. 155ft. 5 5ft. 8' 2// 1//'. 2. I-2. 15ft. 4' 10// 4//"'. 3. 2ft. 6' 3/ 11"'. 3. 4. 5ft. 10' 7". 4. 21ft. 6' 5" 5/". * The percentage is estimated upon the custom house value. (PAGES 311 —330. ) 403 Ecx. Ans. EEx. Ans. 5. 36ft. 4' 6" 5"//. 5. 232ft. 2' 8" 6. 22ft. 2/ 1" 10"'. 6. 866ft. 8' 3". 7. 2 cords 5 cord feet. ART. 303. 8. 57ft. 4/ 6", 3. 214ft. 1/ 1" 6"'. 9. 4. - 10. 185ft. 6' 4// 3"'. INVOLUTION. 1. 1953125. 9. 2. 343. 10..0001. 3. 3600. 11. 37.4544. 4. 12. 1000000. 5. 1889568. 13. 3421 6. 1. 14. 7. I. 15. 666024768837. 8..001. EXTRACTION OF THE SQUARE ROOT. ART. 309. 6..0321. 3. 462. 7. 2.104. 4. 1506.23+. 8. 6.906. 5. 3897.89-. 6. ART. 311. 7. 4698. 1..5236+. ART. 310. 2. 2. 57.19+. 3..4203+. 3. 69.247+. 4. 1.0682+ 4. 2.091+. 5..86602+. 5. 6..93309+. EXTRACTION OF THE CUBE ROOT. ART. 313. 4. 319. 2 - 5. 439 3. 179. 6. 638. 404 (PAGES 330-339. ) Ex. Ans. Ex. Ans. 7. - 7. 3.026. 68. 3002. ART. 315. ART. 314. 1. s 1..5032+. 2. 2..955. 3. 33. 3. 2.35. 4. 3 4. - 5..829 +. 5..707. 6..822+. 6. 1.505. 7. ARITHMETICAL PROGRESSION. ART. 318. 4. 5 miles 1300 yards. 2. $1,53. 3. $205. GENERAL EXAMPLES. ART. 319. 1. 89. 2. 4. 2. 4 years. 3. 3. 5 miles. $4. ART. 320. 5. 3 miles each day. 2. _ 100 " in all. 3. ast term 34. 6. 3. Sum 162. 7. $272. GEOMETRICAL PROGRESSION. ART. 322. ART. 323. 2. 2. ~204 15s. 3. ~25600. 3. $196,83. $295,24. 4. $61,44. 4. MENSURATION. ART. 326. I3. sA. 1R. 15P. 2. 36 acres. 4. 135A1. ( PAGES 341-350.-) 405.Ex. Ans. Ex. Ans. AiT. 329. 3. 4071.5136. 1; 437A. 2R. 34P.+. 4. 196996571.722104sq.ms. 2. 291A. 2R. 16P. 3, 35A. OR. 25P. 2. 268.0832. 4. 20A 5. 40A. 3. 2144.6656. 5., 40A. 6. 15A. 4. 259992792079.869+. 7. 24A. 1R. 8P. 5. 904.7808sq.ft. 8. 130080sq. yds. ART. 340. ART. 331. 1. 91OOsq. ft. p. 1440 sq.ft. 2. 21A. OR. 39.824P. 3. 921sq.ft. 10' 6". ART. 341. 4. 704.125sq. yds. 2. 110592solid in. 5. 60A. 3R. 12.8P. 6. 270A. 1R. 24P. 42solidft. 4. 3156 gal. ART. 332. 5. 13820solid ft. 2. 584.3376. ART. 343. 3. 125.664. 3. 125.664. 2. 233.333+sq. ft. 4. 3. 2827.44sq. in. ART. 333. 4. 6283.2sq. ft. 2. 7418. ART. 344. 3. 4360.835 +. 2. 36442.56. 3. 13571.712. 4. 9650.9952. 2. 19.636. 5. 7363.125. 8. 153.9384, 4. 1.069016+. ART. 346. ~. 4380. A.;. 336. 3. 2484-. 2. 615.7536. 4. 5620. 406 (PAGES 350-362. ) Ex. Ans.! E~x. Ans. 5. 5760. ART. 349. 6. 14400. 1. 241.86ft. 7. 1800. 2. 17.204ft. 3. 14.142ft. ART.- 348. ART. 350. 2. 9160.9056. 1. 302.9702ft. 3. 8659.035. 2. 28yd. 4. 2827.44. 3. 77.8875ft. MECHANICAL POWERS. ART. 355. 2. 401b. 1. 401b. 3. 201b. 2. 251b. 3. 501b. ART. 362. 4. 201b. 1. 7ft. 5. 401b. 2. ljft. 6. lin., l1in., 2in., 4in., &c. 7. 641b. ART. 363. 8. 1501b. 1. 401b. ART. 361. 2. 1001b. 1. 601b. 2. 601b. PROMISCUOUS QUESTIONS. 1. $1853,131+. Greater number, 3664~. 2. 657 pieces. Less " 16651. 3. 12uwk. 3g4da. 8. 46yr. 11lmo. 20da. 101hr. 4. 4 years. 9. 120 yards. 5. 0_ 10. 4thpartner'shsare$2500. 11 $31,25. 3d " " $3675. $6760pricehewould have. 2d " " $5375. 12. paid. 1st " " $9625. $6890price actally paid. ( PAGES 362-365. ) 407 Ex. Ans. 13. $454.9375. 16. 3 o'clock. 14. 23599680 cubic yards. 17. 140. 15. 18. i The second 6]- days after the third. The first 51o days after the second. 19. $4646,36+. A's share, 20. ~194 16s. 1lMd. 20. 767 27. B's " 129 17s. 4~d. 21. 7500 men. C's " ~97 8s. 04~yd. D's " ~77 18s. 52yd. 22. ~157 10s.7 23. 4 days. 28 2d, $180. (A's stock, ~304 16s. 29. 108 24. B's " ~276 30. C's " ~210 31. 10 hours. 25. 32. $1,75. 624 gal. of the 1st kind. 33. $1,853+. 26. 83 " " 2d 34. 441. 146 " " 3d " 35. 36. 5 13010~ yards of cloth. Price of linen per yard, $0,58-1. 37. $42,34-T. 42. $1724,363+. 38. 4 yards. 43. 356,25. 39. 12cwt. 161b. 15oz. 44. 342 per cent. 40. 45. 41. yr. 1 lmo. 27 6sda. 46. 5 Cost per yard, $4,90T1T. Entire cost, $18378,37H. 47. $8640. $ *920,20 = what the 1st gave. 48*. $2760,60 = " 2d ($5521,20 = " 3d ($282 amount paid each workman. 49. { 1st company cleared 87 acres. Cost of clearing, $88i per acre. 408 (PAGES 365-367. ) Ex. Ans. 50. 51. 6 o'clock 3m. 481l4sec. 52. 140 miles. Share of the 1st, $2019,651+. S"4tre " 2d, $4871,803+. 53. " " 3d, $4815,805 +. | " " 4th, $6467,739 +. " " 5th,$1825. 54. 693 miles from New-Haven. 55. 58. 5hr. 41m. 20sec. THE END.